Macro-, Meso-, Micro- and Nano-Mechanics of Materials [1 ed.] 9783038130376, 9780878499793

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Macro-, Meso-, Micro- and Nano-Mechanics of Materials

Macro-, Meso-, Micro- and Nano-Mechanics of Materials Special Issue Containing the Proceedings of the International Symposium on Macro-, Meso-, Micro- and Nano-Mechanics of Materials (MM2003) 8-10 December 2003, Hong Kong

Edited by

Tong Yi-Zhang and Jang-Kyo Kim Organized by

Department of Mechanical Engineering Hong Kong University of Science and Technology

TRANS TECH PUBLICATIONS LTD Switzerland Germany UK USA

Copyright © 2005 Trans Tech Publications Ltd, Switzerland All rights reserved. No part of the contents of this book may be reproduced or transmitted in any form or by any means without the written permission of the publisher. Trans Tech Publications Ltd Brandrain 6 CH-8707 Uetikon-Zuerich Switzerland http://www.ttp.net ISBN 0-87849-979-2 Volume 9 of Advanced Materials Research ISSN 1022-6680 Full text available online at http://www.scientific.net

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Preface This issue is a compilation of the papers selected amongst those presented at the International Symposium on Macro-, Meso-, Micro- and Nano-Mechanics of Materials (MM2003), held on December 8-10, 2003 at the Hong Kong University of Science & Technology (HKUST), Hong Kong. The principal objective of MM2003 was to create an international forum for the exchange, dissemination and discussion of the state-of-the-art technologies and recent developments in macro-, meso-, micro-, nano-mechanics of materials and structures. While MM2003 focused on the most recent advances in strength, fracture and mechanics of micro- and nano-structured materials, it also served to link the gaps between the conventional macro/mesosystems and the emerging micro/nano-materials technology. 128 papers were presented at MM2003, covering the following areas: Nanotube/Nanofiber; Thin films; Biomaterials & Biomimetics; Polymer/Composites/Interface; Fatigue & Creep; Piezoelectrics & Ferroelectrics; Fracture Mechanics; Fracture Physics; Computational Mechanics; Mechanics of Rock & Concrete; Dynamic Fracture & Damage. Amongst these topical areas, papers in the fields of composite materials; thin films and coatings; as well as fracture, fatigue and strengths of advanced materials are included here. MM2003 was originally incepted to honor Professor Pin Tong, an eminent scholar and educator, who has made many important contributions during his fifteen years at HKUST, on the occasion of his 65th birthday. Along with regular sessions with presentations of contributed papers, many reputed speakers were invited amongst his former/current colleagues, friends and graduate students from different parts of the world. These invited presentations reflect Professor Pin Tong’ s past and present research interests on computational mechanics of a broad range of materials and structures. We wish to thank all authors who contributed their papers to MM2003. Thanks are also due to the members of the International Advisory Committee and the Local Organizing Committee for their efforts in making MM2003 a success. We also appreciate very much for the financial and moral supports by various sponsors, including the K.C. Wong Education Foundation, the US Natural Science Foundation, the US Army Research Office – Far East, the Far East and Oceanic Fracture Society (FEOFS), the Hong Kong Society of Theoretical & Applied Mechanics (HKSTAM), the Micro Materials Measuring Technology, the Guyline (Asia) Ltd., and the Department of Mechanical Engineering, HKUST.

Tong-Yi Zhang Jang-Kyo Kim Editors and Chairs of the MM2003 Organizing Committee

Table of Contents Preface Finite Element Analysis of Particle Reinforced Composite Using Different Cell Models J.P. Fan, C.Y. Tang, C.L. Chow and C.P. Tsui Active Finite Element Method for Simulating the Contraction Behavior of a Muscle-Tendon Complex C.P. Tsui, C.Y. Tang, C.L. Chow, S.C. Hui and Y.L. Hong Microstructure of the Fiber/Matrix of Chafer Cuticle B. Chen, X. Feng, J.H. Fan and X.L. Wu Residual Stress and Stress-Strain Relationship of Electrodeposited Nickel Coatings Y.C. Zhou, Y.P. Jiang and Y. Pan Failure of Thermal Barrier Ceramic Coating Induced by Buckling Due to Temperature Gradient and Creep W.G. Mao and Y.C. Zhou Cyclic Stress-Strain Behavior and Thermomechanical Effect in Metal Matrix Composites H.G. Kim Optimization of Residual Thermal Stress in SiC/C Plasma Facing Material for Future Tokamak Facilities of China C.C. Ge, W.B. Cao and A.H. Wu Influence of Surface Properties on Microscratch Durability of Aluminum Nitride Semiconductor Processing Component L. Chouanine, M. Takano, F. Ashihara, O. Kamiya and M. Nishida Thermal Shock and Thermal Fatigue of Ferroelectric Thin Film due to Pulsed Laser Heating X.J. Zheng, S.F. Deng, Y.C. Zhou and N. Noda Fatigue Damage of Materials with Small Crack Calculated by the Ratio-Method under Cyclic Loading Y.G. Yu Dynamic Strength of Steel Welds under High Strain Rate Loading B. Wang and G. Lu Synthesis and Mechanical Properties of Nanostructured Mg-Al-Nd Alloys C. Yan, L. Ye, L.L. Yan, L. Lu, M.O. Lai and Y.W. Mai Study on Fracture Behaviors of High-Density Polyethylene Pipe Material Based on Local Approach F.J. Qi, L.X. Huo and H.Y. Jing Effects of Gun Tube Profile and Sabot on Stresses and Velocity of Long Rod Penetrator J.B. Kim Numerical Study on Cracking Process of Masonry Structure S.H. Wang, Y.B. Zhang, C.A. Tang and L.C. Li Micromechanical Model for Simulating Hydraulic Fractures of Rock T.H. Yang, L.G. Tham, S.Y. Wang, W.C. Zhu, L.C. Li and C.A. Tang Residual Stress and Surface Molding Conditions in Thin Wall Injection Molding H.K. Lee, G.E. Yang and H.G. Kim Symplectic Solution for a Plane Couple Stress Problem W.X. Zhong, G. Fang and W. Yao Multi-Scale and Finite Element Analysis of the Mixed Boundary Value Problem in a Perforated Domain under Coupled Thermoelasticity Y. Feng and J. Cui Simulation of Multiple Hydraulic Fracturing in Non-Uniform Pore Pressure Field L.C. Li, C.A. Tang, L.G. Tham, T.H. Yang and S.H. Wang Mechanics Framework for Micron-Scale Planar Structures A.C.M. Chong, F. Yang, D.C.C. Lam and P. Tong A Piezoelectric Screw Dislocation Interacting with a Dielectric Crack in a Hexagonal Piezoelectric Material J.X. Liu and X.L. Liu

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Surface Electrode Problems in Piezoelectric Materials K.L. Zhou, Z.D. Zhou and Z.B. Kuang

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Active Finite Element Method for Simulating the Contraction Behavior of a Muscle-tendon Complex C.P. Tsui1,*, C.Y. Tang1, C.L. Chow2, S.C. Hui3 and Y.L. Hong3 1

Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University 2 Department of Mechanical Engineering, The University of Michigan-Dearborn 3 Department of Sports Science and Physical Education, The Chinese University of Hong Kong *[email protected] Keywords: Active finite element, skeletal muscle, isometric contraction, shortening contraction.

Abstract. A three-dimensional finite element analysis was conducted to simulate the effects of the varying material parameters on the contraction behaviors of a muscle-tendon complex using an active finite element method. The material behavior of the skeletal muscle was assumed to be orthotropic and the muscle model consists of two parts: the active and the passive parts. An active finite element method was then used for accommodating both the active and passive behaviors of the muscle into the muscle model. In this active-passive muscle model, the active component is governed by an activation level, a time period, a muscle sensitivity parameter and a strain rate. The material property of the passive component was assumed to be viscoelastic and the tendon is assumed to be linear elastic. The effects of activation amplitude and viscoelastic material parameters on the active, passive and total force-length relationship of the cat muscle under isometric contraction were predicted. The predicted results were found to be close to the experimental data reported in the available literature. Hence, the active-passive muscle model was extended to simulate the stress distribution of the cat muscle subject to shortening contraction and different activation amplitude. By varying the magnitude of the material parameters, different muscle behaviors could be generated. The proposed active finite element method lays a good foundation for simulation of human musculoskeletal motion. Introduction Knowledge on the mechanical behaviours of skeletal muscles is important to industrial design of sports, automobiles, medical and health equipment for exercise, rehabilitation and treatments. Traditionally, simple line models are available to describe the mechanical behaviours of skeletal muscles. However, the information deduced from these models is very limited because the line element does not have volume and mass. Several muscle models including those of Hill and Huxley models have been developed to describe the mechanical behaviors of the skeletal muscle, such as force-length relation, force-velocity relationships and stress relaxation response [1-4]. As skeletal muscle consists of both active and passive properties, it can actively generate a strain without the need of externally applied force or velocity under a certain activation level. Thus, a new method to describe the mechanical properties of active structures like skeletal muscle is required. In our active finite element method previously developed [5], the active muscle behavior and the passive muscle behavior were incorporated into separate finite elements. In this study, our previous model was further developed to incorporate both active and passive muscle behavior into a single finite element. A three-dimensional active finite element model was constructed by using commercial ABAQUS finite element code. The model was applied to predict the effect of the varying material parameters on the active, passive and total force-length relationship of the skeletal muscle subject to an isometric contraction. The predicted results would be compared with the finding reported in the available literature. After the verification of the model, the model would be

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extended to simulate the stress distribution in the skeletal muscle subject to a shortening contraction under the variation of the activation amplitude. Material Properties of Muscle-tendon Complex The schematic diagram of a muscle-tendon complex in one-dimensional space is shown in Fig.1. Spring

Dashpot

k2

Motor Element

σ = η ε
d

k3

k1 a

ε = − Atβε


a

Active-passive muscle element (muscle)

Viscoelastic Element

Elastic Element (Tendon)

Fig.1 One-dimensional representation of the muscle-tendon complex The muscle was assumed to be built of both motor elements and viscoelastic elements, while the tendon was assumed to be represented by the elastic element. Skeletal muscle in human or animal does not shorten without nervous stimulation. When there is a stimulation signal given to the skeletal muscle, shortening of the sacromeres occurs. In the present model, the motor element in the active-passive muscle element serves as the function of the sacromere to generate an active strain whenever the muscle is activated. Thus, the active behavior of the skeletal muscle could be assumed to be governed by the motor element, the strain εa of which is represented by the following function:

ε a = − Atβε
a

(1)

where A, t, β , ε
a are the activation amplitude, the time variable, the muscle sensitivity parameter and the strain rate respectively. When the value of ε
a is set to be positive, the motor element will cause the muscle to actively shorten to produce a strain in the longitudinal direction with expansion in the transverse directions, and vice versa. The transverse retraction and expansion of the motor element was controlled to ensure that the volume of the muscle could be approximately constant. The internal strain generated by the motor element is transferred to the passive muscle element. The passive muscle behavior was assumed to be described by a viscoelastic element consisting of two springs and a dashpot. The extensional relaxation function is given by

E R (t ) = k1 + k 2 e −t / tR

(2)

where k1 is modulus of the parallel elastic element, k2 is the modulus of the series elastic element and tR is the relaxation time constant. With reference to the literature [6-7], the material constants used in this finite element model were chosen: k1 = 0.05~0.5MPa, k2 = 0.05MPa and tR = 1s.

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For a time-varying shear strain γ(t) applied to the muscle, a time-dependent shear stress,τ(t) can be expressed follows (ABAQUS Ver.6.2) [8]: t

τ (t ) = τ 0 (t ) − ∫ g
R (s )τ 0 (t − s )ds

(3)

0

where g r (t ) is the dimensionless shear relaxation modulus. Its approximation with a single term Prony series expansion of the shear relaxation modulus gives n

g R (t ) = 1 − ∑ g iR (1 − e −t / ti ) R

(4)

i =1

where g iR is the shear modulus ratio, t iR , i = 1,2,….n, is the relaxation time in the first term in the Prony series expansion of the shear relaxation modulus. Finite Element Modeling Nonlinear implicit dynamic analysis was carried out with a commercial finite element software ABAQUS. Considering a body is in dynamic equilibrium, the following equation at a time t k can be obtained by using the principle of virtual work [9].

∫ [δε ] σ T

Ω

ij

T

ij

dΩ = ∫ [δui ] [Pi − ρu

i − cu
i ]dΩ + ∫ Ω

Γt

[δui ]T Ti dΓ

(5)

where σ ij is the stress vector, Pi is the applied body force vector, ρ is the mass density, c is the damping parameter, The domain Ω has two boundaries: Γt on which the surface traction vector Ti is applied and Γu on which the displacement vector u k is specified. u
i and u

i are the velocity vector and the acceleration vector, respectively. The virtual strain, δεij could be expressed as

δεij = δε ija + δε ijp where

(6)

δε ija is a virtual strain due to the motor element and δε ijp is a virtual strain due to the passive

component. For modeling of geometric nonlinearity, Lagrangian coordinate system is adopted in this finite element analysis. In a total Lagrangian formulation, Green’s strains are used. The Green’s strain vector ε in can be written as

ε = ε L + ε NL

(7)

L

where ε and ε NL are the linear and non-linear components of the Green’s strain correspondingly. The Green’s strain vector ε can also be expressed as n 1   ε = ∑  BiL + BiNL d i 2  i =1 

(8)

L NL where Bi and Bi are the linear and non-linear strain-displacement matrices respectively, and

d i is the nodal displacement vector.

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The finite element model of the muscle-tendon complex is shown in Fig.2, which is composed of a skeletal muscle with a tendon connected at its both ends. Three-dimensional eight-node linear isoparametric brick elements were used to construct the finite element mesh. The active muscle property was created by incorporating a user-defined sub-program into the finite element code ABAQUS version 6.2. Boundary conditions are imposed on the finite element model so as to prevent free body movements. The mid-planes of the model along the y-direction whose normal is x-axis and z-axis are constrained in the x-direction and z-direction respectively. Each end of the object are fixed in x, y and z directions. For isometric contraction of the muscle, shortening of the muscle is not allowed and thus the muscle ends represented by the faces B and B’ are not allowed to deform freely. For simulating the shortening contraction of the muscle, an elastic object with Young’s modulus equal to 1MPa is added to each end of the tendon with Young’s modulus equal to 500MPa denoted by the faces A and A’. In this case, shortening of the muscle is allowed and thus the muscle ends represented by the faces B and B’ are not allowed to deform freely. Once a suitable combination of the activation amplitude, the time period and the muscle material parameter is applied to the motor element, the motor element can generate a strain, which in turn induces a strain in the whole muscle.

Muscle B’

A’

B

z

A Tendon

y x

Fig.2 Finite element model of the muscle-tendon complex 35.0

28.0

Experiment (Total force) Experiment (Active force) Experiment (Passive force) Prediction (Total force) Predicted (Active force) Predicted (Passive force)

30.0

Force (N)

Force (N)

20.0 21.0

14.0

Experiment (SOL)

10.0

Prediction (SOL) Experiment (PB) Prediction (PB)

7.0

0.0 1.00

1.04

1.08

1.12

Relative length

Fig.3 Comparison between the predicted and experimental total, active and passive forcelength curves of the soleus (SOL) muscle of cat under isometric contraction (Experimental data obtained from [1])

0.0 1.00

1.04

1.08

1.12

Relative length

Fig.4 Comparison between the predicted and experimental total force-length curves of two different muscles of the cat under isometric contraction, SOL represents the soleus muscle while PB stands for peroneus brovis.(Experimental data obtained from [1])

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Results and Discussion In order to verify the validity of the proposed model, an isometric contraction of the cat muscle was considered. An isometric muscle contraction refers to the condition that the muscle is stimulated at a fixed length and there is no shortening of the muscle body. Shortening of the sacromeres occurs but muscle length remains unchanged because the load opposing shortening exceeds the tension that can be generated by the muscle. In this study, the motor element in the active-passive muscle element serves as the function of the sacromere to generate an active strain. The internal strain generated by the motor element is transferred to the passive element. The muscle ends were fixed in this model to oppose the shortening during the isometric contraction. During isometric contraction of the skeletal muscle, the force production from the muscle depends on its initial length before contraction. The predicted and experimental force-length relationship of the cat soleus is compared with the experimental one [1] in Fig.3. The total force is generally accepted to the sum of the passive force and the active force generated by the active muscle tissue due to contraction at different lengths. The action of the active muscle tissue is represented by the motor element in this study. When the relative length of the muscle is zero, the muscle was assumed to be at rest and no passive force can be developed. When the muscle is stretched a little bit, the muscle length changes and the passive force can be developed. The passiveforce curve at different muscle lengths can therefore be obtained. Hence, the active-force curve due to the motor element was obtained by subtracting the passive force curve from the total force curve. In Fig.3, predicted total force and passive force generated by the soleus muscle of the cat increase with increasing relative muscle length while the active force decreases with the increase in the relative muscle length. The predicted results are found to be close to the experimental ones. In order to demonstrate the effect of different material parameters on the force-length relationship of the cat muscle, material constants, k1 and k2, and activation amplitude, A were adjusted to fit the experimental results of the soleus (SOL) muscle and peroneus brovis (PB) muscle of the cat under isometric contraction at different stimulation amplitude as shown in Fig.4. k1 values for SOL and PB were set to be 0.2MPa and 0.5MPa respectively, while k2 value was set to be 0.05MPa in both cases. It can be observed from Fig.4 that the slope of the curve for SOL muscle with higher k1 value is higher than that for PB. As both muscles were stimulated under different amplitude, the values of A for SOL and PB muscles were set to be 180 and 60 respectively. It can be found from Fig.4 that the force-length curve for SOL muscle with higher value of A has a higher starting value because the motor element with a higher activation amplitude could generate a greater active strain, inducing a larger total force in the muscle. By inputting appropriate the values of the material constants, the force-length of different muscles could be predicted as shown in Fig.4. When a positive value of ε
m in combination with a certain value of the activation amplitude, A, a time period and the muscle sensitivity parameter is given to the motor element, the motor element in the muscle is activated to actively shorten in the y-direction with expansion in the x- and zdirections as shown in Fig.5, which was obtained at the end of the time period of 0.1 second. This induces strains onto the passive element, resulting in retraction of the whole muscle longitudinally with a lateral expansion. It can be seen from Fig.5 that the region of the high stress concentration is located in the tendon region rather than the muscle, because the tendon is much stiffer than the muscle. When the activation amplitude is increased, the longitudinal muscle retraction and lateral expansion shown in Fig.6 are found to be more significant than those activated by a lower activation amplitude. Conclusions The effects of varying material parameters on the force-length relationship of the cat muscle-tendon complex under isometric contraction have been successfully predicted using the active finite element method. It was found that the changes in the material constants of the active-passive muscle model are very sensitive to the change in the muscle length. The predicted active, passive and total

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force-length relationship is close to the experimental findings in the available literature. After the verification of the model, the active-passive muscle model was extended to simulate the stress distribution in the cat muscle-tendon complex subject to the shortening contraction. Hence, by suitably altering the parameters in the material model, the desirable muscle behavior could be simulated using the active finite element method.

Fig.5 Von Mises stress distribution in the PB cat muscle-tendon subject to shortening contraction at A=100 (The elastic objects at both ends of the tendons are not displayed.)

Fig.6 Von Mises stress distribution in the PB cat muscle-tendon subject to shortening contraction at A=500 (The elastic objects at both ends of the tendons are not displayed.)

Acknowledgements The authors would like to thank the substantial support from the Research Committee of the Hong Kong Polytechnic University (Project code: G-T645). References 1. H. Gareis, M. Solomonow, R. Baratta, R. Best and R. D’Ambrosia, J. Biomechanics. Vol. 25 (1992), p. 903-916. 2. B. Hete and K. K. Shung, Ultrasound in Med. & Biol. Vol. 21 (1995), p. 343-352. 3. G. H. Shue and P. E. Crago, Annals Biomed. Eng. Vol. 26 (1998), p. 369-380. 4. T. Johansson, P. Meier and R. Blickhan, J. Theor. Biol. Vol. 206 (2000), p. 131-149. 5. C. P. Tsui, C. K. Li, C. P. Leung, Y. F. Ng, H. K. Chow, K. W. Cheng and C. Y. Tang, Proc. IASTED Int. Conf. Biomechanics, Greece, (June 2003), p. 45-48. 6. E. M. H. Bosboom, M. K. C. Hesselink, C. W. J. Oomens, C. V. C. Bouten, M. R. Drost and F. P. Baaijens, J. Biomech. Vol. 34 (2001), p. 1365-1368. 7. SimBio: A generic environment for bio-numerical simulation, (The SimBio Consortium. 2000). 8. ABAQUS/Standard User’s Manual, Version 6.2, (Habbit, Karlson & Sorensen, Inc). 9. D. R. J. Owen and E. Hinton,: Finite Elements in Plasticity: Theory and Practice, (Pineridge, UK 1982).

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Residual Stress and Stress-Strain Relationship of Electrodeposited Nickel Coatings Y. C. Zhou1,2,a,Y. P. Jiang2, Y. Pan2 1

Key Laboratory for Advanced Materials and Rheological Properties of Ministry of Education, Xiangtan University, Hunan, 411105, China 2

Faculty of Material Photoelectron Physics,Xiangtan University, Hunan, 411105, China a [email protected]

Keywords: coating; mechanical properties; stress/strain relationship.

Abstract The uniform nickel coatings on substrate of low carbon steel were prepared by an electrodeposition method. The residual stress in the electrodeposited nickel coating was measured by X-ray diffraction (XRD). It was tensile when the coating was not treated. Laser beam thermal shock was used to modify the mechanical properties of the nickel coating. Laser beam thermal shock could redistribute the residual stress in the nickel coating. The residual stress could be converted from tensile to compressive. A tensile method to determine the stress-strain curve of the coating is proposed where the stress-strain relationship of the substrate without coating was determined for the specimen loaded by an applied tensile force. Introduction Electrodeposited coatings are used as wear-resistant, corrosion-resistant, decorative and functional surface layers in many fields, such as aviation, space flight and electron. Electroplated nickel coating is widely used in the fabrication of metallic microdevices or microsystems due to some outstanding properties, such as soft, ductile, machinable and dense[1]. The electrodeposited nickel coating can also be used as the shell of high quality and high energy-storing battery. As a good corrosive-resistant coating, electrodeposited nickel coating must have high interface strength, good thermal-mechanical match between substrate and coating, and good residual stress distribution. There are many methods to determine the elasto-plastic stress-strain relationship of bulk materials, such as axial tensile test and indentation method [2,3]. However, it is not easy to determine the stress-strain elasto-plastic relationship of thin coatings by tensile method. As we know, in the tensile method of determining stress-strain relationship of thin film, one method [4,5,6] is to conduct the uniaxial tensile tests on free-standing. In this method, the films should be carefully peeled from the substrate. Although the films are not requested to peel from the substrate, the residual stresses are not considered and the failure processing can not be observed in the uniaxial tensile tests [4,5]. Another method is to conduct the unaxial tensile test for the film/substrate composite as proposed by Wojciechowski [7], however, in this method, only the Young’s modulus can be determined and the elastio-plastic stress-strain relationship can not be determined. The intent of this article is to study the residual stresses and the elasto-plastic stress-strain relationship of electrodeposited nickel coatings. In the method, the stress-strain relationship of substrate without coating was first determined for the specimen loaded by an applied tensile stress in scanning

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electron microscopy (SEM) with a microtensile machine. After that, the stress-strain relationship of the coating/substrate composite was determined for the specimen loaded by an applied tensile stress. The elasto-plastic stress-strain relationship can be obtained by a model proposed in the present investigation using the data about the conducted tensile test for substrate and for coating/substrate composite. The surface of the specimen could be observed by scanning electron microscopy (SEM), when the specimen was loaded by applied tensile stress. In this case, the initiation of micro-cracks and the propagation of macro-crack could be observed. The tensile method could take into account the effect of residual stress in the coating on the stress-strain relationship of coating. The tensile method was used to determine the elasto-plastic stress-strain relationship of nickel coating which was electrodeposited on a low carbon steel strip. Determination of the Stress-Strain Relationship of Coating

y

σ1

σ2

σ2

x σ2

σ1

σ2

Fig.1. Theoretical model for the determination of stress-strain equation of coating, where the subscript 1 denotes the substrate and subscript 2 denotes the coating. Considering a continuous system with no cracks, we assume that the system is infinite in x direction, therefore the edge effect may be neglected, and the shear strains are also zero. It is further assumed that the coating and substrate is perfectly adhered. Before any fracture has occurred, the strains of both coating and substrate layers are equal [7]. Figure 1 is the schematic of the specimen loaded by an applied tensile stress for determining the stress-strain relationship of thin coating. When the specimen was loaded as described in Fig.1, the constitutive equations both for coating and substrate can be written as follows[8],

εx =

1 2µ

1   p p σ x − 4 (3 − κ )(σ x + σ y ) + ε x + ηε z .  

(1)

εy =

1 2µ

1   p p σ y − 4 (3 − κ )(σ x + σ y ) + ε y + ηε z .  

(2)

Here, ε and σ denote the strain and stress, respectively. The superscript p describes the plastic strain,

µ

is

shear

modulus.

Moreover, κ = 3 − 4ν , η = ν

for

plane

strain,

and

κ = (3 − ν ) /(1 + ν ) , η = 0 for plane stress, ν is Poisson’s ratio. In Fig. 1, the composite in y-direction is free, and therefore the stresses in the composite are σ y = 0 and σ xy = 0 . The normal stress in x-direction can be written in the following form, 8µ σx = (ε x − ε xp − ηε zp ) . 1+κ

(3)

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where the plastic strains ε xp and ε zp must be the function of total strain ε x . Then, the above equation could be written as,

σx =

8G (ε x ) εx . 1+κ

 µ G (ε x ) =   g (ε x )

(4)

Elastic deformation . Plastic deformation

(5)

where, g (ε x ) denotes a new function of total strain ε x . In equation (4), only the strain ε x is included. The stresses in coating/substrate composite are induced by the combination of applied force and residual stress. Equation (4) for coating and substrate could be written in the following form,

σ s (ε s ) =

8G s (ε s ) ε s = Fs (ε s ) / S s + σ sr . 1+κ s

(6)

σ c (ε c ) =

8Gc (ε c ) ε c = Fc (ε c ) / S c + σ cr . 1+κc

(7)

where, F and S denote the applied force and section area, respectively. The subscripts s and c refer to substrate and coating, respectively. The subscript r denotes the residual stress. The applied force F which is the sum of the forces in substrate and coating equal to the force experienced by the composite, F = Fs + Fc .

(8)

The condition of perfect adhesion between coating and substrate, prior to cracking, is expressed as,

ε = εs = εc .

(9)

Substituting equations (6) and (7) into equation (8), one can obtain, F = (σ s S s + σ c S c ) − (σ sr S s + σ cr S c ) .

(10)

When the coating/substrate composite is not loaded by applied force, the residual stress in coating should make the composite in an equilibrium state. Therefore, one has the following equation,

σ sr S s + σ cr S c = 0 .

(11)

The relationship of residual stress σ sr and residual strain ε sr in substrate should be written as,

ε sr =

1+κ s σ sr . 8µ s

(12)

Substituting Eq. (11) into Eq. (10) gives F (ε ) = σ s (ε ) S s + σ c (ε ) S c .

(13)

If one obtains the stress-strain relationship of substrate σ s (ε ) , the stress-strain relationship of

24

Macro-, Meso-, Micro- and Nano-Mechanics of Materials

coating σ c (ε ) could easily be obtained by Eq.(13) when the relationship F (ε ) of applied tensile force F and strain ε for the coating/substrate composite was known. It is assumed that the residual stress does not exist for the substrate without coating. The residual stress in the substrate with coating will result in an initial strain ε sr existing in the substrate. In the following derivation, the prime denotes the substrate without coating. Therefore, the strain could be written in the following form,

ε = ε s = ε s′ + ε sr .

(14)

The stress-strain relationship of substrate without coating could be written as,

σ s′ (ε s′ ) =

8Gs (ε s′ ) ε s′ = Fs′(ε s′ ) / S s . 1+κ s

(15)

where Fs′ denotes the applied tensile force for the substrate without coating. It is seen that equations (6) and (15) are in the same form, although the strains in equations (6) and (15) are not same. Substituting Eq. (14) into Eq. (6), the stress-strain relationship of substrate with coating is as follow,

σ s (ε ) =

8G s (ε s′ +ε sr ) (ε s′ + ε sr ) = Fs (ε s ) / S s + σ sr . 1+κ s

(16)

One could see that the stress-strain relationship of substrate σ s (ε ) with coating could be obtained by shifting a negative strain ε sr for the stress-strain curve of substrate without coating in the stress-strain plane. If the stress-strain relationship σ s′ (ε s′ ) in Eq. (16) for substrate without coating is obtained, the stress-strain relationship of substrate σ s (ε ) with coating could be obtained via Eq. (16). After that, using equation (13), the stress-strain relationship of coating σ c (ε ) with residual stress could be obtained as,

σ c (ε ) = F (ε ) / S c − σ s (ε ) S s / S c =

8Gc (ε ) ε. 1+κ c

(17)

The strain of coating ε c′ without residual stress could be written as,

ε c′ = ε c − ε cr = ε − ε cr .

(18)

Substituting Eq. (18) into Eq. (17) gives the stress-strain relationship of coating without residual stress as,

σ c′ (ε c′ ) =

8Gc (ε c′ ) 8G (ε − ε cr ) (ε − ε cr ) . ε c′ = c 1+κc 1+κc

(19)

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25

It is seen that the stress-strain relationship of coating σ c′ (ε c′ ) without residual stress could be obtained by shifting a negative ε cr for the stress-strain curve of coating with residual stress in the stress-strain plane. In the case of coating without residual stress, ε sr = ε cr = 0 , ε s = ε c = ε s′ = ε and therefore, equations (15), (16), (17) and (19) are same. Combing equations (15) and (16), one can obtain the following equation,

σ s = Fs / S s = Fs' / S s .

(20)

Combing equations (19) and (17), the following equation could be written as,

σ c′ = σ c = ( F − Fs′) / S c .

(21)

When the deformation is elastic, we can obtain the following equation via equations (15) and (16),

σ s (ε ) = σ s′ (ε s′ ) + σ sr = Fs′ / S s + σ sr .

(22)

According the equations (19) and (16), the following equation could be obtained as,

σ c′ (ε c′ ) = σ c (ε ) − σ cr = F (ε ) / S c − σ s (ε ) S s / S c − σ cr .

