Dynamic Response and Failure of Composite Materials and Structures presents an overview of recent developments in a spec

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*English*
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- Valentina Lopresto
- Antonio Langella and Serge Abrate (Eds.)

*Table of contents : Content: Front Matter,Copyright,List of contributorsEntitled to full textPart One: Dynamic behavior of composite structures1 - 2D thermo-elastic solutions for laminates and sandwiches with interlayer delaminations and imperfect thermal contact, Pages 3-46, H. Darban, R. Massabò2 - Dynamic behavior of composite marine propeller blades, Pages 47-83, S. Abrate3 - Numerical simulations and experimental experiences of impact on composite structures, Pages 85-125, F. Marulo, M. Guida, L. Maio, F. Ricci4 - Modeling low velocity impact phenomena on composite structures, Pages 129-158, A. Riccio, S. Saputo, A. Raimondo, A. Sellitto5 - Study of medium velocity impacts on the lower surface of helicopter blades, Pages 159-181, F. Pascal, P. Navarro, S. Marguet, J.-F. Ferrero6 - Impact on composite plates in contact with water, Pages 183-216, S. Abrate7 - Impact response of advanced composite structures reinforced by carbon nanoparticles, Pages 217-235, S. Laurenzi, M.G. Santonicola8 - Implosion of composite cylinders due to underwater impulsive loads, Pages 239-262, T. Qu, M. Zhou9 - Composite materials for blast applications in air and underwater, Pages 263-295, E. Rolfe, M. Kelly, H. Arora, J.P. Dear10 - Progressive bearing failure of composites for crash energy absorption, Pages 299-334, T. Bergmann, S. Heimbs11 - Thin-walled truncated conical structures under axial collapse: Analysis of crushing parameters, Pages 335-363, S. Boria12 - Lightweight solutions for vehicle frontal bumper: Crash design and manufacturing issues, Pages 365-393, G. Belingardi, A.T. Beyene, E.G. Koricho, B. Martorana13 - Pressure reconstruction during water impact through particle image velocimetry: Methodology overview and applications to lightweight structures, Pages 395-416, M. Porfiri, A. ShamsIndex, Pages 417-425*

Dynamic Response and Failure of Composite Materials and Structures

Related titles Lightweight Composite Structures in Transport (ISBN 978-1-78242-325-6) Modelling Damage, Fatigue and Failure in Composite Materials (ISBN 978-1-78242-286-0) Numerical Modelling of Failure in Advanced Composite Materials (ISBN 978-0-08-100332-9)

Woodhead Publishing Series in Composites Science and Engineering

Dynamic Response and Failure of Composite Materials and Structures Edited by

Valentina Lopresto Antonio Langella Serge Abrate

An imprint of Elsevier

Woodhead Publishing is an imprint of Elsevier The Officers’ Mess Business Centre, Royston Road, Duxford, CB22 4QH, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, OX5 1GB, United Kingdom © 2017 Elsevier Ltd. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-08-100887-4 (print) ISBN: 978-0-08-100902-4 (online) For information on all Woodhead publications visit our website at https://www.elsevier.com/books-and-journals

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List of contributors

S. Abrate Southern Illinois University, Carbondale, IL, United States H. Arora Imperial College London, London, United Kingdom G. Belingardi Politecnico di Torino, Torino, Italy T. Bergmann Airbus Group Innovations, Munich, Germany A.T. Beyene Politecnico di Torino, Torino, Italy S. Boria University of Camerino, Camerino, Italy H. Darban University of Genova, Genova, Italy J.P. Dear Imperial College London, London, United Kingdom J.-F. Ferrero University of Toulouse - Clement Ader Institute, Toulouse, France M. Guida University of Naples “Federico II”, Naples, Italy S. Heimbs Airbus Group Innovations, Munich, Germany M. Kelly Imperial College London, London, United Kingdom E.G. Koricho Michigan State University, East Lansing, MI, United States S. Laurenzi Sapienza University of Rome, Rome, Italy L. Maio University of Naples “Federico II”, Naples, Italy S. Marguet University of Toulouse - Clement Ader Institute, Toulouse, France B. Martorana FCA – FIAT Research Center, Torino, Italy F. Marulo University of Naples “Federico II”, Naples, Italy R. Massabo` University of Genova, Genova, Italy

x

List of contributors

P. Navarro University of Toulouse - Clement Ader Institute, Toulouse, France F. Pascal University of Toulouse - Clement Ader Institute, Toulouse, France M. Porfiri New York University Brooklyn, Brooklyn, NY, United States T. Qu Georgia Institute of Technology, Atlanta, GA, United States A. Raimondo The University of Campania “Luigi Vanvitelli”, Aversa, Italy F. Ricci University of Naples “Federico II”, Naples, Italy A. Riccio The University of Campania “Luigi Vanvitelli”, Aversa, Italy E. Rolfe Imperial College London, London, United Kingdom M.G. Santonicola Sapienza University of Rome, Rome, Italy; University of Twente, Enschede, The Netherlands S. Saputo The University of Campania “Luigi Vanvitelli”, Aversa, Italy A. Sellitto The University of Campania “Luigi Vanvitelli”, Aversa, Italy A. Shams New York University Brooklyn, Brooklyn, NY, United States M. Zhou Georgia Institute of Technology, Atlanta, GA, United States

2D thermo-elastic solutions for laminates and sandwiches with interlayer delaminations and imperfect thermal contact

1

H. Darban, R. Massabo` University of Genova, Genova, Italy

1.1

Introduction

Laminated, sandwich, and layered composites are largely used in different areas of technology and industry. Their application in primary structures of mechanical devices and vehicles, such as turbines, wind blades, aircrafts, automobiles, or ships, is progressively increasing. The interest in the use of composite materials is due to the fact that their mechanical properties can be tailored, by proper selection of the materials and design of the layups, to meet the stringent design requirements of modern mechanical devices. Current applications require withstanding severe mechanical loadings and surviving aggressive environments characterized for instance by very high or very low temperatures. The focus of this chapter is on layered composite beams and wide plates subjected to stationary thermo-mechanical loading. Three-dimensional thermo-elasticity provides exact solutions which can be used to confidently design the plates in the elastic regime or validate approximate structural theories. Exact solutions in the literature are mostly limited to simple geometries and loading and boundary conditions; in addition, these solutions typically require extensive computational work when the number of layers is large and this limits their utilization by the design community. In this chapter, a matrix technique will be used, along with the linear theory of 2D thermo-elasticity, to efficiently solve the thermo-mechanical problem in simply supported multilayered wide plates and beams with an arbitrary number of imperfectly bonded layers and thermally imperfect interfaces. The technique and the explicit formulas derived in the chapter simplify the solution procedure and provide a framework for other researchers to easily generate any desirable benchmark solutions. Section 1.2 presents a literature review on two- and three-dimensional thermo-elasticity models for laminated and sandwich structures with thermally and mechanically perfect and imperfect interfaces. In Section 1.3 the thermo-elasticity model formulated in Ref. [1] for simply supported wide plates with imperfect interfaces is recalled and a matrix technique based on the transfer matrix method is formulated to derive explicit formulas, which allow the efficient closed-form solution of the Dynamic Response and Failure of Composite Materials and Structures. http://dx.doi.org/10.1016/B978-0-08-100887-4.00001-9 © 2017 Elsevier Ltd. All rights reserved.

4

Dynamic Response and Failure of Composite Materials and Structures

problem with an arbitrary number of layers. Exemplary benchmark solutions are presented in Sections 1.4 and 1.5 and some conclusions in Section 1.6.

1.2

Thermo-elasticity solutions for laminated and sandwich plates with imperfect interfaces: state of the art

In an early paper [2], Pagano used the Airy stress function method to obtain an exact solution in the framework of the linear theory of elasticity for simply supported cross-ply laminates composed of perfectly bonded orthotropic/isotropic layers. The solution was given for plates subjected to sinusoidal transverse loads under plane-strain conditions. The theory was extended to include uniformly distributed and concentrated loads described by means of Fourier series in Ref. [3], and in Ref. [4] to treat stationary sinusoidally distributed thermal loads, under the simplifying assumption of linear thickness-wise temperature distribution. Thanks to these exact solutions, the limitations of classical laminated plate theory for the analysis of laminates with low span-to-thickness ratios were first revealed and the solutions are still used nowadays to assess the range of validity of approximate theories and numerical models. Recently, Pagano’s solution was completed in Ref. [5], using the displacement method, for cases where the characteristic equation of the problem has complex conjugate roots, as it occurs in sandwich plates with honeycomb cores having transverse stiffness much higher than the in-plane stiffnesses. An exact stationary thermo-elasticity solution for simply supported plates in plane-strain and subjected to arbitrary thermomechanical loading was obtained in Ref. [6] using the method of the displacement potentials and assuming perfect thermal contact at the layer interfaces. In a later study by Pagano [7], three-dimensional elasticity solutions were obtained for simply supported rectangular bidirectional laminated and sandwich plates composed of perfectly bonded orthotropic/isotropic layers. The characteristic equation of this problem was restated in the form of a cubic equation whose discriminant controls the nature of the solution. Pagano obtained closed-form solutions for the cases of negative and zero discriminants, for example, isotropic layers. Solutions for the case of a positive discriminant were presented later in Ref. [8]. These exact solutions proved the faster convergence to the exact solution of classical plate theory on increasing the number of layers [9]. In parallel with Pagano’s work, Srinivas et al. [10,11] obtained elasticity solutions for simply supported perfectly bonded cross-ply laminates under arbitrary loading by expressing the displacement and stress components in terms of infinite series. The thermo-elastic problem was studied in Refs. [12–14] for plates with perfect thermal contact between the layers, by assuming a prescribed temperature distribution with a through-the-thickness linear variation in Ref. [13] and through the exact solution of the heat conduction problem in Refs. [12,14]. Solutions for plates with boundary conditions other than the simple supports were obtained in the form of infinite series in Refs. [15,16] and through the extended Kantorovich method in Ref. [17]. All the aforementioned thermo-elasticity theories are based on two main steps. First, the general forms of the field variables, which satisfy the edge boundary

2D thermo-elastic solutions for laminates and sandwiches

5

conditions and the governing field equations, are obtained for a generic layer. Then, the unknown constants in the solutions of each layer are calculated by imposing continuity conditions at the layer interfaces and boundary conditions at the top and bottom surfaces of the plate. For a plate composed of n layers, a system of 4 n and 6 n algebraic equations needs to be solved for plane-strain and general plate problems, respectively [2,7]. Solving the system of equations becomes cumbersome when the number of layers increases and this restricts the applicability of the models. The transfer matrix method was originally proposed by Thomson and Haskell [18,19] for the problem of wave propagation in multilayered media. In Ref. [20], the method was employed, along with a mixed formulation of elasticity, to efficiently study statics and dynamics of simply supported bidirectional laminated plates. The basic idea of the method relies on the introduction of a local transfer matrix, which relates the field variables at the bottom of a layer to those at the top; the local matrices of the layers are then coupled through continuity conditions at the interfaces to obtain a global transfer matrix, which relates field quantities of the bottom and top surfaces of the medium. Using the transfer matrix method, the continuity conditions at the interfaces are then satisfied a priori and the field variables in each layer are obtained through the application of the relevant boundary conditions. The number of coupled algebraic equations which need to be solved becomes independent of the number of layers and equal to four or six, for planestrain or general plate problems, as it would be for a single-layer plate solved with the classical approach. The transfer matrix method was applied for the solution of the thermo-elasticity problems in simply supported laminated plates and arches in Refs. [21,22]. Applications of the method in the ultrasonic and seismology fields are presented in the review paper [23]. The thermo-elasticity models mentioned above assume the layers to be perfectly bonded and in perfect thermal contact, which imply continuity of displacements, tractions, temperature, and heat flux at the layer interfaces. This assumption does not describe systems with damaged interfaces or delaminations between the layers or systems where the plies are connected by thin adhesive layers which are not described as regular layers in the formulation. Flaws and delaminations may develop during the manufacturing processes and/or in service due to, for instance, fatigue loads, impacts, or environmental effects, such as temperature or humidity. They modify the continuity conditions at the layer interfaces and may result in stiffness and heat conductivity degradation and in reduction of the load-carrying capacity of the plates [24]. From a mechanical point of view, an imperfect interface or a very thin interlayer can be represented as a zero-thickness surface across which the interfacial tractions are continuous, while the displacements are discontinuous. The interfacial tractions can then be related to the relative displacements of the layers at the interfaces using interfacial traction laws able to describe different interfacial mechanisms. Linear interfacial traction laws have been frequently used in the literature; they assume that the interfacial tractions are proportional to the corresponding relative sliding and opening displacements and the proportionality factors are the interfacial tangential and normal stiffnesses. These laws well describe the response of thin adhesive elastic layers and the initial branch of more general interfacial traction laws, such as those

6

Dynamic Response and Failure of Composite Materials and Structures

which are typically used to model cohesive delamination fracture. In addition, these laws can be used to describe the limiting cases of perfectly bonded and fully debonded layers [1]. To the authors’ knowledge, Williams and Addessio [25] were the first to employ the concept of linear interfacial traction law in conjunction with Pagano’s model [2] in order to obtain exact elasticity solutions for the laminates with imperfect interfaces in plane-strain and subjected to mechanical loading. The extension is straightforward since the general solutions for stresses and displacements in each layer are unchanged, while the interfacial continuity conditions must account for the assumed interfacial traction laws. The same idea was applied in Ref. [26] to verify structural models based on a zigzag homogenization used to improve classical structural theories; in Refs. [27,28] it was used along with the state-space approach, for the bending and free vibrations of simply supported cross-ply laminates and cylindrical panels with imperfect interfaces; in Ref. [29] it was applied to study plates subjected to arbitrary boundary conditions. From a thermal point of view, heat transfer through the layers of a plate with interfacial imperfections is a rather complex process. Microcracks, voids, and delaminations reduce the areas of actual physical contact between adjacent layers and create regions separated by air gaps which prevent the heat flow across the interface. Heat transfer across the imperfect interfaces takes place through conduction at the contact spots and conduction and/or radiation through the air gaps. These mechanisms control and reduce the interfacial thermal conductance, which also depends on other factors, such as the applied pressure and the mean temperature. The consequence of this behavior is a jump in the temperatures of the layers at the interface [30]. Hence for laminates with interfacial imperfections, the assumption of perfect thermal contact between the layers, which implies a continuous temperature at the interface, is not valid. The concept of thermally imperfect interface characterized by an interfacial thermal resistance has been frequently used in the literature to account for the behavior described above [1,31–33]. This model enforces the equality of the heat fluxes, which enter and leave the interface, and assumes that the heat flux through the interface is proportional to the interfacial temperature jump; the interfacial thermal conductance H is the proportionality factor and should then account for the various modes of heat transfer through the interface mentioned previously [30]. An interfacial thermal resistance is then introduced, which is the reciprocal of the interfacial thermal conductance, R ¼ 1=H. If R ¼ 0, the model describes perfect interfaces, where the temperature is continuous at the interface. Impermeable interfaces, where the heat flux vanishes, can be modeled by setting H ¼ 0. The model can also be used to efficiently describe the thermal behavior of thin adhesive layers when they are represented as interfaces and H will then be related to the conductivity and thickness of the adhesive. Pelassa et al. [1] employed the concept of interfacial thermal resistance and assumed the interfaces to be mechanically imperfect and described by linear traction laws to extend the thermo-elasticity model presented in Ref. [14] to multilayered plates with thermally and/or mechanically imperfect interfaces.

2D thermo-elastic solutions for laminates and sandwiches

1.3

7

Two-dimensional elasticity model for simply supported multilayered wide plates with mechanically and/or thermally imperfect interfaces

The plane-strain thermo-elasticity model formulated in Ref. [1] for simply supported multilayered plates with interfacial imperfections is briefly recalled in this section. Explicit formulas are then derived, by means of the transfer matrix method, which allow the efficient, closed-form solution of the problem for an arbitrary number of layers.

1.3.1

Model assumptions

The multilayered plate shown in Fig. 1.1 is assumed to be under plane-strain conditions parallel to the plane x2 x3 , with x1 x2 x3 a system of Cartesian coordinates with origin at the left edge. The plate has global thickness h, is simply supported at x2 ¼ 0 and x2 ¼ L, and subjected to sinusoidal thermo-mechanical loading acting on the upper and lower surfaces: f3 ðx2 Þ ¼ f0 sin ðpx2 Þ are normal surface tractions with p ¼ mπ=L and m 2 ℕ , and T ðx2 Þ ¼ T0 sin ðpx2 Þ are applied temperatures. The laminate is composed of n layers joined by n 1 interfaces which may be mechanically and/or thermally imperfect. The kth layer (k ¼ 1,…, n is numbered from bottom to top) is defined by the coordinates of its lower and upper surfaces, x3k1 and xk3, and has thickness (k)h (the superscript (k) on the left of a quantity shows association with the layer k, while the superscript k on the right identifies the interface).

f3(x2) = fu sin(px2) x3

(n) •• • (2) (1)

T(x2) = Tu sin(px2) kth layer x3k x3k−1 x2

h

f3(x2) = fl sin(px2) L

T(x2) = Tl sin(px2)

Fig. 1.1 Simply supported multilayered plate with thermally and/or mechanically imperfect interfaces subjected to thermo-mechanical loadings. The exemplary surface tractions acting on the upper surface correspond to the case p ¼ π=L.

8

Dynamic Response and Failure of Composite Materials and Structures

The layers are assumed to be linearly elastic, homogenous, and orthotropic with principal material axes parallel to the geometrical axes (e.g., cross-ply laminate). The plane-strain thermo-elastic constitutive equations of the generic layer k are: 9 ðkÞ 2 9 ðkÞ 8 9 3 ðkÞ 8 C22 C23 0 < σ 22 = < ε22 α2 T = < C12 α1 T = ¼ 4 C23 C33 0 5 σ ε α3 T C α T , ; : 33 ; : 33 : 13 1 ; 0 0 C55 σ 23 2ε23 0

ðkÞ 8

(1.1)

where ðkÞ σ ij ¼ ðkÞ σ ij ðx2 , x3 Þ and ðkÞ εij ¼ ðkÞ εij ðx2 , x3 Þ (for i, j ¼ 2, 3) are Cauchy stress and linear strain components, (k)Cij (for i, j ¼ 1, 2, 3, 5) are stiffness coefficients, (k) αi is the coefficient of thermal expansion along the xi direction, and ðkÞ T ¼ ðkÞ T ðx2 , x3 Þ is the temperature increment in the layer k. The material properties are assumed to be independent of the temperature. The constitutive equations neglect coupling between elastic deformations and heat transfer and assume that the temperature distribution is prescribed and does not change over time.

1.3.2

Mechanically and thermally imperfect interfaces

The interfaces are zero-thickness mathematical surfaces which separate the layers and where material properties, displacements, and temperature may be discontinuous while the interfacial tractions and heat fluxes are continuous. In this work, the mechanical constitutive equations of the interfaces are defined by linear uncoupled traction laws which relate the interfacial tractions to the relative displacements of the adjacent layers. The assumed traction laws of the interface at x3 ¼ xk3 are: σ^k2 ðx2 Þ ¼ KSk v^k2 ðx2 Þ σ^k3 ðx2 Þ ¼ KNk v^k3 ðx2 Þ,

(1.2)

where σ^k2 ðx2 Þ and σ^k3 ðx2 Þ are the interfacial tractions in the tangential and normal directions acting on the upper surface of the layer k with unit positive normal vector, KkS and KNk are the tangential and normal stiffnesses, and v^k2 ðx2 Þ and v^k3 ðx2 Þ are the relative displacements between the layers k and k + 1: v^ki ðx2 Þ ¼ ðk +1Þ vi x2 , x3 ¼ xk3 ðkÞ vi x2 , x3 ¼ xk3

(1.3)

with i ¼ 2, 3 and (k)vi(x2, x3) the displacement component in the layer k. The limiting case of a perfectly bonded interface is described by 1=KSk ¼ 1=KNk ¼ 0, which leads to a continuous displacement field and vanishing relative displacements between the layers, v^k2 ¼ v^k3 ¼ 0; an interface which allows relative sliding displacements in constrained contact is defined by 1=KNk ¼ 0 and KSk ¼ 0, which yield σ^k2 ¼ 0 and v^k3 ¼ 0. The thermal behavior of the imperfect interface is described by introducing the interfacial thermal resistance Rk, which is defined as the reciprocal of the thermal conductance Hk, Rk ¼ 1=H k , and controls the heat flux through the interface [30]. In this work it is assumed that the interfacial thermal resistance is constant and independent

2D thermo-elastic solutions for laminates and sandwiches

9

of the interfacial displacement jumps and the heat flux through the interface is proportional to the temperature jump by the factor Hk: qk3 ðx2 Þ ¼

i 1 hðk +1Þ T x2 , x3 ¼ xk3 ðkÞ T x2 ,x3 ¼ xk3 : k R

(1.4)

The heat flux through the interface is related to the flux in the adjacent layers by qk3 ¼ ðkÞ q3 x2 , x3 ¼ xk3 ¼ ðkÞ K3 ðkÞ T x2 , x3 ¼ xk3 , 3 where (k)K3 is the thermal conductivity of the layer k in the x3 direction. Here and throughout the derivation, a comma followed by a subscript denotes a partial derivative with respect to the corresponding coordinate. The limiting cases which correspond to perfect thermal contact, where the temperature is continuous across the interface, and impermeable interface, where the heat flux through the interface vanishes, are described by Rk ¼ 0 and 1=Rk ¼ 0, respectively.

1.3.3

Heat conduction and thermo-elastic problems

In this section, heat conduction and equilibrium equations are presented for a generic layer k along with the associated thermal and mechanical boundary and continuity conditions. The two-dimensional steady-state heat conduction equation for the layer k, in the absence of internal heat generation, is given by: ðkÞ

K2

@ 2ðkÞ T ðx2 , x3 Þ ðkÞ @ 2ðkÞ T ðx2 , x3 Þ + K3 ¼ 0, @x22 @x23

(1.5)

where (k)Ki is the thermal conductivity of the layer k in the xi direction. The solution of the heat conduction equation must satisfy the continuity conditions at the layer interfaces, also accounting for the interfacial thermal laws Eq. (1.4) T x2 , x3 ¼ xk3 ðkÞ T x2 ,x3 ¼ xk3 ¼ Rk qk3 ðx2 Þ ðkÞ q3 x2 ,x3 ¼ xk3 ¼ ðk +1Þ q3 x2 ,x3 ¼ xk3 ðk +1Þ

(1.6)

for k ¼ 1, …,n 1, and the boundary conditions at the upper and lower surfaces of the plate and at the plate edges (Fig. 1.1): T x2 , x3 ¼ xn3 ¼ Tu sin ðpx2 Þ ð1Þ T x2 , x3 ¼ x03 ¼ Tl sin ðpx2 Þ ðkÞ T ðx2 ¼ 0,x3 Þ ¼ ðkÞ T ðx2 ¼ L, x3 Þ ¼ 0, for k ¼ 1,…,n: ðnÞ

(1.7)

The two-dimensional equilibrium equations for the layer k, in the absence of body forces, are: ðkÞ ðkÞ

σ 22 , 2 + ðkÞ σ 32 , 3 ¼ 0 σ 33 , 3 + ðkÞ σ 23 , 2 ¼ 0

(1.8)

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Dynamic Response and Failure of Composite Materials and Structures

and the compatibility equations: ðkÞ

ε22 ¼ ðkÞ v2 , 2

ðkÞ

ε33 ¼ ðkÞ v3 , 3 2ðkÞ ε23 ¼ ðkÞ v2 , 3 + ðkÞ v3 , 2 :

(1.9)

Using the constitutive and compatibility equations (1.1) and (1.9), the equilibrium equations are restated in terms of displacements, (k)v2 and (k)v3: ðkÞ

C22 ðkÞ v2 , 22 + ðkÞ ðC23 + C55 ÞðkÞ v3 , 23 + ðkÞ C55 ðkÞ v2 , 33

¼ ðkÞ ðC12 α1 + C22 α2 + C23 α3 ÞðkÞ T , 2 ðkÞ

C33 ðkÞ v3 , 33 + ðkÞ ðC23 + C55 ÞðkÞ v2 , 23 + ðkÞ C55 ðkÞ v3 , 22

(1.10)

¼ ðkÞ ðC13 α1 + C23 α2 + C33 α3 ÞðkÞ T , 3 : The continuity conditions on interfacial tractions and relative displacements, also accounting for the interfacial tractions laws Eq. (1.2), are expressed in terms of stresses in the adjacent layers as: 1 v2 x2 , xk3 ðkÞ v2 x2 , xk3 ¼ k ðkÞ σ 23 x2 , xk3 KS 1 k ðkÞ k ðk +1Þ v3 x2 , x3 v3 x2 , x3 ¼ k ðkÞ σ 33 x2 , xk3 K ðk +1Þ N k k ðkÞ σ 23 x2 , x3 ¼ σ 23 x2 , x3 ðk +1Þ k ðkÞ σ 33 x2 , x3 ¼ σ 33 x2 , xk3 ðk +1Þ

(1.11)

for k ¼ 1,…, n 1. The boundary conditions at the upper and lower surfaces of the plate and at the plate edges impose: σ 33 x2 ,x3 ¼ xn3 ¼ fu sin ðpx2 Þ ðnÞ σ 23 x2 ,x3 ¼ xn3 ¼ 0 ð1Þ σ 33 x2 ,x3 ¼ x03 ¼ fl sin ðpx2 Þ ð1Þ σ 23 x2 ,x3 ¼ x03 ¼ 0 ðnÞ

ðkÞ

σ 22 ðx2 ¼ 0, x3 Þ ¼ ðkÞ σ 22 ðx2 ¼ L,x3 Þ ¼ 0, for k ¼ 1,…, n

ðkÞ

v3 ðx2 ¼ 0, x3 Þ ¼ ðkÞ v3 ðx2 ¼ L, x3 Þ ¼ 0, for k ¼ 1, …, n:

1.3.4

(1.12)

Solution of the heat conduction problem through the transfer matrix method

The solution of the heat conduction problem in the layer k, which satisfies the thermal boundary conditions at the plate edges in Eq. (1.7), is obtained using the method of separation of variables:

2D thermo-elastic solutions for laminates and sandwiches ðkÞ

T ðx2 , x3 Þ ¼ ðkÞ Fðx3 Þ sin ðpx2 Þ,

11

(1.13)

where ðkÞ

Fð x 3 Þ ¼

ðkÞ

c1 e

ðkÞ

sx3

+ ðkÞ c2 e

ðkÞ

sx3

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðkÞ K2 ðkÞ s ¼ p ðkÞ K3

(1.14)

and p ¼ mπ=L. The thermal boundary conditions at the upper and lower surfaces of the plate, Eq. (1.7), and the thermal continuity conditions at the layer interfaces, Eq. (1.6), lead to an algebraic system of 2 n coupled equations in the 2 n unknown constants (k) c1 and (k)c2, for k ¼ 1,…, n. The transfer matrix method is used here for the efficient closed-form derivation of the 2 n unknown constants. The objective of the method is to derive explicit expressions which relate the unknown constants in the generic layer k, (k)c1 and (k)c2, to those of the first layer. In this way, the problem will be reduced to finding only two unknown constants, (1)c1 and (1)c2, through the application of the boundary conditions at the upper and lower surfaces. The temperature distribution, Eqs. (1.13) and (1.14), and the heat flux in the layer k, ðkÞ q3 ðx2 , x3 Þ ¼ ðkÞ K3 ðkÞ T ðx2 , x3 Þ, 3 , are written in matrix form as: ðkÞ 2

6 4

T ðx2 , x3 Þ K3

3

7¼ @T ðx2 , x3 Þ 5 @x3

(1.15) ðkÞ

Gðx3 Þ sin ðpx2 Þ,

where ðkÞ

ðkÞ

c1 , c2

(1.16)

esx3 esx3 , K3 sesx3 K3 sesx3

(1.17)

ðkÞ

Gðx3 Þ ¼ ðkÞ Dðx3 Þ

Dðx3 Þ ¼

ðkÞ

and (k)D(x3) is a 2 2 matrix whose elements are explicit functions of the length, the shape of the load, the thermal conductivities, and the coordinate x3. In order to establish a relationship between (k)c1 and (k)c2, and the constants of the layer k ¼ 1, the matrix (k)G is firstly related to the matrix (1)G following the procedure outlined below and schematized in Fig. 1.2. The procedure necessitates two main relationships, namely the local transfer matrix, which relates the values of the matrix (k)G at the upper and lower surfaces of the layer, and the interfacial continuity conditions expressed in terms of matrix (k)G.

12

Dynamic Response and Failure of Composite Materials and Structures

Fig. 1.2 Schematic of the transfer matrix method.

1.3.4.1

Local transfer matrix of a generic layer

The matrix of the unknown coefficients (k)[c1, c2]T in the layer k is expressed as function of (k)G using Eq. (1.16) and setting x3 ¼ x3k1 : ðkÞ

c1 ¼ ðkÞ D1 x3 ¼ x3k1 ðkÞ G x3 ¼ x3k1 , c2

(1.18)

where the superscript 1 at the right of a matrix denotes its inverse. The local transfer matrix of the generic layer k is then obtained by substituting Eq. (1.18) into Eq. (1.16) and setting x3 ¼ xk3 : ðkÞ

G x3 ¼ xk3 ¼ ðkÞ D x3 ¼ xk3 ðkÞ D1 x3 ¼ x3k1 ðkÞ G x3 ¼ x3k1 :

(1.19)

The local 2 2 transfer matrix, ðkÞ D x3 ¼ xk3 ðkÞ D1 x3 ¼ x3k1 , relates the values of (k) G calculated at the upper and lower surfaces of the layer.

1.3.4.2

Continuity conditions at a generic interface and global transfer matrix

The thermal continuity conditions at the interface x3 ¼ x3k1 , between the layer k and k 1, Eq. (1.6), are expressed in terms of matrix (k)G as:

2D thermo-elastic solutions for laminates and sandwiches

13

G x3 ¼ x3k1 ¼ J k1 ðk1Þ G x3 ¼ x3k1 1 Rk1 k1 J ¼ , 0 1

ðkÞ

(1.20)

where the 2 2 matrix J k1 depends on the interfacial thermal resistance of the interface k 1 and for perfect thermal contact it becomes the identity matrix since Rk1 ¼ 0. Substitution of ðkÞ G x3 ¼ x3k1 from Eq. (1.20) into Eq. (1.19) gives a relationship between ðkÞ G x3 ¼ xk3 and ðk1Þ G x3 ¼ x3k1 . Repeating this procedure and using the local transfer matrices and the continuity conditions of the layers and interfaces below layer k, the following explicit relationship between ðkÞ G x3 ¼ xk3 and ð1Þ G x3 ¼ x03 is derived: ( ) 1 Y 1 ðkÞ G x3 ¼ xk3 ¼ J k J i ðiÞ D x3 ¼ xi3 ðiÞ D1 x3 ¼ x3i1 ð1Þ G x3 ¼ x03 : i¼k

(1.21) The explicit relationships between (k)c1 and (k)c2, and the constants of the first layer are then derived by simply substituting ðkÞ G x3 ¼ xk3 and ð1Þ G x3 ¼ x03 defined by Eq. (1.16) into Eq. (1.21): ðkÞ

( ) 1 Y 1 c1 ¼ ðkÞ D1 x3 ¼ xk3 J k J i ðiÞ D x3 ¼ xi3 ðiÞ D1 x3 ¼ x3i1 c2 i¼k ð1Þ

D

x3 ¼ x03

ð1Þ c1 c2

(1.22)

with k ¼ 1,…, n. Eq. (1.22) reduces the heat conduction problem to finding the two unknown constants of the first layer, which requires the two boundary conditions at the bottom and top surfaces of the plate, Eq. (1.7). The first condition directly depends on (1)c1 and (1)c2 while the second is restated in terms of (1)c1 and (1)c2 through the global transfer matrix which is defined by setting k ¼ n into Eq. (1.22). Once (1)c1 and (1)c2 have been derived, the constants in each layers are calculated using again Eq. (1.22). Explicit expressions for the constants are given in Ref. [34].

1.3.5

Solution of the thermo-elastic problem

The displacement components in the layer k are obtained by summing particular and complementary solutions of the governing equilibrium equations (1.10), for i ¼ 2, 3: ðkÞ

vi ðx2 , x3 Þ ¼ ðkÞ vip ðx2 , x3 Þ + ðkÞ vic ðx2 , x3 Þ:

In the absence of thermal loads, the particular solution is

(1.23) ðkÞ

vip ðx2 , x3 Þ ¼ 0.

14

Dynamic Response and Failure of Composite Materials and Structures

1.3.5.1

Particular solution for layer k

A particular solution of the equilibrium equations (1.10) for the layer k, with ðkÞ T ¼ ðkÞ T ðx2 , x3 Þ prescribed by the previous solution of the heat conduction problem, which satisfies the edge boundary conditions in Eq. (1.12), is: ðkÞ ðkÞ ðkÞ v2p ðx2 , x3 Þ ¼ ðkÞ B1 e sx3 + ðkÞ B2 e sx3 cos ðpx2 Þ ðkÞ ðkÞ ðkÞ v3p ðx2 , x3 Þ ¼ ðkÞ D1 e sx3 + ðkÞ D2 e sx3 sin ðpx2 Þ, (1.24) where (k)B1, (k)B2, (k)D1, and (k)D2 are unknown constants and (k)s is given in Eq. (1.14). Substituting (k)v2p and (k)v3p from Eq. (1.24) into Eq. (1.10) results in four algebraic equations whose solution yields: ðkÞ B ¼ 18 h i 9 ðkÞ> 2 2 C α + C α s2 C + C 2 2 >

> : ; C22 p2 C33 s2 C55 p2 C23 p2 s2 C23 + 2C55 C33 C55 s4 ! ðkÞ c2 ðkÞ ðkÞ B ¼ B1 2 c1 ðkÞ D ¼ 1 8 h i 9 ðkÞ> 2 2 C α + C α + p2 C + C 2 2 >

> : ; C22 p2 C33 s2 C55 p2 C23 p2 s2 C23 + 2C55 C33 C55 s4 ! ðkÞ c2 ðkÞ ðkÞ D ¼ D1 , 2 c1 (1.25)

which depend on the layer’s material properties and the integration constants of the thermal problem, obtained using Eq. (1.22) and the thermal boundary conditions.

1.3.5.2

Complementary solution for layer k

A solution of the complementary problem, which satisfies the boundary conditions at the plate edges in Eq. (1.12), is obtained using the separation of variables: ðkÞ

v2c ðx2 , x3 Þ ¼ ðkÞ V ðx3 Þcos ðpx2 Þ

ðkÞ

v3c ðx2 , x3 Þ ¼ ðkÞ W ðx3 Þsin ðpx2 Þ

(1.26)

with h

ðkÞ

i h i ðkÞ Vðx3 Þ, ðkÞ Wðx3 Þ ¼ ðkÞ V0 , ðkÞ W0 e tx3 ,

(1.27)

where (k)V0 and (k)W0 are unknown constants and (k)t is the root of the associated characteristic equation which is defined below. Substituting Eqs. (1.26) and (1.27) in the homogenous part of Eq. (1.10) results in the following system of algebraic equations:

2D thermo-elastic solutions for laminates and sandwiches

C22 p2 C55 t2 V0 ðC23 + C55 ÞptW0 ¼ 0

ðkÞ ðC23 + C55 ÞptV0 + C55 p2 C33 t2 W0 ¼ 0: ðkÞ

15

(1.28)

The nontrivial solution of the system is obtained by imposing the determinant of the coefficients to be zero. This yields the characteristic equation for the layer k: ðkÞ

A0 t4 + A1 t2 + A2 ¼ 0,

(1.29)

where A0 ¼ ðkÞ ðC33 C55 Þ h i ðkÞ ðkÞ A1 ¼ ðC23 + C55 Þ2 C22 C33 C255 p2 ðkÞ

ðkÞ

(1.30)

A2 ¼ ðkÞ ðC22 C55 Þp4 :

The characteristic equation is put in the standard quadratic form by introducing ðkÞ γ ¼ ðkÞ t 2 . The discriminant ðkÞ Δ ¼ ðkÞ A21 4A0 A2 then controls the nature of the solution. Here the most common case of positive discriminant and positive roots is examined and explicit formulas for the solution of the problem are derived using the transfer matrix method. The solutions for the other cases are presented in Appendix.

Positive discriminant and positive roots When the discriminant of Eq. (1.29) is positive and the roots (k)γ j are positive, the displacement functions of Eq. (1.27) take the following forms [1]: ðkÞ

V ðx 3 Þ ¼

2 X

ðkÞ

a1j cosh

ðkÞ

mj x3 + ðkÞ a2j sinh ðkÞ mj x3

j¼1 ðkÞ

W ðx 3 Þ ¼

2 X

ðkÞ

βj ðkÞ a2j cosh

ðkÞ

mj x3 + ðkÞ βj ðkÞ a1j sinh ðkÞ mj x3 ,

(1.31)

j¼1

where

ðkÞ

ðkÞ

mj ¼

βj ¼

ðkÞ

qﬃﬃﬃﬃﬃﬃﬃﬃ ðkÞ γ , j "

ðkÞ

tj ¼ ðkÞ mj , for j ¼ 1, 2, and

# ðC23 + C55 Þpmj : C33 m2j C55 p2

The complementary solution for each layer is obtained by substituting (k)V(x3) and W(x3), which depend on four independent unknown constants, (k)a11, (k)a21, (k)a12, and (k)a22, into Eq. (1.26). The displacement components in each layer, Eq. (1.23), then depend on four unknown constants, which leads to a total of 4 n unknowns

(k)

16

Dynamic Response and Failure of Composite Materials and Structures

for the plate. The unknowns are typically obtained using interfacial continuity conditions and boundary conditions, which lead to a system of coupled algebraic equations whose solution becomes computationally cumbersome on increasing the number of layers.

1.3.5.3

Transfer matrix method

The transfer matrix method allows an efficient solution of the system of the equations, as previously shown for the solution of the thermal problem. The first step of the procedure is the definition of explicit relationships between the four unknown constants of the generic layer k, (k)a11, (k)a21, (k)a12, and (k)a22, and those of the first layer. Then, the problem is reduced to finding only four unknown constants (1)a11, (1)a21, (1)a12, and (1) a22 through the application of boundary conditions at the upper and lower surfaces of the plate. From the above derivations and Eq. (1.23), the displacements of the layer k are: i + ðkÞ V x3 , ðkÞ a11 , ðkÞ a21 , ðkÞ a12 , ðkÞ a22 cos ðpx2 Þ h i ðkÞ v ðx , x Þ ¼ ðkÞ D eðkÞ sx3 + ðkÞ D eðkÞ sx3 + ðkÞ W x , ðkÞ a , ðkÞ a , ðkÞ a , ðkÞ a sin ðpx2 Þ, 3 2 3 1 2 3 11 21 12 22 ðkÞ v

2 ðx2 , x3 Þ ¼

h

ðkÞ

B1 e

ðkÞ

sx3

+ ðkÞ B2 e

ðkÞ

sx3

(1.32)

where the constants (k)B1, (k)B2, (k)D1, and (k)D2 are defined in Eq. (1.25). Normal and transverse shear stress components are derived from the equation above using constitutive and compatibility equations (1.1) and (1.9). Displacements and transverse stresses are then collected in the following matrix form: ðkÞ 2

v2 ðx2 , x3 Þ

3

(1.33)

6 7 6 v3 ðx2 , x3 Þ 7 6 7 6 7¼ Cðx2 ÞðkÞ Mðx3 Þ, 6 σ 33 ðx2 , x3 Þ 7 4 5 σ 23 ðx2 , x3 Þ where C(x2) and

(k)

M(x3) are 4 4 and 4 1 matrices defined as follows:

2

3 cos ðpx2 Þ 0 0 0 6 7 0 0 0 sin ðpx2 Þ 6 7 Cðx2 Þ ¼ 6 7, 4 5 0 0 sin ðpx2 Þ 0 0 0 0 cos ðpx2 Þ 3 a11 6 a21 7 ðkÞ 7 Mðx3 Þ ¼ ðkÞ Qðx3 Þ + ðkÞ Eðx3 Þ 6 4 a12 5: a22 ðkÞ

(1.34)

2

(1.35)

2D thermo-elastic solutions for laminates and sandwiches

17

The matrix (k)Q(x3), which is related to the particular solution, is known and given by: ðkÞ Q x 3 3 ðkÞ 2 B1 esx3 + B2 esx3 6 7 6 7 D1 esx3 + D2 esx3 6 7 7: ¼ 6 6 C23 p B1 esx3 + B2 esx3 + C33 s D1 esx3 D2 esx3 c1 esx3 + c2 esx3 C23 α2 + C13 α1 + C33 α3 7 4 5 h i sx sx sx sx 3 +p D e 3 +D e 3 C55 s B1 e 3 B2 e 1 2

(1.36) The matrix (k)E(x3) is a 4 4 matrix related to the complementary solution of the layer k. For the case of positive discriminant and positive roots examined above, Eq. (1.31) yields E11 ¼ cosh ðkÞ m1 x3 , ðkÞ E12 ¼ sinh ðkÞ m1 x3 , ðkÞ E13 ¼ cosh ðkÞ m2 x3 , ðkÞ E14 ¼ sinh ðkÞ m2 x3 , ðkÞ E21 ¼ ðkÞ β1 sinh ðkÞ m1 x3 , ðkÞ E22 ¼ ðkÞ β1 cosh ðkÞ m1 x3 , ðkÞ E23 ¼ ðkÞ β2 sinh ðkÞ m2 x3 , ðkÞ E24 ¼ ðkÞ β2 cosh ðkÞ m2 x3 , ðkÞ E31 ¼ ðkÞ ðC33 β1 m1 C23 pÞ cosh ðkÞ m1 x3 , ðkÞ E32 ¼ ðkÞ ðC33 β1 m1 C23 pÞsinh ðkÞ m1 x3 , ðkÞ E33 ¼ ðkÞ ðC33 β2 m2 C23 pÞ cosh ðkÞ m2 x3 , ðkÞ E34 ¼ ðkÞ ðC33 β2 m2 C23 pÞsinh ðkÞ m2 x3 , ðkÞ E41 ¼ ðkÞ C55 ðkÞ ðβ1 p + m1 Þ sinh ðkÞ m1 x3 , ðkÞ E42 ¼ ðkÞ C55 ðkÞ ðβ1 p + m1 Þ cosh ðkÞ m1 x3 , ðkÞ E43 ¼ ðkÞ C55 ðkÞ ðβ2 p + m2 Þ sinh ðkÞ m2 x3 , ðkÞ E44 ¼ ðkÞ C55 ðkÞ ðβ2 p + m2 Þ cosh ðkÞ m2 x3 : ðkÞ

(1.37) The other forms of (k)E(x3), which correspond to the cases of positive discriminant with negative roots, zero and negative discriminants, are given in Appendix.

Local transfer matrix An expression for the unknown constants of the layer k, (k)[a11, a21, a12, a22]T, is found by setting x3 ¼ x3k1 in Eq. (1.35). The local transfer matrix of the layer k is then derived by substituting the expression into Eq. (1.35) and setting x3 ¼ xk3 : ðkÞ M

h x3 ¼ xk3 ¼ ðkÞ E x3 ¼ xk3 ðkÞ E1 x3 ¼ x3k1 ðkÞ M x3 ¼ x3k1 i ðkÞ Q x3 ¼ x3k1 + ðkÞ Q x3 ¼ xk3 :

(1.38)

Eq. (1.38) establishes a relationship between the values of matrix (k)M at the top and bottom surfaces of the layer.

Interfacial continuity conditions and global transfer matrix The continuity conditions between the layer k and k 1, Eq. (1.11), are written in matrix form: ðkÞ

M x3 ¼ x3k1 ¼ Bk1 ðk1Þ M x3 ¼ x3k1

(1.39)

18

Dynamic Response and Failure of Composite Materials and Structures

with 2

1 60 k1 B ¼6 40 0

3 0 0 1=KSk1 1 1=KNk1 0 7 7: 0 1 0 5 0 0 1

(1.40)

The matrix Bk1 depends on the interfacial stiffnesses and for the case of perfect bonding, with 1=KSk1 ¼ 1=KNk1 ¼ 0, it becomes the identity matrix. The procedure used for the solution of the heat conduction problem (Fig. 1.2) is used to derive a relationship between ðkÞ M x3 ¼ xk3 and ð1Þ M x3 ¼ x03 : ðkÞ

M

x3 ¼ xk3

( 1 n k 1 Y oð1Þ Bi ðiÞ E x3 ¼ xi3 ðiÞ E1 x3 ¼ xi1 M x3 ¼ x03 ¼ B 3 i¼k

( ) k i n o Y X ðiÞ i1 Bj ðjÞ E x3 ¼ xj3 ðjÞ E1 x3 ¼ xj1 Q x ¼ x 3 3 3 i¼1

j¼k

(

k i n o Y X Bi1 ði1Þ Q x3 ¼ xi1 + Bj ðjÞ E x3 ¼ xj3 ðjÞ E1 x3 ¼ xj1 3 3 i¼2

+

ðkÞ

))

j¼k

Q x3 ¼ xk3 :

(1.41)

The explicit expressions relating the four unknown constants, (k)a11, (k)a21, (k)a12, and a22, in Eq. (1.32) to those of the first layer are then derived substituting ðkÞ M x3 ¼ xk3 and ð1Þ M x3 ¼ x03 defined by Eq. (1.35) into Eq. (1.41): (k)

3 a11 ( 1 n 6 a21 7 ðkÞ 1 k k 1 Y o 6 7¼ E x B Bi ðiÞ E xi3 ðiÞ E1 x3i1 ð1Þ E x03 ð1Þ ½ a11 a21 a12 a22 T 3 4 a12 5 i¼k a22 !) k i n on Y X i1 ðiÞ i1 o j ðjÞ 1 j1 j ðjÞ i1 ði1Þ : B + B E x3 E x3 Q x3 Q x3

ðkÞ 2

i¼2

j¼k

(1.42) The equation can be used for k ¼ 2, …, n to define the integration constants of the layers as function of those of the first layer. Eq. (1.42) reduces the thermo-elasticity problem to finding only four unknown constants in the solution of the first layer, which requires four equations. Two equations are the boundary conditions at the bottom surface of the plate, which directly depend on the unknowns. The other two boundary conditions, at the top surface of the plate, are restated in terms of the unknown constants of the first layer using Eq. (1.42) with k ¼ n. The stress components of each layer are then obtained using the constitutive and compatibility equations (1.1) and (1.9). Explicit expressions for stresses and displacements are given in Ref. [34].

2D thermo-elastic solutions for laminates and sandwiches

19

The expressions of the integration constants in Eq. (1.42) are valid also for the other solution cases (different values of the discriminant), provided the matrix (k) E is changed according to the forms given in Appendix.

1.4

Solutions for simply supported plates subjected to sinusoidal transverse loading

In this section, exact solutions for two simply supported cross-ply laminated plates subjected to sinusoidal transverse loading and deforming in cylindrical bending are presented. The results are given in tables using at least four digits in order to generate benchmark solutions with enough precision for verification of approximate structural theories. The results are also presented in graph form to highlight the important influence of the interfacial imperfections on stress and displacement fields. Dimensionless stresses and displacements are given for different length-to-thickness ratios and interfacial stiffnesses. In the examples, the interfaces have the same interfacial stiffnesses and three cases of perfect bonding, sliding interfaces in constrained contact, and partial bonding are examined. In order to avoid interpenetration between the layers, the results presented for the cases with 1=KN 6¼ 0 are valid only for positive applied surface tractions. The model presented in the previous section and the results are valid under the assumption of infinitesimal strains and displacements, which must be verified in each layer. The validity of this assumption and the range of values of the applied load for which the solutions in the tables are correct can be verified by using the maximum dimensionless transverse displacements and stresses given in the tables.

1.4.1

Plate with three layers and a symmetric layup

The first example is a simply supported anisotropic plate with three layers of equal thickness, symmetrically stacked and joined by two interfaces. The length and thickness of the plate are L and h, respectively, and the origin of the coordinate system is placed at mid-thickness of the left edge of the plate. The elastic constants of the layers are EL =ET ¼ 17, GLT =ET ¼ 0:7, GTT =ET ¼ 0:6, νLT ¼ 0:28, and νTT ¼ 0:4 (subscripts L and T indicate in-plane principal material directions), and the stacking sequence is (0, 90, 0). The assumed ratios between the elastic constants of the layers could represent a graphite-epoxy laminate. The plate is subjected to normally applied tractions f3 ¼ f0 sin ðπx2 =LÞ acting on the upper surface. Results are presented in Table 1.1 for three length-to-thickness ratios, L/h, equal to 4, 10, and 20. The results are tabulated for perfectly bonded interfaces, 1=KS ¼ 1=KN ¼ 0, sliding interfaces in constrained contact, 1=KN ¼ 0 and KS ¼ 0, and partial bonding with dimensionless interfacial stiffnesses, KS h=ET ¼ 0:2 and KN h=ET ¼ 0:5 (see Eq. 1.2 for the interfacial traction laws used in the model). The ratio between the interfacial normal and tangential stiffnesses for this latter case is 2.5 and could represent layers joined by thin elastic and isotropic adhesive interlayers.

Simply supported three-layer plate: normal surface tractions f3 5 f0 sin (πx2 =L) acting on the upper surface

20

Table 1.1

Elastic constants: EL =ET 5 17, GLT =ET 5 0:7, GTT =ET 50:6, νLT 5 0:28, and νTT 5 0:4, stacking sequence of (0, 90, 0). Subscripts l5lower layer and u5upper layer. Perfect bonding: 1=KS 5 0 and 1=KN 50 x3/h L/h

21/2

21/3

21/6u

21/6l

1/6l

1/6u

1/3

1/2

0.045 0.064 0.113

0.118 3.595 30.67

0.118 3.595 30.67

0.414 7.602 62.16

0.914 12.12 94.68

4.727 93.02 1269

4.787 93.11 1270

4.870 93.16 1269

4.870 93.16 1269

4.971 93.16 1269

5.072 93.09 1268

0.040 1.063 4.766

0.168 0.183 0.185 x3/h

0.387 1.431 5.138

1.872 19.58 82.58

5.913 41.15 167.1

12.66 65.44 254.4

Dimensionless longitudinal displacements: v2ET/(f0h) at x2 5 0 4 10 20

0.915 12.21 94.88

0.458 7.710 62.38

0.194 3.718 30.90

0.194 3.718 30.90

Dimensionless transverse displacements: v3ET/(f0h) at x2 5 L=2 4 10 20

4.648 92.67 1268

4.686 92.88 1269

4.727 93.02 1269

Dimensionless bending stresses: σ 22/f0 at x2 5 L=2 4 10 20

L/h

12.27 65.52 254.5 21/2

6.111 41.34 167.3 25/12

21/3

2.487 19.84 82.78 21/4

21/6

21/12

0

Dimensionless transverse shear stresses: σ 23/f0 at x2 5 0. For maxima see 4 10 20

0.000 0.000 0.000

0.682 1.549 3.043

1.164 2.783 5.515

1.495 3.720 7.426

1.709 4.376 8.784

1.708 4.395 8.830

1/12

1/6

1/4

1/3

5/12

1/2

1.686 4.386 8.825

1.665 4.356 8.774

1.485 3.707 7.419

1.174 2.775 5.510

0.696 1.546 3.040

0.000 0.000 0.000

a

1.701 4.399 8.844

Dynamic Response and Failure of Composite Materials and Structures

0

L/h

21/2

27/18

25/18

21/6

21/18

1/18

1/6

5/18

7/18

1/2

0.427 0.424 0.423

0.576 0.577 0.577

0.722 0.730 0.731

0.857 0.867 0.869

0.959 0.964 0.964

1.000 1.000 1.000

Dimensionless transverse normal stresses: σ 33/f0 at x2 5 L=2 4 10 20

0.000 0.000 0.000

0.040 0.036 0.036

0.141 0.133 0.131

0.278 0.271 0.269

Partial bonding: KS h=ET 5 0:2 and KN h=ET 5 0:5 (f0 >0) x3/h L/h

21/2

21/3

21/6l

21/6u

0

1/6l

1/6u

1/3

1/2

0.183 0.153 0.161

1.875 11.90 49.86

1.864 5.991 8.286

0.173 6.233 58.90

2.308 18.88 110.5

17.33 249.3 1999

17.45 249.5 2000

17.48 249.5 2000

18.66 250.8 2001

18.80 251.0 2001

18.86 250.8 2000

1.607 3.724 7.805

0.052 0.155 0.178

1.719 4.036 8.161

24.76 31.87 22.51

2.640 33.79 158.4

31.36 101.7 296.8

Dimensionless longitudinal displacements: v2ET/(f0h) at x2 5 0 4 10 20

2.079 18.86 110.6

0.208 6.336 59.11

1.579 5.772 8.559

2.229 12.20 50.18

2D thermo-elastic solutions for laminates and sandwiches

x3/h

Dimensionless transverse displacements: v3ET/(f0h) at x2 5 L=2 4 10 20

16.51 248.4 1998

16.57 248.7 1998

16.58 248.7 1999

Dimensionless bending stresses: σ 22/f0 at x2 5 L=2 4 10 20

27.88 101.2 296.8

2.728 33.95 158.6

21.32 31.09 22.85

Continued 21

22

Table 1.1

Continued x3/h

L/h

21/2

25/12

21/3

21/4

21/6

21/12

0

Dimensionless transverse shear stresses: σ 23/f0 at x2 5 0. For maxima see 4 10 20

0.000 0.000 0.000

1.377 2.199 3.429

1.939 3.519 5.953

1.740 3.982 7.583

0.762 3.594 8.324

0.840 3.666 8.400

1/12

1/6

1/4

1/3

5/12

1/2

0.833 3.658 8.396

0.748 3.578 8.315

1.894 3.982 7.577

2.151 3.528 5.950

1.541 2.208 3.428

0.000 0.000 0.000

b

0.863 3.687 8.424

L/h

21/2

27/18

25/18

21/6

21/18

1/18

1/6

5/18

7/18

1/2

0.445 0.436 0.427

0.520 0.564 0.574

0.590 0.691 0.720

0.725 0.827 0.857

0.907 0.947 0.960

1.000 1.000 1.000

Dimensionless transverse normal stresses: σ 33/f0 at x2 5L=2 4 10 20

0.000 0.000 0.000

0.083 0.052 0.040

0.247 0.173 0.143

0.373 0.309 0.280

Sliding interfaces in constrained contact : KS 5 0 and 1=KN 5 0 x3/h L/h

21/2

21/3

21/6l

21/6u

0

1/6l

1/6u

1/3

1/2

0.258 0.645 1.288

3.384 50.07 397.3

3.269 49.75 396.7

0.024 0.057 0.112

3.233 49.64 396.5

Dimensionless longitudinal displacements: v2ET/(f0h) at x2 5 0 4 10 20

3.222 49.70 396.6

0.008 0.018 0.036

3.210 49.67 396.5

3.889 51.36 399.9

Dynamic Response and Failure of Composite Materials and Structures

x3/h

4 10 20

27.84 969.6 15,230

27.93 970.1 15,232

27.90 969.6 15,230

27.90 969.6 15,230

28.08 970.3 15,233

28.07 969.8 15,231

28.07 969.8 15,231

28.23 970.4 15,233

28.28 970.0 15,231

2.873 16.01 62.91

0.001 0.000 0.000

2.879 16.01 62.91

43.64 266.7 1064

0.030 0.005 0.001

43.76 266.8 1064

Dimensionless bending stresses: σ 22/f0 at x2 5 L=2 4 10 20

43.21 266.7 1064

0.009 0.002 0.000

43.25 266.7 1064

x3/h L/h

21/2

25/12

21/3

21/4

21/6

21/12

0

Dimensionless transverse shear stresses: σ 23/f0 at x2 5 0. For maxima see 4 10 20

0.000 0.000 0.000

2.070 5.215 10.43

2.737 6.944 13.91

2.071 5.215 10.43

0.000 0.000 0.000

0.141 0.314 0.618

1/12

1/6

1/4

1/3

5/12

1/2

0.141 0.314 0.618

0.000 0.000 0.000

2.091 5.216 10.43

2.767 6.946 13.91

2.094 5.217 10.43

0.000 0.000 0.000

c

0.188 0.419 0.823

2D thermo-elastic solutions for laminates and sandwiches

Dimensionless transverse displacements: v3ET/(f0h) at x2 5 L=2

x3/h L/h

21/2

27/18

25/18

21/6

21/18

1/18

1/6

5/18

7/18

1/2

0.489 0.493 0.493

0.505 0.507 0.507

0.514 0.515 0.514

0.640 0.641 0.640

0.873 0.874 0.874

1.000 1.000 1.000

Dimensionless transverse normal stresses: σ 33/f0 at x2 5L=2 4 10 20

0.000 0.000 0.000

0.125 0.126 0.126

0.356 0.359 0.360

0.481 0.485 0.486

Maxima (x3/h, σ 23max/f0): (0.134, 1.710) for L=h ¼ 4, (0.024, 4.399) for L=h ¼ 10, and (0.006, 8.844) for L=h ¼ 20. Maxima (x3/h, σ 23max/f0): (0.317, 2.168) for L=h ¼ 4, (0.246, 3.983) for, L=h ¼ 10, and (0.004, 8.424) for L=h ¼ 20. c Maxima (x3/h, σ 23max/f0): (0.333, 2.767) for L=h ¼ 4, (0.333, 6.946) for L=h ¼ 10, and (0.333, 13.91) for L=h ¼ 20. a

b

23

24

Dynamic Response and Failure of Composite Materials and Structures

0.5

0.5

0.25

0.25

x3

x3

0

h

h

−0.25 −0.5 −4.5

(A)

0

−0.25

−1.5

1.5

v2ET Ⲑ( f0h) Fully bonded

−0.5

4.5

0

(B)

10

20

30

v3ET Ⲑ( f0h)

KSh ⲐET = 0.2

KS = 0

KNh ⲐET = 0.5

1 ⲐKN = 0

Fig. 1.3 (A) Longitudinal displacements at x2 ¼ 0 and (B) transverse displacements at x2 ¼ L=2 shown through thickness in a simply supported three-layer plate with L=h ¼ 4 subjected to normal surface tractions f3 ¼ f0 sin ðπx2 =LÞ on the upper surface. Elastic constants: EL =ET ¼ 17, GLT =ET ¼ 0:7, GTT =ET ¼ 0:6, νLT ¼ 0:28, and νTT ¼ 0:4; stacking sequence, (0, 90, 0). Solutions for fully bonded, fully debonded, and partially bonded layers.

Figs. 1.3 and 1.4 highlight the influence of the interfacial imperfections on the field variables: both longitudinal and transverse displacements are discontinuous at the interfaces and the stress distributions are substantially modified with changes in the location and value of the maxima (Table 1.1).

1.4.2

Plate with five layers and a symmetric layup

The second example is a simply supported symmetrically laminated plate with length L and thickness h. The layup consists of five layers of equal thickness and joined by four identical interfaces; the stacking sequence is (0, 90, 0, 90, 0). The origin of the coordinate system is assumed to be at mid-thickness of the left edge and the elastic constants of the layers are EL =ET ¼ 17, GLT =ET ¼ 0:7, GTT =ET ¼ 0:6, νLT ¼ 0:28, and νTT ¼ 0:4. The plate is subjected to normal surface tractions f3 ¼ f0 sin ðπx2 =LÞ on the upper surface. Results in Table 1.2 are given for three length-to-thickness ratios of 4, 10, and 20 and for laminates with perfect interfaces, imperfect interfaces with KS h=ET ¼ 0:1 and KN h=ET ¼ 0:25, and sliding interfaces in constrained contact with KS ¼ 0 and 1=KN ¼ 0. The related Figs. 1.5 and 1.6 highlight the influence of the interfacial imperfections on the field variables: both longitudinal and transverse displacements are discontinuous at the interfaces and the stress distributions are substantially modified with changes in the location and value of the maxima.

2D thermo-elastic solutions for laminates and sandwiches

x3 h

25

0.5

0.5

0.25

0.25 x3

0

h

−0.25 −0.5 −50

0

−0.25

−25

(A)

0 s22 Ⲑ f0

25

−0.5

50

(B)

0

1

2 s23 Ⲑ f0

3

4

0.5 Fully bonded

0.25 x3 h

KSh ⲐET = 0.2

0

KNh ⲐET = 0.5 KS = 0

−0.25

1 ⲐKN = 0 −0.5

(C)

0

0.25

0.5 s33 Ⲑ f0

0.75

1

Fig. 1.4 (A) Bending stresses at x2 ¼ L=2, (B) transverse shear stresses at x2 ¼ 0, and (C) transverse normal stresses at x2 ¼ L=2 through thickness in a simply supported three-layer plate with L=h ¼ 4 subjected to normal surface tractions f3 ¼ f0 sin ðπx2 =LÞ on the upper surface. Elastic constants: EL =ET ¼ 17, GLT =ET ¼ 0:7, GTT =ET ¼ 0:6, νLT ¼ 0:28, and νTT ¼ 0:4; stacking sequence, (0, 90, 0).

1.5

Solutions for simply supported plates subjected to thermal loading

This section presents exact solutions of temperature, displacements, and stresses in a simply supported symmetrically laminated wide plate subjected to thermal loading sinusoidally distributed along the length and deforming in cylindrical bending. The solution assumes no coupling between heat transfer and elastic deformations. The plate has thickness h and length L and is composed of three layers of equal thickness, joined by two identical interfaces. The origin of the coordinate system is at the mid-thickness. The stacking sequence of the plate is (0, 90, 0) and the thermo-elastic constants of the layers are EL =ET ¼ 17, GLT =ET ¼ 0:7, GTT =ET ¼ 0:6, νLT ¼ 0:28, and

Simply supported five-layer plate: normal surface tractions f3 5 f0 sin (πx2 =L) on the upper surface

26

Table 1.2

Elastic constants: EL =ET 5 17, GLT =ET 50:7, GTT =ET 5 0:6, νLT 5 0:28, and νTT 50:4, stacking sequence of (0, 90, 0, 90, 0). Subscripts l5lower layer and u5upper layer. Perfect bonding: 1=KS 50 and 1=KN 5 0 x3/h L/h

21/2

22/5

23/10l

23/10u

21/10l

21/10u

1/10u

3/10l

3/10u

2/5

1/2

0.129 2.721 22.36

0.129 2.721 22.36

0.426 8.173 67.07

0.426 8.173 67.07

0.664 11.11 89.84

1.011 14.32 113.2

5.106 107.2 1505

5.212 107.2 1505

5.212 107.2 1505

5.272 107.2 1504

5.329 107.1 1504

Dimensionless longitudinal displacements: v2ET/(f0h) at x2 5 0 4 10 20

1.016 14.42 113.4

0.697 11.22 90.08

0.482 8.295 67.31

0.482 8.295 67.31

0.205 2.854 22.60

0.205 2.854 22.60

Dimensionless transverse displacements: v3ET/(f0h) at x2 5 L=2 4 10 20

4.905 106.7 1503

L/h

21/2

4.932 106.8 1504 22/5

4.957 107.0 1505

4.957 107.0 1505

23/10l

5.021 107.2 1505 23/10u

5.021 107.2 1505

5.106 107.2 1505 x3/h

21/10l

21/10u

1/10l

1/10u

3/10l

3/10u

2/5

1/2

0.011 0.749 3.416

2.604 15.17 60.50

1.982 14.85 60.23

0.356 1.114 3.784

0.688 2.933 10.94

6.055 44.20 180.3

9.279 60.00 241.4

13.96 77.21 303.9

Dimensionless bending stresses: σ 22/f0 at x2 5L=2 4 10 20

13.62 77.35 304.2

9.332 60.20 241.6

6.414 44.46 180.5

0.326 2.565 10.57

Dynamic Response and Failure of Composite Materials and Structures

1/10l

L/h

213/30

211/30

23/10

27/30

21/10

21/30

1/30

1/10

7/30

3/10

11/30

13/30

1.624 4.114 8.241

1.640 4.117 8.242

1.577 3.912 7.822

1.528 3.840 7.693

1.495 3.785 7.591

1.128 2.751 5.491

0.641 1.494 2.963

0.454 0.457 0.457

0.540 0.543 0.543

0.624 0.628 0.628

0.787 0.790 0.791

0.866 0.870 0.871

0.936 0.939 0.939

0.982 0.984 0.984

Dimensionless transverse shear stresses: σ 23/f0 at x2 5 0. For maxima see

a

4 10 20

0.630 1.497 2.965

1.120 2.759 5.496

1.502 3.797 7.599

1.516 3.845 7.697

1.528 3.901 7.818

Dimensionless transverse normal stresses: σ 33/f0 at x2 5 L=2 4 10 20

0.017 0.016 0.016

0.064 0.061 0.061

0.133 0.130 0.130

0.212 0.210 0.210

0.371 0.373 0.372

Partial bonding: KS h=ET 50:1 and KN h=ET 50:25 (f0 > 0) x3/h L/h

21/2

22/5

23/10l

23/10u

21/10l

21/10u

1/10l

1/10u

3/10l

3/10u

2/5

1/2

3.241 18.00 57.73

3.813 14.35 16.56

3.425 22.17 99.24

3.437 9.216 27.09

0.291 9.204 85.38

4.072 27.85 144.2

45.14 595.8 3762

47.62 598.3 3764

47.74 598.3 3764

50.71 601.7 3767

50.82 601.8 3767

50.83 601.6 3766

2D thermo-elastic solutions for laminates and sandwiches

x3/h

Dimensionless longitudinal displacements: v2ET/(f0h) at x2 5 0 4 10 20

3.477 27.64 144.2

0.298 9.255 85.54

2.838 8.907 27.39

3.522 22.40 99.56

3.110 13.86 16.11

3.328 18.23 58.09

Dimensionless transverse displacements: v3ET/(f0h) at x2 5 L=2 4 10 20

42.74 593.2 3758

42.80 593.4 3759

42.77 593.4 3759

43.66 594.1 3759

43.72 594.2 3760

45.06 595.7 3761

Continued 27

Continued

28

Table 1.2

x3/h L/h

21/2

22/5

23/10l

23/10u

21/10l

21/10u

1/10l

1/10u

3/10l

3/10u

2/5

1/2

2.590 4.523 2.693

44.51 97.65 155.7

43.72 96.82 155.1

2.757 4.271 2.357

3.004 7.336 16.01

45.81 49.12 73.02

4.252 49.75 229.4

55.02 149.8 387.1

1/30

1/10

7/30

3/10

11/30

13/30

2.188 4.573 8.507

2.208 4.581 8.510

0.705 3.235 7.429

0.793 3.252 7.350

0.686 3.138 7.214

2.196 3.477 5.905

1.972 2.434 3.502

0.415 0.450 0.455

0.537 0.548 0.545

0.620 0.632 0.629

0.702 0.769 0.785

0.741 0.836 0.861

0.824 0.908 0.930

0.941 0.972 0.981

Dimensionless bending stresses: σ 22/f0 at x2 5L=2 4 10 20

46.64 148.3 386.9

3.954 49.63 229.5

38.16 47.85 73.43

2.688 7.004 15.65

x3/h 213/30

211/30

23/10

27/30

21/10

21/30

Dimensionless transverse shear stresses: σ 23/f0 at x2 5 0. For maxima see

b

4 10 20

1.678 2.412 3.500

1.886 3.452 5.903

0.636 3.131 7.216

0.731 3.237 7.348

0.644 3.209 7.420

Dimensionless transverse normal stresses: σ 33/f0 at x2 5 L=2 4 10 20

0.051 0.028 0.019

0.150 0.091 0.069

0.223 0.163 0.139

0.259 0.230 0.215

0.334 0.366 0.371

Sliding interfaces in constrained contact: KS 5 0 and 1=KN 5 0 x3/h L/h

21/2

22/5

23/10l

23/10u

21/10l

21/10u

1/10l

1/10u

3/10l

3/10u

2/5

1/2

5.837 90.92 727.0

6.473 92.50 730.1

5.787 90.77 726.7

5.892 91.02 727.2

0.026 0.062 0.123

5.847 90.90 726.9

Dimensionless longitudinal displacements: v2ET/(f0h) at x2 5 0 4 10 20

5.846 90.96 727.1

0.005 0.012 0.024

5.838 90.94 727.0

6.285 92.06 729.3

5.947 91.21 727.6

5.865 90.99 727.1

Dynamic Response and Failure of Composite Materials and Structures

L/h

4 10 20

78.01 2918 46,377

78.11 2918 46,379

78.04 2918 46,377

78.04 2918 46,377

78.11 2918 46,377

78.11 2918 46,377

78.19 2918 46,377

78.19 2918 46,377

78.32 2918 46,377

78.32 2918 46,377

78.47 2919 46,379

78.46 2918 46,377

x3/h L/h

21/2

22/5

23/10l

23/10u

21/10l

21/10u

1/10l

1/10u

3/10l

3/10u

2/5

1/2

4.830 28.92 115.0

78.53 488.1 1951

78.56 488.1 1951

4.840 28.93 115.0

4.842 28.93 115.0

78.77 488.1 1951

0.012 0.002 0.000

78.82 488.1 1951

1/30

1/10

7/30

3/10

11/30

13/30

2.712 6.803 13.61

2.713 6.803 13.61

0.000 0.000 0.000

0.169 0.404 0.803

0.000 0.000 0.000

2.721 6.804 13.61

2.721 6.804 13.61

0.422 0.423 0.423

0.576 0.577 0.577

0.659 0.660 0.660

0.674 0.674 0.674

0.679 0.679 0.679

0.762 0.762 0.762

0.917 0.917 0.917

Dimensionless bending stresses: σ 22/f0 at x2 5 L=2 4 10 20

78.42 488.1 1951

0.002 0.000 0.000

78.43 488.1 1951

4.829 28.92 115.0

x3/h L/h

213/30

211/30

23/10

27/30

21/10

21/30

Dimensionless transverse shear stresses: σ 23/f0 at x2 5 0. For maxima see 4 10 20

2.708 6.803 13.61

2.708 6.803 13.61

0.000 0.000 0.000

0.169 0.404 0.803

0.000 0.000 0.000

2D thermo-elastic solutions for laminates and sandwiches

Dimensionless transverse displacements: v3ET/(f0h) at x2 5L=2

c

Dimensionless transverse normal stresses: σ 33/f0 at x2 5 L=2 4 10 20

0.083 0.083 0.083

0.236 0.237 0.238

0.319 0.321 0.321

0.325 0.326 0.326

0.339 0.340 0.340

Maxima (x3/h, σ 23max/f0): (0.014, 1.643) for L=h ¼ 4, (0.001, 4.142) for L=h ¼ 10, and (0.000, 8.294) for L=h ¼ 20. Maxima (x3/h, σ 23max/f0): (0.001, 2.386) for L=h ¼ 4, (0.000, 4.746) for L=h ¼ 10, and (0.000, 8.644) for L=h ¼ 20. Maxima (x3/h, σ 23max/f0): (0.400, 3.057) for L=h ¼ 4, (0.400, 7.652) for L=h ¼ 10, and (0.400, 15.31) for L=h ¼ 20.

a

b c

29

30

Dynamic Response and Failure of Composite Materials and Structures

x3 h

0.5

0.5

0.25

0.25 x3

0

h

−0.25 −0.5 −9

(A)

0 −0.25

−3

3

v2ET Ⲑ( f0h) Fully bonded

−0.5

9

0

(B)

30 60 v3 ET Ⲑ( f0h)

KSh ⲐET = 0.1

KS = 0

KNh ⲐET = 0.25

1 ⲐKN = 0

90

Fig. 1.5 (A) Longitudinal displacements at x2 ¼ 0 and (B) transverse displacements at x2 ¼ L=2 through thickness in a simply supported five-layer plate with L=h ¼ 4 subjected to normal surface tractions f3 ¼ f0 sin ðπx2 =LÞ on the upper surface. Elastic constants: EL =ET ¼ 17, GLT =ET ¼ 0:7, GTT =ET ¼ 0:6, νLT ¼ 0:28, and νTT ¼ 0:4; stacking sequence, (0, 90, 0, 90, 0).

νTT ¼ 0:4, αT =αL ¼ 1125, and KT =KL ¼ 0:16 (subscripts L and T indicate in-plane principal material directions). The assumed thermo-elastic constants could represent a graphite-epoxy laminate with a large ratio between the coefficients of thermal expansion in the principal material directions [35]. Results are presented in tables and figures for a plate subjected to thermal loading T ðx3 ¼ h=2Þ ¼ T0 sin ðπx2 =LÞ on the upper surface with T0 a positive constant; the temperature increment at the lower surface of the plate is assumed to be zero. Results in Tables 1.3 correspond to layers in perfect thermal contact, R ¼ 0, and those in Table 1.4 to layers with interfacial thermal resistance, RKL =h ¼ 10. Dimensionless stresses and displacements are presented for three length-to-thickness ratios equal to 4, 10, and 20 and for three cases of perfect bonding, 1=KS ¼ 1=KN ¼ 0, partial bonding with dimensionless interfacial stiffnesses equal to KS h=ET ¼ 0:1 and KN h=ET ¼ 0:25, and sliding interfaces in constrained contact, KS ¼ 0 and 1=KN ¼ 0. The assumed applied temperature implies no interpenetration of the layers at the interfaces for all interfacial stiffnesses, including the case 1=KN 6¼ 0; for constrained contact the results apply also to T0 < 0. Figs. 1.7–1.9 highlight the influence of the mechanical and thermal interfacial imperfections on the temperature distribution in the layers and the field variables. A nonzero interfacial thermal resistance induces temperature jumps at the layer interfaces and modifies displacement and stress distributions. In the limiting case of an impermeable interface, the temperature inthe third layer at the upper interface ð 3Þ 2 sh=3 2ð3Þ sh=3 ð3Þ = 1+e sin ðπx2 =LÞ and tends to is given by Tim x2 , x3 ¼ x3 ¼ 2T0 e

2D thermo-elastic solutions for laminates and sandwiches

x3 h

31

0.5

0.5

0.25

0.25 x3

0

h

−0.25 −0.5 −100

0

−0.25

−50

0

50

s22 Ⲑ f0

(A)

−0.5

100

(B)

0

1

2 s23 Ⲑ f0

3

4

0.5 Fully bonded

0.25 x3 h

KSh ⲐET = 0.1

0

KNh ⲐET = 0.25

−0.25 −0.5

(C)

KS = 0 1 ⲐKN = 0 0

0.25

0.5

0.75

1

s33 Ⲑ f0

Fig. 1.6 (A) Bending stresses at x2 ¼ L=2, (B) transverse shear stresses at x2 ¼ 0, and (C) transverse normal stresses at x2 ¼ L=2 through thickness in a simply supported five-layer plate with L=h ¼ 4 subjected to normal surface tractions f3 ¼ f0 sin ðπx2 =LÞ on the upper surface. Elastic constants: EL =ET ¼ 17, GLT =ET ¼ 0:7, GTT =ET ¼ 0:6, νLT ¼ 0:28, and νTT ¼ 0:4; stacking sequence, (0, 90, 0, 90, 0).

the applied temperature on increasing L/h. For large values of L/h, the temperature in the upper layer is ð3Þ T ðx2 ¼ L=2,x3 Þ ¼ T0 .

1.6

Conclusion

In this chapter, the transfer matrix method has been used to efficiently solve the thermo-mechanical problem of simply supported layered plates with thermally and mechanically imperfect interfaces and an arbitrary number of layers. The matrix technique systematizes the analysis and facilitates the solution of the system of algebraic equations resulting from the imposition of continuity and boundary conditions. The method uses local transfer matrices and continuity conditions at the interfaces to

32

Table 1.3 Simply supported three-layer plate: applied temperature on the upper surface T(x3 5h=2) 5 2T0 sin (πx2 =L) with T0 a positive constant Thermo-elastic constants: EL =ET 5 17, GLT =ET 50:7, GTT =ET 50:6, νLT 50:28, and νTT 50:4, αT =αL 5 1125 and KT =KL 50:16, stacking sequence of (0, 90, 0). Layers are in perfect thermal contact, R5 0. Subscripts l5lower layer and u5upper layer. Perfect mechanical bonding: 1=KS 50 and 1=KN 5 0; perfect thermal contact: R 50 x3/h L/h

21/2

27/18

25/18

21/6l

21/6u

1/18

1/6l

1/6u

5/18

7/18

1/2

17.76 76.38 165.9

17.74 82.75 180.7

17.74 82.75 180.7

23.29 92.29 197.2

40.32 107.8 216.9

73.69 130.2 240.0

29.63 223.0 893.6

114.7 325.3 998.8

114.7 325.3 998.8

219.0 447.1 1123

346.8 589.1 1268

503.5 752.3 1432

493.0 581.2 596.0 x3/h

606.8 702.8 718.8

51.82 228.5 264.2

88.97 242.5 271.5

269.3 286.4 286.5

654.9 365.0 310.3

1/4

1/3

5/12

1/2

54.63 22.48 11.23

47.68 16.02 7.656

31.86 8.692 3.933

0.000 0.000 0.000

Dimensionless longitudinal displacements: v2/(αLT0h) at x2 5 0 31.27 53.52 96.41

20.90 54.20 108.3

15.24 56.91 121.2

14.10 62.20 135.4

14.10 62.20 135.4

Dimensionless transverse displacements: v3/(αLT0h) at x2 5 L=2 4 10 20

149.0 5.185 656.9

143.1 3.273 665.8

123.2 30.15 693.9

88.45 75.60 741.2

88.45 75.60 741.2

Dimensionless bending stresses: σ 22/(αLT0ET) at x2 5 L=2 4 10 20

419.4 287.1 258.7

L/h

21/2

253.7 255.8 254.0 25/12

149.3 235.0 251.9 21/3

102.3 227.3 253.4 21/4

21/6

274.5 339.7 351.0 21/12

1/12

Dimensionless transverse shear stresses: σ 23/(αLT0ET) at x2 5 0. For maxima see 4 10 20

0.000 0.000 0.000

22.97 7.187 3.360

38.61 13.81 6.680

49.12 20.02 9.980

56.60 26.01 13.29

36.00 15.93 8.090

21.28 11.33 5.909

1/6 a

58.18 28.53 14.72

Dynamic Response and Failure of Composite Materials and Structures

4 10 20

4 10 20

0.000 0.000 0.000

0.798 0.095 0.022

2.847 0.371 0.088

5.740 0.815 0.197

9.211 1.417 0.349

12.27 1.972 0.490

13.47 2.133 0.529

10.90 1.617 0.396

7.203 0.949 0.226

3.822 0.443 0.102

1.154 0.117 0.026

0.000 0.000 0.000

Partial mechanical bonding: KS h=ET 50:1 and KN h=ET 50:25; perfect thermal contact: R5 0 x3/h L/h

21/2

27/18

25/18

21/6l

21/6u

1/18

1/6l

1/6u

5/18

7/18

1/2

293.6 305.6 297.6

297.3 315.1 314.4

8.676 71.28 173.4

15.40 83.65 191.9

32.17 101.7 213.5

63.77 126.5 238.6

35.84 298.7 1006

111.5 397.8 1111

83.56 392.0 1109

188.4 514.0 1234

316.6 656.0 1378

473.6 819.2 1542

273.0 508.7 575.3

384.6 629.4 697.7

68.24 167.0 244.6

16.16 196.2 257.2

160.1 253.9 277.5

521.8 345.5 306.4

Dimensionless longitudinal displacements: v2/(αLT0h) at x2 5 0 4 10 20

19.92 43.30 90.10

12.48 46.86 104.0

8.193 52.25 118.8

7.138 60.07 135.0

290.6 286.2 263.2

Dimensionless transverse displacements: v3/(αLT0h) at x2 5 L=2 4 10 20

104.8 81.39 772.7

98.56 89.97 781.6

78.22 116.9 809.7

43.00 162.4 857.0

63.70 157.6 855.7

2D thermo-elastic solutions for laminates and sandwiches

Dimensionless transverse normal stresses: σ 33/(αLT0ET) at x2 5L=2

Dimensionless bending stresses: σ 22/(αLT0ET) at x2 5 L=2 4 10 20

267.2 232.3 241.7

141.0 216.5 242.4

55.46 210.0 245.6

10.54 216.0 252.3

54.72 268.9 330.8

Continued

33

34

Table 1.3

Continued x3/h

L/h

21/2

25/12

21/3

21/4

21/6

21/12

1/12

1/6

Dimensionless transverse shear stresses: σ 23/(αLT0ET) at x2 5 0. For maxima see 4 10 20

0.000 0.000 0.000

14.12 5.907 3.165

22.57 11.54 6.342

26.82 17.05 9.551

28.35 22.62 12.82

22.10 14.40 7.886

1/4

1/3

5/12

1/2

b

28.87 24.38 14.10

32.69 19.79 10.85

32.61 14.49 7.453

23.87 8.054 3.858

0.000 0.000 0.000

8.116 1.870 0.510

6.990 1.437 0.383

4.966 0.858 0.220

2.796 0.408 0.100

0.885 0.110 0.026

0.000 0.000 0.000

Dimensionless transverse normal stresses: σ 33/(αLT0ET) at x2 5L=2 4 10 20

0.000 0.000 0.000

0.497 0.078 0.021

1.725 0.307 0.083

3.360 0.681 0.187

5.176 1.200 0.333

6.856 1.690 0.470

Sliding interfaces in constrained contact: KS 50 and 1=KN 5 0; perfect thermal contact: R 50 x3/h L/h

21/2

27/18

25/18

21/6l

21/6u

1/18

1/6l

1/6u

5/18

7/18

1/2

567.7 1728 3563

574.2 1746 3599

5.392 16.22 44.92

6.027 37.02 82.54

25.77 62.16 122.4

59.16 92.88 165.1

31.28 465.4 2019

97.57 544.4 2100

97.57 544.4 2100

203.0 667.1 2226

331.5 809.8 2371

488.7 973.5 2536

Dimensionless longitudinal displacements: v2/(αLT0h) at x2 5 0 4 10 20

14.56 5.520 27.66

5.931 4.345 4.232

0.756 14.79 36.39

5.855 26.57 69.28

567.3 1698 3494

Dimensionless transverse displacements: v3/(αLT0h) at x2 5 L=2 4 10 20

112.6 281.4 1828

106.1 290.6 1838

85.41 318.1 1867

49.62 364.2 1915

49.62 364.2 1915

Dynamic Response and Failure of Composite Materials and Structures

6.447 9.089 5.573

4 10 20

195.4 29.62 74.19

53.29 11.61 25.15

64.07 9.241 24.53

162.5 36.73 76.05

164.9 177.3 179.2

54.84 59.08 59.73

164.9 177.3 179.2

255.7 127.9 99.95

141.4 53.86 36.22

74.36 41.60 33.09

460.0 164.8 109.3

x3/h L/h

21/2

25/12

21/3

21/4

21/6

21/12

1/12

1/6

1/4

1/3

5/12

1/2

8.087 3.481 1.759

0.000 0.000 0.000

14.39 2.667 1.002

22.16 3.785 1.365

19.31 3.028 1.048

0.000 0.000 0.000

3.617 0.214 0.035

3.911 0.265 0.048

3.412 0.227 0.040

2.170 0.139 0.024

0.742 0.045 0.008

0.000 0.000 0.000

Dimensionless transverse shear stresses: σ 23/(αLT0ET) at x2 5 0. For maxima see 4 10 20

0.000 0.000 0.000

9.119 0.601 0.730

11.60 0.840 0.979

8.328 0.670 0.740

0.000 0.000 0.000

8.086 3.481 1.759

c

Dimensionless transverse normal stresses: σ 33/(αLT0ET) at x2 5L=2 4 10 20

0.000 0.000 0.000

0.337 0.009 0.005

1.049 0.028 0.017

1.731 0.049 0.029

2.029 0.059 0.034

2.323 0.008 0.021

2D thermo-elastic solutions for laminates and sandwiches

Dimensionless bending stresses: σ 22/(αLT0ET) at x2 5 L=2

Maxima [x3/h, σ 23max/(αLT0ET)]: [0.167, 58.18] for L=h ¼ 4, [0.167, 28.53] for L=h ¼ 10, and [0.167, 14.72] for L=h ¼ 20. Maxima [x3/h, σ 23max/(αLT0ET)]: [0.293, 33.40] for L=h ¼ 4, [0.167, 24.38] for L=h ¼ 10, and [0.167, 14.10] for L=h ¼ 20. Maxima [x3/h, σ 23max/(αLT0ET)]: [0.358, 22.64] for L=h ¼ 4, [0.000, 4.640] for L=h ¼ 10, and [0.000, 2.346] for L=h ¼ 20.

a

b c

35

36

Table 1.4 Simply supported three-layer plate: Applied temperature on the upper surface T(x3 5h=2) 5 2T0 sin (πx2 =L) with T0 a positive constant Thermo-elastic constants: EL =ET 5 17, GLT =ET 50:7, GTT =ET 50:6, νLT 50:28, and νTT 50:4, αT =αL 5 1125 and KT =KL 50:16, stacking sequence of (0, 90, 0). Layers are in imperfect thermal contact, RKL =h510. Subscripts l5lower layer and u5upper layer. Perfect mechanical bonding: 1=KS 50 and 1=KN 5 0; imperfect thermal contact: RKL =h 510 x3/h L/h

21/2

27/18

25/18

21/6l

21/6u

1/18

1/6l

1/6u

5/18

7/18

1/2

15.51 75.53 166.4

15.19 82.70 183.6

15.19 82.70 183.6

21.02 93.49 202.8

39.34 111.0 225.4

74.53 135.8 251.8

22.34 248.9 1035

84.64 334.5 1124

84.64 334.5 1124

218.6 492.0 1286

363.5 655.3 1452

525.8 825.5 1624

389.7 520.5 544.7 x3/h

416.8 547.7 571.7

50.24 146.6 188.1

14.79 194.9 230.5

234.8 276.5 281.6

666.1 395.2 342.1

1/4

1/3

5/12

1/2

47.07 21.37 10.86

44.66 16.03 7.768

31.44 9.085 4.169

0.000 0.000 0.000

Dimensionless longitudinal displacements: v2/(αLT0h) at x2 5 0 23.91 45.18 83.21

17.13 48.46 98.38

13.68 53.27 114.3

13.05 59.84 131.2

13.05 59.84 131.2

Dimensionless transverse displacements: v3/(αLT0h) at x2 5 L=2 4 10 20

100.3 74.27 850.3

99.72 75.74 851.9

96.46 81.35 858.0

90.56 91.12 868.5

90.56 91.12 868.5

Dimensionless bending stresses: σ 22/(αLT0ET) at x2 5 L=2 4 10 20

320.8 242.4 223.2

L/h

21/2

224.4 251.9 255.3 25/12

171.8 269.5 289.4 21/3

156.0 296.3 326.0 21/4

21/6

343.3 467.8 490.9 21/12

1/12

Dimensionless transverse shear stresses: σ 23/(αLT0ET) at x2 5 0. For maxima see 4 10 20

0.000 0.000 0.000

18.32 6.421 3.077

32.50 13.07 6.474

44.09 20.06 10.21

54.47 27.53 14.29

31.48 15.02 7.733

17.93 11.53 6.174

1/6 a

44.53 25.60 13.52

Dynamic Response and Failure of Composite Materials and Structures

4 10 20

Dimensionless transverse normal stresses: σ 33/(αLT0ET) at x2 5L=2 0.000 0.000 0.000

0.627 0.084 0.020

2.308 0.338 0.082

4.825 0.771 0.191

8.054 1.393 0.351

10.87 1.951 0.495

11.81 2.051 0.518

9.774 1.566 0.389

6.764 0.950 0.229

3.721 0.457 0.107

1.153 0.124 0.028

0.000 0.000 0.000

Partial bonding: KS h=ET 5 0:1 and KN h=ET 5 0:25; imperfect thermal contact: RKL =h510 x3/h L/h

21/2

27/18

25/18

21/6l

21/6u

1/18

1/6l

1/6u

5/18

7/18

1/2

250.6 296.4 295.7

248.0 301.9 311.9

15.52 79.65 181.6

17.44 88.54 199.8

31.23 104.0 221.4

60.38 126.6 246.7

28.93 189.0 973.6

25.50 271.5 1063

1.901 266.0 1061

136.1 423.6 1223

281.4 587.0 1389

444.1 757.3 1560

202.2 450.7 524.2

231.6 478.4 551.5

44.21 130.4 182.8

32.29 168.3 222.4

126.2 239.0 270.7

476.4 345.5 328.2

1/4

1/3

5/12

1/2

26.67 18.51 10.45

27.83 13.90 7.463

21.18 7.906 4.002

0.000 0.000 0.000

Dimensionless longitudinal displacements: v2/(αLT0h) at x2 5 0 4 10 20

19.50 40.93 80.67

10.88 42.43 94.82

4.430 45.28 109.7

0.901 49.65 125.5

260.0 285.3 262.7

Dimensionless transverse displacements: v3/(αLT0h) at x2 5 L=2 4 10 20

116.3 25.17 792.4

115.5 26.71 794.1

111.9 32.43 800.2

105.4 42.33 810.7

125.6 37.47 809.4

2D thermo-elastic solutions for laminates and sandwiches

4 10 20

Dimensionless bending stresses: σ 22//(αLT0ET) at x2 5 L=2 4 10 20

261.6 219.6 216.4

140.6 219.6 245.8

48.09 226.6 277.1

30.06 241.7 310.8

147.2 396.6 470.2

x3/h L/h

21/2

25/12

21/3

21/4

21/6

21/12

1/12

Dimensionless transverse shear stresses: σ 23/(αLT0ET) at x2 5 0. For maxima see 0.000 0.000 0.000

13.92 5.733 2.975

22.33 11.50 6.242

26.19 17.40 9.817

26.09 23.56 13.72

15.80 12.91 7.429

8.817 9.969 5.941

23.24 22.22 13.03

Continued

37

4 10 20

1/6 b

38

Table 1.4

Continued

Dimensionless transverse normal stresses: σ 33/(αLT0ET) at x2 5L=2 4 10 20

0.000 0.000 0.000

0.489 0.075 0.019

1.703 0.300 0.079

3.313 0.678 0.184

5.045 1.214 0.338

6.423 1.692 0.476

6.941 1.779 0.498

5.899 1.359 0.374

4.267 0.824 0.220

2.457 0.397 0.102

0.794 0.108 0.027

0.000 0.000 0.000

Sliding interfaces in constrained contact: KS 50 and 1=KN 50; imperfect thermal contact: RKL =h 5 10 x3/h 21/2

27/18

25/18

21/6l

21/6u

1/18

1/6l

1/6u

5/18

7/18

1/2

485.9 1677 3526

483.5 1679 3534

6.615 44.70 102.0

10.65 51.06 111.7

25.31 62.86 124.3

54.24 80.62 139.9

54.25 44.39 413.7

7.827 107.4 479.8

7.827 107.4 479.8

126.9 265.6 641.9

272.4 429.5 808.9

435.3 600.4 981.1

14.83 14.38 14.30

44.61 43.16 42.91

162.6 56.68 30.84

122.9 32.56 13.82

46.95 18.48 10.22

394.0 99.04 41.65

Dimensionless longitudinal displacements: v2/(αLT0h) at x2 5 0 4 10 20

17.26 6.508 2.454

5.702 3.565 2.351

4.640 0.829 7.229

15.53 1.675 12.27

501.8 1677 3513

Dimensionless transverse displacements: v3/(αLT0h) at x2 5 L=2 4 10 20

146.6 87.12 275.0

145.7 85.08 277.2

141.8 78.77 284.0

134.8 68.14 295.3

134.8 68.14 295.3

Dimensionless bending stresses: σ 22/(αLT0ET) at x2 5 L=2 4 10 20

231.5 34.92 6.584

71.22 11.11 2.303

73.24 11.65 2.157

225.3 33.29 7.023

44.58 43.16 42.91

Dynamic Response and Failure of Composite Materials and Structures

L/h

L/s

21/2

25/12

21/3

21/4

21/6

21/12

1/12

1/6

1/4

1/3

5/12

1/2

2.188 0.847 0.421

0.000 0.000 0.000

10.38 1.311 0.329

17.35 2.019 0.473

15.96 1.725 0.382

0.000 0.000 0.000

2.978 0.129 0.013

3.058 0.141 0.017

2.711 0.123 0.014

1.772 0.078 0.009

0.622 0.026 0.003

0.000 0.000 0.000

Dimensionless transverse shear stresses: σ 23/(αLT0ET) at x2 5 0. For maxima see 4 10 20

0.000 0.000 0.000

11.04 0.678 0.065

14.52 0.893 0.088

10.89 0.661 0.067

0.000 0.000 0.000

2.187 0.847 0.421

c

Dimensionless transverse normal stresses: σ 33/(αLT0ET) at x2 5L=2 4 10 20

0.000 0.000 0.000

0.405 0.010 0.000

1.281 0.031 0.002

2.151 0.053 0.003

2.549 0.062 0.003

2.628 0.075 0.000

Maxima [x3/h, σ 23max/(αLT0ET)]: [0.167, 54.47] for L=h ¼ 4, [0.167, 27.53] for L=h ¼ 10, and [0.167, 14.29] for L=h ¼ 20. Maxima [x3/h, σ 23max/(αLT0ET)]: [0.312, 28.04] for L=h ¼ 4, [0.167, 23.56] for L=h ¼ 10, and [0.167, 13.72] for L=h ¼ 20. c Maxima [x3/h, σ 23max/(αLT0ET)]: [0.366, 18.10] for L=h ¼ 4, [0.335, 2.056] for L=h ¼ 10, and [0.000, 0.562] for L=h ¼ 20. a

b

2D thermo-elastic solutions for laminates and sandwiches

x3/h

39

40

Dynamic Response and Failure of Composite Materials and Structures

0.5

0.9

–Tim/T0

–T 2u /T0 0.8

0.25 x3 h

0

0.7

Tim/T0

0.6

–0.25 –0.5

(A)

0.5 –1

–0.75

–0.5

–0.25

0

T/T0

Perfect thermal contact

(B)

0

50

100

150

200

RKL/h

Impermeable interface

RKL/h = 10

Fig. 1.7 (A) Through-thickness temperature distribution at x2 ¼ L=2 in a three-layer plate with L=h ¼ 4 subjected to an applied temperature T ðx3 ¼ h=2Þ ¼ T0 sin ðπx2 =LÞ on the upper surface ðT0 > 0Þ and (B) normalized temperature in the upper layer at the interface at x2 ¼ L=2 on increasing RKL=h (RKL=h ¼ 0 perfect contact; h=ðRKL Þ ¼ 0 impermeable interface). The thermal conductivities of the layers are KT =KL ¼ 0:16; stacking sequence (0, 90, 0).

x3 h

0.5

0.5

0.25

0.25 x3

0

h

−0.25

−0.25 −0.5 −60

(A)

0

240 v2 Ⲑ(aLT0h) Fully bonded

−0.5 −600

540

(B)

0 −400 −200 v3 Ⲑ(aLT0h)

KSh ⲐET = 0.1

KS = 0

KNh ⲐET = 0.25

1 ⲐKN = 0

200

Fig. 1.8 (A) Longitudinal displacements at x2 ¼ 0 and (B) transverse displacements at x2 ¼ L=2 through thickness in a simply supported three-layer plate with L=h ¼ 4 subjected to an applied temperature T ðx3 ¼ h=2Þ ¼ T0 sin ðπx2 =LÞ on the upper surface ðT0 > 0Þ. Thermo-elastic constants: EL =ET ¼ 17, GLT =ET ¼ 0:7, GTT =ET ¼ 0:6, νLT ¼ 0:28, and νTT ¼ 0:4, αT =αL ¼ 1125 and KT =KL ¼ 0:16. Layers are in imperfect thermal contact with interfacial thermal resistance RKL =h ¼ 10; stacking sequence, (0, 90, 0).