(23)

Experimental Procedure

Nickel coating was electrodeposited on a low carbon steel with thickness of 0.3mm. Prior to electrodeposition, the substrate was degreased and submerged in 6M hydrochloric acid for 2 minutes and rinsed in distilled water. The coating/substrate samples were prepared with two electrolytes. One was nickel sulphamate electrolyte and another was nickel sulphate electrolyte. The nickel sulphamate electrolyte was composed of 420g Ni(SO3NH2)2·4H2O, 20g NiCl2·6H2O, 40g H3BO3 per liter, while the nickel sulphate electrolyte was composed of 240g NiSO4·6H2O, 40g NiCl2·6H2O, 40g H3BO3 per liter. Pure nickel was used as the anode. Before electrodeposition, H2O2 and active carbon were used to get rid of the impurities. Moreover the electrolyte solution was electrolyzed for 24 hours. In the cases of two electrolytes, the PH value of the solution was 4.0±0.1 and the temperature was 50±0.50C. The current density was 0.02A/cm2. The cell was automatically agitated. The thickness of coatings was measured by a thickness tester of ZWB. In order to precisely measure the thickness of the coating, the section of the coating/substrate composite was observed by (JEOL)JSM-5600LV SEM and the thickness of the coating was measured. Laser beam thermal shock which has many advantages such as the local heating and controllable heating rate was used to modify the properties of the materials[9,10,11]. Comparing with the traditional annealing method, laser beam thermal shock does not cause significant macroscopic deformation of the treated zone. In the investigation, the laser thermal shock was produced by Nd:YAG laser beam (λ = 1.06µm) with adjustable frequency and pulse width. The residual stress in the coating was measured by X-ray diffraction. The X-ray goniometer was operated at 10mA and 40kV, and the radiation source was CuKα with wavelength of λ = 1.5406 A0. The residual stress in the electrodeposited nickel coatings was measured by sin2ψ method. Here we used the {200} plane according to X-ray diffraction pattern. The stress-strain relationship of electrodeposited nickel coating was determined using a SEM. The tensile loading rate was controlled by the displacement rate at a displacement rate of 50µm/s.

26

Macro-, Meso-, Micro- and Nano-Mechanics of Materials

The strain was measured by strain gauges, and the relative error was less than 5%. The surface of the specimen was observed by scanning electron microscopy (SEM) when the specimen was loaded by the applied tensile load. In this case, the initiation of micro-cracks and the propagation of macro-crack were observed. Figure 2 is a schematic of the specimen configuration and dimensions. The strain gauge was on the surface of the sample. The thickness and width of the specimen were 0.32mm and 4.0mm, respectively. Because the sample’s width was much larger than the thickness, the problem could be assumed as a plane strain problem. Therefore, one had κ = 3 − 4ν with Poisson’s ratioν s = 0.34 for substrate and ν c = 0.36 for coating.

Tensile Direction

Tensile Direction

Thickness:

Unit: mm

Fig. 2. Schematic of specimen configuration with dimensions. Results and Discussion

Figure 3 shows the residual stress of the samples prepared with nickel sulphamate electrolyte and nickel sulphate electrolyte, where the samples were not treated. The curves a and b denote the residual stress for electrodeposited nickel coating prepared with nickel sulphate and sulphamate electrolytes, respectively. The residual stress in the electrodeposited nickel coating is tensile. Although the electrolytes have same nickel content, the tensile residual stress in the nickel coating prepared with nickel sulphamate electrolyte was lower than that prepared with nickel sulphate electrolyte when the thickness of the sample was same. Figure 4 shows the relationship between the residual stress in the coating and laser beam’s power density for single pulse laser beam thermal shocking. Here the electrodeposited nickel coating was prepared by nickel sulphamate electrolyte, the thickness of the coating was 1.5µm and the residual stress in the coating before laser beam thermal shock is approximately 262 MPa. The residual stresses in the electrodeposited nickel coating would be converted from tensile to compressive after laser beam thermal shock. The level of the redistribution of residual stress induced by laser beam thermal shock was strongly dependent on laser beam parameters such as power density, pulse frequency. The conversion of residual stress from tensile to compressive after laser beam thermal shock is a very interesting phenomenon. As we know, when the electrodeposited nickel coating is used as the shell of high quality and high energy storing battery, the coating/substrate composite must be deeply drawn. In the numerical simulation of the deep-drawing process [9], the mechanical model should take into account the large elastoplastic strains and rotations that occur in the deep-drawing process [10,11]. A high tensile stresses in the coating will be produced in the deep-drawing process. However, the tensile residual stresses in the coating

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27

combining the applied tensile load will result in interface spallation easily [12,13]. The compressive residual stresses in the electrodeposited nickel coating experienced applied tensile load in the deep-drawing process could prevent the interface spallation. -50

600

Residual Stress σ(MPa)

Residual Stress (MPa)

700

a

500 400 300

b

200

-100 -150 -200 -250 -300

100 1.0

1.5

2.0

2.5

3.0

Thickness of the coating (µm)

Fig.3. Residual stress of original electrodeposited nickel coating versus coating thickness.

Experimental Data Fitting Curve

0.2

2

σ =-480.39031+650.49575P-257.28569 P

0.4

0.6

0.8

1.0

1.2

1.4 2

Laser beam power density P(GW/mm )

Fig.4. The relationship between residual stress and laser beam’s power density for single pulse.

The chemical and mechanical properties of electrodeposited nickel coating could be modified by laser beam thermal shock. The microstructure was not changed before and after laser beam thermal shock, and therefore the thermal stress dominated the redistribution of residual stress in the electrodeposited nickel coating induced by laser beam thermal shock. The heat conduct equation could be described as, 1 ∂T q ∇ 2T − + =0. (27) α ∂t k where T is temperature rise, t is time, α = k / ρC p and q = q0 / H . k , ρ , C p and H are the thermal conductivity, density, specific heat capacity of the materials and the thickness of the sample, respectively. q0 is the laser beam power density absorbed by the sample. Due to the initiatory and the boundary condition, we can obtain the temperature field as, T ( z, t ) =

 z  2 q 0α 1 / 2 1 / 2 . t ⋅ ierfc  1/ 2  k  2(αt ) 

(28)

where ierfc is the error function. The temperature gradient leads to a high thermal stress in coating and substrate. The highest temperature on the surface of electrodeposited nickel coating could be up to 8000C-10000C in the temperature, the thermal stresses in nickel coating should be higher than the yield strength. At the end of cooling, the plastic strain in the coating would be residual strain. In this case, the residual stress in the nickel coating after laser beam thermal shock would redistribute. When the applied tensile load was high, the micro-cracks in the electrodeposited nickel coating could be observed by (JEOL)JSM-5600LV SEM. The typical photos for the surface of the electrodeposited nickel coating with cracks are shown in figure 5. SEM photos in figures 5 (a)-(c) denote the evolution of microcracks with the increase of applied load, and SEM photo in figure 5(d) denotes the macrocrack. The characteristics of the micro-cracks in the tensile test are that the surface cracks are in the direction of 450 angle of loaded axis. However, in the similar experiment for brittle coating such as thermal barrier ceramic coatings, the cracks were normal to the loaded

28

Macro-, Meso-, Micro- and Nano-Mechanics of Materials

axis and the cracks were appeared on the surface of the top coat layer. The phenomenon for surface cracks in brittle coating normal loaded axis was also observed previously [14]. It is shown that the tensile strength of brittle material is lower than the shear strength, and in this case the micro-cracks are produced in the direction of maximum tensile stress. However, the shear strength of ductile coating such as electrodeposited nickel coating is lower than the tensile strength, and in this case the micro-cracks are produced in the direction of maximum shear stress. Therefore, the micro-cracks in ductile coating were produced in the direction of 450 angle of loaded axis. Tensile

Tensile

Direction

Tensile

Direction

Direction

(a)

Tensile Direction

(c)

(b)

Tensile

Tensile

Direction

Direction

(d)

Fig.5 Evolution of cracks in electrodeposited nickel coating The load-strain curves of low carbon steel strip with and without nickel coatings are shown in figure 6. The stress-strain curves of electrodeposited nickel coatings are shown in figure 7, where the residual stress was not taken into account. This means that the stress-strain curves are the combination of applied force and residual stress in the coating. The scatter of the data for the stress-strain in the plastic part could be seen. The scatter should be come from the experimental error. The stress-strain curves of electrodeposited nickel coatings in which the residual stress was taken into account are shown in figure 8. This means that the stress-strain curves in figure 8 are the results only produced by applied force, and the residual stress in coating was moved away. The Young’s modulus and yield strength of the electrodeposited nickel coating can be obtained from the data of figure 8. The results are listed in table 1. In order to check the measured data for electrodeposited nickel coating, the Young’s modulus and yield strength of bulk material are also listed in table 1. It is seen that the difference for Young’s modulus and yield strength between electrodeposited nickel coating and bulk nickel material is very low. In order to conveniently use the data measured in the present investigation to the numerical simulation of deeply drawing for the coating/substrate composite, the linear hardening model of stress-strain could be written in the following form,

Advanced Materials Research Vol. 9

Eε  σ = σ y + E ′(ε − ε y )

29

ε ≤εy . ε >εy

(29)

where E = 223GPa , σ y = 480 MPa , ε y = 0.2% , E ′ = 7.7GPa .

300 250

F(N)

200

6.2µm 8.1µm 9.6µm no coatings

150 100 50 0 0

5000 10000 15000 20000 25000 30000 35000 -6

Microstrain(10 )

Fig.6. Load-strain curves of low carbon steel strip with and without nickel coatings. 700 700

600 600

500

Stress(MPa)

Stress(MPa)

500 400

6.2µm 8.1µm 9.6µm

300 200

400

6.2µm 8.1µm 9.6µm

300 200 100

100

0

0 0

5000 10000 15000 20000 25000 30000 35000

0

5000 10000 15000 20000 25000 30000 35000



-6


  

Microstrain(10 )

Fig.7. Stress-strain curves of nickel coatings without taking into account the residual stress

Fig.8. Stress-strain curves of nickel coatings in which the residual stress was taken into account.

Table 1. Young’s modulus and yield strength of nickel coating and bulk nickel Materials Coating 1 Material 2 (Measured in the 3 present) Bulk Material

Elastic modulus

Yield strength (0.2%)

245 [GPa]

439[ MPa]

223 [GPa]

500 [MPa]

202[GPa]

499[MPa]

Ni200

204 [GPa]

483~724 [MPa]

Ni201

207[GPa]

~

Ni270

207[GPa]

621[MPa]

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Macro-, Meso-, Micro- and Nano-Mechanics of Materials

Conclusions

A uniform nickel coating on the substrate of low carbon steel was prepared with electrodeposited method. The residual stress in the electrodeposited nickel coating was measured by X-ray diffraction method. The laser beam thermal shock was used to modify the mechanical properties of nickel coating. A method to determine the stress-strain curve of coating was proposed. The stress-strain curve of electrodeposited nickel coating was determined by applied tensile load in SEM. The main results obtained in this study are summarized in the following. (1) The residual stress in the electrodeposited nickel coating without any treatment was tensile and it was dependent on the prepared parameters. (2) Laser beam thermal shock converted the residual stress in the nickel coating from tensile to compressive. The residual stress redistribution in the coating depended on laser beam parameters. (3) The mechanism of residual stress’ redistribution induced by laser beam thermal shock is discussed and it is due to the residual thermal plastic deformation in the coating. (4) The stress-strain curve of electrodeposited nickel coating was determined by the method proposed in the present paper. The method is very simple and convenient. In the same time of determining stress-strain curve, the failure processing of coating could be in situ observed by SEM. The method is also suitable for determining elastoplastic relationship or elastic modulus for any coating such as brittle or ductile coating, and any substrate such as brittle or ductile substrate. Acknowledgments

The support from the Key Project of Hunan Province under grant No. 03GKY1021 and the 863 High-Tech Plan under grant No. 2003AA331090, is gratefully acknowledged. References

1. S. Basrour and L. Robert: Mater. Sci. and Eng. Vol. A288 (2000) 270-274. 2. T.A. Venkatesh, K.J. Van Vliet, A.E. Giannakopoulos and S. Suresh: Scripta Mater. Vol. 42 (2000), p. 833-839. 3. A.E. Giannakopoulos and S. Suresh: Scripta Mater. Vol. 40 (1999), p. 1191-1198. 4. M. Macionczyk and W. Bruckner: J. Appl. Phys. Vol. 86 (1999), p. 4922-4929. 5. M. Ignat, T. Marieb, T. Fujimoto and P.A. Flinn: Thin Films Vol. 353 (1999), p. 201-207. 6. L.B. Freund and S. Suresh, Thin Film Materials: Stress, Defect Formation and Surface Evolution. (Cambridge University Press, 2003) 535-539. 7. P.H. Wojciechowski, in Physics of Thin Films, Vol. 16 (1992) edited by M. H. Francombe and J. L. Vossen, Academic Press, pp 271-341. 8. Y.C. Zhou and T. Hashida: Int. J. Solids Struct. Vol. 38 (2001), p. 4235-4264. 9. L.Q. Zhou, Y.C. Zhou and Y. Pan: J. Mater. Sci. Vol. 39 (2004), p. 757-760 10. D.W. Jung and K.B. Yang: J. Mater. Proc. Technol. Vol. 104 (2000), p. 185-190. 11. L.F. Menezes and C. Teodosiu: J. Mater. Proc. Technol. Vol. 97 (2000), p. 100-106. 12. H.M. Jensen and M.D. Thouless: Int. J. Solids Struct. Vol. 30 (1993), p.779-795. 13. Z.G. Suo and J.W. Hutchinson: Int. J. Fract. Vol. 43 (1990), p.1-18. 14. M.S. Hu and A.G. Evans, Acta Matell. 37 (1989) 917-925.

Advanced Materials Research Vol. 9 (2005) pp 31-40 © (2005) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMR.9.31

Failure of Thermal Barrier Ceramic Coating Induced By Buckling Due to Temperature Gradient and Creep W. G. Mao and Y. C. Zhou1 Key Laboratory for Advanced Materials and Rheological Properties of Ministry of Education Faculty of Materials & Optoelectronics Physics, Xiangtan University, Hunan, 411105, P.R. China 1 [email protected] Keywords: thermal barrier ceramic coatings, residual thermal stress, thermal cycle, buckling failure

Abstract. The intent of this article was to study the failure mechanism of thermal barrier coatings (TBCs) induced by buckling. The main content included the following two parts. The first part investigated the thermal residual stresses fields in TBCs with thermal cycles, which induced by the non-linear coupled effect of temperature gradient, thermal fatigue and creep strain of TBCs. One found that the residual stresses in ceramic coating were compressive and accumulated with thermal cycles, which may be high enough to induce the buckling failure of ceramic coating. The second part studied the critical buckling failure loading of the ceramic coating in TBCs under the condition of the compressive loading by use a theoretical model. Finally, a buckling plane, i.e. n − Ts plane, was obtained by combined the above sections and applied to predict the buckling failure of the TBCs system. In this plane, it was divided into the two parts, i.e., non-buckling region and buckling region. Introduction The TBCs system consisting of ceramic coating, bond coat and substrate was now employed in most turbine engines, permitting gas temperature to be raised substantially above those for uncoated systems [1,2]. The ceramic coating (i.e.7-9wt% yttria stabilized zirconia) could protect the turbine components due to its low thermal conductivity and diffusivity as well as its acceptable thermal shock resistance. However, there are many factors that can degrade the thermal function of the TBCs. Zhou and Evans et al. have reviewed the mainly failure mechanisms of the TBCs system [3,4,5,6]. The oxidation at the interface of ceramic coating/bond coat at high temperature may result in the interface delamination [7,8,9]. The accumulated residual stresses and the oxidation in TBCs may result in the buckling and spallation failure of the ceramic coating [10]. In this paper, the thermal residual stresses in TBCs are firstly studied during thermal cycles. After that, a theoretical model is proposed to predict the critical buckling loading of the ceramic coating under the condition of the compressive loading. Thermal Residual Stresses Field in TBCs The deformation of the ceramic coating is assumed to be elastic and creep. The deformation of bond coat and substrate are regarded as elastic [3,11]. Furthermore, due to the fact that the material properties of bond coat and substrate are very close, they will be regarded as a layer and named as substrate, as shown in Fig.1.

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Macro-, Meso-, Micro- and Nano-Mechanics of Materials

z

neutral axis

hc

ceramic coating hr

hs

bond coat

x

substrate geometric center

Fig.1. Schematic of the crossing-section view of the TBC system. The stress-strain fields of each layer in TBCs are determined incrementally during the heating, holding temperature and cooling period. They can be written as follows,

dε ij ( z , t ) = deije ( z , t ) + δ ij dε + δ ijα (T )dT ( z , t ) + dε ijc ( z , t ) .

(1)

where the deije represents the deviator strain increment components, δ ij and dε m represent the Kronecker delta and mean strain increment, respectively. α (T ) denotes thermal expansion coefficient. In equation (1), dε ijc ( z , t ) is the creep increment of strain component. It can be written as [3,11], dε ijc ( z , t ) = F (t ) f 1 ( z ) A × exp(−Q / RT ( z , t )) × ( 3J 2 / σ 0 ) n × (t / t 0 ) − s dt .

(2)

where 1 0 < t < t 2 F (t ) =  . 0 t 2 < t < t 3

(3)

0 − (hs − hr ) < z < hr f1 ( z ) =  . hr < z < hr + hc 1

(4)

In the above equations, A, Q, n and s are, respectively, material constant, creep activation energy, stress exponent and time exponent. J 2 is the second invariant of the deviator stress. R, T , t

denote, respectively, gas constant, temperature and time. σ 0 and t 0 are the initial stress and reference time, respectively. In equation (3), the physical meaning of t i (i = 1,2,3) denotes the different moment during the thermal cycle history, i.e. the heating time, holding time and cooling time, respectively. In equation (4), hc and hs are, respectively, the thickness of the ceramic coating and that of substrate. Due to the thermal expansion coefficients and other material properties mismatch in TBCs, the geometric axis and neutral axis don’t superpose. The physical meaning of hr is given in Fig.1 and it can be obtained as [12], hr = ( E s hs − E c hc2 ) / 2(E s hs + E c hc ) . 2

(5)

where E s and E c denote, respectively, the Young’s modulus of substrate and that of ceramic coating. Note that the material properties, such as thermal expansion coefficient, Young’s modulus and Poisson’s ratio are dependent on temperature. Moreover, temperature distribution in TBCs is

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33

assumed to be the linear function of the coordinate axial z [3]. Therefore, the temperature fields, T ( z , t ) , can be written in the following form,

(T1 ( z ) − T0 ) × t / t1 + T0  T ( z , t ) = T1 ( z ) T ( z ) − (T ( z ) − T ) × (t − t ) /(t − t ) 1 0 2 3 2  1

(0 ≤ t ≤ t 1 ) (t 1 ≤ t ≤ t 2 ) .

(6)

(t 2 ≤ t ≤ t 3 )

where, (Ts − Tb ) × z / hc + (Tb − h1 × (Ts − Tb ) / hc ) T1 ( z ) =  Tb

(h1 ≤ z ≤ h1 + hc ) (h1 − hs ≤ z ≤ h1 )

.

(7)

Here T0 and T1 ( z ) denote, respectively, the ambient temperature and the temperature distribution in TBCs during the holding period, which is the function of position z. Ts and Tb denote, respectively, the highest temperature on the surface of ceramic coating and that of substrate. The problem is assumed to be plane stress and the stresses are equi-biaxial stress state in the following analysis. The effect of bending moment on the stress distribution is neglected [11]. The deformation of each layer is assumed to satisfy the strain compatibility, mechanical equivalent and boundary condition at the edges of the TBCs system segment during the thermal cycles. Moreover, stress-strain relationship in TBCs is assumed to obey the Hooke’s law during the cooling period. Eventually, thermal residual stress in TBCs at the end of cooling can be deduced by using the thermal stress theory and equivalent boundary method. The analytical solution can be obtained as follows,

σ residual ( n ) ( z , t ) = σ residual ( n −1) ( z , t ) +

t E (T )  × − ∫ 2 B( z , t ) f1 ( z ) F (t )(t / t 0 ) − s + α (T )dT ( z , t ) / dt dt 1 − v(T )  0

(

)

h1 + hc t   t E(T )  Ec (T ) −s −s α + ( ) + ( ) T dT z t dt F t f z B z t t t dt dz F t f z B z t t t dt dz ( ) ( , ) / ( ) ( ) ( , ) / ( ) ( ) ( , ) /    1 0 1 0 ∫ ∫ 1 −ν (T ) ∫ ∫ 1−νc (T )  −hs + h1  0 h1 0  + h1 + hc  E(T )  dz ∫ 1 −ν (T )  − hs + h1 h1 + hc

(

  t E (T )  + × − α (T )(dT ( z , t ) / dt )dt + 1 − ν (T )  t∫2   

)

 E (T ) ( )( ) T dT ( z , t ) / dt dtdz × α  ∫ ∫ 1 − ν (T ) − hs + h1 t 2 . h1 + hc  E (T )  dz ∫  ( ) 1 T − ν − hs + h1  h1 + hc t

(8)

where B( z , t ) = A × exp(−Q / RT ( z , t )) × ( 3J 2 / σ 0 ) n .

(9)

Note that the subscript n in equation (8) denotes thermal cycle number. In the second and subsequent cycles, the effect of the residual stress in TBCs produced in the last thermal cycle must be considered. Model of the Buckling Failure

The calculated results in the following section will show that the accumulated residual stress in the ceramic coating is compressive and increases with thermal cycle. In addition, it is well known that

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Macro-, Meso-, Micro- and Nano-Mechanics of Materials

the interface oxidation may result in interface delamination or crack during the thermal cycle. The buckling failure phenomenon of the ceramic coating may occur under the influence of the compressive residual stresses when the interface delamination/crack lengthens with thermal cycles. In this section, in order to predict the buckling failure of the ceramic coating, more attention will be focused on the relationship of the residual stress, interface delamination length, critical buckling loading and thermal cycle number. Basic Assumptions. Fig. 2 is the schematic of the buckling failure of the ceramic coating. The interface is assumed to be an across-the-width delamination and the TBCs system is divided into four regions. Therefore, the simple beam theory can be used to study the buckling failure mechanism. Moreover, the materials are assumed to be homogeneous and isotropic. A mechanical force P is applied at the two edges of TBCs system and the load is used to approximately replace the compressive residual stress in the ceramic coating accumulated during thermal cycles. The interface between regions 2 and 3 could be delaminated, which means the weakest adhesion region. The interfaces in regions 1 and 4 represent the strong adhesion region, which can be regarded that the delamination/crack will not propagate along these interfaces in this paper. The delaminated TBCs system is assume to be clamped at the two edges. Z

l1

l4 Region 3 A

a

TBC

B

Region 2

Region 4

H

Region 1

P

σ

X

P

substrate

Fig.2. Schematic of the buckling failure of the ceramic coating in TBCs system under the condition of compressive stresses. Balance Equations. The first order shear deformation theory is introduced to describe the displacement of each region and they can be written as follows [13],

ui ( xi , zi ) = ui ( xi ) + ziψ i ( xi ) .

(10)

wi ( xi , zi ) = wi ( xi ) .

(11)

where ui ( xi , zi ) and wi ( xi , zi ) are, respectively, the displacement in the x -direction and the deflection in the z -direction. u i ( x i ) and ψ i ( xi ) in equation (10) denote, respectively, the displacement of each region along the x direction and the rotation of the normal to the beam midplane. In equations (10) and (11), xi ∈ [0, l i ] and z i ∈ [− hi / 2, hi / 2] denote, respectively, the local axial coordinate and transverse coordinate of each region, where

zi = z − di

and

di = H / 2 − hi / 2 . Using Hooke’s law and geometry equation, the strain energy and potential energy

Advanced Materials Research Vol. 9

in each region can, respectively, be written as follows [13], 1 2 2 U i = ∫ [ D0i u i , x + D2iψ i , x + B0i (ψ i + wi , x ) 2 + 2 D1i u i , xψ i , x ]dxi . 2

(

)

Vi = − Pi ∫ u i , x + z iϕ i , x + wi2, x / 2 dxi .

35

(12) (13)

where D0i , D1i , D2i , B0i denote the stiffness coefficient matrix and the details are given in the reference [13]. In order to obtain the balance equations by potential variational principle, the displacement variables are divided into two parts, 0

a

u i ( x i ) = u i ( xi ) + u i ( xi ) .

(14)

ψ i ( xi ) = ψ i 0 ( xi ) + ψ i a ( x i ) .

(15)

0

a

wi ( xi ) = wi ( xi ) + wi ( xi ) .

(16)

where {ui 0 (xi ) ψ i 0 (xi ) wi 0 (xi )} is the displacement function of the initiate balance state and a a a {u i ( x i ) ψ i ( x i ) wi ( x i )} is the small variables of the displacement function when the buckling

failure phenomenon occurs in ceramic coating. The total potential energy is done by the two order variation, i.e. 4

4

i =1

i =1

δ 2 ∏ = ∑ δ 2 ∏ i = ∑ δ 2 (U i + Vi ) .

(17)

Therefore, the balance equations in each region can easily be obtained, a

a

a

a

D0i u i , xx + D1iψ i , xx = 0 .

(18) a

a

D1i u i , xx + D2iψ i , xx − B0i (ψ i + wi , x ) = 0 . a

a

a

Pi wi , xx − B0i (ψ i , x + wi , xx ) = 0 .

(19) (20)

Boundary Conditions and Continuity Conditions. The boundary conditions on both edges can be given as follows, a

a

u1 = ψ 1 = w1a = 0, a

a

u 4 = ψ 4 = w4a = 0,

for ( x1 = 0) .

(21)

for ( x 4 = l 4 ) .

(22)

The displacement continuity at positions A and B shown in Fig. 2 must be satisfied and can be written as follows, a

a

a

ui (0) = u1 (l1 ) + diψ 1 (l1 ) .

(23)

ψ i a (0) = ψ 1 a (l1 ) .

(24)

a

a

wi (0) = w1 (l1 ) .

(25)

36

Macro-, Meso-, Micro- and Nano-Mechanics of Materials

a

a

a

u i (l i ) = u 4 (0) + d iψ 4 (0) .

(26)

ψ i a (l i ) = ψ 4 a (0) .

(27)

a

a

wi (li ) = w4 (0) .

(28)

where the subscript i equals 2 and 3, respectively. To simplify the analysis in the following, the generalized forces of each region are defined as, a

a

X i ( xi ) = D0i u i , x + D1iψ i , x .

(29)

φ i ( xi ) = D1i u i , x a + D2iψ i , x a . a

(30)

a

a

Z i ( xi ) = B0i (ψ i + wi , x ) − Pi wi , x .

(31)

The continuity conditions of generalized forces at positions A and B can be written as follows,

∑X

X j (l14 ) =

i

(l 23 ) .

(32)

i = 2, 3

φ j (l14 ) =

∑ [d X (l i

i

23

) + φi (l23 )] .

(33)

i = 2,3

∑ Z (l

Z j (l14 ) =

i

23

).

(34)

i = 2,3

Here, j = 1 and j = 4 denote positions A and B, respectively. For point A, l14 = l1 , l 23 = 0 and for point B, l14 = 0, l 23 = l 2 . State-Space Method (SSM). The state space method is used to study the critical buckling failure loading of the TBCs system. The space state variable in each region is defined as,

{η i } = {uia

ψ ia

wia

u ia, x ψ ia, x

wia, x }T .

(35)

where ''T '' denotes the transposition of matrix in equation (35). Therefore, the governing balance equations (18), (19) and (20) in each region can be denoted by using the following form, dη i / dxi = S iη i .

(36)

where S i denotes the coefficients matrix and the detail is given in the reference [13]. The solution of the equation (36) can be obtained as follows,

η i = e S x η i0 = K i ( xi )η i0 .

(37)

i i

where η i0 is an integral invariant matrix. All of the boundary conditions (21)-(22) and continuity conditions (i.e. from Eq. (23) to Eq. (28), from Eq. (32) to Eq. (34)) are rewritten in the state-space. Finally, the linear algebraic equations can be obtained as follows [13], Mη 0 = 0 .

{

where η 0 = η10 ,η20 ,η30 ,η40

(38) T

}

and the coefficient matrix M is defined in reference [13]. If an

untrivial solution exists in equation (38), the determinant of its coefficient matrix must be equal to

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zero, Det M = 0 .

(39)

The lowest eigen value of the determinant of Eq. (39) is just the critical buckling loading. Results and Discussion Thermal Residual Stresses in TBC. The materials properties used in the calculation are

temperature-dependent and taken from reference [3]. hc and hs are, respectively, 0.35mm and 2.1mm. The thickness of bond coat is 0.1mm. t1 , t 2 , t 3 are equal to, respectively, 600s, 4200s, 4800s. Substrate temperature, Tb , equals 700 0 C . The highest surface temperature of the ceramic coating, Ts , is assumed to vary from 1000 0 C to 1600 0 C . During the calculation, a finite difference approach was used to solve the equation (8). The temperature fields and stress fields in TBCs are determined incrementally during thermal cycles. After TBCs experiences many thermal cycles, the thermal stresses distribution in ceramic coating are shown in Fig.3. It can be seen that the thermal stress in x -direction, i.e. σ 11 ( z, t ) , at the end of holding period is tensile, where Ts equals 1000 0 C . However, due to the effect of the last residual compressive stress, the tensile thermal stress at the end of holding time decreases with the increase of thermal cycles. The thermal tensile stress at ceramic coating/bond coat interface is far larger than that on the top surface of ceramic coating. Moreover, when thermal cycles n equals 15, the thermal stress on the top surface of ceramic coating equals approximately zero and this decrease is due to the influence of creep deformation and stress relaxation in ceramic coating. The results are in a good agreement with the experimental results as reported by Zhu et al.[11]. 0

200

N=1 N=5 N=10 N=15

160

Tensile Stress (MPa)

140 120

-20

Residual Stress σ r (MPa)

180

100 80 60 40 20 hc=0.35mm t1=600 s 0 t2=4200 s

-20 -40

0

-40

-60

-80

N=1 N=5 N=10 N=15 N=20 N=25 N=30

-100

-120

Ts=1000 C 0

-60 Tb=700 C

-140

0.03 0.06 0.09 0.12

0.15

0.18

0.21

0.24

0.27

0.30

0.33

0.36

The distance from the interface z (mm)

Fig.3. The relationship of thermal stress in ceramic coating at the end of holding time with thermal cycles.

0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24 0.27 0.30 0.33 0.36

The distance from the interface z (mm)

Fig.4. The relationship of residual stress in ceramic coating at the end of cooling time with thermal cycles.