2D thermo-elastic solutions for laminates and sandwiches

x3 h

41

0.5

0.5

0.25

0.25 x3

0

−0.25 −0.5 −800

(A)

0

h

−0.25

0 400 −400 s22 Ⲑ(aLT0 ET)

800

−0.5 −70

(B)

0.5

0 35 −35 s23 Ⲑ(aLT0 ET)

70

Fully bonded

0.25 x3 h

KSh ⲐET = 0.1 KNh ⲐET = 0.25

0

KS=0

−0.25

1 ⲐKN = 0 −0.5

(C)

0

10 5 s33 Ⲑ(aLT0 ET)

15

Fig. 1.9 (A) Bending stresses at x2 ¼ L=2, (B) transverse shear stresses at x2 ¼ 0, and (C) transverse normal stresses at x2 ¼ L=2 through thickness in a simply supported three-layer plate with L=h ¼ 4 subjected to an applied temperature T ðx3 ¼ h=2Þ ¼ T0 sin ðπx2 =LÞ on the upper surface ðT0 > 0Þ. Thermo-elastic constants: EL =ET ¼ 17, GLT =ET ¼ 0:7, GTT =ET ¼ 0:6, νLT ¼ 0:28, and νTT ¼ 0:4, αT =αL ¼ 1125, and KT =KL ¼ 0:16. Layers are in imperfect thermal contact with interfacial thermal resistance RKL =h ¼ 10; stacking sequence, (0, 90, 0).

establish explicit matrix relationships between the unknown integration constants in the solution of a generic layer and those of the first layer. In this manner, all field variables in each layer are defined only in terms of the unknown constants of the first layer. Once the integration constants have been derived, the thermo-elasticity problem of a plate with many layers is reduced to that of a single-layer plate whose solution can be obtained by the imposition of the boundary conditions: the number of equations which needs to be solved becomes independent of the number of layers and is equal to four (two-dimensional problem).

42

Dynamic Response and Failure of Composite Materials and Structures

Explicit formulas are presented for the integration constants of the problem. A set of benchmark solutions is presented in tabular and graph forms for plates with different layups, length-to-thickness ratios and interfacial stiffness and thermal resistance, to highlight the efficacy of the method and the important effect of the imperfections on the field variables. Any other desirable benchmark solution can be conveniently generated by the given explicit formulas.

Appendix: Displacement functions of the layer k and

(k)

E

Positive discriminant and negative roots The displacement functions of Eq. (1.27) take the following forms: ðkÞ

V ðx3 Þ ¼

2 X

ðkÞ

ðkÞ

a1j cos

mj x3 + ðkÞ a2j sin ðkÞ mj x3

j¼1 2 X ðkÞ W ðx3 Þ ¼ ðkÞ λj ðkÞ a2j cos ðkÞ mj x3 + ðkÞ λj ðkÞ a1j sin ðkÞ mj x3 ,

(1.43)

j¼1

where

ðkÞ

mj ¼ ðkÞ

ðkÞ

λj ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðkÞ γ j and

ðkÞ

tj ¼ iðkÞ mj , and

! ðC23 + C55 Þpmj : C33 m2j + C55 p2

The components of the matrix

m1 x 3 ,

ðkÞ

m2 x 3 ,

ðkÞ

E are:

ðkÞ

E11 ¼ cos

ðkÞ

E14 ¼ sin

ðkÞ

E23 ¼ ðkÞ λ2 sin

ðkÞ

E31 ¼ ðkÞ ðC33 λ1 m1 C23 pÞ cos

ðkÞ

E33 ¼ ðkÞ ðC33 λ2 m2 C23 pÞ cos

ðkÞ

E41 ¼ ðkÞ C55 ðkÞ ðλ1 p m1 Þ sin

ðkÞ

E43 ¼ ðkÞ C55 ðkÞ ðλ2 p m2 Þ sin

ðkÞ

(k)

ðkÞ

ðkÞ

E12 ¼ sin

ðkÞ

E21 ¼ ðkÞ λ1 sin

m2 x3 ,

ðkÞ

m1 x3 ,

ðkÞ

m1 x3 ,

E24 ¼ ðkÞ λ2 cos

ðkÞ

ðkÞ

E13 ¼ cos ðkÞ

ðkÞ

m2 x3 ,

E22 ¼ ðkÞ λ1 cos

ðkÞ

m1 x3 ,

m2 x 3 ,

m1 x 3 ,

ðkÞ

E32 ¼ ðkÞ ðC33 λ1 m1 C23 pÞsin

m2 x 3 ,

ðkÞ

E34 ¼ ðkÞ ðC33 λ2 m2 C23 pÞsin

ðkÞ

ðkÞ

ðkÞ

m1 x 3 ,

ðkÞ

E42 ¼ ðkÞ C55 ðkÞ ðλ1 p m1 Þ cos

ðkÞ

m2 x 3 ,

ðkÞ

E44 ¼ ðkÞ C55 ðkÞ ðλ2 p m2 Þ cos

ðkÞ

m1 x 3 ,

ðkÞ

m2 x 3 ,

ðkÞ

m1 x3 ,

ðkÞ

m2 x3 :

(1.44)

2D thermo-elastic solutions for laminates and sandwiches

43

Zero discriminant This case occurs when the layer is isotropic, ðkÞ γ 1, 2 ¼ ðkÞ ðA1 =2A0 Þ ¼ p2 , with p ¼ mπ=L. Then, ðkÞ tj ¼ p for j ¼ 1, 2 and the displacement functions are: ðkÞ

ðkÞ

V ðx 3 Þ ¼

ðkÞ

a11 + ðkÞ a21 x3 epx3 + ðkÞ a12 + ðkÞ a22 x3 epx3

W ð x3 Þ ¼

ðkÞ

a11 +

4ðkÞ ν 3 ðkÞ 4ðkÞ ν 3 ðkÞ a21 + ðkÞ a21 x3 epx3 ðkÞ a12 + a22 ðkÞ a22 x3 epx3 , p p

(1.45) where ν and E are the Poisson ratio and Young’s modulus of the layer. The components of the matrix (k)E in this case are: ðkÞ

2

3 x3 epx3 epx3 x3 epx3

7 4ν 3 4ν 3 epx3 + x3 epx3 x3 epx3 7 7 p p 7 px3 + 2ðν 1Þ px3 pE px3 px3 + 2ð1 νÞ px3 7 7: Ee e Ee 7 7 1+ν 1+ν 1+ν 5 px3 + 2ν 1 px3 pE px3 px3 2ν + 1 px3 Ee e Ee 1+ν 1+ν 1+ν (1.46)

epx3

6 6 epx3 6 6 ðkÞ Eð x 3 Þ ¼ 6 6 pE px3 e 6 61 + ν 4 pE px3 e 1+ν

Negative discriminant This case occurs when the transverse stiffness of the layer is much higher than the in-plane stiffnesses. For this case, ðkÞ γ 1, 2 ¼ ðkÞ ðμr iμc Þ¼ðkÞ ½r ð cos θ i sin θÞ pﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ jΔj=2A0 , ðkÞ r ¼ ðkÞ μ2c + ðkÞ μ2r , and where ðkÞ μr ¼ ðkÞ ðA1 =2A0 Þ, ðkÞ μc ¼ ðkÞ θ ¼ arctan ðkÞ ðμc =μr Þ. The displacement functions take the following forms, with pﬃﬃ

pﬃﬃ

ðkÞ t1, 2, 3, 4 ¼ ðkÞ ðρ1 iρ2 Þ, ðkÞ ρ1 ¼ ðkÞ r cos ðθ=2Þ , and ðkÞ ρ2 ¼ ðkÞ r sin ðθ=2Þ : ðkÞ

ðkÞ

V ðx3 Þ ¼ ðkÞ a11 e

ðkÞ

ρ 1 x3

+ ðkÞ a12 e ðkÞ

W ðx 3 Þ ¼

ρ1 x3

ðkÞ

ðkÞ ρ2 x3 + ðkÞ a21 e ρ1 x3 sin ðkÞ ρ2 x3

cos

ðkÞ

ðkÞ ρ2 x3 + ðkÞ a22 e ρ1 x3 sin ðkÞ ρ2 x3

ðkÞ r1 ðkÞ a11 ðkÞ r2 ðkÞ a21 e ρ1 x3 cos ðkÞ ρ2 x3

ðkÞ

+

ðkÞ

cos

ðkÞ

ðkÞ r2 ðkÞ a11 + ðkÞ r1 ðkÞ a21 e ρ1 x3 sin ðkÞ ρ2 x3

ðkÞ + ðkÞ r1 ðkÞ a12 ðkÞ r2 ðkÞ a22 e ρ1 x3 cos ðkÞ ρ2 x3 +

ðkÞ

ðkÞ r2 ðkÞ a12 ðkÞ r1 ðkÞ a22 e ρ1 x3 sin ðkÞ ρ2 x3 ,

(1.47)

44

Dynamic Response and Failure of Composite Materials and Structures

where ! ðkÞ ρ C p2 C ρ2 + ρ2

22 55 1 ðkÞ 1 2 r1 ¼ pðC23 + C55 Þ ρ21 + ρ22 ! ðkÞ ρ C p2 + C ρ2 + ρ2

22 55 1 2 2 ðkÞ : r2 ¼ pðC23 + C55 Þ ρ21 + ρ22 The components of the matrix ρ2 x3 , ðkÞ ðkÞ E13 ¼ e ρ1 x3 cos ðkÞ ρ2 x3 , ðkÞ

ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ

E11 ¼ e

E21 ¼ e E22 ¼ e E23 ¼ e

ðkÞ

ρ1 x3

ðkÞ

ðkÞ

E in this case are:

E12 ¼ e

ðkÞ

ðkÞ

ρ1 x3

E14 ¼ e

ðkÞ

ρ1 x3 ðkÞ

½r1 cos ðρ2 x3 Þ + r2 sin ðρ2 x3 Þ,

ðkÞ

ρ1 x3 ðkÞ

½r1 sin ðρ2 x3 Þ r2 cos ðρ2 x3 Þ,

ðkÞ ρ1 x3 ðkÞ

ðkÞ

ðkÞ

ðkÞ

sin

ρ2 x 3 , ðkÞ

ρ2 x3 ,

½r2 sin ðρ2 x3 Þ r1 cos ðρ2 x3 Þ,

ρ1 x3 ðkÞ

ρ1 x3 ðkÞ

sin

ρ1 x 3

ðkÞ

E24 ¼ e E31 ¼ e

cos

(k)

(1.48)

½r2 cos ðρ2 x3 Þ + r1 sin ðρ2 x3 Þ,

½pC23 cos ðρ2 x3 Þ + C33 ðρ1 r1 cos ðρ2 x3 Þ ρ2 r1 sin ðρ2 x3 Þ

+ ρ1 r2 sin ðρ2 x3 Þ + ρ2 r2 cos ðρ2 x3 ÞÞ, ðkÞ

E32 ¼ e

ðkÞ

ρ1 x3 ðkÞ

½pC23 sin ðρ2 x3 Þ + C33 ðρ1 r2 cos ðρ2 x3 Þ

+ ρ2 r2 sin ðρ2 x3 Þ + ρ1 r1 sin ðρ2 x3 Þ + ρ2 r1 cos ðρ2 x3 ÞÞ, ðkÞ

E33 ¼ e

ðkÞ

ρ1 x3 ðkÞ

½pC23 cos ðρ2 x3 Þ + C33 ðρ1 r1 cos ðρ2 x3 Þ + ρ2 r1 sin ðρ2 x3 Þ

ρ1 r2 sin ðρ2 x3 Þ + ρ2 r2 cos ðρ2 x3 ÞÞ, ðkÞ

E34 ¼ e

ðkÞ

ρ1 x3 ðkÞ

½pC23 sin ðρ2 x3 Þ + C33 ðρ1 r2 cos ðρ2 x3 Þ + ρ2 r2 sin ðρ2 x3 Þ

+ ρ1 r1 sin ðρ2 x3 Þ ρ2 r1 cos ðρ2 x3 ÞÞ, ðkÞ

ρ1 x3 ðkÞ

C55 ðkÞ ½ρ1 cos ðρ2 x3 Þ ρ2 sin ðρ2 x3 Þ + pðr1 cos ðρ2 x3 Þ + r2 sin ðρ2 x3 ÞÞ,

ðkÞ

ρ1 x3 ðkÞ

C55 ðkÞ ½ρ1 sin ðρ2 x3 Þ + ρ2 cos ðρ2 x3 Þ + pðr1 sin ðρ2 x3 Þ r2 cos ðρ2 x3 ÞÞ,

ðkÞ

E41 ¼ e

ðkÞ

E42 ¼ e

ðkÞ

E43 ¼ e

ðkÞ

ρ1 x3 ðkÞ

C55 ðkÞ ½ρ1 cos ðρ2 x3 Þ ρ2 sin ðρ2 x3 Þ + pðr1 cos ðρ2 x3 Þ + r2 sin ðρ2 x3 ÞÞ,

ðkÞ

E44 ¼ e

ðkÞ

ρ1 x3 ðkÞ

C55 ðkÞ ½ρ1 sin ðρ2 x3 Þ + ρ2 cos ðρ2 x3 Þ pðr1 sin ðρ2 x3 Þ + r2 cos ðρ2 x3 ÞÞ: (1.49)

Acknowledgment Support by the U.S. Office of Naval Research, contract no. N00014-14-1-0254, program manager Dr. Y.D.S. Rajapakse, is acknowledged. The authors wish to thank Prof. Serge Abrate who introduced them to the transfer matrix method and provided references.

2D thermo-elastic solutions for laminates and sandwiches

45

References [1] Pelassa M, Massabo` R. Explicit solutions for multi-layered wide plates and beams with perfect and imperfect bonding and delaminations under thermo-mechanical loading. Meccanica 2015;50:2497–524. [2] Pagano NJ. Exact solutions for composite laminates in cylindrical bending. J Compos Mater 1969;3:398–411. [3] Pagano NJ, Wang ASD. Further study of composite laminates under cylindrical bending. J Compos Mater 1971;5:521–8. [4] Murakami H. Assessment of plate theories for treating the thermomechanical response of layered plates. Compos Eng 1993;3:137–49. [5] Kardomateas GA, Phan CN. Three-dimensional elasticity solution for sandwich beams/ wide plates with orthotropic phases: the negative discriminant case. J Sandw Struct Mater 2011;13:641–61. [6] Tauchert TR. Thermoelastic analysis of laminated orthotropic slabs. J Therm Stresses 1980;3:117–32. [7] Pagano NJ. Exact solutions for rectangular bidirectional composites and sandwich plates. J Compos Mater 1970;4:20–34. [8] Wu ZJ, Wardenier J. Further investigation on the exact elasticity solution for anisotropic thick rectangular plates. Int J Solids Struct 1998;35:747–58. [9] Pagano NJ, Hatfield SJ. Elastic behavior of multilayered bidirectional composites. AIAA J 1972;10:931–3. [10] Srinivas S, Rao AK, Goga Rao CV. Flexure of simply supported thick homogeneous and laminated rectangular plates. Z Angew Math Mech 1969;49:449–58. [11] Srinivas S, Rao AK. Bending, vibration and buckling of simply supported thick orthotropic rectangular plates and laminates. Int J Solids Struct 1970;6:1463–81. [12] Savoia M, Reddy JN. Three-dimensional thermal analysis of laminated composite plates. Int J Solids Struct 1995;32:593–608. [13] Bhaskar K, Varadan TK, Ali JSM. Thermoelastic solutions for orthotropic and anisotropic composite laminates. Compos B Eng 1996;27:415–20. [14] Tungikar VB, Rao KM. Three dimensional exact solution of thermal stresses in rectangular composite laminate. Compos Struct 1994;27:419–30. [15] Srinivas S, Rao AK. Flexure of thick rectangular plates. J Appl Mech 1973;40:298–9. [16] Vel SS, Batra RC. The generalized plane strain deformations of thick anisotropic composite laminated plates. Int J Solids Struct 2000;37:715–33. [17] Kapuria S, Kumari P. Multiterm extended Kantorovich method for three-dimensional elasticity solution of laminated plates. J Appl Mech 2012;79:061018. [18] Thomson WT. Transmission of elastic waves through a stratified solid medium. J Appl Phys 1950;21:89–93. [19] Haskell NA. The dispersion of surface waves on multilayered media. Bull Seismol Soc Am 1953;43:17–34. [20] Fan J, Ye J. An exact solution for the statics and dynamics of laminated thick plates with orthotropic layers. Int J Solids Struct 1990;26:655–62. [21] Qian H, Zhou D, Liu WQ, Fang H, Lu WD. 3-D elasticity solutions of layered rectangular plates subjected to thermo-loads. J Therm Stresses 2015;38:377–98. [22] Qian H, Zhou D, Liu W, Fang H, Lu W. Elasticity solutions of simply supported laminated cylindrical arches subjected to thermo-loads. Compos Struct 2015;131:273–81. [23] Lowe MJS. Matrix techniques for modeling ultrasonic waves in multilayered media. IEEE Trans Ultrason Ferroelectr Freq Control 1995;42:525–42.

46

Dynamic Response and Failure of Composite Materials and Structures

[24] Abrate S. Impact on laminated composite materials. Appl Mech Rev 1991;44:155–90. [25] Williams TO, Addessio FL. A general theory for laminated plates with delaminations. Int J Solids Struct 1997;34:2003–24. [26] Massabo` R, Campi F. Assessment and correction of theories for multilayered plates with imperfect interfaces. Meccanica 2015;50:1045–71. [27] Chen WQ, Cai JB, Ye GR. Exact solutions of cross-ply laminates with bonding imperfections. AIAA J 2003;41:2244–50. [28] Chen WQ, Wang YF, Cai JB, Ye GR. Three-dimensional analysis of cross-ply laminated cylindrical panels with weak interfaces. Int J Solids Struct 2004;41:2429–46. [29] Kapuria S, Dhanesh N. Three-dimensional extended Kantorovich solution for accurate prediction of interlaminar stresses in composite laminated panels with interfacial imperfections. J Eng Mech 2015;141:04014140. [30] Boley BA, Weiner JH. Theory of thermal stresses. New York: Wiley; 1960. [31] Chen TC, Jang HI. Thermal stresses in a multilayered anisotropic medium with interface thermal resistance. J Appl Mech Trans ASME 1995;62:810–1. [32] Blandford GE, Tauchert TR. Thermoelastic analysis of layered structures with imperfect layer contact. Comput Struct 1985;21:1283–91. [33] Levy A, Heider D, Tierney J, Gillespie JW. Inter-layer thermal contact resistance evolution with the degree of intimate contact in the processing of thermoplastic composite laminates. J Compos Mater 2014;48:491–503. [34] Darban H, Massabo` R. Thermo-elastic solutions for multilayered wide plates and beams with interfacial imperfections through the transfer matrix method. Meccanica, 2017. http://dx.doi.org/10.1007/s11012-017-0657-6 (in press). [35] Tsai SW, Hahn HT. Introduction to composite materials. Lancaster: Technomic Publishing Company, Inc; 1980.

Dynamic behavior of composite marine propeller blades

2

S. Abrate Southern Illinois University, Carbondale, IL, United States

2.1

Introduction

Traditionally, marine propellers are made out of manganese aluminum bronze (MAB) or nickel aluminum bronze (NAB). A recent trend is to design propeller blades made out of composite materials that present several advantages including: reducing weight, improving corrosion resistance, and reducing cost through net shape manufacturing. The earliest publications on composite marine propeller blade appear to be those of Lin [1,2]. Since then, many articles have appeared on this subject and this chapter provides an assessment of the current state of knowledge and offers suggestions for future studies. Section 2.2 addresses the dynamics of rotating beams including mathematical models with various degrees of refinement and basic results regarding the effect of the centrifugal force, the pitch angle, pretwisting, skew angle, and elastic end restraints. Since the blade operates in water a significant added-mass effect is also present. Rotating beams’ models are used to model helicopter blades, turboprops, and turbine blades. Marine composite blades have more complex geometries so Section 2.3 deals with the dynamics of rotating plates. Composite materials are usually highly anisotropic and the blade will be designed as laminate with many layers. The choice of fiber orientation in each layer introduces more variables in the design process and, at the same time, it gives the opportunity to tailor the layup to produce desirable effects usually suggested in the phrase “blade adaptability.” Section 2.4 presents a comprehensive approach to select layups and presents simple examples showing elastic couplings due to material anisotropy. Coupled fluid-structure analyses of composite marine propellers are discussed in Section 2.5, which also points out the limitation of some of these investigations and issues needing to be addressed.

2.2

Dynamics of rotating beams

Many theories are available to analyze the behavior of beams including the Bernoulli-Euler beam theory and the Timoshenko beam theory. Because of applications in turbomachinery, aircraft propellers, and helicopter rotor blades, a vast literature on the dynamics of rotating blades is available. This section starts by examining the effect of the rotation on the dynamics of the beam. That is, the effect of the centrifugal forces generated and that of the Coriolis acceleration. Complications occur when (1) the shear center and the mass center of blade shape cross-sections do not coincide; (2) the blade is Dynamic Response and Failure of Composite Materials and Structures. http://dx.doi.org/10.1016/B978-0-08-100887-4.00002-0 © 2017 Elsevier Ltd. All rights reserved.

48

Dynamic Response and Failure of Composite Materials and Structures

set at an angle relative to the plane of rotation; (3) the blade is pretwisted; and (4) the axis of the blade is inclined relative to the radial direction. Marine propellers operating in water; the inertia of the water opposes the motion of the propeller, which results in added mass and damping. This section discusses how all of these issues can be included in a mathematical model of a rotating blade. This knowledge will be beneficial in understanding the behavior of rotating plates, shells, or composite marine propeller blades. The literature on the dynamics of rotating beams up to 1991 is reviewed in Ref. [3] and Houbolt and Brooks [4].

2.2.1

Inertia loads

As a rigid body rotates about an axis, the acceleration at an arbitrary point consists of a radial and a tangential component. With a deformable body, there is motion relative to a rotating coordinate system and this results in two additional components: the acceleration relative to the rotating frame and a term called Coriolis acceleration that accounts for the combined effects of the rotation and the relative motion. This section examines these effects on a rotating beam and derives the equation of motion for a tapered rotating beam using three beam theories. A simple formula is derived to predict the effect of rotation on the natural frequencies of the beam and that formula is identical to Southwell’s equation for spinning discs.

2.2.1.1

Acceleration

For a body rotating with an angular velocity ω, the position of an arbitrary point is defined by the vector r in the xyz rotating frame fixed to the body. The velocity is v ¼ r_ + ω r and the acceleration by a ¼ r€ + ω_ r + 2ω r_ ω ðω rÞ

(2.1)

where r€ is the relative acceleration in the rotating frame, ω_ r is the effect of the angular acceleration of the rotating frame, 2ω r_ is the Coriolis acceleration, and ω ðω rÞ is the centripetal acceleration. After deformation, a point initially at ro ¼ xi+ yj+ zk is located at r ¼ ro + u where u is the displacement vector. For a Bernoulli-Euler beam the displacement field is u ¼ ðuo zwo, x yvo, x Þ i+ ðvo zφÞ j+ ðwo + yφÞ k

(2.2)

where uo is the axial displacement, wo is the bending deflection, and φ is the angle of the centripetal acceleration is given twist. If the beam rotates with a velocity ω ¼ Ωk, by ω ðω rÞ ¼ ðx + uo zwo, x yvo, x ÞΩ2 i+ ðy + vo zφÞΩ2 j

(2.3)

The centripetal acceleration creates a force per unit volume equal to ρω ðω rÞ. The centrifugal force per unit length

Composite marine propeller blades

ðð

49

ðð

ρ ðx + uo zwo, x yvo, x ÞΩ2 i+ ðy + vo zφÞΩ2 j dA

ρω ðω rÞ dA ¼ A

A

If ðð goes through the center of mass of the cross-section, ðð the x-axis ρy dA ¼ ρz dA ¼ 0 so A

A

ðð

ðð

ρ ðx + uo ÞΩ2 i+ vo Ω2 j dA

ρω ðω rÞ dA ¼ A

(2.4)

A

Usually the displacements uo and vo are small compared with x and can be neglected. For the portion of the blade between an arbitrary location ξ ¼ x and the end ξ ¼ L, the centrifugal force T is given by T¼

ðL

mξΩ2 dξ ¼

x

mΩ2 2 L x2 2

(2.5)

where m is the mass per unit length, Ω is the rotational speed, and ξ is the coordinate of an arbitrary point between x and L, the end of the string. The Coriolis acceleration (2.6) 2ω r_ ¼ 2Ω vo, t zφ, t i+ ðuo, t zwo, xt yvo, xt Þ j The force per unit length due to the Coriolis acceleration is ðð Fc ¼ 2ρ ω r_ dA ¼ 2ρAΩ ðvo, t i+ uo, t jÞ

(2.7)

A

if the x-axis goes through the center of mass of the cross-section. If the beam is simply flapping in the x–z plane, Fc ¼ 0. The equation of motion for a tapered rotating beam is that of a beam under initial tension due to the centrifugal force [5–11] ðEI ðxÞw,xx Þ, xx ðT ðxÞw, x Þ, x + mðxÞw€ ¼ 0

(2.8)

with T from Eq. (2.5). There are two extreme cases: (1) when the centrifugal force is small, the behavior is that of a stationary beam; (2) when the bending rigidity EI is small, the beam behaves like a rotating string.

2.2.1.2 Rotating string The equation of motion for such a rotating string is ðTw,x Þ,x ¼ mw€

(2.9)

which is Legendre’s differential equation. For pinned-free boundary conditions the mode shapes are the odd Legendre polynomials of order 2p 1 starting at p ¼ 2. The first three mode shapes are

50

Dynamic Response and Failure of Composite Materials and Structures

P3 ¼ 5x3 3x =2, P5 ¼ 63x5 70x3 + 15x =8, P5 ¼ 429x7 693x5 + 315x3 35x =16 and the corresponding natural frequencies ω2p ¼ pð2p 1ÞΩ2 so ω22 ¼ 6Ω2 , ω23 ¼ 15Ω2 , and ω24 ¼ 28Ω2 . This special case is interesting because it has an exact solution for the natural frequencies and mode shapes. It can be used as a benchmark when the bending rigidity of the beam is negligible.

2.2.1.3

Rotating tapered beam

Natural frequencies for rotating beams governed by Eq. (2.8) are given in several publications (e.g., [12–18]). A general result on the effect of centrifugal effects on rotating structures is derived here for a beam. Multiplying Eq. (2.8) by a test function η and integrating from 0 to L give the weak formulation ðL

ðL EIη,xx w,xx dx + 0

0

η, x T ðxÞw, x dx +

ðL

€ ¼0 mη wdx

(2.10)

0

provided that η is sufficiently differentiable and satisfies the boundary conditions. For a one-term approximation where ϕi is the ith mode shape of the stationary beam, we find that the natural frequency of the rotating beam is given by ðL ω2iR ¼

0

EIϕi, xx ϕi, xx dx + ðL mϕi ϕi dx 0

ðL 0

ϕi,x T ðxÞϕi, x dx ðL mϕi ϕi dx

(2.11)

0

The first term on the right-hand side is the ratio of the effective stiffness and the effective mass of the stationary beam for mode i. That is, the ith frequency of the stationary beam. Since T is proportional to Ω2, Eq. (2.11) can be written as ω2iR ¼ ω2iS + Si Ω2

(2.12)

is the frequency of the stationary beam and where ωiS ðL ð 2 1 L Si ¼ mϕi, x L x2 ϕi, x dx= mϕi ϕi dx. Rotation increases the natural frequencies 2 0 0 due to the stiffening effect of the centrifugal force. Eq. (2.12) is called Southwell’s equation developed by Lamb and Southwell [19,20] in 1921 to account for the stiffening effect of rotation for spinning discs. The same equation was proposed by Berry [21] to account for centrifugal stiffening in aircraft propellers. Eq. (2.8) is based on the Bernoulli-Euler beam theory. Mei [22] considers a rotating The equations of Timoshenko beam with displacement field u ¼ ðuo + zφÞi+ wk. motion are

Composite marine propeller blades

51

ðTw,x Þ,x + κGA w, xx ϕ, x ¼ ρAw, tt EIϕ,xx + κGA ðw, x ϕÞ + ρIΩ2 ϕ ¼ ρIϕ, tt

(2.13)

These are the usual Timoshenko beam equations with two additional terms: the first underlined term accounts for the stiffening effect of the centrifugal force T as it affects the transverse motion. The term underlined twice accounts for the effect of the beam’s rotation on the rotation of the cross-section. This theory includes the effect of rotary inertia so, starting with the centrifugal acceleration ac ¼ ω ðω rÞ, integrating ρ r ac over the cross-section gives the twice underlined term in Eq. (2.13). This ρIΩ2ϕ term is not included in Ref. [23]. Eq. (2.13), also used in [24], govern the motion of a Timoshenko beam rotating with a constant velocity Ω. This model is extended to analyze doubly tapered rotating Timoshenko beam made out functionally graded materials in Ref. [25,26]. White and Heppler [27] give the equation of motion for a Timoshenko beam rotating with a variable angular velocity. Refs. [28,29] adapted the Rayleigh beam theory to include the effect of rotation and derive the following equation of motion ðEIw,xx Þ,xx + ρAw, tt ðTw, x Þ, x ðρIw, xtt Þ, x + Ω2 ðρIw,x Þ, x ¼ 0

(2.14)

where the first three terms are those of the rotating Bernoulli-Euler beam, the third term accounts for the effect of rotary inertia in the Rayleigh beam theory, and the last term accounts for the effect of the centripetal acceleration on the rotation of the cross-section.

2.2.2

Effect of preset angle

Propeller blades usually make an angle ψ with the plane of rotation (Fig. 2.1). This Lo and Renbarger angle is called the preset angle of the pitch angle. Then, ω ¼ Ωk. [30] show that ωRi,ψ , the ith natural frequency of a beam with a setting angle ψ is related to that of a beam with a zero setting angle ωRi,0 by ω2Ri, ψ ¼ ω2Ri,0 Ω2 sin 2 ψ

(2.15) Z

z y

y Y

Fig. 2.1 Preset angle or initial pitch angle for a rotating beam.

52

Dynamic Response and Failure of Composite Materials and Structures

This means that the introduction of a preset angle reduces the natural frequency of the rotating beam. The same result is mentioned in Kosko [31] who also showed that the torsional vibrations also follow Southwell’s formula. the centrifugal acceleration For a Bernoulli-Euler beam with u ¼ ðuo zw, x Þ i+ wk, 2 ac has an axial component Ω ðx + uo zw, x Þ that will give the same centrifugal force T in Eq. (2.5). The component in the transverse direction is Ω2 ðz + wÞsin 2 ψ integrating of the cross-section gives an inertia force per unit area equal to mðxÞ Ω2 wsin 2 ψ in addition to the m(x)w,tt term resulting from the r€ term in Eq. (2.1). Then, the equation of motion of a rotating Bernoulli-Euler beam with a preset angle is [32–35] ðEðxÞI ðxÞw,xx Þ,xx ðT ðxÞw, x Þ, x + mðxÞw€ mðxÞΩ2 sin 2 ψw ¼ 0

(2.16)

where the last term accounts for the effect of the preset angle. Multiplying this equation by a test function η and integrating over the length of the beam, after some manipulations we obtain ðL ðL ϕi, xx EIϕi, xx dx ϕi, x L2 x2 ϕi, x dx 2 Ω o + Ω2 sin 2 Ψ (2.17) ω2i ¼ oð L ðL 2 mϕi ϕi mdx ϕi ϕi mdx o

o

The first term on the right-hand side is the natural frequency of the nonrotating beam, the second term is the effect of the centrifugal force, and the third term represents the effect of the preset angle. Thus we have recovered the result of Lo and Renbarger (Eq. 2.15). The motion of a rotating tapered Timoshenko beam with a preset angle θ is governed by [32,36] EIyy ðxÞθy, x ,x + GAðxÞ w, x θy ρIyy ðxÞθy, tt ¼ 0 (2.18) GAðxÞ w, x θy , x ρAðxÞw, tt + ½T ðxÞw, x , x ρAðxÞΩ2 sin 2 ψw ¼ 0 The effects of Coriolis acceleration have included in the formulation for rotating Bernoulli-Euler beams [37] and rotating Timoshenko beam [38].

2.2.3

Effect of pretwist

Propeller blades usually are pretwisted so that the pitch angle increases from the root to the end. For beams with a rectangular cross-section, pretwisting couples bending in the xz- and yz-planes. For more general cross-sectional shapes, the shear center and the mass center do not coincide and bending-bending-twisting coupling is introduced. These issues and the effect of rotation are discussed next.

2.2.3.1

Nonrotating pretwisted beams

The vibration of pretwisted beams is studied in many publications starting with Mendelson and Gendler [39] and Flax and Goland [40] in 1951 followed by Rosard [41] in 1953 and Carnegie [42] in 1959. The equations of motion for pretwisted beams with Bernoulli-Euler kinematics and a rectangular cross-section are [43]

Composite marine propeller blades

EIxx v,zz + EIxy u, zz

, zz

+ mv, tt ¼ 0,

53

EIyy u, zz + EIxy v, zz

,zz

+ mu,tt ¼ 0

(2.19)

where assuming linear pretwist α ¼ αL z=L and the moments of inertia vary along the axis of the beam as Ixx ¼ IYY sin 2 α + IXX cos 2 α, Iyy ¼ IYY cos 2 α + IXX sin 2 α IYY IXX sin 2α Ixy ¼ 2

(2.20)

in terms of the principal values of the cross-section: IXX and IYY. This formulation is used in [44,45]. Similarly, pretwisted Timoshenko beams with rectangular cross-section are studied in Ref. [46–49] and this model results in four coupled equations for bending-bendingcoupled motion. Leissa and Jacob [50] used the Rayleigh-Ritz method to study the free vibration of twisted parallelepipeds starting from three-dimensional elasticity. For beams with rectangular the center of mass coincide with the shear center and the pretwisting of the cross-section couples the two bending deformations but does not produce any torsion of the beam. In 1962, Carnegie [51] developed the equations of motion for pretwisted blades for which the center of mass does not coincide with the shear center. This results in three coupled equations of motion

+ mv, tt ¼ mry θ, tt , zz EIyy u, zz + EIxy v, zz , zz + mu, tt ¼ mrx θ, tt EIxx v,zz + EIxy u, zz

(2.21)

C1 θ, zzzz Cθ,zz + Iθ, tt ¼ mry u, tt + mrx v, tt where rx and ry are the distances between the center of flexure and the centroid in the x- and y-directions. C is the torsional stiffness, I is the polar moment of inertia about ð E 2 3 b t db. The C1θ,zzzz term accounts for the warping of the center of flexure, C1 ¼ 12 A the cross-section. Carnegie [52] presents one-term approximations for the first natural frequency of pretwisted beams with coupling between bending and torsional modes.

2.2.3.2 Rotating pretwisted beams Surace et al. [53] study the coupled bending-bending-torsion vibration analysis of rotating pretwisted blades. The model can be described as the bending-bending model of Eq. (2.19) coupled with the wave equation for Saint Venant torsion. It includes the effect of the centrifugal force, the preset angle, and inertia coupling effects due to the fact that the elastic and mass centers do not coincide. Chen and Keer [54], Chen and Ho [55], and Lin et al. [48] study the coupled bending-bending vibrations of rotating twisted Timoshenko beams with rectangular

54

Dynamic Response and Failure of Composite Materials and Structures

cross-section. Like for nonrotating pretwisted rectangular beams the motion is governed by four coupled equations of motion.

2.2.4

Complicating factors

Several complicating factors should be considered for rotating beams’ models of propeller blades or turbine blades. Often, the axis of the blade is not oriented in the radial direction and it may not even be straight. While the studies cited earlier and many others deal with cantilever rotating beams, the root of the blade is not perfectly clamped. This section discusses the effects of the skew angle and that of elastically restrained ends.

2.2.4.1

Effect of skew angle

Generally the axis of the blade is inclined relative to the radial direction (Fig. 2.2). Huang [56] derived the equation of motion for an inclined Bernoulli-Euler beam EAu,xx ¼ ρA u,tt Ω2 ðR cos α + x + uÞ (2.22) EIv,xxxx + ρAv,tt ¼ EA ðu, x v, x Þ, x + ρI v, xxtt Ω2 v, xx Combining the effect of preset angle and the skew angle, the motion of an inclined rotating, Bernoulli-Euler beam, is [57] EAu, xx + 2ρAΩsin ψv, t ¼ ρA u, tt Ω2 ðRcos α + x + uÞ EIv,xxxx + ρAv,tt ðTv, x Þ, x + 2ρAΩ sin ψu, t ρAΩ2 vsin 2 ψ ¼ ρAΩ2 R sin α sin ψ

(2.23)

The equations of motion for an inclined rotating Timoshenko beam are [58] EAu, xx + 2ρAΩv,t ¼ ρA u, tt Ω2 ðR cos α + x + uÞ EIϕ,xx + κGA ðw,x ϕÞ + ρIΩ2 ϕ ¼ ρIϕ, tt κGA v,xx ϕ,x + ðTv, x Þ, x + ρAΩ2 v ρAv, tt 2ρAΩu, t ¼ 0

Y

Ω a

Fig. 2.2 Skew angle for an inclined beam.

X

(2.24)

Composite marine propeller blades

55

The underlined terms in Eqs. (2.23), (2.24) account for the effect of the Coriolis acceleration, which are usually small.

2.2.4.2 Elastically restrained ends In most analyses the blade is assumed to be clamped at the root and free at the other end. Idealized cantilever boundary is not always realized in practical applications and a several references assumed that the blades are elastically restrained at the root [33,59–66].

2.2.5

Spatial beams

The analysis of complex composite beams generally consists of two steps: one to determine the properties of the cross-section using a detailed two-dimensional analysis and the other is the analysis using a one-dimensional beam theory [67].

2.2.5.1 Timoshenko beam kinematics It is assumed that the cross-section remains plane (no out-of-plane warping) and does not deform in its plane. The displacements of an arbitrary point with coordinates x, y, and z are given by u ¼ uo + zθy yθz v ¼ vo zφ w ¼ wo + yφ

(2.25)

where uo(x), vo(x), wo(x) are the displacements of the point on the x-axis. φ, θy, and θy are the rotations about the x, y, and z-axes, respectively. With the usual definitions, the strains are εxx ¼ uo, x + zθy, x yθz, x , εxy ¼ θz + vo, x zφ, x , εxz ¼ θy + wo, x + yφ, x

(2.26)

For an anisotropic material the stress-strain relations are taken as 8 9 2 38 9 c11 c12 c13 > > = = < σ xx > < εxx > 6 7 σ xy ¼ 4 c12 c22 c23 5 εxy > > ; ; : > : > σ xz c13 c23 c33 εxz

(2.27)

The forces and moments acting on a cross-section are defined as 9 8 9 8 9 8 9 8 > > = ð > = = ð > = < Nx > < σ xx > < Mx > < yσ xz zσ xy > dA σ xy dA, fM g ¼ My ¼ zσ xx fN g ¼ Vy ¼ > > > A> A> ; ; ; ; : > : > : > : Vz σ xz Mz yσ xx

(2.28)

The In vector notation, the tractions acting on a cross-section are t ¼ σ xx i+ σ xy j+ σ xz k. force acting on a small element of area dA is t dA and its moment about the origin of the

56

Dynamic Response and Failure of Composite Materials and Structures

This leads to the definition of {M} in ¼ r t dA, where r ¼ yj+ zk. cross-section is d M Eq. (2.28). The deformations are defined by two vectors. 8 >

=

8 9 > < φ, x > = fεo g ¼ θz + vo, x , fκg ¼ θy, x : > > > > : ; : ; θy + wo, x θz, x uo, x

(2.29)

{εo} contains the axial strain and the two transverse shear strains while includes the angle of twist per unit length and the two bending curvatures. With these definitions, the constitutive equations for the beam are

N

M

εo

¼ ½K

(2.30)

κ

KA KB is a 6 6 symmetric matrix with The stiffness matrix ½K ¼ K TB K D 2

3 2 3 c11 c12 c13 ð zc12 + yc13 zc11 yc11 6 7 6 7 K A ¼ 4 c12 c22 c23 5dA, K B ¼ 4 zc22 + yc23 zc12 yc12 5 dA A A c13 c23 c33 zc23 + yc33 zc13 yc13 ð

(2.31)

and 2

ð

6 4

KD ¼

z2 c22 + y2 c33 2yzc23 yzc13 + z2 c12 yy2 c13 yzc12

A

yzc13 z2 c12

z2 c11

yzc11

yzc12 y c13

yzc11

2

2

3 7 5dA

(2.32)

+y c11

Constitutive equations of this form (Eq. 2.30) with a 6 6 stiffness matrix are obtained in several publications [67–70]. When there is no elastic coupling ðc12 ¼ c13 ¼ c23 ¼ 0Þ, 2

K11

6 KA ¼ 4 0

0 K22

0

0

0

3

2

0 K15 K16

6 7 0 5, K B ¼ 4 0 K33 0

0 0

3

2

K44

7 6 0 5, K D ¼ 4 0 0 0

0

0

3

7 K55 K56 5 K65 K66

Therefore there is no extension-shear coupling since KA is diagonal. The extension-bending coupling in these equations can be eliminated if the x-axis goes through the “modulus weighted centroid” so that ð

ð zc11 dA ¼ A

yc11 dA ¼ 0 A

Composite marine propeller blades

57

and K15 ¼ K16 ¼ 0. The line joining the modulus weighted centroids is usually called the elastic axis. The K56 ¼ K65 terms in KD introduce a bending–bending coupling that can be eliminated from these equations by a rotation of the coordinate system about the x-axis.

2.2.5.2 Elastic center, shear center, and mass center The elastic center is defined as the point where a force applied normal to the cross-section will produce no bending curvatures [71]. Inverting Eq. (2.25),

εo κ

¼ ½F

N

(2.33)

M

Applying a force F in the x-direction at (ye, ze) results in N ¼ F, My ¼ ze N and My ¼ ye N. If point E is the elastic center then Eq. (2.33) implies that 9 8 N >

> = < θy, x 0 F51 F55 F56 (2.34) ¼ ¼ ze N > 0 θz,x F61 F56 F66 > ; : ye N Solving for the coordinates of the elastic center ye ¼

F55 F61 F56 F51 , F55 F66 F256

ze ¼

F56 F61 F66 F51 F55 F66 F256

(2.35)

The shear center is defined as the point at which a load applied parallel to the plane of the section will produce no torsion [71]. Applying a force P at ysc, zsc in the y-direction at the end of the beam (Fig. 2.3) produces a shear force Vy ¼ P, a torque Mx ¼ zsc P at an arbitrary location x, and a bending moment Mz ¼ ðL xÞP. Substituting into Eq. (2.33) gives φ, x ¼ F42 P zsc PF44 + ðL xÞPF46 ¼ 0

(2.36)

zc

SC C

SC

Fig. 2.3 Shear center and bend-twist coupling.

yc

58

Dynamic Response and Failure of Composite Materials and Structures

an equation from which to obtain zsc. Similarly, applying a force in the z-direction, Vz ¼ P, Mx ¼ ysc P, My ¼ ðL xÞP. Substituting into Eq. (2.33) gives an equation yielding ysc. Then, φ,x ¼ F43 P + ysc PF44 ðL xÞPF45 ¼ 0

(2.37)

The coordinates of the shear center are ysc ¼

F43 F45 F42 F46 + ðL xÞ, zsc ¼ + ðL xÞ F44 F44 F44 F44

(2.38)

This shows that when F45 6¼ 0 and F45 6¼ 0, the shear center is not a property of the cross-section as indicated in [71]. The mass center can be defined as the point about which a uniform acceleration in the axial direction does not create a moment about the y or z-axes. The location of the mass center is given by ð ym ¼

ρydA A

.ð

ð ρdA,

zm ¼

A

ρzdA A

.ð

ρdA

(2.39)

A

When the mass center does not coincide with the elastic center, the centrifugal force will create bending moments too.

2.2.5.3

Determining the elastic properties of the beam

The properties of the cross-section of a beam are hard to determine. A comprehensive review of the existing approaches is available in Ref. [72]. One approach is to model a small slice of it using a detailed finite element model, apply a given type of deformation, and infer the forces and moments required to produce that deformation. This approach studies the bending-twisting coupling of initially twisted beams [73], prismatic beams with coupling [74], laminated composite beams [75], curved beams [76], and the torsion on bars with square cross-sections, circular cross-sections with cracks and I-sections [74]. A similar approach is used in Branner et al. [71] to determine the constitutive equations (Eq. 2.30) for the cross-section of wind turbine blade. The approach introduced by Cesnik and Hodges [77] to determine the constitutive equations of helicopter blades based on a detailed finite element analysis of small section is widely used.

2.2.5.4

Special cases of elastic coupling

Elastic coupling between bending and torsion is examined in Refs. [78–82]. The constitutive equations relating the ending moment M and the torque T to the curvature w,xx and the angle of twist per unit length φ,x written as

M T

¼

EI Kps Kps GJ

w, xx φ, x

are a special case of Eq. (2.30).

(2.40)

Composite marine propeller blades

59

Extension-torsion coupling [79,83] occurs when the normal force N and the torque T are related to the axial displacement uo and the angle of twist φ by

uo, x EA Kpa N ¼ φ, x Kpa GJ T

(2.41)

where Kpa is the extension-twisting coupling. With extension-bending coupling N [84] and the bending moment M are related to uo, to the rotation of the cross-section θ and to the transverse displacement w by

N M

EA K ¼ K EI

uo, x , V ¼ GAðw, x θÞ θ, x

(2.42)

Again, K is a constant accounting for the coupling between the two types of motion. Those are simple cases that have been treated in several publications. In the most general case, the motion of the cross-section is described by six degrees of freedom and there will be coupling between extension, two bending modes, and torsion. More degrees of freedom will be introduced if in-plane and out-of-plane warping of the cross-section is accounted for.

2.2.6

Added-mass effect

When a body moves in a fluid, the inertia of the fluid opposes the motion and that effects are equivalent to having a virtual mass added to the mass of the solid. The study of the added-mass effect was traced to the experimental work of Dubua (1776) on a spherical pendulum and the analytical work of Green (1833) and Stokes (1843) to determine the added mass of a sphere [85]. Parsons et al. [86] study the dynamics of ship’s propulsion system. The propeller is modeled as a rigid body with six degrees of freedom. The motion of the propeller is governed by ð½M + ½Ma Þ fx€g + ½Ca fx_g ¼ ffs g + ffe g

(2.43)

T where fxg ¼ δx , δy , δz , θx , θy , θz is the displacement vector with three translations and three rotations. The matrix [M] represents the inertial properties of the propeller. It is a diagonal matrix with the m m m Jxx Jyy Jzz , where m is the mass of the propeller, Jxx is the mass moment of inertia of the propeller about the axis of rotation, Jyy ¼ Jzz are the diametral moments of inertia. [Ma] is the added-mass matrix, [Cp] is the added-damping matrix, {fs} is the forces applied by the shaft of the propeller, and {f } is the vector of excitation forces. The hydrodynamic force ffh g ¼ ½Ma fx€g e Cp fx_g is the force due to the motion the propeller. [Ma] and [Cp] are 6 6 matrices with off-diagonal elements so the equations of motion are coupled. The motion of a deformable propeller is governed by [87–93] ð½M + ½Ma Þ fu€g + ð½C + ½Ca Þ fu_g + ½K fug ¼ fFce g + fFco g + fFR g

(2.44)

60

Dynamic Response and Failure of Composite Materials and Structures

where {u} is the displacement vector in the finite element model of the propeller, [K] is the stiffness matrix, {Fce} is the force due to centrifugal effects, {Fco} is the force due to Coriolis acceleration, and {FR} is the force applied if the blade were rigid.

2.3

Dynamics of rotating plates

Often the geometry of marine propeller blades is better approximated by plates or shells. This section starts by deriving the tools necessary to analyze rotating plates on shells and presents an overview of the literature dealing with their vibrations and their response to impact.

2.3.1

Mechanics of plates

In the following, the equations of motion for the three most frequently used plate theories are derived starting from the equations of motion from three-dimensional elasticity. Expressions are derived for the strain energy and the kinetic energy of Kirchhoff-Love plates, which are needed in order to obtain solutions using the Rayleigh-Ritz method or the finite element method (FEM).

2.3.1.1

Equations of motion in terms of force and moment resultants

In three-dimensional elasticity, the equations of motion are σ xx, x + σ yx, y + σ zx,z + X ¼ ρu, tt σ xy, x + σ yy, y + σ zy, z + Y ¼ ρv, tt

(2.45)

σ xz, x + σ yz, y + σ zz, z + Z ¼ ρw, tt in terms of the stresses σ ij, the forces per unit volume X, Y, Z, and the displacement components u, v, and w. Integrating through the thickness gives Nx, x + Nxy, y + qx ¼ ρhuo, tt Nxy, x + Ny, y + qy ¼ ρhvo, tt

(2.46)

Qx, x + Qy, y + qz ¼ ρhwo, tt In this process, the in-plane force resultants, the transverse force resultants, and the loads applied on the external surfaces are defined naturally as

Nx , Ny , Nxy ¼

qx , q y , q z ¼

ð h=2 h=2

ð h=2

h=2

σ xx , σ yy , σ xy dz, Qx , Qy ¼

σ zx, z , σ zy, z , σ zz,z dz +

The displacement field is often taken as

ð h=2 h=2

ð h=2 h=2

σ xz , σ yz dz

ðX, Y, Z Þdz

(2.47)

Composite marine propeller blades

u ¼ uo + zu1 δR

61

4z3 4z3 ð u + w Þ, v ¼ v + zv δ v1 + wo, y 1 o, x o 1 R 3h2 3h2

w ¼ wo

(2.48)

where uo, vo, and wo are the displacements of points on the midplane, u1 and v1 are the rotations of normal to the midplane for z ¼ 0, and δR is a tracer constant. The displacement field of Reddy’s third-order plate theory is recovered when δR ¼ 1. δR ¼ 0 gives the displacement field of the Mindlin-Reissner first-order shear deformation plate theory. For the classical plate theory, δR ¼ 0, u1 ¼ wo, x , and v1 ¼ wo, y . In all three cases, after integrating the right-hand side of Eq. (2.45) only the uo,tt, uvo,tt and wo,tt remain if, ð h=2 ð h=2 ρzdz ¼ ρz2 dz ¼ 0. as it is generally the case, h=2

h=2

Two additional equations are obtained by multiplying Eq. (2.45a, b) by z and integrating through the thickness gives Mx, x + Mxy, y Qx ¼ I2 u1, tt + δR I4 ðu1,tt + wo, xtt Þ Mxy, x + My, y Qy ¼ I2 v1, tt + δR I4 v1,tt + wo, ytt

(2.49)

ð 4 h=2 ρz4 dz. For the classical plate theory δR ¼ 0, 3h2 h=2 h=2 the rotary inertia is neglected ðI2 0Þ. The bending moments are defined as where I2 ¼

ð h=2

ρz2 dz and I4 ¼

Mx , My , Mxy ¼

ð h=2 h=2

σ xx , σ yy , σ xy zdz

(2.50)

Eqs. (2.46), (2.49) are the five equations of motion in terms of force and moment resultants.

2.3.1.2 Kirchhoff–Love plate theory With the CPT the strain-displacement relations are 9 8 9 8 9 8 > = > = > = < εx > < uo, x > < w,xx > εy ¼ vo, y + z w,yy fεg ¼ fεo g + zfκg or > > > ; > ; > ; : > : : εxy uo, y + vo, x 2w,xy

(2.51)

The stress-strain relations are given by fσ g ¼ Q fεg. After substituting into Eqs. (2.47), (2.50) the plate constitutive equations are fN g ¼ ½Afεo g + ½Bfκ g, fMg ¼ ½Bfεo g + ½Dfκg where

(2.52)

62

Dynamic Response and Failure of Composite Materials and Structures

2

A11 A12 A16

3

2

B11 B12 B16

3

2

D11 D12 D16

3

6 7 6 7 6 7 7 6 7 ½A ¼ 4 A12 A22 A26 5, ½B ¼ 6 4 B12 B22 B26 5, ½D ¼ 4 D12 D22 D26 5 A16 A26 A66 B16 B26 B66 D16 D26 D66

(2.53)

When [B] ¼ 0, the bending moments depend only on the transverse displacement w. Solving Eq. (2.49) for Qx and Qy and substituting into Eq. (2.46c) gives Mx, xx + 2Mxy, xy + My, yy + qz ¼ ρhwo, tt

(2.54)

or D11 w,xxxx + 2ðD12 + 2D66 Þw, xxyy + D22 w, yyyy + 4D16 w,xxxy + 4D26 w,xyyy + qz ¼ ρhw,tt

(2.55)

where the subscript o has been dropped for the transverse displacement of the midplane. For free vibrations, w ¼ W sin ωt. Multiplying Eq. (2.10) by the virtual displacement, integrating over the domain occupied by the plate, and performing the usual operations in variational calculus, we derive the boundary conditions (not reproduced here) and δ ðU T Þ ¼ 0

(2.56)

where the strain energy and the kinetic energy are given by 1 U¼ 2

ðð n Ω

D11 ðW, xx Þ2 + 2D12 W, yy W, xx + 4D16 W,xy W,xx + 4D26 W,yy W,xy 2 2 o + 4D66 W, xy + D22 W, yy dΩ (2.57)

and ðð 1 ðW Þ2 dΩ T ¼ ω2 ρhδ 2 Ω

(2.58)

respectively. This expression can be used to find approximate solutions.

2.3.2

Vibrations of cantilever composite plates

The vibrations of laminated composite plates have been studied extensively. One publication is of particular interest here because it deals with the vibrations of graphite-epoxy cantilever plates and shells and provides extensive experimental and finite element results [94]. Crawley and Dugundji [95] and others show how simple approximate solutions can be obtained using the Ritz method. A full Rayleigh-Ritz method is used by Crawley and Lazarus [99].

Composite marine propeller blades

63

2.3.2.1 Approximate solutions Approximate solutions can be obtained using Eq. (2.12) in two different ways. With the partial Ritz procedure the displacements are taken to follow a simple assumed shape in the chordwise direction while the variation in the axial direction remains unknown. Following Ref. [96] for isotropic plates or Ref. [95] for symmetrically laminated plates, the displacements of a rectangular cantilever plate clamped at x ¼ 0 are assumed to vary as ðxÞ + yθðxÞ W ðx, yÞ ¼ w

(2.59)

After substitution into Eqs. (2.57), (2.58), the Ritz procedure leads to two coupled equations of motion [95]

D11

d4 w d3 θ + 2D ¼ ω2 ρhw 16 dx4 dx3

c3 d 4 θ d3 w d2 θ c2 2D16 3 4D66 2 ¼ ω2 ρh θ D11 4 12 dx 12 dx dx

(2.60)

¼ 0, θ, x ¼ 0, θ ¼ 0 at x ¼ 0 , x ¼ 0, w with the essential boundary conditions w , xx + 2D16 θ,x ¼ 0, D11 w ,xxx + 2D16 θ,xx ¼ 0, and the natural boundary conditions D11 w c2 c2 , xx + 4D66 θ, x D11 θ, xxx ¼ 0 at x ¼ L. D11 θ, xx ¼ 0, 2D16 w 12 12 Eq. (2.60) shows that a nonzero value of D16 induces bend-twist coupling. When D16 ¼ 0, Eq. (2.60b) indicates that the torsional motion is not governed by the wave equation due to the presence of the d4θ/dx4 term. Other displacement approximations have been proposed:

c2 2 ðxÞ + yθðxÞ + 4y ϕð x Þ W ¼w 3

(2.61)

in Jensen and Crawley [97] or 1 ðxÞ + yθðxÞ + y2 ϕðxÞ W ¼w 2

(2.62)

in Bernhard and Chopra [98], for example. Eq. (2.59) does not account for curvature in the y-direction while Eqs. (2.61), (2.62) do. The other way to obtain approximate solutions is to use the full Ritz procedure starting with a displacement approximation like W ¼ C1 x2 + C2 xy + C3 xy2 as suggested by Crawley and Lazarus [99] or, more generally,

(2.63)

64

Dynamic Response and Failure of Composite Materials and Structures

W¼

M X N X Cmn xm yn

(2.64)

m¼2 n¼0

Using the Ritz minimization process @ ðU T Þ=@Cmn ¼ 0

(2.65)

results in an eigenvalue problem to be solved for the natural frequencies and mode shapes (e.g., see [100] laminated cantilever plates).

2.3.2.2

Pretwisted plates

The middle surface of a shallow shell is defined by [101–114]

1 x2 2xy y2 + + z¼ 2 Rx Rxy Ry

(2.66)

Twisted plates can be seen as shallow shells with infinite radius of curvature in the xand y-directions 1=Rx ¼ 1=Ry ¼ 0 . The twist radius Rxy is related to the angle of twist by tan ψ ¼ L=Rxy , where L is the length and ψ is the angle of twist between the clamped end at x ¼ 0 and the free end at x ¼ L. In 1984, Leissa et al. [115] reviewed the results of previous work on the vibration of isotropic twisted cantilever plates of rectangular planform. The Rayleigh-Ritz method is used to study the free vibration of twisted cantilever plates in [50,115–119]. The material is isotropic in [50,116,119] and a laminated composite in [117,118]. A three-dimensional elasticity model is used in Refs. [50,116], the classical plate theory in Refs. [117,118], and the first-order shear deformation theory in Refs. [119]. The vibration of rotating composite cantilever plates has been studied using a 3D elasticity approach [120], the classical plate theory [121], the FSDT [122], or a quadratic layerwise theory [123]. The vibration of pretwisted rotating cantilever plates is studied in Refs. [124–127].

2.3.3

Impact on composite blades

Laminated composite structures are likely to experience significant damage when subjected to foreign object impact damage [128–133]. Composites are used in the construction of fan blades in modern jet engines, which can be subjected to impacts by gravels, rivets, bolts, ice balls, and birds. The tip velocity of these blades can reach 244 m/s [134]. Some of the early research on that topic includes [134–141]. Refs. [142] and [143] are examples of recent research on composite fan blades. Typically, helicopter blades are thin-walled beams with composite skins stabilized by polymeric foams. These blades are also subjected to impacts by debris,

Composite marine propeller blades

65

hailstones, and birds. Research on the effect of impacts on helicopter blades includes [144–152]. Pretwisted conical shell panels subjected to impact are analyzed using the FEM in [102,104,106,153–155]. These are laminated composite panels [153] or panels made out of functionally graded materials [104].

2.4

Designing with anisotropic materials

Designing structures made out of composites is more challenging because in addition to selecting a material and a thickness, one must also choose the number of layers in the laminates and the fiber orientations in each layer. This provides opportunities that are not available with isotropic materials. The layup can be tailored to provide the necessary stiffness and strength for the expected loads and also to produce special effects by introducing couplings between the different deformation modes. Section 2.2 discussed couplings introduced by the geometry (pitch angle, pretwist, and skew angle). Here we will see that anisotropy can also introduce couplings. An efficient approach to finding an optimal design for laminated composites is to use a stiffness invariant formulation that effectively separates the effects of thickness, material properties, and stacking sequence. A layup is described by 12 nondimensional parameters regardless of the number of plies. This formulation is recalled next and then we consider three examples in which a layup is used to generate: extension-shear, bend-twist, and extension-bending coupling.

2.4.1

Stiffness invariants and lamination parameters

The stress-strain relations for a single ply in material principal coordinates are 8 9 2 8 9 38 9 Q11 Q12 0 > > > = = = < σ 11 > < ε11 > < α1 > 6 7 σ 22 ¼ 4 Q12 Q22 0 5 ε22 ΔT α2 > > > ; ; ; : > : > : > 0 0 Q66 σ 12 ε12 0

(2.67)

where ΔT is the temperature change and α1 and α2 are the coefficients of thermal expansion in the two principal directions. The constitutive equations at the laminate level are (

N + NT M + MT

)

" ¼

A B B D

#(

εo κ

) (2.68)

where N and M are the force and moment resultants, NT and MT are the thermal force and moment resultants to be defined later, εo is the mid-surface strain vector, and κ is the vector of plate curvatures.

66

Dynamic Response and Failure of Composite Materials and Structures

9 8 2 1 ξ1 A11 > > > > > > > > 6 > A22 > > > 6 1 ξ1 > > > 6 =

60 0 12 ¼ h6 60 0 > A66 > > > 6 > > > > 6 > > > A16 > 4 0 ξ3 =2 > > > > ; : 0 ξ3 =2 A26

3 0 0 8 9 U1 > > > > > > 0 07 > 7> > U2 > > 7> = < 7 1 0 7 U3 7 > 0 1 7> > > > U4 > > 7> > > 0 0 5> ; : > U5 0 0

ξ2 ξ2 ξ2 ξ2 ξ4 ξ4

9 8 2 B11 > 1 ξ5 > > > > > 6 > > > B22 > > > 6 1 ξ5 > > > = h2 6

60 0 12 ¼ 6 > > 46 B66 > > 60 0 > > > > 6 > > > > 4 0 ξ7 =2 B > > 16 > > ; : B26 0 ξ7 =2

3 0 0 8 9 > > > U1 > 0 07 > > > 7> > U > > 7> 2 1 0 7< = 7 U3 0 17 > > > 7> > > U4 > 7> > > 0 0 5> ; : > U5 0 0

ξ6 ξ6 ξ6 ξ6 ξ8 ξ8

9 8 2 D11 > 1 ξ9 > > > > 6 1 ξ > >D > > > 22 > 9 > 6 > > > > 6 < D12 = h3 6 0 0 ¼ 6 > 12 6 > D66 > > 60 0 > > > > 6 > > > > 4 0 ξ11 =2 D16 > > > > ; : 0 ξ11 =2 D26

ξ10 ξ10 ξ10 ξ10 ξ12 ξ12

3 0 0 8 9 > > > U1 > 0 07 > > > 7> U2 > > > 7> = < 1 07 7 U3 > > > 0 17 7> U4 > > > 7> > > > 5 0 0 : > ; U5 0 0

(2.69)

(2.70)

(2.71)

For convenience, the elastic properties can be expressed in terms of five stiffness invariants U1–U5 that are functions of Q11, Q22, Q12, and Q66. 8 9 2 1 U1 > > > > > > > > 6 > > 2 > = 16 < U2 > 6 6 U3 ¼ 6 1 > 46 > > > > U4 > > > 41 > > > ; : > 1 U5

3 4 8 9 Q11 > 7> > > 0 7> > 7< Q22 = 7 1 2 4 7 > > 7> > > Q12 > 1 2 4 5: ; Q66 1 2 0

1 2 2 0

(2.72)

The effect of the layup is accounted for by 12 lamination parameters ðξ1 , ξ2 , ξ3 , ξ4 Þ ¼ ðξ5 , ξ6 , ξ7 , ξ8 Þ ¼

1 h

ð h=2

4 h2

h=2

ð h=2

cos 2θ, cos 2 2θ, sin 2θ, sin 2θ cos 2θ dz

h=2

12 ðξ9 , ξ10 , ξ11 , ξ12 Þ ¼ 3 h

cos 2θ, cos 2 2θ, sin 2θ, sin 2θ cos 2θ zdz

ð h=2 h=2

cos 2θ, cos 2 2θ, sin 2θ, sin 2θ cos 2θ z2 dz

(2.73)

Composite marine propeller blades

67

The first four lamination parameters control the in-plane stiffness ([A] matrix), the last four parameters control the bending rigidities ([D] matrix), and the parameters control the extension-bending coupling terms ([B] matrix). The optimal design requires a maximum of 12 variables regardless of the number of layers. Most often, symmetric layups are used so ξ5 ¼ ξ6 ¼ ξ7 ¼ ξ8 ¼ 0 and only eight variables remain. For symmetric and balanced laminates ξ3 ¼ ξ4 ¼ 0 and, if the number of layers is large enough, the effect of ξ11 and ξ12 is negligible so only four parameters need be considered: ξ1 and ξ2 for in-plane motion and ξ9 and ξ10 for bending. The thermal force and moment resultants are defined as 8 9 8 9 q1 + q2 cos 2θ > q1 + q2 cos 2θ > ð h=2 > ð h=2 > < = < = T q1 q2 cos 2θ ΔT dz MT ¼ q1 q2 cos 2θ ΔT zdz N ¼ > > h=2 > h=2 > : ; : ; q2 sin 2θ q2 sin 2θ (2.74) where q1 ¼ ½α1 Q11 + Q12 ðα1 + α1 Þ + α2 Q22 =2 and q2 ¼ ½α1 Q11 + Q12 ðα1 α1 Þ α2 Q22 =2. If ΔTt and ΔTb are the temperature changes on the top and bottom surfaces of the laminate and the temperature is assumed to vary linearly through the thickness, 9 9 8 8 q2 ξ 5 > q1 + q2 ξ 1 > > > > > > > = = < T h2 < q2 ξ 6 N ¼ ΔTa h q1 q2 ξ1 + S > > > 4> > > > > ; ; : : q2 ξ 3 q2 ξ 7 (2.75) 9 9 8 8 q q ξ + q ξ > > > > 2 1 2 5 9 > > > > = = T h2 < h3 < M ¼ ΔTa q2 ξ 6 + S q 1 q2 ξ 9 > > 4> 12 > > > > > ; ; : : q2 ξ 7 q2 ξ11 where ΔTa ¼ ðΔTt + ΔTb Þ=2 and S ¼ ðΔTt ΔTb Þ=h. Thus, the thermal force and moment resultant depend on the layup and this dependence is accounted for by the same lamination parameters. Moisture absorbed by composites has a similar effect of their behavior and that can be accounted for by replacing the temperature change ΔT by ΔM, the percentage of water uptake by weight, and the coefficients of thermal expansions by the corresponding coefficients of moisture expansion β1 and β2 in the constitutive equations for a single ply. This formulation of the design problem using lamination parameters was used to find the optimum layups for composite shafts [156,157] and plates [158–163].

2.4.2

Elastic coupling with anisotropic laminates

The following provides just examples of couplings that can be introduced in a laminate by different choices of layups. The bend-twist coupling has been used in the design of composite marine propellers but there are many other types of coupling.

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Dynamic Response and Failure of Composite Materials and Structures

2.4.2.1

Extension-shear coupling

For a symmetric layup ½B ¼ 0. Applying a force resultant Nx with Ny ¼ Nxy ¼ 0, Eq. (2.67) gives the in-plane shear strain εxy ¼ εxx ðA22 A16 A12 A26 Þ=ðA22 A66 A26 A26 Þ

(2.76)

Therefore, a tensile load induces both an extension in that direction and shear deformation unless A16 ¼ A26 ¼ 0. A16 and A26 are called extension-shear coupling terms.

2.4.2.2

Bend-twist coupling

Applying a bending moment Mx on a symmetric laminate with My ¼ Mxy ¼ 0 results in curvatures κxx and κ yy but also in κ xy ¼ κxx ðD22 D16 D12 D26 Þ= D22 D66 D226

(2.77)

The twisting is characterized by κ xy, and D16 and D26 are the bending-twisting coupling coefficients. In the general case κ xy 6¼ 0 except for the special case of cross-ply laminates for which D16 ¼ D26 ¼ 0 so there is no coupling. This example is used to introduce the concept of bend-twist coupling for a laminate [164]. Murray et al. [165] show that the bend-twist coupling depends on the number of layers by considering three stacking sequences: [θ], ½θ, θ, θ, and ½θ, θ,θ, θ, θ. In addition to these we also consider the ½θ, θ, θ, θ,θ, θ, θ seven ply layup. Parameters ξ9 and ξ10 depend on how cos2θ varies through the thickness so they will be the same for these three laminates and as a result, D22, D12, and D66 are the same for all three. It can be shown that ξ11 ¼ Ci sin 2θ and ξ12 ¼ ðCi =2Þ sin 4θ for these layups and Ci is a constant that depends on the number of layers. Ci ¼ 1, 0.925926, 0.584, 0.423 for i ¼ 1, 3, 5, 7, respectively, so ξ11 and ξ12 decrease significantly with the number of plies. In Eq. (2.77), the numerator decreases with the number of layers and the denominator increases so that κ xy decreases rapidly with the number of layers. ξ11 and ξ12 are both zero when θ ¼ 0. ξ11 is maximum when θ ¼ 45degrees and ξ12 is maximum when θ ¼ 22:5degrees. These observations are consistent with the results obtained in Ref. [165] for the bending of a cantilever plate showing no bend-twist coupling for θ ¼ 0, a maximum for θ slightly lower than 30 degrees, and a sharp decrease in κxy with the number of plies.

2.4.2.3

Extension-bending coupling

For laminates with 0 degree and 90 degrees layers, applying an in-plane load Nx 6¼ 0 and Ny ¼ Nxy ¼ Mx ¼ My ¼ Mxy ¼ 0, Eq. (2.67) becomes

Composite marine propeller blades

69

38 9 38 9 2 εxx > B11 0 0 > > > > > > κxx > 7< = 6 7< = 6 6 7 6 7 Ny ¼ 0 ¼ 4 A12 A22 0 5 εyy + 4 0 B22 0 5 κyy > > > > > > > > > > ; ; ; : > : > : Nxy ¼ 0 εxy κxy 0 0 A66 0 0 0 9 2 8 38 9 2 38 9 Mx ¼ 0 > εxx > B11 0 0 > D11 D12 0 > > > > > > > > κxx > = 6 < 7< = 6 7< = 6 7 7 My ¼ 0 ¼ 6 ε κ + 0 B D 0 D 0 22 4 5> yy > 4 12 22 5> yy > > > > > > > ; ; ; : : > : > Mxy ¼ 0 εxy κxy 0 0 0 0 0 D66 8 > >

> =

2

A11 A12

0

(2.78)

Nx produces the bending curvatures κxx ¼ D22 B11 εxx D12 B22 εyy =ðD11 D22 D12 D12 Þ κyy ¼ D22 B22 εyy D12 B11 εxx =ðD11 D22 D12 D12 Þ

(2.79)

but no twisting is induced κxy ¼ 0 . Substituting into the first set of equations gives the axial strains εxx and εyy in terms of Nx and εxy ¼ 0. Applying an in-plane load Nx, for example, generates in-plane and out-of-plane deformations.

2.5

Composite marine propeller blades

Traditionally, marine propellers were made out of NAB or MAB. Replacing these materials by composite offers advantages such are weight savings, net shape manufacturing, corrosion resistance, and the ability to tailor the behavior of blade to generate deformations that cannot be obtained with isotropic materials. Since composite materials are anisotropic and have significantly different properties, accurate methods of analysis are needed. Two difficulties quickly become apparent: (1) the shape of the blades is complex so it cannot be modeled as a rotating beam, plate, or shallow shell; and (2) composite marine propellers are significantly more flexible than metallic propellers. As a result, the blades are modeled using the FEM. If the deflections are large enough, they will affect the fluid flow which will change the loads applied on the blades and in turn change the deflections. This requires that both the blades and the surrounding fluid be modeled numerically for this coupled fluid-structure interaction (FSI) problem to be solved. The goal is to determine how the thickness of the blade varies, the number of layers, and the fiber orientation in each layer. Several objectives should be met: (1) The blades should not fail under load; (2) under design conditions, the pitch angle should be that which provides maximum efficiency; and (3) if possible, the layup should be tailored to improve efficiency over a range of operating conditions. To meet the first objective, stresses in each ply must be determined accurately and should satisfy an appropriate failure criterion. Das and Kapuria [166] showed that in several earlier publications, this analysis is not performed and that designs that address only the second

70

Dynamic Response and Failure of Composite Materials and Structures

and third objectives would actually fail. If the blade deforms significantly, its unloaded shape should be such that under load, after deformation, its shape is the one that yields the best efficiency. Under different operating conditions, different pitch angles are required so the hope is that a proper tailoring of the blade would produce deformations that better approximate the requirements for efficiency. The third objective is then to design passive shape-adaptive propellers by allowing the blades deform with load changes to gain higher efficiency over a greater range of operating conditions compared to conventional “rigid” propeller [167]. Other criteria might be to reduce cavitation, noise, etc. The thrust coefficient, the torque coefficient, and the propeller efficiency are defined as KT ¼

T Q JKT , KQ ¼ 2 5 , ηo ¼ 2πKQ ρn2 D4 ρn D

(2.80)

in terms of the thrust T, the torque Q, the propeller’s advance ratio J ¼ Va =nD, the density of water, the mean in-flow velocity Va, the rotational speed n in revolutions per second (rps), and the propeller’s diameter D. These parameters are widely used to evaluate the performance of various designs.

2.5.1

Propeller size

In the existing literature, most studies are conducted on scale model propellers with diameters of 0.22 m [168], 0.25 m [169,170], 0.305 m [171,172], 0.35 m [173,174], 0.4 m [175,176], 0.61 m [164,177–179], 0.7 m [180], and 1.4 m [181,182]. A 0.68-m diameter composite blade was also designed for a small fishing boat [183]. A few studies consider larger size composite propellers: 2.12 m [184], 4.2 m [185], 4.4 m [167], and 5.18 m [89]. Large isotropic propellers with diameters of 6.091 m [186] and 7.01 m [187] have also been analyzed. Considering scaled models is problematic because the hydrodynamics problem and the structural dynamics problem do not scale the same way [166].

2.5.2

Methods of analysis

The most common modeling approach is to use the FEM to model the propeller and the boundary element method (BEM) for the fluid [87,92,93,173,177,178,164,179, 180,185–189]. This BEM-FEM approach is usually used to analyze composite propellers but it is also used for propellers made out of NAB [186], iron-chromiumnickel-copper alloy [189], or some unspecified isotropic material [187]. Other studies use the FEM to model the blade and other methods for the fluid: finite volume method [166,171] and lifting-surface theory [172,174,181,182]. The blades are discretized using solid elements [168,180,185] but also shell elements [166,174,181,182,184,185]. In “one-way” FSI analysis, the pressures determined from the fluid model are applied to the structure so that its deflections can be determined. It assumes that the deflections are so small that the fluid flow problem is not affected. When the

Composite marine propeller blades

71

deflections of the structure become large, they affect the fluid flow so that the pressures applied are different that those initially predicted. In that case a “two-way” FSI is needed. For example, knowing the displacement ui1 at step i 1 and the estimated force vector fi, the structural problem is solved for the displacement ui in step 1. In step 2, fi is reevaluated accounting for the new geometry. Some iterations may be needed to achieve convergence for ui. In step 3, the fluid flow problem is solved to estimate f i + 1 , the forces acting on the structure at step i + 1. The process is repeated again for the next step. This approach is used in many studies including [169] and [185].

2.5.3

Layups

Only symmetric laminates are used in previous studies on composite propeller blades so there is no extension-bending coupling. Attempts to design adaptive blades that deform is a desirable way to improve in off-design conditions rely on the D16 and D26 terms to provide bend-twist coupling. Often, the choice of layups is limited at the outset. For example in [185] three configurations are considered: [(θ)32], [(θ1)8, (θ2)8]s, and [(θ1)8, (θ2)8, (θ3)8, (θ4)8]. The optimization process shows that for the example considered, the layup that produces the largest reduction in fuel consumption is the [(40°)32] unidirectional layup. Refs. [181] and [182] consider symmetric balanced laminates of the form ½…, θ, θ, 90, 0s and unbalanced ½…, θ2 , 90, 0s . Equations indicate that the bending rigidities D11, D22, D12, and D66 depend only on two lamination parameters ξ9 and ξ10. In the ξ9–ξ10 plane (Fig. 2.4A), all possible layups fall in the domain bounded by the ξ10 ¼ ξ29 parabola representing all angle-ply laminates and the horizontal-line segment 1 ξ9 1, ξ10 ¼ 1 that represents all cross-ply laminates. The first laminates considered in Ref. [185] are angle-ply laminates so only the lower boundary of the feasible domain is explored. In that case, the other two parameters ξ11 and ξ12 vary as shown in Fig. 2.4B. Ref. [185] also considers [(θ1)8, (θ2)8]2 laminates. Fig. 2.5 shows that this type of laminates covers only a narrow region of the feasible space in the ξ9–ξ10 plane: the region between the lower boundary and the dotted lines. In order to show how this region is swept by [(θ1)8, (θ2)8]2 Cross-ply

[90°] 1

1 x10

[0°]

0.5

0.5 Angle-ply

(A)

x12

0 –90

–1

x11

0

1

0 [±45°]

–45

0

45

q 90

–0.5

x9

–1

(B)

Fig. 2.4 Lamination parameters controlling the bending rigidities. (A) Feasible domain in the ξ9–ξ10 plane; (B) parameters ξ11 and ξ12 for a single layer.

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Dynamic Response and Failure of Composite Materials and Structures

1

x10

90°

0°

0.5 30°

60° 45°

x9

0 0

–1

1

Fig. 2.5 [(θ1)8, (θ2)8]2 laminates. 1

x10 C1 C2 C3

0.5

0 −1

−0.5

0

0.5

1

x9

Fig. 2.6 Lamination parameters ξ9 and ξ10 for layups C1–C3 in [168].

laminates, Fig. 2.5 also shows how ξ9 and ξ10 vary for five such layups where θ1 is taken to be 0, 30, 45, 60, or 90 degrees and θ2 varies from 0 to 90 degrees. A BEM-FEM simulation for composite marine propellers discusses the effect of fiber orientation on the deflections and the vibrations of the blades [168]. Three types of layups called C1, C2, and C3 are considered: [30°2, 45°2, 0°4, 45°6, 0°6, 45°6, 0°6, …]s, [0°2, 45°2, 0°4, 45°6, 0°6, 45°6, 0°6, …]s, and [0°2, 45°2, 90°4, 45°6, 90°6, 45°6, 90°6, …]s. The lamination parameters ξ9 and ξ10 for layups [30°2, 45°2, 0°4, 45°6, 0°6, 45°6, 0°6]s, ° [02, 45°2, 0°4, 45°6, 0°6, 45°6, 0°6]s, and [0°2, 45°2, 90°4, 45°6, 90°6, 45°6, 90°6]s identified as C1, C2, and C3 are shown in Fig. 2.6. Replacing the 30°2 outside layer by a 0°2 layer is seen to increase both ξ9 and ξ10 which leads to D11, the bending rigidity in the spanwise direction. Replacing 90 degrees layers by 0 degree layers in layup C3 keeps ξ10 constant but greatly reduces ξ9. Parameters ξ11 and ξ12 for those three cases are given in Table 2.1. It was shown that switching to the C2 and C3 layups provides reductions to the hydrodynamic loads on the blades by 49.5% and 70.6%, respectively. In Refs. [164] and [188], the optimal stacking sequence for a 10-layer symmetric laminate is determined to be [15°/30°/15°/0°/30°]s. It was asserted “that a multilayer composite laminate can be modeled using an equivalent unidirectional fiber angle, θeq, which results in approximately the same load-deformation characteristics.” For this particular layup, the equivalent angle θeq ¼ 5° in Ref. [188]. In Ref. [90] the optimal layup was found to be [30°/30°/90°/90°/30°]s and it is said that it can be

Composite marine propeller blades

Table 2.1

in [168] ξ11 ξ12

73

Lamination parameters ξ11 and ξ12 for layups C1–C3 C1

C2

C3

0.5965 0.0762

0.4441 0

0.4441 0

modeled by a single layer with θeq ¼ 32° . Calculations showed that the parameters ξ9 ξ12 are of the equivalent layer are significantly different than those of the laminate, indicating a very different behavior. In several other publications [89,177,179,190,191], it assumed from the outset that a laminate can be replaced by a single layer with an equivalent orientation angle θeq. θeq was taken to be 5 degrees in Ref. [89] and 20 degrees in Ref. [177] while its optimal value is found to be between 31 degrees and 34 degrees in Ref. [179]. Some studies consider a larger number of plies. For example for 48-ply laminates, the optimal layup was found to be [456, 06, 452, 04, 452, 02, 452]s in Ref. [174] and [452, 902, 458, 010, 452]s in Refs. [172]. The use of curved fiber paths has also been investigated [185]. In Paik et al. [170] the efficiency of the three composite propellers is nearly identical to those of a metallic blade and increased until J ¼ 0.9 at which point the efficiencies of the composite blade became significantly lower than that of the metallic blade. This examination of the type of layups shows that the full range of possibilities offered by the use of highly anisotropic materials to tailor the properties of a laminate has not been explored so far.

2.5.4

Outstanding issues

After a thorough review of the literature, several issues can be seen as requiring further attention. To list just four of them: 1. With a few exceptions, existing studies deal with scaled model propellers under 1 m in diameter, typically. On large ships, diameters could be of the order of 5–6 m. Such a large jump in size may not be possible with composite materials. 2. In previous studies, composite marine propeller blades are solid laminates. For larger propellers transmitting more power, a sandwich construction with spars and a core may be necessary. 3. The study of the hydrodynamic problem requires that the flexibility of the blade be included in the model since deformations can be large enough to affect the flow. It has been shown that with certain layups, blades can be designed to deform in such a way that the efficiency of the propeller remains near the optimum over a range of operating conditions. This is called (passive) blade adaptability in the sense that the applied stresses cause deformations that lead to improved efficiency. Numerical simulations suggesting that this could be achieved are performed modeling that the blade as a single layer of anisotropic material. Then, the elastic bend-twist coupling is maximum. However, it is well known that to resist the loads applied a

74

Dynamic Response and Failure of Composite Materials and Structures

multidirectional laminate is needed and that the bend-twist coupling quickly vanishes. Analyses should also consider the strength of the blade. 4. The thickness of the blades varies from the root to the tip. With metal blades a continuous variation can be achieved by casting and traditional machining processes. With composites, changes can only occur in steps by reducing the number of plies. The loads carried by the dropped plies are transferred to the continuous plies through thin resin-rich interfaces inducing high shear and peeling stresses near the ply drops. These interlaminar stress concentrations can lead to delaminations and extensive damage. This issue has been studied extensively for composite structures in aerospace applications, helicopter blades, and wind turbine blades but not for marine composite propeller blades. Similarly, interlaminar stress concentrations occur along near the edges of a laminate.

2.6

Conclusion

This chapter deals with the dynamic behavior of composite marine propeller blades. The use of composites requires revisiting all the issues involved in the mechanics of such blades. The effect of the centrifugal forces and the Coriolis acceleration have to be accounted for as well as the effect of the centrifugal force, the pitch angle, pretwisting, skew angle, and elastic end restraints. In addition one has to model laminates with anisotropic layers, select the thickness, the number of layers, and the fiber orientation in each layer. This chapter provides the background necessary to understand the issues involved and a detailed analysis of the current state of the art in the analysis of composite propellers. A number of significant shortcomings and issues that have yet to be addressed are pointed out. This introduction, assessment, and recommendation for future work combination are not currently available and should be valuable to many readers.

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Numerical simulations and experimental experiences of impact on composite structures

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F. Marulo, M. Guida, L. Maio, F. Ricci University of Naples “Federico II”, Naples, Italy

3.1

Introduction

This chapter deals with composite materials and their application in the aeronautical field. Considering the fact that the aerospace industry has a strong certification and compliance requirement, with consequences on development cost and technology solutions, it is necessary to use the ability of powerful computational methods to resolve material behavior at different scales and communication across them is fostering unprecedented advances in multiscale modeling. These models provide in-depth understanding of material deformation and failure that can revolutionize integrated structure-material design, and an added value is represented by the experimental tests on small-scale models. The widespread use of composite materials in aeronautical or aerospace industry in general, is not the only challenge of our times. The need of composites is born from the first time that an airplane has disconnected the wheels off the ground. The development of composite materials was accelerated by the outbreak of World War II, since 1940 the fuselage of the Spitfire was designed and built by the company Aero Research Ltd in Duxford, from a material made from flax fibers not twisted, dipped in phenolic resin and was developed by Norman de Bruyne and Malcolm Gordon [1]. Nowadays the news is not what composites the big aeronautical manufacturers use but how much composite materials they have used. The novelty is that the Boeing company makes the 787 Dreamliner aircraft where 50% of the structure is made up of composite material. Airbus with its flight A350XWB airplane now boasts a 53% usage of composite material among its long products. A composite is a macroscopic (visible to the naked eye) combination of two or more materials which results in a material possessing structural properties that none of the constituent materials possess individually. Because the materials are not soluble in one another, they retain their identity. The composites have amply demonstrated their functionality in increasingly diverse fields. The recent development and the development of new technologies have allowed composites to compete with metals, even more, in the aeronautical field Dynamic Response and Failure of Composite Materials and Structures. http://dx.doi.org/10.1016/B978-0-08-100887-4.00003-2 © 2017 Elsevier Ltd. All rights reserved.

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where the composite materials are widely used especially thanks to the savings in weight combined with high fatigue and corrosion resistance. More parts of an aircraft use composite materials: parts of wings and tails, fuselages, antennas, landing gear, floors seats, interior panels, tanks, and helicopter blades. Not only the light weight aspect is a peculiarity of the composite materials, but the strength is often a defining feature, as well as high-temperature resistant composite materials allow to reduce fuel consumption, improve efficiency, and reduce direct operating costs of aircrafts. The most important characteristic of the composite materials is that they can be layered, with the fibers in each layer running in a different direction. This allows an engineer to design structures with unique properties, furthermore a structure can be designed so that it will bend in one direction, but not another. The aeronautical structures certification requirements are basically the same regardless of the material used to produce them. The certification of composite component is more complex than that of metal elements (aluminum alloys, titanium, and steels); this difficulty is due to the wide variability of the properties of composites and not to the current total mastery of design technique with these materials. The composites are produced from perishable raw materials, such as polymeric resins; for this reason, they require a greater quality control. The approach that the authorities use to composite materials is of the type: the structures made of composite should not submit the risks to aircraft operators are higher than those that accept to giving themselves to metallic materials. It is the responsibility of the designer to ensure these security levels. In 1978 the Federal Aviation Administration (FAA) Advisory Circular AC20-107 issued the certification of aeronautical structures made of composite materials [2]. It is a short document in which it is specified that the composite design must reach a level of security at least equivalent to that required by the metal structures. Analysis, whether performed by closed form solutions or finite element models, can be useful in design and certification of impact-damaged composite structures. The state of the art is such that analysis cannot stand by itself but can be useful in directing and analyzing test results and expanding test data to untested configurations by semiempirical methods. Impact damage is one of the main problems that composite structures face, hence, there needs to be a way of reducing that damage when it occurs, reducing it enough so that the integrity of the structure is not compromised. There are a number of solutions to improve impact resistance and damage of composite materials, such as fiber toughening, matrix toughening, interface toughening, through-the-thickness reinforcements, and selective interlayers and hybrids. One of the possible ways to reduce impact damage in composite structures is to embed the shape memory alloy wires inside the polymer composites due to their superelastic behavior allowing remarkably high strain to-failure and recoverable strain and their capability to generate recovery tensile stresses and hence reduce the deflections and the in-plane strains and stresses of the structure [3]. Then, honeycomb material offers a great possibility to optimize the energy distribution during the impact, for example the cores made from continuous fiber reinforced composites shows better response to impact loading compared to that of short fiber reinforced, exhibiting a large elastic region and higher peak loads [4].

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Also the importance of the fruition of FML development marks a step in the long history of research starting in 1945 at Fokker, where earlier bonding experience at de Havilland inspired investigation into the improved properties of bonded aluminum laminates compared to monolithic aluminum. Later, NASA got interested in reinforcing metal parts with composite materials as part of the Space Shuttle program, which led to the introduction of fibers to the bond layers, and the concept of FMLs was born. Although GLARE is a composite material, its material properties and fabrication are very similar to bulk aluminum metal sheets [5]. It has far less in common with composite structures when it comes to design, manufacture, inspection, or maintenance. GLARE parts are constructed and repaired using mostly conventional metal material techniques.

3.2 3.2.1

Computational damage mechanics for composite Damage progression

3.2.1.1 Challenging issue in designing composites A challenging issue in designing composites is the prediction of various failure modes, such as fiber breakage, matrix cracking, fiber/matrix debonding, fiber kinking, and delamination between adjacent plies [6]. The difficulty of the problem was evidenced by the World Wide Failure Exercise, an international activity launched by Hinton, Kaddour, and Soden [7] to establish the status of currently available theoretical methods for predicting material failure in fiber reinforced polymer composites materials in the course of which 12 of the leading theories for predicting failure in composite laminates have been tested against experimental evidence. This event revealed that very few theories successfully predicted failure of composite coupons deformed quasi statically. In general, the load carrying capacity of a structure does not vanish as soon as either failure or damage ensues at a material point and the structure can support additional load before it eventually fails. Thus it is important to quantify damage caused by the initiation of a failure mode and study its development and progression and the eventual failure of a structure with an increase in the applied load. Therefore, in many structural applications, the progressive failure analysis is required to predict composite structure mechanical response under various loading conditions. Failure and damage in laminated structures can be studied by using a micro-mechanics approach [8] (see Ref. [6] for more information) but the damage studied at the constituent level is only computationally expensive for a real size problem. The alternative is an approach based on continuum damage mechanics (CDM) [9], in which material properties of the composite have been homogenized and both damage and failure are studied at the ply/lamina level; e.g., see Refs. [9,10]. However, a micromechanical approach can be used to deduce effective properties of a ply and CDM approach to study failure and damage at the lamina level, Ref. [8].

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Dynamic Response and Failure of Composite Materials and Structures

Material models

The inhomogeneous and not-isotropic nature of composite materials results in fracture and failure behaviors unlike that of conventional metallic alloys. As seen before, the damage can initiate and propagate in many ways: many different mechanisms can occur (interfacial debonding, fiber microbuckling, matrix cracking, fiber breakage, delamination, etc.) and damage growth is not self-similar. In multidirectional composite laminates, damage accumulates during the loading process and the final failure occurs as a result of damage accumulation and stress redistribution. The ultimate failure load is higher than the damage onset load. Therefore, it is useful to have models capable of predicting the onset of material degradation, the effect of the noncritical damage mechanisms on the stiffness of the laminate, and ultimate structural failure. Traditional finite element approach cannot capture details of all individual failure modes, but needs to make further approximations; therefore, the key is to know how to make the right approximations. Material failure is treated simply and suitably on mesoscale and macroscale, but in this way it is not possible to account for detailed differences between failure mechanisms neither is it possible to consider other phenomena on microscale. Today, typically, numerical models based on lamina-level failure criteria are used to simulate the damage of the fiber-reinforced composite material, although with well-accepted limitations. In these constitutive models, composites are modeled as orthotropic linear elastic materials within the failure surface; the failure surface is defined by the failure criterion (damage onset criterion) as maximum stress/strain criterion, Hashin’s criterion, Christensen’s criterion, Chang-Chang’s criterion, Puck’s criterion, LARC, etc.; they are based on stress or strain components; beyond the failure surface, elastic properties are degraded according to laws defined by the material model. There are two degradation schemes used to reduce the material properties once failure is initiated: (a) ply properties are reduced to a value close to zero in the finite element that satisfied the criterion of damage; (b) damage parameters are used to degrade ply properties in a continuous form (CDM) in the finite elements that satisfied the damage onset condition (Fig. 3.1). The first scheme uses a ply discount method to degrade the elastic properties of the ply from its undamaged state to a fully damaged state; in the specific, the scalar components of the stiffness tensor are reduced to approximately zero when damage is predicted (null stiffness can lead to numerical instability). Elastic properties are dependent on field variables. After a failure index has exceeded 1.0, the corresponding user-defined field variable is made to transit from 0 (undamaged) to 1 (fully damaged) instantaneously and it continues to have the value 1.0 even though the stresses may reduce significantly, which ensures that the material does not “heal” after it has become damaged. The material models based on this scheme are easy to use and require few input parameters. However, they cannot represent with satisfactory accuracy the progressive reduction of the stiffness of a laminate as a result of the accumulation of damage modes (Table 3.1). Failure modes in laminated composites are strongly dependent on geometry, loading direction, and ply orientation. One distinguishes for convenience in-plane failure

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89

Damage

Elastic property

Initial value (before damage)

Residual value (after damage)

0

1 Field variable

Fig. 3.1 Nonprogressive degradation of the elastic properties.

Table 3.1 Chang/Chang (material model MAT_54 in LSDYNA) damage onset criteria Tensile fiber mode: 2 2 σ xy σ xx +β 1 σ xx > 0 e2ft ¼ Xt Sc 2 Set eft 0: damaged, Ex ¼ Ey ¼ Gxy ¼ vyx ¼ vxy ! 0; se e2ft < 0: elastic behavior Compressive fiber mode: 2 σ xx 1 σ xx < 0 e2fc ¼ Xc Set e2fc 0: damaged, Ex ¼ vyx ¼ vxy ¼ !; se e2fc < 0: elastic behavior Tensile matrix mode: 2 2 σ yy σ xy + 1 σ yy > 0 e2mt ¼ Yt Sc Set e2mt 0: damaged, Ey ¼ vyx ¼ 0 ) Gxy ! 0; se e2mt < 0: elastic behavior Compressive matrix mode: # 2 " 2 σ yy σ yy σ xy 2 Yc 2 + + 1 1 σ yy < 0 emc ¼ 2Sc 2Sc Yc Sc Set e2mc 0: damaged, Ey ¼ vyx ¼ vxy ! 0 ) Gxy ! 0; se e2mc < 0: elastic behavior The stresses are compared to the measured strength values by applying different relations depending on the damage mode. After a layer has experienced matrix failure, for example, the material properties Ey and Gxy of the damaged layer are multiplied by a factor according to the degradation rules, Refs. [11,12]. eft, efc, emt, emc are failure indices of the considered damage modes. “x” is the fiber direction; “y” is transverse. Xc is the longitudinal compressive strength (absolute value is used). Xt is the longitudinal tensile strength (absolute value is used). Yc is the transverse compressive strength, b-axis (positive value), see below. Yt is the transverse tensile strength. Sc is the shear strength. β is a weighting factor for shear term in tensile fiber mode.

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Dynamic Response and Failure of Composite Materials and Structures

modes and transverse failure modes (associated with interlaminar shear or peel stress). However, when the composite is loaded in-plane, only in-plane failure modes need to be considered, which can be done for each ply individually. For a unidirectional ply, at least four failure modes can be considered: matrix tensile cracking, matrix compression, fiber breakage, and fiber buckling. All the mechanisms, with the exception of fiber breakage, can cause compression failure in laminated composites. The issue of damage growth in fiber reinforced plastic (FRP) laminated composites has been addressed by an ever-increasing number of researchers through the use of CDM. It is generally recognized that CDM started with the papers by Kachanov (1958) and Rabotnov (1968). However, the use of CDM for the simulation of composite behaviors has been popularized in the 1990s by Ladeveze [63] and Talreja [13]. The CDM approach focuses on the effect of the presence of micro-failures in the material. In detail, it attempts to predict the effect of microscale defects and damage at a macroscale by making assumptions about the nature of the damage and its effect on the macroscale properties (e.g., elastic moduli) of the material. This damage theory describes the damage, i.e., the appearance of cracks, as a state variable that can be expressed as a scalar or as a tensor to quantify the isotropic or anisotropic damage. Therefore, the CDM theories capture effects of microscopic damage by using the theory of internal variables. Different models have been developed to permit the damage prediction in composite structures under loading. Ladeveze and Le Dantec [14], formulated mesomechanical damage model for single-ply laminate considering as composite damages fiber/matrix debonding and matrix microcracking; these damage modes are represented by two internal (damage) state variables; the damage evolution is then governed by a law assumed to be a linear function of equivalent damage energy release rate. Xiao et al. [15,16] used this approach in modeling energy absorption of composite structures in crashworthiness applications and to study damage during quasistatic punching of woven fabric composites; Williams and Vaziri [17], used damage mechanics principles along with matrix and fiber failure criteria to model damage due to low-velocity impacts; Yen and Caiazzo [18], implemented a damage model (MAT 162) in LS-DYNA by generalizing a layer failure model that already existed (MAT 161); their damage model is based on damage mechanics approach due to Matezenmiller et al. [19], and it incorporates progressive damage and softening behavior after damage initiation. This model will be analyzed subsequently and then implemented as material subroutine in FORTRAN code for the commercial finite element software ABAQUS for single integration point brick elements only. It will be used in progressive failure analyses on single element.

3.2.1.3

The CDM: a damage constitutive model

As seen previously, progressive failure models are a combination of failure criteria (which indicates if failure has occurred and if, what is the mode of failure) and postfailure degradation rules. The simple rule uses knockdown factors close to zero; the more complex rule uses gradual unloading based on CDM. A schematic presentation on the different degradation models and their influence on the finite element level stress strain curve is shown in Fig. 3.2.

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91

Stress

Constant stress

Gradual unloading

Instantaneous unloading

Strain

Fig. 3.2 A schematic presentation of different degradation rules. The simple rule uses knockdown factors close to zero; the constant stress model assumes that the damaged element will carry its failure load but no additional loads; in real case the ply behavior is something between these two models, therefore, models that use gradual unloading have been developed.

The CDM theory allows to represent the damage state of a material in terms of properly defined state variables (or damage variables) and to describe the mechanical behavior of the damage material and the further development of the damage by the use of these state variables [20]. The term damage is used to indicate the deterioration of the material capability to carry loads. From a general point of view, damage develops in the material microstructure when nonreversible phenomena such as microcracking, debonding between the matrix and the second phase particles, and microvoid formation take place [21]. Kachanov [22], pioneered the subject of damage mechanics by introducing the concept of effective stress. This concept is based on considering a fictitious undamaged configuration of a body and comparing it with the actual damaged configuration. The damage variable is defined in terms of both the damaged and effective cross-sectional areas of the body. Kachanov [22], originally formulated his theory by using simple uniaxial tension. Following its work, a cylindrical bar subjected to a uniaxial tensile force F, as shown in Fig. 3.3, is now considered. The cross-sectional area of the bar is A and it is assumed that both voids and cracks appear as damage in the bar. The uniaxial stress σ in the bar is found easily from the formula F ¼ σA. In order to use the principles of damage mechanics, a fictitious undamaged configuration of the bar, as shown in Fig. 3.3 on the right, is considered. In this configuration, all types of damage, including both voids and cracks, are removed from the bar. The effective cross-sectional area of the bar in this configuration is denoted by A and the effective uniaxial stress is σ. The bar in both the damaged configuration and the effective undamaged configuration is subjected to the same

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Dynamic Response and Failure of Composite Materials and Structures

A

F

F

F

F

As

Damaged state (0 < d < 1) Equivalent fictitious undamaged state (d = 0)

Fig. 3.3 Cylindrical bar subjected to uniaxial tension.

tensile force F. Therefore, considering the effective undamaged configuration, it is possible to have the formula F ¼ σ^As . Equating the two expressions of F obtained from both configurations, the following expression for the effective uniaxial stress σ^ is formed: σ^ ¼

A σ As

(3.1)

Following Kachanov’s work, a widely accepted definition in the macroscopic scale is the geometrical description of material damage in this way: d¼

Ad A As ¼ A A

(3.2)

where A is the nominal cross-sectional; As is the net area of the damaged specimen which excludes the area held by the damage entities (discontinuities) or in other words it is the effective resisting section area of the specimen reduced by the presence of microdefects and by their mutual interaction and which carries the applied load; Ad ¼ A As is the “damaged” cross section. Thus, the damage variable is defined as the ratio of the total area of voids and cracks to the overall area; in its simplest form, it may be considered as a reduction of area relative to the initial area of a specimen. Its value ranges from zero (for the case of an undamaged specimen) to one (for the case of complete rupture). The introduction of a damage variable, Eq. (3.2), which represents a surface density of discontinuities, leads to the concept of effective stress that is stress calculated over the effectively resisting section. Lemaitre, using the effective stress definition proposed by Kachanov, determined the constitutive equations for ductile damaged material. According to the principle of strain equivalence of Lemaitre [23,24], the strain constitutive equations for a damaged material can be derived using the same formulations used for an undamaged material except that the stress is

Experimental experiences of impact on composite structures

93

replaced by the effective stress. So, the mechanical behavior of a damaged material is usually described by using the notion of the effective stress, together with the hypothesis of mechanical equivalence between the damaged and the undamaged material. The “effective” area is As or ð1 dÞA and substituting this into Eq. (3.1), one obtains the following expression for the effective uniaxial stress: σ^ ¼

σ 1d

(3.3)

Eq. (3.3) represents the relationship between macroscopic stress σ and the corresponding effective stress σ^ in a damaged material and it is clear from this equation that the case of complete rupture (d ¼ 1) is unattainable, because the damage variable d is not allowed to take the value 1 in the denominator. The corresponding strain of an effective stress is called effective strain. It is important to note that the presence of cracks in the material diminishes the relative area of material capable of withstanding loads, thereby increasing the stress in the undamaged material under a given strain. This is also a further justification to the concept of effective stress, which implies that a damaged material subject to a load σ under a certain strain E can be modeled as an equivalent undamaged material also subject to the same strain E, but under a modified load or effective stress state σ^, see Fig. 3.4, and this can be expressed mathematically in tensorial form by simply replacing the scalar transformation between nominal and effective stress by a tensorial one as: fσ^g ¼ ½MðdÞfσ g

(3.4)

where [M(d)] is a transformation tensor or damage effect tensor containing the damage variables which is a function of the damage state, d, and {σ}, fσ^g are respectively

s s sˆ å

s

[M]

sˆ å

0

å

å

Fig. 3.4 Hypothesis of effective stress. Adapted from Simo J, Ju, J. Strain and stress based continuum damage models. I. Formulation. Int J Solids Struct 1987;23(7):821–40.

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Dynamic Response and Failure of Composite Materials and Structures

the actual stress and equivalent stress tensors. The form of the [M(d)] adopted in this model is: 2

1 0 0 0 0 6 ð 1 d1 Þ 6 6 6 1 6 0 0 0 0 6 ð 1 d2 Þ 6 6 6 1 6 0 0 0 0 6 ð1 d3 Þ 6 ½ M ðd Þ ¼ 6 6 1 6 0 0 0 0 6 ð 1 d4 Þ 6 6 6 1 6 0 0 0 0 6 ð 1 d5 Þ 6 6 4 0 0 0 0 0

3 0 0 0 0 0 1 ð 1 d6 Þ

7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5

(3.5)

Note that here and throughout the paragraph the symmetric second order and fourth-order tensors are written in Voigt matrix notation. Using the effective stress–strain relationship: (3.6) fσ^g ¼ C0 fεg then the above equation becomes: fσ g ¼ ½M1 C0 fεg ¼ ½CðdÞfεg

(3.7)

where [C0] is the stiffness tensor of the undamaged material and [C(d)] is the damaged or effective stiffness tensor (denotes the so-called “damaged” nonsymmetric stiffness tensor). The elements of matrix [C(d)] are the elastic coefficients, which are functions of the undamaged (or initial) elastic material constants: 3

2

1 v21 v31 0 0 6 ð1 d1 ÞE1 E2 E3 6 6 6 v12 1 v32 6 0 0 6 E ð ÞE E3 1 d 1 2 2 6 6 6 v13 v23 1 6 0 0 6 E E ð 1 d3 ÞE3 1 2 6 ½Cðd Þ ¼ 6 6 1 6 0 0 0 0 6 ð 1 d 4 ÞG12 6 6 6 1 6 0 0 0 0 6 ð 1 d 5 ÞG23 6 6 4 0 0 0 0 0

0 0 0 0 0

7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5

1 ð1 d6 ÞG31 (3.8)

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95

The extent of damage-induced stiffness loss is quantified by six damage variables dj with j ¼ 1, …, 6, one for each of the six elastic moduli. Therefore, the damaged stiffness tensor depends on six damage parameters, each associated to respective elastic constant. The elastic modulus reduction can be expressed in terms of associated damage parameter d in the following way: Ereduced ¼ ð1 dÞEinitial

(3.9)

From Eq. (3.9), therefore, it is evident that the postfailure mechanisms in a composite material is characterized by a reduction in material stiffness. The role of a CDM model is to provide a mathematical description of the dependence of the elastic coefficients on the damage state and the change in the damage state with load state. For this purpose, similar to the theory of plasticity, a loading function f, specifying elastic domain and the states at which damage grows, is introduced. In the damage theory, it is natural to work in the strain space and therefore the loading function is depending on the strain and on an additional parameter r, which controls the evolution of the elastic domain and so describing the evolution of the damage. Physically, r is a scalar measure of the largest strain level ever reached in the history of the material. The loading function usually is postulated in the form: fi ¼ F i ri

(3.10)

where i is the subscript which identifies the damage type (e.g., fiber and matrix tensile/ compressive failure modes), Fi is a function adopted in the form of Hashin’s failure criterion (e.g., function of the strain components, elastic moduli and strengths); ri is the damage threshold corresponding to failure mechanism. States for which f < 0 are supposed to be below the current damage threshold (i.e., the condition represents a set of states for which damage does not grow). Damage can grow only if current state reaches the boundary of elastic domain, which is represented from condition f ¼ 0 (damage loading). Therefore, the variable r describes the evolution of elastic domain. So it is possible to say that f < 0 is the elasticity criterion and f ¼ 0 is the failure criterion [25]. According to the Hashin-type failure criteria, the loading functions for different failure mechanisms are given in Table 3.2. For simplicity, rate effects on the values of the parameter in the loading function f are ignored. This limits the initial application of the model to rate insensitive materials such as CFRP and, to a lesser degree, glass fiber-reinforced plastic composites. The function f and the rate of scalar variable r_: have to satisfy the Kuhn-Tucker loading-unloading conditions: f 0, r_ 0, f r_ ¼ 0

(3.11)

where the overdot denotes differentiation with respect to time. The first condition means that r can never be smaller than F and the second condition means that r cannot decrease. Finally, according to the third condition, r can grow only if the current values of F and r

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Dynamic Response and Failure of Composite Materials and Structures

Table 3.2

Chang examples of failure criteria

E1 ε1 2 r12 ðε1 0Þ tensilefiberfailure mode Xt E1 ε1 2 r22 ðε1 < 0Þ compressivefiberfailure mode Xc E2 ε2 2 G12 ε12 2 G23 ε23 2 + + r32 ðε2 0Þ tensile=shearmatrixfailure mode Yt S12 S23 2 E2 ε2 r42 ðε2 < 0Þ compressivematrixfailuremode Yc All the failure criteria are expressed in terms of stain components based on ply level strains (ε1, ε2, ε3, ε12, ε23, ε31) and associated elastic moduli (E1, E2, E3, G12, G23, G31).

are equal [26]. It remains to link the variable r to the damage parameter d. As both r and d grow monotonically, it is convenient to postulate an explicit evolution law: dj ¼ g ð r i Þ

(3.12)

In Eq. (3.12), the generic damage parameter may depend on several i types of damage. The generic damage threshold, ri, which controls the size of the damage surface and depends on the loading history, initially is taken as one. The 3D damage model until now described may contain the formulation for different modes of failure, e.g., fiber breakage, fiber buckling or kinking, matrix cracking, matrix crushing, etc. These damage types are modeled by means of a combination of growth functions ϕi and damage coupling coefficients qji. In detail, as suggested in Matzenmiller et al. [19], the growth rate of generic damage variable d_i (or rate of damage evolution), is defined by the following type of evolution law: d_j ¼

X

ϕ_ i qji

(3.13)

i

where the scalar functions ϕ_ i define the growth rate of damage mode, i, and the coefficients qji (j associated to damage variable, i associated to damage mode) provide the coupling between the individual damage variables, dj, and the various damage modes provided by the damage criteria fi. In general, the damage increases (ϕ_ i will be nonzero) if deformation path crosses the corresponding loading surface (or damage surface, f ¼ F r ¼ 0) or in other words, if the strain rate vector ε_: forms an acute angle with the gradient rε f at the given state of strain ε on the loading surface: @f ε_ > 0 loading @ε @f f ¼ 0, ε_ ¼ 0 neutralloading @ε @f f ¼ 0, ε_ < 0 neutralloading @ε

f ¼ 0,

(3.14)

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97

The prediction of damage growth thereby relies on obtaining the strain gradient when the strain path crosses the loading surface. In loading case (damage growth), the associated strain-vector increment has a positive component along the outward normal to the damage surface and in Ref. [27]. It is shown that ϕ_ i can assume the form: m

1 1 ϕ_ i ¼ ð1 ϕi Þfi2 f_i 2

(3.15)

where ϕi is a variable representing the extent of mode-i damage, and mi is a material constant that quantifies sensitivity of the material stiffness to the extent of damage. Integrating appropriately the above Eq. (3.15), it follows that: 1

ϕi ¼ 1 emi

ð1ri mi Þ

(3.16)

from which, utilizing the damage coupling matrix, the expression of the damage variable is derived. Therefore, the damage variable grows with the development of generic damage mode i according to equation that follows: 1

dj ¼ 1 emi

ð1ri mi Þ

, j ¼ 1,…, 6

(3.17)

where dj is the damage variable; mi is the strain softening parameter, and ri is the damage threshold. The damage variable dj varies from 0 to 1.0 as ri varies from 1 to ∞, respectively. The number of strain softening parameters can depend from number of damage modes and how these are connected in terms of material softening. These damage parameters provide the softening response in the postfailure regime of the stress-strain curve. The effect of the exponent m on the stress-strain response of the element is such that high values of m result in brittle failure of the material; low values of m clearly indicate a ductile failure response resulting in more energy absorption prior to complete damage with a gradual loss of stiffness after failure. The key to the success of all CDM models is to maintain a coherent link with the physical observations of damage growth and material response. So, it is important to note that in keeping with the thermodynamic constraints in damage mechanics, damage is considered to be an irreversible process; therefore, the damage evolution rate should satisfy the following condition d_j 0, implicitly contained in Eq. (3.11), and this leads to say that dj is a monotonically increasing function of time t such that: h i dj ¼ max djτ jτ t, djt , j ¼ 1,…, 6

(3.18)

where djt is the damage variable calculated from Eq. (3.17) for the current load state, and djτ represents the state of damage at previous times τ t.

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Dynamic Response and Failure of Composite Materials and Structures

3.2.1.4

The constitutive response

The discussed damage model has been implemented into ABAQUS/Explicit within hexahedron solid elements and its ability to represent progressive damage is evaluated by simulating a tension test on one eight-node brick element. Therefore, a single element uniaxial stress test, Fig. 3.5, was conducted to verify the constitutive response of the damage model. The predicted reaction/displacement curve of the cubic element subjected to monotonic tension is shown in Fig. 3.6. The one-element test is used to observe even the effect of the exponent m on the constitutive response. This exponent determines the brittle/ductile response of the element. In fact it can be seen that smaller values of m make the material more ductile that is a material that absorbs more energy prior to complete damage, with significant stiffness degradation prior to failure and a more gradual loss of stiffness after failure; conversely higher values give the material more brittle behavior with little or no loss in stiffness prior to failure and full damage corresponding to zero stiffness shortly after failure. These qualitative observations were used later as a guide in selecting values of the parameters mi in Eq. (3.17).

Traction load

1 mm Zero displacements in plane y = 0

1 mm

Input data: • Elastic moduli • Strength • Softening parameter

1 mm

Y – Matrix direction Element C3D8R :

Y

Z

X

X – Fiber direction

• Linear brick (eight nodes) • Reduced integration • Hourglass control

Fig. 3.5 One-element test. Material data source in Choi HY, Chang FK. A model for predicting damage on graphite/epoxy laminated composites resulting from low-velocity point impact. J Compos Mater 1992;26 (14):2l34–69.

Experimental experiences of impact on composite structures

Damage onset

99

Adjustable material softening m high Æ brittle behavior m low Æ ductile behavior

45. 40.

Gradual

35.

post failure (material softening)

Reaction (N)

30. m low

25. 20. 15. 10. m high

5. 0. 0.000

0.002

0.006 0.004 Displacement (mm)

0.008

0.010

Fig. 3.6 The constitutive response.

3.3

Material testing and component validation

3.3.1

Material calibration

Material allowable are used to assess the mechanical behavior and to determine if a structure is able to resist the external loads. Because of the need to compare measured properties and performance on a common basis, users and producers of materials use standardized test methods such as those developed by the American Society for Testing and Materials (ASTM) and the International Organization for Standardization (ISO). These standards prescribe the method by which a test specimen should be prepared and tested, as well as how the test results will be analyzed and reported. The following sections contain information about mechanical tests in general as well as tension, compression, shear, and impact tests in particular. The mechanical characteristics of the composite material are influenced by humidity and temperature that modify the matrix and related properties [28]. Low velocity impacts (LVIs) also affect the residual strength of composite material and the allowable to be used in the structural design [29]. Composite structure must be able to maintain the ultimate load even in the case of nonvisual impact damage as, for example, in the case of tool drop. The allowable design strength for a typical laminate is so defined: Fa ¼ FBASIS Kc Kd where Fa is the design allowable strength of the laminate (tension, compression, or shear); FBASIS allowable strength in environmental temperature I dry condition; Kc

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Dynamic Response and Failure of Composite Materials and Structures

decay factor due to the temperature and humidity; Kd decay factor due to the impact by visual damage. Numerical simulation of cracked structures is an important aspect in structural safety assessment. In recent years, there has been an increasing rate of development of numerical codes for modeling fracture procedure. The subject of this investigation is implementing automated fracture modes in nonlinear explicit finite element code to simulate crack growth procedure. LS-DYNA [25], is a finite element code, developed by Livermore Software Technology Corporation (LSTC) and is used for high nonlinear problems like crash analysis. The most powerful aspect is the explicit solution because it is used for “high velocity dynamic” problems. Furthermore, prior to start of the simulations on the detailed FE model, it is necessary to calibrate several aspects of the model. In particular, the following topics will be studied: l

l

Material calibration Impact Calibration

In the first calibration process, the attention is focused on the type of the material to use. In particular, the simulations are performed according to the ASTM procedure, regarding traction and compression tests on unnotched and open hole coupons. Then, simulation’s results are compared with experimental tests previously performed. The composite material used is a fabric type with plies stacked with 0 degree and 45 degree orientations (Fig. 3.7).

3.3.2

Micromechanical calibration: FE modeling

This section describes an extensive series of materials and their experimental and numerical characterization to determine stiffness and strength properties on different layups of thermoplastic composite made to PPS/CF, carbon fiber/polyphenylene sulfide.

Fig. 3.7 Example of a fabric ply.

Experimental experiences of impact on composite structures

Table 3.3

101

Several layups with thickness of the single ply at 0.013 in

Ply number

Stacking sequence

Thickness (in.)

5 8 10 16

[45/0/0/0/45] [45/0/45/0]s [45/0/45/0/45]s [45/0/45/0/0/45/0/45]s

0.0628 0.1005 0.1256 0.2010

Static tests were performed to determine the stress-strain curve, while the impact tests done to determine the threshold for impact energy, which corresponds to visible impact damage (Table 3.3). In the FEA, the coupon is modeled with shell elements. The number of integration points through the section thickness has been chosen utilizing the rule NIP ¼ N + 1, where NIP are the number of integration points and N are the ply number of the coupon. The material properties of the lamina used for the simulations are listed in Table 3.4. Models based on lamina-level failure criteria have been used, to predict the onset of damage within laminated codes. A degradation scheme is used to reduce the material properties once failure is initiated. With today’s computational power it is not possible to capture each of the failure mechanisms. Explicit finite element code solves equations of motion numerically by direct integration using explicit methods. LS-DYNA uses material models (or MAT cards): progressive failure (PFM) and CDM material models (examples are: MAT 22, MAT54/55, MAT58, MAT162). Each material model utilizes a different modeling strategy: failure criterion, degradation scheme, mat props, and set of parameters that are needed for computation but do not have an immediate physical meaning [12]. The MAT_54 material card is valid only for shell formulation. The difference in failure mechanism between MAT_54 and MAT_58 is in the predamage and postfailure process. The MAT_54 reduces the fiber strength to account for the matrix failure and implements a progressive failure model after yield. In this case a brittle failure with no yield is implemented. Otherwise the MAT_54 material card cannot model nonlinearity near the failure as it is performed by the MAT_58 material card. With the material card MAT_58 the failure mechanism is different, micro cracks and cavities are introduced into the lamina causing the stiffness degradations. Such reduction in the elastic module introduces nonlinearity to the deformation. This Table 3.4

Composite material properties (lamina)

Ea (psi)

Eb (psi)

νab

G12 (psi)

Xc (psi)

Xt (psi)

Yc (psi)

S12 (psi)

8.45 106

8 106

0.5

0.5 106

99 103

110 106

99 106

12 103

Ea, Young’s modulus in longitudinal direction; Eb: Young’s modulus in transverse direction; νab, Poisson’s ratio; G12, Shear modulus; Xc, longitudinal compressive strength; Xt, longitudinal tensile strength; Yc, transverse compressive strength, b-axis; Yt, transverse tensile strength, b-axis; S12: shear strength, ab plane.

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Dynamic Response and Failure of Composite Materials and Structures

material may be used to model composite materials with unidirectional layer, complete laminates, and woven fabrics, and is implemented for thin and thick shell elements.

3.3.2.1

Tensile test

The coupon modeled in LS-DYNA is shown in Fig. 3.8. One end is fixed and the other one moves with constant velocity, according to the ASTM standard. Simulations are performed on specimens of different ply number but taking the number of elements constant. The mesh of the open hole coupon, Fig. 3.9, is optimized in the hole proximity due to the fact that in this region there is a complex stress intensification factor distribution depending on both the stacking sequence and the circumferential position around the hole. The mesh on the open hole compression test is shown in the following figure. In this case the mesh of the “numerical” coupon is intensified in the hole proximity (Fig. 3.10). The reference value for the results is the coupon ultimate load average method. The ultimate load is read through the sum of the SPC_FORCE (Single Point Constrained Force) that represent the constraints reactions.

Y X

Prescribed motion Fixed

Fig. 3.8 Coupon modeling and boundary conditions.

Rw

Fig. 3.9 Mesh on the open hole tension coupon.

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103

Fig. 3.10 Mesh on the open hole compression coupon.

The LS-DYNA results are compared with experimental data obtained in the lab of Department of Industrial Engineering University of Naples. Experimental tests are performed on specimen with different ply number. For each coupon type and test type six tests have been performed as prescribed by the ASTM standard to obtain a statistical relevance. In Table 3.5 the coupon configurations are listed. The notched tensile and compression tests are performed with the same setting of the MAT_54 card. As presented in Table 3.6, there is a high data dispersion for the coupon result through different ply number. The material calibration process needs to ensure that the numerical ultimate load falls within the bands delimited by the minimum and maximum ultimate load of the experimental tests as near as possible to the mean value. Table 3.5

Coupon ply number used in the experimental tests

Test type

No. of plies

Open hole tensile

Compressive

Open hole compressive

Tensile

5, 8, 10, 16

8

8

6

Table 3.6

Experimental data dispersion

No. of plies Distance % between bands

5 12.54

8 11.90

10 8.94

16 9.41

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All the coupons with different ply number show a good comparison of the upper and lower limit band; on the other hand an iterative approach has been used varying the parameters of the material card as DFAILX to tune the results of the experimental data. DFAIL is a postfailure parameter [12]. It can be seen in MAT 54 and MAT 58 and it is the maximum strain, this could be for fiber tension (DFAILT), fiber compression (DFAILC), matrix straining in tension or compression (DFAILM), and shear (DFAILS). The layer in the element is totally deleted when the value of DFAILx is reached. Note that the input value for DFAILC should have a negative sign. The maximum value of DFAILx is 1 which means the element can withstand 100% strain. Fig. 3.11 shows a comparison between numerical output and relevant test failure load. The calibration process shows a good result through the different thickness. Results obtained with the numerical simulations are quite good and have a maximum deviation from the mean value of about 5%.

3.3.2.2

Compression, tensile open hole and compression open hole results

The results of the test for the eight ply coupon are shown in Table 3.5. The results for compression and open hole compression tests [30], are very close to the experimental data with a maximum gap of 4%. Some problems are present for the open hole tensile test; in fact, the percentage error is about 15%. More experimental results are necessary for an optimum material

Mat_54 — enhanced composite material 23,000 21,000 Ultimate load [lbf ]

19,000 17,000 15,000 13,000 11,000 9000 7000 5000 4

6

8

10

12

14

16

n° plies Mean value

Min. value

Max. value

LS-Dyna results

Fig. 3.11 Comparison between experimental results and LS-DYNA results using material model MAT_54.

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Table 3.7 Comparison between experimental and numerical results using material card MAT_54 Ply number

MAT 54 (lbf )

Average experimental results

Error (%) to MAT_54

10,242

2.1

5329

15

Ultimate load compression 8

10,463

Ultimate load open hole tension 8

4499

Ultimate load open hole compression 6

22,200

23,150

4

calibration. The goal is to start an experimental campaign to obtain the ultimate loads also at several ply numbers in the same manner done for the tensile test [31,32] (Table 3.7). A similar approach is used for MAT_58. As presented in Fig. 3.12, a great deviation is appreciable for the numerical model. A practical difficulty has been experienced in model calibration when the MAT_58 material model has been adopted. For this reason, the tests are made only for tensile tests. To calibrate material, ASTM tests have been numerically simulated using LS-DYNA explicit solution to reproduce CFRP.

Ultimate load [lbf ]

Mat_58 — laminated composite fabric 23,000 21,000 19,000 17,000 15,000 13,000 11,000 9000 7000 5000 4

6

8

10

12

14

16

n° plies Mean value

Min. value

Max. value

LS-Dyna results

Fig. 3.12 Comparison between experimental results and LS-DYNA results using material model MAT_58.

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Dynamic Response and Failure of Composite Materials and Structures

1. Tensile and compressive strength (ASTM-D3039, Refs. [33,34]) lay-up [45/0/45/0]s 2. Shear strength (ASTM-D3518, Refs. [35]) lay-up [45/45/45/45]s

Tests have been performed using equivalent shell approach with ICOMP Flag activated for orthotropic/anisotropic layered composite material. Gauss quadrature rule define integration point through different plies of coupon. According to the ASTM-D3039 stress/strain curves have been calculated. Figs. 3.13–3.15 show different behaviors of materials due to different damage approach. As it can be seen in the previous diagrams, stress in fiber direction increases and at the maximum value it remains constant. Then fibers break and the stress in fiber direction vanishes. The material behavior is elasto-plastic up to failure. MAT58 is a so-called elastic damage model, where it is assumed that the deformation introduces micro-cracks and cavities into the material. These effect causes stiffness degradation with small permanent deformation unless the material undergoes rather high loading and is not close to deterioration. When their maximum stress values are reached, strain softening starts and, due to minor numerical differences, several elements are further deformed, whereas other elements are unloaded.

3.3.2.3

Simulation of low energy impact

LVI is one of the most critical events for composite laminates. Indeed, laminated structures submitted to low energy impacts or small dropped objects, such as tools during assembly or maintenance operations, can undergo significant damage in terms of matrix cracks, fiber breakage, or delamination [36]. The interfacial damage is particularly dangerous [37] because it drastically reduces the residual mechanical 120,000 Traction 100,000

Stress [lbf ]

80,000

60,000

40,000 MAT 54 MAT 58

20,000

0 0

1000

2000

3000

4000

5000

Strain [µm/m]

Fig. 3.13 Traction stress (σ x)-strain (εz) curve; laminate [45/0/45/0]s.

6000

7000

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140,000 Compression 120,000

Stress [lbf ]

100,000 80,000 60,000 40,000 MAT 54 20,000

MAT 58

0 0

2000

4000

6000

8000

10000

12000

Strain [µm/m]

Fig. 3.14 Compression stress (σ x)-strain (εz) curve; laminate [45/0/45/0]s.

14,000 Shear 12,000

Stress [lbf ]

10,000 8,000 6,000 4,000 MAT 54 MAT 58

2,000 0 0

2000

4000

6000

8000

10,000

12,000

Strain [µm/m]

Fig. 3.15 Shear stress (σ x)-strain (γ xy) curve; laminate [45/45/45/45/45/45].

characteristics of the structure, and at the same time can leave very limited visible marks on the impacted surface [38]. Although it has been extensively investigated, the delamination mechanism of composite panels is far from being fully understood. One method to understand this damage mode is by the use of finite element analysis. The simulation of delamination using the finite element method is normally performed by means of the virtual crack closure technique (VCCT) [39] or using cohesive finite elements [40]. However, there are some difficulties when using the VCCT in the

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Dynamic Response and Failure of Composite Materials and Structures

simulation of progressive delamination. The calculation of fracture parameters requires nodal variables and topological information from the nodes ahead and behind the crack front. Such calculations are tedious to perform and may require remeshing for crack propagation. The use of cohesive finite elements can overcome some of these difficulties. They can be used to simulate both the onset and the propagation of delamination, Refs. [41,42], allowing to describe the nonlinear behavior of the surface of interlaminar adhesion and in particular to model the complex phenomena that occur in the so-called process zone, in the vicinity of the apex of the fracture. In this paragraph, the delamination damage resulting from LVI is examined using cohesive elements. The application of cohesive model is performed by finite element analysis in which the candidate delamination surfaces are modeled by interface elements obeying cohesive zone constitutive relationship. As understood from the name “interface,” the elements are located between the adjacent plies where the delamination occurs. They are positioned in places where potential cracks can develop, for example where there is a mismatch of material properties (discontinuity) across the interface as in layered structures between two adjacent plies with different fiber orientations, see Fig. 3.16. Cohesive elements are characterized by a traction-relative displacement relationship (typically nonlinear) which describes the evolution of the process zone and the formation of free surfaces by means of tractions. They combine aspects of strength-based analysis to predict the onset of damage at the interface and fracture mechanics to predict the propagation of a delamination. The interface element consists of two surfaces that are connected to adjacent solid elements; initially the two surfaces coincide, but they may be driven apart mechanically. The interface elements, in which the process zone is concentrated, are supposed to be in zero or near-to-zero thicknesses because they are located in the interfaces without affecting the real thickness of structure. This is the reason why the constitutive relations depend on displacements, not strains. The displacements in a cohesive zone model (CZM) are relative displacements

Composite ply

Cohesive interface

Fig. 3.16 Cohesive interfaces between two adjacent plies.

Experimental experiences of impact on composite structures

109 8

Top face

Thickness direction

n

Cohesive element node

Thickness direction

5

4

1

3 6

z

2 y

Bottom face

Top face 7

Bottom face

Midsurface

(A)

x

(B) F6

F5

8

7 F2

5 6 F3

F4

4

3

3 2

(C)

1

Stack direction = 1 From face 6 to face 4

F1

2

1 Stack direction

Stack direction = 2 From face 3 to face 5

Stack direction = 3 From face 1 to face 2

Fig. 3.17 COH3D8 finite element characteristic available in ABAQUS: (A) representation, (B) thickness direction, and (C) stack directions [43,44].

between the top and bottom interface surfaces, which represent the adjacent delamination surfaces (in other words, the lower and upper surfaces are adjacent laminas). In order to predict the initiation and three-dimensional growth of delamination, an eight-node cohesive element, available in most finite element commercial software packages, such ABAQUS for example, see Fig. 3.17, is typically adopted. By adopting this technique, the behavior of the material is divided into two parts: l

l

continuous undamaged part with an arbitrary material law (modeled with standard solid elements) as 3D eight-noded linear brick (hexahedral elements); the cohesive interface between the elements of the continuum, which specify only the damage of the material with cohesive properties (modeled with cohesive solid elements), see Fig. 3.18.

Therefore, the elements of the cohesive zone do not represent any physical material, but describe the cohesive forces that occur when elements of material are pulled from

Solid elements

Cohesive elements

Fig. 3.18 Eight-node cohesive and standard elements. Cohesive elements are positioned between the ones of bulk material.

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Dynamic Response and Failure of Composite Materials and Structures

nts

me

le lk e

Bu

zone sive Cohe ents elem Bulk

ents

elem

Fig. 3.19 The opening of the cohesive element [45].

opposite sides. The cohesive elements are connected with the layers above and below the interface, sharing nodes or by an interface bond (interface constraint or tie constraint). When the damage occurs or grows up, in the case of opening stress, the cohesive elements are opened in order to simulate the crack onset or growth, see Fig. 3.19. As previously anticipated, because the crack path can only follow these elements, the crack propagation direction depends on the presence (or absence) of cohesive zone elements, which means that the crack path is mesh dependent. If the crack propagation direction is not known in advance, the mesh generation must provide different possible crack paths. The behavior of interface elements for delamination simulation can be based on CZMs available in most commercial finite element software or on a cohesive law implemented into analysis code through an user subroutine. This last possibility is considered here. The CZM chosen is based on the study in Refs. [46,47], including the friction effect, and it is implemented in the explicit finite element code ABAQUS by using a vectorized user material subroutine, called VUMAT, written in Fortran [48] in order to study computationally the consequences of a low energy impact on interface between plies of dissimilar orientation. The user material is associated with the cohesive elements available in ABAQUS software library (COH2D4 for 2D problems, COH3D8 for 3D applications). For the definition of the interface behavior, the knowledge of the interface cohesive properties is necessary. The macroscopic properties of the interface material, such as the critical fracture energy in pure mode I, II, and III (previously referred to as GcI, GcII, and GcIII), can be measured experimentally and used directly in the cohesive model; instead, the damage-driving force at damage onset in pure mode I, II, and III may be obtained by experimental-numerical calibration procedure of model and the undamaged interface stiffness by means of empirical formulas. However, here, the fracture modes II and III will not be distinguished themselves and the cohesive properties associated with the mode III will be considered the same as those relating to the mode II. For the purposes above, the impact damage study on composite laminate reported in Ref. [49] will be the reference. In the following the cohesive-frictional model with interface elements discussed in Refs. [46,47] will be first calibrated with the data

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111

available by means of simulations of standard fracture toughness tests and finally employed to model the impact response of cross-ply graphite/epoxy laminated plate analyzed in Ref. [49].

3.3.2.4 Cohesive input properties In composite structures the damage onset and evolution usually consist of combined interlaminar (delamination) and intralaminar ply damaging mechanisms. In fact, the two physical damage forms are strongly coupled. Therefore, it is necessary to identify experiments which isolate interlaminar behavior (i.e., tests that result in minimal or no intralaminar material failure) in order to use experimental data to define cohesive material properties. Moreover, it is necessary to identify tests to characterize interfacial behavior which separate the normal and shear modes of delamination. In this regard the double cantilever beam (DCB) test is designed to produce pure normal mode delamination without any intralaminar material damaging and it is commonly used for obtaining pure modeI loading condition; while the end-notched flexural (ENF) specimen is designed to produce pure shear mode delamination without any intralaminar material damaging and it is commonly used for obtaining pure mode-II loading condition. Utilizing experimental data to characterize cohesive behavior, the number of cohesive properties that must be determined is minimized [50]. Cohesive input properties for the previously discussed CZM include parameters that define the stiffness, elastic energies at damage onset, and fracture energy of the cohesive material layer in each of its three deformation modes: a normal mode (denoted by a subscript n) and two shear modes (denoted by subscript s and t respectively), see Table 3.8. In order to determine the input properties of cohesive elements that must be used in finite element models to fit numerical results of DCB [51] and ENF simulations with experimental ones derived from the same tests, three steps are necessary: 1. to determine mesh size; 2. to calculate cohesive stiffness: kn, ks; 3. to calibrate elastic energy at damage onset: G0I, G0II

Table 3.8

Cohesive input properties

Normal mode: kn ¼ Stiffness (MPa/mm) G0I ¼ Elastic energy at damage onset (mJ/mm2) GcI ¼ Fracture energy (mJ/mm2) Shear mode: ks ¼ Stiffness (MPa/mm) G0II ¼ Elastic energy at damage onset (mJ/mm2) GcII ¼ Fracture energy (mJ/mm2) Note. The cohesive properties for the mode II and III are assumed to be equal.

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Dynamic Response and Failure of Composite Materials and Structures

3.3.2.5

Mesh size

Initially it is necessary to create DCB and ENF finite element models with loading and dimensions that match the experimental conditions. The DCB model is used to determine the normal cohesive properties (kn, G0I, and GcI), and the ENF model is used to determine the shear cohesive properties (ks, G0II, and GcII). The cohesive properties for the mode II and III are assumed to be equal. When a finite element model is created one must be aware that predicted cohesive behavior is mesh-dependent, i.e., using cohesive properties across a wide range of cohesive mesh densities a considerable range of predicted delamination responses will be obtained, Refs. [49–52]. Since cohesive solutions are mesh dependent, it is important that the meshes for the DCB and ENF specimens use cohesive elements that are approximately the same size as the cohesive elements that are anticipated to be used in subsequent progressive failure analyses of composite structural components. The influence of effect of mesh density on the simulated response of DCB and ENF tests was investigated during the first phase of the study but it will not be discussed here.

3.3.2.6

Cohesive stiffness

The cohesive stiffness should be determined before the elastic energy at damage onset is obtained. It is important to realize that one cannot determine a definitive value of stiffness for cohesive layers when it is used to simulate the delamination between plies. The stiffness of the cohesive layer needs to be stiff enough so that it provides adequate load transfer between the bonded layers, but if it is too stiff, then spurious stress oscillations can occur. As such, the following equation, presented in Ref. [52], can be used to estimate the stiffness of the cohesive layer in the mode-I direction: kn

αE3 t

(3.19)

According to the above formula, the interface stiffness is the elastic modulus of the resin per unit thickness, where E3 is the Young’s modulus of the laminate in the thickness direction, t is the larger of the sublaminate thicknesses above or below the cohesive layer, and α is a parameter that is much larger than 1 with a suggested value of 50 in Ref. [52] to obtain a stiffness of the cohesive layer which is small enough to avoid numerical problems (spurious oscillations of the tractions in an cohesive element) and also large enough to prevent the laminate from being too compliant in the thickness direction. Therefore, a value of α equal to 50 was used in the current study. In calculating the stiffness ks and kt in the shear directions associated respectively to mode II and mode III, the normal modulus of the composite material E3 is replaced with the shear moduli G12 and G13 of the laminate, respectively. However, for considerations made above, these stiffnesses are considered to be equal.

Experimental experiences of impact on composite structures

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3.3.2.7 Energies at damage onset After setting the stiffness of the cohesive material, the finite element models of the DCB and ENF specimens can be used to iteratively determine the elastic energies at damage onset of the cohesive material (G0I, G0II ¼ G0III). The DCB finite element model is used to calibrate G0I, and the ENF finite element model is used to calibrate G0II (or G0III). The energy values predicted by the DCB and ENF finite element models are dependent on both the cohesive mesh density and the stiffnesses chosen for the cohesive material (previous steps), thus it is likely that the initial damage energy estimates need to be adjusted in order for the DCB and ENF models to match the experimental results obtained by the same tests.

3.3.2.8 Calibration of the cohesive properties Numerical simulations of mode I and mode II delamination are now discussed. General guidelines to create finite element models of each test are given in Ref. [49]; in this are reported essential mechanical properties and geometries of testing specimens. The dimensions of each specimen are 20 mm in width, 150 mm in length, with a total thickness of 3 mm and an initial crack (crack-like defect) of 35 mm. The elastic properties of materials and the fracture properties of interface are reported in Table 3.9. In Table 3.2, E11, E22, and E33, are the Young’s modules in the direction of the fibers, in direction orthogonal to fiber and situated in the lamina plane and in the direction normal to the plane of the composite lamina respectively; ν12, ν13, and ν23 are the values of the Poisson’s ratio and G12, G13, and G23 are the shear modules; ρ is the composite material density and similar value is considered for the cohesive zone (SERVE LA DENSITA’?); finally, GcI, GcII ¼ GcIII are fracture energy associated to fracture modes obtained from experimental tests (Figs. 3.20 and 3.21). The experimental determination of critical energy release rates was conducted in Ref. [50], by means of DCB and the ENF specimens. DCB and ENF tests are performed on unidirectional composite laminates, which means that delamination growth occurs at a [0/0] interface and crack propagation is parallel to the fiber orientation. However, this kind of delamination growth will rarely occur in real structures [54].

Table 3.9

Cohesive input properties

Properties for composite material [53] E11 ¼ 93.7 GPa; E22 ¼ E33 ¼ 7.45 GPa G12 ¼ G23 ¼ G13 ¼ 3.97 GPa ν12 ¼ ν23 ¼ ν13 ¼ 0.261 ρ ¼ 1.5 109 ton/mm3 Properties for cohesive interface [35] GcI ¼ 0.520 mJ/mm2 GcII ¼ GcIII ¼ 0.970 mJ/mm2

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Dynamic Response and Failure of Composite Materials and Structures

a

L

Fig. 3.20 Double cantilever beam specimen geometry.

h

a

L

Fig. 3.21 End-notched flexure specimen geometry.

Bidimensional finite element models are used in simulations because with this simplification a simulation produces results practically coincident with those obtained by three-dimensional models; this was noted also in Ref. [51]. The composite arms are modeled with continuum plane stress elements with four nodes and reduced formulation (CPS4R). The behavior of these elements is linear elastic without any possibility of damage. The elastic parameters for these elements are reported in Table 3.9. Between the two composite arms, there is the cohesive layer with the exclusion of precracked zone. The interface between the sublaminates (the cohesive zone) is modeled as a zero-thickness cohesive layer and discretized using four nodes cohesive elements (COH2D4) available in the commercial finite element code ABAQUS. Cohesive elements are formulated in terms of traction vs. relative displacement relationship. They were not introduced for all the length of the specimen FE models associated to tests, but only from the end of the crack length, where the tip of the initial crack is, to

Experimental experiences of impact on composite structures

115

the end of the specimen. A convergence analysis was preliminary conducted to determine the appropriate element size but it will not be discussed here. The chosen length of the cohesive element is 0.5 mm because with it the responses computed via finite element analyses do not exhibit false instabilities caused by coarse meshes, as reported by several authors Refs. [55,56]. Contact pair surfaces were introduced only in ENF tests, in order to avoid interpenetration of sublaminates in the precracked zone (contact between composite arms and contact between bottom ply and rigid supports). Finally, an appropriate displacement was prescribed at nodes of load application according to considered experimental test. The cohesive model with interface elements adopting an exponential cohesive law discussed in the previous section is used to simulate delamination by means of user material subroutine implemented in ABAQUS. The onset of delamination is determined based on the interlaminar quadratic nominal stress criterion and the delamination growth is based on a critical fracture energy criterion. Damage is modeled as an irreversible process by including a damage parameter. The first numerical results presented regard the DCB simulation, which is obtained at the end of calibration. In Fig. 3.22 it is possible to see the deformed shape of the DCB specimen captured during the simulation; moreover the process zone, in progressive damage condition and resulting from opening displacements applied to composite arms, is in evidence. In Fig. 3.23 the force-opening curve is shown and this is compared with the same curve obtained in the experimental test. There is a good agreement between the elastic branch of the curve and also the onset prediction of the crack propagation looks adequate. The second numerical results presented regard the ENF simulation, which is obtained at the end of calibration. In Fig. 3.24 it is possible to see the deformed shape of the ENF specimen captured during the simulation; moreover, the contact zones between ply and rigid supports and between ply and ply in the precracked region are in evidence. In these zones a contact formulation is necessary to avoid the interpenetration. In Fig. 3.25 the force-sliding curve is shown and this is compared with the same curve obtained in the experimental test. There is a good agreement between the elastic branch of the curve and also the onset prediction of the fracture propagation looks adequate.

Fig. 3.22 Deformed shape of the specimen captured during the DCB simulation test.

Fig. 3.23 Calibration of the model with experimental result [50] for mode I: force vs. opening curve for the DCB test.

Fig. 3.24 Deformed shape of the specimen captured during the ENF simulation test.

Fig. 3.25 Model calibration with experimental result [51] for mode II: force vs. sliding curve for the ENF test.

Experimental experiences of impact on composite structures

117

3.3.2.9 Numerical simulation A numerical study was conducted to investigate the predictive capabilities of the cohesive-frictional model, discussed in this chapter, when applied to a practical impact problem. In the present investigation, the explicit solver of finite element software ABAQUS is used to calculate the transient response of the impact on composite laminates. The transient response of the impact is analyzed on the basis of the following assumptions: frictionless between the impactor and composite structure; neglecting the damping effect in the composite structure; ignoring the gravity force during the impact period; ignoring strain rate effect; rigid body for the impactor. Numerical simulations are carried out to test the implemented model ability to predict the interfacial damage for LVI on composite plate in terms of orientation, shape, and dimension of delamination. For the above purposes, the studies reported in Refs. [49,57] are the reference sources; from these are obtained geometrical data of the laminate, boundary conditions, mechanical properties of composite material, and the experimental results as X radiography image of specimen after impact and resulting delamination dimensions. The authors of indicated references conducted numerical and experimental investigations on impact-induced delamination in cross-ply composite laminates. The graphite/epoxy composite material forming the laminate, object of study is rectangular (45 mm 67.5 mm) and simply supported with laminate stacking sequence of [03/906/03]; therefore, two potential interfaces for the development of delamination are present: [0/90] and [90/0]. In fact, it has been demonstrated that impact-induced delamination in composite laminates is strongly dependent on the stacking sequence: the larger the difference of fiber angles between two adjacent laminae, the larger the bending stiffness between them, and hence the larger the delamination area at their interface [57,58]. Moreover several additional studies have revealed the characteristic “peanut” shape delamination in unidirectional composite laminate [59–61]. The delamination of the peanut shape is of particular interest here and will be considered in the present study (Fig. 3.26). For Coulomb’s friction coefficient a generic value of 0.5 taken from the literature, Refs. [15,62], is assumed. The impactor is a spherical body in steel of radius 6.25 mm and mass 2.3 kg. The impactor is modeled using an analytical rigid body which means that there is no deformation. The modeling performed for each ply consists of eight-node brick elements with single integration point (C3D8R); only one element through the thickness is considered for each group of plies. Moreover, because of symmetry, only one half of the model was built and analyzed. At the interface eight-node three-dimensional cohesive elements (COH3D8) are placed between layers with different fiber orientations. However, in Refs. [50,57] only the experimental results for the bottom interface [903/03] are available; therefore, only for this interface numerical results will be presented. The introduction of cohesive elements requires partitioning of the model and this phase is shown in Fig. 3.27; in this way the cohesive layers are created. After meshing, cohesive elements are assigned to the resin sections (interfaces). In order to create zero thickness cohesive elements, the collapse of the cohesive element nodes on to each other is necessary. However, it should be noted that zero thickness

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Dynamic Response and Failure of Composite Materials and Structures

Fig. 3.26 Model of composite plate in phase of partitioning. Initially the cohesive layer has finite thickness; after meshing the distance between the nodes on the opposite faces of the cohesive elements is reduced to zero.

Fig. 3.27 Model of composite plate in phase of partitioning. Initially the cohesive layer has finite thickness; after meshing the distance between the nodes on the opposite faces of the cohesive elements is reduced to zero.

Experimental experiences of impact on composite structures

119

cohesive elements are just an approximation. Fig. 3.27 shows the finite element mesh adopted for this study which represents a simplified model of the impact test set-up; the mesh size increased from the center toward the edges because a fine grid in the impact region of the target allows to obtain a smooth stress gradient. For proper contact definition between impactor and laminate, the “Surface-to-surface contact” formulation in ABAQUS is used. Impact energy levels between 1 and 2.5 J were considered. However, the limits of this range are sufficient to show the most important results. It is natural to expect that a significant delamination occurs at the highest impact energy. In Fig. 3.28 it is shown that the development of the damage interface to the interface farthest from the impact area when the impactor’s kinetic energy is maximum. Several studies [60,61] have found that each delamination oriented itself along the fiber direction of the bottom ply of the delaminated interface. In the analyzed case, the damage evolution and the delamination lobe occur precisely in the direction of the fibers oriented at 0 degree. Furthermore, the overall pattern of damage was similar to

Fig. 3.28 Development of delamination in the interface region [90/0].

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Dynamic Response and Failure of Composite Materials and Structures

that seen in carbon fiber reinforced composites used in the experimental investigation, Fig. 3.28. Therefore, the material model provides a good prediction of both the delamination peanut shapes (that is area characteristic of the interface damage caused by impact loading in fiber-reinforced composite laminates) and their orientations (Fig. 3.29). Finally, in Fig. 3.30, a numerical experimental comparison about the delamination length is presented. Good agreement between the finite element results and experimental data was achieved. Furthermore, from the two presented limiting cases it is possible to observe that until the impact energy is low, the effect of friction is not visible.

Fig. 3.29 X-radiography of composite panel: peanut-shaped delamination on bottom interface in evidence.

25

Delamination length (mm)

Experiment 20

15

10

5

0 0.0

0.5

1.0

1.5

2.0

2.5

Impact energy (J) Without friction With friction

Fig. 3.30 Numerical experimental comparison about the delamination length.

Experimental experiences of impact on composite structures

121

As the speed of the impactor increases (because the mass does not change) the delaminated area increases and therefore the friction between the damaged plies in contact becomes appreciable. However, this study does not aim to extinguish all doubt but especially proposes the simple introduction of the frictional component in a model of cohesive zone always neglected in the formulation of the cohesive approaches designed to simulate impact tests and whose ability in the prediction of delamination is just demonstrated here. Furthermore, the importance of friction component shall be validated by means of targeted and detailed experimental investigations that this study aims to promote. For example, it is not possible to conduct important research on the effect of friction at the interface closer to the region of impact because of the lack of data; in this interface it is expected that the frictional component plays a more important role being the compression stresses of greater magnitude. Moreover, the response sensitivity to the friction coefficient was not investigated leaving these studies for future works.

3.4

Conclusions and future trends

The growing widespread use of composite materials in the aerospace industry and, more generally, in all industrial applications requires the continuous evaluation of their mechanical properties through experimental characterization tests. The results obtained are the starting point for the determination of eligible project, which in turn, constitute the input data of the analysis of the structures in composite material. This chapter provided a comprehensive representation of the characterization of composite materials and equipment and measurement techniques currently employed, and how to resort to numerical simulations to avoid repeating expensive tests with the implementation of the model that can predict the time and manner of breaking elements that have followed a calibration process. Throughout the report they have called up the main ASTM regulations on testing. The objectives of the chapter are to develop a test methodology and to perform an extensive campaign of tests to provide a prediction of the deformation response of composite materials. The material model can capture the nonlinear material behavior of composites due to strain softening, so asto be implemented for various FE dynamic analysis system dedicated to evaluate a crash test of the small aircraft seat. A real possibility is to create a database which provides high speed dynamic material properties in order to perform the simulation of impact behavior with results close to the real behavior. Numerical simulation models are increasingly replacing physical testing because it provides a more rapid and less expensive way to evaluate design concepts and design details. In the aerospace industry, crashworthiness numerical simulation methods are primarily used at the very end of the product development process. Often they are applied to confirm the reliability of an already existing design, or sometimes for further design improvements by means of optimization methods. The database about high speed dynamic material characteristics could be a necessary output to gain an understanding of the fundamental modeling methods and a feeling for the comparative usefulness of different approaches.

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Dynamic Response and Failure of Composite Materials and Structures

Finally, to evaluate new materials and process to optimize the behavior of the structure at impact performance a very promising technological solution, that could be investigated, is represented by additive manufacturing processes. These processes allow users to both create one-off prototypes without requiring expensive tooling and create parts that are impossible or extremely challenging to build using alternative methods. With this technology, 3D parts are “printed” by drawing a molten plastic filament along a predefined path in a single layer in the x-y plane. The next layer is printed on top of the preceding layer. This stack of printed layers is somewhat analogous to a composite lamina contained within a laminate. Due to imperfect bonding between the filaments both in the x-y layers and between the y-z layers, the material strengths are reduced in both the x-y plane and z-plane as compared to a part manufactured with injection molding or subtractive manufacturing, such as milling.

References [1] Kinlock A. Norman Adrian de Bruyne. 8 November 1940 to 7 March 1997. Biogr Mems Fell R Soc 2000;46:127–43. [2] Federal Aviation Administration. Advisory circular AC 20-107B, composite aircraft structure; 2009. [3] Meo M, Marulo F, Guida M, Russo S. Shape memory alloy hybrid composites for improved impact properties for aeronautical applications. Compos Struct 2013;95:756–66. http://dx.doi.org/10.1016/j.compstruct.2012.08.011. [4] Petrone G, Rao S, De Rosa S, Mace BR, Franco F, Bhattacharyya D. Behaviour of fibre-reinforced honeycomb core under low velocity impact loading. Compos Struct 2013;100:356–62. http://dx.doi.org/10.1016/j.compstruct.2013.01.004. [5] Guida M, Marulo F, Meo M, Russo S. Experimental tests analysis of fiber metal laminate under birdstrike. Mech Adv Mater Struct 2012;19(5):376–95. http://dx.doi.org/ 10.1080/15376494.2010.542273. [6] Batra RC, Gopinatha G, Zheng JQ. Damage and failure in low energy impact of fiber-reinforced polymeric composite laminates. Compos Struct 2012;94:540–7. [7] Soden PD, Kaddour AS, Hinton MJ. Recommendations for designers and researchers resulting from the world-wide failure exercise. Compos Sci Technol 2004;64:589–601. [8] Moncada AM, Bednarcyk BA. Micromechanics-based progressive failure analysis of composite laminates using different constituent failure theories. J Reinf Plast Compos 2012;31(21):1467–87. [9] Maa R, Cheng J. A CDM-based failure model for predicting strength of notched composite laminates. Compos Part B 2002;33(6):479–89. [10] Van Der Meer FP, Sluys LJ. Continuum models for the analysis of progressive failure in composite laminates. J Compos Mater 2009;43(20):2131–56. [11] Livermore Software Technology Corporation. LS-DYNA® Keyword user’s manual, Version 971; July 2006. [12] LS-DYNA Theory Manual; March 2006. [13] Talreja R. Damage characterization by internal variables. In: Talreja R, editor. Damage mechanics of composite materials. Amsterdam: Elsevier; 1994. [14] Ladeveze P, Le Dantec E. Damage modeling of the elementary ply for laminated composites. Compos Sci Technol 1992;43:257–67.

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[15] Xiao Y, Wang WX, Takao Y, Ishikawa T. The effective friction coefficient of a laminate composite, and analysis of pin-loaded plates. J Compos Mater 2000;34(1):69–87. [16] Xiao X. Modeling energy absorption with a damage mechanics based composite material model. J Compos Mater 2009;43(5):427–44. [17] Williams K, Vaziri R. Application of a damage mechanics model for predicting impact damage in composite structures. Compos Struct 2001;79:997–1011. [18] Yen, CF. Caiazzo, A. Innovative processing of multifunctional composite armor for ground vehicles. ARL technical report arL-CR-484. Aberdeen Providing Ground, MD: US Army Research Laboratory; 2001. [19] Matezenmiller A, Lublinear J, Taylor RL. A constitutive model for aniusotropic damage in fiber-composites. Mech Mater 1995;20:125–52. [20] Murakami S. Continuum damage mechanics—a continuum mechanics approach to the analysis of damage and fracture. Dordrecht, Heidelberg, London, New York: Springer; 2012. [21] Bonora N. A nonlinear CDM model for ductile failure. Eng Fract Mech 1997;58(1):11–28. [22] Kachanov L. On the creep rupture time. Izv Akad Nauk SSSR 1958;8:26–31. [23] Lemaitre J. Evaluation and dissipation of damage in metals submitted to dynamic loading, In: Proceedings I.C.M.I. Kyoto, Japan; 1971. [24] Lemaitre J. A continuous damage mechanics model for ductile fracture. J Eng Mater Technol 1985;107:83–9. [25] Sluzalec A. Theory of thermomechanical processes in welding. The Netherlands: Springer; 2005. [26] Jira´sek M. Damage and smeared crack models. Numerical modeling of concrete cracking. CISM international centre for mechanical sciences, vol. 532. Vienna: Springer; 2011. pp. 1–49. [27] Haqu BZ. A progressive composite damage model for unidirectional and woven fabric composites. User manual v. 10.0. Materials Sciences Corporation & University of Delaware Center for Composite Materials; 2011. [28] Hebda DA, Whitlock ME, Ditman JB, White SR. J Intell Mater Syst Struct 1995;6:220. [29] Esteban D, Moncavo J, Wagner H, Drechsler K. Benchmarks for composite delamination using Ls-Dyna 971: low velocity impact. LS-DYNA Anwenderforum, Frankenthal; 2007. [30] Curtis, PT. Crag test method for the measurement of the engineering properties of the fibred reinforced plastic. RAE technical report 88012; 1988. [31] ASTM D5766/D5766M-11. Standard test method for open-hole tensile strength of polymer matrix composite laminates; 2011. [32] ASTM D6484/D6484M-14. Standard test method for open-hole compressive strength of polymer matrix composite laminates; 2014. [33] ASTM D3039/D3039M-14. Standard test method for tensile properties of polymer matrix composite materials; 2014. [34] ASTM D695-15. Standard test method for compressive properties of rigid plastics; 2015. [35] ASTM D3518/D3518M-13. Standard test method for in-plane shear response of polymer matrix composite materials by tensile test of a 45° laminate; 2013. [36] Bouvet C, Rivallant S, Barrau JJ. Low velocity impact modeling in composite laminates capturing permanent indentation. Compos Sci Technol 2012;72(16):1977–88. [37] Van der Sluis O, Yuan CA, van Driel WD, Zhang GQ. Advances in delamination modeling. Nanotechnologies and electronics packaging. New York: Springer US; 2008. pp. 61–91. [38] Abrate S. Impact on composites structures. Cambridge: Cambridge University Press; 1998.

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[39] Krueger R. The virtual crack closure technique: history, approach and applications report2002-211628. NASA/Contractor; 2002. [40] Chen J, Crisfield MA, Kinloch AJ, Busso EP, Matthews FL, Qiu Y. Predicting progressive delamination of composite material specimens via interface elements. Mech Compos Mater Struct 1999;6:301–17. [41] Camanho PP, Da´vila CG, de Moura MFSF. Numerical simulation of mixed-mode progressive delamination in composite materials. J Compos Mater 2003;37(16):1415–38. [42] Da´vila CG, Camanho PP, Turon A. Cohesive elements for shells. NASA/TP-2007214869; 2007. [43] Rudawska A. Adhesive joint strength of hybrid assemblies: titanium sheet-composites and aluminium sheet-composites-experimental and numerical verification. Int J Adhes Adhes 2010;30(7):574–82. [44] Simulia (2008) Abaqus analysis user’s manual, version 6.5. Simulia Inc, Providence. https://www.simulia.com [45] Marzi S, Ramon-Villalonga L, Poklitar M, Kleiner F. Usage of cohesive element in crash analysis of large, experimental tests and simulation. German LS-Dyna Forum, Bamberg; 2008. [46] Valoroso N, Champaney L. A damage-mechanics-based approach for modelling decohesion in adhesively bonded assemblies. Eng Fract Mech 2006;73(18):2774–801. [47] Valoroso N, Champaney L. A damage-friction formulation for the de-cohesion analysis of adhesive joints. In: Proceedings of the fifth international conference on engineering computational technology; 2006. p. 103. [48] Dassault systems, Abaqus User Subroutines Reference Manual, Version 6.8. Simulia Inc, Providence; 2007. https://www.simulia.com. [49] Aymerich F, Dore F, Priolo P. Prediction of impact-induced delamination in cross-ply composite laminates using cohesive interface elements. Compos Sci Technol 2008;68:2383–90. [50] Guidelines for determining finite element cohesive material parameters. Firehole Composites, https://www.scribd.com/document/228643141/Guidelines-for-CoheseiveParameters-eBook-3. [51] ASTM D5528-13. Standard test method for mode I interlaminar fracture toughness of unidirectional fiber-reinforced polymer matrix composites; 2013. [52] Turon A, Davila CG, Camanho PP, Costa J. An engineering solution for mesh size effects in the simulation of delamination using cohesive zone models. Eng Fract Mech 2007;74 (10):1665–82. [53] Maimı´ P, Camanho PP, Mayugo JA, Davila CG. A continuum damage model for composite laminates: part II – computational implementation and validation. Mech Mater 2007;39 (10):909–19. [54] Kruger R. Three dimensional finite element analysis of multidirectional composite DCB, SLB and ENF specimens. ISD-report no. 94/2; 1994. [55] Schellekens JCJ, de Borst R. On the numerical integration of interface elements. Int J Numer Methods Eng 1992;36:43–66. [56] de Borst R, Remmers JJC, Needleman A. Computational aspects of cohesive zone models, In: 15th European conference on fracture. Stockholm, Sweden: European Structural Integrity Society; 2004. [57] Aymerich F, Dore F, Priolo P. Simulation of multiple delaminations in impacted cross-ply laminates using a finite element model based on cohesive interface elements. Compos Sci Technol 2009;69:1699–709.

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[58] Liu D. Impact-induced delamination – a view of material property mismatching. J Compos Mater 1988;22(7):674–91. [59] Liu D, Kokolakis M, Raju BB. Effects of stacking sequence on perforation resistance of composite laminates, In: Proceedings of the American Society for Composites, fourteenth annual conference, Dayton, OH; 1999. p. 121–30. [60] Takeda S, Minakuchi Y, Okabe Y, Takeda N. Delamination monitoring of laminated composites subjected to low velocity impact using small-diameter FBG sensors. Compos Part A 2004;36:903–8. [61] Choi HY, Chang FK. A model for predicting damage on graphite/epoxy laminated composites resulting from low-velocity point impact. J Compos Mater 1992;26 (14):2134–69. [62] Sch€on J. Coefficient of friction of composite delamination surfaces. Wear 2000;237 (1):77–89. [63] Ladeveze P. Inelastic strains and damage. In: Talreja R, editor. Damage mechanics of composite materials, Composite materials series, vol. 9. Amsterdam: Elsevier Science; 1994.

Further Reading [1] Simo J, Ju J. Strain and stress based continuum damage models. I. Formulation. Int J Solids Struct 1987;23(7):821–40. [2] ASTM D7136/D7136M-15. Standard test method for measuring the damage resistance of a fiber-reinforced polymer matrix composite to a drop-weight impact event; 2015. [3] Loikkanen MJ. Guidelines for modeling delamination in composite materials using Ls-Dyna, Boeing Commercial Airplanes, Seattle, Washington (unpublished data). [4] Camanho CG, Davila MF, Moura DE. Numerical simulation of mixed-mode progressive delamination in composite materials. J Compos Mater 2003;37(16). http://dx.doi.org/ 10.1177/0021998303034505. [5] Maimı´ P, Camanho PP, Mayugo JA, Davila CG. A continuum damage model for composite laminates: part I – constitutive model. Mech Mater 2007;39(10):897–908. [6] Lapczyk I, Hurtado JA. Progressive damage modeling in fiber-reinforced materials. Compos A: Appl Sci Manuf 2007;38(11):2333–41. [7] Turon A, Camanho PP, Costa J, Davila CG. A damage model for the simulation of delamination in advanced composites under variable-mode loading. Mech Mater 2006;38 (11):1072–89. [8] Hashin Z. Failure criteria for unidirectional composites. J Appl Mech 1980;47:329–34.

Modeling low velocity impact phenomena on composite structures

4

A. Riccio, S. Saputo, A. Raimondo, A. Sellitto The University of Campania “Luigi Vanvitelli”, Aversa, Italy

4.1

Introduction

Composite materials are being used in several industrial fields such as railways, aerospace, naval, and automotive for their excellent strength and outstanding specific stiffness and strength properties. However, the weight saving obtainable by this class of materials is still limited by overconservative design approaches partially related to their poor damage tolerance to low velocity impact (LVI) phenomena [1]. During their operative life composite components are exposed to LVIs, such as runway debris and dropped tools. In most cases, this type of impact can induce damage formations which are not easily detectable by visual inspection (BVID) and can cause a substantial reduction of component structural strength [2–4]. In-plane damage, such as fiber breakage and matrix cracking, and out-of-plane damage, such as delaminations, are the main failure modes induced by LVIs [5]. In particular, delaminations are among the most dangerous failure mechanisms because, especially under compressive load, they can propagate (undetected) during service, leading to a substantial premature strength reduction of the whole structure. The heterogeneous nature of composite materials jointed to an uneven resin distribution, fiber discontinuities, and micro-gaps can cause, as already remarked, several failure mechanisms (fiber breakage, matrix failure, and delamination) interacting each other. Furthermore, the damage onset and evolution can be deeply influenced by physical impact parameters, environmental condition, impactor properties, and shape and structural configuration. Because of all these parameters, the development of a reliable numerical tool able to exhaustively describe and predict impact induced failure is a challenging task still in progress even if the prediction of the impact response and the characterization of impact induced damage are widely treated literature topics. Indeed, analytical, experimental, and numerical approaches have been explored to exhaustively characterize the effects of LVIs on composite structures. In particular, the analytical approaches [6–8] can be used only under strong physical simplifications. Consequently, the solution obtained with these approaches is effective for a limited range of impact configurations. On the other hand, numerical approaches, thanks to advances in computer science, are almost able to reproduce composite failure mechanisms even for complex impacted structures. The majority of constitutive damage models available in Dynamic Response and Failure of Composite Materials and Structures. http://dx.doi.org/10.1016/B978-0-08-100887-4.00004-4 © 2017 Elsevier Ltd. All rights reserved.

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literature, for numerical approaches, assumes the behavior of a single lamina as linear elastic up to the failure, while nonlinear models are adopted to take into account the in-plane and out-of-plane shear behavior [9,10]. The several numerical models proposed in literature papers usually differ in terms of damage onset and propagation criteria. Damage onset criteria can be classified in terms of interaction between stress and strain components. Maximum stress and strain criteria are characterized by no interaction between strain and stress components; indeed, these approaches assume that failure takes place when at least one stress or strain component along the principal material axis exceeds the corresponding strength or ultimate strain value. On the other hand, interactive criteria evaluate the equivalent failure load by contributions of load in all directions, taking into account the interaction between strain and stress components but not the modes separation. Finally, more refined criteria consider failure modes separation by evaluating the single mode-related failure load taking at the same time in account the interaction between stress and strain components. Numerical models, adopting the aforementioned damage criteria to take into account the damage progression, can be classified in “heuristic models” based on ply-discount material degradation [11] and “Continuum Damage Mechanics (CDM) based models” based on constitutive equations of composite materials [12,13]. The Heuristic models use a degradation factor β to reduce instantaneously the elastic stiffness moduli after the damage event to take into account the damage accumulation. This approach which considers the single lamina as a brittle material, even if very trivial to implement, gives a strongly unrealistic prediction when evaluating the stress redistribution in undamaged neighboring plies [14]. According to “Continuum Damage Mechanics (CDM) based models,” the material stiffness is reduced in a gradual way. These models reduce the numerical convergence problems by introducing several Internal State Variables (ISV) to describe the material damage state which is related to the mechanical constitutive laws. The ISV can find a reasonable physical interpretation if related to the density of microcracks in a damaged Representative Volume Element (RVE) of material with the assumption that the propagation of microcracks, evaluated thanks to micromechanical theories, can be used macroscopically as a measure of the damage over the RVE [12]. Several damage ISV are available in literature; Talreja [15] and Krajcinovic and Fonseca [16] evaluate the anisotropy of material introducing a damage variable for each damage mode. Pickett et al. [17] proposed a bilinear curve to define the material softening phenomenon reduction. Matzenmiller et al. [18] introduce in his damage model two damage variables associated to longitudinal and transverse normal stress components and one for the in-plane shear stress component. Damage propagation numerical models introduce mesh dependency issues, due to the presence of finite elements which, actually, cannot fully represent the continuity of damage propagation phenomena. To alleviate the mesh dependency issue, a number of theories have been developed. Heimbs et al. [19] introduce the smeared crack formulation based on a length parameter (connected to finite element discretization), which, joined to the constitutive law, allows to obtain a constant energy release value per unit area of crack [20]. Maimı´ and Lopes [21,22] adopted the smeared crack formulation using an exponential damage evolution law. Iannucci and Willows [23] introduced a

Modeling low velocity impact phenomena on composite structures

131

novel approach to evaluate the damage evolution on an impacted reinforced composite plate. Pinho et al. [24] and Falzon and Apruzzese [25] suggested a 3D failure theory based on matrix compression failure modes of Puck’s Theory [26] accounting for a nonlinear shear formulation [27]. As already remarked, a relevant role in impact-induced damage on set and evolution is played by the delaminations. Several methods for predicting delaminations have been proposed in literature. The virtual crack closure technique (VCCT) investigated, among others, by Krueger [28] provides an accurate evaluation of the energy release rate associated with delamination growth even with a coarse mesh. This technique has several relevant limitations such as the mandatory presence of a preexisting delamination and a severe mesh dependency. Cohesive Zone Model-based finite elements (CZM) [29–32] are almost able to overcome the VCCT limitations, by introducing proper damage initiation and propagation laws zero or quasi zero thickness in order to simulate delamination onset and propagation. The cited intralaminar and interlaminar damage onset and propagation models have been used to investigate the interlaminar and intralaminar damage evolution in composites [33–36]. For both the damage categories, as already pointed out, the mesh size influences the crack propagations, invalidating the results [32]. To reduce the limitations related to mesh size dependency, a variety of energy criteria-based approaches have been introduced [37,38]. The aim of this work is to investigate the damage phenomena, associated to LVIs, by adopting a sophisticated FEM (Finite Element Method) model based on CDM and CZM, respectively, for intralaminar and interlaminar damage onset and propagation simulation. A first validation of the proposed numerical model has been attempted by comparing the force, displacement, and energy vs time numerical curves with experimental data [39,40]. The influence of the complexity of the model, which can selectively take into account intralaminar and interlaminar damage simulation, on the numerical results has been also evaluated. Force and energy vs time and phenomenon duration obtained by means of the numerical simulations has been evaluated. Results obtained by means of the proposed intralaminar failure model (introduced by adopting the ABAQUS user subroutine capability) have been compared to numerical results obtained by means of the standard Hashin’s criteria failure formulation, available in the ABAQUS explicit FEM code and to experimental data. In Section 4.2 the theoretical background is presented, while in Section 4.3 the numerical applications are introduced, compared to the experimental data, and discussed.

4.2

Theoretical background

LVI events on a composite laminate, usually, cause concurring failure mechanisms such as matrix cracks, fiber failure, and delaminations. Generally, matrix cracking is the first occurring failure mechanisms induced by LVIs and, even if not able to considerably reduce the laminate material properties, it acts as a delamination initiation trigger. Delaminations occur at the interface between different oriented layers driven by interlaminar shear stresses, stiffness variation between the adjacent plies, and

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structural deflection. Consequently, the key point for a correct impact phenomenon prediction is the accurate simulation of the interaction between intralaminar damage (matrix cracks and fiber failure) and interlaminar damage (delamination). In the next sections a survey about the damage models employed in this work to simulate the impact-induced failure phenomena is given. The interlaminar damage onset and propagation has been predicted by using CZM-based elements while for the intralaminar damage modeling a CDM-based approach with ISV acting as damage variables has been implemented in Fe code Abaqus/Explicit [41].

4.2.1

Interlaminar damage model

Cohesive elements have been used to simulate the interlaminar damage formation and propagation. Cohesive constitutive response is based on two different phases, the initial damage phase (representative of the damage onset) and the evolution damage phase (representative of the propagation of the damage up to the complete failure of the element), with the chance to remove the fully damaged elements (Fig. 4.1) Abaqus FE code allows to define two different stress-based initiation criteria for the traction-separation law, maximum nominal stress criterion (MAXS), and quadratic nominal stress criterion (QUADS). The MAXS criterion does not assume any relation between the different stress directions while the QUADS criterion considers concurring quadratic ratios between nominal stress and allowable stress acting in different directions. In Table 4.1 the relations associated to the initiation criteria are shown. Traction

Nmax, Tmax, Smax,

Failure onset

En heff

GIC, GIIC, GIIIC Fracture energy

Complete failure d initial

d final

Separation

Fig. 4.1 Traction-separation law for the cohesive material.

Table 4.1 MAXS QUADS

Damage initiation criteria

σn σt σs ¼1 , , Nmax Tmax Smax σn 2 σt 2 σs 2 + + ¼1 Nmax Tmax Smax MAX

Modeling low velocity impact phenomena on composite structures

Table 4.2

133

Failure evolution laws

Power law (PW) Benzeggagh-Kenane (BK)

α GII α GIII α + + ¼1 GIIc GIIIc GSHEAR η ¼ GTC GIc + ðGIIc GIc Þ GT GI GIc

In this work the QUADS approach has been adopted. With reference to Table 4.1, Nmax is the nominal stress in the pure normal mode, Tmax is the nominal stress in the first shear direction, and Smax is the nominal stress in the second shear direction. Abaqus FE code allows to define different failure evolution laws for cohesive element, the “Power law PW,” the “Benzeggagh-Kenane BK,” and a degradation law implemented by the user with a tabular form. In Table 4.2 the PW and BK failure evolution laws are reported. With reference to Table 4.2, Gj is the Energy Release Rate associated to the fracture mode j, Gjc is the critical Energy Release Rate associated to the fracture mode j (with GIc Normal mode fracture energy; GIIc Shear mode fracture energy first direction; GIIIc Shear mode fracture energy second direction). For isotropic fractures GIc ¼ GIIc the material response is independent by η value. Besides GSHEAR ¼ GII + GIII; GT ¼ GI + GSHEAR. α and η are empiric values, usually ranging between 1 and 1.6, defined by means of ad hoc experimental tests.

4.2.2

Intralaminar damage models

Two numerical models for the simulation of the intralaminar damage onset and evolution have been adopted in this paper. Indeed, the ABAQUS standard approach has been compared with a newly developed procedure implemented in ABAQUS by the user subroutine feature.

4.2.2.1 ABAQUS standard approach for intralaminar damage prediction The ABAQUS considers, as standard formulation for the simulation of intralaminar damage onset and propagation, criteria associated to composite failure modes (Hashin’s criteria formulation) [42,43]. According to these criteria, it is possible to distinguish among different failure modes such as matrix tensile and compression cracking and fiber tensile and compression breakage. The adopted criteria, able to identify the failure initiation, are reported in Table 4.3. The constitutive relation adopted for each failure mode is graphically shown in Fig. 4.2. Again (as seen for the intralaminar damage constitutive law), two different phases can be identified, the initiation phase, where the material is considered undamaged, and the propagation phase, where the material is considered partially damaged up to element’s final failure condition (point B). Point A in Fig. 4.2 identifies the limit

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Dynamic Response and Failure of Composite Materials and Structures

Table 4.3

Hashin failure criteria (ABAQUS standard formulation) _ 2 _ 2 Fft ¼ σX11 + σS12 ¼1 T L _ 2 ¼1 Ffc ¼ σX11 C _ 2 _ 2 + σS12 ¼1 Fmt ¼ σY22 T L " # _ _ 2 2 _ 2 σ 22 YC σ 22 σ 12 + 1 + ¼1 Fmc ¼ 2ST 2ST YC SL

_

Fiber tension σ 11 > 0 _

Fiber compression σ 11 < 0 _

Matrix tension σ 22 0 _

Matrix compression σ 22 0

Equivalent stress

A

0 seq

ing

ad

lo Un kd

(1–di) kd

0 deq

Gc B t Equivalent deq

displacement

Fig. 4.2 Constitutive relation adopted for each failure mode (interlaminar damage—ABAQUS standard formulation).

stress value according to Hashin’s criteria; starting from this point the increase of the stress causes the gradual degradation of the material properties. The degradation coefficient, dI, for the failure mode I is given by relation (4.1). This coefficient ranges between 0 (undamaged state) and 1 (fully damaged state) for each failure mode. δ fI,eq δI,eq δ0I, eq ; δ0I, eq δI, eq δ fI, eq ; I 2 ðfc , ft , mc , mt Þ dI ¼ δI,eq δ fI,eq δ0I, eq

(4.1)

δI,eq0 is the equivalent displacement at which the initiation criterion is satisfied, and f is the equivalent displacement at which the material is completely damaged δI,eq f , introduced in Eq. (4.1), can be obtained from the following (dI ¼ 1). The variable δI,eq relation, assuming that the fracture energy Gc is known and the softening is linear: δ fI,eq ¼

2GIc σ 0I,eq

(4.2)

Modeling low velocity impact phenomena on composite structures

Table 4.4

135

Equivalent displacement and stress definitions

Failure mode Fiber tension ðσ^11 0Þ

δeq qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Lc hε11 i2 + αε212

Fiber compression ðσ^11 < 0Þ

Lc hε11 i

Matrix tension ðσ^22 0Þ

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Lc hε22 i2 + ε212

Matrix compression ðσ^22 < 0Þ

Lc

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ hε22 i2 + ε212

σ eq Lc ðhσ 11 ihε11 i + ασ 12 ε12 Þ δ fteq Lc hσ 11 ihε11 i δ fc eq Lc ðhσ 22 ihε22 i + σ 12 ε12 Þ δmt eq Lc ðhσ 22 ihε22 i + σ 12 ε12 Þ δmc eq

σ I,eq0 is the equivalent stress satisfying the initiation criteria. The equivalent stress and displacement expressions for the four different failure modes are reported in Table 4.4. The damage evolution law is based on the fracture energy dissipated during the damage process, Gc. This evolution law is a generalization of the approach proposed by Camanho and Davila [44] for modeling interlaminar delamination using cohesive elements. With reference to Table 4.4, Lc is the characteristic length of the element. In ABAQUS, this parameter is introduced to alleviate mesh dependency issue during the material softening phase. is the Macaulay bracket operator defined as: hγ i ¼

ðγ + j γ j Þ 8γ 2 ℜ 2

(4.3)

Different methods are suggested in literature for the computation of the characteristic length. Among others, Raimondo et al. [45] propose the following relation for square elements: pﬃﬃﬃﬃﬃﬃ Aip (4.4) Lc ¼ cos θ where Aip is the area associated to an integration point and θ is the angle between the element edge and the crack direction. In this paper the characteristic length has been assumed as the square root of the area associated to the integration point. This method is very efficient computationally, especially for elements with aspect ratios close to 1. However, as already remarked, even if desirable, characteristic length does not completely eliminate the mesh dependency issue.

4.2.2.2 Developed user subroutine (VUMAT) approach for intralaminar damage prediction In this section, the newly developed user subroutine (VUMAT) for the simulation of the intralaminar damage onset and propagation is described. Five intralaminar failure modes have been taken into account: tensile fiber failure, compressive fiber failure, tensile

136

Dynamic Response and Failure of Composite Materials and Structures

matrix failure, compressive matrix failure, and shear (nonlinear formulation failure). For each failure mode, the damage onset criteria and evolution have been described.

Tensile fiber failure For the fiber tensile failure, the failure onset criterion is identified by relation (4.5), while the failure evolution law is represented by Eq. (4.6)

FTrac 1

σ1 ¼ XT

2 1 with σ 1 0

(4.5)

0 δ fail δ δ eq eq eq d1Trac ¼ fail δeq δ eq δ0eq

(4.6)

where δ fail eq ¼

2GTIC σ 0eq

(4.7)

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðε11 LC Þ2 δ0eq ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ FTrac 1

(4.8)

Fig. 4.3 shows the constitutive law representing the fiber tensile failure mode. The damage onset condition (point A in Fig. 4.3) and the fully damaged element condition (point B in Fig. 4.3) can be appreciated. Equivalent displacement can generally be written as δ ¼ Lc ε [46].

Compressive fiber failure For fiber compression, the failure onset, and evolution are described by the following equations. Equivalent stress

A

0 seq

g

din

a nlo

U kd

(1–di) kd

Gc B

0 deq

Fig. 4.3 Constitutive relation adopted for fiber traction failure mode.

t Equivalent deq

displacement

Modeling low velocity impact phenomena on composite structures

σ1 XC

FCompr ¼ 1

137

2 1 with σ 1 < 0

(4.9)

δ fail δeq δ0eq eq d1Compr ¼ 0 δeq δ fail eq δeq

(4.10)

where 2GCIC σ 0eq

(4.11)

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðε11 LC Þ2 0 δeq ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ FCompr 1

(4.12)

δ fail eq ¼

In order to introduce the effect of fragment interaction under compression within the damaged area, a fiber compression residual strength, assumed equal to the matrix compressive strength and graphically shown in Fig. 4.4, has been considered.

Tensile matrix failure Damage initiation and evolution laws for tensile matrix cracking have been defined according to the following relations: ¼ FTrac 2

σ2 YT

2 2 2 τ12 τ23 + + 1 with σ 2 0 S12 S23

Equivalent stress XT

A

Ei 0

deq Residual strength

(4.13)

(1–di)Ei

Gc

0

deq

Fig. 4.4 Constitutive relation for fiber failure in traction and compression.

B t deq Equivalent displacement

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Dynamic Response and Failure of Composite Materials and Structures

0 δ fail δ δ eq eq eq d2Trac ¼ fail δeq δ eq δ0eq

(4.14)

where qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðε22 LC Þ2 + ðε12 LC Þ2 + ðε23 LC Þ2 0 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ δeq ¼ FTrac 2 δ fail eq ¼

(4.15)

2GTIIC σ 0eq

(4.16)

Compressive matrix failure Experiments show that unidirectional laminates under transverse compressive loads fail by shear [26], with a fracture plane oriented at θf ¼ 53° 2° from the laminate plane (see Fig. 4.5). To correctly describe this phenomenon, the candidate criteria should thus evaluate the stress/strain state along the real fracture plane instead of the nominal coordinate plane (with θf ¼ 0°). Considering a general fracture plane L-N-T, with fracture angle θf, the stresses can be expressed in the (LNT) coordinate system, thanks to a transformation from the laminae coordinate system 1-2-3 (as shown in Fig. 4.5) according to the following relations: 8 2 2 > < σ nn ¼ σ y cos θf + σ z sin θf + 2τyz cos θf sin θf τnl ¼ τyz cos θf + τzx cos θf >

: τnt ¼ σ z σ y cos θf sin θf + τyz cos 2 θf sin 2 θf

(4.17)

3

stn

qf

snn 2

sln T N

1

Fig. 4.5 Compressive matrix failure and fracture plane.

L

Modeling low velocity impact phenomena on composite structures

139

The damage model adopted in this paper for simulating transverse failure under compression adopts the criterion proposed by Puck and Schurman [26] (see Eq. 4.18) to check for the damage initiation.

FCompr 2

τnt ¼ A S23 μnt σ nn

2 +

τnl S12 μnl σ nn

2 1 with σ 2 < 0

(4.18)

where S12 is the longitudinal shear strength and SA23 is the transverse shear strength in the potential fracture plane; μnt and μnl are friction coefficients, respectively, in the transverse and longitudinal directions. These parameters are related to the fracture angle according to the following relations (where YC is the standard transverse compressive strength). ϕ ¼ 2θf 90°

(4.19)

μnt ¼ tan ϕ

(4.20)

SA23 ¼

YC ð1 sin ϕÞ 2 cos ϕ

μnl ¼ μnt

S12 SA23

(4.21)

(4.22)

The damage evolution is evaluated by using the following criterion. δ fail δeq δ0eq eq d2Compr ¼ 0 δeq δ fail eq δeq

(4.23)

where δ fail eq ¼

2GCIIC σ 0eq

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðεln LC Þ2 + ðεtn LC Þ2 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ δ0eq ¼ FCompr 2

(4.24)

(4.25)

Nonlinear shear damage Experimental evidence indicates that composite materials exhibit a nonlinear and irreversible behavior under shear loading [27]. The total shear strain can be decomposed into elastic and damage-related strain contributions according to the following relations:

140

Dynamic Response and Failure of Composite Materials and Structures d γ ij ¼ γ el ij + γ ij i, j ¼ 1, 2,3 i 6¼ j

(4.26)

in γ dij ¼ γ eld ij + γ ij

(4.27)

As shown in Eq. (4.27) the damage-related strain contribution can be further in decomposed in a reversible elastic strain γeld ij and irreversible inelastic strain γ ij . Indeed, the damage-related deformation can be partially or completely recovered upon unloading phase. Hence, during the unloading phase only the elastic strain γel ij and elastic damage-related strain are recovered γeld ij . The nonlinear shear behavior has been introduced in the user material subroutine by adopting a cubic polynomial stress-strain curve according to relation (4.28) and Fig. 4.6 (where c1, c2, and c3 are coefficients determined by fitting the polynomial curve to the experimental shear stress-strain data). τij ¼ c1 γ ij + c2 γ 2ij + c3 γ 3ij

(4.28)

The damage phase can be then evaluated by means of the following relations. el γ fail γ γ ij ij ij ds ¼ fail γ ij γ ij γ el ij

(4.29)

where in γ el ij ¼ γ ij γ ij

(4.30)

2

1

ti, j = c3g i, j + c2g i, j + c1g i, j

ti, j

0

Gi, j Gi, j

in

g i, j

eld

g i, j

0

Gi, j

el

g i, j

Fig. 4.6 Shear stress-strain response.

0

g i, j

fs

g i, j

g i, j

Modeling low velocity impact phenomena on composite structures

141

Start of increment

Abaqus explicit

Calculate integration point field variable from nodal values

eold, De, sold, dold

Abaqus explicit

enew, = eold + De Update stress variables

Check failure creteria

NO

YES Evaluation di

Update stress variables

Fig. 4.7 Intralaminar failure modeling user subroutine (VUMAT) flow chart representation.

γ in ij ¼ γ ij

τij Gij

(4.31)

γ in ij ¼ γ ij

τij Gij

(4.32)

The user subroutine can be schematically represented by the flowchart introduced in Fig. 4.7. At each time step, the subroutine evaluates the field variables starting from the nodal values. Then the interpolation to the integration points is performed to check for failure criteria. If, at least, a failure criterion is satisfied, the subroutine updates the damage variables and the stresses accordingly.

4.3

Numerical application

The mechanical response of composite material under LVI is, generally, very complex and difficult to be predicted. A stiffened composite panel has been used as test case for this work. To evaluate the effectiveness of the proposed numerical approach, comparisons with numerical results from a standard commercial FEM code and experimental

142

Dynamic Response and Failure of Composite Materials and Structures

data taken from literature [47] are presented. Numerical analyses have been performed by adopting the ABAQUS explicit FEM platform. A 25 Joule impact has been simulated and the numerical results in terms of damage onset and propagation during impact have been evaluated. In the following section, the description of the FE model is provided together with the boundary conditions and material properties of the composite laminate used to manufacture the investigated stiffened panel. In the second section, the numerical-experimental comparisons in terms of load, energy, and impactor displacement as a function of time are presented and discussed. The predictions obtained with the standard ABAQUS approach and with the newly developed User Subroutine are then analyzed and assessed.

4.3.1

Geometrical and material description of the investigated stiffened panel

A first geometrical description of the modeled reinforced panel is schematically represented in Fig. 4.8 by exploding the skin and the reinforcements to highlight the approach adopted to manufacture the panel’s parts. Both the parts are characterized by the stacking sequence [0/90]S. The impactor has been modeled as a hemispherical 3-D rigid body. The complete geometrical description of the model, the mass, and the velocity of the impactors assumed to simulate the impact event is introduced in Fig. 4.9. Two FEM models for the investigated stiffened panel have been created. For the first model the standard ABAQUS Hashin criteria have been adopted for intralaminar damage. This standard approach can only be applied to continuum shell (CS) elements; that is why CS elements have been adopted for the discretization of the first model. The second FEM model adopts the newly implemented User Subroutine for intralaminar damage, which can be applied to 3D brick elements. In both cases, one element per ply through the thickness has been used in order to accurately simulate the onset and growth of intralaminar and interlaminar damage. For both the FEM models, cohesive elements have been placed at the interface between plies with different orientations. The contact between the impactor and the panel has been simulated by means of penalty algorithm-based contact elements. The same contact algorithm has been also applied to guarantee the contact between neighboring plies when fully degraded Panel skin stack [0/90]s

Stringer profiles stack [0/90]s

Fig. 4.8 Laminates stack schematization.

Modeling low velocity impact phenomena on composite structures

143

Fig. 4.9 Panel description (impact location and impactor characteristics).

cohesive elements are removed from the model. In this work, an average value of the frictional coefficient (namely 0.5) has been used for all the contact pairs, regardless of the ply’s orientations. All the translational degrees of freedom have been suppressed at the edges of the panel, while the impactor has been allowed to move only perpendicular to the laminate. The FE model and the boundary conditions are presented in Fig. 4.10.

Fig. 4.10 Boundary conditions.

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Dynamic Response and Failure of Composite Materials and Structures

Table 4.5

Material properties

Properties

Value

Density Orthotropic properties

1400 kg/m3 E11 ¼ 130050 MPa; E22 ¼ 11550 MPa G12 ¼ G23 ¼ G13 ¼ 6000 MPa ν12 ¼ ν23 ¼ ν31 ¼ 0.312 XT ¼ 1022 MPa; XC ¼ 614 MPa; YT ¼ 54 MPa; YT ¼ 169 MPa S12 ¼ S13 ¼ 63 MPa; S23 ¼ 28 MPa

Strength In-plane fracture energies

G1CT ¼ 11.5 kJ/m2; G1CC ¼ 4.1 kJ/m2; G2CT ¼ 0.35 kJ/m2; G2CC ¼ 3.2 kJ/m2;

Interlaminar fracture toughness

GIC ¼ 0.18 kJ/m2; GIIC ¼ 0.5 kJ/m2; GIIIC ¼ 0.5 kJ/m2

Cohesive properties

Mode I

Mode II

Mode II

Elastic modulus (MPa/mm) Strength (MPa)

1159 59

600 86

600 86

The laminate material properties of the composite system and the relevant cohesive parameters are reported in Table 4.5.

4.3.2

Numerical results: Comparison with experimental data

The obtained numerical results are introduced in this section. Correlations with experimental data are presented in terms of force vs time, displacement vs time, impact energy vs time, and force vs displacement. Moreover, to better appreciate the quality of the numerical results obtained with the adoption of the user subroutine (VUMAT), with respect to the standard approach applied to CS elements, intralaminar/interlaminar damage shapes obtained with both the numerical approaches are compared. In Fig. 4.11 the numerical-experimental correlations obtained for force vs time, displacement vs time, impact energy vs time, and force vs displacement are introduced. Force vs Displacement curves have been reported in Fig. 4.11A. In the first phase of the impact event both numerical models, VUMAT and CS, underestimate the slope when compared to the experimental curve. Indeed, the CS model shows an initial stiffness much lower than the experimental data while the VUMAT model provides a better fitting of the experimental data. This trend is confirmed also for the absorbed energy (which can be represented by the area circumscribed by the curve): The CS model overestimates the absorbed energy while the VUMAT model is in good agreement with the experimental curve. As a matter of fact, the CS model with ABAQUS standard approach for the intralaminar damage evaluation seems to overestimate the impact induced intralaminar damage. The Force vs Time behavior (reported in Fig. 4.11B) of the VUMAT model seems to be in excellent agreement with the experimental results. On the other hand, the CS

(A)

0

2

4

6 8 Displacement (mm)

Displacement (mm)

12

(C)

10

12

Experimental CS Vumat

10 8 6 4

(B)

145

Experimental CS Vumat

0

2

4

6 8 Time (s)

10

12

14

25 Experimental CS Vumat

20 15 10 5

2 0

5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0

Experimental CS Vumat Force (N)

5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0

Energy (J)

Force (N)

Modeling low velocity impact phenomena on composite structures

0 2

4

6

8

Time (s)

10

12

14

(D)

0

2

4

6

8

10

12

14

Time (s)

Fig. 4.11 Numerical-experimental correlation: (A) force vs displacement, (B) force vs time, (C) displacement vs time, and (D) kinetic energy vs time—VMAT and CS numerical models.

model anticipates the peak force with respect to experimental data; this is probably due to a fiber and matrix damage overestimation. The duration of the impact phenomenon is predicted very well by both the models. From the Displacement vs Time curve (shown in Fig. 4.11C), in terms of maximum deflection, the CS model is characterized by a maximum percentage deviation from the experimental curve of about 11%. The maximum VUMAT deviation from the experimental data was found to be about 2%. According to the Kinetic Energy vs Time curves (reported in Fig. 4.11D), starting from a value of 25 J imposed to the panel, the final experimental data show at the end of the impact event a slight energy dissipation (due to friction and fracture). The CS model overestimates the energy dissipation, and consequently the internal damage in response to an impact phenomenon. The VUMAT model is in good agreement with experimental data in terms of energy dissipation. In Fig. 4.12 the panel deformation at maximum deflection stage has been reported. To better show the deformation, the impactor has been removed and results are presented in section cut according to a plane normal to the stringers direction passing from the impact point. The delamination predictions obtained by means of the two adopted numerical models are comparable both in shape and in dimensions as shown in Figs. 4.13 and 4.14. This result was expected since the same CZM elements have been used in both models. The intralaminar damaged areas extensions obtained by means of the CS and the VUMAT models are compared, for fiber failure modes in Figs. 4.15 and 4.16 (fiber

146

Dynamic Response and Failure of Composite Materials and Structures

Fig. 4.12 Displacement section (deformation induced by the impactor).

VUMAT damage SDEG (Avg: 75%) +1.000e+00 +9.167e−01 +8.333e−01 +7.500e−01 +6.667e−01 +5.833e−01 +5.000e−01 +4.167e−01 +3.333e−01 +2.500e−01 +1.667e−01 +8.333e−02 +0.000e+00

Y X

0°–90° ply Interface

90°–0° ply Interface

0°–90° ply Interface

90°–90° ply Interface

Fig. 4.13 Delamination damage VUMAT model.

0°–0° ply Interface

90°–0° ply Interface

Modeling low velocity impact phenomena on composite structures

147

Continuum shell damage SDEG (Avg: 75%) +1.000e+00 +9.167e−01 +8.333e−01 +7.500e−01 +6.667e−01 +5.833e−01 +5.000e−01 +4.167e−01 +3.333e−01 +2.500e−01 +1.667e−01 +8.333e−02 +0.000e+00

Y

X

0°–90° ply Interface

0°–90° ply Interface

90°–0° ply Interface

90°–90° ply Interface

0°–0° ply Interface

90°–0° ply Interface

Fig. 4.14 Delamination damage CS model.

traction mode) and Figs. 4.17 and 4.18 (fiber compression mode). From these figures, the overestimation of the fiber damages obtained with the CS models can be appreciated. The intralaminar damaged areas extensions obtained by means of the CS and the VUMAT models are compared, for matrix failure modes in Figs. 4.19 and 4.20 (matrix traction mode) and Figs. 4.21 and 4.22 (matrix compression mode). From these figures a substantially good agreement in terms of damaged area size and shape can be noted for the matrix tensile and compressive failure modes. The intralaminar damaged areas extensions, obtained by means of the CS and the VUMAT models, are compared, for shear failure modes, in Figs. 4.23 and 4.24. From these figures the relevant overestimation of the in-plane shear obtained with the CS model can be appreciated. The intralaminar damaged areas extensions obtained by means of the VUMAT model for out-of-plane shear failure modes is introduced in Figs. 4.25 and 4.26. Indeed, the CS model is able to take into account only the shear in plane x-y (due

148

Dynamic Response and Failure of Composite Materials and Structures

VUMAT damage Layer 0°

Layer 90°

Layer 90°

Layer 0°

Layer 0°

Layer 90°

Layer 90°

Layer 0°

SDV1 (Avg: 75%) +9.900e−01 +9.075e−01 +8.250e−01 +7.425e−01 +6.600e−01 +5.775e−01 +4.950e−01 +4.125e−01 +3.300e−01 +2.475e−01 +1.650e−01 +8.250e−00 +0.000e−00

Y

Fibre traction

X

Fig. 4.15 Fiber traction damage—VUMAT approach.

Continuum shell damage DAMAGEFT Multiple section points

(Avg: 75%) +1.000e+00 +9.167e-01 +8.333e-01 +7.500e-01 +6.667e-01 +5.833e-01 +5.000e-01 +4.167e-01 +3.333e-01 +2.500e-01 +1.667e-01 +8.333e-02 +0.000e+00

Y

Fibre traction

X

Fig. 4.16 Fiber traction damage—damage CS approach.

Modeling low velocity impact phenomena on composite structures

VUMAT damage SDV2 (Avg: 75%) +9.900e−01 +9.075e−01 +8.250e−01 +7.425e−01 +6.600e−01 +5.775e−01 +4.950e−01 +4.125e−01 +3.300e−01 +2.475e−01 +1.650e−01 +8.250e−02 +0.000e+00

Y

Fibre compression

X

Fig. 4.17 Fiber compression damage—VUMAT approach.

Continuum shell damage DAMAGEFC Multiple section points (Avg: 75%) +1.000e+00 +9.167e−01 +8.333e−01 +7.500e−01 +6.667e−01 +5.833e−01 +5.000e−01 +4.167e−01 +3.333e−01 +2.500e−01 +1.667e−01 +8.333e−02 +0.000e+00

Y

Fibre compression

X

Fig. 4.18 Fiber compression damage—damage CS approach.