The relationship of residual stress in ceramic coating at the end of cooling with position z for different thermal cycles n is shown in Fig.4, where the temperature on the surface of ceramic coating T s is 1000 0 C . It is seen that the residual stresses at ceramic coating/bond coat interface

38

Macro-, Meso-, Micro- and Nano-Mechanics of Materials

increase with the increase of thermal cycles. Especially, the residual stress is compressive. When the TBCs experiences 30th thermal cycles, the largest compressive residual stress is up to − 134 MPa . However, the residual stresses on the top surface of ceramic coating are close to − 70 MPa . As we know, the surface temperature of the top coating is much higher than that at the interface. Therefore, the creep strain on the surface of the top ceramic coating is much higher than that at the interface. Finally, the stress relaxation induced by the creep strain of ceramic coating may occur on the surface of the ceramic coating [11,13]. Fig.5 shows the relationship of the residual stresses at the ceramic coating/bond coat interface with thermal cycles for different temperature on the surface of the top ceramic coating. The surface temperature on the ceramic coating T s varies from 1000 0 C to 1600 0 C . The substrate temperature remains invariable according to the actual application, i.e., Tb = 700 0 C . When thermal cycle and the temperature on the surface the ceramic coating increase, the residual stresses at the interface are compressive and become higher and higher. As mentioned above, the accumulated compressive residual stresses may be up to a critical value and induce the buckling failure of the ceramic coating. The analysis in the next section will focus on the discussions of the buckling failure behavior of the ceramic coating under the effect of compressive residual stress and thermal cycles. 0

0

Ts=1100 C

0

Ts=1300 C

Ts=1000 C Ts=1200 C

-20

Ts=1400 C

-40

0.7

0 0

Ts=1500 C

β=0.2 β=0.3

0.6

0

Ts=1600 C

-60

0.5

Non-Buckling Failure Region

-80

0.4 -100

P/Pu

Residual Stress σ r (MPa)

0

0

-120 -140

0.3 0.2

Buckling Failure Region

-160

0.1

-180 -200 0

5

10

15

20

25

30

35

40

45

Thermal Cycles Times N

Fig.5 The relationship of residual stresses at ceramic coating/bond coat interface with thermal cycles, where the substrate temperature remains invariable, i.e. 7000C.

0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

γ=a/L

Fig.6 The relationship of the non-dimensional critical loading P / Pu with the non-dimensional crack length γ , where β = hc / H .

Critical Buckling Stress. Generally, as we know, the critical buckling loading is mainly influenced by the interface cracking length a , the total thickness H , the total length L and the ceramic coating thickness hc . The calculated results show that the interface crack length is the key parameter to control the critical buckling load. Fig.6 shows the relationship of non-dimensional

critical buckling loading P / Pu with the non-dimensional interface crack length γ = a / L , where Pu denotes the critical buckling loading for the intact TBCs system. It can be seen that the dependence of the critical buckling load on the interface crack length is not very high when the crack is short. However, when the crack is long, for example, γ equals 0.3, the dependence of the critical buckling loading on the interface crack length is high. When the interface crack is very long,

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39

for example, γ equals 0.65, the critical buckling load is very low and the non-dimensional critical buckling load P / Pu is almost zero. However, as studied by Choi et al. [14], the TBC system has sufficient stiffness to suppress small scale buckling (SSB) of the TGO. Accordingly, the eventual failure often occurs by large scale buckling (LSB) but only after a sufficiently large separation has developed at the interface, typically several mm in length. In this paper, the interface crack length is assumed to be 1mm and 0.1mm . The corresponding critical bucking loads/stresses are, respectively, − 98.78MPa and − 1.2GPa by using the buckling failure model. It is obviously seen that the critical buckling stresses will be quickly decrease when the interface crack length increases during the thermal cycles. For interface delamination length 1mm , the corresponding buckling stress is marked by bold line as shown in Fig 5. Therefore, when the residual stress in ceramic coating accumulates with thermal cycles and it is up to the corresponding critical buckling stress, the buckling failure phenomenon of the ceramic coating will occur. It is obviously seen that if the surface temperature Ts is low, the delaminated TBC system must experience more thermal cycles such that the residual stress accumulated in ceramic coating can be up to the buckling failure stress. When the surface temperature Ts becomes high, that is to say, the temperature gradient becomes large, the critical thermal cycles decrease rapidly. The accumulated residual stresses may be easy to achieve the critical stresses of the buckling failure of the ceramic coating. That is to say, when the temperature gradient along the thickness direction of TBCs increase, the buckling failure of the top coating in TBCs occurs easily.

600 40

500

Thermal Cycles Number

30

Thermal Cycles Number

35

Buckling Region

25 20 15 10

-98.97MPa

Non-Buckling Region

Buckling Region 400

300

-1.200 GPa 200

Non-Buckling Region 100

5

1000 1050 1100 1150 1200 1250 1300 1350 1400 1450 1500 1550 1600 0

The Surface Tempareture of the Top Ceramic Coating Ts ( C)

Fig.7. The buckling plane, i.e. n − Ts is introduced to predict the buckling failure of the ceramic coating, where the interface length is 1.0mm

0 1000

1100

1200

1300

1400

1500

1600 0

The Surface Tempareture of the Top Ceramic Coating Ts ( C)

Fig. 8. The buckling plane is used to describe the relationship of residual stress, thermal cycles and temperature gradient, where the interface length is 0.1mm

In order to have a clearly physical information, the buckling plane, i.e. n − Ts plane, is introduced to predict the buckling failure of the TBC system. Fig. 7 and Fig. 8 denote, respectively, two different kinds of the buckling plane for different interface crack length, i.e. a = 1mm and a = 0.1mm . In the plane, there are two regions, i.e., non-buckling region and buckling region (top).

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Macro-, Meso-, Micro- and Nano-Mechanics of Materials

When Ts is 1000 0 C and 1600 0 C the critical thermal cycles are, respectively, 40 and 12 as shown in Fig.7. Comparing Fig.7 and Fig.8, it can be seen that the interface length and temperature gradient play an important role for the buckling failure of the ceramic coating. Conclusions

The thermal residual stress and the critical buckling failure of ceramic coating have been studied. The following conclusions can be highlighted. (1) An analytical solution of stress field in TBCs is obtained under the non-linear coupled effect of temperature gradient, thermal fatigue and creep strain of TBCs. The calculated results show that the residual stresses in ceramic coating are compressive. The accumulated residual stresses with thermal cycles could be high enough to induce the buckling failure of the ceramic coating. These conclusions are in good agreement with experimental phenomena. (2) A theoretical model is proposed to predict the critical thermal buckling loads for the failure of TBC system. The important parameters for the critical buckling conditions are the interface crack length and the residual compressive stresses in ceramic coating. By calculation, it is found that when the interface crack is too short, the ceramic coating will not buckle. On the other hand, a buckling plane is introduced to predict the buckling failure of the ceramic coating. In the plane, it can be divided into two regions, i.e., non-buckling region and buckling region. Acknowledgements

The project is sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry (No:[2002]247). References

1. E. Celik, E. Avci, F. Yilmaz: Surf. Coatings Technol. Vol. 97 (1997), 361-365. 2. E. Celik, I. A. Sengil, E. Avci: Surf. Coatings Technol. Vol. 97 (1997), 355-360. 3. Y.C. Zhou, T. Hashida: Int. J. Solids Struct. Vol. 38 (2001), 4235-4264. 4. Y.C. Zhou, T. Hashida: JSME Int. J. Vol. 45 (2002), 57-64. 5. A.G. Evans, M.Y. He, J.W. Hutchinson: Progress Mater. Sci. Vol. 46 (2001), 249-271. 6. A.G. Evans, D.R. Mumm: Progress Mater. Sci. Vol. 46 (2001), 505-553. 7. A.M. Karlsson, C.G. Levi, A.G. Evans: Acta Mater. Vol. 50 (2002), 1263-1273. 8. R.J. Christensen,V.K. Tolpygo, D.R. Clarke: Acta Mater. Vol. 45 (1997), 1761-1766. 9. V. Sergo, G. Pezzotti: Acta Mater. Vol. 46(5) (1998), 1701-1710. 10. S.R. Choi, J.W. Hutchinson, A.G. Evans: Mech. Mater. Vol. 31 (1999), 431-447. 11. D.M. Zhu, A.M. Robert: J. Mater. Res., Vol. 14(1) (1999), 146-161. 12. C.C. Chiu, E.D. Case: Mater. Sci. Eng A., Vol. 132 (1999), 39-47. 13. D.K. Li, J.P. Zhou: Acta Mechanica Solida Sinica, Vol. 21(3), (2000), 225-233. 14. S.R. Choi, J.W. Hutchinson, A.G. Evans: Mech. Mater. Vol. 31 (1999), 431-447.

Advanced Materials Research Vol. 9 (2005) pp 41-50 © (2005) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMR.9.41

Cyclic Stress-Strain Behavior and Thermomechanical Effect in Metal Matrix Composites H. G. Kim Department of Mechanical Engineering, Jeonju University, 1200, 3-Ga Hyoja-dong, Wansan-gu, Jeonju, 560-759, Korea. [email protected] Keywords : Short Fiber Composite, Cyclic Stress, Metal Matrix Composite, Thermo-ElastoPlastic Finite Element Analysis (FEA), Residual Stress, Fiber Volume Fraction

Abstract. A micromechanical model based on continuum analysis has been investigated by using finite element analysis (FEA) in discontinuous metal matrix composites (DMMC). To assess the tensile and compressive constitutive responses, a cyclic stress-strain behavior has been performed. For analysis procedure, the elastoplastic FEA and the regularly aligned axisymmetric single fiber model have been implemented to evaluate the internal field quantities. Accordingly, the fiber and matrix internal stresses were investigated for the constrained representative volume element (RVE). Further, the local plasticity in the matrix were described during loading and unloading precesses, which can predict the damage mechanisms as well as strengthening mechanisms. On the other hand, a thermoelasto-plastic analysis has been performed using FEA for the application to the continuum behavior in a discontinuous metal matrix composite. The internal field quantities of composite as well as overall composite behavior and an experiment was demonstrated to compare with the numerical simulation. As the procedure, the reasonably optimized FE mesh generations, the appropriate imposition of boundary conditions, and the relevant postprocessing such as elasto-plastic thermo-mechanical analysis were taken into account. For micromechanical model, the temperature dependent material properties and precipitation hardening effects have been employed to investigate field quantities. It was found that the residual stresses are induced substantially by the temperature drop during heat treatment and that the FEA results give a good agreement with experimental data. Introduction The main objective of DMMCs is the usage of increased service temperature or specific mechanical properties of structural components by replacing existing superalloys. In these DMMCs, mechanisms of strengthening and of microscopic deformation were issues of academic and practical importance [1,2]. In this paper, an attempt to characterize the major composite strengthening mechanism in DMMCs has been given in detail through the constrained unit cell models implementing an elastoplastic FEA. An axisymmetric single fiber model[5] based on incremental plasticity theory using von Mises yield criterion and Plandtl-Reuss equations is employed to evaluate the constrained model. A cyclic stress-strain curve described by a hysteresis loop that gives the information of tensile and compressive constitutive responses in a designated region are obtained using FEA. In the meantime, many researchers reported that the major strengthening of DMMCs stems from the increased dislocation density in the matrix [3,4]. However, the present study indicates that the constraint effect causes the substantial strengthening phenomina in DMMCs. It is also found that the contribution of overall matrix is not so significant though it generates a source of strengthening effects. On the other hand, thermally induced significant residual stresses can arise due to coefficient of thermal expansion(CTE) mismatch between two constituents when the matrix and reinforcement are well bonded [5-8]. Therefore, it can be mentioned that accurate prediction of the magnitude and distribution of residual stress is crucial to the design and analysis of discontinuous composites. In recent numerical studies [9], it was shown that the magnitude of thermal residual stress is significant, adequate to result in substantial plastic yielding around fibers after cooling from the processing temperature though the age hardening effect was neglected. Furthermore, many studies have been reported to predict the overall stress-strain response including residual stress effects. However, a crucial point to investigate

42

Macro-, Meso-, Micro- and Nano-Mechanics of Materials

the residual stresses is to simulate the material properties as temperature dependent behavior, which gives a lot more accurate results compared to temperature independent behavior. In this paper, the more detailed study considering temperature dependent material properties and precipitation hardening effect was performed. Numerically predicted tensile stress-strain behaviors were also compared with the experimental results and the role of residual stress was discussed in detail. It was found that the residual stress strongly effects on the localized deformation evolution though it is not so sensitive to the macroscopic resultants compared with the microscopic field quantities. Analysis Elastoplastic finite element formulations. Elasto-plastic analysis is performed to investigate the tensile and compressive behavior in DMMCs. To solve nonlinearity, Newton-Raphson method has been implemented in this study. Consistent with small strain theory,

{dε el } = {dε } − {dε pl } .

(1)

where {dε }, {dεel}, and {dεpl} are changes in total, elastic, and plastic strain vectors, respectively. The yield criterion determines the stress level at which yielding is initiated. For an elastoplastic materials, a yield function F which is a function of stresses {σ} and quantities {α} and κ associated with the hardening rule can be defined. Yielding occurs when

F ({σ }, {α }, k ) = 0 .

(2)

where κ is the plastic work per unit volume and {σ} is the translation of yield surface. Specifically, the {α} is history dependent. According to von Mises theory, yielding begins under any state of stress when the effective stress exceeds a certain limit, where

σ e =[

1 { (σ x −σ y )2 + (σ y −σ z )2 + (σ x − σ z )2 }+ 3(τ xy2 +τ yz2 +τ xz 2 ) 2

]1/ 2

.

(3)

The flow rule determines the direction of plastic straining. A plastic potential Q which has a unit of stress and is a function of stresses (that determines the direction of plastic straining), Q = Q ({σ}, {α} , {κ}). With λ, a scalar which is called a plastic multiplier (that determines the amount of plastic straining), plastic strain increments are given by

 ∂Q  {dε pl } = λ    ∂σ 

.

(4)

equation for F now becomes: T

T

∂F  ∂F   ∂F  pl {σ }T {dε pl } = 0 .   {dσ } +   C{dε } + ∂κ  ∂σ   ∂σ 

(5)

The stress increment can be computed via the elastic stress-strain relations as follows:

{dσ } = [ D]({dε } − {dε pl }) . Using the definition of {dεpl} and λ in equations (5) and (6):

(6)

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43

{dσ } = [Dep ]{dε }

(7)

where the elastoplastic matrix [Dep] is  ∂Q  [ Dep ] = [ D ]1 − {C λ }T   ∂σ 

.

(8)

Incorporating associated flow rule (Plandtl-Reuss equation) and isotropic hardening rule, Q=F and {α}=0 have been implemented in this study. Thermo-elasto-plastic finite element formulations. The FEM formulations in this work are centered on the thermo-elasto-plastic analysis with small strain plasticity theory [10] using an axisymmetric single fiber model. The model is based on incremental plasticity theory using von Mises yield criterion, Plandtl-Reuss equations and isotropic hardening rule. The strains here are assumed to develop instantaneously. To solve nonlinearity, Newton-Raphson method has been implemented. Based on the thermo-mechanical theory.

{ d ε el } = { d ε } − { d ε

pl

} − { d ε th } .

(9)

where {dε} , {dεel}, {dεpl} and {dεth} are the changes in total, elastic, plastic and thermal strain vectors, respectively. The thermal strain vector {dεth} is

{dε

th

} = { CTE }{ ∆ T } .

(10)

According to Eq(3), The stress increment can be computed via the elastic stress-strain relations as follows. .

(11)

where [D] is the elastic stress-strain matrix and [Dep] is the elasto-plastic stress-strain matrix which is given by Eq. (8) where Q is the plastic potential and {Cλ} is the factor influencing to the plastic multiplier. On the other hand, the principle of virtual work says that a virtual (very small) change of the internal strain energy must be offset by an identical change in external work due to the applied loads, i. e., (12) δU =δ W . where U is the strain energy (internal work), W is the external work, and d is the virtual operator. The virtual strain energy is δU = {δU }T ∫ [ B ]T [ D ][ B ]dV {u} − {δU }T ∫ [ B ]T [ D ]{ε th }dV V

.

(13)

V

Next, the external virtual work by nodal forces is

δW = {δU }T { Fe } .

(14)

where {Fe} is nodal forces applied to the element. On the other hand, component stresses were calculated for each element at its integration points (or Gauss points).

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Macro-, Meso-, Micro- and Nano-Mechanics of Materials

Micromechanical Modeling Approach Al/SiC as short fiber composite system was chosen to investigate thermal stresses in the two different materials. The short fiber was 1.0µm in diameter with an average aspect ratio of 4 and tended to be aligned in the longitudinal direction which corresponds to the axis of tensile samples as shown in Fig. 1. The reinforcements are assumed to be uniaxially aligned with no fiber/matrix debonding. For uniform fiber distribution, a unit cell was selected as RVE. Under this assumption, the RVE is based on a conventional single fiber model. The FE formulations in this work are centered on the elastoplastic analysis with small strain plasticity theory [4] using an axisymmetric single reinforcement model. To solve nonlinearity, Newton-Raphson method has been implemented [5,6]. Elastoplastic stress-strain matrix can be solved iteratively. The constraint boundary condition enforces elastic and plastic constraint by requiring that the radial and axial boundary of the unit cell is maintained in the straight manner during deformation. Thus, the stress-strain characteristics of the matrix are defined by the elastic modulus, yield stress and work hardening rate (tangent modulus). These characteristics were measured at room temperature on the PM 2124 Al alloy and were found to be the yield stress is 336MPa, Em=70GPa, and ET=1.04GPa, Ef=480GPa, respectively. Here, E is Young' s modulus, ET is tangent modulus, the subscripts m and f are matrix and fiber, respectively.

(a)

(b)

Fig. 1. A schematic of simplifying the unit cell in a composite. (a) Micrograph of Al2124/SiC composite material (×1500), (b) Image of assumed fiber arrangement and selection of RVE. Hence, the axisymmetric single fiber model is used as implemented in the previous study[2]. Fig. 2 shows the single fiber model and finite element mesh with constrainted boundary condition. The fiber and matrix materials were assumed to be isotropic and the elastic constants are assumed to be temperature independent. The experimental data for compressive stress-strain curves was not needed for input data because the isotropic hardening rule was implemented in this study. To solve nonlinear analysis, 25 small load steps of which step has maximum 20 iterations were used incrementally by incremental strain 0.04%. Fig. 3(a) shows the fiber and Fig. 3(b) shows the matrix yielding zone at unloaded state (point B).

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(a) (b) Fig. 2. Micromechanical model, (a) Cylindrical RVE of a short fiber reinforced composite, (b) Axisymmetric finite element model Fig. 3 shows an axisymmetric single fiber model in FEM analysis for temperature cooling down. The constrained RVE as Fig. 3(b) was analysed in the present study. The material properties chosen were 4.30×10-6/K as CTE of fiber and 2.36×10-5/K as CTE of matrix. Temperature dependent properties of 2124 Al alloy are shown as Table 1. Presumably, elastic modulus does not change so much while yield stress, ultimate tensile stress (UTS) and plastic modulus change tremendously by temperature goes up.

(a) (b) Fig. 3. A schematic of deformed shape. Dotted line indicates the deformed shape after cooling down. (a) Without constrained boundary Condition, (b) with constrained boundary condition. Results and Discussion Cyclic Stress-Strain Behavior. To obtain the stress-strain hysteresis behavior numerically, the applied far field strain was subsequently loaded from 0% (Origin) to 1% (point E), 1% to 0% (point J), 0% to 1% (point P), -1% to 0% (point U), and 0% to 1% (point Z), as described in Fig. 4.

Fig. 4. Numerically predicted stress-strain response for fully reversed loading between far-field composite strains of 1% and –1% with and without residual stresses

46

Macro-, Meso-, Micro- and Nano-Mechanics of Materials Table 1. Temperature dependent properties of 2124 Al alloy Temperature(K)

767.0

733.6 700.2 666.8 633.4 600.0 550.0 500.0 450.0 400.0 350.0 300.0 293.0

Young’ s Modulus(GPa)

50.4

51.6

52.8

54.0

55.2

Yield Stress(MPa)

15

20

30

40

UTS(MPa)

25

30

42

55

Plastic Modulus(GPa)

0.10

0.10

0.12

0.15

56.4

58.2

60.0

61.8

63.6

65.4

67.2

67.2

60

90

140

190

205

210

215

220

336

80

115

180

280

310

330

360

380

440

0.20

0.25

0.40

0.90

1.05

1.20

1.45

1.60

1.04

In Fig. 5 to 9, the fiber stress and plastic deformation of every five step for case are described. In the large applied strain regime, the plastic deformation evolution of the unconstrained model is saturated which is not included in this paper, whereas that of constrained model develops up to general yielding as the case of monotonic loading. In general, plasticity in compression loading develops from the fiber tip region and is propagated from the matrix gap region as the case of tensile loading.

Fig. 5. Numerically predicted stress-strain response at point O to E, fiber axial stresses as a function of distance normalized by fiber radius at point E and the propagation of Extensive stresses in the matrix (Dark zones depict the high stressed area over matrix yield stress).

Fig. 6. Numerically predicted stress-strain response at point E to J, fiber axial stresses as a function of distance normalized by fiber radius at point J and the propagation of Extensive stresses in the matrix (Dark zones depict the high stressed area over matrix yield stress).

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Fig. 7. Numerically predicted stress-strain response at point J to P, fiber axial stresses as a function of distance normalized by fiber radius at point P and the propagation of Extensive stresses in the matrix (Dark zones depict the high stressed area over matrix yield stress.

Fig. 8. Numerically predicted stress-strain response at point P to U, fiber axial stresses as a function of distance normalized by fiber radius at point U and the propagation of Extensive stresses in the matrix (Dark zones depict the high stressed area over matrix yield stress.

Fig. 9. Numerically predicted stress-strain response at point U to Z, fiber axial stresses as a function of distance normalized by fiber radius at point Z and the propagation of Extensive stresses in the matrix (Dark zones depict the high stressed area over matrix yield stress. Thermo-mechanical Effects. The deformation processes are shown in Fig. 10 for both without residual stresses and with residual stresses. The development of local plasticity in the matrix is demonstrated sequentially. In Fig. 11, the plastic deformation of four steps, 0.2%, 0.4%, 0.6%, 0.8% of

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composite strain, of both without (left column) and with (right column) residual stresses are described. The model with residual stresses shows the earlier initiation of plasticity in the matrix than that of without residual stresses as shown in Fig. 11. However, plasticity develops from the fiber tip region for both cases and is propagated to the whole matrix. It is found that the behavior of local plastic deformation is substantially different due to residual stresses.

Fig. 10. Process of residual stress generation in case of the T-6 heat treatment condition

Fig. 11. Plastic deformation evolution in the matrix without and with residual stresses. The light and dark regions indicate the elastic and plastic matrix respectively. Left column [(a), (c), (e) and (g)] depict without residual stresses and right column [(b), (d), (f) and (h)] depict with residual stresses. Applied strains are 0.2% in the case of (a) and (b), 0.4% in the case of (c) and (d), 0.6% in the case of (e) and (f), and 0.8% in the case of (g) and (h) respectively. The constitutive stress-strain response of the 20 vol. % SiC reinforced 2124 Al alloy stresses is shown in Fig. 12, which is compared with experimental data. The stress-strain curve with residual stresses is substantially lower than that without residual stresses and also agrees favorably with experimental data.

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Fig. 12. Stress-strain response of 20 vol. % SiC reinforced 2124 Al alloy with and without residual stress effects compared with experimental data. Conclusions An elastoplastic stress analysis is performed to predict the cyclic stress-strain response based on FEA for a unit cell model. A detailed process of plasticity evolution in the matrix which describe the local deformation behavior in DMMCs. It is found that the constraint effect plays a very important role in the macroscopic constitutive response as well as the local plastic deformation evolution. It is also found that the fundamental strengthening mechanism of DMMCs is besed on the fiber stress enhancement due to the triaxiality produced in the matrix between fiber ends. A thermo-elasto-plastic analysis were also performed to investigate the thermal residual stresses which can develop in a short fiber composite due to CTE mismatch. The results showed that the magnitude of thermal stresses is so substantial that effects on the internal stress field as well as macroscopic stress-strain relationship. Further, residual stresses result in a decrease of the flow stress of the composite compared to that without residual stresses. It was found that the FEA results give a good agreement with experimental data if the residual stresses are taken into account. References 1. B. Ji et al., Key Eng. Mat., Vols. 177-180 (2000), pp. 297-302 2. H.G. Kim, Int. J. KSME , 25, 1 (2001) p. 2 3. R. J. Arsenault, and N. Shi, Mat. Sci. and Eng., Vol. 81, (1986) pp. 175-187 4. R. J. Arsenault and M. Taya, Acta Meta., Vol. 35, (1987) pp. 650-659 5. M. Taya and R. J. Arsenault, Metal Matrix Composites, Thermo-mechanical Behavior, Pergamon Press, NY (1989), p. 1 6. R. J. Arsenault, and R. M. Fisher, Scripta Metallurgica, Vol. 17 (1983), p. 67 7. M. Vogelsang et al., Metallurgical Transactions A, Vol. 17A (1986), p. 379 8. M. K. Park and S. Bahk, Key Eng. Mat., Vols. 183-187 (2000), p. 529 9. G. L. Povirk et al., Materials Science and Engineering, A132, (1991), p. 31 10. R. D. Cook et al., Concepts and Applications of Finite Element Analysis, 3rd Edition, John Wiley & Sons (1989) p. 163

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Advanced Materials Research Vol. 9 (2005) pp 59-68 © (2005) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMR.9.59

Influence of Surface Properties on Microscratch Durability of Aluminum Nitride Semiconductor Processing Component L. Chouanine1,a, M. Takano1, F. Ashihara1, O. Kamiya1, M. Nishida2 1

Department of Mechanical Engineering, 2Department of Computer Science and Engineering Faculty of Engineering & Resource Science, Akita University, Tegata Gakuen-machi, Akita-city, 010-8502 Japan [email protected]

Abstract The effect of the surface properties on the microtribological characteristics of AlN-based electrostatic chuck (EC) for silicon plasma etching was investigated using automatic microscratch testing technique in combination with SEM examination of the scratch track. The scratch testing was performed by applying a progressive indenter load. The scratch failure model varied systematically with the surface properties of AlN. The data of the onset of brittle fracture were used as characteristic features of the AlN failure. It was found that the critical load, Lc, the smallest applied normal load leading to unacceptable damage such as chipping and cracking, increases with decreasing the average grain size, density and fracture toughness of AlN and decreases with increasing the surface roughness and area density of pre-existing polishing damages. The resistance to cohesion and adhesion failure of AlN with 0.1 µm Al2O3 oxide layer on top was stronger than that of the AlN bulk material. The fracture initiation and ductile to brittle transition in AlNAl2O3(0.1µm) was in form of discontinuous chipping. The results infer the potential of the combination of the scratch data with the material properties for the understanding of the effect of the surface topography on the mechanical properties and chucking performance of AlN-based EC. Introduction The use of AlN, high performance ceramic, as a material that has tightly controlled electrical properties is critical for applications in high temperature semiconductor processing components, which provide fixtures of silicon wafers in vacuum environments without the use of mechanical clamps. AlN-based EC is a device which holds and often heats the silicon wafer during chemical vapor deposition or plasma etching[1,2]. To date, large technological progress has been made by the ceramic manufacturing community in fabricating AlN-based EC, which exploits attractive features such as good thermal uniformity, rapid heat up, easy temperature control, excellent plasma durability and less contamination[3]. However, the performance of commercially available AlNbased EC by means of perfect chuck/release and damage free handling of the silicon wafer is still behind the practical expectations of the semiconductor manufacturing community. In tribology, the main issue of the heat transfer between an AlN-based EC and the chucked surface of the silicon wafer is concerned with the conductance through the micropoints of contact. The electrical and thermal properties of the AlN-based ESC are influenced by the degree of geometry and surface roughness accuracies as well as by the degree of cleanliness of the quality of the finished surface[4]. As the heating temperature of the silicon wafer rapidly increases, the hardness, yield pressure of the AlN-based EC and the real area of contact decrease. Hardness instability affects both thermal and electrical conductance characteristics of the AlN-based EC[4]. Fig. 1 shows typical data of the chucking pressure vs. heating temperature graph of the AlNbased EC in vacuum environments. As the heating temperature of the silicon wafer exceed 300 °C, the chucking force attains 15 kPa in the case of a smoothly planarized AlN-based EC and only 5 kPa in the case of a roughly finished and uneven surface. The micromachining community requires that the surface of the AlN-based EC should be finished to a high precision roughness and flatness accuracies matching with those of fracture-free ultra-precisely polished silicon wafer[4,5]. The rapid heat up of the silicon wafer by the AlN-based EC causes a sharp increase of the thermal expansion and the friction force at the interface with the silicon wafer. Under these circumstances, if the AlN-based EC suffers considerable sub-surface damages (SSD) caused by polishing[6], it is highly possible that particles may detach from parts of the AlN-based EC, which are characterized

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by chipping and inter-granular cracking. In their turn, it is also highly possible that disconnected particles from the AlN-based EC may further contaminate and damage the silicon wafer.