149

150

Dynamic Response and Failure of Composite Materials and Structures

VUMAT damage

SDV3 (Avg: 75%) +9.900e−01 +9.075e−01 +8.250e−01 +7.425e−01 +6.600e−01 +5.775e−01 +4.950e−01 +4.125e−01 +3.300e−01 +2.475e−01 +1.650e−01 +8.250e−02 +0.000e+00

Y

Matrix traction

X

Fig. 4.19 Matrix traction damage—VUMAT approach.

Continuum shell damage DAMAGEMT Multiple section points

(Avg: 75%) +1.000e+00 +9.167e−01 +8.333e−01 +7.500e−01 +6.667e−01 +5.833e−01 +5.000e−01 +4.167e−01 +3.333e−01 +2.500e−01 +1.667e−01 +8.333e−02 +0.000e+00

Y

Matrix traction

X

Fig. 4.20 Matrix traction damage—CS approach.

Modeling low velocity impact phenomena on composite structures

VUMAT damage

SDV4 (Avg: 75%) +9.900e−01 +9.075e−01 +8.250e−01 +7.425e−01 +6.600e−01 +5.775e−01 +4.950e−01 +4.125e−01 +3.300e−01 +2.475e−01 +1.650e−01 +8.250e−02 +0.000e+00

Y

Matrix compression

X

Fig. 4.21 Matrix compression damage—VUMAT approach.

Continuum shell damage DAMAGEMC Multiple section points (Avg: 75%) +1.000e+00 +9.167e−01 +8.333e−01 +7.500e−01 +6.667e−01 +5.833e−01 +5.000e−01 +4.167e−01 +3.333e−01 +2.500e−01 +1.667e−01 +8.333e−02 +0.000e+00

Y

Matrix compression

X

Fig. 4.22 Matrix compression damage—CS approach.

151

152

Dynamic Response and Failure of Composite Materials and Structures

VUMAT damage

SDV31 (Avg: 75%) +9.900e−01 +9.075e−01 +8.250e−01 +7.425e−01 +6.600e−01 +5.775e−01 +4.950e−01 +4.125e−01 +3.300e−01 +2.475e−01 +1.650e−01 +8.250e−02 +0.000e+00

Y

Shear 1-2

X

Fig. 4.23 Shear in-plane (1-2) damage VUMAT approach.

Continuum shell damage DAMAGESHR Multiple section points (Avg: 75%) +1.000e+00 +9.167e−01 +8.333e−01 +7.500e−01 +6.667e−01 +5.833e−01 +5.000e−01 +4.167e−01 +3.333e−01 +2.500e−01 +1.667e−01 +8.333e−02 +0.000e+00

Y

Shear

X

Fig. 4.24 In-plane shear damage—CS approach.

Modeling low velocity impact phenomena on composite structures

VUMAT damage

SDV33 (Avg: 75%) +9.900e−01 +9.075e−01 +8.250e−01 +7.425e−01 +6.600e−01 +5.775e−01 +4.950e−01 +4.125e−01 +3.300e−01 +2.475e−01 +1.650e−01 +8.250e−02 +0.000e+00

Y

Shear 1-3

X

Fig. 4.25 Shear out-of-plane (1-3) damage—VUMAT approach.

VUMAT damage

SDV32 (Avg: 75%) +9.900e−01 +9.075e−01 +8.250e−01 +7.425e−01 +6.600e−01 +5.775e−01 +4.950e−01 +4.125e−01 +3.300e−01 +2.475e−01 +1.650e−01 +8.250e−02 +0.000e+00

Y

Shear 2-3

X

Fig. 4.26 Shear out-of-plane (2-3) damage—VUMAT approach.

153

154

Dynamic Response and Failure of Composite Materials and Structures

to the CS formulation of Abaqus/Explicit), while the VUMAT model takes also into account the out-of-plane 1-3 and 2-3 shear damages providing a more distributed shear damage with respect to the CS formulation. To better appreciate the interlaminar and intralaminar damage shape and position, for the VUMAT model, an overlapped zoomed views have been reported in Figs. 4.27 and 4.28, respectively.

SDEG (Avg: 75%) +1.000e+00 +9.167e–01 +8.333e–01 +7.500e–01 +6.667e–01 +5.833e–01 +5.000e–01 +4.167e–01 +3.333e–01 +2.500e–01 +1.667e–01 +8.333e–02 +0.000e+00

5 0 m m

50

mm

Fig. 4.27 Zoomed overview of the overlapped delaminations obtained with the VUMAT model.

VUMAT damage

Fibre traction

Fibre compression

Matrix traction

Matrix compression

Damage index (Avg: 75%) +9.900e−01 +9.075e−01 +8.250e−01 +7.425e−01 +6.600e−01 +5.775e−01 +4.950e−01 +4.125e−01 +3.300e−01 +2.475e−01 +1.650e−01 +8.250e−02 +0.000e+00

Shear 1-2

Shear 2-3

Shear 1-3

Fig. 4.28 Zoomed overview of overlapped intralaminar damage obtained with the VUMAT model.

Modeling low velocity impact phenomena on composite structures

155

Permanent indentation

Fig. 4.29 Panel permanent indentation.

The model implemented in this study has been found to be able to provide the shear damage distributions in the three different material planes. The implementation of nonlinear shear allows, furthermore, to capture the permanent indentation occurring during the impact phenomenon as highlighted in Fig. 4.29.

4.4

Conclusions

In this paper, a numerical study on an omega-stiffened composite panel, subjected to a LVI, is introduced. The proposed numerical model as a relevant added value with respect to standard approaches takes into account the shear damage distribution and the permanent indentation by means of a newly developed User Subroutine for Intralaminar damage simulation. The numerical results obtained both with the standard ABAQUS approach for intralaminar damage and the newly developed numerical approach have been compared to experimental data. An excellent correlation between experimental data and the proposed numerical results in terms of force-time, force-displacement, and energy-time curves has been obtained. The proposed intralaminar damage model has been found able, for this kind of phenomenon, to describe the global behavior (both in terms of impact force, induced displacement and absorbed energy) and the local impact behavior (in terms of damaged area shape and size). Besides, the nonlinear shear behavior, jointed to the three-dimensional approach, allowed to capture with good agreement the permanent indentation. As a matter of fact, the simplification of failure models included in the standard commercial FE codes (simplified failure mode criteria and CS formulation), while still able to provide acceptable results in terms of peak force and impact duration, fails when detailed and accurate results, in terms of maximum displacements and impact induced damaged area, are desired.

Acknowledgments Authors would like to thank Dr. V. Antonucci and Dr. M. Zarrelli from CNR (Italy) and Prof. A. Langella from The University of Naples (Italy) for the information provided on the panel manufacturing and testing. This information was needed to perform the numerical-experimental correlation activities described in this chapter.

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[22] Lopes CS, Camanho PP, Gurdal Z, Maimı´ P, Gonzàlez EV. Low-velocity impact damage on dispersed stacking sequence laminates. Part II. Numerical simulations. Compos Sci Technol 2009;69:937–47. [23] Iannucci L, Willows ML. An energy based damage mechanics approach to modelling impact onto woven composite materials: part II. Experimental and numerical results. Compos Part A Appl Sci Manuf 2007;38:540–54. [24] Pinho ST, Iannucci L, Robinson P. Physically based failure models and criteria for laminated fibre-reinforced composites with emphasis on fibre kinking. Part II. FE implementation. Compos Part A Appl Sci Manuf 2006;37:766–77. [25] Falzon BG, Apruzzese P. Numerical analysis of intralaminar failure mechanisms in composite structures Part II. Applications. Compos Struct 2011;93(2):1047–53. [26] Puck A, Sch€urmann H. Failure analysis of FRP laminates by means of physically based phenomenological models. Compos Sci Technol 1998;58:1045–67. [27] Feng D, Aymerich F. Finite element modelling of damage induced by low-velocity impact on composite laminates. Compos Struct 2014;108:161–71. [28] Krueger R. Virtual crack closure technique: history, approach, and applications. Appl Mech Rev 2004;57(2):109–43. [29] Riccio A, Raimondo A, Scaramuzzino F. A study on skin delaminations growth in stiffened composite panels by a novel numerical approach. Appl Compos Mater 2013;20 (4):465–88. [30] Suemasu H. An experimental method to measure the mode-III inter-laminar fracture toughness of composite laminates. Compos Sci Technol 1999;59:1015–21. [31] Rivallant S, Bouvet C, Abi Abdallah E, Broll B, Barrau JJ. Experimental analysis of CFRP laminates subjected to compression after impact: the role of impact-induced cracks in failure. Compos Struct 2014;111:147–57. [32] Aoki Y, Suemasu H, Ishikawa T. Damage propagation in CFRP laminates subjected to low velocity impact and static indentation. Adv Compos Mater 2007;16(1):45–61. [33] Bouvet C, Rivallant S. Damage tolerance of composite structures under low-velocity impact. In: Dynamic deformation, damage and fracture in composite materials and structures. Cambridge, UK: Woodhead Publishing; 2016. p. 1–7. [34] Abdulhamid H, Bouvet C, Michel L, Aboissiere J, Minot C. Numerical simulation of impact and compression after impact of asymmetrically tapered laminated CFRP. Int J Impact Eng 2016;95:154–64. [35] de Borst R, Remmers JJC, Needleman A. Mesh-independent discrete numerical representations of cohesive-zone models. Eng Fract Mech 2006;73(2):160–77. [36] Maimı´ P, Camanho PP, Mayugo JA, Da´vila CG. A continuum damage model for composite laminates. Part I. Constitutive model. Mech Mater 2007;39:897–908. [37] Aymerich F, Dore F, Priolo P. Prediction of impact-induced delamination in cross-ply composite laminates using cohesive interface elements. Compos Sci Technol 2008;68 (12):2383–90. [38] Iannucci L. Progressive failure modelling of woven carbon composite under impact. Int J Impact Eng 2006;32(6):1013–43. [39] Riccio A, Ricchiuto R, Saputo S, Raimondo A, Caputo F, Antonucci V, et al. Impact behaviour of omega stiffened composite panels. Prog Aerosp Sci 2016;81:41–8. [40] Riccio A, Caputo F, Di Felice G, Saputo S, Toscano C, Lopresto V. A joint numerical-experimental study on impact induced intra-laminar and inter-laminar damage in laminated composites. Appl Compos Mater 2016;23:219–37. [41] ABAQUS analysis user’s manual, Version 6.11. Dassault Syste`mes Simulia Corporation; 2011.

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Study of medium velocity impacts on the lower surface of helicopter blades

5

F. Pascal, P. Navarro, S. Marguet, J.-F. Ferrero University of Toulouse - Clement Ader Institute, Toulouse, France

5.1

Introduction

This chapter deals with the study of medium velocity impacts on the lower surface of helicopter blades. It aims to propose a representative finite element modeling (FEM) based on experimental observations of these kinds of impacts. Helicopter blades are large complex composite structures that operate in a severe dynamic environment. Typically, a blade is composed of a leading edge and a trailing edge made of unidirectional glass-epoxy composite: a composite skin made of two or three plies of glass-epoxy and carbon-epoxy woven fabrics stabilized by a stiffener made of carbon-epoxy woven composite. The carbon fabric is a 5-harness satin and the glass fabric is an 8-harness satin. The blade is filled with a polyurethane foam core. Generally, a protection in stainless steel covers the leading edge (Fig. 5.1; [1]). This structure can be impacted in flight, by soft impactors (hailstones, birds, etc.) or hard impactors (stones, little metallic parts from the helicopter structure, etc.). The failure of a part of the blade can lead to catastrophic consequences. Thus, understanding and predicting impact damage mechanisms is required. Due to the rotation of the blades, the impact velocities can vary from a few meters per second to 300 meters per second. Two kinds of impacts can occur: frontal impacts on the leading edge and oblique impacts on the lower skin of the blade. The impact angle is due to the inclination of the blade in flight and it varies from 10 to 20 degrees (Fig. 5.2). In this chapter, only impacts on the lower surface are studied, as they are more critical. More particularly, the behavior of the woven composite skin under impact loadings is investigated. Many works concern the impact and damage of composite structures. Comprehensive reviews by Abrate [2–4] discuss impact failure mechanisms for a large range of composite materials. Typically, cracks in the resin, delamination, and fiber breakages can be observed. Damage mechanisms in composite materials highly depend on its structure. In the specific field of woven laminate composites, the characteristic structure of the plies leads to complex behavior [5–7]. Indeed, even for static loadings, the damage first accumulates in the crimp region where the yarns cross each other. Local behavior has an important influence on the global response, damage initiation, and propagation. Dynamic Response and Failure of Composite Materials and Structures. http://dx.doi.org/10.1016/B978-0-08-100887-4.00005-6 © 2017 Elsevier Ltd. All rights reserved.

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Dynamic Response and Failure of Composite Materials and Structures

Skin Glass-epoxy and carbonepoxy woven fabric

Protection Stainless steel

Stiffener Carbon-epoxy woven fabric

Leading edge Unidirectional glassepoxy composite

Core Polyurethane foam

Trailing edge Unidirectional glassepoxy composite

Fig. 5.1 Section of an helicopter blade.

Frontal impact

Impact angle Oblique impact

Fig. 5.2 Definition of oblique impacts.

For impact loading, Nilakantan et al. [8] showed that the local variation of the strength of the yarns can have a major effect on the impact response of woven fabrics. In the specific field of impacts on woven fabric composites, five main modeling strategies can be highlighted. The first one consists in developing analytical modeling based mainly on the balance between the impact energy, the dissipated energy, and the postimpact kinetic energies [9,10]. Even if these analytical models provide a very good approximation of the damage level, more precise prediction of the damage in more complex composite structures is sometimes needed, in order to predict the postimpact behavior or to analyze with more accuracy the impact behavior, for instance. The second one consists in using FEM with customized damageable energy-based material laws [11–13]. In these models, the woven fabric is represented using homogenized shell elements. In the third main strategy, the developed woven composite models are based on continuum properties calculated from a deforming unit cell [14–17]. This unit cell can be a very detailed three-dimensional pattern of the woven fabric [18] or it can be represented with a specific truss structure [19]. This strategy is able to represent the local strain and stress fields due to the weaving pattern, but it can hardly be used for large structures. These last three strategies rely on the assumption that at an appropriate scale, the woven fabric behaves homogeneously and thus it can be approximated as a continuum. This assumption is no longer available when damage occurs.

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161

Thus, a fourth strategy based on FEM, presented by Johnson et al. [20], consists in modeling each woven lamina with a three-dimensional damaging nonlinear orthotropic material connected with cohesive elements. This strategy provides good prediction of delamination area and contact force-time history for low velocity impacts. Nevertheless, this strategy requires a large number of elements, which can be prohibitive for the modeling of representative structures. Finally, a fifth strategy has been proposed [21] which models impacts with higher velocities. It is based on experimental observations of medium velocity tests [22]. When the resin is totally damaged, the woven fabric behaves like a discrete truss structure. FEM that takes into account the discrete state of the woven composite material has been investigated. A semicontinuous approach, where specific shell elements are coupled with rod elements, was developed. This strategy provides a good representation of the damage mechanisms for thin composite structures made of two or three plies with the same orientation and material. In this paper, an extension of this semicontinuous approach is described. The modeling is modified in order to take into account the weaving pattern geometry in the damage mechanisms. Moreover, this strategy is extended to thicker woven composites with different ply orientations, with the introduction of specific cohesive elements. In the first part of this chapter, some specific impact tests are conducted and analyzed in order to define the key issues that have to be accounted for in the development of the model. Then, the modeling strategy is presented. The damage law and failure behavior are described. A specific interface element is presented. Finally the modeling strategy is validated on various impact tests.

5.2

Experimental observations

Low and medium velocity impact tests have been performed in order to observe and analyze the damage mechanisms in the woven composite skins during impact. In this part, analysis concerning visual inspections of the damages is provided. A more complete analysis on similar tests, reinforced with numerical results, is given in Section 5.4.

5.2.1

Medium velocity oblique impact tests

For this study, given that the impact occurs in the lower surface, the blade structure is simplified: the impacted specimens are plane sandwich structures made of two or three plies of woven fabric separated by a Rohacell 51A foam core. Here, the sandwich specimen skins are made of three plies of 5-harness carbon-epoxy woven composite oriented at 0/90 degrees from the impact axis. Each ply has a thickness of 0.36 mm and the foam has a thickness of 20 mm. The specimen is 200 mm long and 240 mm width. It is simply supported on an inclined table. The impactor is a 19 mm diameter steel ball with a mass of 28 g. It is launched by a gas gun at a velocity of 90 m/s with an angle of 15 degrees measured from the surface of the upper skin. The test is schematically represented in Fig. 5.3.

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Dynamic Response and Failure of Composite Materials and Structures

Skin 3 plies

Side view

Impactor Ø19 mm M = 28 g

V = 90 m/s foam

15°

Inclined table

Fig. 5.3 Schematic representation of the oblique impacts tests. Fig. 5.4 Failure surface of a carbon-epoxy woven skin after an oblique impact.

Firing direction

Upper ply

Lower ply

cm 0

1

2

3

4

5

6

7

8

9

10

As described in Ref. [21], the chronology of the impact can be divided into three steps. First, the resin is damaged. Then, the unstabilized bundles of fiber located under the impactor break in tension, which leads to the apparition of the first failure of the skin. Finally, as the impactor moves forward, the crack grows in the firing direction. The final failure surface is given (Fig. 5.4) for the upper ply and for the lower ply of the impacted skin. For a better readability, the crack is highlighted with yellow dots. Two main observations can be made. First, the pattern of the woven skin has an influence on the final crack geometry. Indeed, the crack passes through the crimp regions. Woven pattern cannot be neglected in the case of medium velocity oblique impacts. It has an important influence on the damage initiation and propagation. The second observation is that the length of the final crack measured in the impacted ply is 21% lower than one of the bottom ply. Consequently, the modeling has to represent each ply separately for a better transcription of the damage mechanisms of the woven composite skin.

5.2.2

Low velocity normal impact tests

Here, drop weight impact tests are performed in order to better understand the relation between the damage mechanisms and the woven pattern. The foam core has been removed and the impacted specimens are made with only one ply of carbon-epoxy

Study of medium velocity impacts

163

Impactor Drop tower 2 kg Ø20 mm 3 m/s Specimen

Mirror Circular frame

High-speed camera

Fig. 5.5 Drop weight impact test device.

5-harness satin woven fabric. The specimen is a square of edge length 100 mm and of thickness 0.36 mm. It lays on a rigid table containing an open circle window of 30 mm diameter as illustrated in Fig. 5.5. The steel impactor has a 20 mm diameter hemispheric shape and a mass of 2 kg. It impacts the specimen with an initial velocity of 3 m/s. A high-speed camera (Photron FastCam APX RS) was used to film the bottom face of the plate at a frame rate of 36,000 fps and a resolution of 512 128 pixels. Besides, the face was covered with a very thin layer of white paint to emphasize the matrix cracks. All impact tests have led to a fracture chronology similar to what is represented in Fig. 5.6 Every crack is oriented along a carbon bundle and matches exactly with the 5-harness satin weave pattern. Moreover, the first bundle failure occurs and propagates at a crimp location. Thus, the modeling accounts for the particularities of crimp regions.

5.3 5.3.1

Modeling strategy Woven ply modeling

Based on the previous experimental observations, a modeling strategy of the ply that accounts for the woven pattern is proposed. This strategy extends previously published models that use a “semicontinuous approach” [21] and [22]. In this model, the ply bundles are represented by 1D rod elements connected by nodes to the four edges of a quadrilateral shell element (Fig. 5.7). The element length respects the

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Dynamic Response and Failure of Composite Materials and Structures

Failure initiation at the crimp region

warp

Propagation at the crimp region

weft Time

Fig. 5.6 Failure chronology of a carbon-epoxy woven ply under low velocity impact.

Elementary pattern

Rod elements

Common nodes sh

me ry a t en e em siz

Elementary mesh 10 mm

Damageable shell element

El

Fig. 5.7 Principle of the presented modeling.

woven fabric pattern, that is, the distance between two bundles. The developed shell element is based on Batoz Q4γ24 shell formulation [23] with four Gauss integration points and reduced integration for in-plane shear. No hourglass control is needed for this shell. The membrane and bending behavior of the specific shell is decoupled: for the in-plane loadings, it behaves like the resin. For the out-of-plane loadings, it has got the stiffness of the homogenized ply. The distinction is made by differentiating the values of the corresponding elastic modulus, Em for membrane, Eb for bending and Gi3 for out-of-plane shear. This element is damageable. This strategy allows to represent both the continuous elastic behavior of the undamaged ply and the discrete behavior of the woven ply when resin is damaged and bundle of fibers are in a nonstabilized state. This type of modeling offers good comparison with experimental data regarding drop weight test and gas gun test. Nevertheless, in order to simulate more accurately experimental observations, two main improvements are made. The first change concerns the shell damage formulation. It has been modified in order to represent better

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165

the resin cracks that are oriented along the bundles. The second change concerns the rupture of the bundles. The maximum strain to failure of the rods is now dependent on their physical positions in the woven pattern. It allows taking into account the strain concentrations due to woven pattern geometry.

5.3.1.1 Shell damage model In this section, the finite element formulation and the damage variables used for the shell are presented. The main assumption about damage mechanism is as follows: when resin is damaged, the local structure is lost and the ply bending stiffness is affected. The bundles are temporally and locally relaxed until, carrying all the loads, they fail in tension. For the damage mechanisms of this shell, the idea is to capture the oriented cracks observed experimentally just before the first fiber ruptures. Thus, matrix failure is modeled with two damage variables, namely d1 and d2, for each local fabric direction. In order to model the complete loss of bending stiffness, these two damage variables degrade both membrane (resin) and out-of-plane (homogenized woven) moduli. The stress-strain laws are given (Eq. 5.1) for the membrane behavior, (Eq. 5.2) for the bending behavior, and (Eq. 5.3) for the shear behavior 2 3 ð1 d1 ÞEm ð1 d1 Þð1 d2 ÞυEm 8 m9 8 9 0 6 7 > εm 2 2 > < σ 11 > = 6 1 ð1 d1 Þð1 d2 Þυ 1 ð1 d1 Þð1 d2 Þυ = 7< 11 > 6 7 m σm ¼ 6 ð1 d1 Þð1 d2 ÞυEm ð1 d2 ÞEm 7 ε22 22 > > 0 7> : m> ; 6 4 1 ð1 d1 Þð1 d2 Þυ2 1 ð1 d1 Þð1 d2 Þυ2 5 : εm ; σ 12 12 0 0 2G12 (5.1) where the subscripts 1 and 2 correspond to the perpendicular local bundle directions, σ m are the membrane stresses, εm are the membrane strains, Em is the membrane modulus, G12 is the in-plane shear modulus, and υ is the Poisson ratio. 2 3 ð1 d1 ÞEm ð1 d1 Þð1 d2 ÞυEm 8 m9 0 78 m 9 6 2 2 > = 6 1 ð1 d1 Þð1 d2 Þυ 1 ð1 d1 Þð1 d2 Þυ = < σ 11 > < ε11 > 7> 6 7 m m m m σ 22 ¼ 6 ð1 d1 Þð1 d2 ÞυE ð1 d2 ÞE 7 ε22 > > 0 7> ; 6 : m> 4 1 ð1 d1 Þð1 d2 Þυ2 1 ð1 d1 Þð1 d2 Þυ2 5 : εm ; σ 12 12 0 0 2G12 (5.2) where σ b are the bending stresses, εb are the bending strains, and Eb is the bending modulus. G13 0 γ13 σ 13 ¼ (5.3) 0 G23 σ 23 γ23 where the subscript 3 denotes the out-of-plane direction, σ i3 is the out-of-plane shear stresses, ɣi3 is the out-of-plane shear strains, and Gi3 is the out-of-plane shear moduli.

166

Dynamic Response and Failure of Composite Materials and Structures

The parameters d1 and d2 are function of the energy release rates Y1 and Y2 calculated from the resin material properties.

2 Em m b ε +ε Yi ¼ sup 2ð1 υ2 Þ ii ii, max +

ði ¼ 1,2Þ

b where hεm ii + εii, max i+ denotes the positive part of the sum of the membrane and the maximum positive bending strains in local direction i ¼ 1,2. The damage evolution is given by:

8 pﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ Y0 + > < Yi p ﬃﬃﬃﬃﬃ if di < 1 Yc ði ¼ 1,2Þ di ¼ > : 1 otherwise where Y0 controls the damage initiation and Yc, the damage evolution. Hence, d1 and d2 manage all the mechanical variables of the shell except for the in-plane shear modulus G12. Indeed, it has been observed experimentally that pseudoplasticity was the predominant phenomenon concerning in-plane shear. Thus, a plastic strain ε12p has been introduced. A classic elastic prediction and plastic correction based on a Newton-Raphson iterative scheme has been implemented. The elastic field is defined by β f ¼ jσ 12 j Kplas εp12 σ 0 where σ 12 is the in-plane shear stress, σ 0 is the plastic strength, ε12p is the plastic shear strain and (Kplas, β) are material parameters defining the plastic hardening law. Besides, a third independent damage variable d12 has been implemented to model the final in-plane shear rupture. Its evolution is driven by a brittle law: ( d12 ¼

0 if εp12 < εpmax 1 if εp12 εpmax

where εpmax is the in-plane shear plastic strain to rupture. The material parameters of the 5-harness satin carbon-epoxy woven fabric have been identified with the same approach that was used in Ref. [22], which was concerned with the 8-harness satin glass-epoxy woven fabric. The matrix (and membrane) modulus Em is a material property given by the fabrics manufacturer. Then, quasistatic and dynamic tests on (0/90) degree and 45 degree oriented specimens have been performed. The other parameters are thus determined with a reverse engineering method, which consists in minimizing the difference between experimental and computed load/displacement curves. The evolution of the damage variables (d1, d2) has been calculated from static and dynamic indentation tests. The identified parameters for the shell elements are given in Table 5.1.

Study of medium velocity impacts

Table 5.1

167

Identified parameters for the shell elements

Shell elasticity Em (MPa) Eb (MPa) Gi3 (MPa) v

4500 52,500 4000 0.3

In-plane shear plasticity G12 (MPa) σ 0 (MPa) Kplas (MPa) β

Shell damage Y0 (J) Yc (J) εpmax

2500 80 120 0.5

1.5 0.4 0.07

F11 F *11

Initiation

Faiture t*

t

Fig. 5.8 Brittle rupture of the rods with an exponential decreasing law.

5.3.1.2 Rod failure The bundle rupture is assumed to be brittle in tension. Therefore, a classic maximum tensile strain criterion is used for the rupture of the rods. In order to avoid numerical instabilities, when the maximum strain criterion is reached, rod normal force is smoothly decreased by the use of a characteristic time τ as follows: F11 ¼ F∗11

t t , 1 exp τ

where t is the exact time at which the criterion is reached and F∗11 is the force stored at time t. This law is plotted in Fig. 5.8. τ has been taken equal to 1 μs, which corresponds to the best compromise between stability and parameter sensitivity to the results. The compression failure mechanism, that is, kink band phenomenon is neglected in our study. The weave pattern geometry is demonstrated by several authors to be a key factor in damage initiation and propagation. Many authors [8,15] highlight that damage first tends to accumulate in the crimp region of the warp where the yarns cross each other. Daggumati et al. [7] observed from DIC measurements that maximum local longitudinal strain on the surface of a 5-harness satin woven loaded in tension occurred at the center of the weft yarn at the yarn crimp location. An improvement has to be made in the modeling strategy to well represent the woven pattern influence on strain concentrations and damage initiation. The main

168

Dynamic Response and Failure of Composite Materials and Structures

issue is that the initial 2D modeling cannot catch naturally these stress concentrations. Thus, the proposed approach consists in lowering the tensile failure strain of the rods located at the crimp regions. The idea is to localize the rupture at these specific points without modeling the waviness of the yarns in those regions. Fig. 5.9 shows a schematic representation of the carbon-epoxy woven fabric on study superimposed with the adopted FE meshing. Three types of rods are thus distinguished: warp yarns at the crimp location, weft yarns at the crimp location, and the others (regardless of warp or weft yarn). The behavior of these rods is monitored with five parameters: two parameters for elastic behavior and three parameters for failure. The elastic parameters are the Young modulus of the rods and the section area. Their identification is done from the results of tensile tests on 0/90 degrees plies and from the fiber volume ratio given by the manufacturer.

Fig. 5.9 Introduction of the 5-harness satin weave pattern into the modeling strategy.

Study of medium velocity impacts

Table 5.2

169

Identified parameters for the rod elements Rod elasticity

Young modulus (MPa) Section area (mm2)

Rod failure 120,000 0.106

εmax pwarp pweft

0.015 0.9 0.8

Concerning the parameters that lead to the fiber ruptures, three parameters that correspond to the maximum tension strain of the three types of rod have to be identified: warp εmax for the crimp warp rods, εmax for the rods away from crimp regions, εwarp max ¼ p weft weft warp and εmax ¼ p εmax for the crimp weft rods. εmax and εweft max are expressed in a more convenient manner with a percent of εmax: pwarp and pweft. The maximum tension strain εmax is identified from dynamic tensile tests. The parameters pwarp and pweft are identified from low-velocity impact tests on a single ply; their values are chosen to fit the crack propagation observed experimentally. The identified parameters used for the rod elements are given in Table 5.2.

5.3.2

Modeling of the woven laminate

The object of this part is to define a strategy to represent the laminate skin from the model of a single woven ply presented above. The idea is to use a specific element to connect the plies. Using a “classic” 8-node solid element with three translational degrees of freedom per node is inappropriate. Indeed, as the rotational degrees of freedom of the connected shells cannot be transmitted, the bending stiffness of the laminate is not representative. More, as the interface is connected directly to the midsurface of the specific shells, part of the material represented by this 8-node solid element is superfluous. One direct consequence is the difficult identification of the material properties for the interface: the stiffness value cannot be simply deduced from the material properties of the resin. In order to solve these two problems, a shell-to-shell 8node interface element has been developed. The main idea is to use a cohesive zone strategy at the real surface of the composite ply. To do so, in addition to the three translational degrees of freedom, the nodes of the developed element have three rotational degrees of freedom, which allow a feasible connection to shell elements. The idea is to take into account the thickness of the connected plies. Eight virtual nodes are created at the real surfaces of these plies (Fig. 5.10). The assumption is made that the straight lines normal to the midsurface of the plies remain normal to the midsurface, that is to say the Kirchhoff-Love hypothesis is assumed. Thus, the virtual nodes are considered to be connected to the real nodes with rigid body elements (Fig. 5.10). In a first approach the out of plane straining of each ply, due to Poisson effects, has been neglected (ZZ ¼ 0). The reaction forces and momentum applied to the real nodes are deduced from the reaction forces applied on the virtual nodes.

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Dynamic Response and Failure of Composite Materials and Structures

Real node Thin resin layer

h

hi = 0

h h

Virtual node

h – 2

hi = 0

h – 2

Virtual cohesive element Rigid body

Schematic laminate

Shell elements

Specific interface element

Rotation real node

Deformed virtual cohesive element Deformed laminate

Fig. 5.10 Principle of the developed interface element.

The interlaminar connection is modeled using a bilinear cohesive law as described in Refs. [24–27]. The interface is considered as a thin layer separating the plies in which the damage mechanisms leading to delamination are localized. The out-of-plane stresses (σ xz, σ yz, σ zz) are calculated using a traction-separation law: 8 9 8 9 < σ xz = < δx = σ yz ¼ K δy : ; : ; σ zz δz where δx and δy correspond to sliding displacements (mode II and mode III) and δz to an opening displacement (mode I). K is the stiffness of the interface calculated from the young modulus of the resin Em and the theoretical thickness of the interface eint. K¼

eply Em with eint ¼ eint 5

The behavior is supposed to be similar in pure mode II and mode III loading. In the following, mode II refers to mode II and mode III. The decohesion process starts when

2 σ II 2 + max 1 σ max σ II I σI

Study of medium velocity impacts

where σ I ¼ hσ zz i + and σ II ¼

171

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ σ 2xz + σ 2yz σ max and σ IImax correspond, respectively, to the I

stress thresholds in mode I and in mode II. Thus the sliding displacement for initiation δ0 is calculated: δ

0

¼ δ0I δ0II

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 + β2 0 0 0 2 0 2 if δI > 0; δ ¼ δII otherwise δII + βδI δI β¼ δII

where δ0I and δII0 are the pure mode I and II displacement threshold that correspond to qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ δ2x + δ2y . the two stresses σ Imax and σ max II , δI ¼ δz and δII ¼ The total failure is reached when: GI GII + 1 GIc GIIc where GI and GII are the current energy release rates. GIc and GIIc correspond to the critical strain energy release rate in mode I and in mode II, respectively. From this failure criterion, a critical sliding displacement δc can be calculated: 1 2 1 + β2 1 β2 2GIIc + if δI > 0; δc ¼ otherwise δ ¼ 0 GIc GIIc Kδ0 Kδ c

Finally, during the unloading phase, the stresses reduce linearly (Fig. 5.11).

s

Fig. 5.11 Bilinear traction-separation behavior for the developed interface.

s Imax

s IImax Mode I

GIc Mode II GIIc dI

b

Mode I + II dII

deq

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Dynamic Response and Failure of Composite Materials and Structures

Identified parameters for the bilinear cohesive law

Table 5.3

Initiation σ Imax σ IImax

40 MPa 60 MPa

Propagation GIc GIIc

0.56 kJ/mm2 1.6 kJ/mm2

The damage parameters of the interface are identified from two tests of interlaminar crack propagation: a pure mode I double cantilever beam (DCB) test, and a pure mode II end notched flexure (ENF) test. Identified values are given in Table 5.3.

5.4

Validation and discussion

In this section, the ability of the model to predict the failure of composite structures is investigated by two kinds of tests: drop weight normal impact and gas gun oblique impact.

5.4.1 5.4.1.1

Drop weight normal impact test One ply specimen

Here, low velocity impact tests are performed on thin specimens made of one ply of 5-satin carbon fibers (AS4)/epoxy matrix (913) woven composite. The experimental setup is the same as described in Section 5.2. The specimen is a square of edge length 100 mm and of thickness 0.36 mm. It lays on a rigid table containing an open circle window of 30 mm diameter as illustrated in Fig. 5.5. The steel impactor has a 20 mm diameter hemispheric shape and a mass of 2 kg. It impacts the specimen with an initial velocity of 3 m/s. The reaction load and the displacement of the impactor are recorded. At last, the bottom face of the specimen is recorded with a high-speed digital camera at a frame rate of 36,000 fps and a resolution of 512 128 pixels. Three specimens are impacted in order to check the reproducibility of the experimental protocol. Two modelings of this test, relying on the strategy presented in Section 5.3, are proposed. In the first modeling, the parameters pwarp and pweft that manage the failure of the rods at the crimp region are set to 1. In the second modeling the values of Table 5.2 are taken. The objective is to observe the influence of taking into account the woven pattern on the damage propagation. The boundary conditions are introduced by the use of a contact between the specimen and a rigid wall. The impactor is a rigid sphere with the adequate mass and initial velocity. The mesh size is 1.4 1.4 mm in agreement with the measured woven fabric pattern. All the computations are performed with the explicit finite element code RADIOSS on 20 cores from HPC resources. The computational time is about 20 min. The comparison between the experimental and numerical results is achieved in Figs. 5.12 and 5.13. Fig. 5.12 presents the load versus displacement curves, whereas Fig. 5.13 presents the propagation of the bundle failure and the final damage state.

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173

1000 Num with pweft = pwarp = 1 Num with pweft = 0.8 and pwarp = 0.9 Experimental

A

800

Force [N]

600

400

200 B C

0 0

2

4 Displacement [mm]

6

8

Fig. 5.12 Load versus displacement curves of drop weight impact test.

After impact

During impact

Bottom

Top

Experimental images

Modeling results with pwarp = 1 pweft = 1

Averaged damage

0.5 (d1 + d2) 1.0 0.8

Modeling results with pwarp = 0.9 pweft = 0.8

0.6 0.4 0.2 0.0

A

B

C

Rods

Shell

Fig. 5.13 Experimental and numerical failure chronology of drop weight impact test.

Each damage state is located on the curves by the use of an upper case letter (A, B, and C). As a first observation, the model gives a satisfactory estimation of the force versus displacement response of the specimen. The peak force is in good agreement with the experimental data, with a relative error of 3%. Concerning the final load drop a difference between the two modeling is observed. Indeed, when the values of pweft and pwarp are set to 1, the load drop is divided into two parts. This can be explained by the fact that the first failure propagates too quickly and in only one direction (point B). The second drop load is due to the apparition of the second crack in the perpendicular

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direction (point C). For the second modeling, the load drop well fits the experimental data. The “cross shape” of the failure appears directly so that the load drops in only one step. The residual load calculated with the two modelings well correlates experimental results. Concerning the final fracture shape, Fig. 5.13 shows that contrary to the model with pweft and pwarp set to 1, the second model is able to well represent the curved shape of the cracks. Moreover, it predicts the resin failure perpendicular to the bundle cracks observed experimentally. At the end, for the second modeling, the relative maximal error between experimental and numerical results in terms of crack lengths is 8%. Therefore, the damage strategy proposed in Section 5.3 appears to be able to give a satisfactory prediction of the failure scenario of the woven composite ply.

5.4.1.2

Sandwich specimen with two plies of the skin

Here, low velocity impact tests are performed on sandwich panels. The skins are made of two plies of 5-satin carbon fibers (AS4)/epoxy matrix (913) woven fabrics oriented at (0/90) degrees and 45 degrees. The core is a 20 mm thick Rohacell 51A foam. The panel is simply supported (Fig. 5.14) The other experimental conditions are the same than in the previous section. The skin is modeled using the strategy described in Section 5.3 with the values of Table 5.2. As the meshes of the two plies are not coincident, the upper nodes of the interface are bonded numerically to the surface of the woven shell elements representing the upper ply. Three-dimensional elements have been used to model the 20 mm thick foam core. The foam core is modeled by the “FOAM_PLAS” material law implemented in Radioss. This law provides a good representation of the three steps that characterize the foam in compression [28]. Its mechanical properties have been identified in previous work [21]. The comparison between the experimental and numerical results is achieved in Figs. 5.15 and 5.16. Fig. 5.15 presents the final fracture shapes, whereas Fig. 5.16 presents the load versus displacement curves. Fig. 5.16 shows that the modeling is able to well represent the final fracture surface. Indeed, the fact that the cracks observed experimentally follow the fiber direction of Rigid sphere Impactor Ø20 mm M = 2 kg

Skin (2 plies)

V = 3 m/s 20 mm

(A)

Foam

Simply supported

(B)

Fig. 5.14 (A) Schematic representation of the oblique impacts tests (B) view of the modeling.

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Numerical Experimental Rods failure

Shell damage

Top ply

Bottom ply

Fig. 5.15 Experimental and numerical fracture surfaces.

1200

Num Experimental

B 1000 C

Force [N]

800 A

600

400

200

0

0

2.5

5

7.5 10 Displacement [mm]

12.5

15

Fig. 5.16 Experimental and numerical load/displacement curves.

each ply is well modeled. Moreover, the dimensions of the calculated damages well correlate with the experimental results. The maximal error for the crack length is 6%. Concerning the load displacement curves, the first drop load observed numerically at point A is in agreement with experimental results. It corresponds to the first failure of the bottom ply. The peak force (point B) is slightly overestimated (13%). The load drop is well represented; it corresponds to the rupture of the top ply.

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Dynamic Response and Failure of Composite Materials and Structures

Fig. 5.17 Numerical failures of the two plies of the impacted skin.

In that curve, a high residual load is observed experimentally and numerically after the rupture of the whole skin. In order to understand this behavior, the numerical results are analyzed: Fig. 5.17 represents the fiber breakages of the two plies at the time corresponding to point C in the load/displacement curve. The fact that the two failures of the plies have not the same orientation explains that the residual flexural stiffness of the “petals” is higher.

5.4.2

Gas gun oblique impact test

In this section, oblique impact tests are performed in the same conditions as that in Section 5.2. The specimens are sandwich structures made of three (0/90) degrees plies of 5-satin carbon fibers (AS4)/epoxy matrix (913) woven fabrics separated by a Rohacell 51A foam core. The specimen is 200 mm long and 240 mm width. Each ply has a thickness of 0.36 mm and the foam has a thickness of 20 mm. It is simply supported on an inclined table. The impactor is a steel ball of diameter 19 mm and mass 28 g. It is launched by a gas gun in order to reach the velocity of 90 m/s with an angle of 15 degrees measured from the surface of the upper skin. The test is recorded with the same high-speed camera as in Section 5.4.1.1 under the same conditions. The application of a speckle on the impactor, in conjunction with the use of a digital image correlation algorithm developed at the laboratory [29], enables a fine computation of the velocities of the impactor. The signal is smooth enough to deduce the acceleration of the impactor, and thus the normal impact force FN. Some examples of the tracking of the speckled impactor by the DIC algorithm are given in Fig. 5.18A. The modeling developed for the simulation of this test reuses the same strategy as presented before for the drop weight normal impact test. There is simply one more ply for the skins. A view of the modeling is given Fig. 5.18B. The main results are summarized in Figs. 5.19 and 5.20. In Fig. 5.19, the normal force FN versus time curve is plotted whereas Fig. 5.20 presents the experimental and numerical states of the specimen after the impact. The load versus time curve computed by the model appears to be very close to the measured one during the test. The shape and the level of the two curves are indeed very similar. The relative errors are around 5.5% on the peak force and 6.1% on the dissipated energy. Once again, upper case letters are used for the description of the curve.

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(A) Rigid sphere

(B) Fig. 5.18 (A) example of tracking of the impactor (B) view of the modeling of an oblique impact.

Fig. 5.19 Load versus time curves of oblique impact test.

This time, the failure of the specimen is analyzed through the model since it gives a fine description of what happens. A comparison of the final states of the specimen obtained numerically and experimentally is finally proposed. After a short part of elastic loading by bending of the upper skin under the impactor, point A is reached. It corresponds to a macrocrack (fibers breakage) of the bottom ply

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Dynamic Response and Failure of Composite Materials and Structures

Fig. 5.20 Experimental (top) and numerical (bottom) damaged specimens after the gas gun oblique impact test.

of the upper skin. This crack is due to the high level of tension seen by the fibers that are orthogonal to the impactor trajectory. The crack is also accompanied by microcracks inside the resin. The peak force is visible on both the experimental and numerical curves. From point A to point B, the damage propagates in the thickness direction. It goes up to reach the two top plies of the skin. Points B and C reflect the fiber breakages of these two upper remaining plies. After point C, the upper skin is cut on its whole thickness which results in a sudden fail of the load. From points C to D, the impactor starts to penetrate into the specimen. This penetration is partial, and the impactor does not go through the whole thickness of the sandwich panel. The plateau between points D and E relates the progress, practically parallel to the upper skin, of the impactor inside the specimen. At this stage, the macro- and microcracks propagate along the impactor trajectory. In the last step, point E indicates the beginning of the rebound of the impactor. The load decreases to zero when the contact is broken. This first analysis shows the ability of the model to capture with accuracy the failure events of this complex specimen. The postimpact residual state is now analyzed. The red lines superimposed on the pictures placed at the top of Fig. 5.20 highlight the macrocracks corresponding to fiber failure. The orange lines correspond to the microcracks (resin cracks) located inside and between the yarns of the woven fabric. A schematic representation of the macro and microcracks is given below to illustrate the importance of the 5-harness satin weave pattern on the shape of the cracks. It is worth noticing the distinct appearances of the damages between the lower and the upper plies. The main macrocrack of the lower ply is clear, with few transverse macrocracks initiated at some intersections of warp and weft yarns (right column of Fig. 5.20). This can be

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179

explained by the brittle nature of the failure of the fibers and by the protection given by the upper plies. Indeed, the lower ply is not directly in contact with the impactor. The upper ply exhibits a smaller macrocrack surrounded by a rectangular (in white in Fig. 5.20) shaped diffuse damage whose upper and lower limits are due to the important local friction and bending of the skin in contact with the impactor. This ply, directly in contact with the impactor, suffers from a more important matrix damage as illustrated by the orange lines. They are in accordance with the bleached white appearance of the yarns in the upper left picture of Fig. 5.20.

5.4.3

Discussion

To conclude this section dedicated to the validation of the proposed model, it can be said that on both kinds of tests, drop weight normal impact and gas gun oblique impact tests, the model is in good agreement with experimental observations. The predictions of the load during the tests are coherent with the measures in terms of magnitude and timescale. Moreover, a close look at the failure mechanisms highlights the satisfactory behavior of the model, which seems able to distinguish between the various different failure phenomena, such as fiber breakage, resin microcracking, and delamination, for thin skins made of plies with different orientations. At last, the introduction of the 5-harness satin weave pattern inside the model, by adequate failure strains at the intersections of warp and weft tows, improves the capability of the model to catch the macrocracks bifurcations evidenced by the experimental tests.

5.5

Conclusion

A 2D modeling of the 5-harness satin woven composite material has been presented in this paper. A modeling strategy based on the semicontinuous approach and at the woven pattern mesh scale is used. Bundles of fibers are represented with rod elements and matrix is modeled with a damageable shell element. In order to be more in agreement with experimental observations, two main improvements have been carried out from the original modeling presented in Ref. [21]. First, a work on the shell damage has been performed to represent the typical matrix cracks observed during a drop weight impact test. Then, an original approach has been conducted in order to take into account the damage initiation (stress concentrations) at the crimp regions. Indeed, instead of modeling the waviness of the bundles, which would inevitably lead to more expensive computational cost, the idea was to lower the strain to failure of the rods located at the crimp regions. From this modeling, the woven laminates are built with the use of a specific three-dimensional interface element. This specific interface element is compatible with the rotational degrees of freedom of shell elements. It is also designed to take into account the thickness of the two-dimensional woven elements. The interface is damageable and the damage parameters are identified from DCB and ENF tests. Comparisons with experimental data from drop weight tests and gas gun tests show good accuracy in the force history and damage size and shape. The modeling strategy

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allows to catch the complex macrocrack path observed experimentally and also the resin microcracking and delamination.

Acknowledgment The authors acknowledge the supercomputing centre CALMIP for granting access to the HPC resources under the allocation 2015-P09105.

References [1] Aubry J, Navarro P, Tawk I, Marguet S, Ferrero J-F, Lemaire S, et al. Experimental and numerical study of normal and oblique impacts on helicopter blades. In: Dynamic failure of composite and sandwich structures. Springer Netherlands; 2013. p. 545–75, ISBN: 978-94-007-5329-7. [2] Abrate S. Impact on laminate composites, recent advances. Appl Mech Rev 1997;47:517–44. [3] Abrate S. Impact on composite structures. Cambridge University Press; 1998. [4] Abrate S. Modeling of impacts on composite structures. Compos Struct 2001;51:129–38. [5] Lomov SV, Ivanov DS, Truong TC, Verpoest I, Baudry F, Vanden Bosche K, et al. Experimental methodology of study of damage initiation and development in textile composites in uniaxial tensile test. Compos Sci Technol 2008;68(12):2340–9. [6] Melro AR. Analytical and numerical modelling of damage and fracture of advanced composites. Thesis manuscript, University of Porto; 2011. [7] Daggumati S, Voet E, Van Paepegem W, Degrieck J, Xu J, Lomov SV, et al. Local strain in a 5-harness satin weave composite under static tension: part I. Experimental analysis. Compos Sci Technol 2011;71(8):1171–9. [8] Nilakantan G, Keefe M, Wetzel ED, Bogetti TA, Gillespie Jr JW. Effect of statistical yarn tensile strength on the probabilistic impact response of woven fabrics. Compos Sci Technol 2012;72(2):320–9. [9] Naik N, Shrirao P, Reddy B. Ballistic impact behaviour of woven fabric composites: formulation. Int J Impact Eng 2006;32:1521–52. [10] Reyes G, Sharma U. Modeling and damage repair of woven thermoplastic composites subjected to low velocity impact. Compos Struct 2010;92:523–31. [11] Iannucci L, Willows M. An energy based damage mechanics approach to modelling impact onto woven composite materials—part I: numerical models. Compos Part A Appl Sci Manuf 2006;37(11):2041–56. [12] Iannucci L, Willows M. An energy based damage mechanics approach to modelling impact onto woven composite materials: part II. experimental and numerical results. Compos Part A Appl Sci Manuf 2007;38(2):540–54. [13] Cousigne O, Moncayo D, Coutellier D, Camanho P, Naceur H, Hampel S. Development of a new nonlinear numerical material model for woven composite materials accounting for permanent deformation and damage. Compos Struct 2013;106:601–14. [14] Daggumati S, Van Paepegem W, Degrieck J, Xu J, Lomov SV, Verpoest I. Local damage in a 5-harness satin weave composite under static tension: part II meso-FE modelling. Compos Sci Technol 2010;70(13):1934–41. [15] Melro AR, Camanho PP, Andrade Pires FM, Pinho ST. Numerical simulation of the nonlinear deformation of 5-harness satin weaves. Comput Mater Sci 2012;61:116–26.

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[16] Obert E, Daghia F, Ladevze P, Ballere L. Micro and meso modeling of woven composites: transverse cracking kinetics and homogenization. Compos Struct 2014;117:212–21. [17] Zako M, Uetsuji Y, Kurashiki T. Finite element analysis of damaged woven fabric composite materials. Compos Sci Technol 2003;63(34):507–16. [18] Boisse P, Zouari B, Gasser A. A mesoscopic approach for the simulation of woven fibre composite forming. Compos Sci Technol 2005;65:429–36. [19] Parsons EM, Weerasooriya T, Sarva S, Socrate S. Impact of woven fabric: experiments and mesostructure-based continuum-level simulations. J Mech Phys Solids 2010;58: 1995–2021. [20] Johnson H, Louca L, Mouring S, Fallah A. Modelling impact damage in marine composite panels. Int J Impact Eng 2009;36:25–39. [21] Navarro P, Aubry J, Marguet S, Ferrero J-F, Lemaire S, Rauch P. Semi-continuous approach for the modeling of thin woven composite panels applied to oblique impacts on helicopter blades. Compos Part A Appl Sci Manuf 2012;43(6):871–9. [22] Navarro P, Aubry J, Marguet S, Ferrero J-F, Lemaire S, Rauch P. Experimental and numerical study of oblique impact on woven composite sandwich structure: influence of the firing axis orientation. Compos Struct 2012;94(6):1967–72. [23] Batoz JL, Dhatt G. Modelisation des structures par elements finis: coques. vol. 3. Paris: Hermes Edition; 1992. [24] Davies G, Hitchings D, Ankersen J. Predicting delamination and debonding in modern aerospace composite structures. Compos Sci Technol 2006;66:846–54. [25] Aymerich F, Dore F, Priolo P. Simulation of multiple delaminations in impacted cross- ply laminates using a finite element model based on cohesive interface elements. Compos Sci Technol 2009;69:1699–709. [26] Bianchi S, Corigliano A, Frassine R, Rink M. Modelling of interlaminar fracture processes in composites using interface elements. Compos Sci Technol 2006;66:255–63. [27] Ladeve`ze P, Allix O, De€u J-F, Lev^eque D. A mesomodel for localisation and damage computation in laminates. Comput Methods Appl Mech Eng 2000;183:105–22. [28] Gibson L, Ashby M. Cellular solids: structure and properties. Cambridge University Press; 1999. [29] Passieux J-C, Navarro P, Perie J-N, Marguet S, Ferrero J-F. A digital image correlation method for tracking planar motions of rigid spheres: application to medium velocity impacts. Exp Mech 2014;54(8):1453–66.

Impact on composite plates in contact with water

6

S. Abrate Southern Illinois University, Carbondale, IL, United States

6.1

Introduction

Composite and sandwich structures are used extensively in marine vehicles and they can sustain impact damage during collisions with other crafts or floating debris and objects dropped inside the hull [1]. Composite structures can develop significant damage during impacts [2–8]. The motion of a structure in contact with water is resisted by the inertia of the fluid. The issue to be addressed here is how to account for this effect in the analysis of the impact problem and to determine if it is significant. The impact of floating debris on (fixed) marine structures has been treated for small size debris such as logs, up to utility poles, and shipping containers impacting buildings, bridge piers, and other structures. Often, a stereomechanical analysis based on conservation of momentum and the use of appropriate coefficients of restitution is used to predict applied impulse and the energy dissipated. This approach is also used to predict the energy dissipated during ship collisions [9]. However, it does not predict the contact force history, the maximum force, or the contact duration. Another approach is to use a spring mass-model, which is a one degree of freedom model in which the mass is equal to the mass of the object plus an “added mass” representing the inertia of the fluid [10]. The spring in that model can account for the deformation of the impacted structure and the local deformation in the contact region. For linear springs, a closed-form solution for the contact force history is available. Nonlinear force-deflection relations can also be used to study the impact of barges against bridge piers for example [11]. Longitudinal or transverse impacts of utility poles or shipping containers motivated the development of models in which a rod or a beam impacts a column or a wall [12–14]. This chapter deals with the impact rigid bodies on elastic plates in contact with water and extensions to beams and shells. The analysis should include: (1) an appropriate model for predicting the displacement of the structure to a concentrated force P at the point of impact; (2) a model for the motion of the projectile (Newton’s law for a rigid body); (3) a contact law that relates the contact force to the relative motion at the contact point. Models for the analysis of composite plates are discussed in detail. An introduction to the added mass effect for structures in contact with water is given and we show how the analysis of the impact response can account for the presence of the fluid without a coupled fluid-structure interaction model.

Dynamic Response and Failure of Composite Materials and Structures. http://dx.doi.org/10.1016/B978-0-08-100887-4.00006-8 © 2017 Elsevier Ltd. All rights reserved.

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A nondimensional formulation for the analysis of impacts on a simply supported plate shows a fluid range of possible responses, the effect of plate to projectile mass ratio, and the number of modes participating in the response.

6.2

Plates in contact with water

Here we consider impacts that do not result in complete penetration of the target and for which the contact duration is much larger that the travel time through the thickness of the plate. Bending deflections take place after several deflections through the thickness so an appropriate theory should be selected modeling the deformation of plate. Section 6.2.1 provides an introduction to existing plate theories starting with the classical Kirchhoff-Love theory as a framework to view the vast literature on the subject and to form a basis for making an appropriate selection. The motion of a solid in contact with a fluid is resisted by the inertia of the fluid. The dynamics of a structure interacting can be investigated as a coupled problem with a model for the structure, a model for the fluid, and a compatibility condition applied at the interface. It has been shown that the presence of the fluid has negligible effects on the mode shapes but that the natural frequencies are significantly lower in the presence of a heavy fluid like water. Once the added mass factors (AMFs) have been determined for each vibration mode, the dynamic response of the structure can be analyzed as an uncoupled problem as discussed in Section 6.2.2.

6.2.1

Plate theories

The development of mathematical models to describe the behavior of plates began in the early 1800s. Single layers of elastic isotropic materials were considered for many years using approximations for the displacement field through the thickness. Extensions to this approach were introduced to deal with layered structures using a single displacement field through the thickness but accounting for the variation in material properties from layer to layer. These are called equivalent single layer (ESL) theories. To overcome difficulties stemming from the use of a single displacement field, separate displacement fields can be chosen for each layer. These are usually called multilayer theories. The following will describe the basic assumptions made in the development of ESL and multilayer theories: The derivation of ESL theories usually starts with an assumed displacement field and the most common approach is to expand the three displacement components in a power series of the transverse coordinate u ¼ uo + zu1 + z2 u2 + z3 u3 + z4 u4 + …zm um

(6.1a)

v ¼ vo + zv1 + z2 v2 + z3 v3 + z4 v4 + …zm vm

(6.1b)

w ¼ wo + zw1 + z2 w2 + z3 w3 + z4 w4 + …zn wn

(6.1c)

Such a polynomial expansion leads to what is called a {m,n} theory. There are three normal strain components

Impact on composite plates in contact with water

εxx ¼ u,x ¼

m X

zi ui, x εyy ¼ v, y ¼

m X

i¼0

185

zj vj, y εzz ¼ w, z ¼

j¼0

n X

zk wk, z

(6.2)

k¼0

Two transverse shear components εxz ¼ u, z + w, x ¼

m X

zi1 ði ui + wi1, x Þ εyz ¼ v, z + w, y ¼

i¼1

m X zi1 i vi + wi1, y

(6.3)

i¼1

and one in-plane shear component εxy ¼ u,y + v,x ¼

m X zi ui, y + vi, x

(6.4)

i¼0

A number of assumptions are used to keep the number of variables to a minimum. The first one concerns the transverse normal deformation. Generally, it is assumed that the plate is inextensible in the transverse direction ðεzz ¼ 0Þ so that w ¼ wo ðx, yÞ: However, for the simple case of a plate under in-plane tension in the x-direction, there is not only a strain εxx ¼ uo, xx but also a strain εzz ¼ νεxx ¼ νuo, xx because of Poisson’s effect. Then, w ¼ zνuo, xx : This suggests that a second term is required in the power series expansion for w. The next assumption is about the transverse shear deformation and finally we must decide how many terms to keep in the expansion for u and v.

6.2.1.1 Kirchhoff-Love theory The Kirchhoff-Love theory also known as the classical plate theory (CPT) is based on three basic assumptions. First, the plate is assumed to be inextensible in the transverse direction ðεzz ¼ 0Þ so w ¼ w o ðx, yÞ. Then, it is assumed that the transverse shear deformations are negligible εxz ¼ 0,εyz ¼ 0 . This means that i ui + wi1, x ¼ 0 and vi + wi1, y ¼ 0 for i 1. For i ¼ 1, this gives u1 ¼ wo, x and v1 ¼ wo, y . Since wi ¼ 0 for i 1 these two conditions imply that ui and ui are zero for i 1. In summary, the displacement field for the CPT is u ¼ uo zwo, x v ¼ vo zwo, y w ¼ wo

(6.5)

This implies that the in-plane strains vary linearly through the thickness and the stress components σ xx, σ yy and σ xy will also vary linearly through the thickness. For a three-dimensional elastic solid, the motion in the x-direction is governed by σ xx, x + σ yx, y + σ zx, z ¼ ρu€

(6.6)

For the static case, if the first two terms vary linear with z then the transverse shear stress σ zx will have a parabolic variation through the thickness. Similarly, considering equilibrium in the y-direction, σ yx is seen to vary parabolically through the thickness.

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Dynamic Response and Failure of Composite Materials and Structures

6.2.1.2

Mindlin-Reissner first-order shear deformation theory

The first-order shear deformation theory also assumes transverse inextensibility. The transverse shear strain is assumed to remain constant through the thickness so u1 ¼ εxz wo, x and v1 ¼ εxz wo, y : u ¼ uo + zu1 v ¼ vo + zv1 w ¼ wo

(6.7)

Since u,z ¼ u1 and v ¼ v1 are constant through the thickness, a small line segment initially perpendicular to the reference surface remains straight but it is not constrained to remain normal to the reference surface after deformation.

6.2.1.3

Reddy’s third-order shear deformation theory

With both the CPT and the FSDT, the in-plane stresses vary linearly through the thickness. Then, as indicated previously, the equations of equilibrium suggest that the transverse shear stresses vary parabolically. Keeping the first three terms in Eqs. and, using the transverse inextensibility condition, εxz ¼ u1 + wo, x + 2zu2 + 3z2 u3 , εyz ¼ v1 + wo, y + 2zv2 + 3z2 v3

(6.8)

Usually shear stresses are zero on the top and bottom surfaces of the plates. Requiring that εxz ¼ εyz ¼ 0 when z ¼ h=2 gives u2 ¼ v2 ¼ 0 and u3 ¼

4 4 ðu1 + wo, x Þ, v3 ¼ 2 v1 + wo, y 2 3h 3h

(6.9)

Substituting into Eq. and keeping only the minimum number of terms, the displacement field of this Reddy shear deformation theory (RSDT) is 4z3 4z3 u ¼ uo zwo, x + z 2 ψ x v ¼ vo zwo, y + z 2 ψ y (6.10) 3h 3h w ¼ wo where two new variables ψ x ¼ u1 + wo, x and ψ y ¼ v1 + wo, y have been introduced for convenience. This theory has five variables (uo, vo, wo, ψ x, ψ y) like the FSDT but here a line segment normal to the midplane does not remain straight and does not remain perpendicular to the midplane after deformation. ψ x and ψ y are the transverse shear strains at z ¼ 0 which implies that the normal does not remain perpendicular to the midplane. Both u,z and v,z vary with z indicating that the normal does not remain straight.

6.2.1.4

Nonpolynomial theories

The displacement field of the three theories discussed so far can be written as u ¼ uo zwo, x + f ðzÞψ x v ¼ vo zwo, y + f ðzÞψ y w ¼ wo

3

(6.11)

4z for the 3h2 RSDT. Many ESL theories have been presented with displacement field given by where f ðzÞ ¼ 0 for the CPT, f ðzÞ ¼ z for the FSDT, and f ðzÞ ¼ z

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187

Eq. (6.11) but with nonpolynomial functions f(z). For example, Levy [15] introduced a “trigonometric theory” with f ðzÞ ¼ sin ðπz=hÞ. This approach has been extended and is sometimes associated with the names of Stein [16] or Touratier [17]. Further examples include the hyperbolic theory of Soldatos [18] in which f ðzÞ ¼ h sinh ðz=hÞ z cosh ð1=2Þ h and thei exponential theory of Karama et al. [19] where f ðzÞ ¼ x exp 2ðz=hÞ2 .

6.2.1.5 Multilayer theories ESL theories were developed as two-dimensional approximations for the three-dimensional elasticity problem for isotropic plates. A single kinematic assumption such as Eq. (6.11) leads to continuous strain distributions through the thickness. This is a problem for laminated plates where continuous strains lead to discontinuous transverse and normal stresses at ply interfaces. Those stresses should be continuous at ply interfaces and they should be determined accurately in order to predict interface damage. Multilayer plate theories start with separate kinematic assumptions for each ply. For example, choosing the FSDT with a 5 variables displacement field (Eq. 6.7), the total number of variables N for an n-ply laminate is N ¼ 5n. Three displacement continuity conditions at the interfaces reduce the number of variables to N ¼ 5n 3 (n 1) ¼ 2n + 3. This approach leads to the development of layerwise theories (LWT) and the example considered here shows that the number of variables increases with the number of layers. LWT have been used in the analysis of impacts on composite structures (e.g., Ref. [20]) Enforcing the continuity of transverse normal and shear stresses at the interfaces can be used to reduce the number of variables further. For the current FSDT-based example, the transverse normal stress is zero and transverse shear stress continuity gives 2(n 1) that can be used to reduce the total number of equations to five, the same number as in the original FSDT. This approach resulting in a fixed number of variables independently of the number of layers in the laminate produces what are usually called zigzag theories (ZZTs). Developing ZZT based on higher order theories presents a challenge. For example, if each layer is modeled using a {3,2} theory, the total number of unknowns is N ¼ 11n initially. Three displacement continuity conditions reduce this number to N ¼ 11n 3(n 1) ¼ 8n + 3, three transverse stress conditions reduce it to N ¼ 8n + 3 3(n 1) ¼ 5n + 6. Requiring that the transverse shear stresses vanish on the top and bottom, N ¼ 5n + 6 4 ¼ 5n + 2, the total number of unknowns still increases with the number of plies. Instead of having all the variables in the displacement field of each layer as local variables, let m of these variables be global variables that do not change from ply to ply. Starting with a {3,2} theory again, the initial number of variables is N ¼ m + (11 m)n. Enforcing the continuity conditions reduces the number of unknowns by 2(n 1) and the zero shear stress conditions on the top and bottom surfaces reduces it to N ¼ m + (11 m)n 6(n 1) 4 ¼ (5 m)n + m + 2. Taking m ¼ 5 results in a total of 7 variables regardless of the number of plies.

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Dynamic Response and Failure of Composite Materials and Structures

In this case, with the notation used in Eq. (6.1a), (6.1b), uo, vo, wo, u3, and v3 can be taken as global variables while u1, v1, w1, u2, v2, and w2 are local variables (one per ply). This local-global approach leads to accurate and efficient theories.

6.2.2

Added mass effect

The motion of a solid in contact with a fluid is resisted by the inertia of the fluid. Friedrich Bessel proposed the concept of added mass in 1828 to describe the motion of a pendulum in a fluid (see Ref. [21]). To illustrate the effect of the fluid on the motion of the solid we first consider a simple example. Olson and Bathe presented results for spring-mass systems oscillating in water. The mass has an elliptical shape in Ref. [22] and a spherical shape in Ref. [23]. In both cases, the natural frequency of the system is given by rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ k (6.12) ω¼ m + m0 where m0 is the added mass of fluid. When m0 ¼ 0, Eq. (6.12) gives the natural frequency of the system in air so ω m0 1=2 ¼ 1+ m ωair

(6.13)

The m0 /m ratio is called the AMF or the nondimensional added virtual mass incremental (NAVMI) factor. For the ellipse on spring example, the stiffness of the spring is 200 N/m, 2a ¼ 25.4 cm, 2b ¼ 5.08 cm, the density of the ellipse is ρs ¼775 kg/m3 and that of water is 999 kg/m3. The mass of the ellipse is m ¼ ρs πab and the added mass of water is m0 ¼ ρf πa2 . Eq. (6.12) predicts a 10.9% reduction in the natural frequency when the system oscillates in water. The vibration of plates in contact with water has been studied extensively for nearly 100 years. In 1920 Lamb [24] studied the vibration of clamped circular plates in contact with water and, in 1923, Powell and Roberts [25] presented experimental results in Lamb’s predictions. McLachlan [26] considered free circular plates in 1932. The vibration of rectangular plates in contact with water was considered more recently: Davies [27], Muthuveerappan et al. [28–30] in 1978, Fu and Price [31], Robinson and Palmer [32]. Plates with other geometries were also studied (e.g., Ref. [33]) Refs. [24] through [32] just highlight some of the developments for isotropic plates. The analysis of plates in contact with water includes a model for the dynamics of the plate and a model for the fluid domain. Often the CPT is used to model the plate but other theories have also been used. The water is generally assumed to be incompressible and inviscid so that it follows potential flow theory. Its motion is governed by r2 φ ¼ 0

(6.14)

where φ is the velocity potential. The boundary conditions of the plate have to be satisfied along with the boundary conditions on the fixed and free boundaries of

Impact on composite plates in contact with water

189

the fluid domain. In addition, compatibility at the plate-water interface should also be satisfied: in the transverse direction, the velocity of the water is equal to the velocity of the plate. Closed-form solutions can only be found in simple cases (e.g., [34]). Approximate solutions can be obtained using the Rayleigh-Ritz method. Let U and Tp be the strain energy and the kinetic energy of the plate and Tw be the kinetic energy of the plate 1 Tw ¼ ρw 2

ððð ðrφÞ2 dV

(6.15)

V

where ρw is the density of water. Defining the total energy of the system as E ¼ ðTw Þmax + Tp max ðU Þmax the Rayleigh-Ritz method is used to derive equations that can be used to study the vibrations of the coupled problem [35]. For plates submerged in water, the mode shapes do not change and the natural frequencies change according to Haddara and Cao [36] and many other publications. The AMF was found to be inversely proportional to the frequency for bending modes and to be exactly the same for torsional modes [36]. For rectangular plates parallel to the surface of the water, it was found that the AMF increased with the depth to length ratio h1/a until h1/a reaches 10% [36]. Habault and Filippi [37] determined the first 20 natural frequencies of clamped stainless steel plates in vacuum and in water (Table 6.1). The dimensions of the plate are 0.350 0.500 0.005 m, and the density, elastic modulus and Poisson ratio are 7800 kg/m3, 200 GPa, and 0.3 respectively. The density of water is 1000 kg/m3 and the speed of sound in water is 1500 m/s. Plotting all 20 data points on the same graph (Fig. 6.1) shows that ω increases linearly with ωvacuum. The m0 /m ratios calculated from these data decrease as the frequencies increase (Table 6.1).

Fig. 6.1 Added mass effect in the first natural frequencies of a stainless steel clamped rectangular plate (0.350 0.500 0.005 m). Natural frequencies in water compared versus natural frequencies in air (Hz).

2500 y = 0.7374x – 165.56 R 2 = 0.9972

In water

2000 1500 1000 500 0 0

1000

2000 In air

3000

4000

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Dynamic Response and Failure of Composite Materials and Structures

Table 6.1 Natural frequencies in vacuum and in water (Hz) and added mass factors for the first 20 modes of a rectangular stainless steel plate (0.350 × 0.500 × 0.005 m) Mode number

Frequencies in vacuum

Frequencies in water

Added mass factor

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

275.2 440 663.8 715.2 817.2 1076.8 1093.6 1252.7 1403.3 1443.6 1571.2 1656 1911.6 2017.2 2038.6 2188.5 2437.2 2477.7 2794.5 3247.2

94.6 202.4 329.5 371.2 441.5 624.4 628.6 709.6 841.6 891.2 968.3 1042.1 1237.3 1322.7 1279.5 1418 1629.4 1683.6 1923.4 2292.6

7.463 3.726 3.058 2.712 2.426 1.974 2.027 2.116 1.780 1.624 1.633 1.525 1.387 1.326 1.539 1.383 1.237 1.166 1.111 1.006

6.3

Dynamics of deformable structures

Since the objective is to model impacts on composite plates, this section starts with an overview of the CPT and the derivation of its equations of motion and boundary conditions. Free vibration analysis is used to obtain the natural frequencies and mode shapes of the plate and a modal superposition approach is used to obtain a set of uncoupled differential equations for the response of each mode to external loading. The response to an arbitrary load can be written using the convolution integral (or Duhamel’s integral). This convolution integral is often used in the formulation of models for the analysis of impacts; its application to beams, plates, and shells is described in this section.

6.3.1 6.3.1.1

Classical plate theory Equations of motion and boundary conditions

The basic kinematic assumptions of the Kirchhoff-Love theory resulting in the displacement given by Eq. (6.5) were discussed in Section 6.2. The following is a brief derivation of the equations of motion for that theory. With that displacement field

Impact on composite plates in contact with water

191

T (Eq. 6.5), the strains E ¼ εxx , εyy , εxy can be written as E ¼ εo + zκ

(6.16)

T

in terms of the midplane strains εo ¼ uo, x , vo, y , uo, y + vo, x and the curvatures T κ ¼ wo, xx , wo, yy , 2wo, xy : The strain energy in this case is given by ð 1 T U ¼ E σdV 2

(6.17)

V

T Q is where σ ¼ σ xx , σ yy , σ xy : The stress-strain relations at the ply level are σ ¼ Qε; the ply stiffness matrix in the global coordinate system. ð 1 T ðεo + zκÞ dV U¼ εo Qðεo + zκÞ + zκT Q (6.18) 2 V

Defining the matrices ðA, B, DÞ ¼

ð h=2 h=2

1, z, z2 dz Q

(6.19)

Plate constitutive equations ð h=2

N¼

h=2

M¼

ðεo + zκÞdz ¼ Q

ð h=2 h=2

ð h=2

ðεo + zκÞzdz ¼ Q

h=2

σdz ¼ Aεo + Bκ

ð h=2 h=2

(6.20) σ zdz ¼ Bεo + Dκ

After integrating through the thickness, the strain energy becomes U¼

1 2

ð Ω

T εo Aεo + 2εTo Bκ + κT Dκ dΩ

(6.21)

and its variation ð δU ¼

Ω

T

δεo ðAεo + BκÞ + δκðBεo + DκÞ dΩ

(6.22)

which can be written explicitly as ð δU ¼

Ω

δuo, x Nx + δvo, y Ny + δ uo, y + vo, x Nxy dΩ

ð

Ω

δwo, xx Mx + δwo, yy My + 2δwo, xy Mxy dΩ

(6.23)

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Dynamic Response and Failure of Composite Materials and Structures

Integrating by parts gives δU ¼

ð h Ω

δuo Nx + δvo Nxy

,x

i + δvo Ny + δuo Nxy , y dΩ

ð

δuo Nx, x + Nxy, y + δNy Ny, y + Nxy, x dΩ Ω ð h i δwo, x Mx + δwo, y Mxy , x + δwo, y My + δwo, x Mxy ,y dΩ ðΩ

+ δwo, x Mx, x + Mxy, y + δwo, y My, y + Mxy, x dΩ

(6.24)

Ω

Using integration by parts again for the fourth term, ð h i δuo Nx + δvo Nxy , x + δvo Nx + δuo Nxy , y dΩ δU ¼ Ω ð

δuo Nx, x + Nxy, y + δNy Ny, y + Nxy, x dΩ ðΩ h i δwo, x Mx + δwo, y Mxy , x + δwo, y My + δwo, x Mxy ,y dΩ ðΩ h i + δwo Mx, x + Mxy, y , x + δwo My, y + Mxy, x ,y dΩ ð Ωh i δwo Mx, x + Mxy, y , x + δwo My, y + Mxy, x ,y dΩ

(6.25)

Ω

Using Gauss theorem ð δU ¼

Γ

δuo Nx + δvo Nxy nx + δvo Ny + δuo Nxy ny dΓ

ð

δuo Nx, x + Nxy, y + δvo Ny, y + Nxy, x dΩ Ω ð

δwo, x Mx + δwo, y Mxy nx + δwo, y My + δwo, x Mxy ny dΓ Γ ð

δwo Mx, x + Mxy, y nx + δwo My, y + Mxy, x ny dΓ + Γ ð h i δwo Mx, x + Mxy, y , x + δwo My, y + Mxy, x ,y dΩ

(6.26)

Ω

The kinetic energy of the plate is given by ð T¼ V

n o ρ ðu,t Þ2 + ðv, t Þ2 + ðw, t Þ2 dV

(6.27)

Impact on composite plates in contact with water

193

With the displacement field in Eq. …, T becomes ð

n o 2 ρ ðuo, t zwo, xt Þ2 + vo, t zwo, yt + ðwo, t Þ2 dV

T¼

(6.28)

V

or T¼

ð

n 2 ρ ðuo, t Þ2 + z2 ðwo, xt Þ2 2zuo, t wo, xt + ðvo, t Þ2 + z2 wo, yt V o 2zvo, t wo, yt + ðwo, t Þ2 dV 1 2

(6.29)

The inertial properties are defined as ðm, I1 ,I2 Þ ¼

ð h=2 h=2

ρ 1, z, z2 dz

(6.30)

where m is the mass per unit area, I2 is the rotary inertia and its effects are ignored in this theory. I1 is the inertial coupling between in-plane and transverse motion and it is zero when the density is symmetric with respect to the reference plane. Then, T¼

1 2

ð

n o m ðuo, t Þ2 + ðvo, t Þ2 + ðwo, t Þ2 dΩ

(6.31)

m fδuo, t uo, t + δvo, t vo, t + δwo, t g dΩ

(6.32)

Ω

And ð δT ¼

Ω

The work done by the external forces is given by ð We ¼

p w dΩ

(6.33)

Ω

assuming that p(xy) is a distributed load normal to the surface. Equations of motion Nx, x + Nxy, y ¼ muo, tt Ny, y + Nxy, x ¼ mvo, tt Mx, xx + 2Mxy, xy + My, yy ¼ mwo, tt

(6.34)

For symmetric laminates, B ¼ 0 so, in terms of displacements, the first two equations governing the in-plane motions can be written in terms of uo and vo. The third equation of motion involves only the transverse displacement wo. Therefore, the equations of motion for the in-plane and out-of-plane motions are uncoupled.

194

Dynamic Response and Failure of Composite Materials and Structures

When the D matrix is fully populated, the transverse motion is governed by D11 w,xxxx + 2ðD12 + 2D66 Þw, xxyy + D22 w, yyyy + 4D16 w, xxxy + 4D26 w,xyyy + mw,tt ¼ pðx, tÞ

(6.35)

For a laminate with many layers, the effect of D16 and D26 terms becomes negligible. Therefore, with the Kirchhoff-Love theory, the equation of motion for a laminated plate is D11 w,xxxx + 2ðD12 + 2D66 Þw, xxyy + D22 w, yyyy + mw, tt ¼ pðx, tÞ

(6.36)

Boundary conditions uo ¼ 0 or Nx nx + Nxy ny ¼ 0 vo ¼ 0 or Nxy nx + Ny ny ¼ 0 wo, x ¼ 0 or Mx nx + Mxy ny ¼ 0 wo, y ¼ 0 or My ny + Mxy nx ¼ 0 wo ¼ 0 or Mx, x + Mxy, y nx + My, y + Mxy, x ny ¼ 0

6.3.1.2

(6.37)

Modal analysis

The objective here is to find the natural frequencies and mode shapes and to use a modal superposition approach to derive a set of uncoupled equations to be used to determine the response of each mode to an arbitrary loading. Multiplying Eq. by a function η and integrating by parts ðð

D11 η,xx w,xx + D12 η, xx w, yy + D12 η, yy w, xx + 4D66 η, xy w,xy + D22 η,yy w,yy dΩ ðð ðð + m η w, tt dΩ ¼ ηpðx, y, tÞdΩ (6.38)

For free vibrations, pðx, y, tÞ ¼ 0 and w ¼ ϕij sin ωij t where ϕij(x, y) is the mode shape and ωij is the natural frequency. Taking η ¼ ϕij and substituting into Eq. gives ωij ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ kij =mij

(6.39)

The modal stiffness and modal mass are given by ðð h i D11 ϕ2ij, xx + 2D12 ϕij, xx ϕij, yy + 4D66 ϕ2ij, xy + D22 ϕ2ij, yy dΩ ðð mij ¼ mϕij ϕij dΩ kij ¼

(6.40)

Impact on composite plates in contact with water

195

respectively. The response to an arbitrary load p(x, y, t) is determined using the modal expansion w¼

M X N X

αmn ðtÞ ϕmn ðx, yÞ

(6.41)

m¼1 n¼1

where the modal participation factors αmn(t) are functions of time to be determined. Taking η ¼ ϕij and substituting into Eq. gives a set of M N uncoupled modal equations mij α€ij + kij αij ¼ fij ðtÞ

(6.42)

with ðð fij ðtÞ ¼

ϕij pðx, y, tÞ dΩ

(6.43)

This uncoupling is possible because of the orthogonality of the mode shapes. That is, ðð

D11 ϕij, xx ϕmn, xx + D12 ϕij, xx ϕmn, yy + D12 ϕij, yy ϕmn, xx + 4D66 ϕij, xy ϕmn, xy

(6.44)

+ D22 ϕij, yy ϕmn, yy dΩ ¼ 0 and ðð m ϕij ϕmn dΩ ¼ 0

(6.45)

when ϕij 6¼ ϕmn : Eq. (6.42) can be solved individually and then added together to find the response of the structure (Eq. 6.41).

6.3.1.3 Transient response using the convolution integral First we determine the modal response for a unit impulsive force applied at (xo, yo), p ¼ δðx xo Þ δðy yo Þ δðtÞ fij ðtÞ ¼ ϕij ðxo , yo Þ δðtÞ

(6.46)

The solution in this case is the Green’s function gðtÞ ¼ αij ¼

ϕij ðxo ,yo Þ sin ωij t mij ωij

(6.47)

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Dynamic Response and Failure of Composite Materials and Structures

For an arbitrary impulsive force P(t) applied at (xo, yo), the modal response is obtained using the convolution integral ðt

ϕij ðxo , yo Þ αij ¼ PðξÞgðt ξÞ dξ ¼ mij ωij 0

ðt

PðξÞsin ωij ðt ξÞ dξ

(6.48)

0

Substituting into Eq, we find that the displacement of the plate at the point of application of the force is wT ¼

ð M X N X ϕij ðxo , yo Þ ðxo ,yo Þ t mij ωij

i¼1 j¼1

PðξÞsin ωij ðt ξÞ dξ

(6.49)

0

The subscript T is added here to designate the displacement of the target at the point of impact. This modal superposition approach to obtain Green’s function and based on the CPT is used in many publications [38,39].

6.3.1.4

Large plate approximation

Boussinesq [40] obtained the following expression for the deflection of an infinite isotropic plate subjected to a concentrated transverse 1 wt ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ 8 Dρh

ðt

FðτÞdτ

(6.50)

0

where D ¼ Eh3 = 12 1 ν2 is the bending rigidity of the plate, ρ is the density, and h is the thickness. The integral is the linear momentum applied by the force F(t). Noting that the displacement at the point of application of the force is proportional to the applied momentum, Eq. (6.40) is rewritten as wt ¼ β

ðt

FðτÞdτ

(6.51)

0

pﬃﬃﬃﬃﬃﬃﬃﬃﬃ The proportionality constant for the plate is β ¼ 1=8 Dρh in this case. In Olsson [41] the response of an infinite orthotropic plate is determined using Eq. (6.41) and pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ β ¼ 1=8 D*m where m is the mass of the plate per unit area and pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ D* ¼ ½ðA + 1Þ=2 D11 D22 and A ¼ ðD12 + 2D66 Þ= D11 D22 . For orthotropic plates, Mittal and Khalili [42] Mittal and Jafri [43] found that β¼

Fðπ=2, K Þ pﬃﬃﬃﬃﬃ 4π ρhðD1 D2 Þ1=4

(6.52)

Impact on composite plates in contact with water

197

where F(π/2, K) is the complete elliptical integral of the first order and sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 D3 =D2 K¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ: 2 2 D1 =D2

6.3.1.5 Approximate methods In the preceding, it was assumed that an exact solution to the free vibration problem is known so, using the mode shapes in the expansion (Eq. 6.41) results in a set of uncoupled modal equations. Often, such solutions are not available and approximate solutions are sought using the Rayleigh-Ritz method, the finite element method, or any other numerical approach. With the Rayleigh-Ritz method, the displacement w is expanded in a double series in terms of admissible functions ψ mn instead of the mode shapes ϕmn and η ¼ ψ ij in Eq. (6.38). Admissible functions are functions that should satisfy the essential boundary conditions of the problem. Substitution into Eq. (6.38) results in a set of coupled ordinary differential equations of the form M X€ + KX ¼ F

(6.53)

The finite element method and other approximation methods generate a set of equations in this form. The natural frequencies and mode shapes for this system can be determined numerically and used to obtain the modal equations.

6.3.1.6 FSDT The displacement field for the Mindlin-Reissner first-order shear deformation theory obtained in Section 6.2 is u ¼ uo + zu1 v ¼ vo + zv1 w ¼ wo

(6.54)

Following the derivation of the governing equations for the Kirchhoff-Love theory above, a short derivation of the equations of motion for the FSDT is presented here. This is done with several goals is mind: (1) to show the similarities between the theories; (2) to show the effect of introducing new variables into the displacement field; (3) to show that given any displacement field, the same approach can be used for the derivation of the equations of motion and the boundary conditions. The in-plane strains E ¼ εo + zκ

(6.55) T and the curvatures in terms of the midplane strains εo ¼ uo, x , vo, y , uo, y + vo, x T T κ ¼ u1, x , v1, y , u1, y + v1, x . The transverse shear strains γ ¼ εyz , εxz ¼ T T v1 + wo,y , u1 + wo, x and the corresponding shear stresses τ ¼ σ yz , σ xz ð ð 1 1 T U ¼ U1 + U2 ¼ E σdV + γT τdV (6.56) 2 V 2 V

198

Dynamic Response and Failure of Composite Materials and Structures

Then, ð δU1 ¼

Γ

ð

δuo Nx + δvo Nxy nx + δvo Ny + δuo Nxy ny dΓ

δu0 Nx, x + Nxy, y + δv0 Ny, y + Nxy, x dΩ

ðΩ

δu1 Mx nx + Mxy ny + δv1 My ny + Mxy nx dΓ ðΓ

δu1 Mx, x + Mxy, y + δv1 Mxy, x + My, y dΩ +

(6.57)

Ω

With the following relationship between transverse shear stresses and strains

τ¼

Q45 γ Q55

Q44 Q45

(6.58)

The transverse shear forces

Qx Qy

ð h=2

¼

τdτ ¼

h=2

ð δU2 ¼

ð h=2 h=2

ðð δγ τdV ¼ T

A44 A45 Q44 Q45 γdτ ¼ γ A45 A55 Q45 Q55

δεyz Qy + δεxz Qx dΩ

(6.59)

(6.60)

V

ðð δU2 ¼

ðð ð δv1 Qy + δu1 Qx dΩ + δwo Qx nx + Qy y dΓ δwo Qy, y + Qx, x dΩ (6.61)

The kinetic energy of the plate is T¼

1 2

ð

n o ρ ðuo, t + zu1, t Þ2 + ðvo, t + zv1, t Þ2 + ðwo, t Þ2 dV

(6.62)

V

Integrating through the thickness and assuming that the density is symmetric about the midplane 1 T¼ 2

ðð n

o mðuo, t Þ2 + mðvo, t Þ2 + mðwo, t Þ2 + ρI ðu1,t Þ2 + ρI ðv1,t Þ2 dΩ

(6.63)

The first variation of the kinetic energy is ðð δT ¼

fmuo, t δuo, t + mvo, t δvo, t + mwo, t δwo, t + ρIu1, t δu1, t + ρIv1, t δv1, t gdΩ

(6.64)

Impact on composite plates in contact with water

199

Using Hamilton’s principle and the expressions for δU and δT above, the equations of motion are Nx, x + Nxy, y ¼ muo, tt Ny, y + Nxy, x ¼ mvo, tt

(6.65)

Mx, x + Mxy,y + Qx + ρIu1,tt ¼ 0 Mxy, x + My, y + Qy + ρIv1, tt ¼ 0

(6.66)

Qy, y + Qx, x + mwo, tt ¼ 0

(6.67)

At this stage, comparing the equations of motions obtained for the CPT and the FSDT several points can be made: (1) The number of equations of motion is equal to the number of variables retained in the formulation. The CPT has three variables (uo, vo, w) and three equations of motion (Eq. 6.34). The FSDT has five variables (uo, vo, w, u1, v1) and five equations of motion (Eqs. 6.65–6.67) (2) In terms of the force resultants, the equations governing the in-plane motion (i.e., Eqs. 6.34, 6.65) but they will be different when written in terms of displacement variables. (3) The transverse motion is governed by a single equation (Eq. 6.34) for the CPT, in terms of the moment resultants Mx, My, and Mxy. For the FSDT, three equations of motion are needed (Eqs. 6.66, 6.67) involving the moment resultants and the shear force resultants Qx and Qy. (4) Both in-plane and transverse motions appear to be uncoupled but coupling is introduced through the constitutive equations (Eq. 6.20). (5) If the rotary inertia is neglected, Eqs. (6.66), (6.67) can be combined so that Eq. (6.34) is recovered.

6.3.2

Convolution integral and higher order plate and shell theories

The CPT has well-known limitations that led to the development of many higher order theories that are also used to analyze impact problems. However, the transverse displacement can always be expressed as in Eq. (6.39). The convolution integral approach has been used to determine the transverse displacement when the structure is modeled using the FSDT [44–50], the RSDT [51,52], and many other higher order theories. Refs. [53,54] compare the results obtained using the CST, FSDT, and RSDT theories for doubly curved composite panels. The RSDT and the FSDT yielded similar results when the thickness to side length ratio h=a < 0:2 and the CST was found to produce acceptable results only for quasiisotropic panels with h=a < 0:05: The convolution integral approach is also applied to plates and shells modeled by higher order polynomial theories: a {3,0} plate theory [55], a {3,3} theory for cylindrical shell [56]. Maiti and Sinha [57] compared the results from several plate theories: FSDT, {3,2} with uo ¼ u2 ¼ vo ¼ v2 ¼ w1 , {3,0}, {3,2}, and {3,3} theories. It was found that the choice of theory is important for bending and shear stresses but that it has a small effect on the contact force history and motion of projectile. A similar comparison was performed for doubly curved shells [58] using the FSDT and the {1,2}, {3,0}, {3,2}, and {3,3} polynomial theories.

200

Dynamic Response and Failure of Composite Materials and Structures

The convolution integral approach is also used when the plate is modeled using a three-dimensional elasticity approach [59–61]. Koller and Busenhart [62] use a convolution integral to find the dynamic response of a spherical shell and to study its response to the impact by a mass.

6.3.3

Convolution integral and beam theories

Integral equation formulation for the impact of a mass on a beam is given in Ref. [63] first published in 1928.

6.3.3.1

Bernoulli-Euler beams

The motion of a Bernoulli-Euler beam on a Winkler elastic foundation is governed by EI

@4w @2w + kw + ρA 2 ¼ pðx, tÞ 4 @x @t

(6.68)

With the modal superposition approach, the transverse displacement is expanded as w¼

N X

αi ðtÞϕi ðxÞ

(6.69)

i¼1

in terms of the mode shapes ϕi and the modal participation factors. Following the approach used for plates (Section 6.3.1), we obtain the modal equations mi α€i + ki αi ¼ FðtÞϕi ðxo Þ

(6.70)

The modal mass and the modal stiffness for mode i are given by ki ¼ EI

ðL

d 2 ϕi d 2 ϕi dx + k 2 2 0 dx dx

ðL

ϕi ϕi dx mi ¼ ρA

0

ðL

ϕi ϕi dx

(6.71)

0

To account for internal damping the term ci α_ i can be added to the left side of pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Eq. (6.70). ci =mi ¼ 2ωi η where ωi ¼ ki =mi is the undamped natural frequency and η is the damping ratio. The response to a unit impulse FðtÞ ¼ δðtÞ applied at xo is αi ¼

ϕi ðxo Þ ηωi t e sin ω0i t mi ω0i

(6.72)

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ where ω0i ¼ ωi 1 η2 is the damped natural frequency. Using the convolution integral, the modal response to an arbitrary force F(t) αi ¼

ϕ i ðx o Þ mi ω0i

ðt 0

FðτÞeηωi ðtτÞ sin ω0i ðt τÞ dτ

(6.73)

Impact on composite plates in contact with water

201

The displacement at the contact point is given by wT ¼

ð ∞ X ϕ ðxo Þϕ ðxo Þ t i

i

mi ω0i

i¼1

0

FðτÞeηωi ðtτÞ sin ω0i ðt τÞ dτ

(6.74)

Neglecting the damping effects wT ¼

ð ∞ X ϕ ðxo Þϕ ðxo Þ t i

i

m i ωi

i¼1

FðτÞ sin ωi ðt τÞ dτ

(6.75)

0

This expression is used in many publications. For long beams (no foundation) Schwieger [64,65] and Smith have shown that wT ¼ α

ðt

FðτÞðt τÞ1=2 dτ

(6.76)

0

where d is the width, h is the height of the beam, and 1 1 dρh 1=4 α ¼ pﬃﬃﬃﬃﬃ 2π dρh EI

(6.77)

6.3.3.2 Timoshenko beams For infinite Timoshenko beams (no foundation) Mittal [66] found that 1 wt ¼ 2πρA

ðt 0

Fð τ Þ

ð∞

sin ωðt τÞ dξ dτ ω ∞

(6.78)

where rﬃﬃﬃﬃﬃ EI 2 6EI 2 1=2 ξ 1+ ξ ω¼ A 5GA

(6.79)

Eqs. (6.76), (6.78) show that for beams the displacement is not proportional to the applied impulse as it is the case for plates.

6.3.4

Impact problem formulation

6.3.4.1 Motion of the projectile The equation of motion of the projectile P ¼ mp w€p

(6.80)

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Dynamic Response and Failure of Composite Materials and Structures

is obtained using Newton’s law. Multiplying by ðt τÞ and integrating from 0 to t gives wT ¼ Vt

1 mp

ðt

PðτÞ ðt τÞdτ

(6.81)

0

Both equations are used in the formulation of the impact problem.

6.3.4.2

Local deformations in the contact zone

Determining the local deformation between two solids in contact is a complex and time consuming problem particularly for the dynamic case. In the literature, local indentation effects are accounted for through the use of a contact law relating the relative displacement α ¼ wP wT to the applied force P. Moreover, it is assumed that statically determined contact laws can be used in the analysis of low velocity impacts. Often, experimental results for indenters with spherical tips can be fitted by Meyer’s law P ¼ kc αn

(6.82)

during the loading phase of the impact. Generally n is close to 3/2, the value predicted by Hertz in which case Eq. is Hertz’s contact law. Often, the purpose of the analysis is to determine the maximum force acting on the structure and the stresses induced at that time. Then, the same curve is used for the unloading process. The unloading process is generally modeled by P ¼ Pm

α αo α m αo

2:5 (6.83)

where Pm and αm are the maximum force and indentation reached during the loading phase and αo is a parameter to be determined. UL, the area under the force-indentation curve during the loading phase, is the maximum energy stored into the contact region. UUL, the area under the unloading curve, is the energy restituted during the unloading process. The coefficient of restitution can be defined as e2 ¼ UUL =UL . With Eqs. (6.82), (6.83) we find that 5 αo αo 7 or ¼ 1 e2 (6.84) e2 ¼ 1 αm αm 7 5 In some cases, a linear contact law is used for the loading phase P ¼ k1 x

(6.85)

Assuming a linear unloading with P ¼ k 2 ðx x m Þ + k 1 x m

(6.86)

Impact on composite plates in contact with water

203

we find that the coefficient of restitution is related to the stiffness coefficients k1 and k2 by e2 ¼ 1

k1 k2

(6.87)

6.3.4.3 Integral formulation In general the indentation α ¼ wP wT can be written as ð 1 t PðτÞ ðt τÞdτ wT α ¼ Vt mp 0

(6.88)

The displacement of the target at the impact point wT is obtained using the convolution integral for different models. For impact on a finite plate ð 1 t PðτÞðt τÞdτ wP wT ¼ Vt mp 0 (6.89) ð M X N X ϕij ðxo , yo Þϕij ðxo , yo Þ t PðξÞ sin ωij ðt ξÞdξ mij ωij 0 i¼1 j¼1 For impacts on an infinite plate, α€ ¼ w€P w€T . From Eq. (6.82), w€P ¼ P=mp and from Eq. (6.41), w€T ¼ βP_ so α€ +

P + βP_ ¼ 0 mP

(6.90)

With Hertz’s contact law we obtain a single nonlinear differential equation α€ +

kc 3=2 d 3=2 α α + βkc ¼0 mP dt

(6.91)

with the initial conditions αð0Þ ¼ 0 and α_ ð0Þ ¼ V. With the change of variables τ ¼ at and α ¼ bα, Eq. (6.91) becomes d 3=2 3=2 α00 + λ α (6.92) +α ¼0 dτ with initial conditions αð0Þ ¼ 0 and α0 ð0Þ ¼ 1 provided that a ¼ V 1=5 kc =mP , 2=5

2=5

2=5

2=5

2=5

3=5

and λ ¼ βkc mP V 1=5 . This is Zener’s approach who derived b ¼ mp V 4=5 kc Eqs. (6.90), (6.83) in a slightly different form [67]. Conway and Lee [68] used a linear contact law (Eq. 6.85) and, with the change of variables τ ¼ at and α ¼ bα, Eq. (6.91) becomes α00 + λα0 + α ¼ 0

(6.93)

204

Dynamic Response and Failure of Composite Materials and Structures 1=2 1=2

with αð0Þ ¼ 0 and α0 ð0Þ ¼ 1. In this case, a ¼ kc =mP , b ¼ Vmp kc , and λ ¼ βkc V. Eq. (6.93) is the equation of motion of a damped single degree of freedom system. When λ ¼ 0, it reduces to the case of an impact against a rigid surface. This equation shows that the plate acts like a damper dissipating energy. Similarly, the second term in Eq. (6.90) shows that the deformation of the plate introduces some nonlinear damping during the impact. Phillips and Calvitt [69] extended Zener’s approach to treat the impact of a sphere on a viscoelastic plate. 1=2

6.3.4.4

1=2

Direct formulation

In a more direct formulation, the motion of the structure is governed by a set of N second-order differential equations obtained analytically, using the Rayleigh-Ritz method, the finite element method, or any other method. The motion of a rigid projectile is governed by Eq. (6.80) and these N + 1 equations are coupled by the contact law relating the force P to the displacements.

6.4

Spring-mass impact models

Some insight into the problem can be gained by considering some simpler cases: (1) the impact of a mass against a half-space; (2) the impact between two masses; (3) the impact of a mass against a spring-mass system.

6.4.1

Hertz’s impact problem

The normal impact of a sphere on a half-space is often called Hertz’s problem. The motion of the sphere results from the indentation of the surface of the half-space. The force resisting the motion of the projectile is assumed to follow a static contact law and the motion is governed by mP w€P + kc wnP ¼ 0

(6.94)

with the initial conditions wP ð0Þ ¼ 0 and w_ P ð0Þ ¼ V where V is the initial velocity. P ¼ wP =b Introducing the nondimensional variables τ ¼ at, w 00P + w nP ¼ 0 w

(6.95)

0P ð0Þ ¼ 1 when a and b are taken to be and w 1=ðn + 1Þ 1=ðn + 1Þ a ¼ V n1 kc =mP b ¼ mP V 2 =kc

(6.96)

The nondimensional contact force history for n ¼ 3/2 (Fig. 6.2) shows that the impact nP ¼ 1:155: ends when τ ¼ 3:22 and the maximum value of the nondimensional force w

Impact on composite plates in contact with water

205

Force

1

0.5

0

0

1

2 Time

3

4

Fig. 6.2 Contact force history for Hertz’s impact problem. Nondimensional force versus time curve.

6.4.2

Impact of two masses

For a direct central impact between two rigid bodies, the motion of the projectile is governed by mP w€P ¼ P and for the target mT w€T ¼ P. The contact force P is related to the indentation α ¼ wP wT by the contact law (Eq. 6.82). The indentation should satisfy the equation me α€ + kc αn ¼ 0

(6.97)

where me is the equivalent mass defined by α_ ð0Þ ¼ V:

6.4.3

1 1 1 ¼ + with the initial condition me mP mT

Impact of a mass against a spring-mass system

For the impact problem in Fig. 6.3, the motion of the projectile and the target are governed by mT w€T + kT wT ¼ kc ðwP wT Þ mP w€P ¼ kc ðwP wT Þ

V kc mP

kT mT

uP

Fig. 6.3 Impact on a spring-mass system.

uT

(6.98)

206

Dynamic Response and Failure of Composite Materials and Structures

with the initial conditions wT ð0Þ ¼ wP ð0Þ ¼ 0 and w_ P ð0Þ ¼ V , introducing the nonpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ P ¼ wP =b. When a ¼ kT =mT and T ¼ wT =b, and w dimensional variables τ ¼ at, w b ¼ V=a, the nondimensional form of the problem is 00T + w T ¼ w

kc m T kc P w P w T Þ w 00P ¼ T Þ ðw ðw kT m P kT

(6.99)

P ð0Þ ¼ 0 and w 0P ð0Þ ¼ 1. This shows that this type T ð 0Þ ¼ w with the initial conditions w of impact depends on two parameters: the mass ratio mT/mP and the stiffness ratio kc/kT. For a case when the mass of the target is much larger than that of the projectile, the displacement of the target is much smaller than that of the projectile and the impact ends when those two displacements become equal (Fig. 6.4). Results in this figure show that while mT =mP ¼ 10, the value of kc/kT affects the nondimensional contact duration and indentation. When mT =mP ¼ 0:2, the plate is impacted by a large mass and the motion of the projectile is smooth while that of the target oscillates (Fig. 6.5). These oscillations are small when kc =kT ¼ 0:5, and become significant when kc =kT ¼ 1 but in both cases there is only one impact. When kc =kT ¼ 5, the displacements of the target become larger and at times become larger than that of the projectile, resulting in multiple impacts. 0.2

0.2 0 0 0

1

0.2

0.4

0.6

2

–0.2

–0.2

(A)

(B) 0.5

0.25

0

0

0.5

1

1.5

2

–0.25

(C) Fig. 6.4 Impact of a mass against a spring-mass system. Linear contact law, mT =mP ¼ 10. (A) kc =kT ¼ 5, (B) kc =kT ¼ 1, (C) kc =kT ¼ 0:5. Heavy line: contact force history, thin line: displacement of the projectile, dashed line: displacement of the target.

Impact on composite plates in contact with water

207

3

2

2 1 1 0

0

5

0 0

10

–1

–1

(A)

(B)

2

4

6

8

10

12

5

3

1

–1

0

5

10

15

20

(C) Fig. 6.5 Impact of a mass against a spring-mass system. Linear contact law, mT =mP ¼ 1=5. (A) kc =kT ¼ 5, (B) kc =kT ¼ 1, (C) kc =kT ¼ 0:5. Heavy line: contact force history, thin line: displacement of the projectile, dashed line: displacement of the target.

With a nonlinear contact law, using the same nondimensional parameters a and b, the equations of motion become mT P w 00P ¼ kðw T ¼ kðw P w T Þ3=2 w T Þ3=2 00T + w w mP

(6.100)

1=4 5=4 where k¼ kc V 1=2 mT =kT :

6.5

Impact on structures in contact with water

This section deals with the response of structures in contact with water. First we discuss the bending and torsion of bars with rectangular cross sections for which the added mass is the same for all modes. The dynamics of plates impacted by a spherical tipped projectile are discussed in Section 6.5.2. The formulation is general and results are presented for plates in air. The influence of the added mass of water is discussed in Section 6.5.3.

208

Dynamic Response and Failure of Composite Materials and Structures

6.5.1

Impact on beams immersed in water

For the flexural motion of beams with circular cross sections, the added mass is m0 ¼ ρf A where A ¼ πD2 =4 and D is the diameter [70]. For beams with rectangular cross sections m0 ¼ ρf πb2 =4 where b is the width of the beam [71]. For rectangular cross sections, the AMF is m0 =m ¼ πρf b =ð4 ρs hÞ

(6.101)

Neglecting the warping of the cross section, the torsion of beams is governed by GK @ 2 ϕ @2ϕ ρ I ¼ mðx, tÞ p c L2 @x2 @t2

(6.102)

where ϕ is the angle of twist. In this case, the AMF for the torsion of a rectangular bar is [72] Ip0 =Ip ¼ 3πρf b =ð32 ρs hÞ

(6.103)

The validity of these results is discussed in Refs. [72,73], and it was concluded that they are accurate when the Reynolds number Re ¼ ρωb2 =ð4ηÞ≫1. These results are applicable for beams immersed in water which has a very small viscosity η. With the values of Ip0 and Ip given in Ref. [71], the additional factor 1 + t2 =b2 appears in the denominator of Eq. (6.103). It is negligible for thin beams. Since these AMFs are frequency independent, the natural frequencies of beams pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ immersed in water will all be divided by 1 + m0 =m for the bending modes and by qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 + Ip0 =Ip for the torsional modes.

6.5.2

Impact on plates

Results are obtained for simply supported plates using the direct formulation. For a typical mode of a simply supported plate ϕij ¼ sin

iπx jπy sin a b

(6.104)

The modal stiffness and the modal mass are " 2 2 4 # ab iπ 4 iπ jπ jπ + 2ðD12 + 2D66 Þ + D22 D11 kij ¼ 4 a a b b ðð ab mij ¼ mϕij ϕij dΩ ¼ m 4

(6.105)

Impact on composite plates in contact with water

209

The impact problem is defined by one equation of motion for the projectile, a set of equations for the motion of the plate, and the contact law mp w€p ¼ PðtÞ mij q€ij + kij qij ¼ PðtÞ ϕij ðxo ,yo Þ 3=2 P ¼ kc wp wT

(6.106)

where w¼

M X N X

qmn ðtÞ ϕmn ðx, yÞ

(6.107)

m¼1 n¼1

Many investigations have shown that the presence of water has negligible effect on the mode shapes of the plate and that it simply increases the modal masses. For plates, the AMFs are different for each mode and generally decrease with the frequency increase. In the following the modal mass mij is the sum of the modal mass for the plate in air and the added mass of water for that mode. T ¼ wT =b, and Introducing the nondimensional variables τ ¼ at, qij ¼ qij =b, w P ¼ wP =b, the nondimensional equations of motion are w 3=2 m11 p w T w mp 3=2 k ij p w T ϕij ðxo ,yo Þ q00ij + q ¼ w k11 ij 00P ¼ w

(6.108)

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ when a ¼ k11 =m11 and b ¼ ðk11 =kc Þ2 . The initial condition w_ P ð0Þ ¼ V gives 0P ð0Þ ¼ V and the nondimensional force abw 3=2 3 p w T ¼ w P ¼ Pkc2 =k11

(6.109)

For a square isotropic or laminated quasiisotropic plate, k13 =k11 ¼ k31 =k11 ¼ 25 and k33 =k11 ¼ 81. First we study the response of a plate impacted by a heavy mass m11 =mP ¼ 1=5 and examine the effect of the impact velocity and the number of modes used to model the structure. Fig. 6.6A shows the dynamic response obtained 0P ð0Þ ¼ 1. The motion of the when only one mode (q11) is used for the plate and w projectile is smooth while that of the plate is seen to oscillate and that is reflected in the contact force history. The contact force obtained with this one mode approach 0P ð0Þ ¼ 1, 2,3, 4 is compared in Fig. 6.6B. The contact duration decreases slightly for w and the amplitude increases with the initial velocity of the projectile. Retaining three modes for the plate (q11, q13, and q31) in the analysis of this problem shows that modes q13 and q33 have an effect on the response (Fig. 6.6C). Keeping modes q11, q13, q31, and q33 shows that the solution has already converged for this problem (Fig. 6.7).

210

Dynamic Response and Failure of Composite Materials and Structures

wP

wT

12

Force

1 3

3

2 4

Force

Force

8 2

4

1

0

(A)

0 0

2

4

6

8

10

0

2

4

(B)

Time

6

8

10

Time

12 1.5 2 3

10 Force

8 6 4 2 0

0

2

4

8

6

(C)

10

Time

Fig. 6.6 Dynamic response for an isotropic square plate impacted by a heavy mass m11 =mP ¼ 0P ð0Þ ¼ 1: nondimensional displacements and contact force; 1=5: (A) single mode response for w 0P ð0Þ ¼ 1, 2,3, 4; (C) contact force (B) Single mode response, contact force histories for w 0P ð0Þ ¼ 1,2,3, 4 using modes 11, 13, and 31. histories for w

12 1 mode 3 modes 4 modes

Force

8

4

0

0

2

4

6

8

10

Time

Fig. 6.7 Dynamic response for an isotropic square plate impacted by a heavy mass m11 =mP ¼ 1=5. Comparison of contact force histories obtained using one, three, and four modes 0P ð0Þ ¼ 1. for w

Impact on composite plates in contact with water

211

When such a plate is impacted by a small mass ðm11 =mP ¼ 1=5Þ a single impact is produced. Initially, the displacement of the structure is small and the contact force history resembles that obtained when impacting a half-space as shown in Fig. 6.8A. Fig. 6.8B shows the effect of the initial velocity on the contact force history. These results obtained using a single mode show that the contact duration decreases and the magnitude increases as the contact force increases (Fig. 6.8B). The response is asymmetric and the maximum occurs for smaller and smaller values of the nondimensional time τ. Using three modes for the plate produces only slight differences in the response (Fig. 6.8C).

6.5.3

Impact on plates in contact with water

Few references address the problem of low velocity impact of plates in contact with water. Some of the more relevant examples include Ref. [74] in which a small rectangular steel plate with a mass of 2.50 kg is impacted by a 0.1 kg sphere and Ref. [75] in which the same plate impacts a circular plate with a mass of 1.96 kg. In these

1 wP

wT

Force

1.5 2 2 Force

0.5

3 1

4

0 0

1

2

3

4

–0.5

0

(A)

(B)

Time

0

0.5

1

1.5

Time

Force

2 1.5 2 3 4 1

0

(C)

0

0.5

1

1.5

2

Time

Fig. 6.8 Dynamic response for an isotropic square plate impacted by a small mass 0P ð0Þ ¼ 1, single mode ðm11 =mP ¼ 1=5Þ. (A) Contact force history and displacements when w 0 P ð0Þ ¼ 1:5, 2, 3, 4 , single mode response; (C) three response; (B) effect impact velocity w 0P ð0Þ ¼ 1:5, 2, 3, 4. mode response (modes 11, 13, and 31) for w

212

Dynamic Response and Failure of Composite Materials and Structures

references the mass of the projectile is much smaller than the mass of the plate and the contact force history is similar to that shown in Fig. 6.8A. In Refs. [76–78] impact tests were conducted to determine whether or not hydrodynamic effects are significant during the impact of composite plates in contact with water. Three cases were considered: plates in air, plates in contact with water on one side, and plates in contact with water on both sides. In Refs. [77,78], a 12 kg mass impacts a 12.l7 kg rod that is already in contact with 0.795 kg plate. Similar test conditions were used in Ref. [76]. The same tests were also conducted on sandwich plates with composite facings and balsa cores [79,80]. The results are often inconclusive as damage is introduced and sample-to-sample variability might account for changes attributed to the added mass effect. Based on the results obtained here, when mP ≫ mT , the interaction with the water would not make a significant difference. Similarly when mP ≪ mT , the motion of the plate is already small so it will not affect the contact force history very much. In the intermediate range, the fluid-structure interaction could be significant. Some studies reported AMFs up to 7 for the first mode [37] so a large increase in the mass of the target can completely change the dynamics of the system. With the infinite plate model (Eq. 6.92), mP is included in the parameters a, b, and λ indicating that increases in mP result in increases in the amplitude of the response, a decrease in the impact duration, and a change in the shape of the contact force history.

6.6

Conclusion

This chapter addresses a significant issue for high-speed marine vehicles in general and particularly those with composite structures. Impacts with floating debris or ice floes are likely to occur and both the damage resistance and the damage tolerance of the structure should be adequate. During impact the structure is assumed to undergo flexural deformations so the first step in the development of mathematical model to analyze impacts on plates in contact with water is the selection of an appropriate theory. Section 6.2 presents a comprehensive view of such theories and discusses how fluid-structure interaction leads to an added mass effect. A modal superposition approach with AMFs determined in a previous analysis uncouples the structural problem from the fluid mechanics problem for an efficient analysis of the impact problem. A direct formulation of the problem includes: (1) N uncoupled second-order ordinary differential equations describing the response of the structure to a concentrated force P; (2) another second-order ordinary differential equations describing the motion of the projectile; (3) a contact law that couples the N + 1 equations of motion. Another approach described in this chapter, the impact problem, is formulated as a single integral equation. This chapter makes two significant contributions: (1) the different options in the formulation of the problem are described in detail; (2) an example is treated in nondimensional form and the full range of possible responses to the impact is examined. The ratio between the mass of the projectile and the mass of the plate has an important effect on the type of impact response and the number of modes participating in this response.

Impact on composite plates in contact with water

213

This chapter focuses on the impact of a rigid body on a plate and indicates how to treat the impact of rigid bodies on beams and shells. Both the direct method and the integral equation approach can be used to formulate impacts problems between deformable bodies.

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[68] Conway HD, Lee HC. Impact of an indenter on a large plate. J Appl Mech 1970;37 (1):234–5. [69] Phillips JW, Calvit HH. Impact of a rigid sphere on a viscoelastic plate. J Appl Mech 1967;34(4):873–8. [70] Us´ciłowska A, Kołodziej JA. Free vibration of immersed column carrying a tip mass. J Sound Vib 1998;216(1):147–57. [71] Kramer MR, Liu Z, Young YL. Free vibration of cantilevered composite plates in air and in water. Compos Struct 2013;95:254–63. [72] Green CP, Sader JE. Torsional frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope. J Appl Phys 2002;92 (10):6262. [73] Sader JE. Frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope. J Appl Phys 1998;84(1):64–76. [74] Huang W, Zou Y, Zhu P. On the dynamic response of a fluid-supported elastic plate subjected to a low velocity projectile impact. JSME Int J Ser C Mech Syst Mach Elem Manuf 1997;40(2):346–52. [75] Huang W, Li Y, Chen W. Analysis of the dynamic response of a fluid-supported circular elastic plate impacted by a low-velocity projectile. Proc Inst Mech Eng C J Mech Eng Sci 2000;214(5):719–27. [76] Kwon Y, Conner R. Low velocity impact on polymer composite plates in contact with water. Int J Multiphys 2012;6(3):179–98. [77] Kwon Y, Owens A, Kwon A, Didoszak J. Experimental study of impact on composite plates with fluid–structure interaction. Int J Multiphys 2010;4(3):259–71. [78] Owens AC, Didoszak JM, Kwon AS, Kwon YW. Underwater impact of composite structures. In: ASME 2010 pressure vessels and piping division/K-PVP conference; 2010. p. 231–40. [79] Kwon YW, Violette MA, McCrillis RD, Didoszak JM. Transient dynamic response and failure of sandwich composite structures under impact loading with fluid structure interaction. Appl Compos Mater 2012;19(6):921–40. [80] Kwon YW, Violette MA. Damage initiation and growth in laminated polymer composite plates with fluid–structure interaction under impact loading. Int J Multiphys 2012;6 (1):29–42.

Impact response of advanced composite structures reinforced by carbon nanoparticles

7

S. Laurenzi*, M.G. Santonicola*,† *Sapienza University of Rome, Rome, Italy, †University of Twente, Enschede, The Netherlands

7.1

Introduction

The impact response of structural components is a long-standing issue in many technological fields. Particularly for fiber/polymer composites, the structural resistance under impact events is a critical design requirement in rotorcraft, aircraft, and automotive. In these fields, polymer-based composite laminates are increasingly used due to the significant advantages in terms of weight saving, which directly translates into reduced fuel costs, and to the superior specific energy absorption ability with respect to metals. In general, structural elements are designed to absorb the impact energy in a controlled manner: metals by their plastic deformation, polymer composites according to their stress-strain relationships. The energy dissipation of advanced composite structures occurs through specific failure modes, which include fiber fracture, fiber pull-out, fiber/matrix debonding, crack bridging, and matrix cracking [1,2]. Further expressions of energy absorption in structural composites are internal damping and fracture toughness, which is a measure of the energy required for crack initiation and growth. The introduction of carbon nanoparticles (carbon nanotubes and nanofibers, graphene nanoplatelets) into advanced composite materials offers the potential for a simultaneous improvement of several properties, including fracture toughness and energy absorption during impact. When the dimensions of the filler approach the nanometer scale, the multifunctional properties of the corresponding composite structures are modified and new failure mechanisms arise from the interaction between the nanoparticles and the matrix [3]. In this chapter, we review the current progress in structural behavior of advanced fiber composites with the thermosetting epoxy-based matrix reinforced by carbon nanoparticles (carbon nanotubes and nanofibers, graphene nanoplatelets) under impact. The different energy absorption mechanisms and failure modes of the nano-reinforced composite structures as related to the presence of carbon nanoparticles are analyzed. These mechanisms depend on several characteristics of the nanoparticles, such as stiffness, shape and aspect ratio (AR), volume fraction, surface/interfacial adhesion. Experimental evidence of the enhanced energy absorption by the composite structures due to the presence of the nano-reinforcement is described differentiating the results Dynamic Response and Failure of Composite Materials and Structures. http://dx.doi.org/10.1016/B978-0-08-100887-4.00007-X © 2017 Elsevier Ltd. All rights reserved.

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obtained in low impact and high impact energy tests. Original results on advanced carbon fiber structures based on epoxy resins (marine-grade and aerospace-grade) reinforced by multiwalled carbon nanotubes (MWCNTs) are presented.

7.2

Overview of carbon nanoparticles for advanced composite structures

For the past two decades, academic and research organizations as well as industry have investigated novel synthetic routes and processes for carbon nanoparticles with the expectation of exploiting their relevant multifunctional properties in advanced composite structures [4]. To an extent depending on the allotropic form, carbon nanoparticles show high thermal and electrical conductivities [5–7], superior mechanical properties with respect to other materials with structure in the micro- and nanoscales [8–10], and sometimes exotic (nonlinear) optical properties that make them useful, for example, in the manipulation of laser beams and optical limiting devices [11,12]. However, the exploitation of the nanoscale carbon properties after integration of the nanoparticles in polymer matrices is not straightforward, and composite systems with poor structural properties can be fabricated due to the weak interface between nanoparticles and the surrounding matrix. For these reasons, in the last years much research effort has been devoted to the development of technological processes leading to the effective inclusion of carbon nanoparticles inside polymer matrices to obtain advanced composite structures with tailorable properties.

7.2.1

Carbon nanoparticles: From nanotubes to graphene nanoplatelets

The most investigated allotropic forms of carbon nanoparticles for multifunctional epoxy-based composites are carbon nanotubes (CNTs) and graphene nanoplatelets (GNP) [13–15]. CNTs consist of rolled sheets of graphene forming hollow cylinders. Carbon nanotubes can be fabricated as single-walled nanotubes (SWCNTs) or as multiconcentric hollow nanotubes (MWCNTs) interacting via van der Waals forces [16]. Further, SWCNTs can be armchair, zigzag, and chiral types, depending on the arrangement of hexagons along the wall. These nanoparticles can be synthesized as straight cylinders or in different forms, such as Y-shape or helically coiled carbon nanotubes (HCNTs) [17,18]. MWCNTs have been more widely utilized for the development of enhanced hierarchical laminated composites due to their low cost, commercial availability in large quantities, and ease of dispersion in polymer matrices with respect to the SWCNTs. From one side, SWCNTs offer superior performance due to their smaller diameter, and therefore higher surface area and higher AR. For these nanotubes a more efficient load transfer due to the molecular structure, resulting in superior mechanical properties, can be observed. On the other hand, SWCNTs are difficult to disperse and surface functionalization strategies are often required to facilitate their inclusion in composite materials [19,20].

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Graphene nanoplatelets consist of single- to few-layer sheets of sp2-bonded carbon atoms overlaying to form two-dimensional particles with thickness in the nanometer scale. GNPs and graphene sheets can be produced at low cost from the direct exfoliation of inexpensive graphite flakes, for example by ultrasonication in acid bath, followed by the chemical oxidation and then reduction of the graphite oxide nanoplatelets [21,22]. Other production methods for graphene sheets involve mechanical exfoliation, epitaxial growth, chemical vapor deposition, or bottom-up molecular synthesis [23].

7.2.2

Functionalization strategies for carbon nanoparticles

Despite the exceptional properties of carbon nanoparticles, ranging from high elastic modulus and tensile strength to superior thermal conductivity, enhancing the multifunctional properties of carbon-reinforced epoxy resins, for example by single-walled or multiwalled CNTs, is largely unpredictable due to their strong dependence on the manufacturing process [24]. This aspect is especially crucial in advanced structural applications, where manufacturing processes need to be applied to large-scale components. One of the long-standing issues in this context is related to the efficiency of dispersion of carbon nanoparticles inside the epoxy matrix, which can greatly affect the mechanical performance of final nanocomposites [25–27]. To improve their dispersion in epoxy resins, CNTs are commonly functionalized following covalent strategies [28,29]. Chemical treatments, by which carboxylic and amine functional groups are attached to the nanotube walls, make the CNTs more compatible with the epoxy polymer matrix, thus improving the homogeneity of the dispersion. However, these surface modifications, which are generally accomplished through strong acid attack, inherently modify the carbon structure causing a degradation of the overall nanoparticle properties so that the benefits added by the more homogenous dispersion are in fact lost in the final composite material. A different functionalization approach for carbon nanoparticles consists in the noncovalent binding of dispersing agents, most notably surfactants [30] or other amphiphilic molecules, which provide a resin-compatible shell onto the surface of the nanoparticles. In previous works from our group, we have demonstrated the possibility to obtain a homogeneous dispersion of MWCNTs in an epoxy resin using a noncovalent functionalization based on DNA wrapping [31,32]. The advantage of the noncovalent approach is that MWCNTs maintain the intact features of pristine nanotubes.

7.3

Energy absorption mechanisms related to carbon nanoparticles in composite structures

In the last two decades, we have learnt that conventional failure criteria in structural design cannot be transferred directly from metallic materials to composite materials. In fact, composites are characterized by substantially different failure mechanisms than metals, and therefore new failure models have been developed over time [33–35]. The same consideration must be afforded to the design of composite structures containing carbon nanoparticles. The high specific surface areas (SSAs) and the

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nanoscale dimensions of these particles induce different failure modes with respect to standard composite structures [3], which are dominated by particles-particles, particles-matrix, and particles-fibers interactions. However, such interactions are still not completely clear, and therefore more work is necessary to understand the failure mechanisms associated with carbon-based nanocomposites. The identification of the failure modes will yield information regarding the residual strength of the composite structures. Understanding the interaction between failure modes at the level of microand nanoscales is also very important to model and prevent damage mode initiation and propagation. During impact events, the absorbed energy is defined as the total amount of energy dissipated by the composite structure. The impact energy can be spent through several absorption mechanisms depending on the structured constituents and their interaction. It is well known that composite structures are extremely susceptible to crack initiation and propagation in various failure modes. The principal failure modes of composite materials are matrix failure, delamination, fiber failure, and penetration [1,35–37]. In other words, in composite structures the impact energy is mainly dissipated through damage formation, differently from metals in which energy absorption occurs by plastic deformations. The initial fracture modes, especially at low and medium impact velocities, are related to the matrix nature, in particular for crack initiation and delamination. These failure modes are essentially due to the brittle behavior of thermosetting polymers, leading to a severely poor resistance toward crack initiation and propagation. In fact, delamination is one of the most prevalent crack growth modes in composites and it can cause significant reductions of strength and stiffness that can lead to the catastrophic failure of the entire structure. The increase of the fracture toughness of the matrix will lead to a higher damage initiation threshold and to a long-term reliability of the composite structures. Many researchers have shown that the enhancement of the fracture toughness of the polymer matrix may be achieved by particles’ inclusion [38–40]. Viana [38] and Fu and coauthors [39] summarized in all-encompassing reviews the toughening mechanisms arising from the addition of several micro- and nanoparticles, highlighting the different effects due to the soft or rigid nature of the particles, and to the molecular structure of the polymer (crystalline, semicrystalline, and amorphous). In addition, size, concentration, and AR of the particles have been pointed out as important aspects that influence the matrix toughness and thus the fracture failure modes of the overall composite material [41–43]. For composite structures reinforced with nanoparticles the toughening mechanisms occur at two different dimensional levels. At microscopic level, the toughening mechanisms in the epoxy matrix are crack deflection due to the presence of agglomerates, crack pinning, and crack blunting. Other toughening mechanisms that are usually detected on the microscopic scale are inelastic matrix deformation and nucleation of voids, interfacial debonding of fibers or particles from the matrix, pull-out, and crack bridging related to the reinforcement (fiber or particle). These aspects are also relevant at the nanoscale level, for example in the presence of carbon nanotubes, as illustrated in Fig. 7.1.

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(A)

(B)

(C)

(D)

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(E)

Fig. 7.1 Schematic description of possible fracture mechanisms of CNTs: (A) initial state of the CNT; (B) pull-out caused by CNT/matrix debonding in case of weak interfacial adhesion; (C) rupture of CNT—strong interfacial adhesion in combination with extensive and fast local deformation; (D) telescopic pull-out—fracture of the outer layer due to strong interfacial bonding and pull-out of the inner tube; (E) bridging and partial debonding of the interface— local bonding to the matrix enables crack bridging and interfacial failure in the nonbonded regions. Reprinted with permission from Gojny FH, Wichmann MHG, Fiedler B, Schulte K. Influence of different carbon nanotubes on the mechanical properties of epoxy matrix composites—a comparative study. Compos Sci Technol 2005;65(15–16):2300–13.

Nanoparticles have dimensions of the same order of magnitude as single molecules and, therefore, they interact with the surrounding polymer matrix in specific ways that lead to different properties of the corresponding nanocomposites with respect to more traditional composites. Some toughness mechanisms may be not relevant for nanocomposites, whereas other ones lead to high energy absorption of the composites. The main factors affecting the energy absorbing capability by advanced composite structures loaded with carbon nanoparticles include the mechanical properties of the nanofiller, its size, dispersion, and volume fraction, the nature of the polymer matrix material, the interfacial characteristics between filler and matrix, and the manufacturing process [3]. In the following sections, the different nanoparticle features that influence the energy absorption mechanisms of thermosetting composites reinforced with carbon nanotubes and graphene nanoplatelets are reported.

7.3.1

Role of mechanical properties of carbon nanoparticles

In general, the mechanical properties of nanoparticles in a composite material greatly affect the material energy absorption capability. In particular, the stiffness of the nanoparticles can modify the impact toughness and the modulus of elasticity of composite materials [38]. The fracture toughness of a material represents the amount of

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energy required for propagation of a crack in the structure [44]. Increasing the fracture toughness of the matrix implies that more energy is necessary to create new fracture surface area. As a consequence, the resistance of the composites to delamination and crack propagation is increased as well. Many researchers have demonstrated that carbon-based nanoparticles can act as toughening agents for thermosetting polymers as a consequence of their high ARs and excellent mechanical properties [15,44–50]. The modulus of elasticity of the nanoparticles is the mechanical property that mostly affects the toughening mechanisms in a composite material. In particular, the high stiffness of the nanoparticles increases the possibility to dissipate energy by crack bowing and crack deflection. Crack bowing is a toughening mechanism in which the crack bows among the rigid nanoparticles forming a nonlinear crack front that anchors on them. As a consequence, a crack pinned by two impenetrable particles needs more energy to propagate if its front is lengthened by bowing [39]. An additional toughening mechanism, which increases the fracture toughness of the nanocomposites with respect to the neat matrix, is crack deflection around the nanoparticles with consequent energy dissipation in the damage zone. The elastic modulus and the thermal expansion coefficient of the nanoparticles are the main factors promoting the crack deflection phenomenon. In this mechanism, the crack tip can be deflected as a consequence of the nanoparticles hindrance. The nanoparticles in the matrix force the crack to move out of the initial propagation plane by tilting and twisting, thus changing the stress state from mode I to a mixed mode. In particular, if the crack tilts, the crack propagates under mixed mode I/II, that is with a combination of tensile and in-plane shear stress states. If the crack twists, the crack propagates under mixed mode I/III, that is with a combination of tensile and out-of-plane shear stress states. The crack propagation under mixed mode conditions requires a higher driving force than in pure mode I, which results in a higher fracture toughness of the material. This energy dissipation mechanism is amplified as the nanoparticles form a three-dimensional network in the matrix: tilting and twisting continue in a chain reaction as the crack tip meets new nanoparticles. The crack deflection acts not only on the matrix crack mode but also on the delamination mode. In fact, the mode I and mode II interlaminar fracture toughness are important indicators of the ability of the composite laminate to resist to the interply delamination under a normal force perpendicular to the crack plane (mode I) and under a shear force (mode II). Based on the above considerations, it is clear that carbon nanotubes, which show high AR and very large elastic modulus (order of magnitude of the TPa) with respect to other reinforcing materials, can contribute significantly to improve the energy absorption capability of the advanced fiber-reinforced composite structures. A number of studies have showed improvements in fracture toughness of polymer nanocomposites with small CNT contents, often less than 1 wt% [45,46,50–52]. As an example, Gojny et al. found that 0.1 wt% of pristine SWCNT are sufficient to increase the fracture toughness of an epoxy matrix of 23%, whereas 0.3 wt% of pristine DWCNTs or pristine MWCNTs increase the fracture toughness of about 30% [45]. Recently, Herceg and colleagues obtained an impressive increment of the mode I interlaminar fracture toughness, approximately 152%, using 20 wt% of MWCNTs [42].

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Similar results in improving the fracture toughness of nano-reinforced composite materials were found by adding graphene nanoplatelets to the matrix [53–55]. Bortz and coauthors [54] determined that 1 wt% of graphene oxide increases the mode I fracture toughness of about 111% and up to 158% in uniaxial tensile fatigue life. Further, it was observed that graphene nanoplatelets prevent the matrix crack propagation due to their two-dimensional arrangement forming a network. In particular, the researchers noted the presence of crack pinning, which is a toughening mechanism usually not detectable with nanoparticles and nanofibers, because their relative size is much smaller than the crack tip opening displacement. Crack pinning occurs when a propagating crack encounters a series of impenetrable objects, and so the crack front bows out between the particles while remaining pinned at the particles. Chandrasekaran et al. determined the fracture toughness of MWCNT/epoxy, GNP/epoxy, and graphene oxide/epoxy at different loadings [43]. They found that MWCNTs generate a maximum increment of 8% of fracture toughness at 0.5 wt% loading, whereas GNPs are responsible for an increment of 24%, which is three times higher in percentage when compared to MWCNT-reinforced epoxy. As additional result, they observed that for graphene oxide this increment reaches the value of 40%.

7.3.2

Role of nanoparticle size, shape, and concentration

The nanoparticle size and shape play an important role in the development of the energy absorption capability of nano-reinforced composite structures influencing the formation of different toughening mechanisms. The concentration of the nanofiller dispersed in the polymer matrix is also an important aspect, given that it determines the interparticle distance and, therefore, sets their mutual interactions. These aspects (size, shape, and concentration) act together governing the properties of the composite materials loaded with nanoparticles [56]. Fig. 7.2 illustrates the relationship among particle size, particle number, and total surface area in a composite with homogeneous particle distribution [48]. It can be observed that, if considering spherical microparticles with diameter of 10 μm, only three particles would be present at the loading of 3 vol%. Instead, nanoparticles with diameter of 100 nm will be roughly 3,000,000 at the same particle loading, thus presenting a much larger interface area [57]. It is quite intuitive that the toughening mechanisms and the overall composite properties are quite different in the case of nanoparticles loading. The AR and the SSA are distinctive characteristics of the nanoparticles, which derive directly from their shape and size [58]. Often, researchers refer to the nanoparticle AR and SSA to explain some peculiar behavior of the nanocomposites. Indeed, AR and SSA establish in some ways the nanoparticle capacity to be well dispersed in a polymer matrix. For example, carbon nanoparticles with higher SSA are known to be more difficult to disperse as the hydrophobic interactions that drive particle agglomeration act on a larger surface area [25]. Another consequence of the higher SSA of nanoparticles is that a propagating crack front interacts with more particles in the nanocomposite than in the microcomposites, and therefore the fracture mechanical properties are modified [48]. Further, as consequence of their size, nanoparticles interact with the matrix influencing its microstructure and deformation mechanisms in a significant way.

Dynamic Response and Failure of Composite Materials and Structures

108 107

105 104 103

e

tanc

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~d

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101 100,000

Fig. 7.2 Correlation between the particle size (d), the absolute number of particles (n), the interparticle distance (2c), and the resulting interface area, which depends on the particle surface (O). The filler volume content is 3 vol%. Reprinted with permission from Wetzel B, Rosso P, Haupert F, Friedrich K. Epoxy nanocomposites—fracture and toughening mechanisms. Eng Fract Mech 2006;73 (16):2375–98.

The magnitude of this effect depends on the nature of the interaction between nanoparticles and matrix. Carbon nanotubes and graphene nanoplatelets are particles with large AR, which increases the possibility that certain toughening mechanisms occur. For example, with the increase in the AR, the crack bowing and the crack deflection mechanisms increase. In particular, nanoparticles with higher AR twist the crack front more than nanoparticles with lower AR, as illustrated in Fig. 7.3. As mentioned above, the crack propagates under mixed mode conditions requiring a higher driving force than in pure mode I, which results in a higher fracture toughness of the material. Similarly, the SSA contributes to toughening mechanisms. Graphene nanoplatelets can induce crack pinning mechanisms that usually are not detectable in composite structures reinforced with other nanoparticles such as CNTs [53]. These contributions arise from the size and shape of GNPs, which approach the two-dimensional case with 0-thickness. However, these characteristics (AR and SSA) limit the use of carbon nanoparticles above a certain concentration. In fact, with the increasing of carbon nanoparticles loading into the polymer matrix, the interaction between nanoparticles boosts, producing two negative effects: (1) the formation of micrometer entanglements acting as defects inside the matrix; (2) the rapid increase of viscosity of the nano-reinforced resin, which limits the manufacturing processes [59]. Fig. 7.4 shows a comparative study on the normalized fracture toughness of nanocomposites containing nanoclay, MWCNTs, and graphene as a function of wt% extrapolated from the literature [43]. MWCNT-reinforced epoxy resins show an

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AR = 5 Undeflected crack front

AR = 10 Undeflected crack front

Normalized fracture toughness (KIC/KIC(pure epoxy))

Fig. 7.3 Schematic of dependence of deflection crack front on nanoparticle aspect ratio (AR).

2.0

Graphene cluster Nanoclay & its types

1.8

M.R. Ayatollahi, 2011 K. Wang, 2005 B.C. Kim, 2008 G. Swaminathan, 2011 A.G. Morales, 2009

1.6 1.4

Carbon nanotubes & its types

1.2

H.D. Wagner, 2010 F.H. Gojny, 2005 I.A. Kinloch, 2011 M.R. Ayatollahi, 2011

1.0 0.8 0.6

Nanoclay cluster Carbon nanotube cluster

Graphene & its types D.R. Bortz, 2012 S. Chatterjee, 2012 L.C Tang, 2013 I. Zaman, 2012 M.Rafiee, 2010 S.Chandrasekaran et.al., 2013

0.4 0.2 Nao-filler reinforced epoxy based composite 0.0 0.01

0.1

1

10

Nano-filler content (wt%)

Fig. 7.4 Normalized fracture toughness as a function of filler content for different nano-reinforced epoxy resins. Reprinted with permission from Chandrasekaran S, Sato N, T€ olle F, M€ ulhaupt R, Fiedler B, Schulte K. Fracture toughness and failure mechanism of graphene based epoxy composites. Compos Sci Technol 2014;97:90–9.

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improvement of mode I fracture toughness (KIC) at lower filler content (0.1–1.0 wt%) compared to nanoclay composites. Graphene-based epoxy composites show even larger improvement in KIC and at lower filler content, starting at 0.01 wt%.

7.4

Experimental studies of impact response of nano-reinforced composite structures

In conventional composite structures, the absorption mechanisms after an impact event are generally attributed to the following failure modes: through-thickness matrix cracks, multiple delaminations between adjacent plies, fiber breakage, and fiber pull-out. In the presence of carbon nanoparticles, additional dissipation failure modes, as described in the previous section, are involved in the absorbing energy process. The identification of each contribution to the impact response is quite impracticable. However, general differences in the impact response between carbon nano-reinforced composites and equivalent ones without nanofillers can be noticed. In particular, a different behavior is observed depending on the impact energy level. In the following sections, the impact response of nano-reinforced composite structures, differentiating between the behavior under low and high velocity impact events, is analyzed.

7.4.1

Low velocity impact response

Low velocity impacts are common events for advanced composite structures that are routinely used in several civil and military fields. As an example, aerospace structures can suffer from accidental bird strikes and runway debris, tool drops during assembling and maintenance, and so on [60,61]. In most of the cases, the damage is not detectable by visual inspection on the top surface of the structures because the impact energy is not sufficient to penetrate the composite laminate. However, possible damage may arise in the underlying layers especially due to matrix cracking and delamination initiation. Carbon nanoparticles are very useful elements to increase the energy threshold for crack initiation, thus improving the impact strength and the postimpact behavior of the composite structures. The inclusion of rigid nanoparticles with high AR, such as nanotubes, has opened new avenues in the modification of the structural properties of polymer matrices, which are inherently characterized by a weak response to impact. For instance, by using the low-velocity and low-energy Charpy or Izod tests, many authors have demonstrated that the introduction of carbon nanotubes or fibers and, more recently, graphene nanoplatelets increase significantly the energy absorption capability of nano-reinforced polymer materials [46,47,62–64]. This statement holds true for a certain range of nanoparticle concentration. As the loading increases, different issues arise due to nanoparticle entanglement and agglomeration, which severely affect the homogeneous dispersion of the nanoparticles in the polymer matrix. Large agglomerates do not constitute a reinforcement for the structure, but rather act as points of defect where micro-cracks can form.

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In order to investigate the effect of nanoparticle functionalization on the impact resistance of epoxy-based composite materials, we determined the impact strength of composites made of Prime 20LV epoxy (marine-grade) reinforced with pristine and functionalized MWCNTs using Izod test. Covalent (MWCNT-COOH) and noncovalent (MWCNT-DNA) modifications of the nanotubes were tested. Fig. 7.5 summarizes the average values with corresponding standard deviations of the impact strength for the different nanocomposites. The average impact strength of the pristine MWCNT/epoxy composites is about 12% larger than the impact strength of the neat epoxy. A significant difference can be noted in the values of the standard deviations among the different samples. Izod tests were highly reproducible for the neat resin, whereas the pristine MWCNT/epoxy composites present a larger scattering of the data. This phenomenon is due to the fact that pristine MWCNTs are difficult to disperse in the epoxy matrix. In fact, they form large entanglements that act as point of defects inside the composite instead of reinforcing it. As a consequence, the spatial distribution of both entangled MWCNTs and disentangled MWCNTs, which can be located in the bulk, at the surface, near the surface, or at an edge, may facilitate crack initiation and propagation. Hence, the impact strength may be dramatically different from specimen to specimen. For these samples, test reproducibility depends on the quality of the nanoparticle dispersion, meaning the capacity to create a disentangled MWCNT network inside the matrix. When using nanotubes functionalized with carboxylic groups (MWCNT-COOH), the dispersion process is improved and, as a consequence, the data scattering of the Izod tests is strongly reduced (Fig. 7.5). In the case of epoxy composites containing DNA-functionalized MWCNTs, there is also a good reproducibility of the test data with low values of the standard deviation, Fig. 7.5 Impact strengths by Izod test of neat epoxy resin (Prime 20LV), pristine MWCNT/epoxy composites (0.5/100 by weight), MWCNT-COOH/epoxy composites, and MWCNT-DNA/epoxy composites (0.5/100 by weight) prepared with two different cure steps (50°C for 13 h and 75°C for 8 h).

3.0

Impact strength (KJ/m2)

2.5 2.0 1.5 1.0 0.5 0.0 ) ) e °C °C tin OH ris -CO 50 (75 ( p T T NA NA CN WCN T/D NT/D N MW M C C MW MW

sin

t re

a Ne

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Dynamic Response and Failure of Composite Materials and Structures

indicating that the DNA-functionalized MWCNTs are well dispersed inside the epoxy matrix. On the other hand, at first analysis, the DNA functionalization seems to decrease the mechanical properties of the composite from the point of view of absorbed energy. In fact, the impact strength of the MWCNT-DNA/epoxy composites is lower of about 18% with respect to the case of the neat resin. However, further investigations have shown that such result can be related to the different polymerization grade reached by the specimens during the cure step. A preliminary isothermal analysis conducted by differential scanning calorimetry at 50°C revealed that the MWCNT-DNA/epoxy composites require double time to reach the peak of the reticulation than for the neat resin. Therefore, the tested MWCNT-DNA/epoxy specimens were characterized by a lower grade of polymerization with respect to the neat resin, considering that all specimens had the same thermal history. The pristine MWCNTs also showed a certain delay in the polymerization, but the peak of polymerization is near the point of the neat resin. After the Izod impact test, the fracture surfaces of the nanocomposites and of the neat epoxy samples were imaged by optical microscopy (Fig. 7.6). The neat epoxy shows the distinctive smooth surface of the glassy polymer with radial striations on the fracture surface (Fig. 7.6A). The striations converge to the zone of the crack initiation site, that is the area of the hammer impact. Similar features are visible in the fractured surface of pristine MWCNT/epoxy composites (Fig. 7.6B). On the other hand, the inclusion of DNA-functionalized MWCNTs significantly modified the morphology of the fracture surface, in that the fracture surface appears to have a large degree of roughness. This rough fracture surface can be explained by crack deflection and continual crack propagation occurring on two slightly different fracture planes, which can be due to the presence of a large quantity of nanotubes. The crack tips are then forced to change their propagation direction frequently. As a result, a fracture morphology characterized by short and highly curved patterns of the crack propagation was revealed. The performance of composite structures after low velocity impact can be evaluated using several parameters such as the delamination area, the peak response force,

1 mm

1 mm

(A)

1 mm

(B)

(C)

Fig. 7.6 Optical micrographs of the fracture surface of nanocomposite samples after Izod test. (A) Neat epoxy, (B) pristine MWCNT/epoxy composites (0.5/100 by weight), and (C) MWCNT-DNA/epoxy composites (0.5/100 by weight).

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the absorbed energy, the compression-after-impact (CAI) strength, and the effective modulus and fatigue life after impact. Recently, we investigated the low-energy impact behavior of advanced carbon fiber/RTM6 epoxy laminates, which are typically used in aerospace applications, with and without the addition of functionalized nanotubes MWCNT-COOH to the resin, using the dropped-weight method. For the tests we used an experimental setup consisting of an in-house drop tower system capable of simulating low velocity impact conditions at different low energy levels. The performance evaluation was linked to the observation of the samples behavior during the tests, followed by microscopy inspection of the fracture surfaces. Results of the low-energy impact tests on the reference laminates and the equivalent ones reinforced by 0.5 wt% of MWCNT-COOH are reported in Fig. 7.7, showing that the damage areas of MWCNT-COOH-reinforced laminates are always smaller than the reference plates. In particular, the damage areas are almost identical for energy levels up to 6 J, but they differ significantly at higher energy values, reaching a difference of about 45% at 20 J. The smaller delamination area of the MWCNT-COOH-reinforced composite laminates with respect to the reference composites is a clear indication of the onset of additional energy absorption mechanisms due to the presence of MWCNTs in the resin matrix. Similar results were found by Kostopoulos et al. adopting pristine MWCNTs as reinforcement [65]. In particular, the authors found that the time response under dynamic impact loads in the range of 2–20 J is almost the same for carbon fiber-reinforced laminates without and with additional 0.5 wt% of MWCNTs in the matrix. However, the analysis of the absorbed energy highlighted that the

350 Reference plate Reinforced plate (MWCNT-COOH)

Average damage area (mm2)

300 250 200 150 100 50 0

0

2

4

6

8

10 12 14 Energy (J)

16

18

20

22

Fig. 7.7 Average damage area as a function of the impact energy for aerospace-grade carbon fiber/RTM6 composite laminates. The red line refers to carbon fiber composites with neat RTM6 epoxy (reference plate) and the black line is for carbon fiber composites with RTM6 epoxy reinforced by 0.5 wt% of MWCNT-COOH.

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MWCNT-modified laminates absorb more energy upon orthogonal impact on the surface, and that the difference of absorbed energy between the MWCNT-reinforced and the reference laminates increases with the level of the impact energy. Further, the difference of delamination area between the two sets of laminates also increases with the impact energy level. In the same work, the authors introduced a specific delamination energy parameter in order to discern the contribution of the carbon nanotubes to the absorbed energy from the rest of the composite material. This parameter was defined as the energy required to develop a unit delamination area observing the C-scan images [65]. Comparing this parameter for the CTN-doped composite laminates and for equivalent ones without nanoparticles, the authors found appreciable contribution of the CNT mechanism above a certain level of impact energy (>16 J). The composite structures reinforced by carbon nanoparticles show higher resistance to damage as indicated by the increase in threshold impact energy. The CAI modulus and the CAI strength typically increase with the addition of the carbon nanotubes. For instance, the addition of 0.5 wt% of carbon nanotubes to a carbon fiber composite material can enhance the CAI modulus of 15% after an impact event of 16 J [65]. An additional benefit in terms of reduction of the impact damage area in advanced carbon fiber/epoxy laminates has been observed when adding small amounts of SWCNTs instead of MWCNTs. For example, Ashrafi et al. found that the incorporation of 0.1 wt% of SWCNT resulted in a 5% reduction of the impact damage area, meaning that the impact is more localized [56]. This result was likely due to the higher mechanical properties of the SWCNTs with respect to the multiwalled ones, even though SWCNTs are more difficult to disperse in the polymer matrix due their larger SSA. The authors measured a 3.5% increase in the CAI strength, a 13% increase in mode I fracture toughness, and 28% increase in mode II interlaminar fracture toughness. Further analyses by SEM showed that SWCNTs contribute to the increased fracture toughness by crack bridging in mode I and by the formation of larger hackles in mode II [56]. A similar behavior, that is same time response under low velocity impact but reduced impact damage area, was observed in carbon fiber/epoxy laminates reinforced by additional carbon nanofibers (CNFs) at loadings of 5 and 10 wt% [66]. For this system also, the authors found an improvement of failure modes I and II of the fracture toughness and an increase of 8% of the CAI strength at 10 wt% of CNFs.

7.4.2

High velocity impact response

High velocity and hyper velocity impact events are important phenomena in the military and aerospace fields. For example, spacecraft traveling in the outer space or satellites orbiting around the Earth may encounter orbital debris and micrometeorites with an average velocity of 10 km s 1. At these incident impact energies, the target perforation occurs and the passage of the impactor will generally result in petalling, cracking, and spalling. Experimental work on the high velocity impact response of composite structures containing carbon nanoparticles is really few in the literature. Graphene nanoparticles

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have been recently studied as a part of hybrid composite laminates [67], showing enhancement of the energy absorption capacity. More work is available for carbon nanotubes-reinforced composite materials. Laurenzi et al. [46] investigated the ballistic impact response of Kevlar 29/epoxy laminates loaded with 0.5 wt% of ammine-functionalized MWCNTs. The composite laminates were tested by high energy impact tests with projectile velocity in the range of 490–950 m s 1. The energy absorption capability was investigated by analyzing the evolution of the damage after three consecutive impact events on the same specimen. The nano-reinforced panels showed improved ballistic behavior in comparison with the reference ones. In particular, it was observed that the nano-reinforced composites showed a reduced penetration depth of the first shot. This behavior was explained considering additional dissipation mechanisms related to the dynamic nature of the high velocity impact test. When the contact between the projectile and the target occurs, shock waves propagate through the material and a certain amount of energy is converted into vibration. The MWCNTs act as a network of spring dampers increasing the damping capacity of the composite material and controlling the damages of the laminates. In addition, the nano-reinforced panels showed smaller damage areas than reference panels, and the projectile fragmentations were restricted in a more localized area. Further, Pandya at al. [68] determined that the loading of epoxy matrix with 0.5 wt% of MWCNTs enhanced the experimental ballistic limit velocity V50 up to 5% and the energy absorbed up to 10% with respect to neat epoxy. Shear plugging of the target is the major energy absorbing mechanism for different target thicknesses in the range of parameters considered in that work. More interesting results were obtained by Daraio and coauthors, who investigated the impact response of a forest of vertically aligned coiled carbon nanotubes (HCNTs) under normal impacts [69,70]. The HCNTs responded to dynamic loading as perfect nonlinear elastic springs that fully recover their original lengths when the impact force is removed. In fact, HCNTs were able to efficiently absorb impact energy both at low and high velocity impacts (0.01–5 m s 1) and fully recover deformation on the order of 5% strain. Further, by comparing the impact response of HCNT with vertically aligned carbon nanotubes (VACNTs), it was highlighted that HCNT foams also exhibit better impact absorption characteristics as compared to VACNT, suggesting that their use would greatly improve the performance of advanced protective materials for energy dissipation and impact absorption. This superior absorption capability has been explained by the fact that the individual nonlinear deformation of single HCNT collectively leads to a strong nonlinear and nonHertzian contact interaction with the impactor.

7.5

Conclusions

In summary, developing a clear view of the impact response behavior of advanced composite structures requires an understanding of the load-transfer mechanisms between nanofiller and polymer matrix at nanoscale level. Depending on the specific type of carbon nanoparticles, the fracture toughness of nano-reinforced composites can be related to the geometrical features of the fillers, most notably the AR and the SSA.

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Experimental data on the low-energy impact response of advance composite structures highlights that the time responses of carbon fiber/epoxy laminates reinforced with carbon nanoparticles is almost the same of their equivalent counterpart. However, in general the composites reinforced by CNTs present smaller damage areas suggesting the presence of additional absorbing mechanisms due to the action of the nanoparticles. The lack of matrix toughness, which governs the primary damage initiation modes in a composite structure (matrix cracking and delamination), can be overcome by several toughening mechanisms related to the presence of the nanoparticles. Generally, in the experimental results from high velocity impact events, the role of carbon nanoparticles in the energy absorbing ability by the composite structure is less evident. Nonetheless, it can be observed that, as a result of the presence of the nanoparticles, the impact damage area is more localized in nano-reinforced composite laminates even for high velocity impacts. In the case of ballistic impact, the role of the fiber reinforcement on the impact strength appears to be predominant with respect to the nanoparticle one. This is also due to the relatively low amount of nanofillers (up to few wt%) that can be employed in the current manufacturing processes for advanced composite structures.

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Implosion of composite cylinders due to underwater impulsive loads

8

T. Qu, M. Zhou Georgia Institute of Technology, Atlanta, GA, United States

8.1

Introduction

In recent years, there has been increased interest in using composites for underwater marine structures because of the advantages provided by these materials. Composite materials offer alternatives to metals with reduced weight, improved corrosion resistance, and improved blast mitigation. Additionally, these materials have lower thermal, acoustic, and magnetic signatures compared with traditional metallic structures, which provide better stealth qualities. For these reasons, composites are currently used in several naval structures, such as submersibles, sonar domes, and hull sheathings. Despite recent advances in understanding the dynamic response of composite materials, several issues pertaining to the response of composite structures subjected to extreme loading conditions remain unresolved. One such condition is the implosion of cylindrical structures. Implosion is a process in which a submerged, sealed, thin-walled structure (diameter-to-thickness ratio D/t > 20) collapses inward onto itself and is damaged [1–7]. The primary driving force for the event is the difference between higher external water pressure and lower internal air pressure. Under quasistatic conditions, the external-internal pressure difference must exceed the critical buckling pressure of the structure to cause the sudden loss of stability. Under dynamic conditions, the rate and manner in which impulse is delivered externally are also very important. Impulses with short pulse durations or low peak pressures not capable of causing collapse can excite vibrations and lead to local imperfections in the structure. Implosion can be triggered by subsequent bubble pulses or via further exposure to subcritical loading due to damage or resonance. If the initial impulse is sufficiently strong, collapse may not be preceded by vibrations of the structure. The duration of the implosion process is on the order of milliseconds in underwater environment. At the onset of collapse, the walls of the collapsing structure gain inwards momentum and accelerate, causing adjacent fluid to move inward and the local hydrodynamic pressure to decrease. The eventual contact of the inner surfaces of the structure causes the inward propagating water to be highly compressed, resulting in rapid release of energy in the form of an intense outwardly radiating shock wave. This shock wave can damage nearby structures and, therefore, is of

Dynamic Response and Failure of Composite Materials and Structures. http://dx.doi.org/10.1016/B978-0-08-100887-4.00008-1 © 2017 Elsevier Ltd. All rights reserved.

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great concern. Of particular interest are the critical collapse pressure, failure mechanisms of the structure, and the intensity of the shock wave. Under dynamic conditions, the pressure-time history in the water domain is the superposition of different pressure fields, including initial shock wave, bubble pulses [1,3,8], or the pressure pulse emitted from the collapse of gas bubble generated when an explosive is detonated, reflections of the initial shock, and released impulse from the collapse site. Earlier studies of underwater implosion were performed on glass bottles [9,10] and metallic cylinders (i.e., submarine pipelines) [11–14] under various loading conditions to quantify the pressure pulse and investigate the plastic deformation. Most recent research related to underwater implosion has been driven by naval interest. Hydrostatically induced implosion is caused by quasi-static pressurization to the critical buckling pressure of a structure [3,15]. Experimentally [2,4–7,16,17], a pressure vessel filled with water was used to generate the purely hydrostatic loading and the implosion of composite tubes was studied using digital image correlation (DIC) with dynamic pressure transducers to relate collapse mechanics to the changes in local pressure fields. Explosive-induced implosion occurs through a combination of subcritical pressures and underwater explosive (UNDEX) loading [3,15]. Most recently, experimental studies on the implosion of composite cylinders subjected to a nearby explosion were performed. So far, most studies are limited to hydrostatically induced implosion and the limited instrumentation in the tests does not include any direct measurement of the physical values such as stress, strain, and energy, in an explosive-induced implosion event. Predictions capturing these mechanisms using numerical approaches that account for structural attributes and loading conditions are required. Recent developments in finite element models have focused on highly nonlinear fluid-structure interaction (FSI) problems with account of large deformation, self-contact, and materials damage [1,18–31], and were validated against the experimental data of the implosion testing conducted with thin-walled aluminum tubes [25–28]. This chapter provides a focused overview of studies in both hydrostatically induced and explosive-induced implosion of composite cylinders, followed by a recent effort in developing simple design rules for these materials under such extreme loading conditions.

8.2

Hydrostatically induced implosion

Under purely hydrostatic loading, the composite cylindrical structure tends to buckle and implode in a symmetric fashion into one of several possible mode shapes, as shown in Fig. 8.1. The mode shapes are determined through a linear buckling analysis conducted on carbon/epoxy composite tubes subjected to a uniform external pressure [2,7]. The buckling mode and the critical collapse pressure are affected by the length, diameter, and shell thickness of the cylinder, as well as material properties. Accounting for material anisotropy and nonuniform distribution of stress through the shell thickness, Koudela and Strait [32] derived the following solution, Eq. (8.1), to predict the buckling collapse pressure of laminated composite tubes under hydrostatic pressure,

Implosion of composite cylinders due to underwater impulsive loads

Mode 2 (L/D = 9)

Mode 3 (L/D = 5)

241

Mode 4 (L/D = 2)

Fig. 8.1 Symmetric collapse mode shapes for carbon/epoxy composite tubes subjected to purely hydrostatic loading.

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 0:85 E1 E2 t2:5 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : p¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ E1 E2 t 2 ð1 v12 v21 Þ0:75 L Ri Ro 1 + 12Ri Ro G

(8.1)

Here, Ri and Ro are the inner and outer radius of the tube, t is the shell thickness, L is the length of the tube. E1 and E2 are the longitudinal and transverse moduli, respectively, G is the shear modulus, ν12 and ν21 are Poisson’s ratios. Typically, cylinders with large length-to-diameter ratios (L/D) collapse in mode 2, while those with lower L/D ratios collapse in mode 3 or 4. However, it was observed in experiments that deformation of a composite cylinder with large L/D ratios initiates in a higher-order shape early in an implosion event [5]. Fracture may or may not occur during the implosion process, depending on material behavior, layup arrangements, and loading conditions. Several experimental studies have been performed to understand the failure and damage mechanisms of the collapse of composite cylinders during hydrostatically induced implosion events. Among the pioneers in the field are Kyriakides and Shukla along with their coworkers. These researchers have performed a series of experiments to investigate the collapse mode evolution [5] during implosion as well as the effect of structural attributes, such as different reinforcing fabric architectures [2,6,17], different L/D ratios [7], and different thicknesses of polymeric coatings [4], on the critical collapse pressure, failure behavior, and the damage potential of released pressure waves of carbon-fiber/epoxy and glass-fiber/polyester composite cylinders. The implosion experiments were performed in a spherical (or cylindrical [7]) pressure vessel [2,6], shown in Fig. 8.2, with a 2 m diameter and a pressure capacity up to 7 MPa to provide a constant hydrostatic pressure throughout the collapse event. Four windows were equipped at the midspan of the pressure vessel to provide light sources and allow optical monitoring of the experiment. High-speed cameras were placed outside the windows. The tested composite tube specimens were sealed with aluminum

242

Dynamic Response and Failure of Composite Materials and Structures

High speed camera

Lighting

Lighting

Pressure transducer Composite specimen ~2m

Aluminum caps Pressure vessel

Fig. 8.2 Schematic illustration of the experimental setup for studying hydrostatically induced implosion [2,4–6,17].

end caps and mounted in the center of the vessel to prevent rigid body motion. The speckle-pattern-painted surface of the specimen was faced toward the cameras to allow determination, in real-time, of the full-field deformation of the specimen using DIC. Dynamic pressure transducers were mounted at different locations about the specimen both axially and circumferentially. The vessel, initially filled with water, was subsequently pressurized with air at a rate of 0.083 MPa/min until it collapsed [2,4–6,17]. The specimen was 350 mm long, with a 70 mm inner diameter and 2 mm wall thickness. This dimension was selected to provide specimens with relatively low collapse pressure and a high diameter-to-thickness ratio so that thin-wall assumptions could be applied [2,4–6,17]. Fig. 8.3 shows a typical implosion pressure history measured about the midspan of the specimen. The negative impulse represents the decrease in pressure corresponding to an acceleration of the center of the tube. The positive impulse with a large spike represents the change in momentum to the surrounding fluid caused by the rapid deceleration of the water when the structure reaches its fully collapsed state. This indicates the released pressure pulse, which quantifies the damaging effect on the nearby structures during the implosion event. A normalized impulse was defined to evaluate the damage potential of the pressure wave [2,6]

Implosion of composite cylinders due to underwater impulsive loads

243

4 Pmax

Dynamic pressure (MPa)

3

Positive impulse

2 1

0 -1 -2 -1

Pmin Negative impulse -0.5

0 Time (ms)

0.5

1

Fig. 8.3 Plot showing a typical dynamic history about the midspan of a composite cylinder during hydrostatically induced implosion event [2,4–6,17].

ðt pdt I¼ ð t0 p0 dt

(8.2)

0

where p is the dynamic pressure and p0 is the hydrostatic pressure measured in experiment. To explore the damage mechanisms of the structure, a Fourier series-based solution was used to determine the evolution of the collapse mode number during the implosion event [5,33,34], uR ¼ a + b cos ðnφÞ + c sin ðnφÞ,

(8.3)

where uR is the radial displacement of the cylinder at a certain axial location as function of angular position; φ represents the angular position; a, b, c, and n are curve fitting parameters; and n is the collapse mode number to be determined. Presence of damage was pointed out to significantly influence the change of the mode shape as the structure deformed [5]. Experimental examinations showed that damage would initiate from different locations on tubes with different L/D ratios. For example [7], axial cracks were located on the extrados for larger L/D tubes and on the intrados for smaller L/D tubes. Further experiments [2,6,17] were conducted on both glass-fiber and carbon-fiber tubes with composite layers arranged in different architectures (i.e., filament-wound specimens, roll-wrapped specimens, and braided specimens). It was concluded that carbon-fiber composite tubes released higher pressure pulses during the implosion event as compared with glass-fiber composite tubes. Reinforcing architectures significantly affected the collapse nature of the specimens and its damage potential to nearby

244

Dynamic Response and Failure of Composite Materials and Structures

structures, that is, braided specimens experienced enhanced brittle failure and had a greater damage potential while filament-wound specimens were considered optimal in resisting damage evolution. To mitigate the released pressure pulse, composite tubes were coated with polyuria with different thicknesses [4]. The released pressure pulse was reduced for specimens with thick interior coatings and was enhanced for specimens with exterior coatings. To aid the understanding of the mechanism of the implosion phenomena, a complementary finite element model [20–23] based on coupled Eulerian-Lagrangian (CEL) approach was developed to simulate the experiment and was validated against the measured dynamic pressure profile of aluminum tubes [25–28]. Simulations have successfully captured the structural collapse progression during the implosion process, the resulting fluid motion, as well as the pressure waves’ generation and propagation [25–28]. The computational approach is a surrogate for a more comprehensive instrumentation setup, which can aid the future design of composite materials under such extreme loading conditions.

8.3

Explosive-induced implosion

When hydrostatic loading is combined with UNDEX, the implosion process becomes more complex. UNDEX itself is a complicated process, which consists of an initial shock wave followed by multiple bubble pulse pressure loadings. Structures experiencing explosive-induced implosion tend to exhibit asymmetric collapse modes. The dynamic deformation and damage response depend on many factors, including initial shock, hydrostatic pressure, orientation of the structure relative to the explosive source, loading time (damage accumulation), as well as factors such as the constitutive behavior of materials.

8.3.1

Experimental investigation

R.H. Cole, in his book Underwater Explosions [35], gives a detailed account of the shock waves generated during an explosions and the effect of these waves on structures. Explosive materials are inherently unstable compounds, which undergo chemical reactions to form stable products. Explosive reactions are triggered by imparting sufficient energy to the compound. Heated fuses or frictional heat from impact by a firing pin is most commonly used to initiate these reactions. Once initiated, the original material is rapidly converted into a gas at very high temperatures and pressures. This process is called “detonation” and it creates a shock front which advances at the speed of several thousand meters per second. This shock front is termed “detonation wave” and chemical transformation resulting from detonation occurs simultaneously with the progression of this wave. When this wave reaches the boundary of the explosive material and surrounding medium, the pressure is transmitted through the boundary at a finite pressure and velocity. In the case of underwater explosions, the surrounding medium is water, which can be regarded as a homogeneous fluid incapable of sustaining shear stresses. A shock wave traveling through water has two distinct physical characteristics—shock wave velocity and local particle velocity. At the

Implosion of composite cylinders due to underwater impulsive loads

245

pressures considered here, the speed of sound wave in water is independent of peak pressure and is 1440 m/s. Trinitrotoluene (TNT) is commonly used to generate, characterize, and study underwater explosions. TNT has a specific energy of 4500 kJ/kg and the specific energy released by other explosive compounds is often expressed in the form of equivalent mass of TNT for the purpose of calibration. Upon detonation, TNT forms nitrogen, water, carbon monoxide, and solid carbon and generates a large amount of pressure—on the order of 14,000 MPa [36]. This pressure compresses the surrounding medium and radiates a high-pressure disturbance, which falls off rapidly called “explosive decay.” The velocities commonly observed for TNT are several times the limiting value of 1440 m/s in water. The maximum pressure in this wave falls off rapidly with distance and approaches steady-state behavior at large distances. The temporal profile of the wave broadens gradually as the wave radiates outward. This behavior of the blast wave is shown in Fig. 8.4. Proximity to an underwater explosion plays an important role in the dynamic behavior of a marine structure. If an underwater explosion occurs close to the structure, the resulting pressure wave will rupture the hull and cause significant damage to surrounding equipment. On the other hand, if the explosion occurs far from the ship, the blast wave will have a planar front and the pressure loading will be nonuniform. In this case, each section of the marine structure will respond differently to an incident pressure pulse. The defining characteristics of a pressure pulse are the peak pressure and pressure-time history. The peak pressure resulting from and underwater blast, pm, is given by pm ¼ K1

M1=3 R

α1 ,

(8.4)

where K1 and α1 are material constants (with values K ¼ 5 107 and α1 ¼ 1.15 for TNT), M is the mass of TNT used, and R is the distance from explosive source [35,36]. The blast decay constant, γ, for the pressure pulse created due to an underwater explosion is given by

Pressure (MPa)

235

Source

150 50 25 15

1

10

100

1000

2000

Distance from source (m)

Fig. 8.4 Spatial evolution of blast pulse for a TNT explosion. Figure not to scale [35].

246

Dynamic Response and Failure of Composite Materials and Structures

γ ¼ M1=3 K2

1=3 α2 M , R

(8.5)

where K2 and α2 are material constants (with values K ¼ 92 106 and α2 ¼ 0.22 for TNT) source [36]. The decay constant defines the decay time for the peak pressure. According to Taylor’s analysis [37,38], the pressure in the fluid at a distance from the explosive source follows the relation t , pðtÞ ¼ pm exp t0

(8.6)

where t is time and t0 is the pulse time on the order of milliseconds. Since the mechanical impedance of water is much higher than air, underwater blasts travel large distances before attenuating sufficiently to be harmless. When underwater blast waves interact with marine structures, they can cause significant plastic dissipation and fracture. For large unsupported ship sections, damage is in the form of bending and tensile necking. For supported ship sections, damage is in the form of shear rupture and tearing. For thick-walled composite cylindrical sections, the primary failure is ply failure including interlaminar delamination and intralaminar cracking [33,34,39]. Thin-walled structures are more likely to fail by dynamic instability [33,34]. One such condition is that of implosion. Drop weight and impact indentation testing were performed earlier to understand the dynamic response and damage of composite cylinders intended for underwater applications [40–45]. In recent years, gas gun-based impact loading has been successfully used to generate impulsive loading through water [39,46–54]. Fig. 8.5A shows a novel experimental setup, the underwater shock loading simulator (USLS), developed by Avachat and Zhou [39,46–49,54]. The USLS consists of a projectile impact-based impulsive loading mechanism and clamped supported boundary conditions for the

Extension

Flash light high-speed camera

Projectile Steel anvil Water

(A)

sandwich structure

Pressure (MPa)

200 Flyer-plate

V0 = 80 m/s

100

V0 = 50 m/s 50 0

Flash light support

V0 = 100 m/s

150

(B)

V0 = 30 m/s 0

0.2

0.4

0.6

0.8

1

Time (ms)

Fig. 8.5 (A) Sectional view of underwater shock loading simulator showing the setup for high-speed digital imaging and digital image correlation of impulsively loaded cylindrical sandwich structures, and (B) measured experimental pressure histories in the water chamber for four different projectile velocities.

Implosion of composite cylinders due to underwater impulsive loads

247

specimen. The cylindrical structure is supported on a steel flange because this loading condition closely resembles that found in a naval structure. A force transducer is mounted on the flange to measure impulses transmitted in each case. The location of failure in this configuration allows accurate time-resolved measurements using high-speed digital imaging. The USLS can generate impulsive loads with peak pressures within the range of 40–250 MPa, which can be obtained with different projectile velocities, as shown in Fig. 8.5B. The profiles resemble those created by an underwater explosion event, which approximately correspond to 1 kg of TNT exploding at distance of 1250, 620, 425, and 350 mm, respectively. The USLS was used to characterize the dynamic deformations and damage response of sandwich composite tubes as a function of different core densities subjected to high-intensity underwater impulsive loads [39]. Results showed that the transmitted impulse and overall damage increased as the core density increased. Deflection and warping in the impulsively loaded region are influenced by the radius of curvature and facesheet fiber orientation. Most recently, the implosion of composite cylinders under constant hydrostatic pressure subjected to explosive loadings with varying incident shock wave strengths was studied to determine the effect of explosive loading on the mechanisms of collapse [3]. Experiments were conducted in a pressure vessel with three-dimensional DIC technique and pressure transducers [2,4–6,17] as described in Section 8.2. RP-85 exploding-bridgewire detonators (explosive material used was equivalent to 1778 mg of TNT) at different standoff distances were used to generate varying initial shock wave strengths [3]. Results showed that implosion could be initiated at lower static pressure when the structure is subjected to a nearby explosive loading as compared with hydrostatically induced cases. Additionally, higher strength explosive loading excited higher-order initial vibration which then induced the structural collapse. Later, a more detailed study was conducted in an attempt to investigate the shock initiated structural instabilities of underwater cylindrical structures using aluminum specimens [8]. A relationship between hydrostatic pressure, structural stiffness, and vibration frequency was established to evaluate the structural integrity of submerged cylinders under dynamic loading.

8.3.2

Numerical simulation

An important open research topic for marine composite structures is the lack of design relations that quantify the performance of the structures as a function of structural attributes and loading conditions. So far, the limited instrumentation in the tests does not include direct measurement of the physical mechanism in an explosive-induced implosion event. Numerical modeling is required to obtain such quantitative relations. Fatt and coworkers [33,34] developed the solutions for the redial shell deformation and stability diagrams for composite cylinders subjected to uniform hydrostatic loading and side-on explosions using Mathieu differential equations. To account for fluid-coupling effects, a loading routine was employed in finite element analysis to solve the plane-strain problem of a composite cylinder subjected to an external blast loading [19]. Composite material damage initiation and evolution were predicted based on Hashin damage model [55] and the cohesive zone model [56–58]. Recent

248

Dynamic Response and Failure of Composite Materials and Structures

developments in finite element models have focused on the fully coupled multiphysics FSI simulations where fluid domain, structure domain, and their interactions are explicitly modeled [29,30,39]. One such approach is the CEL framework. In the CEL framework, the cylinder is located within an Eulerian chamber which is used to generate a blast wave as shown in Fig. 8.6A. Simulations through the nonlinear finite element commercial code ABAQUS [24] are performed to characterize the dynamic deformation and damage response of structures subjected to combined loads of hydrostatic pressure and high-intensity explosive loading. In relation to the latest experimental work on explosive-induced implosion of composite tubes [3], the

Z

Outflow Eulerian BC Nonreflecting

Inflow Eulerian BC

CEL framework

Hy d pre rosta ss tic ure

Y X

BC–tangential flow only Nonreflecting outflow Z

Y X

(A)

L

Inflow velocity

140

Pressure (MPa)

120 100

L = 115 mm

L = 70 mm

L = 190 mm

L = 5 mm

80 60 40 20 200

0 0

(B)

0.2

0.4 Time (m

0.6

s)

150 m) , L (m

0.8

1

100 tion 50 loca eam tr s n ow

0 D

Fig. 8.6 (A) Schematic illustration showing the CEL framework for studying the explosive-induced implosion event, and (B) output pressure histories at different downstream locations within the Eulerian domain.

Implosion of composite cylinders due to underwater impulsive loads

249

composite specimen has a [15 degree/0 degree/45 degree/15 degree] layup with a 380 mm length, 60 mm inner diameter, and 1.5 mm wall thickness. The specimen is sealed using two aluminum end caps. Boundary conditions are applied at one of the end caps to prevent translation in all directions and rotations around all three axes. Translation in the axial direction is allowed at the other end. Since the effects of hydrostatic pressure are of interest, the initial stress state in the structure is established in a static analysis. The structural state is then imported as the initial conditions in the dynamic analysis [1,2,7,30], where the Eulerian material (water) can interact with Lagrangian elements (cylindrical structure) through Eulerian-Lagrangian contact to allow fully coupled multiphysics FSI simulations. Simulations are carried out with various combinations of hydrostatic and explosive loading. The applied hydrostatic pressure is based on the critical collapse pressure of the structure, denoted as Pc, which is determined through a linearized buckling analysis conducted using the software package [1,2,7]. In the CEL framework, a high-pressure, exponentially decaying impulse can be generated by specifying a chosen profile of velocity v(t), defined in Eq. (8.7), at the inflow Eulerian boundary condition and then the blast wave p(t) can be observed at some distance L downstream [24,59–61] vðtÞ ¼ v0 + vp exp ½ðt t0 Þ=td :

(8.7)

where v0 is the initial particle velocity, vp is the peak particle velocity, t0 defines the start time, and td defines the duration of the blast velocity profile. A one-dimensional Eulerian model (one string of brick Eulerian elements) is used to quantify blast waves as shown in Fig. 8.6A. The front face of the model (inflow face) is prescribed a chosen particle velocity v(t). This flow field simulates the effects of the incident blast wave on the specimen, as shown in Fig. 8.6B. Zero displacement boundary conditions are applied on the sides of the model to restrict flow normal to the walls but allow tangential flow. A nonreflecting outflow Eulerian boundary condition is specified at the back face of the model. The response of water in the Eulerian domain is described by the Mie-Gr€uneisen equation of state such that p¼

ρ0 c 0 2 η

ð1 sηÞ2

1

Γ0 η + Γ 0 ρ0 E m , 2

(8.8)

where p is the pressure, c0 is the speed of sound, ρ0 is the initial density, Em is internal uneisen’s Gamma at a reference state, s ¼ dUs/dUp is energy per unit mass, Γ 0 is the Gr€ the Hugoniot slope coefficient, Us is the shock wave velocity, and Up is the particle velocity, which is related to Us through a linear Hugoniot relation Us ¼ c0 + sUp :

(8.9)

The parameters of water for the Mie-Gr€ uneisen equation of state are obtained from previous studies [39] and are listed in Table 8.1.

250

Dynamic Response and Failure of Composite Materials and Structures

Table 8.1 Parameters for the Mie-Gr€ uneisen equation of state for water ρ0 (kg/m3)

c0 (m/s)

Γ 0 (dimensionless)

s 5 dUs/dUp (dimensionless)

980

1500

0.1

1.75

A finite-deformation framework is adopted to account for large deformations in the composite. The material is assumed to be orthotropic and linear elastic. Damage initiation and failure of each ply are described by the Hashin damage model [24,55]. This is a homogenized model so that individual fibers and fiber-matrix interfaces are not modeled explicitly. Rather, the model provides a phenomenological representation of the different damage modes in composite structures. This framework incorporates four damage mechanisms: (1) matrix damage in tension; (2) matrix damage in compression; (3) fiber damage in tension; and (4) fiber damage in compression. Damage initiation follows the following equations: (1) matrix tension (^ σ 22 0): FTm ¼

σ^22 T22

2 2 ^τ12 + , S12

(8.10)

(2) matrix compression (^ σ 22 < 0):

FCm

σ^22 ¼ 2S23

# 2 2 " ^τ12 C22 2 σ^22 + + 1 , S12 2S23 C22

(8.11)

(3) fiber tension (^ σ 11 0) : FTf ¼

σ^22 T11

2 2 ^τ12 + S12

(8.12)

(4) fiber compression (^ σ 11 < 0) : FCf ¼

σ^11 : C11

(8.13)

In the earlier relations, for each parameter, FmT, FmC, FfT, FfC, a value of less than 1.0 indicates no damage and a value of 1.0 indicates damage. The subscript “11” denotes longitudinal direction while the subscript “22” denotes the transverse direction; E, T, and C are the tensile modulus, tensile strength, and compressive strength, respectively. The in-plane/longitudinal shear strengths are S12 ¼ S31, while the out-of-plane/transverse shear strength is S23. In addition, σ^11 , σ^22 , and ^τ12 are components of the effective stress tensor in the form of σ^ ¼ Mσ, with σ being the nominal stress tensor and M being the damage operator given by

Implosion of composite cylinders due to underwater impulsive loads

3 1=ð1 Df Þ 0 0 5, M¼4 0 0 1=ð1 Dm Þ 0 0 1=ð1 Ds Þ

251

2

(8.14)

where Df, Dm, and Ds are damage variables in fibers, in matrix, and associated with the shear modes, respectively [62]. The upper bound for all damage variables in is Dmax ¼ 1. Prior to damage initiation, the material is linear elastic. After damage initiation, the response of the material follows σ ¼ Cd ε

(8.15)

where ε is the strain and Cd is the elastic constant accounting for damage in the form of 2 3 ð1 Df ÞE11 ð1 Df Þð1 Dm Þv21 E11 0 14 5: Cd ¼ ð1 Df Þð1 Dm Þv12 E22 ð1 Dm ÞE22 0 D ð1 Dm ÞμD 0 0 (8.16) In the earlier relation, D ¼ 1 (1 Df)(1 Dm)ν12ν21, Df reflects the current state of fiber damage, Dm reflects the current state of matrix damage, Ds reflects the current state of shear damage, E11 is the elastic modulus of the composite in the fiber direction, E22 is the elastic modulus of the composite in the transverse directions, μ is the shear modulus, and ν12 and ν21 are Poisson’s ratios. The components of the damage variables are

9 Dtf , fibertensiledamagevariable, > > > c > , fibercompressivedamagevariable, D = f t Dm , matrixtensiledamagevariable, Dm ¼ > and > Dcm , matrixcompressivedamagevariable, > > ; Ds ¼ 1 1 Dtf 1 Dcf 1 Dtm 1 Dcm :

Df ¼

(8.17)

In the earlier expressions, Dft, Dfc, Dmt, and Dmc are calculated using Gmtc, Gmcc, Gftc, and Gfcc which are fracture energies associated with matrix tension and compression and fiber tension and compression, respectively. The material properties for unidirectional carbon-fiber/epoxy composite used are obtained from previous studies [39,47,48,54] and are listed in Table 8.2. In order to study the effect of the orientation of the impulsive loading on the structural collapse and damage, the tube is loaded from two different directions, that is, a side-on shock wave propagating perpendicular to the axis of the cylinder (transverse loading) and a side-on shock wave propagating parallel to the axis of the cylinder (longitudinal loading). A constant uniform pressure with the magnitude of 0.8 Pc is applied to the cylinder surface throughout the analysis. The evolution of the collapse mode shape is examined using Eq. (8.3) at the midspan and axial offsets 150 mm from the midspan. At the midspan (cut-B in

252

Dynamic Response and Failure of Composite Materials and Structures

Table 8.2 Material properties for unidirectional carbon-fiber composite Parameters

Unit

Value

Density (ρ) Longitudinal tensile modulus (E11) Transverse tensile modulus (E22) Shear modulus (G12, G13) Longitudinal tensile strength (T11) Longitudinal compressive strength (C11) Transverse tensile strength (T22) Transverse compressive strength (C22) Longitudinal shear strength (S12, S13, S23)

kg m3 MPa MPa MPa MPa MPa MPa MPa MPa

1580 119,800 10,500 5200 2470 1062 85 275 89

Fig. 8.7A), the collapse mode number first increases sharply and then decreases sharply when the tube is loaded transversely [5,34]. As the initial shock wave decays, the cylinder tends to flatten in a perfect mode 2 shape [2–5], shown in Fig. 8.7A. This deformation at the midspan then propagates along the length to the ends of the cylinder. The evolution of the mode number at locations near the ends (cut-A and cut-C in Fig. 8.7A) exhibits similar features and the predicted numbers agree well with the Longitudinal loading:

Transverse loading:

Cut-B t = 0.7 ms

Cut-A, C t = 0.7 ms

Cut-B t = 1.0 ms

Cut-A t = 1.0 ms

Cut-B t = 1.0 ms

10

30 Transverse loading:

Longitudinal Loading: 8

20

Cut-C Cut-B Cut-A B

15 A

10

0

0.2

Cut-C Cut-B Cut-A 6 A 4

B C

2

C

5 0

Mode number (n)

Mode number (n)

25

(A)

Cut-C t = 1.0 ms

0.6 0.4 Time (ms)

0.8

1

0

(B)

0

0.2

0.4 0.6 Time (ms)

0.8

1

Fig. 8.7 Evolution of collapse mode shape at different axial locations of the cylinder for (A) transverse loading and (B) longitudinal loading.

Implosion of composite cylinders due to underwater impulsive loads

253

deformation obtained from the simulation. When the tube is loaded longitudinally, the mode number at different axial locations varies with time similarly as that for transverse loading. The structure tends to exhibit a symmetric collapse mode, which is predicted as mode 4, as shown in Fig. 8.7B. To further explore the structural deformation and material damage when a cylinder is subjected to transverse loading, the stresses at the point closest to the explosive source on the midspan of the cylinder (i.e., A) are plotted in Fig. 8.8. The components of interest are axial stress S11 and hoop stress S22. As shown in the plot, the highly stressed regions lie between the inner and outer layers. The variation of the axial stress is associated with bending in axial direction and the variation of the hoop stress is associated with the bending in circumferential direction. The deformation of the structure corresponding to the stress state is shown in Fig. 8.8A and B. For example, the collapse mode shape shown in Fig. 8.8B requires the bending in the circumferential direction at point A. This leads to the higher tensile stress in outer plies and higher compressive stress in inner plies, both of which are represented as hoop stress in the structure. S22 S11

A Z

Y Z

A Y

Z

X

Transverse loading

X

X

t = 0.5 ms

X

A

A t = 0.6 ms t = 0.6 ms 8

4

2

t = 0.6 ms S22 (103 MPa)

S11 (103 MPa)

4

t = 0.6 ms

0

−4

0

−2 t = 0.5 ms

−8 0.3

(A)

0.4

0.5

0.6

0.7

Time (ms) Outer−15° 0° layer

0.8

0.9

−4 0.3

1

0.4

0.5

(B) Outer−15° Inner−15°

0.6

0.7

0.8

0.9

1

Time (ms) −45° layer Inner 15°

45° layer

Fig. 8.8 Histories of stresses at each ply in a laminated composite associated with structural deformation under transverse loading: (A) axial stress and (B) hoop stress.

254

Dynamic Response and Failure of Composite Materials and Structures

The stress state discussed earlier further affects damage evolution in the composite layup. According to Fig. 8.8, outer plies undergo higher axial stress which leads to severe tensile fiber failure and higher hoop stress which leads to severe tensile matrix failure. Inner plies experience higher hoop stress (stated as negative) which accelerates the evolution of compressive fiber damage. The evolution of progressive damage of different plies with the deformation states of the structure is shown in Fig. 8.9. The failure modes initiate and can further lead to damage bands or longitudinal cracks at the front wall (the face subjected to initial shock) of the cylinder. This effectively reduces the stiffness of the structure and results in the rapid acceleration of the contact of the opposing walls of the cylinder, leading to a mode 2 collapse shape. It suggests that it is possible to rearrange the layup or line the cylinder with a flexible coating at certain angular positions to prevent the development of longitudinal cracks. Explosive-induced implosion is affected by factors such as initial shock, hydrostatic pressure, bubble pulse, and material properties [1,3,15]. To mimic the effects of different amounts and standoff distances of the explosives at sources, impulsive loading of different intensities is applied to the structure. Uniform pressure of different magnitudes is applied to the cylinder surface throughout the analysis to explore the effect of the combinations. The impulsive loading dominates the structural performance. Higher impulsive load intensity leads to larger deflection and higher damage dissipation energy, as shown in Fig. 8.10A and B. Structures undergo asymmetric collapse with higher mode numbers initiated by the impulsive loading and tend to settle to a mode 2 shape in all cases. To quantify the potential effect of the implosion on nearby structures, the pressure history in the water domain at the location about the midspan t = 0.6 ms

t = 0.9 ms

t = 0.6 ms

t = 0.9 ms

Outer-15° layer Tensile fiber failure

Outer-15° layer Tensile matrix failure

Damage coef.

t = 0.6 ms Inner-15° layer Compressive fiber failure

t = 0.9 ms

0.10 0.09 0.08 0.08 0.07 0.06 0.05 0.04 0.03 0.03 0.02 0.01 0.00

Fig. 8.9 Evolution of progressive damage with collapse mode for the typical inner and outer plies in a laminated composite cylinder.

90

250

800 Increase explosive load

60 30

d

600

o

se

ea ncr

400

l exp

I

200 Increase hydrostatic pressure

0

(A)

loa sive

0

0.5

1.5 1 Time (ms)

0

2

(B)

Increase

0

0.5

ssure static pre

Dynamic pressure (MPa)

UX

120

Energy (J)

Deflection UX (mm)

1000

hydro

1 1.5 Time (ms)

200 Increase shock wave 150

(C)

Reflections

50 0 −50 −100 −150

2

Max. Ux

100

P 30 mm

Implosion initiates

0

0.5

1 1.5 Time (ms)

2

Implosion of composite cylinders due to underwater impulsive loads

150

Fig. 8.10 (A) Deflections, (B) energy dissipation associated with damage, and (C) pressure at midspan of the composite cylinder with a 30 mm standoff for structures subjected to different combinations of explosive load and initial hydrostatic pressure.

255

256

Dynamic Response and Failure of Composite Materials and Structures

of the cylinder with a 30 mm standoff is plotted in Fig. 8.10C. The magnitude of the released pressure wave is smaller than that of the initial shock wave, however, due to its long duration, the impulse is of comparable strength. The negative pressure spike represents a decrease in pressure which corresponds the initiation of implosion. The subsequent spike in pressure corresponds to the deceleration of water when the deflection of the structure reaches the maximum, as shown in Fig. 8.10A and C. The subsequent peaks in pressure correspond to reflections of the initial wave. An acoustic analysis [3,63] shows that an incident wave with a magnitude of Pi is reflected back with a magnitude of 0.535 Pi from the cylinder, providing an explanation for the attenuation seen in Fig. 8.10C. The negative impulse of the underpressure region is equal in magnitude to the positive impulse of the overpressure region. This result is in agreement with the observations from earlier studies [2,26]. The results in Fig. 8.10C show that higher initial impulsive loading leads to higher released impulse. Cylindrical structures with different levels of internal pressure, i.e., 0, 1, 3, and 5 atm, are imported to dynamic simulations to investigate the effect of initial internal pressurization on the structural performance. Results show that initial internal pressurization of the structure significantly enhances the damage resistance of structures. However, the enhancement diminishes as the internal pressure further increases, as shown in Fig. 8.11B. The released pressure pulse is not significantly affected by variations in the internal pressure, as shown in Fig. 8.11C. Earlier results indicate that the highly stressed regions lie within facesheets (inner and outer layers) of the cylindrical structure subjected to impulsive loads. Tailoring the filament-wound layup by modifying the winding angles, for example, [θ degree/ 0 degree/45 degree/θ degree], can change structural performance. The deformation, impulse delivered up to 2 ms, damage dissipation energy for structures with different θ as well as stress states for the inner θ degree layer and outer θ degree layers are shown in Fig. 8.12 as functions of winding angle, θ. Earlier results in Figs. 8.10 and 8.11 showed that the displacement UX first monotonically increases to a peak value when the collapse in a higher mode shape occurs due to the initial shock, and then decreases to reach a steady-state when the cylinder settles into a mode 2 shape. Therefore, the maximum deflection, UX,max, and the deflection measured at the end of the simulation, UX,final, are recorded to quantify the resistance to the impulsive loads, as shown in Fig. 8.12A. To evaluate the damage potential of the impulse of the overpressure regions, the impulse of the underpressure region is calculated [2,6] as I¼

ðt

Δpdt,

(8.18)

0

where Δp is the magnitude of negative pressure and t is the duration of the underpressure region. The results are plotted in Fig. 8.12B. Earlier discussions pointed out that higher axial stress, S11, would lead to severe tensile fiber failure, higher hoop stress, S22, would lead to severe tensile matrix failure, and higher compressive stress would accelerate the evolution of compressive fiber damage. Consequently, the maximum and minimum stress values of the inner and outer layers for each structure are

60

30

Dynamic pressure (MPa)

600 Energy (J)

Deflection UX (mm)

1 atm 3 atm 5 atm

0 atm

400

200

0

0.5

1

1.5 2 Time (ms)

2.5

0

3

(B)

0

0.5

1

1.5 2 Time (ms)

2.5

30 mm

100 50 0 −50 −100

3

(C)

P

150

0

0.5

1

1.5 2 Time (ms)

2.5

3

Implosion of composite cylinders due to underwater impulsive loads

(A)

0 atm 1 atm 3 atm 5 atm

UX

90

0

200

800

120

Fig. 8.11 (A) Deflections, (B) energy dissipation associated with damage, and (C) pressure at midspan of the composite cylinder with a 30 mm standoff for structures with different levels of internal pressure.

257

Dynamic Response and Failure of Composite Materials and Structures 80

60 UX, max

50 Impulse (MPa·ms)

Deflection UX (mm)

60

40

20

0

15

(A)

30

45

60

20

75

0

90

0

15

0 −5 −10

45

60

300 90

75

Max. stress outer −q ° layer Min. stress

15

S22

30

Winding angle (°) 20

S11

5

400 Impulse

(B)

10

Stress (103 MPa)

Stress (103 MPa)

500 20

Min. stress

10

(C)

600 30

inner q ° layer

Max. stress

−15

40

10

Winding angle (°)

15

700 Energy

UX, final

0

800

Energy (J)

258

S11

5 0 −5

S22

−10 0

15

30

45

60

Winding angle (°)

75

90

−15

(D)

0

15

30

45

60

75

90

Winding angle (°)

Fig. 8.12 (A) Deflections, (B) impulse and energy dissipation associated with damage as a function of θ for structures with a layup of [θ degree/0 degree/45 degree/θ degree], (C) maximum and minimum stress for the innermost layer of each structure as a function of θ, (D) maximum and minimum stress for the outermost layer of each structure as a function of θ.

summarized in Fig. 8.12C and D to quantify the damage potential of different structures. The results can serve as a reference guide for the prediction and design of composite cylindrical structures to enhance their resistance to damage in explosive-induced implosion.

8.4

Summary

Structural design of ships and submersibles is a complex undertaking because the deformations experienced by the naval vessels are a result of the combined effects of multiple loads acting simultaneously. There has been increased interest in using composite materials for such marine structures to reduce weight, improve corrosion resistance, and achieve good blast mitigation in recent years. A major of aspect of composite structures in underwater applications that has not been thoroughly investigated is the implosion of cylindrical structures subjected to blast loads. Geometric (shell thickness, length-to-diameter ratios, fiber orientations, etc.) and material nonlinearities (composite laminates and polymeric foams) create complicated loading

Implosion of composite cylinders due to underwater impulsive loads

259

conditions and often cause unexpected failure. Effective design of these structures requires an intimate understanding of the dynamic deformation and failure of such structures and a capability to predict and control their performance. Recent studies of hydrostatically induced and explosive-induced implosion of composite cylindrical structures are reviewed in this article. The complexity of the issues necessitates realistic experiments that account for the actual service environments and computational simulations that allow a wide range of scenarios to be explored. (1) Hydrostatically induced implosion of composite tubes was studied experimentally using a novel pressure vessel filled with water using DIC to relate collapse mechanics to the changes in local pressure fields. Studies focused on measurement of critical collapse pressure and evaluating of the released pulse, which have damage effects on nearby structures. Analyses pointed out that glass-fiber composite tubes with filament-wound layup would be optimal in resisting damage evolution. Polyuria coatings on the interior surface significantly enhanced the mitigation of the implosion pressure pulse released. (2) Explosive-induced implosion of composite tubes was studied in a pressure vessel with pressure transducers and DIC setup. Real explosives were used to generate the initial shock. Results showed that implosion could be initiated at a lower static pressure when an initial shock was applied. However, explosive loads with short pulse durations or lower peak pressures were not capable of causing a collapse from the initial shock alone, but caused subsequent damage accumulation. (3) The dynamic deformation and damage response of composite cylinders during an explosive-induced implosion event is affected by many factors, such as initial shock, hydrostatic pressure, damage accumulation, as well as material properties. Numerical approaches such as CEL-based finite element models are designed to allow exploration of design scenarios involving simultaneously varying loading, material and geometric variables. The results obtained so far provide design recommendations for reducing the severity of implosion. Measures include internal pressurization and tailoring of fiber orientation of facesheets. For similar total mass, sandwich composites with rate-dependent cores as the energy absorbing material are shown to provide superior blast mitigation as compared with monolithic composite structures. Obviously, more research is needed, especially with regard to quantitative performance-loading-material-geometry relations. To achieve this objective, further developments in experimental, numerical, and analytical approaches are needed and are expected.

Acknowledgments Support by the Office of Naval Research through Grant Numbers N00014-09-1-0808, N0001409-1-0618, and N00014-16-1-2289 (program manager: Dr. Yapa D.S. Rajapakse) is gratefully acknowledged. Some of the calculations reported here are carried out on the Athena HPC cluster in the Dynamic Properties Research Laboratory at Georgia Tech.

References [1] Gish LA. Analytic and numerical study of underwater implosion. Massachusetts: Massachusetts Institute of Technology; 2013.

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Composite materials for blast applications in air and underwater

9

E. Rolfe, M. Kelly, H. Arora, J.P. Dear Imperial College London, London, United Kingdom

9.1

Literature review

Fiber reinforced polymer composites (FRPCs) have been in use in naval vessels since the launch of Her Majesty’s Ship (HMS) Wilton in 1972 by Vosper Thornycroft [1]. FRPCs offer a number of advantages over traditional ship building materials including low-radar signature, lightweight design, corrosion resistance, and the combination of these factors to reduce fuel and maintenance costs. This has led to their continued development and use in the latest modern craft such as the Kockums FLEXpattrol littoral ships [2] and the trimaran ocean patrol vessel (OPV), called the Ocean Eagle 43 [1]. In naval applications the threat of explosive attack is high; therefore this project investigates the resistance of composite materials to air and underwater blasts. Due to the high costs of blast testing and the qualitative nature of the results, blast testing is performed infrequently. A number of authors have carried out blast testing on metal plate geometries. Menkes and Opat [3] subjected clamped aluminum beams to explosive loading and identified the damage modes with increasing load as inelastic deformation, tearing, and finally shearing at the support. Neuberger et al. performed full-scale air blast tests [4] and buried charge tests [4] on clamped circular steel plates to investigate the validity of scaled-down blast experiments. Cantwell et al. extended this research into the field of composite materials by analyzing the response of fiber-metal laminate (FML) plates to air blast loading [5–7]. The experiments identified that FMLs resist blast loads through delamination and debonding between the aluminum and composite layers. Additionally, woven and unidirectional composites respond differently. Full-scale air blast experiments on composite sandwich panels have been carried out by Arora et al. [8,9]. The explosive resilience of glass-fiber reinforced polymer (GFRP) face-sheets with styrene acrylonitrile (SAN) foam cores was compared to that of carbon-fiber reinforced polymer (CFRP) face-sheets with the same cores. The CFRP face-sheet panel was found to deflect less than the GFRP face-sheet panel. Both panels suffered from core damage and face-sheet core delamination while the GFRP face-sheet panel suffered from a large face-sheet crack. This area of research into the resilience of composite sandwich materials has been pursued further. Arora [10,11] performed further research into the effect of the core thickness on the sandwich panel response. Shock tubes offer an alternative method of replicating blast loading on material samples and scaled test samples. The shock wave magnitude can be controlled enabling the experiments to focus solely on the shock wave incident rather than other Dynamic Response and Failure of Composite Materials and Structures. http://dx.doi.org/10.1016/B978-0-08-100887-4.00009-3 © 2017 Elsevier Ltd. All rights reserved.