Chucking Force

(kPa)

20

15 Smooth & planarized surface Rough & uneven surface

10

5

0 0

100

200

300

400

Temperature

500

600

700

800

( C) o

Fig. 1. Typical chucking pressure vs. heating temperature of AlN electrostatic chuck. It is technologically difficult and cost expensive to carry out in-situ observations during of the interface between the silicon wafer and the AlN-based EC in order to verify our above claim and to find out apparent links between detached particles from the AlN-based EC and residual damages on the silicon wafer. Accordingly, the main objective of this paper is to carry out of-situ tribological investigations of the AlN-based EC using microscratch testing in combination with SEM examinations of the scratch tracks[7~17]. To date, microscratch testing is a comprehensive method of quantifying the scratch resistance, cohesion (mechanical durability) and adhesion of a wide range of bulk materials and ceramic films[7,8,9]. The method involves generating a controlled scratch with a diamond tip on the sample under test. The tip, a Rockwell C diamond, is drawn across the coated surface under a progressive load[9,10,11]. With increasing load, the AlN deformation generates stresses which, at a given load, result in permanent damage, such as chipping and cracking of the material (cohesive failure) or/and flacking (adhesion failure) of the film in case of a substrate/film system. The smallest load leading to unacceptable damage is called the critical load, Lc, can detected very precisely by means of an acoustic sensor attached to the indenter holder, the frictional force, penetration depth and by optical microscopy observations as well. Material processing and properties We studied three types of AlN-based EC, two of them are dense technical grade materials and the other one includes additives such as Yttrium. All samples have different average grain size of AlN, grain densities and fracture toughness properties. The mechanical properties (ISO standard) of the AlN-based EC are shown in Table 1. A dense technical grade AlN, covalent bonded materials with hexagonal crystal structure, was produced using hot pressing technology. A high purity powder is processed to create an AlN puck, and embedded electrodes were fabricated using cofiring. AlN-based EC is stable to very high temperatures in inert atmospheres, hydrogen, steam, and carbon oxide atmospheres up to 980 °C. In air, surface oxidation occurs above 700 °C. A layer of aluminum oxide, Al2O3, forms and protects the material up to 1370 °C. Above this temperature bulk oxidation occurs. AlN slowly hydrolyzes in water and dissolves in mineral acids through grain boundary attack, and in strong alkalis through attacks on the AlN grains. Fig. 2 shows the forces acting on the diamond tip during the scratch test, where (Fn) is the normal load applied on the tip of radius (R). The shearing force per unit area (τ) is due to the deformation of the surface. The radius of the circle of contact (r) is the half-width of the track left after the test. The hardness (H) of the AlN-based EC is similar to a uniform hydrostatic pressure

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acting normal to the surface of indentation. Assuming that the AlN-based EC follows plastic deformation in scratch, H can be related to Fn and r by: H=

Fn . π .r 2

(1) Table 1 Mechanical properties of AlN (ISO measurement method)

Mechanical Properties

Physical Properties

Sample A (99.9%) 3.26 360 1158 2.2 3.0

Density (g/cm3) Strength (MPa) Hardness (Hv) Young Modulus (GPa) Fracture Toughness (MPa.m1/2) Average grain size (µm)

AlN bulk material Sample B (with additives) 3.27 372 2.8 5.9

Sample C (99.9%) 3.33 280 943 321 3.8 9.8

Fn [Lc (Critical load)] Stylus tip radius

τ

R

Table motion

θ r d H

Tension H (Hardness of AlN) Fig. 2. Geometry of the scratch test showing the forces acting on the diamond tip. The cohesion shear strength, τ, and the critical applied normal load Lc are related by[7]  L  r . τ = H tan θ = c2  (2) πr  R 2 − r 2 12    or L (3) τ = c if R >> r . πrR An accurate determination of Lc is difficult. Several techniques such as SEM observation of the scratch track, and acoustic emission, are used to measure Lc. The friction force (Ft) is measured during scratching to measure Lc as well. The acoustic emission signals and the amplitude of Ft begin to increase as soon as cracks begin to form perpendicular to the direction of the moving stylus.

(

)

Experimental method The scratch tests, schematically shown in Fig. 2, were carried out using a Micro Scratch Tester manufactured by CSM (Switzerland) and operated in the progressive normal load mode, in the range of 0-3 N. A stylus with a Rockwell C type diamond was used, the tip being ground to a radius

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of 10 µm. While a load was applied, the sample was moved at a constant speed of 0.401 mm/min. The loading rate of 0.2 N/min and the scratch length of 6 mm were kept unchanged for all experiments. All experiments were performed using the same scratch conditions given in Table 2. Table 2 Microscratch conditions used for all experiments Diamond Indenter Rockwell Spherical with 10(µm) tip radius Loading range (min load to max load) 0 to 3(N) Loading rate 0.2 (N/min) Loading type Progressive Scratching speed 0.401 (mm/min) Scratching length 6 (mm) AE sensitivity 9

We have tested three types of AlN-based EC (A, B and C) of 3.5 inches of the size. The mechanical properties of these samples measured in accordance with ISO standard are given in Table 1. Before we conducted out scratch experiments, we polished all samples to a surface roughness equal to Ra ≈ 50 nm using SiO2 slurry type (pH3), and cleaned it in an ultra-sonic washing apparatus. When analyzing the surface topography of the AlN-based EC (B), we have found significant pre-existing microcraks and SSD caused by polishing. Therefore, we once more polished this sample to a surface roughness equal to Ra ≈ 7 nm. AlN-based EC (Ba) is in fact sample (B) but with better surface roughness and less machining SSD. We carried out the same scratch tests on AlN-based EC (Ba). In the second stage of our experimental work, we prepared three more AlN-based EC (A1, B1, C1), which are in fact AlN-based EC (A, B and C) with 0.1 µm Al2O3 oxide layers on top. The thickness of the Al2O3 oxide layers, t ≈ 0.1 µm, is kept nearly the same for all these samples. We performed the same scratch experiments on these AlN-Al2O3(0.1µm) samples. We performed three scratched on the AlN-based EC samples, and we measured the average smallest load, Lc, at which a specific failure event was recorded based on the analysis of the acoustic emission signals and the profile of the friction force in combination with SEM examination of the scratch tracks. The statistical deviations of the values of Lc are within ± 0.01 N. Results and Discussions Influence of the mechanical properties of AlN on the cohesive failure load. Table 3 shows the average measured critical loads per three scratches on the AlN-based EC (A, B and C). An example of the scratch test on AlN-based (B) is presented in Figs. 3, which show the friction force vs. the normal load, and the SEM micrographs of the scratch tracks corresponding to normal applied normal loads of 0.49 N and 3.00 N, respectively. With increasing the normal load, the indenter intersected pre-existing machining crack at low load equal to 0.27 N and caused low cohesion failure before the crack area density extend to initiate high cohesion failure at applied normal load nearly equal to 0.45 N. Table 3 Average measured critical normal loads Lc (three tests on each AlN bulk material). The statistical deviations of the measured values are within ± 0.01 N. AlN Ra Average grain size purities Average cohesive failure load Lc (±0.01N) (nm) (µm) A 50 3.0 99.9% 0.51 B 50 5.9 Additives 0.12 C 50 9.8 99.9% 0.21 a B 7 5.9 Additives 0.45

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AlN (sample B) 0.8

0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

Lc=0.40N

0.4

0

0.15 0.3 0.45 0.6 0.75 0.9

Tangential force Ft (N)

0.6

Lc=0.27N

0.2

(a)

0 3

2.7

2.4

2.1

1.8

1.5

1.2

0.9

0.6

0.3

0

Normal load Fn (N)

15.9 µm

16.2 µm

Scratch track

20.0 µm

(c)

Scratch direction

30.0 µm

(b)

Fig. 3. Results of scratch test on AlN sample B (includes additives such as yttrium and has AlN average grain size of AlN of 5.9 µm). The scratch conditions are shown in Table 2. (a) Tangential force vs. normal load. (b)-(c) SEM images of the scratch grooves corresponding to applied normal loads equal to 0.49 N and 3 N, respectively. The indenter intersected initially preexisting machining cracks at low load equal to 0.27 N.

Fig. 4 shows SEM micrographs of the scratch tracks on AlN-based EC corresponding to applied normal load of 3.00 N. The surface topography around the scratch track shows that the AlNbased EC (A), whose average grain size is equal to 3.0 µm suffers relatively less scratch damages comparing with sample (C) whose average grain size equal to 9.8 µm. The data, summarized in Table 3, indicate that with decreasing the average grain size, fracture toughness, density and increasing the hardness of AlN-based EC, the microscratch durability and the average critical load for fracture (Lc) increase. Influence of AlN’s surface roughness on the cohesion failure. Table 3 shows that Lc is also affected by whether the AlN-based EC exhibits pre-existing machining microcracks and SSD or not: Lc measured for AlN-based EC (B), which was polished to a surface roughness equal to Ra ≈ 50 nm and which is exhibiting polishing microcracks and SSD around the scratch track (see Fig. 5a) is equal to Lc = 0.12 N, while that measured for AlN (Ba), which was polished to a surface roughness equal to Ra ≈ 7 nm, and which does not exhibit pre-existing polishing microcracks and SSD around the scratch track (see Fig. 5b) is equal to Lc = 0.45 N.

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30.0 µm (a) AlN (grain size 3.0 µm)

Scratch Direction

30.0 µm (b) AlN (grain size 9.8 µm)

Fig. 4. SEM images of scratch tests on AlN at an applied normal load of 3.0 N. (a) AlN sample A (average grain size is 3.0 µm). (b) AlN sample C (average grain size is 9.8 µm).

In Fig. 5a, the pre-existing polishing microcracks around the scratch track need only propagation to cause cohesive failure of the AlN-based EC. The required load for crack propagation in Fig. 5a is lower than the critical load for crack initiation in Fig. 5b. Hence, Lc decreases with increasing the area density of the pre-existing polishing microcracks around the scratch track. Influence of Al2O3 oxide layer on the cohesion and adhesion failures of AlN. It is cost expensive to perform superfine polishing of AlN-based EC to achieve high surface roughness accuracy and to polish the surface of AlN without inducing microcracks. Accordingly, one of the approaches we have taken in order to deal with the polishing SSD and to improve the mechanical properties of the AlN-based EC is to stop at polishing the sample one time and to produce an Al2O3 oxide layer on top in order to cover the area of the microcracks and damages induced by polishing. Taking into account this engineering approach, we prepared three AlN-based EC (A1, B1 and C1) with Al2O3 oxide layers on top. The thickness of the oxide layer, 0.1 µm, is kept the same for all samples. The results of the scratch tests carried out on these samples are summarized in Table 4.

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Scratch Direction Pre-existing machining chipping & SSD

50.0 µm (a) AlN sample B (Ra = 50 nm) Smooth AlN surface Yttrium Scratch Direction

50.0 µm (b) AlN sample Ba (Ra = 7 nm)

Fig. 5. SEM images of scratch tests on AlN (average grain size is 5.9µm). The corresponding applied normal load is equal to 3.0 N. (a) AlN sample B (has a surface roughness equal to Ra ≈ 50 nm, and exhibits pre-existing machining microcracks around the scratch groove). (b) AlN sample Ba (has a surface roughness equal to Ra ≈ 7 nm, and does not exhibit preexisting machining microcracks around the scratch groove). Table 4 Average cohesive failure load Lc1 and adhesion failure load Lc2 (three tests on each AlNAl2O3(0.1µm) sample). The statistical deviations of the measured values are within ± 0.01 N. Average critical load Substrate Oxide layer (±0.01N) grain Adhesion Cohesive Roughne Al2 Thicknes Al Purities s size failure Lc2 O3 failure Lc1 ss N % (N) (N) Ra (nm) t (µm) (µm) A1 99.9 3.0 50 0.1 0.43 0.66 Additive B1 5.9 50 0.1 0.27 0.34 s C1 99.9 9.8 50 0.1 0.35 0.50

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Fig. 6 shows the results of a scratch test carried out on AlN-Al2O3(0.1 µm). The substrate is AlN-based EC (A), which has an average grain size of AlN equal to 3.0 µm. The friction force together with the acoustic emission intensity in Fig. 6a, clearly indicates that the critical load for fracture initiation (cohesive failure) is equal to Lc1 = 0.45 N. At this critical load, The shape characteristics of the acoustic emission intensity and the friction force shown in Fig. 6a together with the analysis of the SEM image of the scratch track shown in Fig. 6b demonstrates that the fracture initiation and the ductile to brittle scratch transition within the Al2O3 oxide layer appear in form of discontinuous bulky chipping. The distance between these chippings decreases and the crack largely extends around the scratch track as the applied load increases. The cracks may have reach the interface between the Al2O3 oxide layer and the AlN-based EC (Adhesive failure) at the critical load equal to Lc2 = 0.66 N, where delamination is observed. Substituting the value of Lc1 and Lc2 in Eq. (2), leads to the calculation of the cohesion shear strength and the practical adhesion shear strength of the AlN-Al2O3 system, respectively. The critical load, Lc2 = 0.66 N, for adhesive failure of the AlN-Al2O3 system is relatively stronger than the critical load, Lc1 = 0.51 N, for cohesive failure of the AlN-based EC (A). The practical adhesion failure can be demonstrated by SEM observations combined with EDS analysis of the scratch track. The work of adhesion of the AlN-Al2O3 system is derived as a function of the critical load (Lc2), given in Table 4, at which delamination of the Al2O3 layer from the AlNbased EC occurs[12,13,14]: 1

W =

πr 2  2 ELc 2  2

(4)   . 2  t  where, (W) is the work of adhesion, (E) is the Young’s modulus of elasticity and (t) is the thickness of the Al2O3 oxide layer. Although the use of 0.1µm Al2O3 on top of AlN-based EC improves the tribological properties of the AlN-based EC, it should be better to avoid this engineering solution because it is highly possible that the oxide layer may contaminate and damage the silicon wafer during plasma etching. Conclusions The results of this tribological work are summarized in Fig. 7. They infer the potential of the combination of the scratch data with the material properties for the understanding of the effect of the surface topography on the micromechanical properties and chucking performance of AlN-based EC. For clarification, we note that the results of the tests on sample (B) are not included in Figs. 7a, b, and c, because this sample includes additives and has different purity comparing with samples (A) and (C). The average critical load for fracture (Lc) is measured within a statistical deviation of ± 0.01N. 1. Lc increases with decreasing the average grain size of AlN (Fig. 7a) and the fracture toughness of the AlN-based EC (Fig. 7b). 2. Lc increases with decreasing the area density of pre-existing polishing microcracks and SSD in the AlN-based EC (Fig. 7c). The required load for crack propagation is lower than the critical load for crack initiation. 3. The critical load for the practical cohesion and adhesion failure (Lc2) in the case of AlNAl2O3(0.1µm) is found to be relatively stronger than the critical load for cohesion failure (Lc) in the case of the AlN bulk material. 4. Unlike in the case of the AlN bulk material, where the fracture mechanism appears to be in form of different kinds of lateral cracks and inter-granular cracks, the fracture mechanism in the case of AlN-Al2O3(0.1µm) is in form of discontinuous chipping. With increasing the applied normal load, the distance between these chippings decreases and the area propagation of the chipping extend within and around the scratch track. 5. Although the use of 0.1µm Al2O3 on top of AlN-based EC improves the tribological properties of the AlN-based EC, it should be better to avoid this engineering solution

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because it is highly possible that the oxide layer may contaminate and damage the silicon wafer during plasma etching. Acknowledgments Financial support by the Venture Business Laboratory of Akita University is gratefully acknowledged. Thanks are also due to Mr. Gwenael Bollore of Nanotec Corporation, Prof. Hiroshi Eda and Dr. Jun Shimizu of Ibaraki University for their support and encouragement.

Ft & AE vs Fn, for AlN (sample A1 scr#1)

25

0.2 24

0.7

18

0.18 Lc=0.45N

0.16

17.5 0.6

0.14

17

0.12 0.1 Ft (N) AE (%)

21

16.5

0.5

0.08 16

0.06 0.04

20

0.4

15.5

0.02 0.9

0.3

0.96

0.84

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0.6

0.66

0.54

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0.42

0.3

0.36

0.24

0.18

0.12

15 0

0 19

0.06

AE signal (%)

22

18

0.2

Tangential force Ft (N)

23

17 0.1 16

0

0.09

0.17

0.26

0.35

0.43

0.52

0.61

0.7

0.78

0.87

0.96

1.04

1.13

1.22

1.31

1.39

1.48

1.57

1.65

1.74

1.83

2

1.91

2.09

2.18

2.26

2.35

2.44

2.52

2.7

2.61

2.78

2.87

0 2.96

15

Normal load Fn (N)

(a) Scratch Direction

Brittle mode t h

Ductile mode t h Scratch t k

Chipping

30.0 µm

(b) Ductile to brittle transition mode scratch

Fig. 6. Scratch results of AlN-Al2O3(0.1 µm). The substrate is AlN sample A (average grain size 3.0 µm). (a) Tangential force and acoustic emission intensity vs. Applied normal load. (b) SEM micrograph of the scratch corresponding to critical normal load equal to Lc = 0.45 N. The image shows the fracture initiation and the ductile to brittle scratch transition within the oxide layer in form of discontinuous chipping.

0.6

(A)

0.5 0.4 0.3

(C)

0.2 0.1 0 0

2

4

6

8

10

Average Grain Size ( µm)

Critical Load for Fracture, L (mN) c

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68

0.6

(A)

0.5 0.4 0.3

(C)

0.2 0.1 0 0

0.5

1

1.5

Lc (mN)

(B)

0.4 0.3 0.2

(Ba)

0.1 0 0

5

10 15 20 25 30 35 40 45 50 Surface Roughness, Ra (nm)

(c)

2.5

3

3.5

4

(b)

Critical Load for Fracture,

Critical Load for Fracture, Lc (mN)

(a) 0.5

2

Fracture Toughness, KIC (MPa.m1/2 )

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

(A1) (C1) (A) ♦ AlN ■ AlN-Al2 O3 (0.1 µm) 0

2

4

(C)

6

8

10

Average Grain Size ( µm)

(d)

Fig. 7. Influence of the surface properties on average critical load (Lc) for fracture: (a) Lc vs. Average grain size of AlN. (b) Lc vs. Fracture toughness (KIC). (c) Lc vs. Surface roughness (Ra). (d) Lc vs. Average grain size of AlN for both AlN bulk material and AlN-Al2O3(0.1µm) system. References

1. P.R. Choudhury, Handbook of Microlithography, Micromachining, and Microfabrication, Society of Photo-Optical Instrum. Eng. Washington. Vol. 1 (1997) 11-474. 2. P.R. Choudhury, Handbook of Microlithography, Micromachining, and Microfabrication, The Society of Photo-Optical Instrumentation Engineers. Washington. Vol. 2, (1997), 11-474. 3. D.W. Richerson, Modern Ceramic Engineering: Properties, Processing, and Use in Design, 2nd ed. (Marcel Dekker, New York, 1992) p. 374. 4. T. Tsukizoe and T. Hisakado, ASME Jour. Lubrification Technology, 90F (1968) 81-88. 5. D.J. Whithouse, Handbook of Surface Metrology. (The Institute of Physics, 1994) p. 792. 6. J.C. Lambropoulos, S.D. Jacobs, B. Gillman, F. Yang and J. Ruckman, Ceramic Trans. Vol.82 (1998) 469-474. 7. P. Benjamin and C. Weaver, Proc. R. Soc. London. A254 (1960) 63-76. 8. K.L. Mittal, Adhesion Measurement of Thin Films, Thick Films and Bulk Coatings. ASTM., Philadelphia (1978) pp.5-107. 9. K.L. Mittal, Adhesion Measurement of Films and Coatings, (VSP BV, Netherlands 1995). 10. A.J. Perry, Thin Solid Films, 78 (1981) 77-93. 11. A.J. Perry, Thin Solid Films, 197 (1983) 167-18. 12. P.A. Steinmann, Y. Tardy and H.E. Hintermann, Thin Solid Films, 154 (1987) 333-349. 13. P.J. Burnet and D.S. Rickerby, Thin Solid Films 154 (1987) 403-416. 14. J. Schmutz and H.E. Hintermann, Surface Coatings Technol., 48 (1991) 1. 15. S.J. Bull and D.S. Rickerby, Surf. Coatings Technol., 42 (1990)149-164. 16. T.W. Wu, J. Mater. Res., 6 (1991) 407-426. 17. B. Bushan and B.K. Gupta, Adv. Info. Storage Sys. 6 (1995) 193-208.

Advanced Materials Research Vol. 9 (2005) pp 69-78 © (2005) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMR.9.69

Thermal Shock and Thermal Fatigue of Ferroelectric Thin Film due to Pulsed Laser Heating 1

X. J. Zheng 1,2, S. F. Deng 2, Y. C. Zhou 1,2 and N. Noda 3

Key Laboratory for Advanced Materials and Rheological Properties, Ministry of Education, Xiangtan University, Xiangtan, 411105, China 2 Faculty of Material & Photoelectronic Physics, Xiangtan University, Xiangtan, 411105, China 3 Department of Mechanical Engineering Shizuoka University Hamamatsu, Japan [email protected], [email protected], [email protected], [email protected] Keywords: Piezoelectric thin films; Thermal shock; Thermal fatigue; Pulsed laser irradiation; X-ray diffraction; Grain effect; Ferroelectric degradation; Crack

Abstract. Thermal shock and thermal fatigue of ferroelectric (FE) thin films were investigated by the pulsed laser tests. The power density was gradually increasing in the single pulsed laser heating test which simulated a thermal shock, the part melting threshold of Pb(Zr0.52Ti0.48)O3 (PZT) thin films was found by scanning electron microscopy (SEM). After thermal shock resulted the highest temperature below Curie point at the surface of PZT thin film, X-ray diffraction (XRD), SEM and RT66A standard ferroelectrics analyzer were used to study the microstructure, crystal grain sizes, and ferroelectric failure behavior. It was found that XRD peak of PZT thin film after laser beam heating was stronger than that before laser beam heating, crystal grain sizes decreased, and the ferroelectric properties were degraded. However there was no crack observed by SEM, until PZT thin films were melted. The fined grain effects on ferroelectric properties and XRD patterns of PZT thin film, depolarization due to the single pulsed laser heating were discussed respectively. The pulsed cycles with a certain power density were gradually increasing in the repetition pulsed laser heating test. It was interesting to find that the cracks will initiate and propagate due to the thermal fatigue induced by the repetition pulsed laser. The possible origins of the thermal fatigue cracks were also discussed. Introduction With the tendency of high integration and the development of microelectromechanical systems (MEMS), FE thin films have received considerable attention for their attractive physical properties including high dielectric, pyroelectric, piezoelectric and electro-optic properties. FE thin film material has developed rapidly in recent years due to its wide application [1,2]. However, as we know, it is natural that the film may fail due to the heating, electric load as well as mechanical load during service. It is well known that the thin films operating in many structural components, especially aerospace component, are ineluctably subjected to severe thermal loading which may be produced by aerodynamic heating, by laser irradiation, or by localized intense fire [3-5]. A significant amount of energy is delivered to a thin film surface in short time which heats the film to elevated temperature and thermal stress levels, when the thermal load is absorbed into the thin film and substrate [6,7]. In the reference [8], the thermal stress field in elastic half-space due to a single pulse laser for the general case of a mixed-mode structure beam was derived. A rise of Curie temperature [1] and a tendency to develop critical crack growth because of stress concentrations induced by mechanical and/or electric and/or thermal load, have a dominant influence on the failure of components [9,10]. Wang and Meguid proposed a piezoelectric actuator model to study piezoelectric thin film system [11]. Recently, Zheng and Zhou proposed a cylindrical symmetry model to investigate the thermopiezoelectric responses of a piezoelectric thin film deposited on a crystal MgO substrate due to a continuous laser [12]. It has long been recognized that mechanical, electrical and thermal fields are coupled in most of the physical problems. Because of their inherent complexity, relatively few solutions for such problems are available in the literatures.

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In this paper, the single and repetition pulsed lasers irradiated on PZT thin films produced by metal organic decomposition (MOD) were experimentally used to simulate thermal shock and thermal fatigue, respectively. After the single laser heating, XRD and RT66A standard ferroelectrics analyzer were respectively used to investigate the crystalline structure and ferroelectric degradation. The thresholds of part melted and crack growth for PZT thin films were obtained by SEM. The ferroelectric depolarization and the variation of XRD peaks were explained by the effects due to the pulsed laser heating, such as the fined grains, movement of polarized molecules, formation of initial oxygen vacancies defects, and element diffusions. On the other hand, the mechanism of thermal fatigue crack was discussed from the porous and micro-cracked natures of PZT thin film, stress relaxation, creep, and the interfacial tangent stresses varying oppositely in each pulsed cycle of the thermal fatigue test. The researches are significant for the life prediction and failure mechanism of a FE thin film system operating at the heating environment. Surface Profiles, Crystalline and Ferroelectric Properties in the Pulsed Laser Test FE thin films were prepared by MOD method. Firstly we got the precursor compounds which were Zr(OCH2CH2CH3)4, Ti(OCH2CH2CH3)4 and Pb(CH3COO)2· 3H2O, all in CH3COOH, CH3CH2CH2OH solution. The chemical ratio of metal ions Pb:Zr:Ti =1:52:48 were propitious to form the PZT solutions, because the sosoloid solid solution of binary system PbZrO3-PbTiO3 was of excellent electrical properties in the morphotropic phase boundary [13]. In order to prepare PZT thin films with uniform thickness, the solution molarity 0.4mol/L was determined via many tests. The pyrolyzing temperature 320°C and annealing temperature 700°C were determined by thermogravimetric (TG) and differential thermal analyses (DTA). PZT precursor compound was deposited on Pt/Ti/SiO2/Si(100) substrate using a spin coating operated at 3000 rpm for 30s. The substrate was prepared by sputtering 20 nm Ti and 30 nm Pt onto 2mm oxidized Si substrate. After spin coating, the wet films were roasted for 3-5 minutes on the hot plate in RTP-500 rapid thermal processor made in Beijing China at temperature of 320±2°C to pyrolyze the organic solvent. In this case, the dissolvent with a low boiling point could be volatilized and thus dry films could be obtained. The last step was thermal decomposition of the prepared film. The dry films were put into the processor at the temperature of 700oC for 3 minutes. The thickness of PZT thin films after the spin coating were calibrated by MHT-Ⅲ optical profiler WYKO (Veeco Instruments, USA), in which the perpendicular resolution is less than 0.1 nm. The samples with film thickness around 0.5µm were obtained by controlling the spin coating times. In order to measure the hysteresis loops, a set of Pt top electrode dots with 0.3 mm in diameter was deposited on the their surface by magnetron sputter deposition through a shadow mask. The top electrodes were arranged in order, and the distance between two centers was about 0.6~0.7mm. The poling direction was perpendicular to the isotropic plane of thin film. The pulsed laser tests were carried out using JM-HRI Nd: YAG laser (Institute of Mechanics, Chinese Academy of Sciences), in which the laser beams with Gaussian source have the energy ranged from 0-40 J (the relative error 5%), wavelength of 1.06µm, pulse width 1~8ms, and characteristic beam radius of 0.5~2mm. Before the pulsed laser with a certain characteristic radius irradiated on the samples, an irradiated sheet of white paper was used to determine the characteristic radius of laser beam. Because the energy calibration was completed before leaving factory, we can determine the laser power density in the tests by adjusting characteristic beam radius, pulse width and pulsed cycle time. In order to keep adiabatic condition, the pulsed lasers perpendicularly irradiated on the sample surface from a hole at top of the box, in which the asbestos was filled around the film as thermal barrier.

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(b) The branching micro-cracks

The micro-defects

Fig. 1 The typical surface profile of SEM images: (a) in good condition, and (b) some branching micro-cracks and micro-defects before the single pulsed laser heating. The samples labeled as S-1, S-2, S-3 and S-4 have no top electrode while that of S-T1 and S-T2 have Pt top electrodes. All of them were irradiated by the single laser pulsed with the laser parameters listed in Table 1. In the single pulsed laser test, 500 amplification ratio SEM images were used to characterize micro-structural features before the single pulsed laser heating. The typical surface profile was in good condition as shown in Fig. 1(a), and some branching micro-cracks and micro-defects were obviously observed on the sample surface by SEM image given in Fig. 1(b). After the single pulsed laser heating, the typical surface profiles of PZT thin films were recorded by SEM images given in Fig. 2 for the samples without top electrodes and Fig. 3 for that with top electrodes. When the power density was lower, no changes were observed on the surface profiles described as Fig. 2(a) and Fig. 3(a) for the samples S-2 and S-T1, respectively. From Fig. 2(b) and Fig. 3(b), the irradiated surface of S-3 and S-T2 were partly melted. The power density threshold corresponding to the part melting of S-3 and S-T2 thin films was about determined as 3.822×108W/m2. The irradiated surface of S-4 given in Fig. 2(c) was seriously melted. The repetition laser pulsed with the repetition frequency of 1 Hz and different cycles were used to irradiate PZT thin films with the top electrode, labeled as R-1, R-2 and R-3. The other laser parameters are also listed in Table 1. After the pulsed laser heating, some imprints and crescent cracks were observed on the samples irradiated, and the typical surfaces of R-1, R-3 were shown as Fig. 4.

(a)

(b)

The part melted

(c)

The completely melted crater

Fig. 2 Typical SEM micrograph of PZT thin film without the electrodes after the single pulsed lasers: (a) S-2 samples, (b) part melted for S-3, and (c) completely melted crater for S-4.

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(a)

(b)

The part melted

Fig. 3 SEM micrographs of PZT thin film with the electrodes after the single pulsed lasers: (a) S-T1 and (b) S-T2.

(a)

The imprint

(b)

The crescent cracks

Fig. 4 SEM micrographs of PZT thin film with the electrodes: (a) the imprint on the sample R-1 after 200 pulse cycles; (b) the imprint and crescent cracks on the sample R-3 after 1000 pulse The repetition laser pulsed with the repetition frequency of 1 Hz and different cycles were used to irradiate PZT thin films with the top electrode, labeled as R-1, R-2 and R-3. The other laser parameters are also listed in Table 1. After the pulsed laser heating, some imprints and crescent cracks were observed on the samples irradiated, and the typical surfaces of R-1 and R-3 were shown as Fig. 4. It was observed that there is an imprint with the characteristic radius 2mm on the surface of R-1 after 200 cycles pulsed, but no crack was observed (See Fig. 4(a)). A typical imprint and crescent cracks in the region of the characteristic radius were observed on the surface of R-3 after 1000 pulse cycles, with no crack observed outside of the radiating region of the laser beam (See Fig. 4(b)). A D500 X-ray diffractometer with the Cu-Ka radiation was used to analyze the effect of laser pulsed on the crystalline microstructure of the sample S-1. X-ray diffraction pattern of PZT thin films before and after the laser beam heating were given in Fig. 5. From the figure, X-ray diffraction peak of PZT thin film after laser beam heating is stronger than that before laser beam heating, but the preferential crystal orientation (111) of PZT thin films is not changed. In fact, it is very difficult to

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determine the crystal structure of PZT thin films near the morphotropic phase boundary (MPB). The splitting peaks such as (001)/{(100)(010)}, {(101)(011)}/(110), (002)/{(200)(020)}, and {(211)(121)}/(112) doublets are often observed on XRD patterns of tetragonal structure for PZT thin films. However, the peak splitting disappeared at the different orientations such as (100), (110), (002), and (112) on the XRD diagrams of the PZT thin films Zr/Ti=52/48. Similar to experimental results [14], this splitting can be described as simple peak splitting of the cubic structure appearing at a temperature higher than the Curie temperature. It is obvious that there is a lack of precision in the structural data obtained from PZT52/48 thin film diffractograms due to very low tetragonality of its structure hidden by the effect of size defects (peak broadening). Table 1 Parameters of laser pulsed irradiated on the samples Types Parameters

A single laser pulsed

A repetition laser pulsed

Samples S-1 S-2 S-3 S-4 S-T1 S-T2

R-1

R-2

200 1000

Pulsed cycles (times)

1

1

1

1

1

1

200

Pulsed width (ms)

4

4

4

4

4

4

6

Power density (108W/m2) Characteristic radius (mm)

0.497 2.229 3.822 7.693 0.497 3.822 2

0.5

0.5

0.4

2

0.5

8

R-3

6

0.398 0.388 0.398 2

2

2

After X-ray diffraction analyses, the sample surface was eroded by the diluted hydrochloric acid in order to observe the grain sizes and analyse the phase structures of PZT thin film. The representative SEM micrographs before and after the laser beam heating are shown in Fig. 6. In the figure, the density variation of the tiny white spots indicates the phase transition between the trigonal and the tetragonal phase, and the amount of each phase is close because of the ratio of Zr/Ti= 52/48 in the sosoloid solid solution of binary system PbZrO3-PbTiO3. However, one cannot identify which phase the tiny white spots are from the SEM micrographs. On the other hand, the decrease of crystal grain size after the laser beam heating is visible. This means the grain sizes in PZT thin film decrease due to the laser beam heating, the result is in good agreement with the conclusion [15]. The virtual ground mode of RT66 standard ferroelectrics analyzer made in Radiant Technologies Corporation of America was used to measure the hysteresis loops of PZT thin film S-T1 with a pulse waveform shown in Fig. 7(a). The first down pulse at the applied voltage -Vmax was only used to pole the heated samples to a minus remnant polarization (-Pr) state, and nothing was recorded during the process. After 1 second, the polarization change due to the following one-cycle triangle-wave was recorded, and the hysteresis loops were measured within one pulse cycle with pulse period 200ms. The central point of laser beam perpendicular to the surface of S-T1 was chosen as the measured point. Fig. 7(b) shows the hysteresis loops obtained by RT66 manual procedure at the applied voltage 6V for the sample S-T1 before and after laser beam heating. It is obvious that the coercive voltage ( Vc ) does not change, however, the remnant polarization (Pr) decreases after the laser beam heating.