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factors inherent in blast situations (e.g., burning). A shock tube is made up of a long cylinder with a high-pressure driver section separated by a diaphragm from a low-pressure driven section. The driver section is pressurized until the pressure difference causes the diaphragm to rupture releasing a mass of gas and creating a shock wave which propagates down the cylinder to the specimen [12]. The shock resilience of composite materials and structures has been investigated and the test methods developed to great effect. The incorporation of a poly-urea layer into the layup of the face-sheets of composite sandwich panels has been shown to improve the blast performance of the panels [13–15]. Placing poly-urea either behind the front face-sheet or behind the core reduces back face-sheet deflection and in-plane strain hence maintaining structural integrity. Constructing the core from multiple polymeric foam layers of different densities has been shown to mitigate blast damage [16]. By placing the foams in increasing order of density (low/middle/high), with the lowest density foam on the blast side, the core absorbs blast energy in the front layers early in deformation, reducing back-sheet damage. An alternative configuration (middle/ low/high) suffered from face-sheet cracking and severe core damage. Shock tube experiments were also carried out on sandwich panels with three to five core density gradations [13,14]. The results revealed that increasing the number of core layers, thus reducing the difference in acoustic wave impedance between successive layers, helps to maintain structural integrity of the sandwich panel. The performance of these composite sandwich structures has been extended to underwater blast scenarios. Arora et al. [9] investigated the performance of GFRP sandwich panels and GFRP tubes when subjected to underwater blast. The response of these structures was recorded using strain gauges. Gaining high-quality data from these high-risk expensive tests requires great care. Therefore, the principle of a shock tube has also been applied to water blast using the water-hammer effect. A water filled conical shock tube (CST) has been used by LeBlanc and Shukla to investigate the effects of plate curvature and composite panels with poly-urea coatings [17]. The authors subjected the panels to shock loading with peak pressures of 10 MPa and found that a thick coating on the back face improved panel response whereas a thin layer on the front face degraded the response. Research is being conducted into the different aspects of blast loading over a range of scales and materials. Suitable scaling parameters and equivalence between different test methods have not been established, particularly with regard to the study of composite sandwich structures. Full-scale explosive testing is, therefore, required to provide valuable and representative data for industry and it is the main focus of the study described in this chapter.

9.2

Materials

Nine composite sandwich panel configurations were subjected to blast loading and these are detailed in Table 9.1. For the air blast experiments, the performance of three different polymer cores was investigated in Study A. These polymers were SAN, polyvinyl chloride (PVC), and polymethacrylimide (PMI). In Study B the performance of a single density versus graded density SAN foam core was investigated. The graded density core consisted of 10 mm layers of each M100, M130, and M200 SAN Gurit

Composite materials for blast applications

Table 9.1

Summary of panel types Face-skin fiber type

Air blast

265

Study A Study B

Underwater blast

Glass Glass Glass Glass

Glass Glass Carbon Glass

Carbon

Core material PVC PMI SAN M100 Graded SAN (M100/M130/ M200) SAN M130 SAN M130 SAN M130 Graded SAN (M100/M130/ M200) Graded SAN (M100/M130/ M200)

Core thickness (mm)

Panel areal density (kg/ m2)

40 40 40 30

11 11 11 12

30 30 30 30

11 11 9 12

30

10

foam core and this was compared to a 30-mm thick M130 single density SAN foam core. In the subsequent underwater blast experiments the performance of single density versus graded density SAN cores was investigated along with the effect of GFRP versus CFRP skins. Again, the graded cores were constructed from 10 mm layers of each M100, M130, and M200 SAN foam core.

9.2.1

Face-sheets

The face-sheets chosen in the air blast testing were GFRP because these would provide the sandwich panels with greater strain to failure in the face-sheets, allowing greater shear strains in the foam cores. The GFRP face-sheets were similar in construction to those tested by Arora et al. [8,9] and contained Gurit QE1200 quadriaxial glass-fiber plies. The panels were manufactured via vacuum consolidation. The panels were drawn to vacuum and held at room temperature for 24 hours. Following this they were then heated to 85°C at 1°C/min. The CFRP panels for the underwater blast testing were manufactured using the same method. Gurit biaxial XC411 carbon fibers were used in this study. The schematics of these sandwich panels are shown in Fig. 9.1.

9.2.2

Foam characterization

Characterization tests were performed on the different polymeric foam cores to understand their different behavior under tension and compression at quasi-static and dynamic rates. By carrying out these tests it was possible to determine different failure properties at increased loading rates and correspond these to the results observed in the full-scale blast tests.

266

Dynamic Response and Failure of Composite Materials and Structures

Quadriaxial GFRP/CFRP 10 mm thick 100 kg/m2 SAN 10 mm thick 130 kg/m2 SAN 10 mm thick 200 kg/m2 SAN

Quadriaxial GFRP/CFRP SAN foam core

Quadriaxial GFRP/CFRP

Quadriaxial GFRP/CFRP

(A)

(B)

Fig. 9.1 Schematic of layup of (A) single and (B) graded core sandwich panels.

The characterization tests performed were quasi-static compression, dynamic compression, quasi-static tension, and dynamic tension. The dynamic compressive characterization was to assess the behavior of the SAN foams in underwater blast scenarios. Quasi-static tests were performed to gain the full stress versus strain responses of the foams in tension and compression, as opposed to the individual properties provided by the manufacturers. Failure in the foam is caused by a combination of tensile and shear loading. Due to the complexity of performing dynamic shear tests, a proportional relationship between shear and tension was assumed. Dynamic tension tests were, therefore, carried out to understand the tensile behavior of the foams.

9.2.2.1

Quasi-static compression

The test set up for the quasi-static characterization was influenced by the ASTM C365/C365M test standard [18], and the tests were performed using a 150-kN capacity Instron universal testing machine, with an auxiliary 5-kN load cell inserted for higher precision results. A self-aligning platen was used in the test to ensure deviation in the load train was accounted for in the test. Square samples were used for these tests enabling the use of 2D DIC for strain measurement. DIC was performed using a Canon 350D single lens reflex (SLR) camera; an image was captured every 3 seconds. The elastic modulus, crushing stress, crushing strain, and plateau stress were calculated from the DIC strain and load cell, and the densification was recorded as the strain at which the stress began to increase exponentially with strain. An exponential equation was fitted to this behavior using Eq. (9.1), where σ p is the plateau stress, A and λ are densification constants, and εD is the strain at which densification begins. The densification constants were estimated from plotting the natural log of stress against strain for the data beyond the densification strain, and finding the gradient of the line due to Eq. (9.2). σ ¼ σ p + AeλðεεD Þ

ln σ σ p ¼ lnðAÞ + λ ðε εD Þ

9.2.2.2

(9.1) (9.2)

Quasi-static tension

Dog bone samples were cut from each foam type, with gauge sections of 10 mm wide by 6 mm thick and 65 mm long as shown in Fig. 9.2. DIC was used to measure the

Composite materials for blast applications

267

strain in these specimens as slip at the grips made the machine displacement unusable. An auxiliary 5-kN load cell was placed in the machine during testing to reduce the errors in the results. DIC images were captured every 5 seconds using a Canon 350D SLR camera. The tests were performed at 1 mm/min, which equated to a strain rate of 0.015 s1 and five specimens of each foam were tested. The maximum error in the strain to failure of the specimen is 0.083 mm as it is measured using the final photograph captured before the specimen fractured.

9.2.2.3 Dynamic compression Crushing of the polymeric foam cores is the main method in which blast energy is absorbed during underwater blast scenarios. Split Hopkinson pressure bar (SHPB) tests were performed only on the three grades of SAN foam used in the underwater panels. This was performed in order to understand the extent of energy absorption, and the benefits of using a graded density foam core in underwater blast. The test design and data processing codes were developed by David R. Sory and Professor William G. Proud in the Royal British Legion Centre for Blast Injury Studies, Imperial College London. The design included the use of a thin polypropylene (PP) disk pulse shaper, containing small dimples, to reduce the ramp gradient of initial loading on the specimen allowing stress equilibrium to be achieved. A schematic of the SHPB test setup and of the pulse shaper used during the test is shown in Fig. 9.3. The striker bar was accelerated using compressed air and was fired at varying pressures for different strain rates. Data processing of the SHPB results used strain readings of the incidence pulse, the reflected pulse, and the transmitted pulse, read

170 mm

10 mm

65 mm

20 mm

30 mm

Fig. 9.2 Dimensions of the quasi-static tension dog bone test specimen.

1m

1m

32 mm

Front 200 mm

Side

500 mm

Test sample Semi-conductor 12 mm diameter Strain gauges aluminium transmission bar

Textured PP pulse shaper 12 mm diameter aluminium incidence bar

12 mm diameter aluminium striker bar

Fig. 9.3 Schematic of the split Hopkinson pressure bar set up and the pulse shaper used in the tests [19].

268

Dynamic Response and Failure of Composite Materials and Structures

from the semi-conductor strain gauges. In order to check for equilibrium in the soft foam specimen, DIC was utilized to assess the strain distribution across the test specimen. This was required due to the inaccuracy of checking for equilibrium from the difference between the very high magnitude incident and reflected pulses and the low transmitted pulse. A Phantom V12 high-speed camera was used, with a frame rate of 84,104 fps. DIC was used to measure the strain during elastic loading and to determine the compressive modulus of the foam at dynamic rates.

9.2.2.4

Dynamic tension

Dynamic tension tests were performed on a servo-hydraulic Instron test machine capable of a 400 mm actuator travel at 20 m/s and a load of 15 kN. A 5-kN PCB 221B04 dynamic load cell was placed between the test sample and the stationary machine bed. A loss motion device was attached to the top of the specimen, allowing the actuator to reach the desired test velocity before catching the loss motion device and loading the specimen. Thin rubber washers were placed between the actuator and the flange of the loss motion device to dampen the impact, and allow the test specimen to be in equilibrium for the test. The test setup is shown in Fig. 9.4. DIC was again used To hydraulic accumulators

Actuator

Lost motion device

Rubber washers

Test sample Grips

M6 nuts and bolts

PCB 221B04 load cell

To amplifier and oscilloscope

Machine table

Fig. 9.4 Schematic of the dynamic tension test setup [19].

Composite materials for blast applications

269

to independently measure the strain as slippage at the grips invalidated the machine displacement measurements. A Miro 310 high-speed camera, with resolution of 192 304 pixels and shutter speed of 41,000 fps, was used to capture the DIC data. The samples were tested at 5 and 10 m/s, which correspond to strain rates of 180 and 365 s1. At least six specimens of each foam were tested.

9.3

Air blast testing

This section presents the full-scale air blast testing that was performed on five polymeric composite sandwich panels, all with GFRP face-sheets. The experiments were performed at a DNV GL explosive testing site at RAF Spadeadam in Cumbria, UK. The test setup and instrumentation are outlined along with subsequent postblast damage analysis.

9.3.1

Test design and instrumentation

The experimental configuration is shown in Fig. 9.5. This shows the location of all the instrumentation used outside the test cubicle, including pressure transducers and a high-speed camera situated in a protective housing. High-speed cameras were placed Top view Side-on pressure gauge

High-speed camera for external view

Test cubicle Reflected pressure gauge Explosive charge Stand-off distance

Test panel Side view Test panel Explosive charge Stand-off distance Charge stand Steel base plate Test cubicle

Reflected pressure gauge

Fig. 9.5 Blast configuration for air blast tests showing schematics of the top and side view.

270

Dynamic Response and Failure of Composite Materials and Structures

behind the target panels, within the test cubicle. These cameras recorded the full-field displacement of the back faces for analysis via DIC methods. The size of the explosive charge used in each air blast experiment was 100 kg nitromethane (100 kg TNT equivalent), which was situated at a distance of 15 m from the target panel. This stand-off distance was chosen as it was calculated to cause sufficient core and front face-sheet damage while leaving the rear face-sheet intact thus protecting the high-speed camera equipment from the blast. This was determined through an analytical solution outlined by Andrews and Moussa [20]. Failure was required to provide the maximum comparison of the different configurations of sandwich panels. Two panels were mounted side by side onto a reinforced steel front, bolted to large concrete culverts, providing a rigid foundation to the cubicle front, and support for the test panels. To prevent crushing of the sandwich panels upon bolting them to the test fixture, steel stubs were placed inside the bolt holes and 100 6 mm steel strips were attached to the front and back edges of the panels. The steel strips were attached using Sikaflex 291i marine sealing adhesive. The bolt holes for attaching the panel to the test fixture went through both the steel strips and the composite structure. A diagram of the bolting arrangement is shown in Fig. 9.6. A thick steel plate was placed under the charge to create an elastic foundation for the very high energy initial blast wave from the detonation, but beyond that the reflecting surface was simply the concrete floor. The height of the charge was set to 1.2 m, the center of the target panel, using polystyrene foam, which offered negligible blast energy absorption. A PCB 102A06 reflected pressure gauge was located in the center, underneath the two panels as shown in Fig. 9.5. In order to validate blast calculations, a side-on pressure gauge was situated 15 m from the center of the charge, at the same height as the center of the target panels. As previously mentioned, two pairs of high-speed cameras were housed in the concrete test fixture and positioned behind the speckled rear side of the panels. The high-speed cameras were Photron SA1.1’s and Photron SA5’s. The Photron SA1.1’s sampled at 5400 fps at full resolution (1024 x 1024 pixels) while the Photron SA5’s sampled at 7000 fps at the same full resolution. There were five M16 bolt Back steel plate

Steel tube

Cubicle front

Test panel

Front steel plate M16 nut

Fig. 9.6 Schematic of arrangement for mounting the sandwich panels into the cubicle front.

Composite materials for blast applications

271

high-speed cameras used in total during this set of experiments, two behind each of the two panels and one looking at the front face of the cubicle. The validity of DIC data captured during blast testing has been previously addressed through multiple displacement measurements and validation against numerical models [8,9,21]. The use of good vibration isolation and high-mass mounting systems has been evaluated in these previous experiments to enable the implementation of DIC in full-scale air blast conditions.

9.3.2

Postblast damage assessment

In order to evaluate the blast tolerance of each panel type, the location and extent of debonding and cracking following blast testing were recorded. Each sandwich panel was sectioned into 112 pieces such that postblast flexural and compression testing could be performed; this testing is not reported here. Photographs of every section edge were taken and the core shear cracks and debonding between the face-sheets and core were mapped using a MATLAB script. This information was used to produce damage maps highlighting where most damage occurred during blast loading. The percentage and magnitude of damage were recorded for each panel.

9.4

Underwater blast testing

Underwater blast testing was performed in a test pond at RAF Spadeadam. Four composite sandwich panels were subjected to a charge while underwater. The experimental setup and instrumentation used to capture the response of the panels are detailed in this section.

9.4.1

Test design and instrumentation

The test rig consisted of a welded steel channel box, constructed from a steel channel butt welded together with the flanges outwards. A 10-mm thick steel plate was sealed and bolted onto the back. The 800 mm square sandwich panel was then sealed and bolted onto the front of the channel box. The 10-mm thick steel strips were sealed to the front face of the panel using Sikaflex 291i marine sealing adhesive and bolted around the perimeter. To prevent crushing of the sandwich panels upon tightening of the bolts, steel tubes were again placed inside the holes in the sandwich panel. The setup is shown in Fig. 9.7. The explosive sources were 1 kg plastic explosive 4 (PE4) spherical charges and were situated 1 m from the front face of the sandwich panel. These charges had an equivalent weight of 1.28 kg TNT. A pine frame was constructed to hold the charge in place for testing; this was bolted onto the front of the steel box and was designed to simply break apart upon detonation. A crane was used to lower the structure into the water to a charge depth of 3.5 m. A large steel weight was strapped to the bottom of the steel box to ensure it remained vertical underwater. The pressure during the blast was measured using two Neptune Sonar T11 gauges. One was attached to a 10-mm diameter steel bar such that it was 1 m away from the charge and measured the

272

Dynamic Response and Failure of Composite Materials and Structures 800 mm

Front of sandwich panel with 14 electronic strain gauges Back of sandwich panel with 16 electronic strain gauges

A

A

4 x 10 mm x 75 mm steel plate bolted and sealed to sandwich panel front

Air

Section A-A

Steel plate bolted and sealed to sandwich panel front

650 mm

16 x M16 clearance holes

Steel channel welded to steel backing plate Sandwich panel bolted and sealed between steel plate and frame

Fig. 9.7 Schematic of the underwater blast test frame and location of 30 electronic strain gauges.

side-on hydrostatic pressure. The second gauge was attached to the top of the steel box and this measured the actual loading on the structure. The charge frame and pressure gauge assembly are shown in Fig. 9.8. To measure the response of the composite sandwich panels, electronic foil strain gauges were adhered to the front and rear faces. The sandwich panels were square in shape, so only one quarter of each panel had strain gauges attached. Fourteen strain gauges were adhered to the front face and 16 to the rear face. The strain gauges were located as shown in Fig. 9.7. All of the strain gauges were TML FLA-2-350-11 350 Ω foil gauges, adhered with TML CN adhesive.

Reflected pressure gauge

Sandwich panel Assembly suspended from crane Pine frame made from 50 mm x 25 mm slats

Steel box

10 mm diameter steel bar welded onto steel back 1 kg spherical PE charge 1 m away from sandwich panel front face

Side-on pressure gauge

40 kg mass suspended at 1.5 m from the bottom of the box

Fig. 9.8 Schematic of the underwater blast test charge and pressure gauge setup.

Composite materials for blast applications

273

PMMA tube

Panel sections

Fig. 9.9 Schematic of X-ray CT scanning setup.

9.4.2

Postblast X-ray computed tomography damage assessment

Following the underwater blast experiments, the outer 75 mm perimeter of the panels was removed. The panels were then cut into three strips 217 mm 650 mm in size. The panels were reduced from their original size to increase the scanning efficiency while still capturing the required level of detail. Additionally, the width to thickness ratio of the panels is very large which leads to a disparity in the X-ray power required to generate useful data in both the through-thickness and width directions. To enable the same power to be used throughout the scan, the three strips from each panel were stacked within a clear PMMA tube creating a cuboidal structure, as shown in Fig. 9.9. The panel sections were scanned in the custom design Nikon “hutch” μCT scanner at the University of Southampton. Data were acquired using an accelerating potential of 200 kV and tube current of 390 μA using a flat panel detector with 2000 2000 isometric elements. Three vertical detector positions were used to capture the length of the panels. A total of 3412 equiangular projections were acquired during each scan through 360 degrees. A proprietary filtered back-projection algorithm was used to reconstruct the projections into 3D volume datasets with an isometric voxel resolution of 148 μm. Each dataset comprised a stack of 2000 images, with an image size of 2000 2000 voxels. The scans of the panel sections were fused together using FEI Avizo software. The X-ray CT scans revealed the extent of the damage to each of the panels tested.

9.5

Results

The following section describes the results from the laboratory-based foam characterization along with the full-scale air and underwater blast results. Key results from the postblast damage assessment using visual and X-ray CT techniques for the air and underwater blast, respectively, are detailed.

274

9.5.1

Dynamic Response and Failure of Composite Materials and Structures

Foam characterization

Characterization tests were performed on the different polymeric foam cores to understand their tensile and compressive behavior. The results observed in the blast experiments can be related to the different failure properties determined during material characterization.

9.5.1.1

Quasi-static compression

Due to high repeatability of the test and the long test times, three samples of each the M100 SAN, M200 SAN, PVC, and PMI were tested while four M130 SAN samples were tested due to greater spread in the results. A summary of the crushing and densification stress and strain for each foam material is shown in Table 9.2.

9.5.1.2

Dynamic compression

Dynamic compression experiments were carried out on the three SAN foams used in the underwater blast panels as crushing of the polymeric foam cores is the main method of underwater blast energy absorption. Across the strain rate range tested using the SHPB method there was no variation in the SAN samples. The average compressive moduli for the foams found using the SHPB method were 63.7, 107.4, and 239.3 MPa for the M100, M130, and M200 foams, respectively. The thickness of the specimen in SHPB tests has been shown to have a significant effect on the properties; the specimen used in these experiments had a thickness of 6 mm and diameter of 8 mm. The crushing strength for each foam is shown in Fig. 9.10. The M200 samples show an increase in crushing strength with strain rate, whereas the M130 foam and the M100 foam do not show any increase.

9.5.1.3

Quasi-static tension

A summary of the mechanical properties is shown in Table 9.3. The values in this table are averages of all five specimens of each foam; however, all foams showed a large spread in results.

9.5.1.4

Dynamic tension

The foams were tested at 180 and 365 s1 and for all three SAN foam densities the strain to failure of the foams was decreased by an increase in strain rate. The strain to failure is of high importance in sandwich panel applications, as the stiffness of the face-sheets deems the foam stiffness insignificant, so until failure it merely acts to transmit shear load. The low stiffness of the foams introduces load measurement errors hence the modulus of SAN foam varies significantly in these results. The improvement of testing methods to reduce the inherent errors is a focus of research in this field. The PVC foam did not demonstrate reduced strain to failure between the two high-speed test rates but it was significantly reduced from the quasi-static tension

Foam SAN M100 SAN M130 SAN M200 PVC C70.90 PMI 110SL

Summary of the quasi-static compressive properties of the five foam polymers Crushing strength (MPa)

Crushing strain (%)

Plateau stress (MPa)

Densification strain (%)

Densification constant A (MPa)

Densification constant λ

87

1.8

3.6

1.5

39

1.51

9.98

126

3.0

4.4

1.7

34

1.48

9.77

191

5.2

6.6

4.7

38

2.06

9.57

86

2.0

4.5

1.9

51

1.46

9.97

75

2.2

4.9

2.0

45

1.41

8.93

Compressive modulus (MPa)

Composite materials for blast applications

Table 9.2

275

276

Dynamic Response and Failure of Composite Materials and Structures

16

M200 M130 M100

Crushing strength (MPa)

14 12 10 8 6 4 2 0

0

500

1000

1500

2000

2500

3000

3500

Strain rate (s–1) Fig. 9.10 Crushing strength for SAN foams with different densities [19].

Table 9.3 Summary of the quasi-static tension properties of the five foam polymers Foam

Tensile modulus (MPa)

Yield strength (MPa)

Yield strain (%)

SAN M100 SAN M130 SAN M200 PVC C70.90 PMI 110SL

88 133 220 71 139

2.4 3.2 4.9 2.0 2.7

2.3 2.2 2.0 2.6 1.8

results shown in Table 9.3. Additionally, the reliability of the moduli results for the PVC foam samples was improved due to a higher frame rate used on the high-speed cameras, at the expense of resolution. This resulted in fewer DIC measurement points across the sample but sufficient points for average strain computations to be performed. There was no significant reduction in strain to failure for the PMI foam between the two dynamic tension test speeds but there was a large reduction from the quasi-static tests. The PMI foam was far more brittle than SAN or PVC, which led to frequent failure of the foam at the test grips. The results of the tests are summarized in Table 9.4. The foam characterization results indicate how the sandwich panels will perform under blast loading. The PMI foam shows much lower strain to failure under both quasi-static and dynamic loading rates. Indicating that the PMI foam will have a lower

Foam SAN M100 SAN M130 SAN M200 PVC C70.90 PMI 110SL

Summary of the dynamic tension properties of the five foam polymers Tensile modulus at 180 s21 (MPa)

Tensile modulus at 365 s21 (MPa)

Breaking stress at 180 s21 (MPa)

Breaking stress at 365 s21 (MPa)

Breaking strain at 180 s21 (MPa)

Breaking strain at 365 s21 (MPa)

162

206

5.7

5.5

4.5

4.1

267

321

8.1

7.5

3.8

3.5

436

482

7.3

9.2

2.0

2.1

164

197

7.2

7.7

4.4

4.1

417

434

8.7

7.9

2.2

1.9

Composite materials for blast applications

Table 9.4

277

278

Dynamic Response and Failure of Composite Materials and Structures

blast resistance. The strain to failure of the PVC and SAN M100 foams is quite consistent, suggesting that their performance in air blast Study A will be similar.

9.5.2

Air blast loading

The contour plots of the out-of-plane displacement for the 40-mm thick PVC foam core sandwich panels are shown in Fig. 9.11. This figure shows the DIC results from the time of arrival of the blast wave through to the maximum pull-out of the panel. It was only possible to capture DIC results until 26.6 ms due to the presence of dust and water, after this time the panel oscillated at its natural frequency until coming to rest. The maximum out-of-plane displacement occurs across the horizontal center of the sandwich panel so this displacement is shown as edge contour plots in Fig. 9.12. The initial, positive out-of-plane displacement for the PVC foam is shown in Fig. 9.12A, up until the center of the panel reaches the maximum displacement, and the return of the panel to zero and then the pull-out is shown in Fig. 9.12B. The horizontal contour plots for the SAN, PMI, and graded SAN foam core sandwich panels are shown in Figs. 9.13–9.15, respectively. It is evident from the contour plots that the back face-sheet deflection of the graded density panel is very smooth when compared with the single density panels. This is due to cracking first occurring in the low-density foam layer facing the blast, and then in the medium-density layer, resulting in less overall cracking in the high-density foam at the back. This is supported by the foam Out-of-plane displacement 0.00 ms

1.9 ms

3.7 ms

(mm) 5.7 ms

7.6 ms

100 80 60

9.5 ms

11.4 ms

13.3 ms

15.2 ms

17.1 ms

40 20 0 -20

19.0 ms

20.9 ms

22.8 ms

24.7 ms

26.6 ms

-40 -60 -80 -100

Fig. 9.11 Contour plots of out-of-plane displacement of the 40-mm thick PVC foam core sandwich panel [19].

279

100

100

80

80

Panel displacement (mm)

Panel displacement (mm)

Composite materials for blast applications

60 40 20 0 −20 −40 −60

−80 Δt = 0.19 ms −100 −600 −400 −200

(A)

0

200

400

60 40 20 0 −20 −40 −60

−80 Δt = 0.19 ms −100 −600 −400 −200

600

(B)

Horizontal panel position (mm)

0

200

400

600

Horizontal panel position (mm)

100

100

80

80 Panel displacement (mm)

Panel displacement (mm)

Fig. 9.12 Out-of-plane displacement of the horizontal center section of the 40-mm thick PVC foam core sandwich panel for (A) the initial displacement and (B) the rebound [19].

60 40 20 0 −20 −40 −60

−80 Δt = 0.19 ms −100 −600 −400 −200

(A)

0

200

400

600

60 40 20 0 −20 −40 −60

−80 Δt = 0.19 ms −100 −600 −400 −200

(B)

Horizontal panel position (mm)

0

200

400

600

Horizontal panel position (mm)

100

100

80

80

Panel displacement (mm)

Panel displacement (mm)

Fig. 9.13 Out-of-plane displacement of the horizontal center section of the 40-mm thick SAN foam core sandwich panel for (A) the initial displacement and (B) the rebound [19].

60 40 20 0 −20 −40 −60

−80 Δt = 0.19 ms −100 −600 −400 −200

(A)

60 40 20 0 −20 −40 −60 −80 Δt = 0.19 ms

0

200

400

Horizontal panel position (mm)

600

−100

(B)

−600

−400

−200

0

200

400

600

Horizontal panel position (mm)

Fig. 9.14 Out-of-plane displacement of the horizontal center section of the 40-mm thick PMI foam core sandwich panel for (A) the initial displacement and (B) the rebound [19].

Dynamic Response and Failure of Composite Materials and Structures

100

100

80

80 Panel displacement (mm)

Panel displacement (mm)

280

60 40 20 0 −20 −40 −60

−80 Δt = 0.14 ms −100 −600 −400 −200

(A)

60 40 20 0 −20 −40 −60 −80 Δt = 0.14 ms

0

200

400

Horizontal panel position (mm)

600

−100

(B)

−600

−400

−200

0

200

400

600

Horizontal panel position (mm)

Fig. 9.15 Out-of-plane displacement of the horizontal center section of the 30-mm thick graded foam core sandwich panel for (A) the initial displacement and (B) the rebound [19].

characterization results, in which the elastic modulus of the SAN M200 foam was found to be far greater than either the M100 or M130. For all three foam types cracks occur at around one quarter and three quarters across the width of the panel, causing high displacement gradients in these locations. The pull-out in the PMI and PVC cases was larger than the SAN case due to less core damage in the SAN foam core. This indicates that SAN is more resistant to the momentum and negative pressure bending which causes the pull-out. The deflection contour plots show that the maximum panel displacements occur at the center and at the two quarter points across the center of the sandwich panel. The maximum deflection and pull-out of these three points along with the rebound time can be used to assess the strength of the sandwich panels. More damage will lead to greater deflections and longer rebound times due to the reduced stiffness of the damaged panel. The central (w/2) deflection and left (w/4) and right (3w/4) quarter deflections for the PVC, SAN, PMI, and graded SAN core panels are shown in Fig. 9.16. The deflection peaks are greatest in the PMI case due to increased damage. The graded panel has greater deflections than the single density SAN panel but this can be attributed to its reduced thickness. The graded panel, however, has a shorter rebound time than the single density SAN panel indicating that it has suffered from less damage. Following the blast, the sandwich panels were sectioned to record the damaged suffered. Figs. 9.17–9.19 show the damage sustained by the PVC foam core panel, the 40-mm thick M100 SAN foam core panel, the 30-mm thick graded density SAN foam core panel, and the 30-mm thick M130 SAN foam core panel. The center of the panels has very little damage. It was not possible to section the PMI panel because the panel had suffered from such substantial damage. Table 9.5 summarizes the damage map findings. The graded core has suffered more damage than the single core SAN M130 but less damage than the single core SAN M100. The damage in the graded panel has taken the form of debonding between the foam layers and not foam cracking. This is due to the crack propagation through the foams being arrested at the boundaries, and then traveling along the interface. The single density M130 SAN

Composite materials for blast applications

100

100

Central displacement Left displacement Right displacement

60 40 20 0 −20 −40 −60 −80

−100

5

10

15

20

25

40 20 0 −20 −40 −60

100

0

5

10

15

Panel displacement (mm)

40 20 0 −20 −40 −60

25

Central displacement Left displacement Right displacement

80

60

20

Time after blast wave arrival (ms)

100

Central displacement Left displacement Right displacement

80

−100

(B)

Time after blast wave arrival (ms)

Panel displacement (mm)

60

−80 0

(A)

60 40 20 0 −20 −40 −60 −80

−80 −100

Central displacement Left displacement Right displacement

80

Panel displacement (mm)

80

Panel displacement (mm)

281

−100 0

(C)

5

10

15

20

Time after blast wave arrival (ms)

25

0

(D)

5

10

15

20

25

Time after blast wave arrival (ms)

Fig. 9.16 Out-of-plane displacement of the panel with width w, showing center (w/2), left quarter (w/4), and right quarter (3w/4) displacements of (A) 40-mm thick PVC foam core sandwich panel; (B) 40-mm thick SAN foam core sandwich panel; (C) 40-mm thick PMI foam core sandwich panel; and (D) 30-mm thick graded SAN foam core sandwich panel [19].

Debond between back face-sheet and core

40 mm PVC

40 mm SAN

30 mm graded SAN

30 mm SAN

Fig. 9.17 Damage maps showing the debonding between the back face-sheet and core for the PVC, SAN, graded SAN, and 30-mm thick SAN panels.

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Dynamic Response and Failure of Composite Materials and Structures

Cracks through foam core

40 mm PVC

40 mm SAN

30 mm graded SAN

30 mm SAN

Fig. 9.18 Damage maps showing the core cracks for the PVC, SAN, graded SAN, and 30-mm thick SAN panels. Debond between front face-sheet and core

40 mm PVC

40 mm SAN

30 mm graded SAN

30 mm SAN

Fig. 9.19 Damage maps showing the debonding between the front face-sheet and core for the PVC, SAN, graded SAN, and 30-mm thick SAN panels. Table 9.5 Summary of the damage map results for the SAN and PVC foam core sandwich panels

Fraction of panel containing cracks (%) Fraction of the panel with front face-sheet and core debond (%) Fraction of the panel with back face-sheet and core debond (%) a

40 mm PVC

40 mm M100 SAN

30 mm Graded (M100/M130/ M200) SAN

30 mm M130 SAN

41

17

4.6a

3.3

48

21

12b

2.2

30

19

25c

6.3

Average of the three foam layers. Average of the front face-sheet and low-density foam interface, and the low- and medium-density foam interface. c Average of the back face-sheet and high-density foam interface, and the high- and medium-density foam interface. b

Composite materials for blast applications

283

panel has less damage but the core cracks that are present are through-thickness cracks and would be considered as a critical failure. The energy absorbing potential of the panel is increased due to the interfaces between the graded foam layers while protecting the rear face-sheet damage due to a smoother deflection, as shown in the DIC results. It is evident that the sandwich panels with SAN foam cores suffered significantly less damage than the PVC and PMI cores. This is due to its higher strain to failure. The foam characterization demonstrated that the PVC and SAN foams had similar strain to failure values; therefore, the SAN must have outperformed PVC for another reason. Perhaps due to higher fracture toughness or better bonding between the foam and composite face-sheets. Furthermore, it was found that the M100 SAN foam core deflected and rebounded less than the PVC and PMI cores. SAN foam cores demonstrate the best blast performance out of the foam polymer types tested.

9.5.3

Underwater blast loading of GFRP

2 1 0 –1 –2

Front face-sheet strain Rear face-sheet strain 4 2 0 –2 –4

2 1 0 –1 –2

4 2 0 –2 –4

2 1 0 –1 –2

4 2 0 –2 –4

2 1 0 –1 –2

4 2 0 –2 –4

2 1 0 –1 –2

4 2 0 –2 –4 2.5

0

0.5

1

1.5

2

Rear face-sheet strain (%)

Front face-sheet strain (%)

A low-pass filter was applied to the raw strain gauge data to eliminate the high-frequency noise recorded. This filtered strain gauge data for the front and rear face-sheet gauges on the single SAN core GFRP panel are shown in Fig. 9.20.

Time after detonation (ms)

Fig. 9.20 Variation of strain with time along the horizontal section for the single core GFRP panel [19].

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Dynamic Response and Failure of Composite Materials and Structures

On the left-hand side of the figure is a schematic diagram showing the location of the gauge for the corresponding raw data. The strains were interpolated across each direction to create strain contour plots. The front face-sheet strain contour plots for the single core GFRP panel are shown in Fig. 9.21. Fig. 9.21 shows the horizontal center section strain across the front face; this corresponds to the raw strain gauge data as shown in Fig. 9.20. Fig. 9.21B and C shows the vertical center section and diagonal center section across the front face, respectively. The direction of the strain line is shown on the left-hand side of the Front face

325

Distance from center (mm)

0

Distance from center (mm) 460

0

1 0

100 0

–1 0

0.5 1.0 Time after detonation (ms)

1.5

–2

MPS (%)

2

300

1

200

0 100 0

(B) Distance from center (mm)

0

2

200

(A)

325

MPS (%)

300

–1 0

0.5 1.0 Time after detonation (ms)

400

–2 1.5 MPS (%) 2 1

300 200 100 0

(C)

0 –1 –2 0

0.5 1.0 Time after detonation (ms)

1.5

Fig. 9.21 Variation of strain with time for (A) the horizontal section of the front face, (B) the vertical section of the front face, and (C) the diagonal section of the front face; for the single core GFRP panel along with a photograph of the front of the sandwich panel after blast testing.

0

325

Distance from center (mm)

Back face

(A) Distance from center (mm)

325

0

0

Distance from center (mm)

(B) 460

(C)

MPS (%) 4

300

2

200

0 100

–2

00

0.5 1.0 Time after detonation (ms)

1.5 MPS (%)

300

–4 4 2

200

0 100 0

–2 0

0.5 1.0 Time after detonation (ms)

1.5

–4

MPS (%) 4

400 300 200 100 0

2 0 –2 –4 0

0.5 1.0 Time after detonation (ms)

1.5

Fig. 9.22 Variation of strain with time for (A) the horizontal section of the rear face, (B) the vertical section of the rear face, and (C) the diagonal section of the rear face; for the single core GFRP panel along with a photograph of the rear of the sandwich panel after blast testing.

Composite materials for blast applications

285

0

325

Distance from center (mm)

Front face

Distance from center (mm)

0

460

0

Distance from center (mm)

(B)

(C)

2 1

200

0

100 0

(A) 325

MPS (%)

300

–1 –2 0

0.5 1.0 Time after detonation (ms)

1.5 MPS (%) 2

300

1

200

0

100 0

–1 0

0.5 1.0 Time after detonation (ms)

400 300 200 100 0

–2 1.5 MPS (%) 2 1 0 –1

0

0.5 1.0 Time after detonation (ms)

1.5

–2

Fig. 9.23 Variation of strain with time for (A) the horizontal section of the front face, (B) the vertical section of the front face, and (C) the diagonal section of the front face; for the graded core GFRP panel along with a photograph of the front of the sandwich panel after blast testing.

contour plots for each case. Fig. 9.22 shows the strain contour plots for the back face-sheets. The blast wave arrives at the sandwich panel at approximately 0.7 ms after detonation, as shown in the contour plots. The sandwich panel completely fails at around 0.9 ms. Upon arrival of the blast wave, the front face-sheet is put into tension and the back face-sheet in compression, due to crushing of the foam core. Following the crushing phase, the back face-sheet enters into tension and stays in tension until failure. The front face-sheet strain is approximately zero, due to a mixture of bending and membrane loading. During the deflection phase of the test, the outer corners of the back face-sheets are in compression, as shown in Fig. 9.22b. Photographs of the postblast sandwich panel are shown on the left-hand side of Figs. 9.21 and 9.22. Compressive cracks are present in the corners of the back face-sheet. Figs. 9.23 and 9.24 show the graded core GFRP strain contour plots for the front and rear face-sheets, respectively. The strain on the back face of the graded GFRP sandwich panel builds up to critical failure strain later than for the single core GFRP panel, this is highlighted by the strip of high strain in Fig. 9.24.

9.5.4

Underwater blast loading of CFRP

The single core CFRP sandwich panel front and rear strain contour plots are shown in Figs. 9.25 and 9.26, respectively. The sandwich panel fails at 0.93 ms and the strain response is similar to the GFRP cases. This panel, however, fails due to fracture of the back face-sheet, which is visible in the postblast photograph in Fig. 9.26. Figs. 9.27 and 9.28 show the strain contour plots for the front and rear face-sheet strains of the graded core CFRP sandwich panel. This graded CFRP panel shows a significantly different response to the single core case. The deflection is much flatter

286

Dynamic Response and Failure of Composite Materials and Structures Back face Distance from center (mm)

0

325

0

–2 0

0.5 1.0 Time after detonation (ms)

1.5 MPS (%)

4 0

100 0

–4

2

200

Distance from center (mm)

0

0

300

(B) 460

2

100

Distance from center (mm)

0

4

200

(A)

325

MPS (%)

300

–2 –4 0

0.5 1.0 Time after detonation (ms)

1.5 MPS (%)

400 300

4

200

0

2 –2

100 0

(C)

0

0.5 1.0 Time after detonation (ms)

1.5

–4

Fig. 9.24 Variation of strain with time for (A) the horizontal section of the rear face, (B) the vertical section of the rear face, and (C) the diagonal section of the rear face; for the graded core GFRP panel along with a photograph of the rear of the sandwich panel after blast testing.

0

325

Distance from center (mm) 460

2 1 0

100 0

–1 0

0.5 1.0 Time after detonation (ms)

1.5 MPS (%)

300

–2

2 1

200

0

100

(B) Distance from center (mm)

0

MPS (%)

200

(A)

325

0

Distance from center (mm)

Front face 300

–1

0 0

0.5 1.0 Time after detonation (ms)

1.5

–2

MPS (%) 2

400 300

1

200 100

0

(C)

0

–1 0

0.5 1.0 Time after detonation (ms)

1.5

–2

Fig. 9.25 Variation of strain with time for (A) the horizontal section of the front face, (B) the vertical section of the front face, and (C) the diagonal section of the front face; for the single core CFRP panel along with a photograph of the front of the sandwich panel after blast testing.

in shape which causes very high strain at the boundary. It is at the boundary where the panel ultimately fails in tension. This deflection shape is expected to be due to the graded foam layers absorbing energy, so creating a more uniform deflection of the back face-sheet. The panel fails due to back face-sheet fracture as shown in the postblast photograph (Fig. 9.28). It is possible to get an approximation of the central out-of-plane displacement of each panel by linearly interpolating the strains measured across the panel sections. This has been calculated and is detailed in Table 9.6. This calculation assumes that

Composite materials for blast applications

287

0

325

Distance from center (mm)

Back face

Distance from center (mm)

0

0

2 0

100 0

–2 0

0.5 1.0 Time after detonation (ms)

1.5

–4

MPS (%) 4

300

2

200

0 100 0

(B) Distance from center (mm)

460

4

200

(A) 325

MPS (%)

300

–2

0

0.5 1.0 Time after detonation (ms)

1.5 MPS (%)

4

400

2

300

0

200 100 0

(C)

–4

–2 0

0.5 1.0 Time after detonation (ms)

1.5

–4

Fig. 9.26 Variation of strain with time for (A) the horizontal section of the rear face, (B) the vertical section of the rear face, and (C) the diagonal section of the rear face; for the single core CFRP panel along with a photograph of the rear of the sandwich panel after blast testing.

0

325

Distance from center (mm) 460

2 1 0

100 0 0

–1 0.5 1.0 Time after detonation (ms)

1.5 MPS (%)

300

–2 2 1

200

0

100 0

(B) Distance from center (mm)

0

MPS (%)

200

(A)

325

0

Distance from center (mm)

Front face 300

0

–1 0.5 1.0 Time after detonation (ms)

1.5 MPS (%)

400

1

300

0

200 100 0 0

(C)

–2 2

–1 0.5 1.0 Time after detonation (ms)

1.5

–2

Fig. 9.27 Variation of strain with time for (A) the horizontal section of the front face, (B) the vertical section of the front face, and (C) the diagonal section of the front face; for the graded core CFRP panel along with a photograph of the front of the sandwich panel after blast testing.

no crushing occurs in the panels during the blast which is a simplification; nevertheless the values can be used as an indicator to compare the performance of the different panels. These displacements are calculated using an average of the two face-sheet strains and hence are for the centerline of the sandwich panels. The deflection shown is relative to the edge of the sandwich panel, thus taking into account the deflection of the steel box.

288

Dynamic Response and Failure of Composite Materials and Structures

0

325

Distance from center (mm)

Back face

Distance from center (mm) 460

2

0

0

100 0

–2 0

0.5 1.0 Time after detonation (ms)

300

1.5

2 0

100 0

–2

0

0.5 1.0 Time after detonation (ms)

1.5 MPS (%)

2

300

0

200

(C)

–4 4

400

100 0

–4

MPS (%) 4

200

(B) Distance from center (mm)

0

4

200

(A)

325

MPS (%)

300

–2 0

0.5 1.0 Time after detonation (ms)

1.5

–4

Fig. 9.28 Variation of strain with time for (A) the horizontal section of the rear face, (B) the vertical section of the rear face, and (C) the diagonal section of the rear face; for the graded core CFRP panel along with a photograph of the rear of the sandwich panel after blast testing.

Additionally in Table 9.6 the peak adjusted reflected pressure and the displacement velocity, calculated as the average gradient of the initial deflection of the sandwich panel upon arrival of the blast wave, are listed. The low deflection shows that the CFRP graded core effectively mitigates the blast pressure. It is evident that the implementation of a graded density foam core in a sandwich panel reduces the panel deflection when subjected to blast loading. The graded GFRP displacement at failure was 34 mm compared with 48 mm for the single density GFRP panel and the graded CFRP displacement at failure was 13 mm compared with 50 mm for the single density CFRP panel. Furthermore, the higher stiffness of the CFRP face-sheets results in greater bending moments at the panel edges and hence catastrophic boundary failure.

9.5.5

Postblast damage assessment

Following underwater blast testing, the panels were subjected to X-ray CT scanning. The regions of delamination, cracking, and damage were highlighted in 3D reconstructions of the panels. The reconstruction of the single and graded density GFRP panels is shown in Figs. 9.29 and 9.30, respectively. The solid gray regions are foam or face-sheet material and the red regions are areas of debonding and damage. Table 9.7 provides a summary of the CT scan damage analysis. Out of the GFRP panels there is more damage overall in the panel with the single core. Additionally, the debonding is more severe in the single core panel as shown in Fig. 9.29 where a debond between the front face-sheet and core propagates into a through-thickness crack. The center point of the single GFRP foam core has crushed to 4.0 mm; this intense crushing can be seen in Fig. 9.29 while the graded core crushed slightly less to 6.0 mm as more blast energy was dissipated in debonding.

Summary of the underwater blast test results

Face-sheet type

Core type

Charge size (kg of PE4)

GFRP

Single Graded Single Graded

1 1 1 1

CFRP

Composite materials for blast applications

Table 9.6

Stand-off distance (m)

Peak adjusted reflected pressure (MPa)

1 1 1 1

63 58 64 –

Adjusted reflected impulse at failure (MPa)

Displacement at failure (mm)

Time to failure (ms)

Displacement velocity (m/s)

5.2 5.1 5.2 –

48 34 50 13

0.86 0.81 0.89 0.85

507 699 554 221

289

290

Dynamic Response and Failure of Composite Materials and Structures

(A)

(B)

(C) Fig. 9.29 (A) Three-dimensional reconstruction of single core GFRP panel with debonding highlighted in red, and (B) and (C) cross-section views through the reconstruction.

(A)

(B)

(C) Fig. 9.30 (A) Three-dimensional reconstruction of graded core GFRP panel with debonding highlighted in red, and (B) and (C) cross-section views through the reconstruction.

Composite materials for blast applications

Table 9.7

291

Summary of damage results for the underwater blast

panels GFRP

Fraction of panel containing damage (%) Fraction of the panel with front face-sheet and core debond (%) Fraction of the panel with rear face-sheet and core debond (%) Central point foam thickness (mm)

CFRP

Single

Graded

Single

Graded

7.2 26.9

4.4 32.5

20.6 76.0

10.3 58.1

18.2

9.3

15.2

31.8

4.0

6.0

9.6

13.4

(A)

(B)

(C) Fig. 9.31 (A) Three-dimensional reconstruction of single core CFRP panel with debonding highlighted in red, and (B) and (C) cross-section views through the reconstruction.

The reconstructions of the single and graded density CFRP panels are shown in Figs. 9.31 and 9.32. The single core CFRP panel suffers from almost complete debonding between the front face-sheet and the foam core, as shown in Fig. 9.31A and B, the panel has ultimately failed as it is no longer able to transfer stresses between the face-sheets and core. The graded CFRP panel, shown in Fig. 9.32, suffers from less damage and the core at the central point crushes to 13.4 mm. The single core CFRP panel crushes to 9.6 mm at the center. The X-ray CT scans have revealed that the GFRP face-sheet panels suffer from less debonding but more core crushing and the panels with graded cores suffer from significant damage. This damage, however, takes the form of debonding rather than through-thickness cracking and is a result of the reduced out-of-plane displacement of the panels.

292

Dynamic Response and Failure of Composite Materials and Structures

(A)

(B)

(C) Fig. 9.32 (A) Three-dimensional reconstruction of graded core CFRP panel with debonding highlighted in red, and (B) and (C) cross-section views through the reconstruction.

9.6

Discussion

Study A aimed to demonstrate the blast resilience of three different polymeric foam cores. The panel with SAN foam core had the lowest out-of-plane displacement and pull-out compared with both PVC and PMI. Postblast damage assessment revealed that the SAN core panel suffered from the least amount of damage, the PMI panel was too severely damaged to be sectioned for assessment. This supports the pull-out displacements recorded for the panels as lower core damage results in lower pull-out. SAN foam cores demonstrated the best blast performance out of the foam polymer types tested. In Study B, the performance of a graded density SAN core panel was compared to that of a single density SAN core panel. The implementation of graded cores resulted in a more uniform deflection, and this is expected to be due to core cracks forming in the lower density foam at the front of the panel during bending. This reduces the crack density in the rear of the sandwich panel thus enabling the panel to retain more integrity. However, the graded core suffers from more damage predominantly in the form of debonding. By introducing interfaces between the core layers the energy absorbing potential of the panel is increased as core cracks are arrested at the boundaries and propagate laterally. Although the panel may appear to be more damaged the graded core is not critically cracked and is still able to transfer shear loads thus protecting the skins. A similar relationship was observed for the underwater blast panels. Employing graded cores reduced the out-of-plane displacement of the panels due to their ability to absorb more blast energy via debonding between the core layers. In the absence of a graded core, the single GFRP panel suffered from extensive core crushing and a large displacement while the single CFRP panel suffered from almost complete front face-sheet debonding.

Composite materials for blast applications

293

The sandwich panels were subjected to pressures over 100 times greater in magnitude during the underwater blast experiments than in the air blast experiments. The time period of the underwater pressure impulse was less than one-tenth of that for the air impulse. This accounts for the differing energy absorption and failure mechanisms between the experiments. During underwater blast loading the panels suffered from core crushing (up to 75%) and large strains in the skins (>3%) leading to skin fiber breakage on both front and back skins. During the air blast experiments, the panels suffered from distributed skin to skin core failure but in all cases the back face-sheet remained intact. Additionally, the air blast experiments were far-field whereas the underwater blasts were focused and relatively near-field which further accounts for the shift in damage mechanisms. These large-scale experiments, using commercially available marine composites against actual explosive charges, have revealed the ability of relatively simple composite structures to withstand blast loading. Since these are field experiments; however, there were no repeat experiments meaning that sample data are limited for statistical analysis. The experiments carried out build upon years of composites research within the group ([8,9,21–24]) along with worldwide research and findings between groups that are being confirmed. Nevertheless, the results indicate the advantages possessed by SAN foam and that space and/or weight savings could be made by employing graded density cores rather than thicker single density core materials.

9.7

Summary, trends, and outlook

The three blast studies performed have demonstrated the difference in response of a variety of material combinations to air and underwater explosions. The main findings in summary are as follows: l

l

l

l

SAN foam cored panels exhibited the lowest out-of-plane displacement and lowest panel damage compared to PVC and PMI foam cored panels subjected to the same blast. A stepwise graded density foam core panel reduces core cracking as the interfaces prevent crack propagation, thus reducing the panel out-of-plane displacement. A stepwise graded core in underwater blast loading reduces the out-of-plane displacement of panels with both CFRP and GFRP face-sheets, the effect is more prominent for CFRP face-sheets. In underwater blast loading of graded core panels there is a trade-off between reduced out-ofplane displacement and increased panel damage when selecting the FRPC face-sheets.

Further investigation into the development of blast resilient materials is ongoing including research into the hybridization of composites. Experiments have been carried out on the impact strength of combinations of CFRP, GFRP, and/or other fibrous materials [25–27] and whether the materials can be combined in an optimal configuration to achieve the most desirable properties of each of the constituent materials. It is understood that the two materials can be combined to outperform both of the constituents through a synergistic effect known as the “hybrid effect.” The incorporation of poly-urea into composite sandwich panels has been shown to improve blast performance; its incorporation into hybrid composite panels should be investigated. Furthermore, the choice of resin has an effect on blast performance of composite

294

Dynamic Response and Failure of Composite Materials and Structures

materials [28]. Identifying an optimal resin may improve blast performance for a small and easily implemented manufacturing alteration.

Acknowledgments The authors would like to acknowledge the strong support received from Dr Yapa Rajapakse of the Office of Naval Research (N00014-08-1-1151, N00014-12-1-0403, and N62909-15-12004) in particular for supporting Emily Rolfe, Dr. Mark Kelly, and Dr. Hari Arora during their PhDs; and additionally EPSRC for supporting Emily Rolfe as well during her PhD. Much appreciated is the help and support from GOM UK, LaVision, Slowmo Camera Hire, CPNI, Gurit, DNV, and Paul A. Hooper during the full-scale blast experiments. The authors would also like to acknowledge the micro-VIS center at the University of Southampton for provision of tomographic imaging facilities, supported by EPSRC (grant EP-H01506X).

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[12] Tekalur SA, Shivakumar K, Shukla A. Mechanical behavior and damage evolution in E-glass vinyl ester and carbon composites subjected to static and blast loads. Mar Compos Sandwich Struct 2008;39(1):57–65. Available at: http://www.sciencedirect.com/science/ article/pii/S1359836807000388. [13] Gardner N, et al. Blast mitigation in a sandwich composite using graded core and polyurea interlayer. Exp Mech 2012;52(2):119–33. http://dx.doi.org/10.1007/s11340-011-9517-9. [14] Gardner N, Wang E, Shukla A. Performance of functionally graded sandwich composite beams under shock wave loading. Compos Struct 2012;94(5):1755–70. Available at: http://www.sciencedirect.com/science/article/pii/S0263822311004740. [15] Tekalur SA, Shukla A, Shivakumar K. Blast resistance of polyurea based layered composite materials. Compos Struct 2008;84(3):271–81. Available at: http://www.sciencedirect. com/science/article/pii/S0263822307001985. [16] Wang E, Gardner N, Shukla A. The blast resistance of sandwich composites with stepwise graded cores. Int J Solids Struct 2009;46(18–19):3492–502. Available at: http://www. sciencedirect.com/science/article/pii/S0020768309002303. [17] LeBlanc J, Gardner N, Shukla A. Effect of polyurea coatings on the response of curved E-glass/vinyl ester composite panels to underwater explosive loading. Compos Part B 2013;44(1):565–74. Available at: http://www.sciencedirect.com/science/article/pii/ S1359836812001941. [18] ASTM International. ASTM Standard C365/C365M. Standard test method for flatwise compressive properties of sandwich cores; 2003. Current 2003;i:2–4. Available at: www.astm.org. [19] Kelly M. Comparing the blast tolerance of different composite structures [PhD thesis]. Imperial College London; 2016. [20] Andrews EW, Moussa NA. Failure mode maps for composite sandwich panels subjected to air blast loading. Int J Impact Eng 2009;36(3):418–25. Available at: http://www. sciencedirect.com/science/article/pii/S0734743X08002194 [accessed 21.04.16]. [21] Arora H, Hooper PA, Dear JP. Dynamic response of full-scale sandwich composite structures subject to air-blast loading. Compos A Appl Sci Manuf 2011;42(11):1651–62. Available at: http://www.sciencedirect.com/science/article/pii/S1359835X11002247. [22] Arora H, et al. Compressive strength after blast of sandwich composite materials. Philos Trans R Soc Lond A Math Phys Eng Sci 2014;372(2015). Available at: http://rsta. royalsocietypublishing.org/content/372/2015/20130212.abstract. [23] Kelly M, et al. Sandwich panel cores for blast applications: materials and graded density. Exp Mech 2015;1–22. http://dx.doi.org/10.1007/s11340-015-0058-5. [24] Kelly M, Arora H, Dear JP. The comparison of various foam polymer types in composite sandwich panels subjected to full scale air blast loading. Proc Eng 2014;88:48–53. http:// dx.doi.org/10.1016/j.proeng.2014.11.125. [25] Bouwmeester JGH. Carbon/dyneema intralaminar hybrids: new strategy to increase impact resistance or decrease mass of carbon fiber composites, In: 26th international congress of the aeronautical sciences; 2008. [26] Fallah AS, et al. Dynamic response of Dyneema® HB26 plates to localised blast loading. Int J Impact Eng 2014;73:91–100. http://dx.doi.org/10.1016/j.ijimpeng.2014.06.014. [27] Wonderly C, et al. Comparison of mechanical properties of glass fiber/vinyl ester and carbon fiber/vinyl ester composites. Compos Part B 2005;36(5):417–26. [28] Arikan V, Sayman O. Comparative study on repeated impact response of E-glass fiber reinforced polypropylene & epoxy matrix composites. Compos Part B 2015;83:1–6. Available at: http://www.sciencedirect.com/science/article/pii/S1359836815004813 [accessed 21.04.16].

Progressive bearing failure of composites for crash energy absorption

10

T. Bergmann, S. Heimbs Airbus Group Innovations, Munich, Germany

10.1

Introduction

Energy absorbers are used in numerous occupational safety or transport applications like in passenger cars, railway, or aircraft structures for crash energy absorption. Their purpose is to reduce the loads or decelerations acting on a vulnerable component (e.g., the passenger) typically by controlled deformation or destruction of an energy-absorbing structure. In this context, the crushing and fragmentation of fiber-reinforced polymer matrix materials (FRPs) is well known for its high weight-specific energy absorption capability (SEA) [1]. Consequently, composite crushing absorbers are applied today in many high-performance vehicle structures like in sports cars, high-speed trains, or helicopters. The vast majority of such energy absorbers are loaded in compression. However, load cases also exist that necessitate tensile absorbers to absorb energy under tension, which leads to a completely different loading situation and requires different absorber concepts. Classical examples are fall arrest absorbers for persons or items [2–4]. Existing tensile absorber solutions are mostly based on continuous fracture of stitched seams in textile belts or on highly ductile materials loaded in tension. In this context, composite materials are not competitive due to their brittle nature in tension. Although their off-axis tensile behavior under 45 degrees is slightly more promising [5], they cannot compete with highly ductile polymers or metals. One promising option for using composite materials in tensile absorbers is to pull a bolt in a bearing mode continuously through a composite plate with local fracture in front of the bolt, which enables a relatively constant load level that is beneficial for efficient energy absorption and high SEA values (Fig. 10.1). A first description of this energy absorption concept goes back to Extra [6] for energy-absorbing pilot seat structures made of FRP. Comparable investigations were done by L€utzenburger et al. [7] and Olschinka et al. [8,9] integrating this energy absorption concept into the connection of helicopter and aircraft passenger seats. Pein et al. [10–15] were the first authors investigating the concept systematically in terms of the influence of material, trigger mechanisms, and geometrical parameters. Their target application was the integration into the connections of the overhead stowage bins of commercial aircraft structures for load-limiting purposes under tension during emergency landing conditions.

Dynamic Response and Failure of Composite Materials and Structures. http://dx.doi.org/10.1016/B978-0-08-100887-4.00010-X © 2017 Elsevier Ltd. All rights reserved.

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Dynamic Response and Failure of Composite Materials and Structures

F Ftrig

Fmax

Eideal

II Fm,max

Fm

Fm,min

K

Eabs

III Eelast sabs smax l0

I

s

Eabs Eelast Eideal Fm Fmax Ftrig K l0 sabs smax

… Absorbed energy … Elastic energy … Absorbed energy (ideal) … Mean force … Maximum force … Trigger force … Stiffness … Absorber length … Absorption length … Maximum absorption length

Fig. 10.1 Schematic of typical force-displacement diagram of an energy absorber.

F

F

F

F

F Progressive bearing failure

(1) Transverse shearing

(2)

Laminate bending

(3)

Brittle fracturing

(4)

Local buckling

Fig. 10.2 Concept scheme and failure modes of composite materials under compressive loading.

The general physical phenomena of damage and fracture of composite materials in the local compression zone in front of the bolt have been described by Farley and Jones [16]. They define four principal failure modes (modes 1–4, Fig. 10.2), which are transverse shear failure (mode 1), laminate bending failure (mode 2), brittle failure (mode 3) being a combination of modes 1 and 2, and local buckling and folding (mode 4). Modes 1–3 mostly occur in brittle FRP materials with crack development, local fragmentation, and internal friction as the main energy-absorbing mechanisms. Mode 4, on the other hand, is typical for more ductile FRPs with plastic deformation as an additional energy-absorbing phenomenon. The influence of fiber and matrix material, plate thickness, and bolt diameter as well as loading and environmental conditions on these energy absorption mechanisms in a composite or sandwich bearing-mode absorber has not yet been studied in sufficient detail. Therefore, the aim of this chapter is to present a comprehensive experimental investigation for this purpose and, furthermore, to evaluate analytical and numerical techniques for adequate prediction and simulation of the force-displacement behavior of a bearing-mode absorber focusing on the mean absorber force, which is crucial especially in the early concept and design phase.

Progressive bearing failure of composites

10.2

301

Experimental study

In order to assess and characterize the progressive bearing energy absorption concept in terms of general functionality, influencing parameters, and energy absorption mechanisms, a comprehensive experimental study is presented in the following section focusing on the influence of parameters such as material, geometry, loading, and environmental conditions on the overall force-displacement behavior.

10.2.1 Materials and manufacturing In order to investigate the influence of fiber and matrix material on the energy absorption capability, monolithic composite plates were manufactured consisting of various fiber and matrix material systems. Due to the generally higher weight-specific energy absorption of composite materials made of woven fabrics compared to unidirectional fiber-reinforcement under progressive crushing [17,18] or bearing [15], this study focuses on woven fabric materials. Table 10.1 gives an overview of the fiber and matrix materials used in the experimental study. Six different fiber materials were involved (carbon, glass, aramid, liquid crystal polymer (Vectran®), ultra-high molecular weight polyethylene (Dyneema®), and steel fiber fabrics) as well as different thermoplastic (PEEK, PA) and thermoset (toughened and untoughened epoxy, phenolic) matrix systems. Each of the investigated material configurations A to J in Table 10.2 can be described by the abbreviations of the fiber type, matrix type, weave type (P ¼ plain weave, T ¼ twill weave, S ¼ satin weave), and fiber areal weight (FAW) separated by a dash. Hence, the reference configuration A is defined as “CF-EP-P-200,” indicating the fiber type to be HTA carbon fiber, the matrix type to be untoughened epoxy resin RTM 6, the weave type to be plain weave and the FAW to be 200 g/m2. After plate manufacturing specimens in both 0/90 degree and 45 degree directions relative to the warp and weft direction of the fabric were prepared in order to investigate the influence of the fiber orientation. The test campaign was extended to sandwich plates with phenolic-impregnated aramid paper honeycomb cores (Nomex®) and glass fiber-reinforced skins of different thicknesses (referred to as CP-A to CP-C). The total sandwich thicknesses ranged from about 9.3 to 12.3 mm, the skin thicknesses from 0.4 to 1.6 mm. The lateral core material had to be removed partially and replaced by wooden ledges to enable the clamping of the specimen edges. The investigated sandwich structures are summarized in Table 10.3. Due to the different mechanical behavior of honeycomb core material in L (direction of the parallel cell walls) and W (direction of expansion) directions caused by the manufacturing process, sandwich specimens were prepared in both L and W directions. The dimensions of the monolithic composite and sandwich specimens as well as the trigger geometry are shown in Fig. 10.3.

10.2.2 Testing and evaluation procedure For the first test campaign a special device was developed that allows for the clamping of the composite and sandwich plates with a size of 150 mm 75 mm up to thicknesses of about 20 mm via six fixed bolts (Fig. 10.4). A hardened steel bolt with a

302

Dynamic Response and Failure of Composite Materials and Structures

Details of the fiber and matrix materials used in the experimental test campaign

Table 10.1

Abbreviation

Description

Weave type

FAW [g/m2]

Thread count [cm21]/[cm21]

Yarn count [tex]

Plain weavea Plain weavea Plain weavea 2/2 twill weavea

200

55

200 (3k)

390

6 6.5

340/272

170

6.7 6.7

121

200

66

167

Plain weavea Plain weaveb

130

86

88

1195

6.5 9.5

500/485

Fiber code CF GF AF VF

DF SF

HTA carbon fiber E-glass fiber HM aramid fiber HM Vectran® fiber Dyneema® SK65 fiber Steel fiber 316L

Matrix code EP EPT EPRT PEEK PA PF

Epoxy resin HexFlow® RTM 6 (180°C) from Hexcel Epoxy resin PRISM™ EP 2400 with thermoplastic toughener (180°C) from Cytec Epoxy resin EPIKOTE™ L 20/EPIKURE™ EPH 161 hardener (curing at room temperature) from Momentive Specialty Chemicals Thermoplastic polyether ether ketone (Tm ¼ 340°C) Thermoplastic polyamide 6 (Tm ¼ 220°C) Phenolic resin (135°C)

a

Lange + Ritter GmbH. NV Bekaert.

b

diameter of 8 mm was chosen to be pulled through the plates with the results of this test configuration being directly comparable to the studies by Pein et al. [15]. All quasi-static tests were performed at room temperature (RT) on a Zwick universal testing machine (100 kN load cell) with a loading rate of 200 mm/min. Additional high-rate dynamic tests with a loading rate of 3 m/s were performed on a servo-hydraulic Zwick HTM testing machine (50 kN load cell) in order to evaluate potential strain rate effects and to expose the absorption concept to realistic loading rates of a highly dynamic energy absorption event. The key indicator for the performance of an energy absorption concept is the SEA on material level expressed as Es, which is calculated by dividing the absorbed energy Eabs (area under the measured force-displacement curve, Fig. 10.1) by the mass of the destroyed material mabs (Eq. 10.1). The ideal destroyed material volume Vabs,ideal

Conf.

Material abbreviation

A B C D E F G H I J

CF-EP-P-200 CF-EPT-P-200 CF-PEEK-P-200 GF-EP-P-390 GF-PA-T-600 GF-PF-S-296 AF-EP-P-170 VF-EP-T-200 DF-EPRT-P-130 SF-EP-P-1195

Orientation 0/90 degree 0/90 degree 0/90 degree 0/90 degree 0/90 degree 0/90 degree 0/90 degree 0/90 degree 0/90 degree 0/90 degree

45 degree 45 degree 45 degree 45 degree n/a 45 degree 45 degree 45 degree 45 degree 45 degree

Plies

t [mm]

ρ [g/cm3]

FVF [%]

8 8 8 8 4 6 8 8 8 4

1.80 1.80 1.76 2.70 2.12 1.65 1.77 2.12 2.26 1.62

1.46 1.50 1.55 1.78 1.80 1.77 1.30 1.28 1.06 4.93

51 51 52 45 47 42 50 53 47 55

Progressive bearing failure of composites

Overview of investigated monolithic material configurations with information on average thickness (t), density (ρ), and fiber volume fraction (FVF)

Table 10.2

303

304

Dynamic Response and Failure of Composite Materials and Structures

Overview of investigated sandwich structure configurations with information on total thickness (t) and global density (ρ) Table 10.3

Conf.

Skin material

Core material

t [mm]

ρ [g/cm3]

CP-A

0.36

12.30

0.36

CP-C

GF/EP fabric prepreg

Kevlar® honeycomb, 72 kg/m3, 4.0 mm cell size Kevlar® honeycomb, 72 kg/m3, 4.0 mm cell size Kevlar® honeycomb, 48 kg/m3, 3.2 mm cell size

10.10

CP-B

GF/PF fabric prepreg, GF/EP UD prepreg GF/EP fabric prepreg

9.26

0.20

150 110

A-A (Sandwich)

A-A (FRP)

A

90°

75

Ø8 90° 0°

A

Ø 6,6

Load introduction of the testing device

t

t

Fig. 10.3 Dimensions of the specimens and trigger mechanism (in mm).

(Eq. 10.2) is defined as the plate thickness t multiplied by the bolt diameter d and the displacement sabs. Due to the lateral extent of damage in the composite specimen, which is, based on micrographs (Figs. 10.5 and 10.6), typically 20% or 50% higher than the bolt diameter for brittle and ductile fibers, respectively, the actual destroyed material volume (Eq. 10.3) is increased by a corresponding damage factor fdam of 1.2 or 1.5, resulting in a more conservative definition of the Es value on material level. An alternative definition is the SEA based on final energy absorber level, which takes into account the mass of all material that is necessary to achieve the absorber performance, including the surrounding material, support structures, and load introduction devices.

Progressive bearing failure of composites

305

Bolt (d = 8 mm) Tensile strut Specimen Clamping area

Fig. 10.4 High-speed video images of tensile test with 3 m/s (conf. A (CF-EP-P-200), 0/90 degree orientation).

d = 8 mm

9.60 mm damage zone (+20%)

1 mm

Fig. 10.5 Micrograph of the damage zone in lateral direction of configuration C with brittle carbon fibers (CF-PEEK-L-200).

Eabs 1 Es ¼ ¼ mabs mabs

ð sabs F ds

(10.1)

0

mabs,ideal ¼ ρ Vabs, ideal ¼ ρ t d sabs

(10.2)

mabs ¼ fdam mabs,ideal ¼ fdam ρ t d sabs

(10.3)

Besides the SEA, there are further parameters for a general characterization and comparison of energy absorption concepts that can be evaluated from experimental force-displacement curves. These include among others the load efficiency (AE)

306

Dynamic Response and Failure of Composite Materials and Structures

d = 8 mm

12 mm damage zone (+50%)

1 mm

Fig. 10.6 Micrograph of the damage zone in lateral direction of configuration G with ductile aramid fibers (AF-EP-L-170).

and load band width, the geometric efficiency (GE) also known as stroke efficiency (SE), the energy efficiency (EE), and the volumetric energy absorption (VEA), which are discussed in further detail in Refs. [15,19–22]. In a second test campaign the geometrical concept parameters, plate thickness t and bolt diameter d were investigated. For this purpose, another test device was developed, which is capable of testing different bolt diameters up to 16 mm. Furthermore, the improved test device allows fragments, which are generated during the progressive bearing failure of the composite material, to be removed from the damage zone in front of the bolt. These fragments tended to jam within the test device used for the first test campaign leading to non-reproducible force-displacement behavior at high bolt displacements especially in the quasi-static testing.

10.2.3 Influence of fiber material The comparison of the different fiber materials in a similar plain weave fabric configuration in 0/90 degree orientation leads to the highest Es values for the material configurations A, B, C, and D, consisting of brittle fiber materials. Configurations made of ductile fibers, such as G, H, and I show high Es values as well, but due to the higher lateral damage ( fdam ¼ 1.5, Fig. 10.6), the calculation of the SEA leads to lower Es values compared with material configurations made of brittle fibers (compare Fig. 10.12 in Section 10.2.6). Fig. 10.7 shows an overview of tested material configurations in 0/90 degree orientation and their stress-displacement behavior under quasi-static loading of 200 mm/min. Therein the stress is defined as the force divided by the projected contact area, which is the bolt diameter d multiplied by the plate thickness t.

Progressive bearing failure of composites F (GF-PF-S-296)

500

500

400

400

400

300 200 100 0

(A)

300 200 100 0

25

50

75

100

0

(B)

Displacement [mm]

G (AF-EP-P-170)

300 200 100 0

25

50

75

Stress [MPa]

500

400

Stress [MPa]

500

0

100

0

(C)

Displacement [mm]

H (VF-EP-T-200)

300 200 100 0

25

50

75

100

I (DF-EPRT-P-130)

400

100 0

(E)

0

25

50

75

Displacement [mm]

200 100 0

100

(F)

0

25

50

75

Displacement [mm]

Stress [MPa]

500

400

Stress [MPa]

500

400

200

300 200 100 0

100

(G)

25

50

75

Displacement [mm]

75

100

300 200 100 0

0

50

J (SF-EP-P-1195)

500

300

25

Displacement [mm]

400 300

0

(D)

Displacement [mm]

500 Stress [MPa]

Stress [MPa]

D (GF-EP-P-390)

C (CF-PEEK-P-200)

Stress [MPa]

Stress [MPa]

A (CF-EP-P-200)

307

100

(H)

0

25

50

75

100

Displacement [mm]

Fig. 10.7 Test results of the composite material configurations (A) A, (B) C, (C) D, (D) F, (E) G, (F) H, (G) I, and (H) J in 0/90 degree direction under quasi-static loading (200 mm/min).

10.2.4 Influence of fiber orientation The change in fiber orientation from 0/90 degree to 45 degree relative to the bolt direction resulted in an increase of the Es value of about 15%–25% for the configurations A, B, and C made of carbon fiber (Fig. 10.8). However, a comparison of the force-displacement behavior shows stronger oscillation of the force signal compared with the configurations in 0/90 degree direction. Configurations D and F both made of glass fiber do not show a clear trend for the influence of the fiber direction. For the configurations G, H, and I such a comparison is not possible, due to the uncontrolled failure and the lateral rupture of the material resulting in local buckling of the material in front of the bolt (mode 4, Fig. 10.9). Configuration J made of steel fiber material shows no difference in the Es values between 0/90 degree and 45 degree orientation. This is due to the completely different failure behavior, which equals more a lateral shear damage than a progressive bearing failure, resulting in a plastic deformation and local buckling of the material in front of the bolt. Figs. 10.8 and 10.9 show the stress-displacement behavior of configurations A (CF-EP-P-200) and G (AF-EPP-170) under quasi-static loading of 200 mm/min.

308

Dynamic Response and Failure of Composite Materials and Structures 500 0°/90° ±45°

Stress [MPa]

400 300 200 100 0

0

20

40

60

80

100 0°/90°

Displacement [mm]

±45°

Fig. 10.8 Visualization of the influence of fiber orientation using the example of stress-displacement curve of carbon fiber configuration A (CF-EP-P-200).

500

Stress [MPa]

400 300 200 100 0

0°/90° ±45° 0

20

40

60

80

100

Displacement [mm]

0°/90°

±45°

Fig. 10.9 Visualization of the influence of fiber orientation using the example of stress-displacement curve of aramid fiber configuration G (AF-EP-P-170).

10.2.5 Influence of matrix material The configurations A, B, and C all consist of the same carbon fiber material and weave type, but differentiate in the matrix material. Switching from relatively brittle RTM 6 epoxy resin (conf. A) to toughened epoxy resin EP 2400 (conf. B) led to a strong increase of Es of about 18% for 0/90 degree and 45 degree orientation. Using a thermoplastic matrix such as PEEK in combination with carbon fibers resulted in comparable Es values of about 130 kJ/kg (0/90 degree) and 152 kJ/kg (45 degree) as for the toughened epoxy resin EP 2400. For the configurations D, E, and F, all made of E-glass fabric with different matrix materials, the substitution of RTM 6 epoxy resin (conf. D) by PA 6 (conf. E) led to a strong reduction of Es of about 34%. Other than configuration C, in which the thermoplastic PEEK matrix with mechanical properties comparable to RTM 6 epoxy resin

Progressive bearing failure of composites

309

but with higher fracture toughness resulted in an increase of the SEA, the thermoplastic PA 6 led to a decrease in the SEA due to the lower mechanical properties despite its higher fracture toughness. Configuration F with phenolic resin showed by far the weakest performance resulting in the lowest Es values, which is mainly due to the combination of low mechanical properties and the low plate thickness. The investigation in Ref. [15] using a comparable material configuration made of E-Glass fabric and phenolic resin confirms the generally low Es values of this material configuration but also an increase of the SEA with increasing plate thickness. The experimental study by Thornton et al. [23] using crushing tubes results in an increase of the SEA from phenolic over polyester to epoxy resin in combination with glass fibers, confirming the mechanical properties of the matrix material having a strong influence on the energy absorption capability under progressive crushing.

10.2.6 Influence of loading rate High-rate loading of the composite specimens had a significant influence on the energy absorption capability of the tested material configurations. The change in loading rate from quasi-static (200 mm/min) to high-rate dynamic loading (3 m/s) resulted in a significant decrease in mean crushing stress and hence a reduction in weight-specific energy absorption by 20%–40%. This general trend is shown in Fig. 10.10 comparing the stress-displacement behavior of configuration A (CF-EP-P-200) both in 0/90 500 200 mm/min 3 m/s

Stress [MPa]

400 300 200 100

(A)

0

0

20

40

60

80

100

60

80

100

500 200 mm/min 3 m/s

Stress [MPa]

400 300 200 100 0

(B)

0

20

40

Displacement [mm]

Fig. 10.10 Influence of loading rate of configuration A (CF-EP-P-200, t ¼ 1.80 mm) shown in stress-displacement curves in (A) 0/90 degree orientation and (B) 45 degree orientation (d ¼ 8 mm).

310

Dynamic Response and Failure of Composite Materials and Structures 160 140

SEA [kJ/kg]

120 100 80 60 40 20 0 0.001

0.01

0.1

1

10

100

Loading rate [m/s]

Fig. 10.11 Weight-specific energy absorption over loading rate for configuration A (CF-EP-P-200, t ¼ 1.80 mm) in 0/90 degree orientation.

degree and 45 degree orientation under quasi-static and high-rate dynamic loading. This is in agreement with literature data of dynamic crushing of composite structures which also showed a reduction in crushing stress levels up to 10%–30% at comparable or even higher test velocities [15,17,24–31]. This reduction is mainly due to strain rate-dependent mechanical properties of the polymer matrix systems, which tend to embrittle under high-rate dynamic loading having higher strength but lower strain to failure values resulting in an earlier release of the load carrying fibers. Another reason for the strong reduction of the SEA can be found in the friction between the laminate bundles and the load introduction, which can, according to Ref. [28], also vary with loading rate. Testing configuration A in 0/90 degree orientation with additional loading rates of 0.03, 0.3, and 12 m/s resulted in a linear decreasing trend of the weight-specific energy absorption over the logarithm of the loading rate as shown in Fig. 10.11. A general comparative overview on the influence of fiber and matrix material, fiber orientation, and loading rate on the weight-specific energy absorption under progressive bearing loading of all configurations of this study is shown in Fig. 10.12.

10.2.7 Influence of sandwich configuration As mentioned in Section 10.2.1, the test campaign was extended to sandwich plates in order to investigate a potential integration of the absorber concept not only into monolithic composite but also into sandwich structures. Fig. 10.13 shows the stress-displacement behavior of configuration CP-A both in L and W directions under quasi-static (200 mm/min) and high-rate dynamic (3 m/s) loading. Therein the stress is calculated by the measured force divided by the product of the bolt diameter d and the total thickness t of the sandwich structure. It reveals that neither the orientation of the honeycomb core material nor the loading rate has a significant influence on the overall absorber characteristics. This is mainly due to the weak resistance of the core material, which is simply pushed into its cavities by the bolt providing sufficient space for the fragmented material as well as the thin skin layers of about 1.00 mm compared

Progressive bearing failure of composites

311

200 q-s (0°/90°, L direction)

175

dyn. (0°/90°, L direction) q-s (±45°, W direction)

150

SEA [kJ/kg]

dyn. (±45°, W direction)

125 100 75 50 25 0

Fig. 10.12 Comparative overview of weight-specific energy absorption of the investigated composite material configurations (t 2 mm) and sandwich structures under progressive bearing loading in dependence of the influencing parameters including fiber and matrix material, fiber orientation, and loading rate (d ¼ 8 mm).

50

Stress [MPa]

40 30 20 200 mm/min 3 m/s

10 0

(A)

0

20

40

60

80

100

50

Stress [MPa]

40 30 20 200 mm/min 3 m/s

10 0

(B)

0

20

40

60

80

100

Displacement [mm]

Fig. 10.13 Influence of loading rate shown in stress-displacement curves of configuration CP-A in (A) L direction and (B) W direction (d ¼ 8 mm).

312

Dynamic Response and Failure of Composite Materials and Structures

with 2.70 mm of configuration D (GF-EP-P-390) made of a similar glass/epoxy material resulting in a comparatively low bending stiffness of the laminate bundles. Even though the influence of core orientation and loading rate can be neglected for the sandwich configurations (Fig. 10.12), the SEA of about 50–80 kJ/kg is considerably lower than for monolithic composite structures especially if the low material density of 0.20–0.36 g/cm3 is taken into account.

10.2.8 Influence of bolt diameter and plate thickness In a further test campaign, the influence of bolt diameter d and plate thickness t was investigated using the reference material from configuration A (CF-EP-L-200) made of HTA/RTM 6. This choice was made because neither configuration B with toughened epoxy resin EP 2400 nor configuration C with thermoplastic PEEK matrix had a positive effect on the energy absorption capability under high-rate dynamic loading. The lay-up was chosen to be quasi-isotropic [(0/90 degree)/(45/–45 degree)]2ns with n ¼ 1, 2, 3, 4 resulting in multiples of the reference thickness of 1.80 mm (8 plies, n ¼ 1). Hardened steel bolts with diameters of 8 mm (reference), 12 mm, and 16 mm were used resulting in an experimental test matrix shown in Fig. 10.14. The results of the experimental study are summarized in Fig. 10.15, showing a linear relationship between mean absorber force level and bolt diameter d and a nonlinear trend for the mean absorber force level with increasing plate thickness t. This effect is mainly caused by the potential increase of bending stiffness (area moment of inertia) of the laminate bundles in the damage zone with increasing plate thickness and will be n=1

n=2

n=3

F

n=4

Ø 16

Bolt Trigger (45°chamfer)

Ø 12

Removal of fragments

Ø8

FRP (specimen)

Tensile stuts t = 1.80 mm

t = 3.60 mm

t = 5.40 mm

t = 7.20 mm

F/2

F/2

Fig. 10.14 Experimental test matrix for the assessment of the influence of bolt diameter d and plate thickness t on the energy absorption capability under progressive bearing loading.

Progressive bearing failure of composites d = 16

40 35

313

d = 12

200 mm/min

Mean force [kN]

Mean force [kN]

d=8

20 15

25 20

10

5

5 1.8

3.6

5.4

7.2

0

9.0

t = 3.60

15

10

t = 1.80

0

4

Plate thickness [mm]

35

12

16

20

40 3 m/s

3 m/s

35 30

25

Mean force [kN]

30 d = 16

20 d = 12

15

d=8

10

25 20

t = 5.40

15 t = 3.60

10

5 0 0.0

8

Bolt diameter [mm]

40

Mean force [kN]

t = 5.40

30

25

(A)

(B)

200 mm/min

35

30

0 0.0

t = 7.20

40

t = 1.80

5 1.8

3.6

5.4

7.2

9.0

Plate thickness [mm]

0

0

4

8 12 16 Bolt diameter [mm]

20

Fig. 10.15 Mean absorber force level over plate thickness and bolt diameter of configuration A (CF-EP-P-200) under (A) quasi-static (200 mm/min) and (B) high-rate dynamic (3 m/s) bearing loading (quasi-isotropic, d and t in mm).

discussed within the development of an analytical model for the progressive bearing failure in Section 10.3.

10.2.9 Influence of temperature The influence of temperature was investigated within the range of 20°C and 60°C for the reference configuration A (CF-EP-P-200) with a quasi-isotropic lay-up made of carbon fiber woven fabric material with epoxy resin under high-rate dynamic loading of 3 m/s. Within this temperature interval the SEA showed no temperature dependency. This is in agreement with literature data of cylindrical crushing absorbers made of CFRP and GFRP, which show a consistent trend of an almost constant or slightly decreasing SEA with increasing temperature above RT as well [32–38]. In Ref. [32] tubes made of T300 carbon fiber and 934 epoxy resin, which is comparable to the

314

Dynamic Response and Failure of Composite Materials and Structures

composite material of configuration A, were tested within the temperature interval from 200°C to 200°C. Within the temperature range from 20°C to 60°C the SEA shows an almost constant trend and decreases significantly above 150°C.

10.2.10 Summary of experimental findings An extensive experimental study was conducted to investigate influencing parameters such as material, geometry, and loading conditions, as well as environmental conditions on the energy absorption capability of the progressive bearing-mode absorber concept. The achievable quasi-static Es values on material level of 100–200 kJ/kg for monolithic composite plates are very promising and competitive to classical crushing absorbers [1]. However, considerable strain rate effects led to a significant reduction in SEA of about 20%–40% under high-rate dynamic loading of 3 m/s. The investigation of different fiber and matrix materials as well as reinforcement configurations (0/90 degree vs. 45 degree) led to the conclusion that high fiber strength in combination with low strain to failure (brittle fibers) resulted in the highest Es values. Highly ductile fibers had no major advantage in 0/90 degree orientation and led to an uncontrolled failure under local buckling in 45 degree orientation. Matrix materials with high fracture toughness only showed a superior performance under quasi-static loading that vanished under high-rate dynamic loading due to the embrittlement of polymer materials with increasing strain rate. However, the mechanical properties of the matrix have a strong influence on the energy absorption capability as well. The investigation of geometrical parameters using carbon/epoxy laminates resulted in a nonlinear trend for the mean absorber force with increasing plate thickness while the mean absorber force follows a linear trend with increasing bolt diameter. The temperature effect on the energy absorption capability (SEA) of carbon/epoxy laminates within the considered temperature range of 20°C and 60°C has proven to be negligible. The application of sandwich structures for the energy absorption concept under progressive bearing loading is not competitive to monolithic composite structures due to relatively low Es values of 50–80 kJ/kg.

10.3

Analytical study

As shown in the experimental study, neither quasi-static nor high-rate dynamic progressive bearing loading resulted in a linear trend for the mean absorber force with increasing plate thickness. Thus, a preliminary design of the energy absorption concept solely based on a mean crushing or progressive bearing load, for example, from an experimental characterization with an arbitrarily chosen reference geometry having a specific plate thickness and bolt diameter, is not sufficient. In order to reduce time and costs typically associated with experimental studies, the following section is supposed to develop and assess an analytical approach for the prediction of the energy absorption under progressive bearing loading based on the experimental findings in Section 10.2 involving material and geometrical concept parameters.

Progressive bearing failure of composites

315

10.3.1 Model development A first analytical approach for the prediction of the force components acting within the damaged area in front of the load introduction of crushing structures was presented by Fairfull and Hull [39]. The simplified energy-based approach published by Mamalis et al. [40,41] for the prediction of the energy absorption capability of FRPs under crushing loading is based on the inherent failure modes 1, 2, and 3 (Section 10.2) and can be applied to both cylindrical and conical structures of different cross-sections. The following model for the prediction of the progressive bearing failure is based on the approach by Mamalis et al. [40], which was modified to meet the specific characteristics and boundary conditions of this energy absorption concept. According to Ref. [40] the crushing behavior can be subdivided into three different deformation stages. In stage I (0 < s < sI ) the structure deforms elastically up to the trigger load Ftrig, from which the structure starts to fail under local compression (micro-buckling) resulting in a load drop. The required work WI can be set equal to the energy, which is necessary to create an interlaminar crack of the length lc according to Eq. (10.4) with d being the bolt diameter. WI ¼ GIC lc d ¼

ð sI

1 F ds ¼ Ftrig sI 2

0

(10.4)

With progressive deformation, a fracture zone consisting of a wedge of fragmented material separating the laminate into two bundles is built in stage II (sI < s < sII ). The required work WII can be estimated by Eq. (10.5), with φ being the half angle and ls being the length of the idealized wedge of fragments forming an isosceles triangle as shown in Fig. 10.16. In Eq. (10.5) σ 0 is the contact stress acting on the contact area between the wedge of fragments and the laminate bundles.

F Bolt

Fk

A

FR,2

Detail view of the wedge of fragments

B FR,1 F1

Fk

F2

s

B

C

F2 FR,2 j

ls

h

lc

b/2

A

a t/2

C FRP

Fig. 10.16 Geometry and detail view of the idealized damage zone in front of the load introduction (bolt) according to Ref. [40].

316

Dynamic Response and Failure of Composite Materials and Structures

WII ¼ 2

ð φ

ð sII ls dφ d ¼ σ 0 ls F ds 2 0 sI

(10.5)

The load components used herein are normalized to the bolt diameter d being the first assumption of this approach. This normalization differs from the approach in Ref. [40] describing crushing structures in which the corresponding loads are normalized by the mean circumference of the cylindrical structure. The actual energy absorption takes place in the last stage (sII < s) representing the steady progression of the bolt causing the material to fail under progressive bearing. According to Ref. [40], the energy absorption W can be estimated by the sum of the following four energy-absorbing mechanisms (Eqs. 10.6–10.10), which are already adapted to the progressive bearing energy absorption concept: W ¼ Wi + Wii + Wiii + Wiv l

(10.6)

Friction between the laminate, the wedge of fragments, and the bolt (Wi) Wi ¼ 2 ðμ1 F1 + μ2 F2 Þ ðs sII Þ d

l

Continuous bending deformation of the laminate bundles (Wii) Wii ¼ 2

l

(10.7)

ðs ls d φ + F2 φ ds d F2 2 0 sII

ð φ

(10.8)

Central delamination due to interlaminar crack growth (Wiii) Wiii ¼

ð s + lc

GIC d ds ¼ GIC ððs + lc Þ sI Þ d

(10.9)

sI l

Separation of the laminate bundles from the surrounding material (Wiv) Wiv ¼ n

ðs 0

GC

t 2

ds ¼ n

t 2

GC s

(10.10)

Herein, F1 and F2 are the normalized loads acting in the contact areas between the bolt, the laminate bundles, and the wedge of fragments with μ1 and μ2 being the corresponding coefficients of friction (COF). Assuming a constant interlaminar crack growth with the velocity of the load introduction [40], the corresponding work Wiii can be estimated by the interlaminar critical energy release rate GIC under mode I loading. The work Wiv for the separation of the laminate bundles from the surrounding material can be determined by the intralaminar critical energy release rate GC of the laminate in tension and the number of splits n. A detailed description of the solving of the earlier equations for the mean absorber force F ¼ W=s and underlying assumptions can be found in Refs. [19,40] resulting in the following modified analytical approach (Eqs. 10.11–10.15) for the calculation of the energy absorption under progressive bearing loading in dependence of material

Progressive bearing failure of composites

317

and geometrical parameters. The determination of the necessary model parameters will be discussed in Section 10.3.2. W¼

1 1 μ1 + μ 1

sII ðA B + C + DÞ s

1 t2 A ¼ d σx 6 ls μ2 φ ls μ1 ð tan φ + μ2 Þ + + s sII B ¼ ðs sII Þ cos φ 2 cos φ C ¼ GIC ððs + lc Þ sI Þ d ¼ Wiii D¼n

t 2

GC s ¼ Wiv

(10.11)

(10.12)

(10.13) (10.14) (10.15)

10.3.2 Parameter determination and model validation For the application of the analytical model presented earlier (Eq. 10.11), the coefficients of friction μ1 and μ2; the geometrical relations of the damage zone with lc, n, ls, and α; the bending strength of the laminate σ x; and the intralaminar and interlaminar critical energy release rates GC and GIC must be known. The coefficient of friction between crushing structures made of FRP and steel platen having different surface roughness was investigated by Fairfull and Hull [39]. For a polished steel surface, which is comparable to the surface of the hardened steel bolts used in this study, the coefficient of friction μ1 can range from 0.26 to 0.30, which is in good correlation to similar investigations in Refs. [38,40,42]. The coefficient of friction μ2 between the wedge of fragments and the laminate bundles varies from 0.50 to 0.70 according to Mamalis et al. [40]. Sch€on [43] found similar COF for CFRP being in contact to CFRP, which ranges from 0.65 (without wear) to 0.74 (with wear). Due to the unsatisfactory correlation of the analytical equations presented in Ref. [40] for the determination of lc, ls, and α, the geometry of the damage zone (crack length, wedge of fragments) of the tested specimens was evaluated via micrographs. Fig. 10.17 shows the geometry of the wedge of fragments as a function of plate thickness for the configurations tested quasi-statically with a bolt diameter of 12 mm. In consequence of the high-rate dynamic test setup pulling the bolt through the whole length of the laminate, an evaluation of the damage zone via micrographs was not possible for the specimens tested at 3 m/s. The length ls and the angle α of the wedge of fragments follow a linear trend with increasing plate thickness, as shown in Fig. 10.18. With respect to the bolt diameter the damage zone shows a negligible dependency. Thus the geometry of the wedge of

318

Dynamic Response and Failure of Composite Materials and Structures HTA / RTM 6 (d = 12 mm, t = 1.80 mm)

HTA / RTM 6 (d = 12 mm, t = 3.60 mm)

1 mm

(A)

1 mm

(B) HTA / RTM 6 (d = 12 mm, t = 5.40 mm)

HTA / RTM 6 (d = 12 mm, t = 7.20 mm)

1 mm

(C)

1 mm

(D)

Fig. 10.17 Micrographs of the damage zone in front of the bolt for evaluation of the geometry of the wedge of fragments (conf. A (CF-EP-P-200), quasi-static).

fragments can be estimated for the material configuration A (HTA/RTM 6) using Eqs. (10.16), (10.17), with Cl, l0, Cα, and α0 being the slopes and the y-intercepts of the line of best fit. The corresponding values are listed in Table 10.4. Furthermore, the micrographs show a negligible extent of the central crack, which therefore is omitted in the model. ls ðtÞ ¼ Cl t + l0

(10.16)

αðtÞ ¼ Cα t + α0

(10.17)

The intralaminar critical energy release rate GC was taken as a mean value for tension and compression based on the experimental results of compact tension (CT) and

319

3.5

105

3.0

100

2.5

95 Angle a [°]

Length ls [mm]

Progressive bearing failure of composites

2.0 1.5 1.0 0.5

90 85 80

d = 8 mm d = 12 mm

d = 8 mm d = 12 mm

75

d = 16 mm

0.0 0.0

1.8 3.6 5.4 7.2 Plate thickness t [mm]

d = 16 mm

9.0

70

0.0

1.8 3.6 5.4 7.2 Plate thickness t [mm]

9.0

Fig. 10.18 Length ls and angle α of the idealized wedge of fragments as a function of plate thickness t (configuration A (CF-EP-P-200), quasi-static).

compact compression (CC) tests with a similar carbon/epoxy material T300/913 in Ref. [44]. An experimental determination of interlaminar critical energy release rate for mode I loading via Double Cantilever Beam (DCB) tests according to DIN EN 6033 [45] resulted in a value for GIC of 452 J/m2. The bending strength σ x, which is needed for the approach as well, is typically obtained from three- or four-point bending test according to DIN EN ISO 14125 [46]. Additionally, the classical laminate theory (CLT) can be used in combination with the experimentally determined mechanical properties of the single ply (including strain rate effects) and a failure criterion in order to estimate the bending strength. This latter approach resulted in strength values of 550 and 400 MPa for quasi-static and high-rate dynamic loading, respectively, as shown in Table 10.4. Fig. 10.19 shows the mean absorber force (curves) calculated by the analytical model in correlation to the experimental data (data points 3 standard deviation) as a function of the plate thickness and bolt diameter. A detailed evaluation of the corresponding mean forces results in a deviation of the analytical model of 15% from the experimental mean values, which represents a high accuracy considering the assumptions the model is based on. Furthermore, the load distribution in the area in front of the bolt according to Fig. 10.16 is 65% for Fk and 17.5% for F1 (total of 35%), which is comparable to the experimental load distribution of cylindrical GFRP structures with 55%–70% and 45%–30%, respectively, in Ref. [39].

10.3.3 Summary of analytical method Due to the strong dependency of the geometrical parameters of the wedge of fragments, the use of the analytical model as a preliminary design tool in an early design phase is limited. Based on the single material HTA/RTM 6 (configuration A) used within this study, there is no general statement for the geometrical correlations of

Table 10.4

Analytical model parameters used for configuration A (CF-EP-P-200) (quasi-isotropic)

Loading

σ x [MPa]

GC [kJ/m2]

GIC [J/m2]

μ1 [–]

μ2 [–]

n [–]

Cl [–]

l0 [mm]

Cα [degree/mm]

α0 [degree]

lc [mm]

Quasi-static (200 mm/min) Dynamic (3 m/s)

550

100a

452

0.3a

0.65a

4

0.2

1.4

3.0

75

0.0

400

100a

452

0.3a

0.65a

4

0.5b

0.5b

3.0b

75b

0.0

a

Data from literature. Assumptions made for good correlation to the dynamic results.

b

Progressive bearing failure of composites

d = 16

40 35

321

d = 12

t = 7.20

40

200 mm/min

200 mm/min

35

t = 5.40 30 d=8

25 20 15

Mean force [kN]

Mean force [kN]

30

t = 3.60

15 10

5

5

0 0.9 1.8 2.7 3.6 4.5 5.4 6.3 7.2 8.1

0

t = 1.80 4

8

Plate thickness [mm]

35

12

16

20

Bolt diameter [mm]

40

40 3 m/s

3 m/s

35 30

25

d = 16

20

d = 12

15

d=8

Mean force [kN]

30

Mean force [kN]

20

10

(A)

(B)

25

25 20

10

10

5

5

0 0.9 1.8 2.7 3.6 4.5 5.4 6.3 7.2 8.1

0

Plate thickness [mm]

t = 5.40

15 t = 3.60 t = 1.80 4

8

12

16

20

Bolt diameter [mm]

Fig. 10.19 Comparison of the analytical model (lines) and experimental data (data points with 3 standard deviation) under (A) quasi-static and (B) high-rate dynamic loading for configuration A (CF-EP-P-200) (quasi-isotropic, d and t in mm).

the damage zone in front of the bolt being applicable to a different material system or configuration without exceptions. Thus, the geometrical correlations can be a function of the material properties in general. As shown in this section, they can also vary with the loading rate [40]. However, by the use of experimental sampling points, the analytical model can be applied for an extrapolation to different plate thicknesses or bolt diameters. Even though the complex failure mechanisms within the damage zone, such as transverse shear failure, laminate bending failure, and interlaminar crack growth, have been condensed to the maximum bending strength of the idealized laminate bundles, the model is capable of predicting the linear and nonlinear trend of the mean absorber force with the bolt diameter and the plate thickness if all the geometrical and material parameters are known.