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PZT(200) PZT(200)

Pt(111)

PZT(111)

PZT(110)

300

Pt(111)

PZT(111) PZT(110)

PZT(100)

600

PZT(100)

Intensity(arb.units)

900

PZT(112)

(b) After laser beam heating (a) Before laser beam heating

1200

(b)

PZT(112)

74

(a)

(a)

(b)

0 20

30

40

50

60

2theta(degree)

Fig. 5 XRD pattern of the sample S-1 before and after the single pulsed laser.

Fig. 6 Surface morphology of the sample S-2 after HCL erodent: (a) before and (b) after the single pulsed laser.

75 Before laser thermal shock After laser thermal shock

60

1s

-Vm ax

45

Polarization (µC/cm2)

+Vmax

30 15 0 -15 -30 -45 -60 -75

200ms (a)

-6

-4

-2

0

2

4

6

Applied Voltage (V)

(b)

Fig. 7 (a) Schematic of the signals used for hysteresis measurement, (b) Hysteresis loops at the applied voltage 6V before and after the single pulsed laser heating. Failure Due to Single Pulsed Laser Heating All the samples irradiated by a single pulsed laser show no cracks on the surface of PZT thin films. Why is there no crack observed on the surface of PZT thin film during the single laser pulsed heating? According the numerical results [12], the radial and circumferential stresses of the PZT thin film are compressive in the region of the laser beam and therefore it is impossible to observe any surface crack. The normal stresses are compressive at the interface of PZT thin film/substrate, so they do not result in the generation and propagation of the interfacial crack. There are some compressive radial and circumferential stresses due to the laser heating and the surface creep of PZT thin film, but they are too small to initiate an interfacial crack. Although the reverse tangent stresses at the interface of PZT thin film/substrate are propitious to the propagation of interfacial crack, the number of reverse is too small (only one times) to generate the interface crack due to a single laser pulsed.

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From Fig.5, Fig.6 and Fig. 7(b), the large grained film with the lower X-ray diffraction peak such as the “a” line in Fig.5 shows more steep polarization curve given in Fig. 7(b) before the laser beam heating. In Fig. 7(b), the effect of pulse laser beam heating on the ferroelectric properties of PZT thin films results in the decrease in the polarization of the films. First, the grain sizes in PZT thin film decrease due to the laser beam heating. The small grains result in the degradation of FE thin film, and large grained films show a more steep polarization curve [16-18]. Secondly, the movement of polarized molecules in PZT thin films increase because of the thermal energy of laser pulsed. The polarized molecules in order change into foul-up, therefore the remnant polarization of PZT thin films degrade after the laser beam heating. Fig. 8 is given to illustrate the depolarization of PZT thin films due to the pulsed laser heating. In the figure, the elliptical molecules with electric charge in the top pane are under the negative electrical field and become ordered as the middle pane in the first pole process. In the laser beam heating, the ordered pole molecules change into foul-up as shown in the bottom pane because of the polarization energy increased[19]. It is obvious that the poled molecules underwent the pole process and laser beam heating have lower polarization degree than that underwent the first pole process only. Thirdly, the pulsed laser induces the formation of initial oxygen vacancies defects [20] and the growth of micro-cracks due to the thermal mismatch between film and substrate in PZT thin film, and both of the defects and micro-cracks result in the degradation of the remnant polarization[21,22]. Fourthly, because of the constraint between the multi-layers structure PZT/Pt/TiO2/SiO2/Si(100), the original elastic deformation may be gradually replaced by the creep deformation, and the internal stress in PZT thin films will gradually decrease with the developing time. The stress relaxation due to the laser pulsed heating may lead to the change in the polarization, because the measurement of Roytburd shows that the change in the polarization of PZT FE films is proportional to internal stresses due to film-substrate misfit [23]. At last, the diffusions of the elements such as Pt, Si, O and Ti etc. due to the laser beam heating accelerate the chemical reaction on each interfaces of Pt/PZT/Pt/Ti/SiO2/Si(100), and the interface chemical reaction may also results in the decrease of remnant polarization [20]. It explains why the laser heating will result in the depolarization of PZT thin films (See Fig. 7(b)).

Un-poled Negative electrical field

Pulse laser beam heating

Negatively poled

Depolarized

Fig. 8 Illustration of the depolarization of PZT thin film caused by the laser heating. Mechanisms of Fatigue Crack Initiation and Propagation Sometime PZT thin films contain micro-crack networks. For example, there are some micro-defects and branch micro-cracks as shown in Fig. 1(b). Therefore, initiation of the cracks at the surface of

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thin film will be possible during thermal fatigue testing. The mechanisms of the crack initiation can be surface tensile-stress-induced cracking and interface shear-stress-induced cracking at the heating stage. Since the fatigued sample will promote both the coating surface creep and the surface compressive cracking, the accelerated crack initiation and higher surface crack density at the coating surfaces are expected. On the other hand, ceramic sintering and creep will occur under the given temperature and stress conditions. Due to the porous and micro-cracked nature of the ceramic coatings, the primary creep stage is often observed for these coatings, with the strain rate continuously decreasing with time[24]. This creep behavior is related to a stress-enhanced ceramic sintering phenomenon, as has been observed experimentally [25]. According the optical micrographs of the cross-sections of thermal barrier coatings after CO2 laser thermal fatigue testing, Zhu and Miller proposed a surface-wedging mode to account for the growth of high cycle fatigue cracks, which induced many lateral sub-cracks [26]. The crack branching phenomenon could result in multiple delaminations of the coating under the subsequent laser cycle fatigue loading. Zhu and Miller defined thermal high cycle fatigue (HCF) above 6×106 cycles and low cycle fatigue (LCF) below 3067 cycles, respectively [27]. They pointed out, at lower temperatures in LCF test, the relative boundary sliding of splats and grains, and the stress redistribution around the splats and micro-cracks are probably important mechanisms for ceramic creep deformation. The stress-dependent deformation will lead to coating shrinkage and thus stress relaxation at temperature under the compressive stresses. Finally, there is possibility of stress relaxation and creep in the multi-layers Pt/TiO2/SiO2, because they have the thermal expansion coefficients different from that of PZT thin films. When the film is irradiated at elevated temperature, strain due to misfit between the film and substrate would be developed, and there are four internal stresses in thin films, i.e., epitaxial stress, intrinsic stress, thermal stress, and transformation stress[17,28]. When the total internal stress (tensile stress or compressive stress) is higher than a certain value at a fixed temperature, the plastic deformation will be developed with the increase of time and the creep will affect the complicated crack form. Therefore, there are the typical crescent cracks shown in Fig. 4(b) in the region of the characteristic radius after 1000 pulse cycles, and no crack observed outside the radiating region of the laser beam. The branching micro-cracks and micro-defects shown in Fig. 1(b) may be one of the important mechanisms of fatigue crack initiation and propagation. Zheng and Zhou [12] proposed a theoretical model to simulate the thermopiezoelectric response of a piezoelectric thin film PZT-6B deposited on MgO (100) substrate induced by a continuous laser. According to the simulated results of thermo-electro- elastic fields in PZT thin film heated by a single laser beam, the higher temperature induced by Gaussian source at the center of laser beam result in the imprint within the region of the radiating laser, and the imprint region increases as the pulse cycles and temperature of PZT thin film increase. The typical imprint regions given in Fig. 4(a) and Fig. 4(b) change gradually into an ellipse because of the micro displacement induced by the action of pulsed laser with pulse cycles. On the analogy [12], the radial, circumferential and normal stresses should be compressive and the tangent stresses should be opposite in the radiating region of the single pulsed laser beam. The results [29,30] show that the compressive stress may cause the film delaminated from substrate and the tensile stress in thin film may cause surface crack. They indicated that there are interfacial tangent stresses varying oppositely in each pulsed cycle of the thermal fatigue test, and it will accelerate the crack propagation. Then, one can conclude that the interfacial tangent stress is one of the most important mechanisms of fatigue crack propagation. The thermal stress fields will result in the generation and propagation of the interfacial cracks in the radiating region of the laser beam, therefore the crescent cracks shown in Fig. 4(b) observed on the surface of PZT thin film indicate the interface delamination. Concluding Remarks In the paper, the single and repetition laser pulsed tests of PZT thin film were carry out by Nd:YAG

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laser with a wavelength of 1.06µm, and the failure phenomenon were investigated by SEM, WYKO optical profiler, XRD and RT66A standardized ferroelectric analyzer. The main conclusions can be listed as following as. (1) PZT thin films were melted when the temperature in the film exceeds to melting point. However there was no crack observed on the surface of the thin films during the single pulsed laser heating. (2) After a laser pulsed heating, the small grained films with the high X-ray diffraction peak have more smooth polarization curve because of the fined grain effect of PZT thin film due to the single pulsed laser heating. The depolarization of PZT thin film is depended on the fined grain effect, movement of polarized molecules, formation of initial oxygen vacancies defects, and element diffusions. (3) The cracks induced by the repetition pulsed laser beam can be observed on the surface of PZT thin films in the region of the characteristic radius after 1000 pulse cycles, and no crack observed outside of the radiating region. The initiation of thermal fatigue cracks is due to the porous and branching micro-cracked nature of PZT thin films, and the interfacial tangent stresses varying oppositely in each pulsed cycle of the thermal fatigue test will accelerate its propagation. Acknowledgements This work was supported by NNSF of China (No: 10472099) and Trains-Century Training Program Foundation for the Talents by the State Education Commission of China (No:[2002]48) , and Department Education of Hunan Province (No. 03B038). References 1. R. Kant: Trans. ASME J. Appl. Mech. Vol. 55 (1988), p. 93 2. G. H. Haertling: J Am Ceram Soc. Vol. 82 (1999), p.797 3. M. J. Blisset, P. A. Smith, J. A. Yeomans: J. Mater. Sci. Vol. 32 (1997), p.317 4. Y. Kagawa: Compos. Sci. Technol. Vol. 57 (1997), p. 607 5. Y. C. Zhou, Z. P. Duan, Q. B. Yang: Int. J. Non-Linear Mech. Vol. 33 (1998), p. 433 6. L. P. Welsh, J. A. Tuchman, I. P. Herman: J. Appl. Phys. Vol. 64 (1988), p. 6274 7. I. A. Volchenok, G. I. Rudin: J. Engng. Phys. Vol. 55 (1988), p. 1286 8. L. G. J. Hector, R. B. Hetnarski: Trans. ASME J. Appl. Mech. Vol. 63 (1996), p. 38 9. E. F. Crawley, E. H. Anderson: J Intelligent Mater. Sys. Struct. Vol. 1 (1990), p. 4 10. S. Im, S. N. Atluri: AIAA J. Vol. 27 (1989), p. 1801 11. X. D. Wang, S. A. Meguid: Int. J. Solids Struct. Vol. 37 (2000), p. 3231 12. X. J. Zheng, Y. C. Zhou, M. Z. Nin: Int. J. Solids Struct. Vol. 39 (2002), p. 3935 13. X. H. Du, J. Zheng, U. Belegundu, K. Uchino: Appl. Phys. Lett., Vol. 72 (1998), p. 2421 14. N. Floquet, J. Hector, P. Gaucher: J. Appl. Phys. Vol. 84 (1998), p. 3815 15. Y. Zhou, G. H. Wu: Measurement Technology of Materials analysis –X-Ray Diffraction and Electron Microscope (Harbin institute of technology press, Harbin 1998). 16. J. K. Yang, W. S. Kim, H. H. Park: Appl. Surf. Sci. Vol. 169-170 (2001), p. 544 17. Y. Sakashita, H. Segawa: J. Appl. Phys. Vol. 73 (1993), p. 7857 18. X. J. Zheng, Y. C. Zhou, H. Zhong: J. Mater. Res. Vol. 18 (2003), p. 578 19. M. Kikuchi, T. Yoshimura, K. Hoshino, H. Kokado: Jpn. J. Appl. Phys. Vol. 33 (1994), p. 4003 20. S. Pöykkö, D. J. Chadi: Phys. Rev. Lett. Vol. 83 (1999), p. 1231 21. S. H. Kim, D. J. Kim, S. K. Strerffer, A. I. Kingon: J. Mater. Res. Vol. 14 (1999), p. 2476 22. D. H. Bao, N. Wakiya, K. Shinozaki, N. Mizutani, X. Yao: J. Appl. Phys. Vol. 90 (2001), p. 506 23. A. L. Roytburd, S. P. Alpay, V. Nagarajan, C. S. Ganpule, S. Aggarwal, E. D. Williams, R. Ramesh: Phys. Rev. Lett. Vol. 85 (2000), p. 190

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24. B. P. Johnsen, T. A. Cruse, R. A. Miller: J. Eng. Mater. Technol. Vol. 117 (1995), p. 305 25. K. F. Wesling, D. F. Socie, B. Beardaley: J. Am. Ceram. Soc. Vol. 7 (1991), p. 1863 26. D. M. Zhu, R. A. Miller: Surf. Coat. Technol. Vol. 94-95 (1997), p. 94 27. D. M. Zhu, R. A. Miller: Mater. Sci. Eng. Vol. A245 (1998), p. 212 28. X. J. Zheng, J. Y. Li, Y. C. Zhou: Acta Mater. Vol. 52 (2004), p. 3313 29. J. W. Hutchinson, Z. Suo: Advances in Appl. Mech. Vol. 29 (1992), p. 63 30. A. G. Evans, J. W. Hutchinson: Acta Metall. Mater. Vol. 43 (1995), p. 2507

Advanced Materials Research Vol. 9 (2005) pp 79-86 © (2005) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMR.9.79

Fatigue Damage of Materials with Small Crack Calculated by the Ratio Method Under Cyclic Loading Yangui YU Wenzhou University, Chashan-campus, Wenzhou, 325035 Zhejiang, CHINA [email protected] Keywords: Small crack; ratio; material constant; damage rate, life.

Abstract This paper offers some new calculating equations on the small crack growth rate for describing the elastic-plastic behavior of materials under symmetric or un-symmetric cyclic loading. And it yet suggests the estimating formulas of the life relative to varied small crack size aoi at each loading history. The method is to adopt the ratio ∆ε p / ∆ε e by plastic strain range to elastic strain range as a stress-strain parameter, using some staple material parameters as the material constants in damage calculating expression. And it gives out a new concept of the composite material constant, that it is functional relation with each staple material constants, average stress，average strain and critical loading time. The calculated results are accordant with the Landgraf’s equation, so could avoid unnecessary fatigue tests and will be of practical significance to stint times, manpower and capitals, the convenience for engineering applications. Nomenclature D = localized damage parameter. Dmac= damage value corresponding to forming macro-crack size

D0 = baseline damage value corresponding to micro-crack forming size a0. a = small crack size. amac = macro-crack forming size. σm = localized average stress. ∆σ = stress range. ∆ε p = plastic strain range. ∆ε e =

elastic strain range.

σ'f = ε'f =

fatigue strength coefficient

fatigue ductility coefficient. b1' = fatigue strength exponent c1' = the fatigue ductility coefficient K’ = cyclic strength coefficient n’ = strain harden exponent. ΔH = damage stress factor range, ∆I = damage strain factor range, Introduction

Numerous scientists had suggested various kinds of the calculating expressions on fatigue damage of structures and materials, which include the Dowling's equation, the energy equation, the Landgraf's equation, the equation of local stress and strain and so on. These works have done valuable contributions for experimental research and engineering applications. The Landgraf's equation [1] is as follows:

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Macro-, Meso-, Micro- and Nano-Mechanics of Materials

1

σ 'f .∆ε p b1' − c1' 1 D= = 2( ' ) , ( for σ m = 0) . N ∆ ε f E ⋅ ∆ε e

(1a)

1

σ 'f ∆ε pσ 'f ' ' 1 D = = 2[ ] b1 −c1 , ( forσ m ≠ 0) ' N ε ' f E ⋅ ∆ε e (σ f − σ m )

. (1b) And the equation of localized stress and strain is in the following form, ∆σ p n1' ∆ε ∆σ e ) = +( 2 2E 2k ' . (2) The special feature of these equations is that they had all used the fatigue strength coefficient σ ' f , the fatigue strength exponent b1', the fatigue ductility coefficient ε ' f , the cyclic strength coefficient K’ and the strain harden exponent n’. And these material constants had always been accepted and applied extensively in each engineering domain. But these equations do not include the concrete physical parameters that showed about a structure damage (for instance, the damage parameter D, the micro-crack size a, the dislocation loops size λ). On the other hand, many other scientific workers have suggested the damage evolving equations in connection with the damage parameter D, a andλ in modern fatigue damage discipline. Li[2] provided the damage equation connected to the dislocation loops sizeλof a specimen (eq. (25)); And Murakami and Harada[3] and Margolin and Svechova[4] proposed the small crack initiation and propagation problems corresponding to the small crack size a, in eq (3) and (3), respectively:

da = B ( ∆σ ) m a dN .

(3) n

da / dN = C (∆ε p ) a

.

(4)

lg(da/dN) lg(dD/dN) O2 lg(σf -σm)/E,σ’f /E lgε’f Δσ/2 ,Δεp/ 2 A2 A1 C1 ΔH / 2, ΔI/2 3

1 2 ΔH / 2, ΔI/2 Δσ/2 ,Δεp/ 2

O1

2NT

B1

B

6 4 (for localizedσm=0) 5 (for localizedσm≠0) `

A3 C

A

O lg (Δσ/2 ,Δεp/ 2) lg (2Noi) Fig.1 Two direction curves

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Good qualities of these equations, there are damage parameters D, a, andλhaving specific physical meanings. However these equations include new material constants Af , B, C, m, n, and so on, which are not used by general material handbooks in the near future. Therefore the new equations have not been applied extensively in engineering now, because it must run many tests, until these material constants are quite reliable. In this way, it has to give out a great volume of manpower, materials and bankrolls; On the other hand, in this equations of a good many, they have been selected by way of the loading parameter only using single stress parameter ∆σor strain parameter ∆εp to calculate fatigue damage, so it is not comprehensive, neither is nice. The author of present paper studies and finds again and again, the new material parameters B, C, m and n with that six staple constants σ ' f , b1', ε ' f , c1', K’ and n’, there are function relationships. Therefore it has suggested more new small crack growth rate equations for describing elastic-plastic behavior of material, that they both adopt the above material constants used by engineering far-ranging application; and such damage parameter as small crack size aoi. And it is adopted as the loaded stress parameter to use the ratio of plastic strain range ∆εp to elastic strain range ∆εe. It is conceivable that works of author may avoid unnecessary fatigue tests and will be of practical significance to stint times, manpower and capitals. Analysis of the Small Crack Growth Rate Equations and Its Life Estimation Formulas Under Cyclic Loading

In order to use curves to explain the new damage evolving rate and relating life expression suggested by the ratio-method, here it gives the two directions double logarithmic coordinate system and two directions curves (Figure 1). It is well known that the materials for elastic strain component larger than plastic strain component ( ε e
ε p ), the relation between strain and life (Δ

εe / 2-2Noi) is presented by the negative direction coordinate with curve 1

( A1BA, for localizedσ

=0) and curve 2 (A2B1A3 for localizedσm≠0), and the relation between damage evolving rate and damage stress factor range dD/dN- ∆ H is presented by positive direction coordinate with curve 1 ( ABA1) and 2 (A3B1A2). Its equation is that [5,6,7] dD / dN = A1 • ∆H m1 = A1 (∆σ ) m1 D . (5) 1 / m1 where ΔH is defined as the damage stress factor range, ∆H = ∆σ • D , m1 = −1 / b'1 , and A1 is a composite constant of material; When the average stress is equal to 0 (curve 1), it becomes A1 = 2 ( 2σ ' f ) − m1 (ln D mac − ln D o ), (σ m = 0 ) . (6a) But for the average stress not equal to 0 (curve 2), it becomes A'1 = 2[2σ ' f (1 − σ m / σ ' f )]−m1 (ln Dmac − ln D0 ),( forσ m ≠ 0) . (6b) mac mac Here D is the damage value corresponding to macro-crack forming size a for a material of specimen, as amac =0.7-1.0 mm, Dmac =0.7-1.0 (for σ m＝0 , at point A1 or lgσf/E,; but for σ m ≠ 0 , m

at point A2 or lg(σf -σm)/E,). The D0 is a baseline damage value corresponding to micro-crack forming size a0. On the other hand, curves 3 and 6 (Fig.1) show material the behavior for plastic strain to be main component ( ε p
ε e ). The equation of damage evolving rate for describing positive direction curve 6BC1 is[5,6,7]

dD / dN = B1 • ∆I m ' 1 = B1 ∆ε p

m' 1

D.

(7)

where the ∆I is defined to be the damage strain factor range, ∆I = ∆ε p • D1 / m ' 1 , m ' 1 = −1 / c'1 , B1 is also a composite constant of material, when the average strain is equal to 0, it has the form B '1 = 2( 2ε ' f ) − m ' 1 (ln D mac − ln Do ), ( forε m = 0) .

(8a)

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Macro-, Meso-, Micro- and Nano-Mechanics of Materials

For the average strain not equal to 0, then it becomes '

(8b) B1' = 2[2ε ' f (1 − ε m / ε 'f )] − m1 (ln D mac − ln Do ), ( forε m ≠ 0) . mac But the D in the eq.(8a) and (8b) here is just a damage value corresponding to the point C1( lg ε f ). It should be noted that the small crack growth rate da/dN or the damage evolving rate dD/dN at the first stage (macro-crack forming stage) are all direct proportion-relation with the small crack size a or the damage variable D under a certain condition. It is a fact that is confirmed by a lot of experimental researches. Therefore, if the damage variable D in the eq. (5), (6), (7) and (8) are all substituted for the small crack size a1, then the eq. (5), (6), (7) and (8) should be as follows .

dD / dN = A'1 •∆H m1 = A1 ( ∆σ ) m1 a1 . A '1 = 2 ( 2σ ' f )

− m1

(9)

mac

− ln a o ), ( for σ m = 0 ) . A'1 = 2[2σ ' f (1 − σ m / σ ' f )]−m1 (ln a mac − ln a0 ), ( forσ m ≠ 0) .

(10b)

dD / dN = B'1 •∆I m ' 1 = B1∆ε p m ' 1 a1

(11)

B '1 = 2( 2ε ' f )

−m' 1

(ln a

(ln a

' 1

mac

' f

B = 2[2ε ' f (1 − ε m / ε )]

(10a)

.

− ln a o ), ( forε m = 0) . − m1'

(ln a

mac

(12a)

− ln a o ), ( forε m ≠ 0) .

(12b)

In this way the life N oi at varied history can be obtained by integral-method for eq. (9) and (11): ln a oi − ln a 0 , ( forσ m = 0) N oi = − m1 2(2σ ' f ) (ln a mac − ln a 0 )(∆σ ) m1 . (13a) N oi =

N oi = N oi =

ln a oi − ln a 0 2[2σ ' f (1 − σ m / σ ' f )]− m1 (ln a1

mac

− ln a 0 )(∆σ ) m1

ln a oi − ln a 0 2( 2ε ' f ) − m ' 1 (ln a mac − ln a 0 )(∆ε p ) m 1

, ( forσ m ≠ 0)

.

(13b)

, ( forε m = 0) .

ln a oi − ln a 0

ε mac 2[2ε ' f (1 − m' )]− m ' (ln a1 − ln a 0 )(∆ε p ) m ' εf 1

(14a)

, ( forε m ≠ 0) 1

.

(14b)

where the curve 1 (A1BA) and 2 (A2B1A3) presented by the negative direction coordinate system (Δε / 2-2Noi) in Fig.1 corresponds to the eq. (13a) and (13b); and the curve 3 (C1B6) corresponds the eq. (14a) and (14b),

respectively. The following means of expression (15) can be obtained from eq. (13) 1

1

− ln aoi − ln ao m1 m1 ∆σ = ∆ε e ⋅ E = 2σ ( ) ( 2 N ) ,( forσ m = 0) . oi mac ln a − ln ao ' f

1

1

− σ ln aoi − ln ao m ) (2 N oi ) m , ( forσ m ≠ 0) . ∆σ = ∆ε e E = 2σ ' f (1 − m' )( mac σ f ln a1 − ln ao 1

(15a)

1

(15b)

Similarly, the means of expression (16) can also be obtained from formula (14) 1

1

− ' ln a0i − ln ao m1' m1 ∆ε p = 2ε ' f ( ) ( 2 N ) , ( forε m = 0) oi ln a mac − ln ao .

(16a)

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1

83

1

− ε ln a oi − ln a o m m ∆ ε p = 2 ε ' f (1 − m )( ) ( 2 N ) , ( for ε m ≠ 0 ) oi ε ' f ln a1 mac − ln a o . (16b) It studied and proved again and again that: the ∆ε p / ∆ε e can be used in calculating damage. So the eq. (16) divided by the eq. (15) equals to the following eq.(17) ' 1

' 1

m1 − m1'

∆ε p

m1 − m1'

− ε ' f (1 − ε m / ε ' f ) ln aoi − ln ao m •m m ⋅m ∆ε p / ∆σ = = ' ( ) ( 2 ) , ( forσ m ≠ 0) N oi ∆ε e E σ f (1 − σ m / σ ' f ln a1mac − ln ao .

∆ε p / ∆σ =

∆ε p ∆ε e E

1

m ' − m1'

' 1

1

' 1

(17a)

( m1 − m1' )

− ε ' f ln aoi − ln ao m m (m m ) N ( ) ( 2 ) , ( forσ m = 0) oi mac ' σ f ln a − ln ao . ' 1 1

=

' 1 1

(17b)

Thus, small crack growth size (or damage value) per cycle is given: m m'

σ 'f ∆ε p m11−m11' ln a mac − ln a0 a mac 1mm = = 2( ' ⋅ ) , ( forσ m = 0) N oi N oi ln a oi − ln a 0 Eε f ∆σ m1 ⋅m1'

' f ' f

.

(18a)

m1 .m1'

σ − σ m − m1 −m1' ln aoi − ln ao ' a1mac 1mm = = 2[ ] (∆ε p / ∆σ ) m1 − m1 , ( forσ m ≠ 0) . mac N N oi ε −εm ln a1 − ln ao

(18b)

Whereupon, the life estimation expression (19) of each history is derived by ∆ε p / ∆ε e , which is for describing elastic-plastic behavior of a material. ln a oi − ln a 0

N oi = −

2( Eε ' f / σ ' f )

m1m '1

m1m '1

m1 − m '1

, ( forσ m andε m = 0)

(ln a mac − ln a 0 )(∆σ / ∆ε p ) m1 − m '1

.

ln a oi − ln a 0

N oi =

m1m '1

m1m '1

2[(σ ' f −σ m ) /(ε ' f −ε m )] m1 − m '1 (ln a1

mac

(19a)

, ( forσ m andε m ≠ 0)

− ln a 0 )(∆ε p / ∆σ ) m1 + m '1

.

(19b)

It is notable that the curve 4 (C1A) and 5 (C1C) (Fig. 1) of relation between strain and life presented by the negative direction coordinate system (Δε / 2-2Noi) corresponds the eq. (19). If assuming that −

C* = 2(ε ' f E / σ ' f )

m1m '1 m1 − m '1

(ln a mac − ln a0 ), ( forσ m andε m = 0) . m1m '1

C * = 2[(σ ' f −σ m ) /(ε ' f −ε m )]

m1 − m '1

(ln a1

mac

− ln a0 ), ( forσ m andε m ≠ 0) .

(20a) (20b)

Here it should be pointed out that the influence of average strain is lesser, so it may also be assumed that the ε m = 0 , m1m '1

C * = 2[(σ ' f −σ m ) / ε ' f ]

m1 − m '1

(ln a1

mac

− ln a0 ) .

(21)

Then the homologous equation of small crack growth rate is derived from the eq. (19) m1m '1

dD / dN = C * (∆ε p / ∆σ )

m1 − m1'

a1 .

(22)

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Macro-, Meso-, Micro- and Nano-Mechanics of Materials

And the curve 4(AC1) and 5 (CC1) in positive direction coordinate da/dN- ∆ εp / ∆ εe is also the one corresponding to the eq. (22). At this point, the eq. (22) is just the new equation of small crack growth rate that the present paper has propounded for describing elastic-plastic behavior of some metallic materials under symmetrical cyclic loading. It will be seen from this that if the varying course of the small crack aoi is not thought upon varied loading history, and the livelong process aoi= amac is only considered from the baseline damage value a0 to macro-crack forming for a component. Then, it can be assumed in eq. (23) ln aoi − ln a0 = ln a mac − ln a0 .

(23)

And according to m1 = −1 / b'1 , m' 1 = −1 / c'1 , and as amac =0.7-1.0 mm, Dmac =0.7-1.0, such, the equation (22) and the Landgraf's (1) will be equivalent and in full accord. Example

Figure 2. (a) Curve between nominal stress and time; (b) Curve between local stress and strain. Table 1. The contrasts of varied method to damage calculation σs σ’ f E K’ ' Calculate data

Material Rolled steel Range of nominal stress

[Mpa]

φ

[Mpa]

[Mpa]

n

320

0.67

192000

1125.9

0.193

Δso1 [Mpa] 395.5

Δs12 [Mpa] 699

Δs23 [Mpa] 521

b1

[Mpa]

935.9

-0.09 5

Δs14 [Mpa] 791

Calculating Calculation of stress and strain

Cyclic kind 2-3-2’ 5-6-5’ 1-4-7’

Number of equations (2) (24)

Δσ [Mpa] 780 520 910

Δε

σm

0.0122 0.0038 0.024

[Mpa] -21.7 3.8 3.3

By the Landgraf’s equation Calculation of damage

Cumulate damage Life, B, N Equation (1b), (19)

Cyclic kind 2-3-2’ 5-6-5’ 1-4-7’

number of Calculating results equations (1b) D1=1/N= 3.03*10-4 (1b) D2=1/N= 3.565*10-6 (1b) D3=1/N= 1.835*10-3 D = D1+ D2+D3 =2.142*10-3

Kσ

m1

ε’f

c’1

m’1

10.526

0.26

-0.07

2．22

Δs45 [Mpa] 434

Δs56 [Mpa] 240

2.6

Δs67 [Mpa] 656.7

results

εm

Δεp

Δεe

-0.0017 -0.0047 0

0.0082 0.0010 0.0183

0.0041 0.0027 0.0047

By the ratio-method-equation Number of Calculating results equations (22) (da1/dN)1=3.43*10-4[mm] (22) (d a1/dN)2=3.38*10-6 [mm] (22) (d a1/dN)3=1.801*10-3 [mm] dD/dN=(dD/dN)1+(dD/dN)2+(dD/dN)3=2.147*10-3[mm]

B = 1/D=466.98 (Number of cyclic loading segment) B = 1/D=465.7 ( Number of cyclic loading segment) N= h B. The h is a time of per cyclic loading segment.

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A part in a car is made using rolled steel. Its curves between nominal stress and time and between local stress and strain are as figure 2. When it is loaded, we can calculate the local stress and strain. Here it adopts the Neuber’s equation (24) and the equation (2) of cyclic stress-strain to calculate. ∆σ ⋅ ∆ε = K σ2 ( ∆s ) 2 / E .

(24)

Where the Δs is a range of the nominal stress, Kσis a effective stress concentration. E is an elastic modulus. On the other hand we can also calculate for its damage to figure 2 which is by the Landgraf’s equation (1b) and the ratio-method-equation (19b) and (22). Then the calculating results of varied method put in the table 1, so that it can compare expediently. It is noted that calculating results from the table 1 can also be seen that it is almost coincidental by the ratio-method–equations and the Landgraf’s equation. But calculating value by the ratio-method-equations there is a little error for ε m ≠ 0 , because it has considered the influence and calculated by eq. (20b) to composite material constant C*. Discussion


The peculiarities of these equations suggested in present document consist in: (1) The calculating damage rate and varied history life Noi may be yet converted into another physical parameters besides the calculating way mentioned above, it should and also could concretely describe the damage to a material, and its converting ways and means are simpler. It is suggested in reference [2] that there is a consanguineous relation between the damage parameter D and dislocation loop sizeλ D = 1− (

λ p. ) λo

(25)

where the λ0 is a diameter of the dislocation loop for which the damage value equals 0, and the p is a material constant. Due to the damage parameter D and the small crack size a1 at crack forming stage are all the variables of equivalent under certain condition, the damage parameters D in the equation (5) and (7) here must ordain that D0≡< v 2T , JH , v1 > , where

0 J = − I

I , 0 

(21)

that is, H is the Hamiltonian operator matrix in the symplectic space. Denoting the eigenvector functions ψ i , ψ j corresponding to the eigenvalues µ i , µ j , respectively, the symplectic ortho-normality relationship is proved by using the method given by paper [9,10].

< ψ iT , J , ψ j >= 0,

when µ i + µ j ≠ 0 .

(22)

Based on the ortho-normality relationship and expansion theorem, an arbitrary state vector v can be expanded as ∞

v = ∑ (c i ψ i + d i ψ − i ) ,

(23)

i =1

where c i , d i are coefficients to be determined. Equation (19) is a PDE set for transverse coordinate x . To obtain its solution we should first find the characteristic value λ  −µ   0  0 det   − Qλ2   − Qλ  0 

0

1

−µ

−λ νλ −µ Qλ 0 2 2 Q − (1 −ν )λ E 0 0 0

1Q 0 0 −µ 0 −1

0   2 E (1 + ν ) 0  0 E  = 0. 0 0   νλ  −µ −λ − µ  0

(24)

Expanding the determinant gives

(λ

2

[

]

+ µ 2 ) ⋅ (λ2 + µ 2 )(1 + ν ) − 2QE = 0 . 2

(25)

The roots of the Equation (25) are λ = ± µ i and λ = ± µ ' , where

µ ′ = 2 EQ (1 + ν ) − µ 2 .

(26)

Hence, the general solutions of the Equation (19) are given as q1 = Ap cos( µx) + B p sin( µx) + C p x sin( µx) + D p x cos( µx) + E p cosh( µ ′x) + Fp sinh( µ ′x) q2 = Aϕ x sin( µx) + Bϕ x cos( µx) + Cϕ x x cos( µx) + Dϕ x x sin( µx) + Eϕ x sinh( µ ′x) + Fϕ x cosh( µ ′x) q3 = Aϕ y cos( µx) + Bϕ y sin( µx) + Cϕ y x sin( µx) + Dϕ y x cos( µx) + Eϕ y cosh( µ ′x) + Fϕ y sinh( µ ′x) p1 = Aκ cos( µx) + Bκ sin( µx) + Cκ x sin( µx) + Dκ x cos( µx) + Eκ cosh( µ ′x) + Fκ sinh( µ ′x) p2 = Aγ sin( µx) + Bγ cos( µx) + Cγ x cos( µx) + Dγ x sin( µx) + Eγ sinh( µ ′x) + Fγ cosh( µ ′x)

,

(27)

p3 = Aε cos( µx) + Bε sin( µx) + Cε x sin( µx) + Dε x cos( µx) + Eε cosh( µ ′x) + Fε sinh( µ ′x) where all A, B, C , D, E , F are constants to be determined. The A, C , E group solutions give the symmetric deformations with respect to the x = 0 axis, and B, D, F group gives the

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Macro-, Meso-, Micro- and Nano-Mechanics of Materials

anti-symmetric deformations. The solutions corresponding to these coefficients are not all independent. Totally there are six independent coefficients. Substituting symmetric deformation A, C , E group solutions into (19) derives 2   Aκ = − (1 + ν ) Cγ  1 −ν   Aε = µ (1 + ν ) C r − Aγ   ( EQ − 2µ 2 )Cγ − EQµAγ , A =  p 3 Q ν µ ( 1 + )   E Aγ  Aϕ x = (1 + ν ) µ   ECγ − EµAγ  Aϕ y =  (1 + ν ) µ 2

 Cκ = 0   Cε = Cγ  ECγ  , C p = 2 ν µ ( 1 ) +  ECγ  Cϕ x = (1 + ν ) µ  C = ECγ  ϕ y (1 + ν ) µ

QEµ ′Eγ   Eκ = − EQ − µ 2 (1 + ν )  µµ ′(1 + ν ) Eγ  E = ε  EQ − µ 2 (1 + ν )   . E p = 0  EµEγ   Eϕ x = − EQ − µ 2 (1 + ν )  Eµ ′Eγ E = ϕ  y EQ − µ 2 (1 + ν )

(28)

Only three coefficients Aγ , Cγ , Eγ are independent for symmetric deformation. For anti-symmetric deformation group B, D, F , only three coefficients are independent. With the length limited, they are not listed here. Substituting (27), (28) into the lateral boundary conditions derives the transcendental non-zero eigenvalue equation. Both lateral edge free

The boundary conditions with both lateral edge free can be given as p = 0, ϕx = 0, ϕ y = 0,

when x = −a ,

(29a)

~ ~ ~ ~ ~ p = C 0 + C 1 x + C 2 y , ϕ x = C1 , ϕ y = C 2 ,

when x = a .

(29b)

In equation (29b), the non-homogeneous terms appear, and should be solved first. ~ ~ ~ a) Let C1 = 1 , C 0 = C 2 = 0 , the equation is Hψ 0( 0) = 0 , the solution is obtained as

{

}

T

v 0( 0) = ψ 0( 0) = ( x + a ) 2 4a , ( x + a ) 2a , 0, 0, 0, − ν 2 Ea .

(30)

The corresponding displacement field is

u = − νx 2 Ea , v = y 2 Ea .

(31)

The corresponding stresses are m xz = m yz = 0 , σ x = 0 , σ y = 1 2a , τ xy = τ yx = 0 .

(32)

Obviously, it is a simple tension solution along y . ~ ~ ~ b) For the case of C 0 = 1 , C1 = 0, C 2 = 0 , the solution is found as

{

}

T

v 0(1) = ψ 0(1) = − EQRx3 + 3R( EQa 2 + 2) x + 0.5, 3EQR(a 2 − x 2 ), 0, 0, 0, 6QRνx .

(33)

The corresponding displacement field is

u = 3QR( y 2 + νx 2 ) ， v = −6QRxy . The stresses are

(34)

Advanced Materials Research Vol. 9

m yz = −6 R , m xz = 0 , σ x = 0 , σ y = −6 EQRx , τ xy = τ yx = 0 ,

149

(35)

where

R = 1 4a ( EQa 2 + 3) .

(36)

Obviously, this is a constant couple stress solution. ~ ~ ~ c) The case of C 2 = 1 , C1 = 0, C 0 = 0 . Because of the multiplier y , the solution looked like a Jordan form. Solve the equation Hψ 0( 2) = ψ 0(1) , the solution is found as

ψ 0( 2)

0     0   6 Rmt sinh (mx) − EQRx 3 + 3EQRa 2 x + 0.5 = , 6 QR [ x − mt sinh ( mx )]     0   2 2  6QRt cosh (mx) − 3QR(1 + ν )( x − a ) 

(37)

where

m = 2QE (1 + ν ) , t = a m sinh(ma) .

(38)

and the solution of original PDE (17) is given as v 0( 2) = ψ 0( 2) + yψ 0(1) . The corresponding displacement field is given as

u = QRy[3νx 2 + y 2 − 3a 2 (1 + ν )] , v = QR[−3xy 2 + x 3 (2 + ν ) − 3a 2 (1 + ν ) x − 12 t sinh(mx) m] .

(39)

The stresses are m yz = −6 Ry , m xz = 6 R[ x − mt sinh(mx)] , σ x = 0 , σ y = −6 EQRxy ,

τ xy = 3EQR( x 2 − a 2 ) , τ yx = −6 Rm 2 t cosh(mx) + 3EQR( x 2 − a 2 ) .

(40)

Having found the inhomogeneous special solutions, the homogeneous eigensolution vector is to be found below. The homogeneous boundary conditions for both lateral edges free are p = 0, ϕ x = 0, ϕ y = 0 ,

when x = ± a .

(41)

The solutions of Hψ ( x) = µψ ( x) with condition (41) have only non-zero eigenvalues, which can be classified into two groups, symmetric and anti-symmetric with respect to the y axis. We come to talk about the symmetric eigensolutions. When A,C , E group in equation (27) is substituted into the boundary conditions (41), we obtain a set of homogeneous simultaneous equations about Aγ , Cγ , E γ . Since Aγ , Cγ , E γ in the equations are non-zero, the coefficents determinant must equal zero, which derives to the transcendental equation as

sin(2µa) + 2µa +

4 µ 3 cos 2 ( µa) tanh( µ ′a ) 2µ 2 sin(2µa) − = 0. QEµ ′ QE

After solving eigenvalue µ n , the coefficients are obtained

(42)

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Macro-, Meso-, Micro- and Nano-Mechanics of Materials

2 2 [QE − µ n (1 + ν )][sin(µ n a) Aγ + µ n a cos( µ n a)] µ n a sin( µ n a) 2µ n Cγ = µ n , Aγ = +1− , Eγ = . (43) 2 cos( µ n a ) QE (1 + ν ) µ n sinh( µ 'n a)

The corresponding symmetric deformation eigensolution is    2 E E E Cγ − − + cos( ) A x Cγ x sin( µ n x), µ   γ n 3 2 2 µ n (1 + ν )Q  (1 + ν ) µ n (1 + ν ) µ n  (1 + ν ) µ n   Eµ n E E  Aγ sin( µ n x) + Cγ x cos( µ n x) − Eγ sinh( µ 'n x) 2  (1 + ν ) µ n (1 + ν ) µ n EQ − µ n (1 + ν )    Eµ 'n ⋅ cosh( µ 'n x) E E E Cγ − Aγ  cos( µ n x) + Cγ x sin( µ n x) + Eγ  2 2 ψ n =  (1 + ν ) µ n . (44) (1 + ν ) µ n  (1 + ν ) µ n EQ − µ n (1 + ν )  QEµ 'n − 2 C cos( µ x) − Eγ cosh( µ 'n x) γ n 2  (1 + ν ) EQ − µ n (1 + ν )   Aγ sin( µ n x) + Cγ x cos( µ n x) + Eγ sinh( µ 'n x)   1 −ν C − A  cos( µ x) + C x sin( µ x) + µ n µ 'n (1 + ν ) E cosh( µ ' x) γ  γ γ n n n 2  µ (1 + ν ) r EQ − µ n (1 + ν )   n

From this eigensolution ψ n (x) , the solution of original PDE (17) is

v n = ψ n exp(µ n y ) ,

(n = ±1, ± 2,±3......) .

(45)

Similarly, the anti-symmetric transcendental equation and the eigensolution can be obtained. With the length limited, they are not listed here. A numerical example is given below. A long strip domain with both lateral edges free is under uniform tension at the far end y → ∞ . It is fixed at the end y = 0 , where the normal stress distribution is required. Solution: The deformation is obviously symmetric. Hence the eigenfunction expansion is composed of the solution (30) and the symmetric deformation non-zero eigensolutions. Only the Re( µ n ) < 0 class eigensolutions are needed. ∞

v = f 0 v 0( 0) + ∑ f n exp( µ n y )ψ n .

(46)

n =1

The solution of equation (46) satisfies the PDEs in domain and the boundary conditions at lateral edges x = ± a . f 0 is determined by the boundary condition at the far end y → ∞ . The boundary condition at y = 0 is used to identify f i , (i = 1,2,...) . In computation, only the former k terms in (46) are used, and the variational equation of the boundary condition at y = 0 is

∫

a

−a

[κ xz δp + (− γ xy 2)δϕ x + ε x δϕ y ]

dx = 0 .

(47)

y =0

Since complex eigenvalues appear, the computation is transformed as real equations. Note that EQ = (1 + ν ) / L2 , and in the example it is selected as L = 1 × 10 −6 m ， E = 2 × 10 9 N/m 2 ，ν = 0.3 . The eigenvalue equation is derived as sin(2µa) + 2 µa + [4( µa) 3 cos 2 ( µa) tanh( µ ′a) /( µ ′a) − 2( µa) 2 sin( 2µa)] ( EQa 2 ) = 0 .

(48)

The solution of eigen-root transcendental equation has quite a number of complex valued roots, in finding which the Newton method is used and the complex roots of the corresponding classical plane stress problem [9] are used as the initial values. Using the contour integration method in

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151

theory of complex variables function, no complex root loses. The real valued eigen-roots appear only when µ 2 > 2 EQ (1 + ν ) , and are easier to obtain, the equation is f ( µa ) = sin(2 µa ) + 2 µa + [4( µa ) 3 cos 2 ( µa ) tan( µ c a ) /( µ c a ) − 2( µa ) 2 sin(2 µa )] ( EQa 2 ) = 0

(49)

2.2

2.2

2

2

1.8

1.8

1.6

1.6 MPa

MPa

with µ c = µ 2 − 2 EQ (1 + ν ) . Equation (49) has a number of singular points as µ c a = (i + 1 / 2) ⋅ π . The function f (µa) has eigen-roots near by these singular points, and can be solved by the Newton method. For ν = 0.3 , k = 20 , a = E −4 , E −5 m etc., the normal stress distributions are plotted in Fig. 3, where the far end driven stress is σ y = 1 MPa.

1.4

1.4

1.2

1.2

1

1

0.8 0

0.2

0.4

0.6

0.8

0.8 0

1

m

0.2

0.4

0.6

0.8

m

-4

x 10

2.2

2.2

2

2

1.8

1.8

1.6

1.6

1 -5

x 10

MPa

MPa

k=20 k=25

1.4

1.4

1.2

1.2

1

1

0.8 0

1

2

3

4 m

5

6

7

0.8 0

8

1

2

3

4

5

m

-6

x 10

2.2

2.2 k=20 k=25

2

2

1.8

1.8

1.6

1.6 MPa

MPa

k=20 k=25

1.4

1.4

1.2

1.2

1

1

0.8 0

6 -6

x 10

1

2

3 m

4

5

0.8 0

1

-6

x 10

2 m

3

4 -6

x 10

Fig. 3 Stress distributions of different sizes From the plots in Fig. 3, we can see that when the width is larger than a = 1 × 10 −5 m , the stress is far greater than 1 MPa at the corner. The stress at the corner has obvious singularity. It is seen that with the decreasing of strip width a , the distribution of normal stress at the fixed end changes noticeably, especially within the size range of a = 1×10 −5 m to a = 1×10 −6 m. Then new phenomenon appears: The normal stress distribution at the central point is less different from that at

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Macro-, Meso-, Micro- and Nano-Mechanics of Materials

the corner, and the stress concentration at the corner tends to disappear. When a = 5 × 10 −6 m, it is clear, the normal stress is a finite value, which given by the classical theory is infinity. It seems that in the interior of the plate the stress distribution is really as given by the classical theory of elasticity, but in the vicinage of the plate corner, the effect of couple stress becomes significant. The fluctuation in the stress distribution is as usually the case of Fourier series expansion and it is unavoidable. Concluding remarks

Based on the analogy relationship between plane couple stress and Reissner plate bending theory, the plane couple stress analysis theory is introduced into Hamiltonian system. Analytical approach is used to solve a simple strip extension problem with both lateral sides free, which verifies the effectiveness of symplectic solution system. The solution of plane couple stress theory eliminates the stress singularity at the corner of a strip extension. In fracture mechanics, the classical elasticity model gives also stress singularity at the crack tip. It is believed that the application of the couple stress theory will be very important for the stress singularity problem. References

1. A.R. Boresi and Ken.P. Chong, 1999, Elasticity in Engineering Mechanics，appendix 5A. 2nd ed. J. Wiley & Sons. 2. R.A. Toupin, Arch. Ration. Mech. Anal. Vol. 11 (1962), 385-414 3. R.A. Toupin, Arch. Ration. Mech. Anal., Vol. 17 (1964), 85-112. 4. R.D. Mindlin and H.F. Tiersten, Arch. Ration. Mech. Anal., Vol. 11 (1962), 415-448. 5. Koiter, Proc. K.Ned.Akad. 67 (1964), 17-44 6. L. Zhang, Y. Huang, J.Y. Chen and K.C. Hwang, Int. J. Fracture, Vol. 92 (1998), 325-348. 7. Y. Huang, J.Y. Chen, T.F. Guo, L. Zhang and K.C. Hwang, Int. J. Fracture, Vol. 100 (1999), 1-27 8. P. Tong, D.C.C. Lam, F. Yang, A.C.M. Chong and J.Wang, Solid Mechanics Conference (2002). 9. W. Yao and W.Z. Zhong, Symplectic Elasticity, Beijing: Higher Education Press, 2002 (in Chinese) 10. W.Z. Zhong, A new systematic methodology for theory of elasticity，Dalian: Dalian University of Technology press, 1995 (in Chinese) 11. W.Z. Zhong, W. Yao and C.L. Zheng, J. Dalian Univ. Tech. Vol. 42(2002), 519-521 (in Chinese) 12. W. Yao and Y.F. Sui, Appl. Math. Mech. 25 (2004), 178-185. 13. F. Yang, A.C.M. Chong, D.C.C. Lam and P. Tong, Int. J. Solids Struct,, Vol. 39 (2002), 2731-2743

Advanced Materials Research Vol. 9 (2005) pp 153-162 © (2005) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMR.9.153

Multi-scale and Finite Element Analysis of the Mixed Boundary Value Problem in a Perforated Domain under Coupled Thermoelasticity Yongping Feng1 and Junzhi Cui1,2 1

School of Mathematics and Information Sciences, Guangzhou University, Guangzhou, Academy of Mathematics and System Sciences, Academic Sinica, P. O. Box 2719, 100080 1 [email protected] Beijing, P. R. China

2

Keywords: Thermoelasticity, Multi-scale, Finite element computation.

Abstract: The two-scale asymptotic expression and error estimations based on two-scale analysis (TSA) are presented for the solution of the increment of temperature and the displacement of a composite structure with small periodic configurations under coupled thermoelasticity condition in a perforated domain. The two-scale coupled relation between the increment of temperature and displacement is established.The multi-scale finite element algorithms corresponding to TSA are described and numerical results are presented. Introduction The analysis problems of composite structures containing small periodic configurations are often encountered in the development of new products. These structures often consist of materials whose property varies sharply within a very small periodicity, as shown in Fig 1. The mechanical behavior of these periodic structures can be studied with the help of the homogenization method. In the processes of this method, two approaches have been applied: the energy method, and the two-scale asymptotic expansion method. The homogenization method has been popularly used to study the analytical problems of composite structures, (see e.g [1,7,8]). In these studies, however, the mechanical behavior of composite structures were analyzed based on the application and the effects of coupled thermoelasticity on the perforated domain have not been considered previously. In this paper, the multi-scale asymptotic expansion is taken into account for the problem in a perforated domain under coupled thermoelasticity.

Fig 1. Perforated domain Ω ε , whole domain Ω , respectively in 2-dimension Let ω be an unbounded domain of ℜ n with 1-periodic structures. ω is invariant under the shift by any Z = ( z1 ,..., z n ) ∈ Z n . In this paper, we use the notation: Q = {x : 0 < x j < 1, j = 1,..., n}, εG = {x : ε −1 x ∈ G} , z + G = {x : x = z + y, y ∈ G}, ξ = x / ε , where ε is a small parameter. Special focus is made of a perforated domain Ω ε . Ω ε is a perforated domain of type I, satisfying the condition in [7] ， also satisfying the Lipschitz condition ， ∂Ω = Γε ∪ S ε ， where Γε = ∂Ω ∩ εω ， S ε = (∂Ω ε ) ∩ Ω . Let ω be an unbounded domain of ℜ n with a 1-periodic structures. let S ε0 = ( S ε ) ∩ (ω ∩ Q) and Vω ∩Q = {φ | φ ∈ H 1 (ω ∩ Q),φ is 1 - periodic} ，V is the quotient space of Vω ∩Q by the equivalence relation φ1 ≅ φ 2 ⇔ φ1 − φ 2 = const ; For simplicity we let

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Macro-, Meso-, Micro- and Nano-Mechanics of Materials

V 0 = {φ | φ ∈ Vω ∩Q ,

∫ ω

φ dξ = 0}

In section 2, a new kind of multi-scale asymptotic∩Qexpression of the displacement solution for the composite structure of with a basic configuration is presented, meanwhile, the approximation solution and error estimation is given in section 3. The FE computation, numerical samples and simple conclusion are presented in section 4, section 5 and section 6, respectively. Results When the coupled thermoelasticity is used, the analytical equations for the perforated structure of composite materials with small periodic configuration are given ([5,6,8]).  ∂  ε ∂θ ε  +h = 0  k ij in Ω ε    x x ∂ ∂  i j   ∂ ∂ ε ε  aijhk aijhk bhkε θ ε + f i ( x ) = 0, i = 1,..., n , in Ω ε ε hk (u ε ( x )) −  ∂x j  ∂x j  ε θ ε = T 0 (x) Γ ; σ (θ ε ) = ν k ε ∂ θ = 0 on S i ij ε ε ε  ∂x j  ε ε ε ε ε ε ε u (x) = u 0 (x) Γε ; σ ε (u ) = ν j aijhk ε hk (u ( x )) − ν j aijhk bhk θ = 0 on S ε

(

)

(

)

(1)

where u ε ( x) and θ ε ( x) denote the displacement vector and the increment of the temperature as ε compared with the reference temperature, respectively; aijhk bhkε and k ijε are the stiffness, thermal-expansion and thermal conductivity of materials ， respectively. These factors satisfy P (k1 , k 2 ) and Ε(λ1 , λ 2 ) conditions given in [7]. f (x) and h(x) represent the internal force and internal heat sources, respectively. ε hk (u ε ( x )) is the strain evaluated from displacement u ε (x) . It is assumed that f , h, T 0 ( x) and u 0 ( x) are sufficiently smooth functions. Similar to previous work in [1,2,4,7], we assume that the solution θ ε (x) and u ε (x) be the functions of x, ξ and ε at the same time; θ ε (x) and u ε (x) are given by: ∞  ε 0 l l 0 θ ( x) = θ ( x) + ∑ ε ∑ H α (ξ ) Dα θ ( x ),  = l l =1  ∞ ∞ u ε ( x ) = u 0 ( x ) + ε l N α (ξ ) Dαl u 0 ( x ) + ∑ ε l +1 ∑ M α (ξ ) Dαl θ 0 ( x ) ∑ ∑  l =1 = l l =0 =l

(2)

where(1) α = (α 1 ,..., α n ), < α >=| α 1 + ... + α l |, α j = 1,..., n, j = 1,..., l , H α (ξ ), M α (ξ ) and N α (ξ ) are

the 1-periodic scalar and vector functions defined on ω ∩ Q ，respectively.

Substituting Eq (2) into Eq (1), and comparing the coefficients of powers ε l (l = −1,0,1,...) on both sides of Eq (1) (including the boundary conditions), a series of identities are obtained, Due to the limitation of space, these identities are not presented here. From these identities, for < α >= l = 1 , H α1 (ξ ), M 0 (ξ ) and N α1m (ξ ) are determined using the following equation on ω ∩ Q ：  ∂  ∂H α1 (ξ )  ∂ in ω ∩ Q   kij (ξ )  = − ( k (ξ ) ) , ξ ξ ξi ij ∂ ∂ ∂  i j   ∂H α1  on Sε0 = −ν i kiα1 , σ ( H α1 ) = ν i kij ∂ξ j   is 1-periodic in ξ , ∫ H α1 (ξ )d ξ = 0  H α1 (ξ ) ω ∩Q  

(3)

Advanced Materials Research Vol. 9

∂  ∂  ∂ξ (aijhk (ξ )ε hk (M 0 (ξ ))) = ∂ξ (aijhk (ξ )bhk (ξ ) ), j  j  = = ν j aijhk (ξ )bhk (ξ ), σ M ξ ν a ε M ξ ( ( )) ( ( ))  j ijhk hk 0 0  M 0 (ξ ) is 1 - periodic in ξ , ∫ M 0 (ξ )dξ = 0 ω ∩Q 

in ω ∩ Q

)

(

(4)

on Sε0

∂  ∂  ∂ξ aijhk (ξ )ε hk (Nα 1 m (ξ )) = − ∂ξ aijα1m (ξ ) , j j   σ (Nα 1 m (ξ )) = ν j aijhkε hk (Nα 1 m (ξ )) = −ν j aijα1m (ξ ),  Nα 1 m (ξ ) is 1 - periodic in ξ , ∫ Nα 1 m (ξ )dξ = 0 ω ∩Q 

(

155

)

in ω ∩ Q

(5)

on Sε0

For < α >= l = 2 , H α1α 2 (ξ ), M α1 (ξ ) and N α1α 2 m (ξ ) are determined by equations:  ∂  ∂H (ξ )  ∂H  = − ∂ kiα (ξ ) H α (ξ ) + kˆα α − kα j (ξ ) α 1 − kα α ,  kij (ξ ) α1α 2  2 1 1 2 2 1 2   ξ ξ ξ ∂ ∂ ∂ ∂ξ j  i j i   ∂H α1α 2  on Sε0 = −ν i kiα1 Hα 2 , σ ( H α1α 2 ) = ν i kij ∂ξ j    H α1α 2 (ξ ) is 1 - periodic in ξ , ∫ H α1α 2 (ξ )dξ = 0 ω ∩Q  

(

)

∂  ∂  ∂ξ aijhk (ξ )ε hk (Mα 1 (ξ )) = ∂ξ aijhk (ξ )bhk (ξ ) Hα 1 + aiα1hk bhk − j j   ∂ aijα1k (ξ ) M 0 k − aiα1hk ε hk (M 0 (ξ )) + bˆiα1 , in ω ∩ Q   ∂ξ j  σ (Mα 1 (ξ )) = ν j aijhk ε hk (Mα 1 (ξ )) = ν j aijhk (ξ )bhk (ξ ) Hα1 − ν j aijα1k (ξ ) M 0 k ,  Mα 1 (ξ ) is 1 - periodic in ξ , ∫ Mα 1 (ξ )dξ = 0 ω ∩Q 

(

)

(

(

in ω ∩ Q

(6)

)

(7)

)

on Sε0

∂  ∂  ∂ξ aijhk (ξ )ε hk (Nα 1α 2 m (ξ )) = − ∂ξ aijα1m (ξ ) Nα 2 hm + aˆiα 2 mα1 − aiα 2 mα1 − aiα 2 hk ε hk (Nα 1 m ), in ω ∩ Q j  j  ( ( )) ( ( )) on Sε0 N a N σ ξ ν ε ξ = = −ν j aijhα1 (ξ ) Nα 2 hm ,  α 1α 2 m j ijhk hk α 1α 2 m  Nα1α 2 m (ξ ) is 1 - periodic in ξ , ∫ Nα1α 2 m (ξ )dξ = 0 ω ∩Q 

(

)

(

)

< α >= l ≥ 3 ， H α1 ...α l (ξ ), M α1 ...α (ξ ) and

For

l −1

(8)

N α1 ...α l m (ξ ) are determined by equations,

respectively,  ∂  (ξ )  ∂H ∂H  = − ∂ k iα (ξ ) H α ...α (ξ ) − kα j (ξ ) α1 ...α l −1 − kα α H α ...α ,  k ij (ξ ) α1 ...α l  l l −1 l l 1 1 2 3   ∂ξ i ∂ξ j ∂ξ j  ∂ξ i    ∂H α1 ...α l  on S ε0 = −ν i k iα1 H α 2 ...α l , σ ( H α1 ...α l ) = ν i k ij ξ ∂ j   H ( ξ ) is 1 periodic in ξ , ∫ H α1 ...α l (ξ )dξ = 0  α1 ...α l ω ∩Q  