322

10.4

Dynamic Response and Failure of Composite Materials and Structures

Numerical study

Besides all experimental studies, a strong interest in numerical methods for assessing the energy absorption capability of composite structures under crushing or progressive bearing failure exists to accurately predict the mean absorber force in an early design phase. In this context, a numerical study was performed in order to investigate if state-of-the-art modeling methods, available in commonly used commercial explicit finite element codes such as Abaqus/Explicit are able to accurately predict the behavior of the energy absorption concept. For that purpose the modeling methodologies of the composite intralaminar and interlaminar material and failure behavior have been investigated and will be addressed in the following section.

10.4.1 Composite material modeling and model development As mentioned in Section 10.1, different damage and failure mechanisms, identified as matrix cracking, interlaminar and intralaminar failure, laminate bending as well as frictional effects, can occur in composite structures under progressive bearing loads, which result in the energy absorption of this absorber concept. In numerical simulations, which are typically a simplification of real phenomena, it is desired that all these potential failure modes and energy absorption mechanisms are covered enabling their occurrence and contribution to the energy absorption in the model as well. Phenomenological approaches such as CZone in Abaqus/Explicit [47] or crash-front algorithm in LS-DYNA [48] are not capable of predicting the force-displacement behavior because they are based on experimental results, such as mean crushing loads taken from a quasi-static or high-rate dynamic compression test of a corresponding crushing structure.

10.4.1.1 Intralaminar modeling State of the art of intralaminar stiffness and failure modeling of the individual composite ply is either the assumption of orthotropic linear elastic stiffness behavior in combination with failure criteria or the utilization of a continuum damage mechanics (CDM)-based model with continuous stiffness degradation after reaching damage initiation. In the current study using Abaqus/Explicit, the woven fabric material was modeled via a user-defined material model (VUMAT) [49] for woven fabric composite materials based on the Ladeve`ze CDM model [50], which uses orthotropic linear elasticity in fiber direction and non-linear description of the in-plane shear deformation. The postdamage behavior (damage evolution) until complete deletion of the ply is controlled by fracture energies in warp and weft direction for compression and tension. In terms of discretization of the composite structures, two different approaches are typically used today. The structure can be modeled either with 2D 4-node shell elements or with 3D 8-node continuum shells, also referred to as thick shells, which are beneficial for transverse shear and contact calculations. As the plates’ length and width dimensions are large compared to the thickness, a 2D modeling approach with

Progressive bearing failure of composites

323

conventional shell elements is appropriate. Even though stresses in thickness direction are neglected, this modeling approach is efficient for modeling thin-walled structures under crushing loads [51] and will be used in this study. Several plies can be modeled within one shell element (“layered shell approach” [52]) by the definition of specific integration points across the element thickness with adequate ply orientations and thicknesses. However, in order to model the delamination of two plies or laminate bundles consisting of several plies, specific delamination interface models are needed between individual 2D or 3D shell element layers, resulting in a “stacked shell approach” [52].

10.4.1.2 Interlaminar modeling Delamination failure as the separation of two plies or laminate bundles plays a significant role in crushing or progressive bearing simulation of composite structures for the interlaminar separation of the laminate by the wedge of fragments and should therefore be implemented in the numerical model. Because delamination cannot be represented inside the shell elements, the laminate needs to be divided into a certain number of sublaminates with cohesive interfaces in-between, which can fail during the simulation according to a specified failure law. Such cohesive interfaces can be accomplished either by contact formulations with failure law (cohesive contact), or by cohesive elements with an adequate material law with failure. Due to higher modeling effort and computational costs of cohesive elements compared to a cohesive contact formulation, the cohesive contact formulation is used in this study. The failure law of the cohesive interface similarly for both approaches is based on the classical cohesive zone model (CZM) [53] with a typically bilinear traction-separation approach, characterized by the critical energy release rates for mode I (GIC) and mode II (GIIC) as the area under the bilinear curves [54]. Besides these two parameters, the maximum interface stresses and the stiffness values Kxx for the elastic interface behavior need to be defined. An interaction between normal (mode I) and shear loading (mode II) can be implemented by the mixed-mode failure criterion from Benzeggagh-Kenane (BK) [55], assuming the behavior in mode II and mode III shear loading to be equal for fiber-reinforced composite materials.

10.4.1.3 Modeling approaches Within this study, four consecutive modeling approaches on both macro- and meso-scopic modeling levels have been investigated numerically. Based on a simple stacked shell approach (approach 1, Fig. 10.20), in which the laminate is modeled by several tied layers of conventional shell elements with an element length of 1 mm, different modeling enhancements and their effect on the global absorber characteristics have been assessed. At that point, modeling of the first approach as a stacked shell approach has two reasons. On the one hand it provides a potential integration of a CZM in a consecutive modeling approach and on the other hand offset meshing in in-plane direction results in more stable force-displacement behavior in the numerical simulation, which tends to show strong oscillation with every element that is deleted [35,56,57]. In approach 2, the CZM was implemented by a cohesive contact between the

324

Dynamic Response and Failure of Composite Materials and Structures Load introduction

Bolt CZM

Laminate

(1) Stacked shell

(2) Stacked shell + CZM

Wedge

(3) Stacked shell (4) Stacked shell + CZM + CZM + Wedge (fragments) + Wedge (fragments) + Load introduction

Fig. 10.20 Modeling approaches (1–4) for progressive bearing failure of composite structures.

sublaminates. Due to the lack of numerical generation and representation of a wedge of fragmented material by using a Lagrangian modeling approach, an additional wedge modeled as rigid body was added to the approaches 3 and 4 changing the model behavior from a simple element deletion approach (approaches 1 and 2) to a more realistic implementation of the real failure and energy absorption mechanisms, such as interlaminar failure, laminate bending, and friction [58–62]. In approach 4 the load introduction parts were included in the model in order the assess additional frictional effect not only between the laminate bundles and the bolt and wedge of fragments, respectively, but also between the laminate bundles and the load introduction, while the fragments are removed from the fracture zone in front of the bolt. For comparison between experimental and numerical results, the reaction force at the bolt modeled as rigid body over bolt in-plane displacement was evaluated.

10.4.1.4 Modeling parameters In order to keep the influence of assumed material parameters within this study low, an extensive experimental characterization of the intralaminar (VUMAT for fabric-reinforced composites) and interlaminar (CZM) material parameters has been conducted for the reference material configuration A made of HTA carbon fiber fabric (plain weave, FAW ¼ 200 g/m2) with RTM 6 epoxy resin. A detailed overview of the material parameters and their determination and validation can be found in Ref. [19]. The intralaminar and interlaminar material models used in this study do not allow for a strain rate-dependent description of the material properties. Thus two different parameter sets have been derived from the experimental characterization, one for a quasi-static loading with 200 mm/min and one for a high-rate dynamic loading with 3 m/s [19]. The intralaminar fracture energies in warp and weft direction for tension and compression GC,1 + , GC,1 , GC,2 + , and GC,2 are essential modeling parameters for the numerical modeling of the energy absorption capability of composite structures. Pinho et al. [44] provide an experimental method for the determination of the critical fracture energies for both fiber tension and compression based on ASTM E 399 [63]. Due to differences in the definition of the experimental fracture energy (relation to the fracture/crack surface) and the numerical fracture energy used in the material model

Progressive bearing failure of composites

325

(relation to the in-plane characteristic element dimension/area), the experimental parameters cannot be transferred to the numerical model. Thus, a mean fracture energy for tension and compression has been derived from the volume-specific energy absorption Ev of the reference material configuration A with t ¼ 1.80 mm and d ¼ 8 mm by normalization with the single ply thickness of 0.225 mm. This parameter is valid for a characteristic element length Lchar of 1 mm and must be adjusted for a deviating characteristic element length by Eq. (10.18). GC,i ¼

Lchar, i GC, ref Lchar, ref

(10.18)

Additional modeling parameters necessary for the numerical representation of the bearing-mode absorber concept are the coefficients of friction of the surfaces in contact during the simulation (according to Section 10.3.2) and the geometry of the wedge of fragmented material (from micrographs, see Fig. 10.17).

10.4.2 Simulation results In the simplified modeling approach 1, the bolt displacement leads to a successive failure and deletion of elements in front of the bolt, comparable to the phenomenological modeling approaches for the crushing behavior of FRP in Refs. [47,48]. In order to reduce the discretization influence on the numerical oscillation of the force-displacement behavior, the sublaminates consist of four plies (m ¼ 4), resulting in two sublaminates having an offset of half of an element length for the reference configuration with eight plies and 1.80 mm thickness, as shown in Fig. 10.21. Due to the strong simplification of the real failure and energy absorption mechanisms, modeling approach 1 is not capable of predicting the energy absorption capability based on geometrical and material parameters. Compared to the experimental findings, this approach shows a linear trend for the mean absorber force as a function of plate thickness and bolt diameter [64], which is in Quasi-static

Dynamic

20

16

Mean force [kN]

Experiment Simulation

Experiment Simulation

18

14 12 10 8 6 4 2 0 0

1.8

3.6

5.4

Plate thickness [mm]

7.2 0

1.8

3.6

5.4

7.2

Plate thickness [mm]

Fig. 10.21 Numerical versus experimental results for approach 1 (d ¼ 8 mm, t ¼ 1.80 mm).

326

Dynamic Response and Failure of Composite Materials and Structures

Quasi-static

20

Experiment Simulation

18

Mean force [kN]

16

Dynamic Experiment Simulation

14 12 10

8 6 4 2

0

0

1.8 3.6 5.4 7.2 0 1.8 3.6 5.4 7.2 Plate thickness [mm] Plate thickness [mm]

Fig. 10.22 Numerical versus experimental results for approach 2 (d ¼ 8 mm, t ¼ 1.80 mm).

contrast to the progressive curve of the experiment. All in all, this simplified approach does not allow for an accurate numerical prediction of the absorber characteristics. The enhancement of modeling approach 2 by a CZM between the sublaminates enables interlaminar failure, which is illustrated by the gray shading (Abaqus visualization parameter CSDMG) in the damaged area next to the bolt in Fig. 10.22. This interlaminar damage is primarily caused by the successive element deletion in front of the bolt and results in an increased mean absorber force of about 5%–10% compared with modeling approach 1 without CZM. However, the mean absorber force still follows a linear trend with increasing plate thickness and therefore it is not capable of predicting the nonlinear relation. Approach 3, combining a CZM between the sublaminates with a wedge of fragmented material in front of the bolt, changes the numerical representation of the composite material deformation and failure from successive element erosion to a laminate bending deformation and failure. The interlaminar failure is caused both by mode I and mode II loading. Compared to the first two approaches, approach 3 enables a more realistic deformation behavior of the composite structure, even though the interlaminar damage is considerably overpredicted based on the laminate discretization as shown in Fig. 10.23. An evaluation of the mean absorber force with increasing plate thickness is shown in Fig. 10.24. It can be seen that with more plies per sublaminate (m ¼ 4) the numerical mean force gets closer to the experimental results compared with the more detailed modeling approach with m ¼ 2. This is mainly due to the lower number of interlaminar interfaces which showed a premature failure leading to a higher bending resistance and thus reaction forces. Nevertheless, both thickness discretization approaches suffer from a weak representation of the interlaminar behavior compared with the experimental findings in Section 10.2, which show almost no interlaminar failure by delamination. While in approaches 1 and 2 the energy absorption was mainly based on the internal energy due to intralaminar and interlaminar failure (evaluation of the energy fractions

Progressive bearing failure of composites

327

t = 1.80 mm m=2

t = 3.60 mm m=4

t = 3.60 mm m=2

Fig. 10.23 Laminate bending deformation in the area in front of the bolt for different laminate discretization approaches over the thickness, with m ¼ number of plies per sublaminate (d ¼ 8 mm).

Quasi-static

20

18

m=2

Mean force [kN]

16 14

Dynamic

Experiment Simulation

Experiment Simulation

(m = 4)

(m = 4)

Simulation

Simulation

(m = 2)

(m = 2)

12 10 8 +

6 +

4 +

2 0 0

1.8

3.6

5.4

Plate thickness [mm]

7.2 0

1.8

3.6

5.4

7.2

Plate thickness [mm]

Fig. 10.24 Numerical versus experimental results for approach 3 (d ¼ 8 mm, t ¼ 1.80 mm).

in Abaqus), approach 3 approves an amount of about 45% of the total energy absorbed due to frictional effects, which correlates to the findings in Section 10.3. Fig. 10.25 shows a comparison of the experimental and numerical force-displacement behavior for quasi-static and high-rate dynamic loading using modeling approach 3 and the corresponding parameter sets for loading rates 200 mm/min and 3 m/s. Beside the numerical representation of the influence of the loading rate, modeling approach 3 is capable of representing the influence of the fiber orientation qualitatively, comparing the mean absorber force, which is about 15.9% (experimental) and 22.5% (numerical) higher for the configuration with 45 degree fiber orientation with respect to the configuration with 0/90 degree orientation. Approach 4, which is similar to approach 3 with additional load introductions modeled as rigid bodies with a corresponding COF, also does not result in a nonlinear relationship between the mean absorber force and the plate thickness (Fig. 10.26). However, the additional surfaces lead on the one hand to a change in deformation

328

Dynamic Response and Failure of Composite Materials and Structures Quasi-static

Dynamic

5.0

5.0 4.0

Experiment Simulation

Experiment Simulation

(m = 2)

(m = 2)

4.5 4.0 Mean force [kN]

4.5 3.5 Force [kN]

Quasi-static (d = 8 mm, t = 1.80 mm)

3.0 2.5 2.0 1.5

3.5 3.0 2.5 2.0 1.5

1.0

1.0

0.5

0.5 0.0 0°/90°

0.0

0 5 10 15 20 25 0 5 10 15 20 25 Displacement [mm] Displacement [mm]

Experiment Simulation (m = 2)

QI Lay-up

45°/−45°

Fig. 10.25 Comparison of experimental and numerical results for approach 3 showing the influence of loading rate and fiber orientation.

Quasi-static

Dynamic

20 18

Mean force [kN]

16

Experiment Simulation

Experiment Simulation

(m = 2)

(m = 2)

14 12 10 8 6 4 2 0

0

1.8

3.6

5.4

Plate thickness [mm]

7.2 0

1.8

3.6

5.4

7.2

Plate thickness [mm]

Fig. 10.26 Numerical versus experimental results for approach 4 (d ¼ 8 mm, t ¼ 1.80 mm).

behavior of the laminate bundles and on the other hand to an increase in energy absorption by frictional effects of about 10%–15% compared with approach 3 without load introduction parts. For clarity reason, the deformation behavior of modeling approach 4 in Fig. 10.26 is shown without visualization of the CZM failure.

10.4.3 Summary of numerical method The simplified modeling approaches 1 and 2, in which the energy absorption is based on a successive element deletion, are not capable of representing real energy absorption mechanisms, such as intralaminar and interlaminar failure due to laminate bending as well as frictional effects. Hence, this simplified approach does not allow for predicting the absorber behavior based on material and geometrical parameters in an early design phase.

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With the enhanced modeling approaches 3 and 4, the general deformation behavior of the composite structure can be represented depending on the laminate discretization. However, the experimental findings of a nonlinear relation between mean absorber force and plate thickness cannot be represented due to the unrealistic and premature interlaminar failure resulting in a reduced bending stiffness and the oversimplification of the real failure mechanisms on micro level (interlaminar crack initiation and growth, frictional effects). Nevertheless, the enhanced modeling approaches are capable of predicting the influence of the fiber orientation and can be used for analyzing the deformation and failure mechanisms in the damage zone in front of the bolt. Even though the intralaminar fracture energies GC can be adjusted in accordance to characteristic element length by Eq. (10.18), not only the simplified modeling approaches 1 and 2 based on a successive element deletion, but also the enhanced modeling approaches 3 and 4 with CZM, wedge of fragmented material, and load introduction modeling, show a strong influence of the element length on the mean absorber force. In conclusion, the predictive capabilities of the modeling approaches presented earlier are limited, even though the adjustment of model parameters such as intralaminar fracture energy GC allows the general simulation of the progressive bearing loading according to test results. However, further improvements of the modeling approaches, for example, enhancement of the interlaminar behavior, have the potential to increase the predictability of the numerical simulation of such energy absorption concepts, even though the numerical simulation will not be capable of representing all the real failure and energy absorption mechanisms in detail due to model discretization limitations.

10.5

Conclusion

The energy absorption capability of composite materials and composite sandwich structures under progressive bearing loading by a bolt being pulled continuously through the plate has been assessed in an extensive experimental, analytical, and numerical study. This is a new use of composites for energy dissipation, which has not been studied much in the past in contrast to crushing of composite tubes. The influence of fiber and matrix material, fiber orientation, plate thickness, bolt diameter, and test temperature on the weight-specific energy absorption capability has been evaluated for woven fabric-reinforced composites and honeycomb sandwich structures under quasi-static loading conditions as well as high-rate dynamic loading with a test velocity of 3 m/s. Such an extensive test campaign for this new kind of composite energy absorbers has not been available in the literature before. The experimental results revealed that brittle fiber materials like carbon or glass in combination with epoxy resin can lead to high SEA values on material level of 100–200 kJ/kg under quasi-static loading conditions. The temperature effect on the absorber characteristics in the tested range of 20°C to 60°C was negligible. The influence of high loading rates was considerable, though, with a reduction of SEA values by 20%–40% for the test velocity of 3 m/s. The relationship between bolt diameter and mean absorber force level tends to be linear, while it is nonlinear between

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Dynamic Response and Failure of Composite Materials and Structures

plate thickness and mean absorber force. This effect is caused by the nonlinear increase of bending stiffness of the laminate bundles in the damage zone with increasing plate thickness. This relationship could also be verified within an analytical model for the prediction of the mean absorber force level based on the major energy absorption mechanisms, that is, laminate bending failure, intralaminar and interlaminar crack development, and friction. The assessment of numerical finite element simulation approaches and their predictive capabilities for this progressive bearing loading with continuous local crushing of the composite plates led to the conclusion that simple models just based on the successive deletion of elements are not able to accurately predict the absorber characteristics. This is mainly due to the fact that the major physical energy absorption mechanisms “mentioned above” are not represented in such a model. Enhanced models incorporating cohesive interfaces and a wedge of fragmented material allow for a more accurate representation of crack development, laminate bending, and complex friction phenomena. Further model improvements have the potential to enable increased predictive quality of simulations of such energy absorbers. All in all, it can be concluded that this energy absorber concept based on the progressive bearing failure of composite plates by pulling a bolt in in-plane direction through the laminate shows the benefit of stable, constant crushing load levels, comparably high-specific energy absorption values of 100–200 kJ/kg and, moreover, the important capability to adjust the absorber load level according to individual requirements by the appropriate selection of composite material, plate thickness, and bolt diameter. Even inconstant absorber force curves can be obtained by selective increase of the plate thickness in front of the bolt. Another great advantage of this absorber concept is the potential of integration into existing structures, for example, of composite vehicles for weight minimization. Existing bolted joint connections or existing composite panels can be used for this energy absorber reducing additional parts and weight.

Acknowledgments This study was performed within the LuFo IV-4 project INCCA, funded by the German Federal Ministry for Economic Affairs and Energy (BMWi). The financial support is gratefully acknowledged. Additionally, the authors wish to thank Brian Bautz, Christoph Breu, and Georg Tremmel from the Airbus Group Innovations for supporting the design and manufacturing of the hardware; Kurt Pfeffer from the Airbus Group Innovations for performing the quasi-static tests; and Sebastian Schmeer, Benedikt Hannemann, and Stefan Gabriel from the Institute for Composite Materials (IVW) at TU Kaiserslautern for the high-rate dynamic testing activities.

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[3] Goh YM, Love PED. Adequacy of personal fall arrest energy absorbers in relation to heavy workers. Safety Sci 2010;48:747–54. [4] Baszczynski K. Influence of weather conditions on the performance of energy absorbers and guided type fall arresters on a flexible anchorage line during fall arresting. Safety Sci 2004;42:519–36. [5] Bergmann T, Heimbs S, Maier M. Mechanical properties and energy absorption capability of woven fabric composites under 45° off-axis tension. Compos Struct 2015;125: 362–73. [6] Extra W. Mechanische Absorptionsvorrichtung. Patent, DE 19926085 A1; 2000. [7] L€utzenburger M, Johnson A. Helisafe D33-2a—development of a compo-site seat absorber element [Technical report]. Deutsches Zentrum f€ur Luft und Raumfahrt (DLR); 2002. [8] Olschinka C, Schumacher A. Dynamic simulation of flight passenger seats. In: 5th LS-DYNA users conference, Ulm, Germany, October 12–13; 2006. p. 41–58. [9] Olschinka C, Schumacher A. Flight passenger seats—research on crash load cases. Mobiles – Fachzeitschrift f€ur Konstrukteure 2006;32:62–4. [10] Pein M, Krause D, Heimbs S, Middendorf P. Hybrid composite materials for a highly integrated energy-absorbing concept for A/C cabin interior. In: 5th triennial international aircraft fire and cabin safety research conference, Atlantic City, USA, October 29–November 1; 2007. [11] Pein M, Krause D, Middendorf P. Mechanische Absorptionsvorrichtung. Patent, DE 102007014464 A1/B4; 2008. [12] Pein M, Laukart V, Feldmann DG, Krause D. Concepts for energy absorbing support structures and appropriate materials. In: 25th International Congress of the Aeronautical Sciences (ICAS 2006), Hamburg, Germany, September 3–8; 2006. p. 1–9. [13] Pein M, Krause D, Heimbs S, Middendorf P. Innovative energy-absorbing concepts for aircraft cabin interior. In: International workshop on air-craft system technologies (AST 2007), Hamburg, Germany, March 29–30, 2007; 2007. p. 375–84. [14] Pein M, Gumpinger T, Krause D. Innovative approaches for integration of functions in composite sandwich structures by the example of cabin interior. In: 1st European air and space conference (CEAS 2007), Berlin, Germany, September 10–13; 2007. p. 747–51. [15] Pein M. Entwicklungsmethode f€ur Hochleistungswerkstoffe am Beispiel von Energieabsorbern f€ur Flugzeugkabinen [Dissertation]. Technische Universit€at Hamburg-Harburg; 2009. [16] Farley GL, Jones RM. Crushing characteristics of continuous fiber-reinforced composite tubes. J Compos Mater 1992;26(1):37–50. [17] Marsolek JS. Energieabsorptionsverhalten zylinderschalenf€ ormiger Strukturelemente aus Metall und Faserverbundwerkstoff [Dissertation]. Rheinisch-Westf€alische Technische Hochschule Aachen; 2002. [18] Kim JS, Yoon HJ, Lee HS, Kwon TS. Energy absorption characterization of composite tubes for railway application. In: 17th international conference on composite materials (ICCM17), Edinburgh, Scotland, July 27–31; 2009. [19] Bergmann T. Beitrag zur Charakterisierung und Auslegung zugbelasteter Energieabsorberkonzepte mittels experimenteller, analytischer und numerischer Methoden [Dissertation]. Technische Universit€at Kaiserslautern; 2016. [20] Ezra AA, Fay RJ. An assessment of energy absorbing devices for prospective use in aircraft impact situations. In: Herrmann G, Perrone N, editors. Dynamic response of structures. New York: Pergamon Press Inc.; 1972. p. 225–46.

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Thin-walled truncated conical structures under axial collapse: Analysis of crushing parameters

11

S. Boria University of Camerino, Camerino, Italy

Nomenclature R, H T Φ h dc α, β Wb, Wh Wc Wf Wt, We M σ0 ε G μ1, μ2 P δ, s x vid Ra ω c1, c2 pbest gbest

11.1

top mean radius and axial length of tube wall thickness of tube slope of the conical shell internal and external frond length frustum radius at the crack tip external and internal bending angle work required for petals bending and for petals formation work required for circumferential delamination energy dissipated due to friction total energy dissipated and external work bending moment of the laminate ultimate stress in uniaxial tension of the laminate hoop strain critical strain energy release rate per unit interlaminar delaminated crack area coefficient of friction between frond and platen, and between the wedge and the fronds mean crushing load displacement in a single progression and total crushing variable in D-dimension dth component of the velocity vector of the ith particle uniformly distributed random variable inertia weight acceleration coefficients component of the local best component of the global best

Introduction

The collapse behavior of cylindrical and conical shaped shells of round, square, hexagonal, and elliptical cross sections has received attention in the recent years, because of their possible application to the design of safety structures able to absorb a large Dynamic Response and Failure of Composite Materials and Structures. http://dx.doi.org/10.1016/B978-0-08-100887-4.00011-1 © 2017 Elsevier Ltd. All rights reserved.

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amount of energy when axially crushed. Moreover the necessity to reduce the weight in order to contain the gas emissions and reduce pollution implies the use of lightweight engineering materials, such as composites. Such tendency is also strengthened from the greater efficiency as regards the energy absorption capability of such materials with respect to conventional ones [1]. These tubes, if well designed, are able to collapse progressively in stable and controlled manner, through brittle crushing, dissipating large amounts of energy and keeping the load fluctuation during deformation close to the average value, as an ideal absorber. Therefore the mean load of crushing becomes an important factor to estimate while choosing a material and geometry for an impact energy absorbing application. Experimental studies [2–5] indicated that the energy absorbing ability of composites is higher than that of light metal structures. Composites have an advantage in terms of their specific energy absorption (SEA) and they are easy to manufacture and maintain. The energy absorbing mechanism of composites mainly depends on the reasonable structure configuration, ply design, loading rate, and trigger mechanisms. Thornton and Edwards [6] presented a representative work on the crush behavior of composite tubes submitted to axial load. They studied the behavior of various composite tubes, taking into account different fiber types, lay-ups, and thickness to diameter ratio. Their experimental results showed that rectangular and square sections are less effective in energy absorption than circular ones. Failure mechanisms for composite tubes and the influence of geometry and material composition on structural performance were discussed in detail by Hull [7]. Farley and Jones [8] also studied the effect of crushing velocity on the energy absorbing characteristic of composite tubes with different lay-ups. In the Mamalis et al. book [9] many of their results, done on composite tubes with different sections under different loading conditions, were summarized. Based on experimental observations by Mamalis et al. [9,10] thin-walled structures under axial loading can deform in four different modes: deformation confined at impact wall (Mode I), longitudinal crack progression (Mode II), centrally confined circumference crack (Mode III), and large hinge progressive folding (Mode IV). The challenge of design is to arrange the column of material such that the destructive zone can progress in a stable manner (Mode I) due to the large amount of crush energy absorption. Recently many experimental studies have been combined with numerical analysis to predict the final deformation of composite structure for various applications: crushing process of composite materials [11], progressive damage in braided composite tubes [12,13], structural components of a Formula 1 racing car [14], crash-boxes for automotive application based on advanced thermoplastic composite [15], BAR Honda rear impact structure [16], certification of the composite rimp energy absorber for a star Mazda series [17], and composite frontal crash-box for a Formula SAE car [18,19]. In crashworthiness design, the finite element simulations play a very important role thanks to its possibility to predict behavior in early design phases, thus enabling cost reduction and shortening of the design phase [20]. Compared to metallic materials, the modeling of composite ones is much more complicated. There are approaches using a very fine discretization of the composite structure, modeling details of individual tows. Currently these approaches consume too much

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337

computational resources to be applied to analysis of complex structures. So it becomes necessary to develop models that are simple enough to be employed in practical analysis situations but at the same time capable of providing results with a suitable level of accuracy. Very few authors have analyzed the collapse mechanism of composite shells from the theoretical point of view [9,21,22], due to the difficulty to model analytically the brittle behavior and heterogeneity of these composite structures. Mamalis et al. modeled the crumpling and bending process of thin-walled components of fiberglass materials, taking into account the energies involved in total axial crushing. Velmurugan et al. [21] adopted a simpler analytical approach considering only the first load cycle. Each analysis simplified the energy formulation using experimental evidences. The present study exposes an analytical, numerical, and experimental investigation on the failure mechanism, pertaining to the stable mode of collapse (Mode I), of thin-walled composite truncated conical tubes subjected to axial loading in order to predict the mean loads and total displacements during collapse. The analysis is based on previous research results [9,21–24] with the attempt to eliminate some simplifications dictated by experimental evidence and improve the modeling from the mathematical point of view. The theoretical modeling identifies the main internal absorbing energy contributions and equals their sum to the work done by the external load. From this it is possible to explicit the average load, that is a function of several variables. From a minimization process it is possible to obtain the values of the variables that identify the crushing behavior. Together with the theoretical analysis, a finite element modeling is conducted through an explicit dynamic code, such as LS-DYNA. It is possible to capture also numerically the crushing phenomenon with a particular attention during the setting of the numerical parameters and card definitions. The model’s accuracy can be estimated only through an experimental campaign; therefore circular frusta in CFRP material varying some geometrical parameters, such as wall thickness, mean radius, and slope, were tested under axial dynamic loadings. Comparison between analytical, numerical, and experimental investigation as regards mean load and final crushing is good, indicating that the proposed strategy can be a valid approach to predict the energy-absorbing capability of the axially collapsing composite shells, despite the complexity of the phenomenon.

11.2

Theoretical modeling

During the crushing of a composite conical structure under axial impact, after the initial peak, the load tends to oscillate around a mean load P. The formation of a main circumferential intrawall crack of height h at the top end parallel to the axis of the shell wall is responsible for the first sharp drop in the load. As the deformation proceeds further, the externally formed fronds curl downwards with the simultaneous development, along the circumference of the shell, of a number of axial splits followed by splaying of the material strips. The post crushing regime is characterized by the

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Dynamic Response and Failure of Composite Materials and Structures

formation of two lamina bundles bent inwards and outwards due to the flexural damage; they withstand the applied load and buckle when the load or the length of the lamina bundle reaches a critical value. At this stage, a triangular debris wedge of pulverized material starts to form; its formation may be attributed to the friction between the bent bundles and the platen of the hammer mass. Only the first cycle of progressive crushing was taken into account in the theoretical model, because the total deformation is characterized by successive cycles that are repeated with a similar trend. The idealized model of the crush zone is shown in Fig. 11.1, where R is the mean radius, H the axial height, T the thickness of the shell, and ϕ the slope in degree of the tube. In order to simplify the deformation mechanism, the following hypotheses were adopted: the length h of the internal frond is equal to the external one; the transition between straightened and bended zone is sudden, so the central crack is placed in A as shown in Fig. 11.1B; no elastic energy associated with the first impact phase was considered, because of very low respect to the other contributions. Therefore internal energy is absorbed in four principal modes during the formation of crush zone in progressive crushing of tubes [25]: work required for bending of petals (Wb), work required for petal formation (Wh), work required for circumferential delamination (Wc), and energy dissipated due to friction between the debris wedge and fronds and between fronds and platen (Wf). Follow in detail the expressions used for the various energy contributions.

11.2.1 Bending energy As the crushing process initiates, fibers bend both inside and outside the shell radius. Let t1(t2) and β(α) be the thickness of the fiber layers bending inside (outside) the shell radius and the internal (external) bending angle, respectively. By construction t2 is equal to T t1 and the angle β is α 2ϕ. Assuming that the fiber layers are perfectly

P R T

R

h

t2

a

X f

A

H

b

P4 T4 a

t1 h

h

P3 b

T3

h A T

f

(B)

P1

B

T

(A)

Pw

P2

B

d

f

(C)

Fig. 11.1 Conical shell model: (A) shape before loading, (B) idealized crush zone configuration, and (C) force model.

dc

Thin-walled truncated conical structures under axial collapse

339

plastic during bending, the work required to bend the fiber inside and outside the shell radius can be expressed as n h i o π Wb ¼ Wb1 + Wb2 ¼ 2πdc ðM1 α + M2 βÞ ¼ σ 0 dc α t21 + ðT t1 Þ2 2ϕt21 , 3

(11.1)

where dc :¼ R T=2 + t1 + h sin ϕ

(11.2)

is the frustum radius at the crack tip (Fig. 11.1C), M1 and M2 are the bending moments of the internal and external laminate respectively and σ 0 is the ultimate stress in uniaxial tension of the laminate.

11.2.2 Hoop energy Let us assume that the hoop strain varies linearly between A and B (see Fig. 11.1) that is it is null in A and maximum in B. The expression for hoop strain energy in a single crush is Wh ¼ Wh1 + Wh2 ¼

ðh

σ 0 jε1 jdV1 +

0

ðh

σ 0 jε2 jdV2

0

¼ σ 0 πh2 ½T ð sin ðα ϕÞ + sin ϕÞ 2t1 sin ϕ,

(11.3)

where ε1 and ε2 are the hoop strains corresponding to the layers bending inside and outside the shell radius, and dV1, dV2 are the differential volume for the inside and outside layers, respectively.

11.2.3 Crack energy The energy required for circumferential delamination in a single stroke is Wc ¼ 2πhGdc ,

(11.4)

where G is the critical strain energy release rate per unit interlaminar delaminated crack area and dc verifies Eq. (11.2). G is determined experimentally through DCB test as per ASTM D5528-01 [26].

11.2.4 Friction energy After the formation of the internal and external fronds, normal stresses develop on the sides of the debris wedge followed by shear stresses along the same sides due to friction at the interface between the wedge and the fronds. Moreover additional normal and shear stresses develop at the interface between the steel plate and the deforming shell as the formed fronds slide along the interface.

340

Dynamic Response and Failure of Composite Materials and Structures

The energy dissipated in frictional resistance for a crush distance δ is Wf ¼ 2πdc δ½μ1 ðP1 + P2 Þ + μ2 ðP3 + P4 Þ,

(11.5)

where μ1 is the coefficient of friction between frond and platen, μ2 the coefficient of friction between the wedge and the fronds. P1 and P2 are the normal force per unit length applied by the platen to the fronds. P3 and P4 are the normal force per unit length applied to the sides of the wedge, as shown in Fig. 11.1. Due to feasibility μ1, μ2 2 ]0,1[ and by construction the deformation is given by δ ¼ h½ cos ϕ cos ðα ϕÞ:

(11.6)

Static equilibrium at the interface yields P ¼ 2πdc ðPw + P1 + P2 Þ,

(11.7)

where Pw is the normal force per unit length applied by the platen to the debris wedge given by Pw ¼ P3 sin ðβ + ϕÞ + T3 cos ðβ + ϕÞ + P4 sin ðα ϕÞ + T4 cos ðα ϕÞ: T3 and T4 are the frictional forces per unit length developed between wedge and fronds. Assuming that Coulomb friction prevails between the debris wedge and fronds, Ti ¼ μ2Pi for i ¼ 3, 4. Note that Pi ¼ σ 0 h for i ¼ 3,4,

(11.8)

Pw ¼ 2σ 0 h½ sin ðα ϕÞ + μ2 cos ðα ϕÞ:

(11.9)

Therefore the friction energy (11.5) is given by Wf ¼ μ1 h½ cos ϕ cos ðα ϕÞP + 4πσ 0 dc h2 ½ cos ϕ cos ðα ϕÞ ½μ2 μ1 sin ðα ϕÞ μ1 μ2 cos ðα ϕÞ:

(11.10)

11.2.5 Mean load P The total energy dissipated for the deformation of the shell is given by the sum of the bending energy (11.1), the hoop energy (11.3), the energy required for circumferential delamination (11.4), and friction energy (11.10), i.e., Wt :¼ Wb + Wh + Wc + Wf ,

(11.11)

and it is equal to the work done by the external load P on the crushing displacement δ in a single progression, i.e.,

Thin-walled truncated conical structures under axial collapse

We :¼ Pδ ¼ Ph½ cos ϕ cos ðα ϕÞ:

341

(11.12)

Expliciting the mean load, it is obtained that P is a function of three variables h, t1, α and depends on three geometric parameters ϕ, R, and T, i.e., Pðh, t1 , α; ϕ, R, T Þ

π 1 σ 0 dc α t21 + ðT t1 Þ2 2ϕt21 + ¼ ð1 μ1 Þhð cos ϕ cos ðα ϕÞÞ 3 +2hGdc + σ 0 h2 ðT ð sin ðα ϕÞ + sin ϕÞ 2t1 sin ϕÞ +

+4σ 0 h2 ð cos ϕ cos ðα ϕÞÞdc ðμ2 μ1 sin ðα ϕÞ μ1 μ2 cos ðα ϕÞÞ : (11.13) Note that the domain of the function P is given by D ¼ 0, H ½0, T 2ϕ, π=2 ϕ

(11.14)

indeed for feasibility the crush length is strictly positive, the thickness of the plies bending outside belong to the interval [0,T] and the external bending angle α is larger than 2ϕ, because β ¼ α 2ϕ 0 and the denominator of (11.13) has to be strictly positive. From a minimization process it is possible to obtain h, t1, and α values that correspond to the final deformation and therefore deduce the average load during crushing.

11.3

Numerical modeling

Together with the theoretical analysis, a finite element (FE) modeling was conducted on truncated conical structures in order to reproduce their crush phenomena also numerically. Among the various explicit dynamic codes LS-DYNA solver was used. In literature, a variety of methodologies for modeling composite structures can be found [20]. The first classification is made according to the degree of mechanical detail: micro approach with respect to macro one [27–30]. The FE models related to the first group try to simulate the composite crushing phenomenon through a detailed modeling of its micro-mechanical behavior. To reproduce the matrix crack propagation a very fine solid mesh is required, involving an increase into computational efforts, often impractical for complex systems. This approach is used mainly to perform simulations able to capture delamination, in which the growth behavior of a single crack is studied in a very detailed way. The macro-mechanical group, instead, provides a general description of the material collapse [31–34]. It is much more computationally effective and, consequently, it is a suitable choice for engineering crash analysis. However, it is not capable to model precisely all the main failure modes that occur during a crush event. The macro-mechanical group, moreover, can be divided into two main different types of models: the single shell layer models and

342

Dynamic Response and Failure of Composite Materials and Structures

the stacked shell models. The first models use a single layer of shell elements to model the specimen; no interlaminar collapse can be modeled, but they are simple with low computational cost and are used when only load and energy level predictions are required. In such cases a careful calibration of the numerous material parameters is necessary to obtain acceptable global results. In the stacked shell models, instead, more layers of shell elements are modeled, with specific elements for joining (such as cohesive elements, tied contact or springs). Such model allows a compromise between accuracy and efficiency depending on the number of layers used; it is capable of providing a better physical representation of the stratified composite structure but at the same time it keeps the simplicity inherent of the macro-mechanical approach. The considered specimens were modeled using the macro-mechanical approach; in particular single shell layer model was implemented. In such cases only the middle surface of the structure was meshed with shell elements of 2.5 mm. Underintegrated shell elements of the Belytschko-Tsai type (ELFORM ¼ 2) with the stiffness-based hourglass control (IHQ ¼ 4) were used for the modeling. This choice was a posteriori justified since the hourglass energy was negligible, less than 1% of the total energy. As regards material implementation, composite constitutive models implemented in LS-DYNA code are continuum mechanics ones. Composites are modeled as orthotropic linear elastic material within a failure surface. The exact shape of the failure surface depends on the failure criterion adopted in the model. Beyond the failure surface, the appropriate elastic properties are degraded according to the assigned degradation laws that can be divided into two main categories: progressive failure (MAT 22, 54/55, 59) or continuum damage (MAT 58, 161, 162). Progressive failure models have shown success [14,29] in axial crushing of composites exhibiting brittle fracture; therefore, for this study, the linear-elastic model #MAT_ENHANCED_COMPOSITE_DAMAGE was used. This material uses the Chang-Chang failure criterion [35] to determine individual ply failure. When all the layers fail, the element is deleted. Elements which share nodes with the deleted element become “crashfront” elements and can have their strengths reduced by using the SOFT parameter [36] with TFAIL (time-step failure parameter set equal to 0.8 in such simulations) greater than zero. Using a single shell layer approach, the laminate lay-up can be defined by one integration point for each single ply with the respective ply thickness and fiber orientation angle. This can be done easily using the card *PART_COMPOSITE, where the laminate has a thickness defined by the sum of each individual layer with the proper fiber orientation. Once all single layers of the shell element fail, the whole element is eroded and it simply disappears from the calculation. Laminate theory is also activated with LAMSHT parameter in *CONTROLL_SHELL card, to correct for the assumption of a uniform constant shear strain through the shell thickness. A “rigid wall planar moving forces” card with a finite mass comparable with that of the impactor and an initial velocity equal to test were adopted. Master-surface to slave-node and self-contact were defined between the impact mass and the nodes of the sacrificial structures and on the structure, respectively. Because measuring the actual friction coefficient between the tube and the rigid plate and between the fronds and the specimen directly by the experiment is difficult, the values of the

Thin-walled truncated conical structures under axial collapse

343

friction coefficients were obtained from literature [5]; in particular values of 0.3 and 0.22 were used for external and internal contact, respectively. As regards the boundary conditions, the same used for the real tests were implemented on the numerical model in order to reproduce similar conditions of loading. The corresponding FEM is shown in Fig. 11.2.

11.4

Experimental observation

In order to analyze the real behavior of composite impact attenuators, tests on circular frusta specimens under dynamic loadings were conducted [37]. The specimens were manufactured using a carbon-epoxy preimpregnated fabric material. While unidirectional material can be more efficient in energy absorption, fabric reinforced materials are often preferred in impact structures because of their in-plane symmetry, which favor the onset of a stable crush. In particular, the carbon fabric CF290 was used with the high strength carbon fiber T800 both for the warp and for the weft; for the matrix the toughened epoxy resin ER450 was adopted. The static mechanical properties for this material are summarized in Table 11.1. As regards the geometry, thin-walled truncated conical tube was chosen (Fig. 11.3A). The specimens have a uniform length of 200 mm with inner top diameter of 25, 35, and 50 mm, wall thickness of 1.5, 2.5, and 4 mm, and wall inclination of 5, 10, and 15 degrees. Subsequently they will be referred by the following nomenclature:

W2- PLANAR_MOVING_FORCES

RIGID PLATE

SPECIMEN

FIXED CONSTRAINT Z Y

X

Fig. 11.2 Finite element model of composite circular frusta.

344

Dynamic Response and Failure of Composite Materials and Structures

Table 11.1

Prepreg static mechanical properties

Property

Composite CF290/ER450 carbon-epoxy fabric prepreg

Density Tensile modulus Tensile strength Compressive modulus Compressive strength Flexural modulus Flexural strength

1.6 106 kg/mm3 70 GPa 700 MPa 60 GPa 400 MPa 70 GPa 700 MPa

T

d

f H

(A)

(B)

(C)

(D)

(E)

Fig. 11.3 (A) Geometry varying parameter, (B) male and female mold, (C) lamination process, (D) specimen before cutting, and (E) final specimen.

“inner top diameter”_“wall thickness”_“wall inclination” (for example 25_1.5_5 corresponds to the specimen with 25 mm in the top inner diameter, 1.5 mm thick, and 5 degrees in inclination). The various geometries were realized by arranging the laminae by means of a “quasi-isotropic” lamination, achieved by interleaving the orientation of the fabric from 0, 90 to +45, 45 degrees. The production technology used is the high pressure autoclave curing process, commonly used by the aerospace industry for manufacturing composite material. In Fig. 11.3B are shown the male and female molds suitably realized for the purpose, a detail of the skins drawing on the conical male mold in aluminum alloy 6082 T6 and the final specimen after cutting. The female mold is made with glass-epoxy prepreg to ensure that under the pressure of the vacuum bag is flexible and able to close on the mold, adhering perfectly to the laminate. Dynamic impact tests were conducted based on a drop test system (Fig. 11.4) available in Picchio SpA. The weight of the drop hammer was set as 301 kg and the fall height was fixed in order to obtain the desired transient velocity of about 4 m/s and ensure that each tube was able to absorb all the impact energy of about 2400 J. The images and the data were obtained using a 1000 frame/s Mikrotron high-speed camera (Fig. 11.4D), a FA3403 tri-axial accelerometer with 500 g full scale (Fig. 11.4C), and a E3S-GS3E4 photocell (Fig. 11.4B). The corresponding test procedure in the dynamic impact is as follows. First, the initial kinetic energy was

Thin-walled truncated conical structures under axial collapse

345

Fig. 11.4 Drop tower (A) and instrumentation, as photocell (B), accelerometer (C), and high-speed camera (D).

obtained by the drop hammer device. A specimen was placed at the center of the drop hammer system and located at the bottom of the mass center of the drop hammer in the vertical direction. An accelerometer was embedded in the top of the weight of the drop hammer and a photocell was fixed on a rail to measure the initial velocity. Second, the drop hammer is released from the fixed height where it starts and falls on the specimen to trigger the deformation of the circular frusta. The axial crushing deceleration data in the crushing process were measured by the accelerometer and filtered with a CFC60 filter. The acquisition data were collected through a digital oscilloscope and were ultimately stored in a computer thanks to an acquisition system. An integration process was conducted for each test in order to obtain the velocity and displacement versus time. Finally, the force-displacement curves were computed from the recorded data. Fig. 11.5 shows, therefore, the diagrams of load versus displacement for all specimens; the value of the force was obtained multiplying decelerations, expressed in m/s2, for impacting mass. From the diagrams it is evident that the rise into the wall thickness imply the increase of the axial stiffness in the first impact phase and the mean load, while a minor variation is appreciated when the only inclination changes. The area under each curve is equal to the absorbed energy and therefore, given similar test conditions except for initial velocity that tends to vary slightly, it is a nearly constant value. Finally some parameters characterizing the crush resistance were calculated (Table 11.2); these include: the peak load Fpeak, the average crushing load Fav, the total crushing δ, the SEA, the average crushing stress σ av, the load efficiency (η), and the crushing efficiency (SE). The average crushing force is evaluated after the initial peak of force, during the stable crush zone. The SEA is the energy absorbed per unit of mass of crushed materials, that is SEA ¼ E/ρAδ where E is the energy absorbed

346

(2)25_2.5_5 (3)25_4_5 (9)25_1.5_10

50 40

(8)50_4_5

80

(6)35_4_5

160

(15)50_1.5_10

(11)25_4_10

60

(19)25_2.5_15 (20)25_4_15

30

180

70

(18)25_1.5_15

(12)35_1.5_10 (13)35_2.5_10 (14)35_4_10

50

(21)35_2.5_15

40

(22)35_4_15

40

10

20

0

0

(A)

Fig. 11.5 Load-displacement dynamic diagrams.

(B)

0

20

40 60 80 100 120 140 Displacement [mm]

(24)50_2.5_15 (25)50_4_15

80

10 40 60 80 100 120 140 Displacement [mm]

(23)50_1.5_15

100

60

20

(17)50_4_10

120

20

0

(16)50_2.5_10

140

30

20

(7)50_2.5_5

(5)35_2.5_5

(10)25_2.5_10

Load [kN]

60

200

(4)35_1.5_5

90

0

(C)

0

20

40

60

80

100

Displacement [mm]

120

Dynamic Response and Failure of Composite Materials and Structures

70

Load [kN]

100

(1)25_1.5_5

Load [kN]

80

Dynamic results for the specimens tested

# Specimen

Nomenclature

Impact velocity [m/s]

Fpeak [kN]

Fav [kN]

δ [mm]

SEA [kJ/ kg]

σ av [MPa]

η [%]

SE [%]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

25_1.5_5 25_2.5_5 25_4_5 35_1.5_5 35_2.5_5 35_4_5 50_2.5_5 50_4_5 25_1.5_10 25_2.5_10 25_4_10 35_1.5_10 35_2.5_10 35_4_10 50_1.5_10 50_2.5_10 50_4_10 25_1.5_15 25_2.5_15 25_4_15 35_2.5_15 35_4_15 50_1.5_15 50_2.5_15 50_4_15

3.85 3.8 3.7 3.8 3.9 3.7 4 3.6 3.9 4 3.8 3.9 3.7 3.7 3.9 3.7 3.65 3.95 3.9 3.65 3.8 3.55 3.8 3.6 3.55

13.9 4.1 42.2 15.7 29.8 88.8 35.4 177.4 13.1 23.3 58.6 16.2 25.2 67.1 18.2 31.9 162.5 10.8 19.6 38.0 23.1 46.6 13.7 21.1 57.4

16.9 24.4 42.0 18.9 32.9 60.0 36.9 100.0 18.4 28.2 48.9 20.0 30.9 60.0 21.9 38.2 76.9 19.2 28.7 48.8 31.4 54.3 20.9 31.7 54.3

130 90 50 116 70 35 65 19 125 85 45 115 68 35 105 55 26 125 80 41 70 35 105 60 35

52.0 48.7 54.8 47.4 53.1 62.3 44.9 78.7 44.2 46.9 57.7 41.1 44.1 58.1 36.5 43.4 57.6 37.3 41.5 53.3 39.5 49.1 30.7 32.7 37.6

97 90 108 84 96 108 84 149 79 84 108 73 82 96 69 77 101 67 75 95 71 93 57 63 73

121 595 99 120 110 68 104 56 140 121 83 123 123 89 120 119 47 178 146 128 136 116 153 150 94

65 45 25 58 35 17.5 32.5 9.5 62.5 42.5 22.5 57.5 34 17.5 52.5 27.5 13 62.5 40 20.5 35 17.5 52.5 30 17.5

Thin-walled truncated conical structures under axial collapse

Table 11.2

347

348

Dynamic Response and Failure of Composite Materials and Structures

obtained by integrating the load over the total displacement δ, ρ is the material density, and A the cross section area. The average crushing stress is the average crushing force divided by the specimen cross sectional area. The load efficiency is defined as the percentage of the peak force to the average crushing load. The crushing efficiency is instead the ratio in percentage between the crushing displacement and the original length of the specimen.

11.5

Results and discussion

The collapse modes observed throughout the dynamic axial compression tests correspond to that mentioned in literature [9]. In particular, all truncated conical tubes tested have absorbed impact energy through a process of gradual crushing, with a complex combination of fracture mechanisms including fiber breakage, pull-out, and matrix cracking, keeping the remaining part completely intact. In particular, two different failure morphologies were noted: splaying with axial splitting and fronds both external and internal (Fig. 11.6A) and total reversal of the laminate internally to the specimen (Fig. 11.6B). Such configurations relate to the first stable failure mode (mode Ia and Ib [38]) of thin-walled composite structures, characterized by a better energy absorption capacity combined with a progressive and controlled crushing. In both cases the collapse initiated at the narrow end of the frustum. The externally formed fronds inverse freely outwardly with also the development of axial splits followed by splaying of the material strips. The internally formed fronds, instead, are turned inside the wall; firmly compacted hoops of material are developed, constraining in this manner the frond to fold inwards. No axial tears are apparent in the internal fronds, which are more continuous than their external counterparts. The major macroscopic crushing mechanisms governing the axial compression of tubes are identified as [24,39]: bending of the inward and outward fronds, hoop breakage of the outward frond, hoop compression of the inward frond, delamination, and friction between steel plate and fronds and between fronds and wedge. In particular, the formation of two lamina bend inside and outside the specimen was detected for all frusta with a wall inclination of 5 degrees and on those at 10 degrees with a thickness of 4 mm. In all other cases a perfect internal inversion was observed, bringing the progressively fractured material during crushing completely into the remaining part. Despite the different deformation modes, no great change in the

r

(A)

(B)

Fig. 11.6 Different failure modes in the specimens tested: (A) Mode I-a; (B) Mode I-b.

Thin-walled truncated conical structures under axial collapse

349

load-crushing behavior can be noted (Fig. 11.5). Therefore both modes Ia and Ib can be considered efficient crushing mechanisms, able to absorb high energy levels. Figs. 11.7–11.9 show the variation of crushing load with shell shortening for both collapse modes, varying wall thickness, and wall inclination and top inner diameter, respectively. It is evident that average load increases with the wall thickness (Fig. 11.7). An increase in thickness of about 60% involves a congruent growth also into the load value. Such behavior is not found varying only the wall inclination; in such case an almost similar trend can be seen (Fig. 11.8). The crushing force varies also with the top inner diameter. Increasing the diameter there is a growth into the load and stiffness values with a reduction of total displacement (Fig. 11.9), although it is less marked than a thickness variation. Fig. 11.10 shows the variation of average load with ratio between top inner diameter and thickness. The growth of load value is much more relevant, increasing thickness respect to diameter, as mentioned before.

Force [kN]

80 70

(1)25_1.5_5

60

(2)25_2.5_5

50

(3)25_4_5

40 30 20 10 0

0

50 100 Displacement [mm]

150

Fig. 11.7 Load trend varying wall thickness only for specimens 25 mm in diameter and 5 degrees in inclination. 35

(1)25_1.5_5

Force [kN]

30

(5)25_1.5_10

25

(12)25_1.5_15

20 15 10 5 0

0

50 100 Displacement [mm]

150

Fig. 11.8 Load trend varying inclination only for specimens 25 mm in diameter and 1.5 mm thick.

350

Dynamic Response and Failure of Composite Materials and Structures

60

Force [kN]

50

(10)25_2.5_10

40

(13)35_2.5_10 (16)50_2.5_10

30 20 10 0

0

20

40 60 Displacement [mm]

80

100

Fig. 11.9 Load trend varying top inner diameter only for specimens 2.5 mm thick and 10 degrees in inclination.

100

Fav[kN]

80 60 40 20 0

0

10

20 Di /t

30

40

Fig. 11.10 Average force values varying diameter/thickness ratio.

Fig. 11.11 shows how the SEA varies with the top inner diameter/thickness ratio (Di/t). It is clear that increasing the inside diameter (Di) of the upper base or by decreasing the wall thickness (t) the SEA value is not influenced much; only a weak linear growth is recorded. The same cannot be said for the slope influence; the SEA tends to decrease with the increase of the wall inclination (Fig. 11.12). It is possible to conduct the analysis also from the average crushing stress point of view. Figs. 11.13–11.15 show how such function tends to change, varying the geometrical parameters. Only with higher wall thickness values the average crushing stress tends to increase (Fig. 11.13); for the other cases (Figs. 11.14 and 11.15) there is a turnaround. Therefore classifying the specimens from the SEA and average crushing stress point of view, the one with 5 degrees in inclination, the smallest top diameter and

Thin-walled truncated conical structures under axial collapse

351

100

SEA [kJ/kg]

80 60 40 20 0

0

10

20 Di /t

30

40

Fig. 11.11 SEA values varying diameter/thickness ratio.

60

SEA [kJ/kg]

50 40 30 20 10 0

5

10 Inclination [Degree]

15

Average crushing stress [MPa]

Fig. 11.12 SEA varying wall inclination.

200 150 100 50 0

0

1

2

3 t [mm]

Fig. 11.13 Average crushing stress varying wall thickness.

4

5

Dynamic Response and Failure of Composite Materials and Structures

Average crushing stress [MPa]

352

200 150 100 50 0

0

5

10 Slope [Degree]

15

20

Average crushing stress [MPa]

Fig. 11.14 Average crushing stress varying wall inclination.

150 130 110 90 70 50

20

25

30

35

40 45 Di [mm]

50

55

60

Fig. 11.15 Average crushing stress varying top internal diameter.

the largest wall thickness seems to be the most efficient. Therefore the specimen no. 3, 25_4_5, represents the best solution to fill the role of impact attenuator. The collapse mode reproducible with the simulation, due to the type of numerical implementation adopted, corresponded to an internal bending of the laminate with axial splitting and formation of small debris. Fig. 11.16 represents the load-displacement trends for all specimens from the numerical point of view. The energy versus stroke diagrams are also shown for all cases. The simulation results of progressive failure for tube 25_2.5_15 is shown in Fig. 11.17 at different times. The simulated damage morphology remains consistent with the test results. The main damage of the composite frusta concentrates on the area where the hammer and the specimen come into contact with each other. The stress concentration of this region can also be observed. The adjacent regional structure is affected and results in reducing the mechanical performance. Some detached debris splash out of the tube wall. Several fiber bundles with curled petals are also observed with the increase in impact load. Overall, despite the simplification adopted, the simulation result shows how the finite element model proposed in this study is able to reflect the anisotropic damage properties of composite materials. Using the theoretical approach suggested, moreover, it is possible to determine the critical values of the length h, the thickness t1, and the opening angle α belonging to the domain D in which the mean load is minimum. The knowledge of the average force allows also to identify the final crushing of the specimen after test. The nonlinear

Thin-walled truncated conical structures under axial collapse

353

Fig. 11.16 Load-displacement and energy-stroke diagrams from the numerical analysis.

function (11.13) is not easy to analytically minimize because the mean load gradient is difficult to nullify, so a numerical optimization method is necessary. In view of generalizing the model discussed in Section 11.2, considering more than just three independent variables, a numerical algorithm, based on the particle swarm optimization (PSO) method [40,41], was implemented. PSO attempts to simulate the “collective intelligence” in animal societies that have no leader in their group. The algorithm starts from a set of randomly generated solutions sharing and exploring the design space wherein an optimal solution will be found. Each particle of the swarm flies through hyperspace basing its search on the memory of its own best position and the knowledge of its neighborhood’s best at the current time step. PSO system combines local search method (through self-experience) with global search methods (through neighboring experience), attempting to balance exploration and exploitation. Nowadays PSO is greatly applied into both scientific research and engineering use due to its computational efficiency, effectiveness, and ductility. Nevertheless PSO method is still

354

(continued)

Dynamic Response and Failure of Composite Materials and Structures

Fig. 11.17 Simulation results of impact damage morphology of the specimen #19 with different crushing displacements.

355

Fig. 11.17, continued

Thin-walled truncated conical structures under axial collapse

356

Dynamic Response and Failure of Composite Materials and Structures

unsatisfying when the optimization problem has a large number of local optima or when it is high-dimensional. Therefore, in recent times, numerous PSO variants have been introduced in order to enhance the global convergence capability and robustness of this method when chosen to solve complex multimodal problems. Given the variables x(k) ¼ (x1(k), x2(k), …, xd(k), …, xD(k)), the basic PSO algorithm is min f ðxÞ where xmin x xmax

(11.15)

and consists in three steps. Firstly, the positions xi(0) and velocities vi(0) of the initial swarm are randomly generated using upper and lower bounds on the design variable values: xdi ðkÞ ¼ xdmin + Ra xdmax xdmin , xdmin + Ra xdmax xdmin position d , ¼ vi ðk Þ ¼ update Δk

(11.16)

where Ra is a uniformly distributed random variable that can take any value between 0 and 1. The second step is the velocity update of all the particles at time k + 1: vdi ðk + 1Þ ¼ ωd vdi ðkÞ + c1 Rad1 pbestdi ðkÞ xdi ðkÞ + c2 Rad2 gbestd ðkÞ xdi ðkÞ :

(11.17)

Here, k is the generation number, vid(k) and xid(k) represent the dth component of the velocity and position vectors of the ith particle, respectively, ω is termed inertia weight, c1 and c2 are acceleration coefficients, Ra1(k) and Ra2(k) are two vectors randomly generated within [0,1]D, and pbestid(k) and gbestd(k) denote the dth component of the local and global best. The last step in each iteration is the position update of each particle using its velocity (Fig. 11.18): xdi ðk + 1Þ ¼ xdi ðkÞ + vdi ðk + 1Þ:

(11.18)

The three steps of velocity update, position update, and fitness calculations are repeated until a desired convergence criterion is met. For such problem, with only three variables, the basic PSO algorithm was able to detect the global minimum of the function; therefore no PSO variants were considered for the purpose. Once the numerical code was implemented in MATLAB, each case was analyzed with the optimizer in order to validate the method adopted. Fig. 11.19 shows the representation of the mean load for the specimen #8 where in the plane the fold length and the internal thickness were visualized. The red point represents the minimum solution, while the other black ones are the other particles in the last step of the simulation.

Thin-walled truncated conical structures under axial collapse

Xk + 1

357

i

pi

pk g Vk + 1

i

Swarm influence Particle memory influence Vk i Current motion influence

Xk i

Fig. 11.18 Depiction of the velocity and position updates in PSO.

700 600 P [kN]

500 400 300 200 100 0 5 10 h [mm]

15 20

0

0.5

1

1.5

2

2.5

3

3.5

4

t1 [mm]

Fig. 11.19 Load function in the variables h and t1 and representation of the minimal values for specimen #8.

11.6

Comparison between FEM simulation, experimental data, and analytical solution

In order to evaluate the effectiveness of the theoretical model, implemented with the PSO approach, specific cases, available from experimental tests, were analyzed. Table 11.3 reports the mean crushing loads, the crush length, and the errors in percentage between the described method and the experimental results for the geometrical cases taken into account.

358

Dynamic Response and Failure of Composite Materials and Structures

Table 11.3

Geometrical and crushing characteristics during dynamic

loading Specimen geometry

Dynamic

#

R [mm]

T [mm]

ϕ [degrees]

P [kN]

Error [%]

s [mm]

Error [%]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

13.25 13.75 14.5 18.25 18.75 19.5 26.25 27 13.25 13.75 14.5 18.25 18.75 19.5 25.75 26.25 27 13.25 13.75 14.5 18.75 19.5 25.75 26.25 27

1.5 2.5 4 1.5 2.5 4 2.5 4 1.5 2.5 4 1.5 2.5 4 1.5 2.5 4 1.5 2.5 4 2.5 4 1.5 2.5 4

5 5 5 5 5 5 5 5 10 10 10 10 10 10 10 10 10 15 15 15 15 15 15 15 15

19.15 24.15 45.13 22.90 30.71 60.31 40.17 112.51 18.55 25.47 47.67 23.99 32.29 59.32 22.99 42.13 82.50 19.52 26.88 56.49 34.02 67.65 23.51 34.12 59.46

13 1 7 21 6 1 8 12 1 9 2 19 4 1 4 10 7 1 6 15 8 24 12 7 9

125.3 99.3 53.2 104.8 78.1 39.7 59.7 21.3 129.4 94.2 50.3 100.0 74.3 40.4 104.3 56.9 29.0 122.9 89.3 42.3 70.5 35.4 102.0 70.3 40.3

3 10 6 9 11 13 8 12 3 10 11 13 9 15 1 3 11 1 11 3 1 1 2 17 15

From the tables it is clear, despite the simplifications adopted, the proposed analysis is able to predict within 21% the mean load, which is absorbed for about 55% from frictional effects, for about 37% from fronds bending, for about 6% from hoop strain, and for only 2% from crack propagation. Also according to Mamalis et al. the distribution of the main energy sources can be estimated with the same order. As mentioned before, the model refers to the first cycle of deformation; therefore the crush length s can be obtained by multiplying the minimum displacement δ for the ratio between the experimental energy to absorb and the minimum energy obtained from the model. Table 11.4 reports instead the discrepancy between numerical and experimental tests under impact load as regards the average load and the final crushing. It is evident

Thin-walled truncated conical structures under axial collapse

359

Comparison between numerical and experimental dynamic tests

Table 11.4

Average load

Crush stroke

# Specimen

Test [kN]

Simulation [kN]

Error [%]

Test [mm]

Simulation [mm]

Error [%]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

16.9 24.4 42.0 18.9 32.9 60.0 36.9 100.0 18.4 28.2 48.9 20.0 30.9 60.0 21.9 38.2 76.9 19.2 28.7 48.8 31.4 54.3 20.9 31.7 54.3

17.7 24.9 43.3 19.6 35.6 62.6 38.9 104.9 18.2 29.3 52.4 20.4 31.8 61.3 22.7 40.6 83.7 18.7 29.6 46.5 30.2 52.1 21.5 33.5 58.7

4.8 1.9 3.2 3.9 8.3 4.4 5.6 4.9 1.1 3.9 7.2 1.8 2.9 2.1 3.8 6.2 8.9 2.2 3.2 4.7 3.7 4.1 3.1 5.8 8.2

130 90 50 116 70 35 65 19 125 85 45 115 68 35 105 55 26 125 80 41 70 35 105 60 35

129 89 49 115 65 33 63 19 123 83 42 117 67 34 104 53 26 124 80 42 70 35 104 59 32

0.8 1.1 2.0 0.9 7.1 5.7 3.1 0.0 1.6 2.4 6.7 1.7 1.5 2.8 0.9 3.6 0.0 0.8 0.0 2.4 0.0 0.0 0.9 1.7 8.6

that, despite the simplification adopted, the numerical models are able to capture the main parameters with maximum relative errors of 10%. Therefore, when it is not necessary to reproduce the micro-mechanical behavior or delamination under crush and when low computational costs must be guaranteed the single shell layer model can be a valid method to adopt. Also from the point of view of deformation the numerical modeling is able to reproduce the phenomenon. Fig. 11.20 shows the final crushing under dynamic loading for specimen #25, from the point of view both real and simulated. In such cases the ability to reproduce the complete reversal of the laminate internally to the specimen is evident.

360

Dynamic Response and Failure of Composite Materials and Structures

Fig. 11.20 Comparison in terms of deformation for real and simulated specimen.

11.7

Conclusions

A crashworthiness problem was investigated using an analytical, numerical, and experimental approach. In particular the energy absorption of composite circular frusta subjected to dynamic axial loading was analyzed, defining analytically the external load as a function of three variables and identifying the minimum through a optimization approach. Only the first deformation cycle, as Velmurugan et al., and a more realistic formulation for the friction energy, as Mamalis et al., were considered. Differently from other authors no empirical formulations were used in such model. Despite some simplification, the method adopted is able to predict the mean load and the total crushing of conical shell made of composite, once known the parameters of the used material. The kinetic energy is absorbed for 55% from frictional effects, for about 37% from fronds bending, for about 6% from hoop strain, and for only 2% for crack propagation. Also according to Mamalis et al. the distribution of the main energy sources can be estimated with the same order. Therefore the methodology can be used as the first approach to follow during the design of specific impact attenuators. Moreover numerical modeling was conducted using the explicit dynamic code LS-DYNA. The single shell layer model can be considered a valid and efficient methodology to follow in a first step design where no micro-mechanical behavior and delamination phenomenon are necessary to reproduce. In order to validate both approaches, theoretical and numerical, an experimental campaign was conducted using a drop tower suitably instrumented in order to acquire the deceleration versus time of each specimen and therefore, using an integration process, obtain the load-displacement trends. The tests carried out have shown the two typical crushing failure modes of thin-walled composite structures, characterized by the highest energy absorption capacity and by a progressive and controlled deformation. In particular, the formation of two lamina bend inside and outside the specimen was detected for all frusta with a wall inclination of 5 degrees and on those at

Thin-walled truncated conical structures under axial collapse

361

10 degrees with a thickness of 4 mm. In all other cases a perfect internal inversion was observed, bringing the fractured material during crushing completely into the remaining part. The load versus displacement curves have a typical increasing trend with respect to the crushing due to the progressive increase of the area of the cone transverse section. No great change in load-crushing behavior can be noted for the different deformation modes. Analysis of the results in terms of SEA and average crushing stress clearly puts in evidence that to improve the energy absorption capacity of the component it is certainly useful to increase the thickness of the laminate and reduce the wall inclination.

Acknowledgments The authors would like to thank the Picchio S.p.A., for the financial support provided to this research work, and Prof. Belingardi and Ing. Alessandro Scattina of the Politecnico di Torino, for their collaboration in the experimental characterization of the circular frusta.

References [1] Belingardi G, Chiandussi G. Vehicle crashworthiness design—general principles and potentialities of composite material structures. In: Abrate S, editor. Impact engineering of composite structures. vol. 526. Wien: Springer-Verlag Wien; 2011. p. 193–264. [2] Ma J, Yan Y. Quasi-static and dynamic experiment investigations on the crashworthiness response of composite tubes. Polym Compos 2013;34(7):1099–109. [3] Tabiei A, Aminjikurai SB. A strain-rate dependent micro-mechanical model with progressive post-failure behavior for predicting impact response of unidirectional composite laminates. Compos Struct 2009;88(1):65–82. [4] Joosten MW, Dutton S, Kelly D, Thomson R. Experimental evaluation of the crush energy absorption of triggered composite sandwich panels under quasi-static edgewise compressive loading. Compos Part A 2010;41:1099–106. [5] Luo H, Yan Y, Meng X, Jin C. Progressive failure analysis and energy-absorbing experiment of composite tubes under axial dynamic impact. Compos Part B 2016;87:1–11. [6] Thornton PH, Edwards PJ. Energy absorption in composite tubes. J Compos Mater 1982;16:521–45. [7] Hull D. Axial crushing of fibre-reinforced composite tubes. In: Jones N, Wierzbicki T, editors. Structural crashworthiness. Guilford: Butterworths; 1983. p. 118–35. [8] Farley GL, Jones RM. Prediction of the energy-absorption capability of composite tubes. J Compos Mater 1992;26(3):338–404. [9] Mamalis AG, Manolakos DE, Demosthenous GA, Ioannidis MB. Crashworthiness of composite thin-walled structural components. CRC Press, Boca Raton, FL; 1998. [10] Mamalis AG, Manolakos DE, Demosthenous GA, Ioannidis MB. Analytical modeling of the static and dynamic axial collapse of thin-walled fiberglass composite conical shells. Int J Impact Eng 1997;5–6:477–92. [11] Pinho ST, Camanho PP, De Mura MF. Numerical simulation of the crushing process of composite materials. Int J Crashworthiness 2004;9(3):263–76. [12] McGregor CJ, Vaziri R, Poursartip A, Xiao X. Simulation of progressive damage development in braided composite tubes under axial compression. Compos Part A 2007;38 (11):2247–59.

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[13] Xiao X, Botkin Mark E, Johnson Nancy L. Axial crush simulation of braided carbon tubes using MAT58 in LS-DYNA. Thin-Walled Struct 2009;47:740–9. [14] Bisagni C, Piero G, Fraschini L, Terletti D. Progressive crushing of fiber-reinforced composite structural components of a formula one racing car. Compos Struct 2005;68:491–503. [15] Hormann M, Wacker M. Simulation of the crash performance of crash boxes based on advanced thermo-plastic composite. In: 5th European LS-DYNA users conference; 2005. [16] Savage G, Bomphray I, Oxley M. Exploiting the fracture properties of carbon fibre composites to design lightweight energy absorbing structures. Eng Fail Anal 2004;11:677–94. [17] Feraboli P, Norris C, McLarty D. Design and certification of a composite thin-walled structure for energy absorption. Int J Veh Des 2007;44(3/4):247–67. [18] Obradovic J, Boria S, Belingardi G. Lightweight design and crash analysis of composite frontal impact energy absorbing structures. Compos Struct 2012;94(2):423–30. [19] Boria S, Obradovic J, Belingardi G. Experimental and numerical investigations of the impact behavior of composite frontal crash structures. Compos Part B 2015;79:20–7. [20] Bussadori BP, Schuffenhauer K, Scattina A. Modelling of CFRP crushing structures in explicit analysis. Compos Part B 2014;60:725–35. [21] Velmurugan R, Gupta NK, Solaimurugan S, Elayaperumal A. The effect of stitching on FRP cylindrical shells under axial compression. Int J Impact Eng 2004;30:923–38. [22] Solaimurugan S, Velmurugan R. Progressive crushing of stitched glass/polyester composite cylindrical shells. Compos Sci Technol 2007;67:422–37. [23] Boria S, Pettinari S, Giannoni F. Theoretical analysis on the collapse mechanisms of thin-walled composite tubes. Compos Struct 2013;103:43–9. [24] Boria S, Pettinari S. Mathematical design of electric vehicle impact attenuators: metallic vs composite material. Compos Struct 2014;115:51–9. [25] Boria S, Pettinari S. Energy absorbed by composite conical structures in axial crushing. Adv Compos Lett 2014;23(1):11–6. [26] ASTM D5528-01. Standard test method for mode I interlaminar fracture toughness of unidirectional fiber-reinforced polymer matrix composites. [27] Chatiri M, Gull T, Matzenmiller A. An assessment of the new LS-DYNA layered solid element: basics, patch simulation and its potential for thick composite structures analysis, In: Proceedings of the 7th European LS-DYNA conference, Salzburg, 14–15 May; 2009. [28] Wagner W. FE—modeling of fiber reinforced polymer structures, In: Proceedings of the 5th world congress on computational mechanics, Vienna, 7–12 July; 2002. [29] Feraboli P. Development of a corrugated test specimen for composite material energy absorption. J Compos Mater 2008;42(3):229–56. [30] Greco F, Luciano R. A theoretical and numerical stability analysis for composite micro-structures by using homogenization theory. Compos Part B 2011;42(3):382–401. [31] Borovkov A, Palmov V, Banichuk N, Saurin V, Barthold F, Stein E. Macro-failure criterion for the theory of laminated composite structures with free edge delaminations. Comput Struct 2000;76:195–204. [32] Johnson A, Pickett A. Impact and crash modelling of composite structures: a challenge for damage mechanics, In: Proceedings of the 9th user conference EURO-PAM, Darmstadt, 7–8 October; 1999. [33] Greve L, Andrieux F. Deformation and failure modelling of high strength adhesives for crash simulation. Int J Fract 2007;143(2):143–60. [34] Tang CY, Tsui CP, Lin W, Uskokovic PS, Wang ZW. Multi-level finite element analysis for progressive damage behavior of HA/PEEK composite porous structure. Compos Part B 2013;55:22–30.

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[35] Chang FK, Chang KY. A progressive damage model for laminated composites containing stress concentration. J Compos Mater 1987;21:834–55. [36] Boria S, Belingardi G. Numerical investigation of energy absorbers in composite materials for automotive applications. Int J Crashworthiness 2012;17(4):345–56. [37] Boria S, Scattina A, Belingardi G. Axial energy absorption of CFRP truncated cones. Compos Struct 2015;130:18–28. [38] Mamalis AG, Manolakos DE, Demosthenous GA, Ioannidis MB. Axial collapse of thin-walled fiberglass composite tubular components at elevated strain rates. Compos Eng 1994;4(6):653–77. [39] Mamalis AG, Yuan YB, Viegelahn GL. Collapse of thin-wall composite sections subjected to high speed axial loading. Int J Veh Des 1992;13(5–6):564–79. [40] Eberhart RC, Kennedy J. A new optimizer using particle swarm theory. In: Proceedings of 6th international symposium on micromachine and human science; 1995. p. 39–43. [41] Kennedy J, Eberhart RC. Particle swarm optimization. In: Proceedings of IEEE international conference on neural networks; 1995. p. 1942–8.

Lightweight solutions for vehicle frontal bumper: Crash design and manufacturing issues

12

G. Belingardi*, A.T. Beyene*, E.G. Koricho†, B. Martorana‡ *Politecnico di Torino, Torino, Italy, †Michigan State University, East Lansing, MI, United States, ‡FCA – FIAT Research Center, Torino, Italy

12.1

Introduction

12.1.1 Need of lightweight solutions in vehicle development Due to ever-increasing fuel cost, environmentally incompatible growth of Green House Gases (GHG) emission, and to evolving, progressively stringent environmental and safety regulations, lightweight design of vehicle structures has attracted many researchers. As stated in the open literatures, vehicle lightweight structures can be achieved through a combination of three different approaches, namely, structural optimization, advanced lightweight manufacturing technologies, and material replacement. Structural optimization can be classified into three groups: size optimization, shape optimization, and topology optimization. The choice of the optimization method for each particular case in the automotive industry is depending on the desired target for the particular subsystem that is supposed to be reengineered: passive safety, static and dynamic stiffness, weight reduction, acoustic and riding comfort, service life, and serviceability [1]. Researches show that structural optimization can give up to 7% weight reduction and improves the vehicle efficiency. On the other hand, advanced manufacturing process is gaining a significant acceptance in automotive sector in trimming the weight of the vehicle. For instance, Ford Motor Co. has used an advanced forming technology, known as hydroforming, on the steel structural pillars of its new 2013 Fusion body. The process cuts 18 pounds from each car. General Motors (GM) has used resistance spot welding to further reduce the weight of an already lightweight aluminum structure, thus eliminating the use of a very big number of rivets, which can cut up to 2 pounds in each hood, rear liftget, and door. However, the most effective way to achieve a lightweight vehicle structure is material replacement approach. Material replacement, such as with a full aluminum body, can lead to a weight reduction up to 50% [2,3]. If further weight reduction is needed, the ideal candidate materials are fiber-reinforced materials. Beside relevant weight reduction, fiber-reinforced materials have good corrosion resistance, impact cushion, noise attenuation, and allow for relevant part consolidation. This is the direction adopted by Fiat Chrysler Automobiles (FCA) with the design and manufacturing of the recent AlfaRomeo 4C. This model makes Dynamic Response and Failure of Composite Materials and Structures. http://dx.doi.org/10.1016/B978-0-08-100887-4.00012-3 © 2017 Elsevier Ltd. All rights reserved.

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extensive use of composite material not only for the whole passenger cell but also for the rest of the body structure and panels. Recent related studies have shown that 10% of vehicle weight reduction yields 6%– 8% reduction in fuel consumption while keeping the required performance as it is or better than the reference model [4–8]. For the same amount of vehicle weight reduction, 10%, studies have shown that up to 8% of reduction of CO2 emission can be achieved [4,9]. According to one US Department of Energy report [4], today’s average vehicle weighs, in the US market, 3350 pounds (about 1520 kg) without passengers or luggage and are constituted of several different types of materials: 54% iron or mild steel, 10% high strength steel, 9% aluminum, 7% plastic, 4% glass, 1% magnesium, while the remaining 15% is a mix of copper, paint, carpet, padding, insulation, and rubber. The report also indicated that, since 1996, lightweight materials have seen significant increases in vehicle application. For instance aluminum increased by 70%; magnesium increased by 64%; medium and high strength steel increased by 70%; and composite increased by 45%. While there is a wide consensus among researchers and automotive companies on the increasing trend in replacement of conventional materials by lightweight materials, about 19% of the weight of the vehicle can be classified as special-function materials that would be very difficult to substitute with lightweight materials. Hence, the potential weight reduction mainly involves replacement of 68% of the weight constituted by ferrous materials [10]. Considering the entire vehicle structure, the ideal candidate for potentially significant lightweight material replacement is the Body-In-White (BIW), which is usually made of mild steel. In the automotive engineering language, the BIW is the main skeleton of the body structure with the floor and the roof but without external panels and movable parts. On subsystem level, front and rear subsystems are also getting attention to attain considerable weight reduction without detrimentally affecting the required safety performance. Particularly bumper beams can improve not only weight reduction but also structural energy absorption to meet occupant and pedestrian safety standards.

12.1.2 Brief description of vehicle bumper task and state of the art for existing lightweight solution Globally, the vehicle fleet on the road has shown steady growth from 130 million in 1940 to more than 450 million in 2013. It is believed that, over the next 50 years, this number is likely to be 3–5 times greater. Beside fuel cost and CO2 emission concerns, the biggest challenge for regulators and engineers is how to maintain/ improve the existing passive safety standards as the number of vehicles increases on the road. Vehicle crashworthiness can contribute to solve some of the occupant and pedestrian safety issues. Reports show that vehicle crashworthiness has been improved significantly during the past three decades, as indicated, for example, by the reduction in the occupant death rate per million vehicle registrations of 1–3 year old cars from 265 in 1979 to 98 in 2007 [11]. Despite these improvements, 28,869 vehicle occupants

Lightweight solutions for vehicle frontal bumper

367

were killed in road crashes in the United States in 2007. Frontal crashes accounted for half of these fatal events even though new cars have mandatorily passed the frontal crash tests. To mitigate this persistent problem, the design of the frontal crumple zone has played a significant role by managing crash energy, absorbing it within the frontal parts of the vehicle structure, and carefully avoiding its direct transfer to the vehicle occupants, the intrusion into, or the damage of the passenger compartment. In this regard, bumper subsystem is one of the critical design spaces to tune the performance of the crumble zone. Generally, the bumper subsystem includes main components such as bumper beam, rails, and cooling system support, as shown in Fig. 12.1, that are mostly made of steel. Other bumper nonstructural components such as fascia and foam cushions are made of mostly from polypropylene material. Looking at Fig. 12.2, the function of some of the bumper subsystem components that are highly associated with the occupant and pedestrian safeties can be listed as follow: l

l

l

Fascia (Bumper cover): designed for efficient aerodynamic performance, light weight, and esthetically pleasant look to the consumer. Usually fascia is made of polypropylene, polyurethane, or polycarbonate [12]. Energy absorbers (Bumper foam): designed to absorb a portion of the kinetic energy from vehicle collision. Reinforcing beam (Bumper beam): The main key component of the bumper subsystem that absorbs the kinetic energy and provides protection to the rest of the vehicle.

Bumper subsystem

Bumper beam

Fascia

Cooling support

Fig. 12.1 Bumper subsystem components [12].

Fig. 12.2 Bumper component.

Rails

Foam cushions

368

Dynamic Response and Failure of Composite Materials and Structures

Considering the exiting and widely accepted practice in the automotive industry, the bumper beam is one of the potential candidates for significant weight reduction by applying lightweight materials, such as composite materials. Different researchers and car manufacturers have implemented different types of composite materials such as carbon fiber-reinforced plastic (CFRP), glass fiber-reinforced plastic (GFRP), sheet molding compound (SMC), and glass mat thermoplastic (GMT) for bumper beam to improve the bumper subsystem performance as it can offer lightweight as well as reduce the energy consumption [13–15]. Currently, SMC and GMT are widely used because of easy of formability, low material, and manufacturing costs, even though CFRP and GFRP can offer better mechanical performance. After development of SMC mainly by Bayer AG, Germany, during the early 1960s, several automotive manufactures showed their interest owing to the fact that this technology allowed composite materials to be manufactured in mass production for the first time. GM implemented SMC bumper beam in Pontiac Bonneville, Cadillac Seville, and Cadillac Eldorado by replacing the convectional steel material [16]. Also, in the early 1970s Renault and FIAT used SMC for bumper application instead of steel. Ford also introduced SMC for integrated front-end system (IFES) on Taurus and Sable [17]. IFES demonstrates a 14% cost reduction for a platform with a production volume of about 600,000 vehicles. Besides, the consolidation of 22 steel different parts into only two SMC parts and a 22% weight reduction were obtained by joint effort of Ford’s IFES design team and Budd Plastic [18]. It is quite clear to understand that all these improvements are derived from parts consolidation and modular assembly owing to the nature of composite materials that can be tailored according to the available design space and manufacturing capability.

12.1.3 Coupling between material design and manufacturing technology A vehicle development is influenced by the interactions among materials selection, system design, and manufacturing technology, as shown in Fig. 12.3. Most of the lightweight engineering materials can be considered as potential solutions to reduce the Fig. 12.3 Vehicle development.

n

ig

M

es

at

td

er

ia

l

r Pa

Vehicle development

Process

Lightweight solutions for vehicle frontal bumper

369

vehicle weight; however, the challenge comes when manufacturing processes are involved to produce the conceptual design. For example, to develop a steel-based bumper beam, the sheet metal is subjected to particular manufacturing processes, such as deep drawing, materials removal (drilling and machining), joining (bonding and welding), and finishing process (polishing and painting). Assume that there is a need to replace the existing steel-based bumper by aluminum alloy; in this scenario, the entire manufacturing process would be altered to fulfill the desired process requirements for such particular material. Aluminum alloy forgings, particularly closed die forgings, that are generally used to produce highly refined final forging configuration are completely other than those used with hot forged carbon and/or alloy steels [19]. Depending on the type of aluminum alloys, the pressure requirements in forging can vary widely, depending on the alloy chemical composition, forging strain rate, lubrication conditions, and the workpiece and die temperatures. When we come to material removal process, aluminum alloy needs far less specific cutting force than the one required for steel. Owing to the properties of the aluminum alloy, the tooling, the advance rate and depth cut, and lubrication are also quite different when compared again with steel. One critical issue related to multimaterials usage in vehicle application, as also addressed by the US Department of Energy [4], is multimaterials joining. Implementing multimaterial solutions at this bumper subsystem level introduces an additional layer of technology challenge associated with multimaterial joining, corrosion prevention at interface, design tools, and performance prediction. For example, conventional laser welding works great to join the bumper beam with the rail/crash box, but can be problematic when applied to aluminum due to the reactivity of molten aluminum to air. Introducing the aluminum alloy would ask to change the existing joining process to a different robust method that would allow to join multimaterials as for instance, adhesives, self-pierce rivets, clinching, and flow drill screws. In automotive industry, as an important benefit over resistance spot welding, all of the above-mentioned methods allow for joining of dissimilar materials, such as aluminum to steel or aluminum to composite. On the other hand, it would be quite easy to imagine how the level of complexity and the flow of the manufacturing process would be changed when composite materials are involved in lightweight vehicle structure. For instance, in Fig. 12.4, results of the Tecabs research project leaded by Volkswagen show how the standard metallic floor pan, composed of 28 steel parts may be substituted by 8 preforms and 5 cores, which are processed in just one shot by means of the resin transfer molding process [20]. When using carbon fiber, the raw materials cost is higher than steel but “one shot” composites processing is more advantageous in terms of integration and assembly. In many cases, the balance results to be positive for composite materials against metals.

12.2

Our initial approach: The pultruded solution

Pultrusion is a rapidly growing, cost-effective, and fully automated manufacturing process for producing constant cross-section composite profiles. Generally, pultruded profiles have straight axis; however, some recent experience [21] shows that profiles

370

Dynamic Response and Failure of Composite Materials and Structures

Fig. 12.4 Advantage of use of composite materials. Source: Volkswagen.

with curved axis could also be produced. For the structural component of interest, pultrusion could be an appropriate composite manufacturing technology able to obtain high production rate with reasonable low manufacturing cost and very high quality in terms of geometry accuracy and degree of consistency of mechanical property due to process automation. Pultruded pieces have also quite high compression strength, due to precise filament alignment and high fiber volume fraction (up 85% by weight) [22,23], as a consequence of material curing under tension. Published experimental studies [24,25] show that pultruded composites (PFRP) have comparable and sometimes even higher values of energy absorption capability and impact strength with respect to steel and aluminum. Like any composite material, the material characterization study made on coupons cut from pultruded tubes wall shows a brittle behavior; however, a pultruded tube is capable of absorbing significant impact energy amount by behaving in a pseudo-ductile structural mode. This pseudo-ductile response results from material fragmentation. The tubes cross-section undergoes large geometry changes when the tube is subjected to flexural deformation [26]. To substitute the existing metallic vehicle components with similar components made of new materials like composite, a number of factors needs to be considered. First, for the intended load type, the failure modes of the new material as well as the subsystem assembly as a whole need to be understood. This is because crashworthiness of safety components is the fundamental characteristic that depends on the material energy absorption mechanisms. Furthermore, the energy absorption mechanisms are directly linked with the structure failure behavior. The second important factor is the component geometry that has to be optimized as per the material failure behavior.

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371

A wide literature is available regarding the energy absorption mechanisms of fiber-reinforced composite material tubes subjected to axial loading and the failure mechanism is quite well understood and documented, see for instance [27–30]. In general, for axially loaded composite tube, the energy absorption is due to the concurrence of different failure mechanisms such as wall delaminations, delamination penetration with axial cracks formation, and bending of petals followed by fiber fracture. These fiber fractures give place to fracture line formation across the tube wall. On the contrary, the energy absorption mechanisms for composite material tubes subjected to transverse loading, i.e., in bending and shear, which is the usual loading conditions for bumper beams, have not been extensively investigated yet and the failure mechanism is not completely understood. There are few published research results, which suggest that flexural loaded composite components, like beams, absorb energy by progressive tearing along the beam corners [25,31,32]. Therefore, while it is now rather well known how to design composite material tubes to develop a stable or controlled crushing response under axial impact loading, there is not enough and clear information regarding how to design a composite material tube for stable load-carrying capacity under flexural impact loading. In this work, a E-glass/epoxy pultruded bumper beam has been considered at first for flexural impact loading. Its energy absorption capability is compared with steel and E-glass/epoxy fabric composite solutions. Finite element simulations of low-velocity impact events have been performed using ABAQUS, in order to optimize the beam section profile and the beam curvature for crashworthiness. In general, when the material is subjected to any type of loading, failure starts at the point where there is a maximum stress concentration. Therefore, we assume that introducing longitudinal stress concentration zones will lead to localized failure initiation followed by progressive failure development and collapse. The beam is a thin-walled beam and different section profiles are being considered. By introducing a number of properly positioned stress concentration zones (longitudinal grooves), a number of progressive tearing lines are generated, and this results in the expected progressive energy absorption. Eight different beam section profiles have been selected for beam section profile optimization. The profiles of the first row of Fig. 12.5 are characterized by an increasing number of longitudinal grooves along the section width of the beam while the second row shows some possible geometry variation of the profile 3. The failure mode and energy absorption characteristics of all the proposed section solutions have been analyzed, while maintaining the same material properties, overall dimensions, loading, and boundary conditions. Finally, following a similar optimization procedure five different beam curvatures have been analyzed. During the optimization process of the beam section profile and curvature, changes in geometrical perimeters result in variation of load-carrying capacity of the beam, but at the same time in variation of material amount (i.e., mass). These two variations are not strictly correlated, although as a general trend [31] profiles having more material will be able to carry larger load. Moreover results reported by Charoenphan et al. [25] indicate that large section size and wall thickness is giving rapid crack growth in the tubes (i.e., brittle behavior that is undesirable), while substantial energy absorption by

372

Dynamic Response and Failure of Composite Materials and Structures

1

2

3

4

5

6

7

8

Fig. 12.5 Considered bumper beam profiles.

progressive fragmentation can be achieved with appropriate cross-sections. Therefore, in the present work, in order to identify the role of each geometrical parameter a constant amount of material is maintained through modifying the section profile by slight varying the wall thickness. For the eight-beam section profiles, nonlinear finite element simulations, with a simplified bumper beam model (Fig. 12.6), have been carried out using the commercial code ABAQUS/Explicit version 6.14-1. The model comprises four parts, three rigid parts (i.e., the two longitudinal crash boxes and one rigid wall), and one deformable part (i.e., the transverse beam). A discrete rigid surface is used for both rigid bodies to create higher mesh density at critical contact areas. The end profile optimization has been conducted considering E-Glass fiber-epoxy matrix pultruded composite material. However, two other materials were also considered for material comparison (i.e., one conventional material [mild steel] and two composite materials). The basic mechanical properties for the reference steel material are Young modulus E ¼ 206 GPa, density ρ ¼ 7830 kg/m3, Poisson ratio ν ¼ 0.3, and the true stress-plastic strain property as reported in Table 12.1. The mechanical properties for the two composite materials are reported in Table 12.2. In both cases, a 1000 kg mass was rigidly attached at the two rear extremities of the crash boxes, Fig. 12.6 FE model of simplified bumper beam.

Lightweight solutions for vehicle frontal bumper

Table 12.1

373

True stress-plastic strain data for steel

σ (MPa)

305

345

386

425

450

470

εp

0

0.0244

0.0485

0.0951

0.1384

0.1910

Mechanical properties of E-Glass/epoxy pultruded and glass/epoxy fabric

Table 12.2

ρ (kg/ E11 E22 G12 G23 Property m3) (GPa) (GPa) (GPa) (GPa) ν21 Pultruded 1850 Fabric 1850

31.2 29.7

9.36 29.7

5 5.3

5.5 5.3

Xc Xt Yc Yt Sc (MPa) (MPa) (MPa) (MPa) (MPa)

0.29 409 0.17 549

483 369

92.2 549

34.9 369

73.3 97

in order to simulate the vehicle mass, and allowed to move with an initial velocity of 15 km/h toward the rigid wall. Section profile optimization has been conducted in three stages. The first step is by increasing the number of grooves along the width of the beam. The grooves are the main crack initiation mechanism leading to transverse progressive tearing. In the analysis, the number of grooves is increased until the strength of the beam begins to decrease leading to direct impact of the rigid crash box and resulting in very high (not acceptable) load peaks. In this particular case the acceptability limit is less than three grooves. As shown in Fig. 12.7, when the number of grooves becomes three (profile number 4), the bumper beam impact behavior changes from progressive failure to catastrophic failure and results in high-peak reaction force (i.e., three times higher than the other three profiles) (Fig. 12.7A). This implies that, at the chosen impact velocity, the energy cannot be completely absorbed by the beam and the crash boxes and other nearby components have to be involved in the energy absorption process.

Fig. 12.7 Reaction force versus time (A) and energy versus time (B) diagrams of different section profiles with 15 km/h impact velocity.

374

Table 12.3

Dynamic Response and Failure of Composite Materials and Structures

Results of energy and peak loads for different profiles

Profile

Absorbed energy (kJ)

Peak load (kN)

1 2 3 4

8.65 8.70 8.66 8.67

150.0 110.0 96.7 420.8

Comparing the results reported in Fig. 12.7A and Table 12.3 for the four considered profiles, it can be noted that, with comparable energy absorption, profile 3 shows a progressive failure mode, with minimum peak load. To achieve good energy absorption, the diagram of the bumper reaction force should be flat as much as possible during the whole crash time and close to the maximum allowable load. Analyzing the earlier four curves, Profile 3 has the mean load value that is close to the maximum peak load. Similarly, from the energy versus displacement curves of Fig. 12.7B, profile 3 has better linearity (i.e., more progressive failure behavior). Therefore, with regard to the number of grooves, profile 3 results to be the optimized end-section profile. Keeping the number of grooves constant, a variable length and width of each groove was considered in the second optimization step. Profile 3 is modified by varying the groove width and depth so that the rear fold is deeper than the front one (profile 5); this leads to a reduction in the profile strength. Its mode of failure and energy absorption has been compared with the reversed profile geometry, profile 6, where the front groove is deeper than that of the rear one. As it is shown in Fig. 12.8A, profile 5 gives an improved progressive failure whereas profile 6 experiences a local failure. A structure hinge comes out at the beam mid-span, which results in a catastrophic fracture that leads to higher peak load and reduction in energy absorption capability with respect to profile 5.

Fig. 12.8 Reaction force versus time plot with different fold length and depth (A) and varying thickness (B) with 15 km/h impact velocity.

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Again, the selected profile 5 was further modified through varying the thickness along the width. The wall thickness was gradually reduced per fold while moving from the front toward the crash boxes, as shown in Fig. 12.8B (profile 7). Similarly, a comparison was made with the reversed profile geometry. Comparing the diagrams of Fig. 12.8A and B, profile 8 shows a local failure, due to the formation of structural hinge at the beam mid-span, which results in a catastrophic fracture and leads to higher load peak and reduction in energy absorption with respect to profile 7. To sum up, profiles 7 and 8 have a similar trend of failure mode with respect to the previously considered profiles 5 and 6, i.e., when the bumper transverse beam impacts at its weaker side, its failure mode becomes unstable and catastrophic. On the other hand, when the bumper transverse beam impacts against the rigid wall at its stronger side, a progressive failure is obtained. Fig. 12.9 shows the crash performance comparison between profiles 5 and 7 with profile 3, which is the optimal result from the first optimization step. Through such further geometry modifications, it is possible to improve both the mode of failure (i.e., failure progressivity) and the total energy absorption with small increment of peak load as it is visible from the data reported in Table 12.4. The last optimization parameter considered was beam curvature. Five beam curvatures (i.e., beam curvature radius of 2400, 2862, 3200, 3600 mm, and straight beam) are considered, and an investigation of impact event characteristic data is done with a procedure similar to the one adopted for the previous analysis. Maximum peak load values and energy absorption results are presented in Table 12.5 for all profiles.

Fig. 12.9 Reaction force versus time and energy versus displacement plots for three profiles at 15 km/h impact velocity.

Table 12.4

Results of energy and peak loads value

Profile

Energy absorption (kJ)

Peak load (kN)

3 5 7

8.66 8.65 9.85

96.7 105.1 109.4

376

Table 12.5

Dynamic Response and Failure of Composite Materials and Structures

Results of energy and peak loads value

Radius (mm)

Energy absorption (kJ)

Peak load (kN)

2400 2862 3400 3600 Straight

8.0 9.85 9.86 9.85 9.85

276.7 109.4 156.6 381.1 1389.5

Increasing the beam curvature reduces the chance of localized stress formation. Larger zones of the bumper beam are in contact with the flat rigid wall at the same time. This leads to higher peak load that promotes the formation of diffuse fractures on the large portions of the folds, as parts of the beam with same stress level will fracture at the same time. With a similar reasoning, the worst case is when the bumper beam is straight (see Fig. 12.10C), which is currently used by some vehicles. In this situation, the portion of the beam extremities just in front of the crash box with length equal to the crash box width will fracture at the same time. One has to consider that the crash boxes are axially stiffer and stronger with respect to the transverse beam section. When a large portion of the beam is subjected to equal stress level at the same time, little progressive crack propagation can take place and consequently energy absorption does not evolve in a progressive way. Therefore, for this type of structural profile and loading condition, inappropriate replacement of the existing steel beam with composite beam may cause a catastrophic failure. On the contrary, when the beam curvature radius is reduced below some critical value, 2862 mm in this particular case, there will not be time for crack propagation; instead a high local stress line will be developed at the apical portion of the beam. Such high local stress line will result in unstable failure as shown Fig. 12.10A. Therefore, for straight beam, the contact between the rigid modeled crash boxes and the rigid barrier gives the maximum reaction force (see Table 12.5). While for smallest radius of curvature, local structural hinge at the beam mid-span gives place to unstable

Fig. 12.10 Reaction force versus time diagram and mode failure for beam curvatures 2400 (A), 2862 (B), and straight (C) curvatures.

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failure, which leads to a reduction of the energy absorption and even possible damage of the nearby components. At the end of the earlier optimization procedure, it turns out that the best combination is the beam section profile 7 and 3200 mm curvature radius for the proposed E-glass/epoxy pultruded composite beam. This optimized beam solution was further used to compare the pultruded composite bumper beam solution with the steel and the glass fabric/epoxy composite solutions in terms of impact energy absorption and weight. The load versus displacement and energy versus displacement curves are shown in Fig. 12.11 and the amount of energy absorbed by each material solution is reported in Table 12.6. The bumpers made of the earlier three materials were subjected to equal impact energy; however, they result in completely different failure modes. It should be noted that there is a big variation in both the trend and the peak reaction force values. The better solution for the design of vehicle passive safety components is to develop a frontal and rear crumpling zone that can absorb most of the impact energy with a limited value of the deceleration of the occupant inside. A reduction of load values, and in particular of the load peak, is yielding to lower decelerations and this positive for the vehicle passive safety component design. In this perspective, pultruded bumper beam shows a maximum deformation with a minimum mean crash load and an almost constant force diagram whose value is close to the maximum peak load. Thus this solution is giving a good interpretation of the ideal energy absorber and can be considered as the most interesting between the considered three.

Fig. 12.11 Reaction force versus displacement and absorbed energy versus displacement diagrams for the three different material solutions.

Results of energy and peak loads for different material solutions

Table 12.6 Material

Energy absorption (kJ)

Peak load (kN)

Steel Fabric Pultruded

8.69 8.67 8.66

251.56 206.36 109.4

378

12.3

Dynamic Response and Failure of Composite Materials and Structures

Our evolutionary approach: The die forming solution with thermoplastic matrix

Although the developed pultruded solution has clearly a number of advantages, it has also some weak points. First of all, being based on thermo-setting matrices, recyclability at the end of life is a problem. This stimulated us to consider different material system (with thermoplastic matrix) and a different manufacturing process. Our choice was to develop a new design of the bumper system, taking into account the possibility offered by the GMT family of materials and its manufacturing technology (i.e., the die forming). As usual, the first step in the design development has been the careful material characterization.

12.3.1 Mechanical characteristic of the selected materials The two principal responses of the materials to external forces, of interest for the present study, are deformation and fracture. Depending on the material type, the deformation may be elastic, visco-elastic (time-dependent elastic deformation), elasto-plastic, eventually with creep effect (time-dependent plastic deformation), or with strain-rate sensitivity (dependence on the loading velocity) and fracture may occur catastrophically or after repeated application of loads (fatigue). Material engineering characteristic parameters are the base input for the numerical analyses. For isotropic materials, i.e., materials whose mechanical property does not depend on the particular loading direction, only three independent engineering constants (E, G or ν, and α) are needed, sufficient to describe the elastic response of the material during loading. On the contrary, for advanced composite materials (i.e., materials having a directional dependant mechanical behavior), more constants are needed to describe their elastic behavior. For most composite materials, 12 engineering constants (E1, E2, E3, G12, G13, G23, ν12, ν13, ν23, α1, α2, and α3) are required for the elastic regime. Where 1, 2, and 3 are the material axis. To measure these values requires the test of numerous samples with several different test procedures. These procedures are designed to generate a constant or nearly constant state of stress throughout the material, reducing the number of elastic constants needed to describe the deformation for the given loading condition to the smallest possible number. This is usually done by using simple geometries, usually planar samples, to measure 1–3 engineering constants simultaneously (as in longitudinal modulus test—E1, ν12, and ν13 can be measured simultaneously). In an ideal situation, with the proper test matrix and procedures, all the material constants can be obtained, given enough tests and sample materials. A classic glass mat-reinforced thermoplastic (GMT) and two other configurations of the GMT materials were considered for the desired application. The classical GMT consists of an endless fiberglass mat that reinforces a polypropylene (PP) laminate with randomly oriented glass fibers. In the first of these supplementary configurations, the classical GMT panel was reinforced with two additional layers of unidirectional fiber on both sides, obtaining what we will call the GMT-UD. In the second of these

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supplementary configurations the same classical GMT panel was reinforced with one layer of fabric at the center, obtaining what we will call the GMTex. As the classical GMT is already in use for many industrial applications, the information regarding its mechanical property and fracture behavior can easily be found on material company technical data sheet and open literature. While the reinforcement made on the classical GMT using an additional layer of unidirectional or fabric layers are new concepts proposed by the material manufacturer. Therefore, the study has included the full material characterization of the three materials for four different loading conditions. In comparison with traditional metals, composites material in general has enhanced mechanical behavior. They have high strength-to-weight ratio, better chemical/ corrosion resistance, and excellent impact properties. Therefore, with respect to light weightiness and crashworthiness, it has been proved that composite materials have superior advantages over traditional metallic materials. Furthermore, composite materials offer a number of manufacturing advantages, such as design flexibility, lower tooling costs, and opportunities for part consolidation. Furthermore, comparing the GMT family with thermo set composite materials, recyclability (suitable for reuse) and elimination of controlled-storage requirements during manufacturing, are additional advantages of this thermoplastic composite material. This additional feature fosters their wide acceptance in automotive and other industries and currently makes GMT the preferred material for interior and structural applications. The above-mentioned improved mechanical behavior and manufacturing advantage of GMT grasp the attention of the research group to further investigate the material performance. Experimental tests have been conducted to obtain the values of the mechanical characteristic parameters and to identify the fracture mechanisms of the selected materials. The test program included tensile tests (both longitudinal and transverse directions), compressive tests (both longitudinal and transverse directions), plane shear tests (both longitudinal and transverse direction), perforation tests, and drop dart impact tests. In all cases, samples with the standard geometry and dimensions have been cut from the same panel of the respective material to eliminate possible panel-to-panel variability. All tests have been performed at room temperature in the as-fabricated condition. No environmental conditioning has been performed. The quasi-static tests were conducted with a 100-kN capacity servo-hydraulic testing machine (INSTRON-8801), as shown in Fig. 12.12A. During the mount phase of the specimen, the maximum preload was carefully controlled and set lower than 0.2 kN in order to avoid specimen damage. The machine was equipped with a standard load cell and a crosshead displacement-measuring device. Strain gages were used for tensile, compressive, and shear tests. The strain gage data were acquired using NI WLS-9163 data acquisition board while the load and crosshead displacement data were acquired from the testing machine by means of NI DAQCard-6062E card with a sampling rate of 1 kHz. The specimen geometries followed the specifications outlined in ASTM D3039 for the tensile specimens, ASTM D6641/D6641M for compressive specimens, ASTM D5379/D5379M for shear specimens, and ASTM 5628 for drop dart impact specimens.

380

Dynamic Response and Failure of Composite Materials and Structures

Fig. 12.12 Experimental setup for material characterization. (A) Testing machine, (B) shear test fixture, (C) penetration test setup.

Table 12.7 collects the main mechanical properties of the three considered materials in the longitudinal and transverse directions. From the table, one can draw a conclusion that, the modification made on classic GMT using unidirectional (UD) and fabric (Tex) layers, indeed enhances the stiffness of the material for all selected loading scenarios. Similarly, the strength of the material increases (with a multiplying factor bigger than 2) along the longitudinal direction, with a little loss in strength in the transverse direction. Table 12.8 and Fig. 12.13 show the penetration strengths (the maximum force before the first failure of the material occurs) and the corresponding failure displacement of the considered material. Here it is worth to mention that the thickness of the GMTex sample was 3 mm while for the other two materials was 4 mm. Therefore, the lower penetration strength of GMTex is explained by this thickness difference. The dynamic drop dart impact tests were conducted on the CEAST/Instron machine (Fig. 12.14A). The impactor has a hemispherical head with a radius of 10 mm and it has a maximum falling height of 2 m. The drop-weight apparatus is equipped with a motorized lifting track. The collected data were stored after each impact and the impactor returned to its original starting height. The other important dynamic variables were then calculated using the scheme of Fig. 12.14B.

Lightweight solutions for vehicle frontal bumper

Table 12.7

381

Mechanical property of the three materials considered GMTex

Tens. strength (MPa) Tens. modulus (GPa) Poisson’s ratio Compr. strength (MPa) Compr. modulus (GPa) Poisson’s ratio Shear strength (MPa) Shear modulus (GPa)

GMT-UD

GMT

Longitudinal

Transverse

Longitudinal

Transverse

81 6 0.3 66

175 9.3 0.3 69

71 6 0.2 57

180.4 11 0.3 82.2

59 6.5 0.2 59

3.3

4.2

2.7

6

2.6

0.3 45 2.4

0.2 48 3

0.2 47 2.6

0.4 56 3.5

0.2 55.7 3

Average values of the penetration strength, of the displacement at failure, and of the absorbed energy for each type of tested samples for quasi-static indentation tests

Table 12.8

Materials

Penetration strength (N)

Failure displacement (mm)

Energy (J)

GMT GMTex GMT-UD

2000 1400 2100

3.6 3 4

36.6 29.5 37.8

Fig. 12.13 Load versus displacement curve for quasi-static penetration tests.

382

Dynamic Response and Failure of Composite Materials and Structures

Fig. 12.14 Drop dart testing machine (A) and specimen fixture (B).

The dynamic drop dart impact tests were conducted to measure the impact resistance and damage tolerance of the targeted material. The samples were also submitted to repeated impact loading for selected energy levels. Damage variable called “Damage Index” (DI), proposed by Belingardi et al. [32], is used to assess the evolution of the material damage. As shown in Eq. (12.1), damage variable DI is a function of the impact energy Ei, absorbed energy Ea, maximum dart displacement measured during a quasi-static perforation test SQS (see Fig. 12.13), and the maximum dart displacement registered during dynamic impact test Smax. Fig. 12.15A–C shows the force versus time curves for the three considered materials and Fig. 12.15D shows the Damage Index versus impact number comparison of the three materials at the same energy level. DI ¼

Ea Smax Ei SQS

(12.1)

The lower value of the damage index means the better material impact performance. GMTex results to fracture more rapidly with the selected energy level with respect to the other two types of GMT because of the plate thickness. However, comparing the base GMT and GMT-UD, for each impact number GMT-UD has lower damage index (i.e., better damage resistance). Therefore, similarly with the mechanical properties,

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Fig. 12.15 (A–C) Load versus time curves for repeated dynamic impact tests and (D) damage index versus impact number.

the reinforcement made on the classic GMT indeed enhances the fracture performance of the material.

12.4

Bumper beam innovative design: Results and discussion

When metallic components have to be substituted with composite component, depending on the failure mode of the material, a new design approach has to be followed, so that the usage of the important feature of the new material will optimize the results. With the use of the GMT and of the die forming process, it is possible to develop a completely different design of the bumper beam integrated with the crash boxes, so that instead of a number of pieces that have to be manufactures separately and then assembled together, it is possible to produce the subassembly just with one forming process. In order to take advantage from this exciting possibility, the following important design decisions have been taken. –

In the traditional bumper system, the connection between the bumper beam and crash box is obtained by mechanical fastening (i.e., by welding or bolting). Looking at the crash box, the failure mode of axially loaded composite tube commonly initialized by means of properly conceived end-triggering mechanism. Such crash triggering shape is generally located at

384

–

–

–

Dynamic Response and Failure of Composite Materials and Structures

frontal extremity of the tube. However, if one tries to adapt such traditional connection scheme to the new proposed composite bumper beam-crash box solution, in the hypothesis of a nonintegrated solution, this will result in an early detachment of the beam and the crash box and therefore will not meet the intended energy-absorbing goal. New design approaches need to be followed for this group of materials. The proposed models may have a free frontal crash box end with the required crash trigger, while the strength on the connection between the beam and the crash box can be maintained through optimizing the trickiness of the connecting rim to withstand the shearing load resulting from frontal impact loads. As per the international safety regulations, the impact energy resulting from low-velocity collision has to be completely absorbed by the bumper beam without a direct involvement of the crash box and without physical damage on the nearby components. Sufficient clearance between the front surface of the beam and the tip of the crash box has to be considered in such a way that the contact with the impacted obstacle is initially involving the transverse beam while the crash boxes are involved later if the applied load is beyond the structural elastic limit of the beam. The energy absorption for the crashed object is the product of the mean force and the crash length or the area under force displacement curve. The crash stroke of the crash box component is an important parameter for crash component design. During integrated bumper system design, improper placement of the connecting rim will affect the crash length and consequently the energy absorption of the system by blocking the progressive failure of the crash box. The relative position of the crash box and bumper beam rim needs to be optimized in order to get the intended target. The final factor considered during the design stage was manufacturability. Generally, a closed beam profile has better strength and energy-absorbing capacity than open beam profile. However, from the manufacturing point of view, comparatively, it is more difficult to manufacture a closed section beam with the current technology. However, regardless to this fact, a closed section solution has been considered by proposing a back cover for the beam and assembled using adhesives.

The proposed GMT materials have been compared with the reference material (steel) by two ways. At first by the simple substitution of the current steel beam-crash box integrated solution with the proposed materials. At the second stage by adopting the equal bending stiffness approach (i.e., for a given thickness and stiffness of the reference material), the thickness of the targeted material can be calculated using Eq. (12.2): rﬃﬃﬃﬃﬃ 3 ES h x ¼ hs Ex

(12.2)

where hs and hx are thickness of the steel and the targeted material, respectively, and Es and Ex are the elastic modules of the steel and the targeted material, respectively. The thickness of the reference steel beam is 2 mm and the thickness for the GMT beam 8 mm. With this thickness, the combined system (i.e., the integrated beam-crash box of the composite solution) has a mass of 4.69 kg, 4.65 kg, and 4.82 kg for the GMT, GMTex, and GMT-UD, respectively. The comparison of these masses with the reference steel material and configuration one (beam-crash box combined) (i.e., 7.67 kg), yields approximately 35% weight saving. International safety regulations demand that after a low-velocity impact, the bumper subsystem must function

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properly with small cosmetic defect. The expected outcome is a closed load versus displacement curve after the test. However, as shown in Fig. 12.16, the load versus displacement curve for the three GMT proposed materials are open. This indicates that the direct substitution results in a permanent severe damage or a complete fracture of the beam. The proposed solution does not withstand the applied load and need to be optimized accordingly. To be more specific, the GMT-UD and GMTex solutions show an early sharp break at the center of the beam while the GMT solution shows relatively higher elastic deformation (see Fig. 12.17). The second stage of the investigation was made by improving the section wall thickness and height of the base plate. The plate section dimension has been optimized

Fig. 12.16 Force versus time and force versus displacement curve for the steel and the three considered material solutions.

Fig. 12.17 Failure mode of the three GMT material solutions.

386

Table 12.9

Dynamic Response and Failure of Composite Materials and Structures

Thickness and mass of the combined beam–cashbox

Materials

Thickness (mm)

Mass(kg)

Steel GMT GMTex GMT-UD

2.2 7.1 6.2 5.8

7.7 3.7 3.3 3.2

to improve the structural performance of the beam. The wall thickness of the integrated beam-crash box was determined on the basis of the reference material solution, as stated by Eq. (12.2). The resulting wall thickness and the mass of the integrated beam-crash box solutions are reported in Table 12.9. Internal energy, kinetic energy, and total energy versus time curves are important impact characteristic curves to understand the failure mechanism of the impacted structure. For instance, by superimposing, the internal energy versus time and kinetic energy versus time curves, one can have complete insight of the failure extent of the structure. Fig. 12.18 shows three different responses for the impacted structure. Fig. 12.18A shows a complete recovery case in which the double crossing of the lines means that the material is still within its elastic limit or with no relevant damage in the structure. Fig. 12.18B shows a partial recover case, where some fracture takes place. Such a type of curve indicates fracture in the structure for the case of composite material and permanent plastic deformation for the case of ductile metallic materials. Fig. 12.18C is the case of a complete fracture. International safety regulations demand that all the vehicle safety features need to function properly after low-velocity collision. Therefore, considering the earlier three cases, the full rebounce case is the only acceptable case. The proposed beam end-section profile and curvature optimization have been conducted by using of such energy versus time curves as one of the monitoring parameters. Here the energy curves have been used also for material comparison purpose. The tensile modules (see Table 12.7) of GMT-UD solution is approximately 50% higher than GMT and 25% higher than GMTex material configuration. Therefore using equal bending stiffness approach, GMT-UD is expected to have relatively smaller thickness. Besides, the introduction of unidirectional fiber at both sides of the classical GMT material, even if it results in an improvement of the tensile modules, I.energy K.energy

T

(B)

I.energy K.energy E

E

E

(A)

I.energy K.energy

T

(C)

T

Fig. 12.18 Energy versus time curves. (A) Elastic rebound, (B) penetration with material damage, (C) perforation.

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particularly along the fiber direction, it significantly reduces the strength of the material along the transverse direction. Furthermore, once the crack appears at the most stressed part of the beam, most probably at the first contact point of the beam against the rigid wall, it has the freedom to propagate along the fiber direction or along the length of the beam. However, such phenomenon will not be the case for the other two material solutions. This is because in the other two configurations (base GMT and GMTex), there is an equal distribution of the fiber in the longitudinal and transverse directions. Consequently, the stress distribution will be uniform in both axes and the resulting cracks will inhibited from propagation by the perpendicular fiber. Such fracture behaviors of the GMT-UD material has been observed on energy versus displacement curve during experimental dynamic drop dart test and quasi-static penetration test. Fig. 12.19 shows the fracture mechanism of the three materials resulting from the dynamic drop dart test. Through the comparison of the force versus time and displacement versus time curves of Fig. 12.20, for the four considered materials, it comes out that the GMT-UD solution has the minimum peak load (i.e., 25 kN) (i.e., positive, because the peak load is one of the important parameters to be controlled) but it has the maximum intrusion (i.e., 37 mm) (i.e., highly negative because the intrusion must be within a limited allowable value), further the beam results to be totally fractured at its mid-span (see Fig. 12.24A), even with considered impact velocity. The failure outcome can also

Fig. 12.19 Modes of failure for the dynamic impact loading.

Fig. 12.20 Force versus time and displacement versus time curves for the modified bumper system.

388

Dynamic Response and Failure of Composite Materials and Structures

Fig. 12.21 Energy versus displacement and force versus displacement curves for the modified bumper system.

be monitored using the load-displacement curve or the energy versus displacement curves of Fig. 12.21. Both an open curve for the load-displacement (see Fig. 12.16) and energy time curve (see Fig. 12.22) for the GMT-UD solution (which shows the behavior of the impacted system during energy dissipation) confirm that the material experiences a fracture before the end of the impact test, even at the selected velocity. When metallic materials are loaded a little beyond their elastic limit, due to their large margin of plasticity, the extra energy can be absorbed through limited plastic deformation. As a consequence the beam replacement is not required. Whereas, composite materials have very limited or no plastic range, therefore, the open curve shown in Fig. 12.21B for the case of GMT-UD depicts that the beam already passed its elastic limits. The energy dissipation resulted from the material fracture. Fig. 12.21B shows the energy versus displacement curves. These curves provide important information regarding the energy dissipation mechanism along the crashing process. The displacement increment with a constant energy value for the case of GMT-UD shown in the figure is the result of the advancement of the barrier into the nearby structure, and may be that harder components, the engine for instance, come in contact without any form of energy dissipation. Therefore, it results in a drastic increase of the deceleration and may cause the occupant impact against the passenger compartment internal structure with possible serious injury. GMT-UD_4v

Fig. 12.22 Energy curve for GMT-UD.

700

Energy (J)

600 500

KE

400

IE

300

TE

200 100 0 0

0.02

0.04

0.06 Time (S)

0.08

0.1

0.12

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It has been discussed on the topic of material characterization part that the reinforcement made on a classical GMT material using unidirectional fiber lamina indeed improves the strength and the modules of the material particularly along the fiber direction. However, when the beam made of GMT-UD is subjected to transverse loading, the occurrence of crack propagation along the length of the beam (i.e., along the fiber direction) results in a reduction of the structural performance. With a similar configuration and loading, a beam-crash box solution made of GMT or GMTex materials absorbed the impact energy without showing a fracture clue (i.e., remaining within the elastic range). The solution structural behavior has been monitored from the load versus displacement and energy curves of Figs. 12.21 and 12.23 and their failure modes in Fig. 12.24. The energy versus displacement curves for the two materials, shown in Fig. 12.21A, mean a linear trend of the absorbed energy (i.e., the structure remains within its elastic limit). Furthermore, the energy curve shows complete rebounce case. Under low-velocity collision (i.e., with impact velocity of 4 km/h, which is close to the parking load), the bumper has to operate within elastic range, like what is observed for the GMT and GMTex material solutions. With the proposed beam configuration and loading, an integrated beam-crash box solution using GMT or GMTex material can be considered for material replacement in place of traditional metallic material with significant weight saving.

0

0.02

0.04

0.06

0.08

GMT_4V

KE IE TE 0.1 0.12

Energy (J)

Energy (J)

GMtx_4v 700 600 500 400 300 200 100 0

700 600 500 400 300 200 100 0

KE IE TE

0

Time (S)

Fig. 12.23 Energy curve for GMTex and GMT.

Fig. 12.24 Failure modes of the modified bumper system.

0.02

0.04

0.06 Time (S)

0.08

0.1

0.12

390

12.5

Dynamic Response and Failure of Composite Materials and Structures

Conclusions

An automobile bumper subsystem has been considered for material substitution and innovative design. A number of different composite materials such as pultruded E-Glass/epoxy, classical GMT that consist an endless fiber glass mat-reinforced PP laminate with randomly oriented glass fibers; GMT-UD (i.e., a chopped fiber glass mat-reinforced PP laminate with randomly oriented glass fibers additionally reinforced with layers made of a unidirectional-oriented glass fibers); and GMTex (i.e., a chopped fiber glass mat-reinforced PP laminate with randomly oriented glass fibers additionally reinforced with fabric) are used for developing bumper beam. While designing these composite solutions of the bumper transverse beam, their related manufacturing technologies have been taken into account and design optimization process has been adopted, by considering as design variables both the shape to the beam cross-section, the wall thickness in the optimized section, and the beam curvature. The study has two parts. The first part is dedicated to the development of pultruded composite car bumper beam with the aim to maintain or improve its mechanical properties and energy absorption capability with respect to the steel beam that is the reference normal production solution. The material change is mainly aimed to mass reduction. For this part of the study the material mechanical properties used for the simulation were collected from literatures. For the design development, numerical simulations have been conducted and structural results for the bumper beam have been compared between the new composite optimized solution and the normal production one. The following important conclusions can be drawn: l

l

l

For transversally loaded composite components like automotive bumper beam, properly optimized and located stress concentration zone such as beam longitudinal groves, can serve as crash triggering mechanism, to initiate crack formation and to develop progressive tear along beam longitudinal axis. As the main portion of impact kinetic energy is absorbed through this longitudinal tearing, the number and location of groves need to be optimized. When the existing metallic vehicle bumper beam is substituted with composite beam, the beam curvature should be modified accordingly, because in composite bumper beam development, optimal choice of bumper beam curvature radius, besides improving vehicle aerodynamic and architecture, actually can give a relevant contribution to vehicle safety. However, it should be noticed that the production of a curved beam through pultrusion is not consistent with the standard pultrusion process and ask for evolved advanced technology. Finally, material comparison study shows that E-Glass pultruded bumper beam has comparable energy absorption with steel and E-Glass fabric solutions but has better progressive failure mode with reduced peak load. Therefore, this pultruded composite part, being a safety component in the vehicle architecture, besides weight and manufacturing benefits, does improve general vehicle safety behavior.

In the second part, a classic GMTs and two modified materials (i.e., GMTex and GMT-UD) have been used for material substitution. Since the modification made of the classical GMT material is resulting in novel materials, the performed investigation also covers the material behavior study. The considered material being base on

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thermoplastic matrix has better recyclability properties with respect to the previously considered material. Further this type of materials have die stamping as preferred manufacturing process, thus opening the design to completely new innovative architectures. In particular, taking advantage from the outlined characteristics, for this part, a novel beam-crash box integrated bumper subsystem has been designed. Keeping the geometrical parameters and the loading conditions constant, the performance of the integrated bumper beam made of the earlier three materials has been compared with the traditional steel solution. The following important conclusions can be drawn: l

l

l

l

From the point of view of the material characterization it is observed that the modification made on classical GMT through the reinforcement that make use of woven or unidirectional fibers is indeed improving both the tensile and impact performance of the original material. The dynamic drop dart impact tests show that GMT-UD has better impact characteristics and in particular, in case of repeated impact test, can withstand larger number of impacts before perforation than GMT and GMTex. The failure mode is somehow different between the three. At the perforation impact GMT is pierced with similar crack propagation along both transverse and longitudinal axes. This mode of failure is expected since GMT has almost similar mechanical property along the two axes. GMTex has a similar behavior but with higher resisting loads, as it is characterized by a fabric reinforcement. The perforation hole is localized at the specimen impacted center. Finally GMT-UD exhibits a full width crack along the axis of the unidirectional reinforcement (i.e., a completely different failure mode with respect to the other two materials). This failure behavior of GMT-UD combined with its higher fracture and perforation energy performance can be of interest for energy-absorbing components because and a proper geometry optimization may serve for developing a progressive failure. From numerical study, it can be observed that, for low-velocity impact (i.e., 4 km/h), as bumpers are expected only to bump without damage, both the GMT and GMTex bumpers absorb the applied energy trough elastic deformation, with no material damage and with a acceptable displacement (i.e., intrusion into the engine compartment). Whereas, for the selected beam configuration, the GMT-UD bumper solution shows fracture at the mid-span of the beam. The modification made on the classical GMT by introducing unidirectional reinforcement both sides of GMT, particularly at the skin, may result in such a brittle failure mode. The die stamping process is giving the possibility of a large part consolidation, what in the metallic solution is manufactured as tenths of separated pieces to be assembled to obtain the bumper beam and crash box subsystem, can be design as one single piece. This is resulting in a dramatic simplification of the assembly process and related costs.

Generally, in both GMT-UD and GMTex, the modification made on classical GMT is indeed improving both the tensile and impact performance of the original material and these can be used for structural purpose for some application in place of steel and aluminum. However, coming to energy-absorbing components, composite materials have completely different failure behavior than conventional metallic material; their energy-absorbing performance strongly is affected by the geometry of the component and direct adoption of the traditional metallic geometry may lead to a catastrophic failure. Generally higher peak load is resulting in the load-displacement diagram. Therefore to improve the energy absorption contribution of the beam during higher-velocity impact, it is recommended to reconsider the geometry of the beam configuration.

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References [1] Pfestrof M. Manufacturing of high strength steel and aluminum for a mixed material body-in-white. In: Proc of the 11th int. conf. on sheet metal, Erlangen, 5–8 April; 2005. [2] Koricho EG, Belingardi G. An experimental and finite element study of the longitudinal bending behavior of T-joints in vehicle structures. Compos Part B 2015;79:430–43. [3] Jambor A, Beyer M. New cars-new materials. Mater Des 1997;18(4/6):203–9. [4] US Department of Energy. Chapter 8: Advanced clean transportation and vehicle systems and technologies. Quadrennial technology review; 2015. [5] Cheah L. Cars on a diet: The material and energy impacts of passenger vehicle weight reduction in the U.S. Ph.D. thesis, Massachusetts Institute of Technology; 2010. [6] Casadei A, Broda R. Impact of vehicle weight reduction on fuel economy for various vehicle architectures. Ricardo Inc; 2007. [7] Bandivadekar A, Bodek K, Cheah L, Evans C, Groode T, Heywood J, et al. On the road in 2035: Reducing transportation’s petroleum consumption and GHG emissions. Cambridge, MA: MIT Laboratory for Energy and the Environment; 2008. [8] Lutsey N. Review of technical literature and trends related to automobile mass-reduction technology. Davis, CA: University of California; 2010. [Online]. Available: http://pubs. its.ucdavis.edu/publication_detail.php?id¼1390. [9] Koricho EG. Implementation of composites and plastics materials for vehicle lightweight. PhD thesis, Politecnico di Torino, Department of Mechanical and Aerospace Engineering; 2012. [10] Insurance Institute for Highway Safety. Fatality facts. Arlington, VA. Available: http:// www.iihs.org/research/fatality_facts_2007/default.html. [11] Koricho EG. Implementation of composites and plastics materials for vehicle lightweight. PhD thesis, Torino, Italy: Politecnico di Torino, Department of Mechanical and Aerospace Engineering; May 2012. [12] Davoodi MM, Sapuan SM, Ahmad D, Aidy A, Khalian A, Jonoobi M. Concept selection of car bumper beam with developed hybrid bio-composite. Mater Des 2011;32(10):4857–65. [13] Marzbanrad J, Alijanpour M, Kiasat MS. Design and analysis of an automotive bumper beam in low-speed frontal crashes. Thin-Walled Struct 2009;47(9–10):902–11. [14] Cheon SS, Choi JH, Lee DG. Development of the composite beam for passenger cars. Compos Struct 1995;32(1):491–9. [15] General Motors. SMC bumper beams improve productivity for GM. Reinf Plast 1992;36 (4):1–7. [16] Maine EMA. Future of polymers in automotive application. Cambridge, MA: Massachusetts Institute of Technology; 1997. [17] Young JA. Major SMC project enhances Taurus look. Plastic World 1996;24. February. [18] Kuhlman W. Forging of aluminum alloys, metalworking: Bulk forming. ASM handbook, Vol 14A. Materials Park, OH: ASM International; 2005. p. 299–312. [19] Milwich M. Thermoplastic braid pultrusion. In: Proc. of the ICCM17 – XVII int. conf. composite materials, Edinburgh (UK), 27–31 July; 2009. [20] Koricho EG, Belingardi G, Beyene AT, Martorana B, Enrico M. Crashworthiness analysis of a composite and thermoplastic foam structure for automotive bumper subsystem. In: Elmarakbi A, editor. Advanced composite materials for automotive applications: Structural integrity and crashworthiness. Chichester, UK: John Wiley & Sons; 2013, ISBN: 978-1-118-42386-8.

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[21] Campbell FC. Manufacturing processes for advanced composites. Amsterdam, The Netherlands: Elsevier; 2004. [22] Jimenez A, Miravete A, Larrode E, Revuelta D. Effect of trigger geometry on energy absorption in composite profiles. Compos Struct 2004;48:107–11. [23] Suranaree J. Study on impact responses of pultruded GFRP, steel and aluminium beams by using drop weight impact test. Sci Technol 2008;15(3):191–9. [24] Palmer W, Bank LC, Gentry TR. Progressive tearing failure of pultruded composite box beams: Experiment and simulation. Compos Sci Technol 1998;58(8):1353–9. [25] Davoodi MM, Sapuan SM, Ahmad D, Aidy Khalina A, Jonoobi M. Concept selection of car bumper beam with developed hybrid bio-composite material. Mater Des 2011;32:4857–65. [26] Kim KJ, Won ST. Effect of structural variables on automotive body bumper impact beams. Int J Automot Technol 2008;9(6):713–7. [27] Boria S, Belingardi G. Numerical investigation of energy absorbers in composite materials for automotive applications. Int J Crashworthiness 2012;17(4):345–56. http://dx.doi.org/ 10.1080/13588265.2011.648516. [28] Hirao NT, Kotera K, Nakamae M, Inagaki K, Kenafm H. Reinforced biodegradable composite. Compos Sci Technol 2003;63(9):1281–6. [29] Palanivelu S, Van Paepegem W, Degrieck J, Kakogiannis D, Van Ackeren J, Van Hemelrijck D, et al. Parametric study of crushing parameters and failure patterns of pultruded composite tubes using cohesive elements and seam, part I: Central delamination and triggering modelling. Polym Test 2010;29:729–41. [30] Charoenphan S, Bank LC, Plesha ME. Progressive tearing failure in pultruded composite material tubes. Compos Struct 2004;63(1):45–52. [31] Tabiei A, Svenson A, Hargarvec M, Bankd L. Impact performance of pultruded beams for highway safety applications. Compos Struct 1998;42:231–7. [32] Belingardi G, Cavatorta MP, Paolino DS. A new damage index to monitor the range of the penetration process in thick laminates. Compos Sci Technol 2008;68:2646–52.

Further Reading [1] http://altairenlighten.com/2012/11/advanced-manufacturing-techniques-help-automakers-saveweight/. [2] Lee KH, Bang IK. Robust design of an automotive front bumper using design of experiments. J Automotive Eng 2006;220(9):1199–207. [3] Li Y, Lin Z, Jiang A, Chen G. Experimental study of glass-fibre mat thermoplastic material impact properties and lightweight automobile body analysis. Mater Des 2004;25 (7):579–85. [4] Maahs WD, Janowiak AR. Composite bumper beams: Comparing material choices. In: Proceedings of the third annual conference on advanced composites, Detroit, MI, ASM Int., 15–17 September; 1987. p. 11–21. [5] Stodolsky F, Vyas A, Cuenca R. Lightweight materials in the light-duty passenger vehicle market: Their market penetration potential and impacts. In: Proc. of the Second World Car Conference, UC Riverside, March 1995; 1995.

Pressure reconstruction during water impact through particle image velocimetry: Methodology overview and applications to lightweight structures

13

M. Porfiri, A. Shams New York University Brooklyn, Brooklyn, NY, United States

13.1

Introduction

As lightweight composites are increasingly integrated in naval and aerospace structures, there is a more pressing need to precisely quantify fluid-structure interactions during hull slamming [1,2]. Hull slamming is responsible for severe, impulsive loading conditions, which could lead to sustained deformations and ultimately failure [3–5]. The slamming event may take place in only a few milliseconds, with peaks in the pressure as large as few MegaPascal [6]. Experimental and numerical observations on full-scale ship models [7–11] have demonstrated a number of co-existent physical phenomena, associated with hull slamming, including cavitation, air pockets, ventilation, structural vibrations, buckling, and delamination. Although full-scale experiments are the cornerstone against which new structural design should be tested, laboratory-scale experiments can unravel and quantify specific physical phenomena occurring during hull slamming. The most common experimental setup used to study hull slamming consists of an instrumented wedge-shaped body, rigid or elastic, which is dropped from a fixed height on the water surface of a meter-size tank. Pressure and strain gages are simultaneously integrated in the structure to measure the hydrodynamic loading and mechanical deformation in selected locations. This setup has been used to study the role of deadrise angle [12–14], entry velocity [15–17], geometric asymmetry [18,19], and structural deformations [19,20] on hull slamming. To offer a higher degree of control over the impact speed, the use of active pneumatic systems has also been investigated in a number of studies [21–23]. Beyond the simple wedge-like model, some authors have investigated other geometries such as cylinders [24,25], pyramids [23], and axisymmetric bodies [26]. A few efforts have also been devoted to address the role of three-dimensional (3D) phenomena in water entry problems [21,27].

Dynamic Response and Failure of Composite Materials and Structures. http://dx.doi.org/10.1016/B978-0-08-100887-4.00013-5 © 2017 Elsevier Ltd. All rights reserved.

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However, these experiments only provide partial information on the physics of the impact. While knowledge of structural deformations and hydrodynamic loading is essential for understanding hull slamming, the current experimental practice suffers from two main limitations: (i) data are only collected at few locations on the structure, leading to an incomplete spatial resolution of the structural dynamics, and (ii) no information is collected in the fluid, such that only a skewed understanding can be garnered about bidirectional fluid-structure interactions associated with hull slamming. Not only do these limitations hamper our understanding of hull slamming, but also they restrain progress on the development of semianalytical and computational models [28–33]. These models are central to enable systematic structural design and rapid testing of new design ideas, but, for them to be trusted, extensive validation should be performed against experimental results across structural and fluid mechanics. To address these issues, we have recently proposed the integration of particle image velocimetry (PIV), a common technique in experimental fluid mechanics, in hull slamming research. PIV measures the displacement vectors of particle groups in the fluid by cross-correlating pairs of subsequent images [34,35]. The fluid velocity vectors can be estimated from the displacement field using the frame rate of the recorded images. Although PIV does only output an estimate of the velocity field, the pressure field can be indirect reconstructed everywhere in the fluid by hypothesizing a constitutive behavior for the fluid and integrating pertinent balance equations [36,37]. Despite significant progress in PIV-based pressure reconstruction, most of the research has focused on steady fluid flows [38] and unsteady phenomena have only been marginally investigated [36]. An even more elusive area is the analysis of fluid flows in the presence of moving boundaries, which are critical to understand hull slamming. Therein, the fluid domain continuously evolves in time due to both the presence of the moving free surface and the motion of the hull, potentially deforming during the impact. The dominant parameters which control the accuracy of PIV measurements are the spatial and temporal resolutions of the images [37]. Assessing the error propagation in the reconstructed pressure field from PIV data is an important issue, due to inherent uncertainties in PIV velocity measurements, which are in fact dominated by spatial and temporal resolutions [35,37]. Several techniques have been proposed in the literature to reduce error propagation from the velocity field to the reconstructed pressure, such as averaging the pressure along different integral paths [39] and omni-directional integration methods [40]. In addition, some authors have explored the use of Poisson equation to transform the numerical problem in more robust form [41]. A recent critical assessment of PIV pressure reconstruction schemes can be found in [42]. In this chapter, we summarize our recent work on the use of PIV toward an improved understanding of hull slamming. We focus on results from three specific studies which have appeared in the last three years, selected to offer a self-contained presentation of our research. In particular, we focus on results from the first effort from our group where we demonstrated the feasibility of PIV-based pressure reconstruction during water impact [43]. Then, we turn to investigate the role of 3D phenomena in complex hull models and hydroelasticity in flexible wedges by referring to findings

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in Refs. [43] and [44]. Beside highlighting results from these three exemplary studies, we succinctly summarize methodologies and insight developed throughout the years toward attempting at: investigating the role of a number of physical parameters (hull curvature [45], geometric asymmetries [46], and water column height [47]); extending the scheme to study water exit and 3D impact [48,49]; and validating our approach against synthetic data in 2D and 3D [50,51]. Throughout the chapter, we bring to light-specific elements, which are expected to play an important role in the study of lightweight composite materials, and a fundamental objective of our ongoing research. The latter is carefully detailed in the conclusions of this chapter, through the presentation of preliminary results on the impact of lightweight glass-vinyl ester panels. The chapter is organized as follows. In Section 13.2, we detail the key methodological aspects of our research, including the experimental setup, data acquisition system, process of image masking to identify the wetted boundary of the impacting object and the fluid-free surface, and image correlation scheme to analyze images. In Section 13.3, we present our approach to reconstruct the pressure from the flow field, through the integration of Navier-Stokes equations. In Section 13.4, we concisely illustrate the application of the proposed approach to the three of our studies mentioned earlier. In Section 13.5, we summarize key limitations of the approach and present ongoing work.

13.2

Experimental methods

Here, we describe the experimental setup used to perform drop test in our laboratory, the data acquisition system utilized for PIV, and image correlation schemes to analyze PIV images. The setup has been used in a number of our experiments [43–46,52,53]. For brevity, we concentrate on the setup used for 2D impact; modifications needed to upgrade to 3D impact can be found in Ref. [44].

13.2.1 Experimental setup and data acquisition system Our PIV acquisition system consists of high-speed cameras, a laser source, and tracer particles. Specifically, in most of our studies two high-speed Phantom cameras V.9.1, a 5W Nd:YAG Ray Power laser source with a wavelength of 532 nm, and silver-coated hollow glass spheres with a mean diameter of 44 μm as tracer particles. Two cameras are typically used to closely resolve the fluid domain and properly impose the free surface boundary condition during pressure reconstruction. While, we have generally focused on symmetric impacts where both cameras can be located on the same half of the domain, we have also considered asymmetric impacts [46] in which each camera acquires a half of the fluid domain. Seldom, these cameras are not sufficient to accurately resolve the early stage of the impact, especially for hulls with low deadrise angles due to large velocity gradients. In this case, we should resort to higher frequencies as demonstrated in Facci et al. [50]. Experiments are executed in the drop tower shown in Fig. 13.1. The apparatus consists of a transparent tank with dimension of 800 320 350 mm3 that is fixed

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Fig. 13.1 Water entry experimental setup with overlaid nomenclature of all the components, including drop tower apparatus, high-speed cameras, laser, and hull (wedge). A detailed view of the hull and rails is shown in the inset. Reproduced from A. Shams, M. Jalalisendi, M. Porfiri. Experiments on the water entry of asymmetric wedges using particle image velocimetry. Phys Fluids 2015;27(2): 027103 (1994-present).

High speed cameras

Frame

Tank

Rail

Mirror Sledge

Wedge

Laser source

on an aluminum frame above the floor. The tank size is selected such that experimental results are independent of wave reflections at the water surface, and interaction with the bottom wall is controlled for, as shown in Ref. [47]. We constrain the vertical motion of the hull during the fall using two aluminum rails. Rigid hulls have often a rectangular base to proxy a ship hull. The frame of the hull is fabricated in acrylonitrile butadiene styrene material using a rapid prototyping 3D printer. Balsa wood is glued to the frame using an epoxy adhesive to cover the frame. A thin layer of epoxy is utilized to waterproof the wood surface. The hull is connected to a sledge that slides along the aluminum rails that control the vertical motion of the hull. More recently, we have explored the use of a pneumatic system to investigate exit problems and shallow water impact [47,49]. A mirror is mounted below the tank to reflect the laser sheet, see Fig. 13.1. The laser location is adjusted to illuminate the particles at the mid-span of the hull, where 3D phenomena are minimized [48]. The cameras are located orthogonal to the laser sheet to capture images of the impact from the side of the transparent water tank. The acquisition rate of the cameras ranges from 4 to 8 kHz and the cameras are synchronized through a timer box. The parameters of the acquisition system are specified through the DynamicStudio software [54]. We use a target with a reference grid size

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of 8 8 mm2 to calibrate the images. The acquired images from both cameras are merged at each frame for PIV analysis through a MATLAB code. We have often instrumented the hull with accelerometers manufactured by Measurement Specialties Inc. to measure the acceleration of the falling body and dynamic ICP pressure sensors from PCB Piezotronics sensor type 113B27 to directly measure pressure. In addition, we have instrumented one of the aluminum rails with position sensor manufactured by Spectra Symbol to measure the position of the impacting body during the impact. Beside the data from the position sensor, we typically track the motion of the hull during impact using Xcitex Proanalyst version 1.5.2.9. The entry depth ξ of the hull obtained from either tracking of the hull motion or position sensor is eventually used to estimate the hull velocity and acceleration.

13.2.2 PIV analysis Fig. 13.2A shows an area of the fluid domain, scanned by the high-speed cameras and illuminated by the laser sheet during the impact of a hull with deadrise angle of 35 degrees. An open-source MATLAB code “PIVlab” [55] is utilized to analyze the PIV images. A fast Fourier transform cross-correlation algorithm and a multigrid scheme with a 50% interrogation window overlap is leveraged in the MATLAB code. A decreasing interrogation area size technique is adapted with window size 64 64, 32 32, and 16 16 pixels [34,35]. The output of PIV analysis is a temporal sequence of a uniformly spaced velocity grid in the fluid domain (Fig. 13.2B and C). We conduct a manual image masking on the PIV images to identify the wetted boundary of the impacting object and the fluid-free surface at each frame starting from the onset of the impact, when the hull touches the water-free surface, see blue line in Fig. 13.2C. This image mask is also used to estimate the wetted width r? and reference wetted width r, see Fig. 13.2B. In particular, the horizontal projection of the vector Water jet Wedge

Pile-up

r* r Free surface

y

Image masking

x

A

x v

u

e1

(A)

(B)

(C)

e2

Fig. 13.2 (A) A representative PIV image during water entry of a rigid hull with deadrise angle of 35 degrees with overlaid notation. (B) PIV image with overlaid velocity vectors, and wetted and reference wetted widths. (C) Schematic of a uniformly spaced grid obtained from PIV analysis, illustrating: the image masking; location of the Cartesian coordinate system; velocity components u and v along the x- and y-directions; respectively; path for pressure integration; and entry depth ξ.

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connecting the keel and the last wetted point of the hull is defined as the wetted length. The reference wetted length is computed by neglecting the water pile-up and utilizing the free surface away from the impacting object. The pile-up coefficient can be computed by calculating the ratio between the wetted width to the reference width, c ¼ r?/r.

13.3

PIV-based pressure reconstruction

In this section, we present our approach to reconstruct the pressure field from planar PIV data through incompressible Navier-Stokes equations [43] and briefly discuss pressure reconstruction from Poisson equation to reduce experimental uncertainties [46]. We also demonstrate the extension of planer PIV measurements to obtain 3D velocity and pressure fields.

13.3.1 Pressure reconstruction using Navier-Stokes equations We have proposed an entirely data-driven approach to reconstruct the pressure field from PIV data. The governing equations for 2D ideal fluid neglecting compressibility and gravity are as follows [56]: @pðx, y, tÞ @uðx, y, tÞ @uðx, y, tÞ @uðx, y, tÞ ¼ ρ + uðx, y, tÞ + vðx, y, tÞ @x @t @x @y 2 @ uðx, y, tÞ @ 2 uðx, y, tÞ + +μ @x2 @y2 (13.1a) @pðx, y, tÞ @vðx, y, tÞ @vðx, y, tÞ @vðx, y, tÞ ¼ ρ + uðx, y, tÞ + vðx, y, tÞ @y @t @x @y 2 2 @ vðx, y, tÞ @ vðx, y, tÞ + +μ @x2 @y2 (13.1b) @uðx, y, tÞ @vðx, y, tÞ + ¼0 @x @y

(13.1c)

where ρ and μ are constant density and viscosity of the fluid, respectively; t is the time variable; u and v are the velocity components along the x- and y-directions, respectively; and p is the pressure. The velocity components in Eqs. (13.1a) and (13.1b) are estimated from PIV data. The pressure is reconstructed from Eqs. (13.1a) and (13.1b) through the following steps [43]: (i) the pressure at point A on the free surface, away from the hull, is set to zero; (ii) the pressure is integrated along lines ε1 and ε2 using a forward integration; and (iii) the pressure field is calculated in the whole fluid domain from the pressure on the boundaries using a spatial eroding scheme [37]. This procedure is repeated at each time frame. As a result, scanning the fluid flow through planar PIV during the impact

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allows for estimating the spatio-temporal evolution of the hydrodynamic loading. The impact force per unit length can be calculated by integrating the hydrodynamic loading along the wetted width of the hull. We note that viscosity effect has a negligible effect on the reconstructed pressure as discussed in Refs. [43,49]. This spatial integration method is likely more sensitive to noise in the velocity field as compared to integration based on the Poisson equation due to error propagation in the velocity [37].

13.3.2 Pressure reconstruction using Poisson equation The Poisson equation is obtained by differentiating Eqs. (13.1a) and (13.1b) with respect to x and y, respectively, summing the equations, and using Eq. (13.1c) as follows:

@ 2 pðx, y, tÞ @ 2 pðx, y, tÞ + ¼ ρ @x2 @y2

@uðx, y, tÞ @x

2

! @uðx, y, tÞ @vðx, y, tÞ @vðx, y, tÞ 2 + +2 @x @y @y (13.2)

Both Dirichlet and Neumann boundary conditions are imposed on the pressure field to integrate Eq. (13.2), if the portion of the boundary is in contact with the hull or not. The former boundary condition is imposed by integrating the pressure on the free surface, S SR and S SL , and the boundary of the recorded image in the fluid, S BR , S BB , and S BL , see Fig. 13.3. The pressure along these lines is integrated similar to step (i) of the integration scheme in the earlier section by imposing zero pressure at A and B.

SIR

SIL A

i + 1, j Q 1 G i, j + 1 O SIR i, j Q2 i − 1, j

SSL

SSR

∂Ω

B

SIL

i + 1, j Ω SBL

h

b1

i, j + 1

i, j

i, j − 1

h

b2

SBR

i − 1, j SBB

(A)

(B)

Fig. 13.3 (A) Schematic of the stencil used for the discretization of the Poisson equation, along with the definition of all symbols used in the numerical solution of the Poisson equation. (B) Schematic of the procedure used to treat Neumann boundary conditions: the fluid points Q1 and Q2 are utilized to reconstruct the pressure at ghost point G close the hull boundary. In (A) and (B), the black dots represent points in the fluid or its boundary, and open circles refer to ghost points. Reproduced from A. Shams, M. Jalalisendi, M. Porfiri. Experiments on the water entry of asymmetric wedges using particle image velocimetry. Phys Fluids 2015;27(2):027103 (1994-present).

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Dynamic Response and Failure of Composite Materials and Structures

The latter boundary condition is imposed by calculating @p/@n at the intersection of the impacting body and fluid S IR , where n is the normal to the impacting body at each frame. @p/@n is calculated by projecting Eqs. (13.1a) and (13.1b) on n and using PIV data. We have implemented a time-marching algorithm to solve Eq. (13.2). The selected grid is consistent with the dimension of the PIV data matrix, see Fig. 13.3. Ghost points are introduced outside the fluid domain close to the hull boundary to apply the Neumann boundary conditions, Fig. 13.3. The grid points are comprised of points inside the fluid domain Ω, on the boundaries @Ω, and the ghost points close the boundary @Ω, see Fig. 13.3.

13.3.3 Extending 2D pressure reconstruction to 3D We have extended our PIV-based technique to elucidate the role of 3D effects on the flow physics and hydrodynamic loading during slamming of rectangular base hulls [48] and more complex geometries described by multiple curvatures [44]. To estimate the full 3D velocity field in the fluid, PIV is performed on several planes along both the length and width of the hull. Fig. 13.4 shows the model which is designed to proxy the complex shape of a ship hull, a local Cartesian coordinate system defined to prescribe a specific geometry, and PIV measurement planes. More specifically, we perform two sets of PIV experiments on multiple cross-sections of the impacting body. In the first set, PIV measurements are conducted along the width to estimate the cross-sectional velocity components u and v along the x- and y-directions (Fig. 13.4). In the second set, experiments are performed along the length of the hull to measure the axial velocity component w along the z-axis (orthogonal to the xy-plane) and v (Fig. 13.4). We combine the planar PIV analyses along the length and width of the hull to estimate the 3D velocity field of the fluid. Specifically, we use an interpolation scheme

6 20 cm

5

Z

X

5.6 cm

Y (cm)

4 Y

3 2 1

18 cm

(A)

0

(B)

0

1

2

3

4 5 X (cm)

6

7

8

Fig. 13.4 (A) Computer-aided design of the hull with overlaid dimensions and representative laser planes for cross-sectional and axial PIV. (B) Cross-section profiles of the hull. Cross-sections are plotted for different values of Z from the mid-span (Z ¼ 0 cm) to the end (Z ¼ 10 cm) of the body. Reproduced from M. Jalalisendi, S.J. Osma, M. Porfiri. Three-dimensional water entry of a solid body: a particle image velocimetry study. J Fluids Struct 2015;59:85–102.

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consisting of the following steps: (i) we utilize the time history of the location of pile-up in the axial PIV experiments to identify the specific time instants at which the water touches the hull in the other cross-sectional PIV measurement (z 6¼ 0 cm); (ii) we follow the same procedure to identify the specific time instants when the water wets the hull in the other axial PIV measurement planes (x 6¼ 0 cm) using the time history of the location of pile-up in cross-sectional PIV experiments; (iii) from the information obtained in steps (i) and (ii), we correlate the frames of each points of experiments for axial and cross-sectional PIV; (iv) we enrich each set of cross-sectional grid points with the out-of-plane velocity component w; (v) we use a cubic spline interpolation scheme to interpolate the axial components in the whole grid; and (vi) we repeat steps (iv) and (v) for each frame (see Fig. 13.5).

13.4

Demonstration of the approach

Here, we concisely illustrate the application of the proposed approach to a number of problems, from our published work in the field. Specifically, we touch on water entry of rigid hull in 2D and 3D and briefly discuss the role of flexibility.

13.4.1 2D water entry of rigid hulls Fig. 13.6 shows PIV images overlaid with velocity vector maps along with contour plots of the velocity magnitude at t ¼ 5 and 20 ms for a rigid hull with 25 degrees deadrise angle, impacting the water surface in free fall from a height of 50 cm. The fluid velocity increases at the onset of the impact and decreases as time advances from t ¼ 5 to 20 ms. The fluid velocity is always maximized at the pile-up region during water entry, while it noticeably decreases further from the hull. Fig. 13.7 shows the reconstructed pressure field from Navier-Stokes equations for three falling heights of 25, 50, and 75 cm at different time instants. For clarity, we normalize the pressure by 1=2ρ ξ_ 2 , where ξ_ is the entry velocity at each frame. As expected, the maximum pressure is attained in the pile-up region. The pressure sharply decreases away from the pile-up location and it is zero on the free surface. The light reflection at the water surface causes uncertainties in the pressure reconstruction which lead to nonzero pressure values on the free surface. The pressure is positive in the early stage of the impact over the entire wetted surface of the hull, and may attain negative values, below atmospheric pressure, at the keel, as time progresses. A negative pressure value is obtained after 15 ms for the lowest entry velocity (see Fig. 13.7A), while the pressure reaches negative values after 10 ms for moderate entry velocities. For the highest entry velocity, the pressure field becomes negative during the early stages of the impact (see Fig. 13.7C). These negative pressure values are always larger than vapor pressure, such that cavitation is not observed in these laboratory-scale tests [57]. To assess the accuracy of PIV-based pressure reconstruction, we have compared the reconstructed pressure with the classical Wagner’s solution [29] in Fig. 13.8. The comparison between Wagner’s solution and experimental results offers evidence for the possibility of using PIV in predicting the hydrodynamic loading (see Fig. 13.8A).

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z=0

z=6

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(C) Fig. 13.5 Interpolation steps (ii), (iii), (iv), and (v); the units of all numbers are cm. PIV enrichment is illustrated for a representative cross-section at z ¼ 6 cm. Cross-sectional PIV data are enriched with the axial velocity component available from x ¼ 0 to 5 cm. (A) The velocity grid obtained after PIV analysis for z ¼ 6 cm is overlaid with the hull cross-section and the pile-up shape at z ¼ 0 cm, highlighting the difference between the pile-up shape and the area occupied by the hull at each cross-section. (B) Perspective view of the hull, illustrating the grid points where the axial velocity component is estimated. (C) The red grid points are enriched with the estimation of the axial velocity. The overlapped region between the velocity grid points at z ¼ 6 cm and the cross-section at z ¼ 5 cm, shown as a shaded area, is where the extrapolation of the velocity gradients @u/@z and @v/@z is performed. Reproduced from M. Jalalisendi, S.J. Osma, M. Porfiri. Three-dimensional water entry of a solid body: a particle image velocimetry study. J Fluids Struct 2015;59:85–102.

Specifically, a remarkable agreement is observed in both the pile-up region where the maximum pressure is reached and the keel region where negative pressures are attained. Further insight into accuracy of the hydrodynamic measurements is obtained by comparing the time evolution of the maximum and the minimum pressures calculated form Wagner’s solution and experimental results in Fig. 13.8B. Results demonstrate the accuracy of our approach in estimating the maximum and minimum values of the pressure over time. In particular, the accuracy of PIV measurements seems to improve as the wedge further penetrates into the water surface. This should be ascribed to underestimation of the velocity field in the first stage of the impact, especially in the pile-up region, due to the presence of large velocity gradients [43,48].

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1.81.4 1.20.8 0.6 0.4 0.2 0.1

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Fig. 13.6 High-speed images overlaid with PIV velocity vectors and magnitude of the velocity in m/s at (A, B) t ¼ 5 ms and (C, D) t ¼ 20 ms. Adapted from data published in R. Panciroli, M. Porfiri. Evaluation of the pressure field on a rigid body entering a quiescent fluid through particle image velocimetry. Exp Fluids 2013;54 (12):1630.

In Fig. 13.9, we compare the impact force obtained from Wagner’s solution, PIV measurement, and direct measurement of the acceleration of the hull. Results hint at the accuracy of the PIV-based pressure reconstruction for different impact heights. Specifically, PIV is able to anticipate both the rise time and the peak value of the impact force compared with Wagner’s solution. Noticeable differences between PIV and Wagner’s solution are observed in the initial stage of the impact. Such discrepancies could be ascribed to experimental uncertainties; underestimation of the velocity field in the first stage of the impact; and inherent simplification of the Wagner solution, which tends to overestimate the hydrodynamic loading of the hull for deadrise angles larger than 10 degrees [30,58]. After this first, preliminary validation of the approach, we have pursued several avenues to assess the accuracy of PIV-based pressure reconstruction. For example, validation of the approach against more refined theoretical solutions was presented

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in Shams et al. [46], and comparison against direct numerical simulations was undertaken in Facci et al. [50].

13.4.2 Role of structural flexibility on 2D water entry To demonstrate the quantification of hydroelastic phenomena, which have been shown to be critical for lightweight composites [1,5], we have studied the impact of an aluminum flexible hull with thickness of 0.5 mm, deadrise angle of 22 degrees, and length of 130 mm. Experiments are conducted for three different impact heights of 25, 50, and 75 cm to vary the hydroelasticity factor between 0.068 and 0.124. The

20

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x (mm)

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Fig. 13.8 (A) Comparison between the normalized pressure obtained from classical Wagner’s solution (solid line) and the experimental evaluation (dashed lines) for a hull falling from 75 cm. Five experimental measurements are shown from 12.5 to 13.5 ms with a time increment of 0.25 ms. Wagner’s solution is evaluated at 13 ms for clarity. (B) Maximum- and minimum-normalized pressures. The solid lines are theoretical predictions from Wagner’s solution and the dashed lines are experimental results. Adapted from data published in R. Panciroli, M. Porfiri. Evaluation of the pressure field on a rigid body entering a quiescent fluid through particle image velocimetry. Exp Fluids 2013;54 (12):1630. 1000 Force per unit depth (N/m)

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Fig. 13.9 Impact force per unit depth obtained from the classical Wagner’s solution (dashed line), indirect measurement based on acceleration (dotted line), and PIV-based pressure reconstruction (solid line). Results are shown for different falling heights: 25 cm (A), 50 cm (B), and 75 cm (C). Adapted from data published in R. Panciroli, M. Porfiri. Evaluation of the pressure field on a rigid body entering a quiescent fluid through particle image velocimetry. Exp Fluids 2013;54 (12):1630.

Dynamic Response and Failure of Composite Materials and Structures 15 12.5 10 7.5 5 2.5 0 –2.5 –5 –7.5

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Fig. 13.10 Normalized hydrodynamic pressure distribution over the normalized wetted width of the hull at several instants of the impact: (A) Wagner’s solution for a rigid hull and (B) PIV-based reconstructed pressure. Three repetitions of the same experimental condition are shown. Reproduced from R. Panciroli, M. Porfiri. Analysis of hydroelastic slamming through particle image velocimetry. J Sound Vib 2015;347:63–78.

hydroelasticity factor is a parameter that is generally used to quantify the extent of hydroelastic phenomena [33,53,59]. In Fig. 13.10, we compare the hydrodynamic loading reconstructed from PIV using Navier-Stokes equations and the classical Wagner’s solution neglecting the role of flexibility at various time instants during water entry for an impact height of 25 cm. We find that the flexibility of the hull has an important effect on the hydrodynamic loading. Specifically, the spatial distribution of the pressure significantly changes as time increases; different from Wagner’s solution for a rigid hull in Fig. 13.10A. Notably, the flexibility of the hull increases the region of negative pressure as the hull penetrates the water surface and diminishes the peak value of the pressure in the pile-up. To further dissect the role of hydroelastic phenomena and validate the PIV measurement, we compare the impact forces obtained from experimental results, the classical Wagner’s solution for a rigid hull, and the semianalytical solution proposed by our group in Ref. [33] for flexible hulls in Fig. 13.11. Our solution [33] builds on Wagner’s approach, but explicitly models the role of dynamic, elastic deformations during the impact.1 Results are presented for three different impact heights. Fig. 13.11 shows that PIV-based pressure reconstruction is in good agreement with the model and captures the oscillations in the hydrodynamic loading. In particular, PIV is successful in anticipating the peak value and the rise time, similar to our previous findings on the impact of rigid hulls. It also predicts the possibility of suction during the impact, in good agreement with model predictions. As the entry velocity increases, PIV predictions consistently underestimate the impact force, which is likely due to the underestimation of the velocity field in the early stage of the impact. 1

Since the submission of this chapter, we have extended the modeling framework to the study of impact problems for water-backed panels [62].

PIV-based pressure reconstruction during water impact 1000 PIV Wagner Shams and Porfiri (2015)

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Fig. 13.11 Force per unit length on the wetted width of the hull for three different entry velocities: (A) 25 cm, (B) 50 cm, and (C) 75 cm. Reproduced from A. Shams, M. Porfiri. Treatment of hydroelastic impact of flexible wedges. J Fluid Struct 2015;17:229–46.

13.4.3 3D water entry PIV has been utilized to characterize the 3D hydrodynamic loading of a complex geometry that, designed to proxy a miniature ship hull, impacting the water surface from a height of 50 cm. Specifically, we illustrate cross-sectional PIV images overlaid with contours of the cross-sectional velocity magnitude at different locations: z ¼ 0, 3, 6, and 9 cm at t ¼ 17.5 ms. Consistent with the results of 2D impact in Fig. 13.6, the velocity maximum is attained in the pile-up independent of the location of the PIV measurement in Fig. 13.12. The cross-sectional components of the velocity decrease by moving away from the mid-span of the model (z ¼ 0 cm), such that the magnitude of the velocity is reduced to half of original value at z ¼ 9 cm. This can be associated with the 3D nature of the flow field, where the axial velocity increases far away from the mid-span. Fig. 13.13 shows the hydrodynamic loading exerted on the hull for three time instants t ¼ 7.5, 12.5, and 17.5 ms. Our results demonstrate the ability of PIV to capture the spatio-temporal evolution of the hydrodynamic loading. Fig. 13.13 shows that the hydrodynamic loading in the initial stage of impact is maximized in the contact line between the hull, reaching its minimum close to the keel near the mid-span. As time advances and the hull penetrates the water surface, the

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hydrodynamic distribution dramatically changes, whereby the pressure at the keel can attain, or even exceed, the values measured in the pile-up. To further elucidate the role of the axial velocity on the hydrodynamic loading, we have compared the peak value of the impact force per unit length obtained from PIV for the all cross-sectional planes in Fig. 13.14. Therein, we also present the impact forces obtained by neglecting the axial velocity component. Fig. 13.14 shows that the force is approximately constant for planes close to the mid-span (0 z 3 cm), while it starts to decrease for z > 3 cm. These results suggest that the cross-section at the mid-span experiences a larger load as compared to the end of the hull. We find a highly nonlinear behavior for the maximum impact force per unit length by fitting the data through third-order polynomial. Finally, our results show that the axial

X (cm)

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Fig. 13.13 Normalized pressure distribution on the surface of the hull at three time instants: (A) t ¼ 7.5 ms, (B) t ¼ 12.5 ms, and (C) t ¼ 17.5 ms. Reproduced from M. Jalalisendi, S.J. Osma, M. Porfiri. Three-dimensional water entry of a solid body: a particle image velocimetry study. J Fluids Struct 2015;59:85-102.

Maximum force per unit length (N/m)

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Fig. 13.14 Maximum force per unit length for all the measurement locations along the hull length. Black triangles and red circles refer to average values of the three repetitions using the estimation of the axial velocity component (3D) or not (2D), respectively. Error bars indicate the maximum difference between the repetitions and the average. The blue line is a third order polynomial fit of the averaged 3D data (1.59z3 27.72z2 + 61.03z + 600.10 where Z is in cm). Reproduced from M. Jalalisendi, S.J. Osma, M. Porfiri. Three-dimensional water entry of a solid body: a particle image velocimetry study. J Fluids Struct 2015;59:85-102.

velocity component moderately influences the maximum force per unit length with a maximum difference of 16.3% between 2D and 3D values. A detailed validation of PIV-based pressure reconstruction in the study of 3D water impact can be found in Facci et al. [51], where we have extended the computational framework proposed in Facci et al. [50] to study 3D impact.

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Conclusions and outlook on ongoing research

In this chapter, we have demonstrated an alternative, new approach to experimentally investigate hull slamming. The methodology revolves around PIV, a classical nonintrusive technique used in experimental fluid mechanics to measure the velocity field in fluid flows. In our approach, PIV is used to measure the flow velocity during water impact, which is, in turn, leveraged to reconstruct the pressure field. Specifically, we have proposed an entirely data-driven approach to reconstruct pressure from the integration of Navier-Stokes equations. The reconstructed pressure has been ultimately utilized to estimate the impact force on the wetted width of the impacting body. The proposed experimental scheme applies to the study of 2D and 3D impact, and can be used to treat rigid as well as flexible hulls. Our results illustrate the robustness of PIV-based pressure measurements. Specifically, comparison of PIV-based results with available analytical results demonstrated that PIV is able to successfully estimate the spatio-temporal evolution of the hydrodynamic loading generated by 2D impact of rigid hulls. PIV is also successful in anticipating the effect of the structural flexibility on the hydrodynamic loading. We found that structural flexibility, depending on the hydroelasticity factor, significantly modulates the peak value and rise time of the impact force. Further, we contemplated the extension of the approach to analyze the role of 3D phenomena on the hydrodynamic loading. Results show that PIV is able to elegantly elucidate the role of axial velocity on the hydrodynamic loading. The key parameters controlling the accuracy of the PIV measurements are spatial and temporal resolutions, which are particularly challenging to control in the pile-up region. Specifically, we found that PIV measurements underestimate the hydrodynamic loading compared to analytical results due to underestimation of the velocity field in the early stage of the impact. However, this discrepancy is mitigated as time evolves and more vectors are available to estimate large velocity gradients, such that after few milliseconds results should be considered valid. Building on this experimental framework, we are currently seeking to provide guidelines for designing composite panels for marine vessels, by dissecting the physical principles underpinning material degradation. Our research is focusing on syntactic foams, which are a class of closed-porosity particulate composites with tremendous potential in underwater applications [60,61]. We have designed a metallic frame to hold flat vinyl ester matrix-glass microballoons syntactic foam panels during impact (see Fig. 13.15). We have analyzed the response of syntactic foam panels during water entry and investigated the effect of different microballoon type reinforcement and impact heights on the panels’ failure. The hydroelastic model developed by our group [33] was utilized to inform the design of the experiments, that is, selecting the geometric dimensions of the panels and the impact height. Preliminarily results indicate that the failure of the syntactic foam panels is controlled by the impact height and can be mitigated through careful selection of the embedded microballoons.2

2

Since the submission of this chapter, we have completed these experiments which have been recently published [63].

PIV-based pressure reconstruction during water impact

VE220-60 type Syntactic foam panel

Fixture

413

Fig. 13.15 Illustration of our recently developed experimental fixture to investigate hull slamming of syntactic foam panels.

Failed specimen

Beyond the analysis of syntactic foam panels, we also plan to expand our PIV-based approach to analyze and predict the response of composite structures to ice-impact loading. We specifically seek to establish a physically based understanding of the dynamic behavior of composite structures interacting with particle-laden flows, composed of a solid phase (ice) dispersed in a liquid (water), under extreme environmental conditions. Further, we are investigating asymmetric impacts considering both roll and heave motions, using both direct and indirect measurements for hydrodynamic loading. Estimating the pressure field from PIV data during hull slamming alleviates the drawbacks of direct pressure measurements and facilitates a thorough validation of theoretical and computational approaches. Not only does PIV afford spatio-temporal evolution of the hydrodynamic loading experienced during hull slamming, but also it provides information about the resolution of the flow physics, which is central toward an improved understanding of fluid-structure interactions.

Acknowledgment This work has been supported by the Office of Naval Research (Grant N00014-10-1-0988) with Dr. Y.D.S. Rajapakse as the program manager. The authors would like to thank Dr. Andrea Facci, Mr. Mohammad Jalalisendi, Mr. Steven J. Osma, Dr. Riccardo Panciroli, Dr. Stefano Ubertini, and Mr. Sam Zhao, who have contributed to the research efforts summarized in this chapter.

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[43] Panciroli R, Porfiri M. Evaluation of the pressure field on a rigid body entering a quiescent fluid through particle image velocimetry. Exp Fluids 2013;54(12):1630. [44] Jalalisendi M, Osma SJ, Porfiri M. Three-dimensional water entry of a solid body: a particle image velocimetry study. J Fluids Struct 2015;59:85–102. [45] Panciroli R, Shams A, Porfiri M. Experiments on the water entry of curved wedges: high speed imaging and particle image velocimetry. Ocean Eng 2015;94:213–22. [46] Shams A, Jalalisendi M, Porfiri M. Experiments on the water entry of asymmetric wedges using particle image velocimetry. Phys Fluids 2015;27(2):027103 [1994-present]. [47] Jalalisendi M, Zhao S, Porfiri M. Shallow water entry: modeling and experiments. J Eng Math 2016;1–26, http://dx.doi.org/10.1007/s10665-016-9877-3. [48] Jalalisendi M, Shams A, Panciroli R, Porfiri M. Experimental reconstruction of three-dimensional hydrodynamic loading in water entry problems through particle image velocimetry. Exp Fluids 2015;56:1–17. [49] Shams A, Zhao S, Porfiri M. Hydroelastic slamming of flexible wedges: modeling and experiments from water entry to exit. Physics of Fluids 2017;29:037107. [50] Facci AL, Panciroli R, Ubertini S, Porfiri M. Assessment of PIV-based analysis of water entry problems through synthetic numerical datasets. J Fluids Struct 2015;55:484–500. [51] Facci AL, Porfiri M, Ubertini S. Three-dimensional water entry of a solid body: a computational study. J Fluid Struct 2016;66:36–53. [52] Panciroli R, Porfiri M. Hydroelastic impact of piezoelectric structures. Int J Impact Eng 2014;66:18–27. [53] Panciroli R, Porfiri M. Analysis of hydroelastic slamming through particle image velocimetry. J Sound Vib 2015;347:63–78. [54] Dantec. DynamicStudio V3.12 user’s guide. Skovlunde, Denmark: Dantec Dynamics; 2010. [55] Thielicke W, Stamhuis EJ. PIVlab–towards user-friendly, affordable and accurate digital particle image velocimetry in MATLAB. J Open Res Softw 2014;2(1):e30. [56] Panton RL. Incompressible flow. New York: John Wiley & Sons; 1994. [57] Truscott TT, Epps BP, Belden J. Water entry of projectiles. Annu Rev Fluid Mech 2014;46:355–78. [58] Mei X, Liu Y, Yue DKP. On the water impact of general two-dimensional sections. Appl Ocean Res 1999;21(1):1–15. [59] Faltinsen OM. Water entry of a wedge by hydroelastic orthotropic plate theory. J Ship Res 1999;43(3):180–93. [60] Sagi-Mana D, Narkis M, Siegmann A, Joseph R, Dodiuk H. The effect of marine environment on a vinyl ester resin and its highly filled particulate quartz composites. J Appl Polym Sci 1998;69:2229–34. [61] Gupta N, Pinisetty D, Shunmugasamy VC. Reinforced polymer matrix syntactic foams: effect of nano and micro-scale reinforcement. New York: Springer; 2013. [62] Shams A, Lopresto V, Porfiri M. Modeling fluid-structure interactions during impact loading of water-backed panels. Composite Structures 2017;171:576–90. [63] Shams A, Zhao S, Porfiri M. Water impact of syntactic foams. Materials 2017;10(3):224.

Index Note: Page numbers followed by f indicate figures, and t indicate tables. A ABAQUS, 109–110, 109f, 114–115, 117, 131, 133–135, 134f, 134t, 141–142 Added mass effect, 59–60, 188–189, 189f Added mass factors (AMFs), 184, 190t Air blast loading damage maps, 280–283, 281–282f, 282t out-of-plane displacement, 278–280, 278–281f Air blast testing, 263 blast configuration, 269f face-sheets, 265 postblast damage assessment, 271 test design and instrumentation, 269–271, 270f Airy stress function method, 4 Aluminum alloys, 368–369 AMFs. See Added mass factors (AMFs) Anisotropic laminates, elastic coupling with, 67–69 Anisotropic materials designing, 65–69 B Beam elastic properties of, 58 theories, 200–201 Bearing-mode absorber, 300, 325 BEM. See Boundary element method (BEM) Bending energy, 318–319, 338–341, 338f Bend-twist coupling, 68 Benzeggagh-Kenane (BK) law, 133t Bernoulli-Euler beam, 200–201 theory, 47–48, 50–52, 54 Bilinear cohesive law parameter, 172t Bilinear traction-separation approach, 323 behavior, 171f Blast testing, 263 DIC data, 271 GFRP, 265 Body-In-White (BIW), 366

Boundary element method (BEM), 70 BEM-FEM approach, 70, 72 Bumper beam innovative design beam-cashbox, 386t die forming process, 383–384 dynamic impact loading, 387f energy vs. time curves, 386f failure modes, 389f GMT, 383–384, 385f GMTex, 389f GMT-UD, 388, 388f load vs. displacement curve, 385f mechanical fastening, 383 modified bumper system, 387–388f

C Cantilever composite plate vibration, 62–64 Carbon-epoxy woven composite, 159 ply, 164f Carbon fiber composite, 252t impact behavior of, 228–229 Carbon-fiber reinforced polymer (CFRP) face-sheet panel, 263, 368 influence of temperature, 374 three-dimensional reconstruction, 291f underwater blast loading, 285–288, 301 Carbon nanofibers (CNFs), 230 Carbon nanoparticles in composite structure, 217–218 energy absorption mechanism, 219–226, 221f functionalization strategy, 219 from nanotubes to graphene nanoplatelet, 218–219 role of mechanical properties of, 221–226 Carbon nanotubes (CNTs), 218–219 fracture mechanism, 221f Cartesian coordinate system, 402

418

CFRP. See Carbon-fiber reinforced polymer (CFRP) Chang/Chang damage onset criteria, 89t Classical laminate theory (CLT), 318–319 Classical plate theory (CPT). See Kirchhoff–Love theory Classical Wagner’s solution hydrodynamic pressure distribution, 408f vs. PIV-based pressure reconstruction, 403–404, 407f CNFs. See Carbon nanofibers (CNFs) CNTs. See Carbon nanotubes (CNTs) Coefficients of friction (COF), 316–317 COH3D8 finite element, 109f Cohesive element, 108–109, 109–110f, 132 Cohesive-frictional model, 110–111 Cohesive material, traction-separation law for, 132f Cohesive zone model (CZM), 108–112, 323–324, 327–328 FEM model on, 131 Compact compression (CC) test, 318–319 Compact tension (CT) test, 318–319 Component geometry, 370 Composite blade, impact on, 64–65 Composite constitutive models, 342 Composite marine propeller blades, 69–74 analysis method, 70–71 layups, 71–73 propeller size, 70 tailoring of, 69–70 Composite materials, 3, 64, 85–86 advantages, 370f air and underwater blasts, 263 calibration of cohesive properties, 113–116 CDM for, 87–98 challenging issue in designing, 87 cohesive input properties, 111–112, 111t, 113t cohesive stiffness, 112 constitutive response, 98, 99f crushing process of, 336–337 damage mechanism in, 159–160 elastic behavior, 378 energies at damage onset, 113 energy absorption mechanisms, 371 failure modes, 300 graded density core, 293

Index

heterogeneous nature of, 129–130 laminate material properties of, 144t material calibration, 99–100 material models, 88–90 mechanical properties, 373t, 379 mesh size, 112 micromechanical calibration, 100–121 nonprogressive degradation of elastic properties, 89f numerical simulation, 117–121 plate model, 118f properties, 101t scheme and failure modes, 300f shock resilience, 263–264 X-radiography, 120f Composite plates, impact on beams immersed in water, 208 in contact with water, 184–189, 207–212 direct formulation, 204 dynamics of deformable structures, 190–204 integral formulation, 203–204 local deformations, 202–203 motion of projectile, 201–202 Composite structure impact behavior of, 226 modeling approach, 323–324, 324f Composite tube, 240–244, 241f, 246–249, 259, 336 Compressive fiber failure, 136–137, 137f Compressive matrix failure, 138–139, 138f Computational models, 396 Conical shock tube (CST), 264 Constitutive damage model, 129–130 Continuum damage mechanics (CDM), 87–98, 322 damage constitutive model, 90–97 FEM model on, 131 model, 95, 97, 130 theory, 91 Continuum shell (CS) elements, 142 fiber traction damage, 148f in-plane shear damage, 152f matrix compression damage, 151f matrix traction damage, 150f model, 147f Convolution integral approach, 199–200 Coriolis acceleration, 47–49

Index

Coupled Eulerian-Lagrangian (CEL) approach, 244 CPT. See Classical plate theory (CPT) Crashworthiness design, 336–337 Critical strain energy, 339, 343f Crushing process conical shell model, 338–339, 338f critical strain energy, 339, 343f friction energy, 338f, 339–340 geometrical and crushing characteristics, 358t hoop strain energy, 338f, 339 male and female molds, 343–344, 344f numerical vs. experimental dynamic tests, 359t CZM. See Cohesive zone model (CZM) D Damage constitutive model, CDM, 90–97 Damage Index (DI), 382 Damage initiation criteria, 132t Damage mechanism, in composite materials, 159–160 Damage propagation numerical models, 130–131 Data acquisition system, 397 DCB. See Double cantilever beam (DCB) Deformable propeller, 59–60 Deformation in contact zone, 202–203 theory, Mindlin-Reissner first-order shear, 186 Delamination, 129–132 Delamination damage CS model, 147f VUMAT model, 146f DFAIL, 104 Digital image correlation (DIC), 240–242, 247, 259 Displacement functions, 42–44 negative discriminant, 43–44 positive discriminant and negative roots, 42 zero discriminant, 43 Double cantilever beam (DCB) finite element model, 113 model, 112 simulation test, 115f

419

specimen geometry, 114f test, 111–113, 116f, 318–319 Drop test system, 344–345, 345f Drop weight normal impact test, 163f, 172–176, 173f Dynamic axial compression test, 348 Dynamic compression, 267–268, 267f, 274, 276f Dynamic impact test, 344–345, 347t load vs. displacement, 345, 346f DynamicStudio software, 398–399 Dynamic tension test, 268–269, 268f, 274–278 E Effective stress hypothesis, 93f E-glass/epoxy pultruded beam bumper, 371 composite, 377 Elastic center, 57 Elastic coupling, 58–59 with anisotropic laminates, 67–69 Elastic damage model, 106 Elasticity model, two-dimensional, 7–19 Elastic properties, of beam, 58 End-notched flexure (ENF) finite element models, 112 simulations, 111–112 specimen, 111–113, 114f test, 114–115, 116f Energy absorbers, 299, 329 bolt diameter, 312–313, 312–313f crushing behavior, 315 fiber material, 306, 307f fiber orientation, 307 force-displacement diagram, 300f high-rate loading, 309–310, 309–310f laminate bending deformation, 327f materials and manufacturing, 301, 302–303t matrix material, 308–309 plate thickness, 312–313, 312–313f sandwich configuration, 310–312, 311f, 314 temperature influence, 313–314 testing and evaluation procedure, 301–306, 305f Energy absorption, 336, 370 fiber-reinforced composite material, 371

420

Energy absorption, analytical approach, 314, 319–321 vs. experimental data, 319, 321f model development, 315–317, 315f model validation, 317–319, 318f parameter determination, 317–319, 318f Energy absorption, numerical method, 322, 328–329 composite material modeling, 322–325 model development, 322–325 simulation results, 325–328 ENF. See End-notched flexure (ENF) Equation of motion for rotating string, 49–50 in terms of force and moment resultants, 60–61 Equivalent displacement and stress definitions, 135t Equivalent single layer (ESL) theories, 184, 187 Explosive-induced implosion, 240, 244 experimental investigation, 244–247 numerical simulation, 247–258, 248f, 253–255f, 257–258f Extension-bending coupling, 68–69 Extension-shear coupling, 68 Extension-torsion coupling, 59 F Face-sheet, 265, 266f Failure criteria, Chang examples of, 96t Failure evolution law, 133t FEM. See Finite element method (FEM) Fiber compression damage, VUMAT approach, 149f Fiber-metal laminate (FML) plates, 263 Fiber-reinforced material, 365–366 Fiber reinforced plastic (FRP) laminated composites, 90 Fiber reinforced polymer composites (FRPCs), 263 Fiber-reinforced polymer matrix materials (FRPs), 299, 325–326 Fiber tensile failure, 136 Fiber traction damage CS approach, 148f VUMAT approach, 148–149f Fiber traction failure mode, 136f

Index

Finite element method (FEM), 60 Finite element model (FEM), 100–121, 161, 330, 341–342 and boundary conditions, 143f on CDM and CZM, 131 composite circular frust, 343f DCB, 113 First-order shear deformation theory (FSDT), 197–199 Mindlin-Reissner, 186, 197 Fluid-structure interaction (FSI) analysis, 69–71 problem, 240 simulations, 247–249 Fluid velocity, 403 Foam characterization, 265–266, 274 dynamic compression, 267–268, 267f, 274, 276f dynamic tension, 268–269, 268f, 274–278 quasi-static compression, 266, 274, 275t quasi-static tension, 266–267, 267f, 274, 276t Fracture toughness, 217, 220–222 of nanocomposites, 222–226 Free vibration analysis, 190 Friction energy, 338f, 339–340 FSDT. See First-order shear deformation theory (FSDT) FSI. See Fluid-structure interaction (FSI) Full-scale air blast test, 263, 269, 271 Full-scale ship model, 395 G Gas gun oblique impact test, 176–179, 178f GFRP. See Glass-fiber reinforced polymer (GFRP) GLARE, 87 Glass-fiber reinforced polymer (GFRP), 368 explosive resilience, 263 face-sheets, 265 strain contour plots, 284–285f underwater blast loading, 264, 283–285 Glass mat-reinforced thermoplastic (GMT), 368, 378–379 energy curve, 389f fabric (Tex) layers, 380 failure mode, 385f unidirectional, 380

Index

Graphene nanoplatelets (GNP), 218–219 Green House Gases (GHG) emission, 365 H Hashin failure criteria, 134t HCNTs. See Helically coiled carbon nanotubes (HCNTs) Heat conduction problem, 9–10 and thermo-elastic problems, 9–10 through transfer matrix method, 10–13 Heat transfer, 6 Helically coiled carbon nanotubes (HCNTs), 218, 231 Helicopter blades, 64–65, 159, 160f impacts, 159–160, 160f Hertz’s impact problem, 204, 205f Heuristic model, 130 Hoop strain energy, 338f, 339–341 Hull falling contour plots, 406f normalized pressure distribution, 411f solid line vs. dashed lines, 407f Hull slamming, 395, 413f fluid-structure interactions, 395 hydrodynamic loading, 396 structural deformations, 396 Hydrodynamic loading, 396, 402 Hydroforming, 365–366 Hydrostatically induced implosion, 240–244, 242–243f Hydrostatic loading, 240–241, 241f I Impact behavior, 155 of composite structure, 226 Impact damage, 159 Implosion, 239–240. See also Explosiveinduced implosion hydrostatically induced, 240–244, 242–243f In-plane shear damage, CS approach, 152f Integrated front-end system (IFES), 368 Interface element, 170f Interfacial thermal resistance, 6, 8–9, 30–31, 40f Interlaminar damage model, 132–133 Interlaminar modeling, 323–324 Internal energy, 338

421

Internal state variables (ISV), 130 Intralaminar damage models, 133–141 Intralaminar damage prediction ABAQUS standard approach for, 133–135 user subroutine (VUMAT) approach for, 135–141 Intralaminar failure mode, 135–136 Intralaminar failure model, user subroutine (VUMAT) flow chart, 141f Intralaminar modeling, 322–324 Isotropic square plate, dynamic response for, 209, 210–211f K Kirchhoff–Love theory, 169, 184–185, 190–191, 197–198 approximate methods, 197 equations of motion and boundary conditions, 190–194 FSDT, 197–199 large plate approximation, 196–197 modal analysis, 194–195 plates, 60–62 transient response using convolution integral, 195–196 Kuhn-Tucker loading-unloading conditions, 95–96 L Ladeve`ze CDM model, 322 Lagrangian modeling approach, 323–324 Laminated composite, 3, 64, 253–254f failure modes in, 88–90 FRP, 90 Laminated plate, thermo-elasticity solutions for, 4–6 Laminates stack schematization, 142f Lamination parameters, 65–67, 71f Layerwise theories (LWT), 187 Lightweight composite structures, 395 Lightweight material replacement, 366 Lightweight solution bumper beam innovative design, 383–389, 385–389f, 386t material characterization, 378–383, 380–383f, 381t pultruded solution, 369–377, 370f, 372–377f, 373–377t

422

Lightweight solution (Continued) thermoplastic matrix, 378 vehicle development, 368–369, 368f for vehicle frontal bumper, 365–368, 367f Linear interfacial traction laws, 5–6 Local transfer matrix, 5 Low energy impact simulation, 106–111 Low velocity impacts (LVIs), 99, 106–108, 117, 129–132 Low velocity normal impact test, 162–163, 174 LS-DYNA, 100–103, 104–105f LVIs. See Low velocity impacts (LVIs) M Macro-scopic modeling, 323–324 Marine propellers, 47 Mass center, 58 MAT54, 104f MAT58, 105f, 106 Material replacement approach, 365–366 Matrix compression damage CS approach, 151f VUMAT approach, 151f Matrix traction damage CS approach, 150f VUMAT approach, 150f Maximum nominal stress (MAXS) criterion, 132, 132t Medium velocity oblique impact test, 161–162, 162f Meso-scopic modeling, 323–324 Meyer’s law, 202 Micro-mechanics approach, 87 Mie-Gr€uneisen equation, 250t Mindlin-Reissner first-order shear deformation theory, 60–61, 186, 197 Multilayer plate theories, 187–188 Multi-walled nanotubes (MWCNTs), 218, 222–230 N Nanocomposites, fracture toughness of, 222–226 Nano-reinforced composite structure, impact response of, 226 high velocity impact response, 230–231 low velocity impact response, 226–230, 227f, 229f

Index

Navier-Stokes equation, 400–401 Newton-Raphson iterative scheme, 166 Nonlinear shear damage, 139–141 Nonpolynomial theories, 186–187 Nonrotating pretwisted beam, 52–53 Numerical modeling, 341–342 damage propagation, 130–131 energy-stroke diagrams, 353f load-displacement, 345, 346f, 353f PSO method, 352–356 real and simulated specimen, 360f O Oblique impact, 159, 160f, 162f modeling, 177f Oblique impact test, 162f, 174f load vs. time curves of, 177f medium velocity, 161–162 One-element test, 98f P Pagano’s model, 6 Particle image velocimetry (PIV), 396 analysis, 399–400, 399f data acquisition system, 397–399, 398f experimental setup, 397–399, 398f syntactic foam, 412 Particle image velocimetry (PIV)-based pressure reconstruction, 396, 400 vs. classical Wagner’s solution, 403–404, 407f error propagation, 396 force per unit depth, 407f Navier-Stokes equation, 400–401 normalized hydrodynamic pressure distribution, 408f Poisson equation, 401–402, 401f 3D water entry, 409–411, 410–411f 2D to 3D, 402–403, 402f 2D water entry, 403–408, 405–409f Particle swarm optimization (PSO), 352–356 velocity and position updates, 357f PIV. See Particle image velocimetry (PIV) Plate theory, 184–188 Kirchhoff-Love theory, 184–185, 197–198 Mindlin-Reissner first-order shear deformation theory, 186, 197 multilayer theories, 187–188

Index

nonpolynomial theories, 186–187 Reddy’s third-order shear deformation theory, 186 Poisson equation discretization, 401f time-marching algorithm, 402 Polynomial theory, 199 Postblast damage assessment CFRP panels, 291–292f GFRP panels, 290f X-ray CT scanning, 288 Power law (PW), 133t Pressure gage, 395 Pretwisted beam nonrotating, 52–53 rotating, 53–54 Pretwisted plate, 64 Progressive bearing failure, 307 energy absorption, 315–316 experimental test matrix, 312f modeling approaches, 324f weight-specific energy absorption, 311f Propeller blades, 51–52 composite marine, 69–74 Propeller, deformable, 59–60 PSO. See Particle swarm optimization (PSO) Pultruded composites (PFRP), 370 Pultrusion, 369–370 energy and peak loads value, 375–376t energy vs. displacement, 377, 377f load vs. displacement, 377, 377f reaction force vs. time, 373–376f simplified bumper beam model, 372–373, 372f Young modulus, 372–373 Q Quadratic nominal stress (QUADS) criterion, 132, 132t Quasi-isotropic lamination, 343–344 Quasi-static compression, 266, 274, 275t Quasi-static tension, 266–267, 267f, 274, 276t Quasi-static test, 379 penetration strength, 381t R Rayleigh beam theory, 51 Rayleigh-Ritz method, 60, 62, 188–189

423

Reddy shear deformation theory (RSDT), 60–61, 186, 199 Reinforced panel, 142 Representative volume element (RVE), 130 Rod element, parameters for, 169t Rod failure, 167–169, 168f Rotating beams added-mass effect, 59–60 complicating factors, 54–55 dynamics of, 47–48 effect of preset angle, 51–52, 51f effect of pretwist, 52–54 inertia loads, 48–51 spatial beams, 55–59 Rotating plates dynamics of, 60 impact on composite blades, 64–65 mechanics of plates, 60–62 vibrations of cantilever composite plates, 62–64 Rotating pretwisted beam, 53–54 Rotating string, equation of motion for, 49–50 Rotating tapered beam, 50–51 RSDT. See Reddy shear deformation theory (RSDT) RVE. See Representative Volume Element (RVE) S Sandwich plates, thermo-elasticity solutions for, 4–6 Sandwich specimen, with two plies of skin, 174–176, 176f SEA. See Specific energy absorption (SEA) Semianalytical models, 396 Shear center, 57–58 and bend-twist coupling, 57f Shear deformation theory Mindlin-Reissner first-order, 186 Reddy’s third-order, 186 Shear in-plane (1-2) damage, VUMAT approach, 152f Shear out-of-plane (1-3) damage, VUMAT approach, 153f Shear out-of-plane (2-3) damage, VUMAT approach, 153f Shear stress-strain response, 140f Sheet molding compound (SMC), 368

424

Index

Shell damage model, 165–166 Shell element parameter, 167t Shock tubes, 263–264 Single shell layer models, 341–342, 358–359 Single-walled nanotubes (SWCNTs), 218, 230 Sinusoidal transverse loading, 19–24 five layer plate, 24, 26–29t three layer plate, 19–24, 20–23t, 32–39t Skew angle effect, 54–55, 54f Southwell’s equation, 50 Spatial beams, 55–59 Spatial integration method, 401 Specific energy absorption (SEA), 299, 336 diameter/thickness ratio, 351f wall inclination, 351f Specific surface areas (SSAs), 219–220 Split Hopkinson pressure bar (SHPB), 267–268, 267f Spring-mass impact models, 183, 204 Hertz’s impact problem, 204, 205f mass against, 205–207, 205–207f two masses, 205 SSAs. See Specific surface areas (SSAs) Stacked shell approach, 322–324 Stainless steel plate modes, 190t Steady-state thermo-elasticity solution, 4 Steel material, stress-plastic strain property, 373t Stiffened composite panel, 141–142 FEM models for, 142 geometrical and material description of investigated, 142–144 Stiffness invariant, 65–67 Strain gages, 395 Stress-strain laws, 165–166 Stress-strain relations, 55, 65–66, 94–95 Styrene acrylonitrile (SAN), 263 SWCNTs. See Single-walled nanotubes (SWCNTs) T Tensile Tensile Tensile Tensile

absorbers, 299 fiber failure, 136 matrix failure, 137–138 test, 102–104, 102f

Theoretical modeling, 337–338, 338f bending energy, 338–339 crack energy, 339 friction energy, 339–340 hoop energy, 339 mean load P, 340–341 Thermal continuity condition, transfer matrix, 12–13 Thermal loading, 25–31 Thermal resistance, interfacial, 6, 8–9, 30–31, 40f Thermo-elasticity models, 5 solutions for laminated and sandwich plates, 4–6 theory, 4–5 three-dimensional, 3 Thermo-elasticity problem, 9–10 complementary solution for layer k, 14–16 particular solution for layer k, 14 positive discriminant and positive roots, 15–16 through transfer matrix method, 13–19 Thermoplastic matrix, 378 drop dart testing machine, 382f material characterization, 378–383, 380f quasi-static penetration tests, 381f specimen configuration, 382f Thin-walled truncated conical structures, 336, 343–344 energy absorption capacity, 348 fiberglass materials, 337 Three-dimensional thermo-elasticity, 3 3D water entry, 409–411, 410f Time-marching algorithm, 402 Timoshenko beam, 54–55, 201 equation, 51–52 kinematics, 55–57 theory, 47–48 Traction-separation law, 170 for cohesive material, 132f Transfer matrix method, 5, 12f, 13–19 continuity conditions, 12–13, 17–19 of generic layer, 12, 17 heat conduction problem through, 10–13 thermo-elastic problem through, 13–19 Trinitrotoluene (TNT), 245 explosion, 245f, 246–247

Index

Two-dimensional elasticity model, 7 assumptions, 7–8, 7f mechanically and thermally imperfect interfaces, 8–9 2D modeling approach, 322–323 2D pressure reconstruction Cartesian coordinate system, 402 interpolation steps, 404f to 3D, 402–403 2D water entry of rigid hulls, 403–406, 405–407f structural flexibility, 406–408, 408–409f U Underwater blast loading of CFRP, 285–288, 286–288f, 289t of GFRP, 283–285, 283–285f Underwater blast testing, 263 composite sandwich panels, 271 schematic of, 272f test design and instrumentation, 271–272, 272f X-ray CT scanning setup, 273, 273f Underwater shock loading simulator (USLS), 246–247, 246f UNDEX, 244 V VACNTs. See Vertically aligned carbon nanotubes (VACNTs) VCCT. See Virtual crack closure technique (VCCT)

425

Vehicle development, 368–369, 368f lightweight structures, 365 Vehicle bumper task components of, 367, 367f composite materials, 368 crashworthiness, 366–367 Velocity normal impact test, low, 162–163 oblique impact test, medium, 161–162, 162f Vertically aligned carbon nanotubes (VACNTs), 231 Virtual crack closure technique (VCCT), 106–108, 131 VUMAT approach, 110 fiber compression damage, 149f fiber traction damage, 148f matrix compression damage, 151f matrix traction damage, 150f shear in-plane (1-2) damage, 152f shear out-of-plane (1-3) damage, 153f shear out-of-plane (2-3) damage, 153f VUMAT model, 144–145, 146f, 147, 154f W Woven composite, 159, 161, 179 carbon-epoxy, 159 model, 160 Woven laminate modeling, 169–172 Woven ply modeling, 163–169, 164f Z Zigzag theories (ZZTs), 187–188