(

(

)

)

(

in ω ∩ Q

)

∂  ∂  ∂ξ aijhk (ξ )ε hk (M α1 ...α l −1 (ξ )) = ∂ξ aijhk (ξ )bhk (ξ ) H α1 ...α l −1 + aiα l −1hk bhk H α1 ...α l −2 − j  j  ∂ in ω ∩ Q aijα1k (ξ ) M α 2 ...α l −1k − aiα1hk ε hk (M α 2 ...α l −1 (ξ )) − aiα l −1hα l − 2 M α1 ...α l −3h (ξ ),   ∂ξ j  0 σ (M α1 ...α l −1 (ξ )) = ν j aijhk ε hk (M α1 ...α l −1 (ξ )) = ν j aijhk (ξ )bhk (ξ ) H α1 ...α l −1 −ν j aijα1k (ξ ) M α 2 ...α l −1k , on S ε  M α1 ...α l −1 (ξ ) is 1 - periodic in ξ , ∫ M α1 ...α l −1 (ξ )dξ = 0 ω ∩Q 

(

)

(9)

(10)

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Macro-, Meso-, Micro- and Nano-Mechanics of Materials

∂  ∂  ∂ξ aijhk (ξ )ε hk (Nα 1 ...α l m (ξ )) = − ∂ξ aijα 1m (ξ ) Nα 2 ...α l hm − aiα 2 hα 1 Nα 3 ...α l hm − aiα 1hk ε hk (Nα 2 ...α l m ), in ω ∩ Q j  j  σ N ξ ν a ε N ξ ( ( )) ( ( )) = −ν j aijhα 1 (ξ ) Nα 2 ...α l hm , on Sε0 =  α 1 ...α l m α 1 ...α l m j ijhk hk  Nα 1 ...α l m (ξ ) is 1 - periodic in ξ , ∫ Nα 1 ...α l m (ξ )dξ = 0 ω ∩Q 

(

)

(

)

(11)

Comparing the coefficients of powers ε 0 on both sides of Eq (1), we obtain the following identities:  ∂H α1α 2 (ξ )  ∂ 2θ 0 ( x) ∂  ∂θ 0 ( x)  ∂   k ij (ξ ) + + k ij (ξ )  ∂x ∂x ∂x i  ∂x j  ∂ξ i  ∂x i   α1 α 2  ∂H α1  ∂ 2θ 0 ( x )  ∂ ∂ 2θ 0 ( x )   = h, + k ij (ξ ) k ij (ξ ) H α1 (ξ )  ∂xα1 ∂x j  ∂ξ i  ∂xα1 ∂x j  ∂ξ i   ∂H α1α 2 (ξ ) ∂ 2θ 0 ( x) ∂ 2θ 0 ( x)    = 0, + H α1 (ξ ) υ i k ij (ξ ) ∂ξ j ∂xα1 ∂xα 2 ∂xα1 ∂x j   

(

)

in

Ωε

on ∂S ε

2 0 2 0  ∂ 2 u m0 ∂ 2 u 0 ( x) 1 ∂ 1  ∂ u h ( x) ∂ u k ( x)  a ijhk  (a ijhk N α1hm ) + + + a ijhk ε hk N α1 (ξ ) +   2  ∂x k ∂x j ∂x k ∂xα1 ∂x h ∂x j  ∂xα1 ∂x j 2 ∂ξ j   ∂ 2 u m0 ∂ 2 u 0 ( x) 1 ∂ ∂  2 ∂ξ (a ijhk N α1km ) ∂x ∂x + ∂ξ a ijhk ε hk N α1α 2 (ξ ) ∂x ∂x + j h α1 j α1 α 2   0 0 0 a ijhk ε hk (M 0 (ξ ) ) ∂θ ( x) + 1 ∂ (a ijhk M 0 h ) ∂θ ( x) + 1 ∂ (a ijhk M 0 k ) ∂θ ( x) +  2 ∂ξ j 2 ∂ξ j ∂x j ∂x k ∂x h  0 0 0  ∂ ∂θ ( x) ∂θ ( x) ∂θ ( x) ∂  ( a ijhk bhk H α1 ) a ijhk ε hk M α1 (ξ ) − a ijhk bhk − = − f i , in Ω ε ∂xα1 ∂x j ∂ξ j ∂xα1  ∂ξ j  1 1 υ a  a ε N (ξ ) Dα21α 2 u m0 + ε hk M α1 (ξ ) Dα1 1 θ 0 + N α1hm Dα21k u m0 + N α1km Dα21h u m0  j ijhk  ijhk hk α1α 2 m 2 2  1 1  1 0 1 0 1 0 on Sε + 2 M 0 k D hθ + 2 M 0 h D k θ  − υ j a ijhk bhk H α1 Dα1 θ ( x) = 0,  

(

)

(

(

(

(12)

(

))

(13)

))

(

)

(

)

We can integrate both sides of identities in Eqs (12) and (13) with respect to ξ in Q ∩ ω , thus from the 1-periodicity of H α1 (ξ ), H α1α 2 (ξ ) , M 0 (ξ ), M α1 (ξ ) , N α1 (ξ ), N α1α 2 (ξ ) , it follows that  ∂  ∂θ 0 ( x)   kˆij  = h( x )  ∂x j   ∂xi   0 0 θ ( x) = T ( x)

on ∂Ω

(

)

∂ ˆ 0  ∂ 0  ∂x aˆ ijhk ε hk (u ( x)) + ∂x bijθ ( x) = − f i ( x) j  j u 0 ( x) = u ( x) 0 

(

(14)

in Ω

)

in Ω

(15)

on ∂Ω

where aˆ ijhk = [ mes(Q ∩ ω )] −1

∫ [a

ijhk

+ aijlm ε lm ( N hk )]dξ ,

Q ∩ω

bˆij = [mes (Q ∩ ω )] −1

∫ [−aijhk bhk + aijlm ε lm (M 0 )]dξ ,

(16)

Q ∩ω

kˆij = [mes (Q ∩ ω )] −1

∫ω [ k

Q∩

ij

+ k il

∂H j ∂ξ l

]dξ

aˆ ijhk , bˆij and kˆij are the homogenization coefficients. Eqs (14) and (15) generate two new problems, so-called the homogenization problems. θ 0 ( x) and u 0 ( x) are the homogenization solution corresponding to θ ε (x) and u ε (x) , which can be determined based on Eqs (14) and (15) with

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157

homogenization coefficients that reflect the equivalent properties of composite materials. From formulas of computing aˆ ijhk , bˆij and kˆij , one can notice that they not only depend on coefficients a ijhk (ξ ) , bij (ξ ) and k ij (ξ ) of composite material, but also on H α1 (ξ ) , N α1 (ξ ) and M 0 (ξ ) , which represent the interaction caused by the difference of composite coefficients of different composition in each basic cell. Similar to the previous study [7], one can prove that aˆ ijhk , − bˆij and kˆij satisfy P(k1 , k 2 ) and Ε(λ1 , λ 2 ) conditions presented in [7], respectively. To summarize, we have the following results. Theorem 1: The structure problem Eq (1) has following multi-scale asymptotic solution ∞  ε l l 0 0 θ ( x) = θ ( x) + ∑ ε ∑ H α (ξ ) Dα θ ( x),  l =1 =l  ∞ ∞ u ε ( x) = u 0 ( x) + ε l N α (ξ ) Dαl u 0 ( x) + ∑ ε l +1 ∑ M α (ξ ) Dαl θ 0 ( x) ∑ ∑  l =1 =l l =0 =l

(17)

(1) u 0 ( x) and θ 0 ( x) are the solutions of homogenization problems Eqs (14) and (15) with the homogenized coefficients, which is the homogenized solution of 1 . (2) aˆ ijhk , bˆij and kˆij are the homogenization coefficients. (3) The problems in Eqs (3)-(11) have unique solutions in perforated basic cell Q ∩ ω , and for a sufficiently smooth perforated domain we can extend these solutions to define on Q ∩ ω for the whole basic cell Q , Eqs (14) and (15) have unique solution in the whole domain Ω . For a sufficiently smooth perforated domain Ω ε , we can extend the solution u ε ( x) and θ ε ( x) to define Ω ε for a bounded domain Ω . Approximate Solution and Error Estimation The approximate numerical solution of the first L items in Eq (17) is given: L  ( L) 0 0 l l θ ε ( x) = θ ( x) + ∑ ε ∑ H α (ξ ) Dα θ ( x),  l =1 =l  L L −1 u ( L ) ( x) = u 0 ( x) + ε l N α (ξ ) Dαl u 0 ( x) + ∑ ε l +1 ∑ M α (ξ ) Dαl θ 0 ( x) ∑ ∑ ε  l =1 l =0 =l =l

(18)

then L

ε hk (u ε( L ) ( x)) = ε hk (u 0 ( x)) + ∑ ε l −1ε hk (N α ...α m (ξ )) 1

l =1

l

∂ l u 0 ( x) + ∂xα1 ...∂xα l

L −1 ∂ u ( x) ∂ l +1θ 0 ( x) ∂ l +1θ 0 ( x)  1 + ∑ ε l +1  M αh + M αk ε ε hk (M α1 ...α l (ξ ))  ∑ ∂xα1 ...∂xα l l =1 ∂xα1 ...∂xα l ∂x k ∂xα1 ...∂xα l ∂x h  2  l =1 L −1

l

0

(19)

l

L

+ ∑ε l l =1

(

∂ l +1u 0 ( x) ∂ l +1u 0 ( x)  1 + N αkm  N αhm  ∂xα1 ...∂xα l ∂x k 2  ∂xα1 ...∂xα l ∂x h 

)

x

(

σ ij u ( L ) ( x) = a ijhk ( )ε hk u ( L ) ( x) ε ε ε

)

(20)

Generally taking the first L items for L = 1,2,3. Especially for L = 1 , we have the following asymptotic error estimation between 1-order multi-scale solution θ ε(1) ( x) , u ε(1) ( x) and weak solution θ ε (x) , u ε (x) of problem Eq (1). Due to limitation of space the detail proof of this theorem is not provide in this paper.

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Macro-, Meso-, Micro- and Nano-Mechanics of Materials

ε

ε

Theorem 2: Assuming θ ( x) ， u ( x) are weak solution of Eq (1)，if T ( x ), u 0 ( x ) ∈ H (Ω) ， h( x), f (x) ∈ H 1 (Ω) ，then following estimation holds,  θ ε ( x) − θ (1) ( x) ε   ε  u ( x) − u ε(1) ( x) 

≤ cε 1 / 2 ( T 0

H 1 (Ωε )

≤ cε 1 / 2 ( T 0

ε

H (Ω ) 1

H 5 / 2 (Ω)

H

5/2

(Ω)

0

+ h( x ) + h( x )

H 1 (Ω) H (Ω) 1

3

) , + u0

H

5/2

(Ω)

+ f

(21) H

1

) (Ω)

where c is independent of ε From this theorem we see that the infinite series in Eq (17), especially L -order multi-scale asymptotic solution defined by finite series Eq (18) is convergent and it converges to the weak solution of problem Eq (1). For the perforated domain, it is very difficult to obtain the higher order error estimation because of the difficulties involved in dealing with boundary conditions. For perforated domain, however, one can obtain the higher order asymptotic error estimation using the boundary layer method. Finite Element Solution and Algorithm

(1) FE computation of Hα (ξ ), M 0 (ξ ), Nα (ξ ) and homogenization coefficients of aˆ ijhk , bˆij and kˆij x x x corresponding to a ijhk ( ) , bij ( ), k ij ( ) . ε ε ε Let Th be a quasi-uniform triangulation of perforated basic cell ω ∩ Q with mesh h , and V h1 ⊂ V 2 , V h2 ⊂ V be the linear conforming finite element space defined on Th ，the finite element approximation of Hα1 (ξ ), M 0 (ξ ), Nα 1 (ξ ) are to find Hαh1 ∈ Vh2 , M 0h ∈ Vh1, Nαh 1 ∈ Vh1 such that: 1

1

∂H αh1 ∂v ∂v k ij dξ = − ∫ k iα 1 d ξ , ∂ ξ ∂ ξ ∂ i j ∩Q ω ∩Q ξ i

∫ ω ∫ ω

ε ij ( v )a ijhk ε hk (M 0h )dξ =

∩Q

∫ε ω

∫ε ω

ij

∀v ∈ Vh2

( v ) a ijhk bhk dξ ,

(22)

∀v ∈ Vh2

∩Q

h ij ( v ) aijhk ε hk ( Nα 1 ) dξ

=−

∩Q

∫ε ω

ij ( v ) aijα 1m dξ ,

(23)

∀v ∈ Vh1

∩Q

The FE approximation of homogenization coefficients of composite material can be computed as follows: h aˆ ijhk = [mes(Q ∩ ω )]−1

∫ω[a

ijhk

+ aijlmε lm (N hhk )]dξ ,

Q∩

bˆijh = [mes(Q ∩ ω )]−1

(24)

h ∫ [−aijhk bhk + aijlmε lm (M 0 )]dξ ,

Q ∩ω

kˆijh = [mes(Q ∩ ω )]−1

∫

[k ij + k il

Q ∩ω

∂H hj ∂ξ l

]dξ

FE computation of homogenization solution u 0 ( x),θ 0 ( x) Let Th0 be a uniform triangulation of Ω with mesh h0 , and Vh1 ⊂ ( H 01 (Ω)) 2 ,Vh2 ⊂ H 01 (Ω) be the usual quadratic Lagrange finite element space defined on Th0 ， the FE computation of u 0 ( x ), θ 0 ( x ) are functions of u (0, h0 ) ( x), θ (0, h0 ) ( x) ， such that u (0, h0 ) ( x) − u0 ∈Vh10 (0, h0 ) 0 θ ( x) − T ∈Vh20 and the following FE virtual work equations with FE homogenization material coefficients aˆ ijhk , bˆij and kˆij hold, 0

∂v ˆ h ∂v ˆ h ∂θ ( 0,h0 ) k ij dx = − ∫ k iα dx, ∂ xi 1 ∂ x i j Ω

∫ ∂x

Ω

∫ε

Ω

ij

h ( v ) aˆ ijkh ε hk (u ( 0,h0 ) ) dx = ∫ ε iα1 ( v )bˆihα1θ ( 0,h0 ) dx, Ω

(2) Multi-scale approximation solution

∀v ∈ Vh20 ∀v ∈ Vh20

0

(25)

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By definition of θ ε(1) ( x) , u ε(1) ( x) , the multi-scale FE solution θ ε(1,h,h0 ) ( x) , u ε(1,h,h0 ) ( x) anywhere on Ω can be given by:

∑

θε(1, h, h0 ) ( x) = θ ( 0, h0 ) ( x) + ε Hαh (ξ ) Dα1 θ ( 0, h0 ) ( x),  =1  (1, h, h ) ( 0, h0 ) 0 ( ) ( ) u x u x ε N1α (ξ ) Dα1 u( 0, h0 ) ( x) + ε 2 Mαh (ξ ) Dα1 θ (0, h0 ) ( x), = +  ε =1 =1 

∑

∑

(26)

then ε hk (u ε(1,h,h0 ) ( x)) = ε hk (u (0,h0 ) ( x)) + ε hk (N αh m (ξ )) εε hk (M αh1 (ξ )) +ε

∂ u (0,h0 ) ( x) + ∂xα

1 ∂ u 0 ( x) ∂ 2θ (0,h0 ) ( x) ∂ 2θ ( 0,h0 ) ( x)  + ε 2  M αhh + M αhk  2  ∂xα1 ∂xα ∂x k ∂xα ∂x h 

(27)

1  h ∂ 2 u (0,h0 ) ( x) ∂ 2 u ( 0,h0 ) ( x)  + N αhkm   N αhm 2  ∂xα ∂x h  ∂xα ∂x k

(

)

(

x

σ ij u (1, h, h0 ) ( x) = a ijhk ( )ε hk u (1,h ,h0 ) ( x) ε ε ε

)

(28)

The procedure of multi-scale algorithms is in the following: (1) Determine the properties of basic cell of composite materials, composition and interface, and then prescribe the coefficients a ijhk (ξ ), bij (ξ ) and k ij (ξ ) for the basic cell of composite materials. (2) Compute the FE solutions H αh1 (ξ ), M 0h (ξ ), N αh 1 (ξ ) of H α1 (ξ ), M 0 (ξ ), N α1 (ξ ) by solving the FE virtual work equations Eqs (22), (23), and evaluate the homogenization material parameters h , bˆijh and kˆijh based on Eq (24). aˆ ijhk (3) Compute the FE solution of homogenized displacement u (0,h0 ) and the increment of temperature θ (0,h0 ) by solving virtual Eqs (25), (26) on the whole structure Ω . (4) Compute the high order partial derivatives of homogenization solution u (0,h0 ) ( x) and θ (0,h0 ) ( x) . (5) Compute the 1-order multi-scale solution θ ε(1,h,h0 ) ( x) , u ε(1,h,h0 ) ( x) of displacement and increment of temperature at arbitrary point of structure, strains and stresses. Numerical Example To show the interaction between the different compositions produced by the increment of temperature and displacement for the basic configurations, we consider the following simple problem:  ∂  ε ∂θ ε   k ij +h = 0 in Ω ε  ∂x j   ∂xi   ∂ ε ε  ∂ aijhk ε hk (u ε ( x)) − aijhk bhkε θ ε + f i ( x) = 0, i = 1,..., n, in Ω ε  ∂x j ∂x j  ε  ε 2 2 ε ε ∂θ = 0 on S ε θ = 100 + x + y − x − y Γε ; σ ε (θ ) = ν i k ij ∂x j   ε 1 2 ε ε xi Γε ; σ ε (u ε ) = ν j aijhk ε hk (u ε ( x)) − ν j aijhk bhkε θ ε = 0 on S ε u i = u 0i ( x ) = 300 

(

)

(

)

(29)

where Ω ε and ω ∩ Q as shown in Figs 2 and 3. For simplicity, let ε = 0.1 . In each basic cell, the white color denotes a cavity, the black color denotes material 1, and the remained part denotes material 2. f i ( x) = 5000, h( x) = 20000

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Macro-, Meso-, Micro- and Nano-Mechanics of Materials

Fig. 2 Domain Ω ε

Fig. 3 Basic cell Q ∩ ω

In following discussion we let u ε (x) and θ ε (x) be finite element solutions u εh ( x) and θ εh ( x) in the very fine mesh since we can not obtain the exact solution of the problem in Eq (29). Owing to the space limitation, we list only the solution of the 1st component of displacement on fine mesh, the homogenized solution and 1-order multi-scale solution. The Lame coefficients of different materials and homogenized coefficients are: 1

 27857  D1 =  5571 0 

5571

 315000    D =  2 157500 0 11142  0

27857 0 0

1

0  155314 63294 0     315000 0  Dˆ =  63294 17919 0  0 0 78750 0 45763 

157500

Similarly, the thermal conductivity coefficients of different materials and the homogenized thermal coefficients are listed, respectively, 0  2.24 K1 =   2.24  0

0  ˆ  21.92 0  39.66 K1 =   K =  39.66  21.92  0 0

The thermal modules of different materials and the homogenization thermal module are given:   57.700 1.6714305 0  , B 2 =  B1 =  0 0 1.6714305   

   − 25.584 0   , Bˆ =  0 29.068 57.700   

0

We solved the problem in Eq (29) using FE (finite element) method. Table 5.1 compares the approximate running time for calculation based on the traditional FE method and the multi-scale FE method. It is founded that the multi-scale FE algorithm required much less time to solve the same problem. Table 5.1 Comparison of the number of elements and nodes in computation Multi-scale FE Computation

Classical FE Computation with Refined Meshes Approximate computational capacity Approximate running time

1.2 × 10

14

45 Min

Basic Cell

2 × 10

11

14 Sec

Homogenization problem

2 × 1011 16 Sec

Figs 4 to 6 show the 1st component of classical FE solution u1h1 ( x) based on a very fine mesh, homogenization solution u 0h0 ( x) and multi-scale FE solution u1(1,h,h0 ) ( x) based on a coarse mesh, respectively. Fig. 7 shows the asymptotic error of u1h1 ( x) and u1(1,h,h0 ) ( x) .

Advanced Materials Research Vol. 9

Fig. 4 1st component of displacement

Fig.

5

component

of

homoge

h displacement u 0 0 ( x )

on fine mesh

Fig. 6 1st component of 1-order solution of displacement

1st

161

(1, h , h ) h Fig. 7 Error of error of u1 1 − u1 0

(1, h , h ) u1 0 ( x )

From these figures, it is seen that the multi-scale FE solution u ε(1,h,h0 ) ( x) , θ ε (1,h,h0 ) ( x) are the very fine approximation to the displacement u ε (x) and the increment of temperature θ ε (x) . Conclusions The new multi-scale asymptotic expression for the solution of displacement and increment of temperature is presented. From the infinite series, the asymptotic expression can be divided into two parts: the first part formed by the homogenization solution on normal domain Ω , and the second part produced by the interaction solution M α (ξ ) , N α (ξ ) , H α (ξ ) describing the interaction of different compositions within the basic cell. The multi-scale analysis and its FE algorithm given in this paper are an effective method to analyze the problems of composite materials with a small periodic configuration under condition of coupled thermoelasticity by introducing the coupled items M α (ξ ) , N α (ξ ) , H α (ξ ) . From M α (ξ ) , the homogenization coefficients of composite materials can be evaluated by local coefficients a ijhk ( x / ε ) , bij ( x / ε ) , k ij ( x / ε ) . By evaluating u ε( L ,h ,h ) ( x) , θ ε( L ,h ,h ) ( x) , N α (ξ ) , H α (ξ ) and inter-coupled item M α (ξ ) , the detail information about the effects of coupled thermolelasticity, such as strains, thermal strains, stresses and thermal stresses in different materials can be evaluated. It should be pointed out that the two-scale analysis (TSA) previously could be extended to composite materials with multi-scale configuration. 0

0

References 1. A.Bensoussan, J. L. Lions, and G. Papanicolou, Asymptotic Analysis of Periodic Structures, North-Holland, Amsterdan, 1978. 2. L.Q. Cao and J.Z. Cui, J. Appl. Math, Vol. 22 (1999), 38-46. 3. J.Z. Cui and H.Y. Yang, Intern. J. Comp. Math. Vol. 14 (1996), 157-174. 4. J.Z. Cui and T.M. Shin, Struct Eng. Mech. Vol. 7 (1996), 601-614.

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5. A.G. Mclellan, The Classical Thermodynamics of Deformable Materials, Cambridge Univ. Press, 1980. 6. J.L. Nominski, Theory of Thermoelasticity with Applications, Sijthoff and Noordhoof, 1978. 7. O.A. Oleinik, A.S. Shamaev and G.A. Yosifian, Mathematical Problems in Elasticity and Homogenization, North--Holland, Amsterdan, 1992. 8. L.N. Sneddon, The Linear Theory of Thermoelasticity, Springer Verlag Wien, 1974.

Advanced Materials Research Vol. 9 (2005) pp 163-172 © (2005) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMR.9.163

Simulation of Multiple Hydraulic Fracturing in Non-uniform Pore Pressure Field L.C. Li1, a, C.A. Tang1,b, L.G. Tham2, T.H. Yang1 and S.H. Wang1 1

Center for Rock Instability and Seismicity Research, Northeastern University, Shenyang, China 2 Department of Civil Engineering, The University of Hong Kong, Hong Kong, China a [email protected], [email protected]

Keywords: Numerical simulation, Heterogeneous material, Hydraulic fracturing, Seepage mechanics, Failure process.

Abstract. A series of numerical simulations of hydraulic fracturing were performed to study the initiation, propagation and breakdown of fluid driven fractures. The simulations are conducted with a flow-coupled Rock Failure Process Analysis code (RFPA2D). Both heterogeneity and permeability of the rocks are taken into account in the studies. The simulated results reflect macroscopic failure evolution process induced by microscopic fracture subjected to porosity pressure, which are well agreeable to the character of multiple hydraulic fracturing experiments. Based on the modeling results, it is pointed out that fracture is influenced not only by pore pressure magnitude on a local scale around the fracture tip but also by the orientation and the distribution of pore pressure gradients on a global scale. The fracture initiation, the orientation of crack path, the breakdown pressure and the stress field evolution around the fracture tip are influenced considerably by the orientation of the pore pressure. The research provides valuable guidance to the designers of hydraulic fracturing engineering. Introduction Except for being used in determining in-situ stresses in rock masses, hydraulic fracturing is also a method used by oil companies to stimulate reservoir production enhance petroleum recovery. In the last decade, there are several different interpretation theories for hydraulic fracturing, and many scholars have make lots of investigation on the hydraulic fracturing by using all kinds of method [1,2]. In hydraulic fracturing engineering, in order to increase the permeable communication between injection and production wells for enhanced petroleum recovery, it may be desirable to have multiple fractures. Then the simultaneous hydraulic fracturing from two neighboring wells has been used to establish communication between wells. The interaction of two fractures passing through rocks is of interest to the petroleum industry. A review of hydraulic fracturing research shows that most of the published studies [1,2] demonstrated that the failure of porous rock is controlled by the conventional effective stress, equal to the total macroscopic stress σij plus pore pressure p. However, it is less clear what role pore pressure plays during the failure process of rock. Pore pressure may affect the crack propagation on both a local and global scale. For the orientation and distribution of pore pressure, a few researchers have investigated the influence of uniform or symmetric pore pressure distribution on straight fracture propagation [3,4,5]. But more complex questions concern the orientation of fracture propagation within non-uniform pore pressure fields. Due to the difficulty to gain a complete solution of a multiple hydraulic fracturing problem within non-uniform pore pressure fields, a numerical tool incorporate with the flow-stress-damage (FSD) coupling model [6], Rock Failure Process Analysis code (RFPA2D) was employed to obtain an insight into the propagation and interaction of multiple hydraulic fractures in the non-uniform pore pressure field. Mathematically, RFPA2D is completely a continuum mechanics method fro numerically processing nonlinear and discontinuum mechanics problems in rock failure. The code has been successfully applied in failure process analysis and hydraulic fracturing problems of heterogeneous and permeable rocks materials [6,7].

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Macro-, Meso-, Micro- and Nano-Mechanics of Materials

Numerical Method Rock Failure Process Analysis (RFPA) code, developed by CRISR [8] is a numerical tool capable of handling the failure process of heterogeneous and permeable materials. It can give an effective description to the microscopic damage mechanism and macroscopic failure of rock medium. A model, implemented with RFPA, of coupling between flow, stress and damage has been proposed [6,7]. Assignment of Material Properties. Here the failure process simulation is attained using FEM as the basic stress analysis tool, where the four-node isoparametric element is used as the basic element in the finite element mesh. The mechanical parameters of rock, such as Young’s modulus, strength and Poisson’s ratio, are heterogeneous and assumed to conform to the Weibull distribution [8]. The mesoscopic element is assumed to be homogeneous and isotropic, and its damage evolution conforms to the specific elastic damage constitutive law. The material properties (such as failure strength σ c and elastic modulus Ec) for mesoscopic elements are randomly distributed throughout the rock sample by following a Weibull distribution [8], taking the σ c as an example: m σ  f (σ ) = σ 0  σ 0

  

m −1

 σ exp −  σ0

  

m

(1)

f(σ )

where σ is the strength parameter of the element; the scale parameterσ0 is related to the average strength of element and the parameter m defines the shape of the distribution function. The parameter m defines the degree of material homogeneity and is called the homogeneity index. We can numerically produce a heterogeneous material in a computer simulation for a material composed of many mesoscopic elements. Here, this heterogeneous material produced by computer is used to simulate the real specimen used in the laboratory, so it is called numerical sample in this paper. Fig. 1 shows the variations of f with respect to m and it obvious that a higher m value represents a more homogeneous material.

σ /σ 0 Fig.1 Weibull’s distribution for mechanical properties of rock samples with different homogeneity indexes m. Basic Equation. Coupled seepage and stress processes in saturated geological media could be interpreted with Biot's theory of consolidation [6, 9]. By extending Biot's theory of consolidation to include stress effects to permeability, the following governing equations could be obtained: The basic balance equation can be expressed as:

Advanced Materials Research Vol. 9 ∂σ

ij

∂ x ij

+ ρX

j

= 0

( i , j = 1 ,2 ,3 )

165

(2)

where σ ij is the stress tensor, ρ is the mass density and Xj is the body force. Strains and displacements relationship, namely geometrical equation, can be defined as:

ε ij =

1 ( ui , j + u j ,i ) 2

ε v = ε 11 + ε 22 + ε 33

(3)

where ε ij represents the individual components of strain tensor and u is the displacement And the coupled equilibrium equation is:

σ ' ij = σ ij − α p δi = λ δi ε v + 2Gεi j

j

j

(4)

where p is the pore pressure, α is the coefficient of pore-fluid pressure, λ is the Lame coefficient; G is the shear modulus and δij is the Kronecher constant. The seepage equation can be expressed as: k∇ 2 p =

1 ∂p ∂εv −α Q ∂t ∂t

(5)

where k is the coefficient of permeability; Q is the Biot’s constant. Coupling equation:

k (σ , p) = ξ ko e− β (σ ii / 3−αp )

(6)

where ki is the initial coefficient of permeability; β and ξ are material constants. The above Eq. 2 to Eq. 5 are from the Biot's theory of consolidation. An additional Eq. 6 was introduced to represent the influence of stress on permeability [6,9]. Furthermore, experimental results show that the permeability cannot be a constant but a function of stresses since the fracture aperture is most likely to change when the stress conditions vary. Based on this observation, various permeability-stress relationships have been established [6], and the relationship between permeability and stress is assumed to follow a negative exponential function. Constitutive Relations of Mesoscopic Element. When we carry out numerical simulations with this model, the numerical sample was discretized into large number of small square elements. The seepage and stress analysis were carried out by using the finite element approach. According to the stress level, the mesoscopic elements could be classified into four phases, as shown in Fig. 2. 1 + sin φ (1) Elastic phase: If σ 1 − σ 3 < f c or σ 3 > − ft (where σ 1 and σ 3 are the major and minor 1 − sin φ principal stresses. φ is the friction angle and fc is the compressive strength), the element is in the elastic phase and Eo and ν are the elastic modulus and Poisson’s ratio of the element, respectively. As the permeability will decrease with the stress, it is assumed that α = 0 and ξ = 1. Therefore, the permeability is given by

k( σ , p ) = k o e − β ( σ ii / 3 )

(7)

A comparison of the experimental results shows that Eq. 7 can approximately reflect the change in permeability fairly well [6].

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Macro-, Meso-, Micro- and Nano-Mechanics of Materials

1 + sin φ ≥ f c , the element is in the damage phase and the failure is 1 − sin φ due to compression-shear. The elasticity modulus of the element will decrease according to the constitutive law for brittle failure (Fig. 2). Mathematically, the modulus can be written as: (2) Damage Phase: If σ 1 − σ 3

E = ( 1 − D )E o

(8)

where D represents the damage variable. E is the elastic modulus of the damaged elements. The damage variable can be defined as:

D = 1−

f cr E oε

(9)

where fcr is the residual compressive strength. On the other hand, its permeability will undoubtedly increase as fractures begin to form and develop. To reflect the increase in permeability, ξis taken to be bigger value, ξD , and the permeability is calculated by

k (σ , p) = ξ D ko e − β (σ ii / 3)

(10)

If σ 3 ≤ − ft , the element fails in tensile failure mode. The elastic modulus can still be given by Eq. 8 but the damage variable has to be defined in terms of the residual tensile strength ftr, that is D = 1−

f tr Eoε

(11)

As the change in permeability should be independent on the mode of failure, it is assumed that the change in permeability after damage can also be given by Eq. 6. (3) Cracked phase: In the cracked phase, macro fractures begin to form and the element will loss its capacity and stiffness. Therefore, the elastic modulus is assigned a very small value. Due to the existence of macro fractures [10], the permeability will increase significantly and it can be obtained by assuming a very big ξ=ξC and α＝1, that is

k (σ , p) = ξC ko e − β (σ ii / 3− p)

(12)

(4) Crack close phase: When the compressive strain of the crack element ε is greater than ε cu , the crack can be considered to be closed. The element can transfer stress again and its modulus will increase as the compressive strain increase. The modulus can be calculated by E=

f cr

ε cu

×

ε ε cu

(13)

To consider the permeability change in this phase [10], we assume thatξ becomes small again ξ CC and α＝0, that is

k (σ , p) = ξ CC ko e − β (σ ii / 3)

(14)

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167

σ1

fc

εtu

ε t0

fcr

ε3

εcu

ε c0

- f tr

ε1

- ft σ3 fractureded phase

Tensi le damage

Elastic phase

C ompression phase

phase

Closed-crack phase

Fig. 2 Damage constitutive law for mesoscopic element. As the elements may change from one phase to another, an iteration process is required to modify the elastic modulus and permeability by taking into account such changes. The FEM iteration calculation continues until there are no changes in phase at an equilibrium strain field [6]. Numerical Simulation and Discussions Hydraulic Fracturing Simulation of Circular Sample with Double Holes. In this section, the hydraulicfracturing simulation is conducted on a numerically circular sample with double holes. The diameter of the numerical sample is 180mm and has been discretized into a 180×180 (32,400 elements) mesh. The sample sketch is shown in Fig. 3.

Pw Pc

Fig. 3 Numerical sample with double holes subjected to hydraulic. The homogeneity index is chosen to be 2.5. The hydraulic fracturing is simulated by synchronously increasing injection pressure at two prescribed holes within the numerical sample and maintaining constant total stress at the sample boundary. An initial pressure of 1 MPa is applied in the two holes respectively and an incremental pressure of 0.05Mpa maintained at each hole. Other parameters adopted in this simulation are listed in Table 1

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Table1. Mechanical parameter for hydraulic fracturing sample with double holes Confining pressure Characteristic Young's modulus Characteristic Poisson's ratio Characteristic internal friction angle Characteristic uniaxial compressive strength Characteristic direct tesnsile strength Residual strength ratio Characteristic coefficient of permeability Increment of hydraulic pressure per step Pore pressure coefficient Biot's constants

Pc E0 v φ fc ft λ k0 ∆p α β

1.0[MPa] 10000[MPa] 0.25 30[°] 80[MPa] 8[MPa] 0.1 0.1 [m/sec] 0.05 [MPa] 0.1 0.01

The stress distribution across the central line across the sample before the fracture initiation is presented in Fig. 4. Unlike the most conventional analysis on hydraulic fracturing problems, which is restricted to homogeneous materials, the presented model can easily consider the heterogeneity of rock materials. The influence of material heterogeneity on the stress distribution in the ring is illustrated very clearly. The major principal stresses mainly concentrate along the boundary of the two injection holes. But due to the variation in the mechanical properties of the elements, there are obvious fluctuations in the stresses. Max principal stress Min principal stress

1.50E+00

Stress(MPa)

1.00E+00 5.00E-01 0.00E+00 -5.00E-01 -1.00E+00 -1.50E+00 0

30

60

90

120

150

180

Distance(mm)

Fig 4 Numerically obtained stress distribution along the central line across the sample at first loading step. Fig 5 show the evolution of fracture and stress field during the hydraulic fracturing process. In these figures the gray degree indicates magnitude of the minimum principal stress. At first, the stresses concentrate along the boundary of the hole wall (Fig 5 a). The bright areas at the immediate vicinity of the wall are the zones of highest tensile stresses. Generally speaking, in the numerical sample subjected to hydraulic stress in the far-field, there is no preferential location along the borehole wall for the fracture to initiate, since the geometry of the sample is symmetrical. However, due to the heterogeneity existed in the sample, stress fluctuation occurs in the sample. Therefore, the location and orientation of the fracture initiation is unpredictable. With the increasing of internal pore pressure in the two holes, some of weak elements along the boundary of injection holes are tensioned to failure, then cracks begin to initiate and stresses concentrate at the tip of crack (Fig 5 b). As soon as the fracture initiates, the local heterogeneity will also affect the evolving fracture patterns. The

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fractures radially propagate from the injection holes. If one carefully studies Fig 5, it can be noted that the fractures are influenced by the far-field stresses at fist; as they propagate and enter each other’s zone of influence, they changed their directions of propagation due to the influence of the stress fields on each other and it looks as if both of them approach each other (Fig 5 c, d). Fig 5 e, f, g, h show the distribution of waterhead isoline within the sample. It is clear that the waterhead gradient move ahead as the fractures propagate and a non-uniform pore pressure field come into being gradually. Finally, the waterhead merge together within the zone between the two injection holes.

(a)

(e)

(b)

(f)

(c)

(g)

(d)

(f)

Fig. 5 Numerically obtained fractures propagation process and pore pressure field evolution of sample with double holes (min. principal stress, a, b, c, d and waterhead, e, f, g, h). Hydraulic Fracturing Simulation of Square Sample with Three Holes. To examine fracture propagation under simpler boundary conditions and better controll stress states in the more non-uniform pore pressure field, additional test was conducted on a square sample without confining loading. The geometry of the numerical sample is 150mm×150mm and has been discretized into a 160×160 (25,600 elements) mesh. Fig 6 shows the arrangement of three holes (H1, H2 and H3) in the sample. The pore pressure gradient is established across the lower portion of the numerical sample by injecting water into the upper hole(H1), its initial pore pressure in H1 is 1Mpa and the increment of which is 0.05Mpa at each step. In order to study the effect of non-uniform pore pressure, pressure varying from 0.3 MPa to 1.5 MPa is applied at H2, while the third hole(H3) is maintained at atmospheric throughout the whole process. In addition, a vertical notch approx 1mm wide and 5mm long is cut into the H1 to provide a staring direction for the hydraulic fracture and it is called Guide Crack. The other parameters adopted in the analysis are same as those adopted in the numerical samples with double-holes. Fig. 7 presents the results for the case conducted with 0.8MPa pressure maintained at the lower hole H2. It is noted that a fracture initiates at the bottom of H1 when it is pressurized to 3.2Mpa (Fig. 7 a). And it was seen to propagate towards the region of higher local pore pressure. This fracture continues to grow as the load increases and a new fracture also opens at the crown of H1 when the pressure is raised to 3.5MPa (Fig. 7 b). Due to the non-uniform pressure field created by pressurizing H2, the fracture initiated from Guide Crack does not follow a straight path. Instead, it has traced a curved path curving towards H2 (Fig. 7 c). It is also noted that the fractures grew rapidly and they extended over 38mm when the pressure is 3.9 MPa. The direction of fractures is controlled by the

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non-uniform pore pressure field between the hole H1 and hole H2. Propagating rock fractures generally tend to orient themselves parallel to the maximum compressive far field stress, minimizing the energy required for extension. Differences in pore pressure around the crack tip can influence fracture direction on a local scaled while pressure gradient induced body forces can affect fracture orientation on a more global scale.

H1 Φ=4mm Guide crack Guide crack 38mm

H2

150mm

38

H3

45mm

150mm 150mm

Fig. 6 Hydraulic fracturing model in more non-uniform pore pressure field

(a)

(b)

(c)

Fig. 7 Numerically obtained fractures propagation process of sample with three holes (min. principal stress) Varying the pressure in H2, one can study its effect on the path of propagation of the fractures. Several simulations were conducted with a pressure at H2 ranging from0.4MPa to 1.5MPa. The results are shown in Fig. 8. It is obvious that the higher the pressure at H2, the stronger is the tendency for the path to curve more towards H2, with the greatest deviation occurring for the higher constant pressure case. In the case (d) (pressure at H2 = 1.5MPa), the fracture actually ends in H2. The simulated results agree fairly well with the experimental results (Fig. 9) [4]. In Bruno’s laboratory experimental, case for Fig. 9 a-2 and Fig. 9 b-2, the injection pressure at the lower hole H2 is 1.40MPa and 1.73MPa respectively. For the case a-2, the major fracture has bypassed the hole H2 although it is partial to H2. While for case b-2, the major fracture propagates to H2 and keep on extending. The fracture is influenced by both pore pressure magnitude on a local scale around the crack tip and by the orientation and distribution of pore pressure gradients on a global scale.

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(b)

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(c)

(d)

Fig 8. Crack trajectories for cases with different hydraulic pressures in H2.

(a-1)

(a-2)

(b-1)

(b-2)

Fig 9.Comparison of macroscopic failure form between simulation(a-1,b-1) and experiment(a-2,b-2) results.

Hydraulic pressure (MPa)

The variation of the fracture initiation pressure with the pressure at H2 is shown in Fig. 10. It is noted that the initiation pressure for all cases is same while the breakdown pressure decreases gradually as the pressure in H2 increases. When the pressure in the hole H2 is lower, local stress of crack tip provide all the energy that caused the crack to extension. While the pressure in the hole H2 is high enough, the body stress caused by the global pore pressure field will contribute to crack tip, enhancing the effective stress and the strain energy available for crack extension, and cause the deviation of crack.

6

y = -1.3476x + 5.6059 2

5

R = 0.9129

4 3 2

Breakdown pressure Initiation pressure=3.2MPa

1 0 0. 2

linear

0. 4

0. 6

0. 8

1

1. 2

1. 4

1. 6

Hydraulic pressure in bore H2(MPa) Fig. 10 Relationship between fractured pressure and the pressure applied in H2

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Concluding Remarks

In this paper, the RFPA2D incorporate with FSD model clearly reveal the evolutionary nature of the multiple hydraulic fracturing process associated with porosity pressure. The simulation of circular sample with double holes demonstrates the interaction mechanism of multiple hydraulic fractures subjected to hydraulic pressure. The fractures, resulted from tensional rupture, affect each other within the influence zone of stresses field, which will change their directions of propagation. The simulation conducted on the square sample found that fracture behavior is influenced by both hydraulic pressure magnitude on a local scale around the crack tip and the orientation of hydraulic pressure on a global scale. The fractures will hardly be interfered when the far-field stresses level is lower, but when the far-field stresses level is high enough, the orientation of fractures and breakdown pressure for hydraulic fracturing are all to be remarkably influenced by far-field stresses. The fractures are shown to propagate towards regions of higher local hydraulic pressure. All of the numerical results presented in this paper prove that flow-stress-damage coupled RFPA2D is a valid tool in understanding the physical essence of hydraulic fracturing problem, especially in heterogeneous and permeable materials. It provides us with a more sensible physical intuition and a more accurate mathematical for the behavior of tensile rupture with hydraulic fracturing. Acknowledgements

The study presented in this paper was supported by grants from the China National Natural Science Foundation (No. 50134040, No. 50204003 and No. 50174013) and the Hong Kong Research Grants Council (No. HKU7029/02E). References

1. M.K. Hubbert and D.G. Willis, Mechanics of Hydraulic Fracturing. (Trans. AIME, 1957) 2. D. Lockner and J.D. Byerlee, J. Geophy. Res. Vol. 82(1977), p, 2018-2026 3. S. Brown, A. Caprihan and R. Hardy, J. Geophy. Res. Vol. 103(1998), p. 5125-5132 4. M. S. Bruno, Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. Vol. 28(4)(1991), p. 261-27 5. W.L. Medlin and L. Masse, Soc. Petrol. Engrs. J. Vol.19(1979), p. 129-144 6. C.A. Tang, Int. J. Rock Mech. & Min. Sci. Vol. 39(2002), p. 477-489 7. T.H. Yang, L.G. Tham, and C.A. Tang, Rock Mech. Rock Eng. Vol. 37(4)(2004), p. 251-275 8. W. C. Zhu and C. A. Tang, Rock Mech. Rock Eng. Vol. 37(1)(2004), p. 25–56 9. M. A. Biot, J Appl Phys. Vol. 12(1941), p. 155-164 10. W.L. Zhu and T. F. Wong, J. Geophy. Res. Vol. 104(1999), p. 2963-2971

Advanced Materials Research Vol. 9 (2005) pp 173-182 © (2005) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMR.9.173

Mechanics Framework for Micron-scale Planar Structures A.C.M. Chong1,*, F. Yang1,2, D.C.C Lam1 and P. Tong1 1

Department of Mechanical Engineering, The Hong Kong University of the Science and Technology, Clear Water Bay, Kowloon, Hong Kong, P.R. China 2 Institute of Computational Engineering and Science, Southwest Jiaotong University, Chengdu 610031, Sichuan, P.R. China * Currently with ASM Assembly Automation Ltd. 4/F, Watson Centre, 16 Kung Yip Street, Kwai Chung, Hong Kong. [email protected], [email protected], [email protected], [email protected] Keywords: plane strain, plane stress; strain gradient elasticity; MEMS, thin films analysis.

Abstract: Structures are assemblies of planar and three-dimensional objects. Planar components and parts are commonly because the deformation behaviors of plates and beams can be analyzed within the plane problem framework. For micron-scale structures, patterning processes in microfabrications are intrinsically planar and the resulting fabricated structures are also planar. These planar micron-scale structures have been designed and analyzed using conventional mechanics, but increasingly as the sizes of these structures become smaller, higher order effects become significant. In nanometer-scale, surfaces were recognized to play significant roles in affecting the physical behavior. Size dependent elastic and plastic deformation behaviors in micron-scale structures were also observed. Size dependence is an intrinsic part of higher order theory of mechanics and has been used successfully to explain scale dependent behavior in threedimensional structures. In this paper, two-dimensional higher order elastic relations in plane stress and plane strain for compressible solids are developed. The difference between the higher order and conventional elasticity theories is compared Introduction Size dependences in elastic and plastic deformation behavior in micron-scaled structures were experimentally observed in metals [1,2,3,4,5,6,7;8] and polymers [9,10,11]. Size dependence in deformation is an intrinsic part of higher order theories of mechanics and a number of theoretical frameworks were proposed. Higher order elastic and couple stress theories have been studied by Toupin [12], Mindlin and Tiersten [13], Koiter [14] and Mindlin [15,16]. Fleck and Hutchinson [17] reformulated and developed higher order elastic theories first in the form of couple stress (CS) theory and later in the form of strain gradient (SG) theory of plasticity. The available higher order theories are three-dimensional and a framework for analyses of planar structures is not available. In this paper, a higher order framework of mechanics for planar problems is presented. We first summarize the general three-parameter strain gradient elastic theory [11] and then decompose the equilibrium equations and boundary conditions into in-plane and out-of-plane formulations. This is followed by the development of the governing equations for plane stress and plane strain. The kinematic relations, the constitutive equations, the equilibrium equations and the boundary conditions for plane problems are summarized and compared. Strain gradient theory for elasticity In the three-parameter general strain gradient elastic theory [11], the deformation measures are defined as, 1 εij = (u j ,i + ui , j ), 2 γi = εmm,i , ηijk = uk ,ij ,

(1)

174

Macro-, Meso-, Micro- and Nano-Mechanics of Materials  1 1 δij (ηmmk + 2ηkmm ) (1) ηijk = (ηijk + η jki + ηkij ) −  , 3 15 +δ jk (ηmmi + 2ηimm ) + δki (ηmmj + 2η jmm )

1 4 where εij is the strain tensor, γi is the dilatation gradient vector, ηijk(1) is the deviatoric stretch

χijs = (eipqη jpq + e jpqηipq ) ,

gradient tensor, χ ijs is the symmetric rotation gradient tensor, δij and eijk are the Kronecker delta and the alternating tensor, respectively, and ηijk is the second order displacement gradient tensor. Denoting w as the elastic strain energy density, the corresponding stress measures respectively are σij, pi, τ ijk(1) and mij,

∂w ∂w ∂w ∂w (1) , pi = , τijk = (1) , mij = s , (2) ∂εij ∂γi ∂ηijk ∂χij which are the work-conjugates to the deformation measures. For linear elastic center-symmetric isotropic materials, the constitutive function is a quadratic function of the invariant strain metrics, w = 12 kε iiε jj + µε ij′ ε ij′ + µ l02ε mm ,iε nn ,i + µ l12ηijk(1)ηijk(1) + µ l22 χijs χijs (3) σij =

where ε ij′ is deviatoric strain,

ε ij′ = ε ij − 13 ε mmδ ij .

(4)

The constitutive relations for linear elastic materials are ν (1) (1) σij = 2µ(εij + ε mm δij ), pi = 2µl02 γ i , τijk = 2µl12 ηijk , mij = 2µl22 χijs , (5) 1 − 2ν where µ is the shear modulus, ν is Poisson’s ratio, and l’s are the three material length scale parameters. Application of the principle of virtual work gives the equilibrium equations, 1 (1) σik ,i − e jlk mij ,il − pi ,ik − τijk (6) ,ij + f k = 0 , 2 where f k is the body force and e jlk is the permutation tensor. The boundary conditions on a smooth boundary surface Si are as follows, 1  (1) n j (σ jk − 2 e jkl mil ,i − δ jk pi ,i − τijk ,i )  (1) (1) + ( Dl nl )(n p p p nk + ni n j τijk + n p nq nr τ pqr nk )  − 1 e D (n n m n ) − D ( p n ) − D (n τ(1) + n n τ(1) n ) = t k p p j i ijk i l ijl k k  2 jlk l p q pq j  or uk = uk , ni mik − (n p m pq nq )nk + 2eklj nl n p nq τ(1) pqj = qk  or (δik − ni nk )θi = θk , (1)  ni pi + ni n j nk τijk =r  or ε n = ni n j εij = εn ,

(7)

where ni is a unit normal to the boundary surface, Di = (δij − ni n j )∂ j is the surface gradient operator, tk , q j and r are the prescribed force, torque and double force tractions, and uk , θ j and

ε n are the prescribed displacements, tangential rotations and normal strain, respectively. For a non-smooth boundary surface, the strain gradient theory requires additional conditions along the sharp edges to be satisfied. The boundary conditions along the sharp edge Ci are

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 1 (1) (1) ∆[ (np mpq nq )e jlk n j kl + kk ( pi ni ) + k j ni τijk + ni n j kl τijl nk ] = pk , (8)  2 or uk = uk where, ∆[ ] represents the sum of the bracketed quantities at both sides of the edge, ki is a unit vector on the boundary surface normal to the edge, and pk is line traction along the edge. These boundary conditions contain higher order terms, which are unaccounted for in conventional mechanics. Higher order deformation of compressible elastic solids

In conventional theory, the assumption of the zero stress components σ z , τ zx and τ zy reduces three-dimensional deformation into a plane-stress deformation, whereas the assumption of zero displacement in z-direction reduces a three-dimensional deformation into plane strain deformation. For higher order plane problems with displacement-prescribed boundary conditions, the governing equations can be directly obtained from the three-dimensional governing equations. For higher order plane problems with traction-prescribed boundary conditions, the solution is more complex owing to the presence of higher order stresses. In the following sections, the general threedimensional equations in Section 2 are first decomposed into in- and out-plane directions. This is followed by development of the plane strain equations and the plane stress equations. In the following derivations, lower-case Roman subscripts (e.g., i, j, k) range from 1 to 3 and Greek subscripts (e.g., α, β, γ ) from 1 to 2. For two-dimensional problems in the x1- and x2-plane, the three-dimensional equilibrium equations Eq. (6) decomposes to the in-plane equilibrium equations σαγ ,α + σ3 γ ,3 − 12 e3δγ ( mα 3,αδ + m33,3δ ) − 12 eβ 3 γ ( mαβ,α 3 + m3β,33 ) (9) (1) (1) − pα ,αγ − p3,3 γ − τ(1) − 2 τ − τ = 0 αβγ ,αβ 3 αγ , α 3 33 γ ,33 and the out-of-plane equilibrium equations σα 3,α + σ33,3 − 12 eβδ3 (mαβ ,αδ + m3β,3δ ) (1) (1) (1) − pα ,α 3 − p3,33 − ταβ 3,αβ − 2τ α 33, α 3 − τ333,33 = 0

.

(10)

Replacing the unit normal nk in Eq. (7) by nγ and n3 = 0 , we obtain the independent in-plane traction boundary conditions, (1) (1)  tγ = nβ  σβγ − 12 eβγ 3 ( mα 3, α + m33,3 ) − δβγ ( pα , α + p3,3 ) − ταβγ , α − 2τ3βγ ,3  (1) 1 + ( Dζ nζ ) ( nα nγ pα + nα nβ τ(1) αβγ + nα nβ nδ nγ τ αβδ ) − 2 eβ 3 γ nα nψ nβ mαψ ,3 + Dγ ( pα nα ) (1) (1) − Dβ ( nα τ(1) αβγ + nα nδ nγ ταβδ ) − nα nδ nγ τ3 αδ ,3 ,

(11)

(1) (1)  t3 = nβ σβ 3 − 12 eβ3δ ( mαδ, α + m3δ,3 ) − τ(1) αβ 3, α − 2τβ 33,3  + ( Dζ nζ ) ( nα nβ τ αβ 3 ) (1) − 12 eβδ3 Dδ ( nα nψ nβ mαψ ) − D3 ( pα nα ) − Dβ ( nα ταβ 3 ),

and (1) q = nα nβ eγ 3β mαγ − 2nα nβ ταβ 3, (1) q3 = nα mα 3 + 2e3δβ nδ nα nψ ταψβ ,

(12)

where q is the applied tangential torque obtained by multiplying qγ in Eq. (7)2 by nβ eγ 3β . The double forces are (1) r = nα pα + nα nβ nγ ταβγ (13) Replacing the unit normal nk in Eq. (7) by nα = 0 and n3 = 1 yields the independent out-of-plane traction boundary conditions,

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Macro-, Meso-, Micro- and Nano-Mechanics of Materials (1) tγ = σ3 γ − 12 e3 γδ ( mαδ,α + m3δ,3 − m33,δ ) − 2τ3(1)αγ ,α − τ33 γ ,3 + p3, γ ,

(14)

(1) t3 = σ33 − pα ,α − p3,3 − 3τα(1)33,α − τ333,3 ,

and (1) qγ = m3 γ + 2eγ 3α τ33 α.

r = p3 + τ

(15)

(1) 333

(16) There are two independent edge boundary conditions. One corresponds to the line traction pk at the intersection of the lateral surface with unit normal n = (0, 0,1) and the in-plane boundary with unit normal n = (nα , 0) . The in-plane and out-plane components of the line traction are (1) pα = ( 12 m33e3βα nβ − nα p3 + nβ τ3(1)βα ) + ( 12 nγ nβ nδ mγβ eδ3α + nγ τ(1) γ 3α + nγ nβ nα τ γβ3 ) ,

(17) p3 = 3nα τα(1)33 + pα nα . The other corresponds to the line traction pk at the intersection of the two lateral surfaces with unit normal n = ( nα1 , 0 ) and n = ( nα2 , 0 ) . The in- and out-plane components of the line traction are (1) 2 2 2 2 (2) 2 2 2 2 (1) pα = ( kα1 pβnβ1 + kγ1nβ1τβγα + n1γ nβ1nα1 kδ1τ(1) γβδ ) + ( kα pβnβ + kγ nβ τβγα + nγ nβ nαkδ τγβδ ) , (1) 2 2 (1) 1 2 2 2 2 p3 = ( 12 nα1 nβ1nδ1kγ1mαβeδγ3 + nα1 kβ1ταβ 3 ) + ( 2 nαnβ nδ kγ mαβeδγ3 + nα kβ ταβ3 ) .

(18)

The resulted in- and out-plane governing equations are used in the derivation of plane-stress and plane-strain solutions in the following sections. Plane-stress derivation

Considering for traction free conditions such that the stress, the couple stress and the higher-order tractions in the x3-direction are vanishing. Therefore, Eq. (14), (15) and (16) become (1) σ3 γ − 12 e3 γδ ( mαδ ,α + m3δ,3 − m33,δ ) − 2τ3(1)αγ ,α − τ33 x3 = ± h / 2 γ ,3 + p3, γ = 0 , (1) σ33 − pα ,α − p3,3 − 3τα(1)33,α − τ333,3 =0,

x3 = ± h / 2

(1) m3 γ + 2eγ 3α τ33 α = 0,

x3 = ± h / 2

(19)

(1) p3 + τ 333 =0 x3 = ± h / 2 where h is the thickness in the out-of-plane direction and is much smaller than the characteristic length in the x1-x2 plane. A normalized coordinate s is introduced to describe the displacement variations along the x3-direction: x s= 3 (20) h Hence the derivative with respect to the out-of-plane direction can be written as ∂ / ∂x3 = 1/ h(∂ / ∂s ) . The upper and lower lateral surfaces correspond to s = ±1/ 2 , respectively. The displacement u3 in the out-of-plane direction is linearly proportional to x3. Therefore, the inplane and out-of-plane displacements can be expanded in a power series’ of thickness h as [18] uα ( x1 , x2 , x3 ) = uα(0) ( x1 , x2 , s ) + h 2 uα(2) ( x1 , x2 , s ) + O(h4 ), (21) u3 ( x1 , x2 , x3 ) = hu3(1) ( x1 , x2 , s ) + h3u3(3) ( x1 , x2 , s ) + O(h5 ), where the superscript denotes the order of h in the power series’. The in-plane displacement uα

and the out-of-plane displacement u3 are even and odd powers of h, respectively. This expansion is consistent with that in classical elasticity (Love, 1927). The strains are obtained from the straindisplacement equation Eq. (1)1 as

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(0) ∂u (1) 1  ∂u (0) ∂u  εαβ =  α + β  + O(h 2 ), ε33 = 3 + O(h 2 ), ∂xα  ∂s 2  ∂xβ (22) (0) (2) (1)   1 ∂uα h ∂u ∂u εα 3 = +  α + 3  + O(h3 ). 2h ∂s 2  ∂s ∂xα  Since strains must be finite as the out-of-plane thickness h approaches 0 in a plane-stress problem, Eq. (22)3 requires ∂uα(0) =0 or uα(0) = uα(0) ( x1 , x2 ) (23) ∂s This means that the leading terms of the in-plane displacement uα(0) are independent of s (or x3). The strains in Eq. (22) can then be written as (0) 1  ∂uα(0) ∂uβ  (0) 2 2 εαβ = ε αβ + O(h ) =  +  + O(h ),  2  ∂xβ ∂xα  ∂u (1) (0) ε33 = ε33 + O(h 2 ) = 3 + O(h2 ), (24) ∂s εα 3 = O(h).

Here the zeroth-order terms in strains ε(0) αβ are independent of the out-of-plane coordinate s (or x3). The dilatational gradients are ∂ 2uβ(0) ∂ 2u3(1) (0) 2 γ α = γ α + O(h ) = + + O(h 2 ) ∂xα ∂xβ ∂xα ∂s (25)  ∂ 2uβ(2) ∂ 2u3(3)  1 ∂ 2u3(1) 1 ∂ 2u3(1) γ3 = + h + = + O ( h) 2   ∂x ∂s  h ∂s 2 h ∂s 2 s ∂ β   Since dilatational gradient must be finite as the out-of-plane thickness h approaches 0 in a planestress problem, Eq. (25)2 requires ∂ 2u3(1) ∂u3(1) = 0 or = f ( xα ) (26) ∂s 2 ∂s The symmetric rotation gradients can be determined by using Eqs. (1)3 and (1)5 as S S χαβ = O(h), χ33 = O(h),

(27)Using  ∂ 2uβ(0)   ∂ 2u3(1) ∂ 2uδ(2)  2 e O ( h ). χ = χ + O(h ) = e3αδ  − + +   3δβ ∂s 2  ∂xα ∂xδ   ∂xδ ∂s  the strain gradient-displacement relation Eq. (1)3, we can write the deviatoric stretch gradients Eq. (1)4 as (0)(1) 2 (1) (0)(1) 2 η(1) η(1) αβγ = ηαβγ + O ( h ), α 33 = η33α = ηα 33 + O ( h ), (28) (1) (1) η(1) = η = O ( h ), η = O ( h ). αβ 3 αβ 3 333 S α3

(0) S α3

2

1 4

The leading terms of the deviatoric stretch gradients, η(0)(1) and η(0)(1) are in terms of uα(0) and αβγ α 33 ∂ 2uα(2) ∂ 2u3(1) and . Using the constitutive equation, Eq. (5) the stresses are as ∂s 2 ∂xα ∂s ν  (0)  (0) 2 σαβ = σαβ + O(h 2 ) = 2µ  εαβ + ε (0) γγ δαβ  + O ( h ), 1 − 2 ν   ν  (0)  (0) 2 σ33 = σ33 + O(h 2 ) = 2µ  ε33 + ε (0) γγ  + O ( h ), 1 − 2ν   σα 3 = hσα(1)3 = O(h),

(29)

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and the work conjugates to the dilatational gradients are as pα = 2µl02 γ α(0) + O(h 2 ) = pα(0) + O(h2 ) p3 = p3(1) = O(h). The deviatoric higher-order stresses are as (0)(1) 2 2 (0)(1) 2 τ(1) αβγ = τ αβγ + O ( h ) = 2µl1 ηαβγ + O ( h ) , (1) (0)(1) 2 2 (0)(1) 2 τ(1) α 33 = τ33 α = τα 33 + O ( h ) = 2µl1 ηα 33 + O ( h ) , (1) (1)(1) (1) (1)(1) τ(1) τ333 = hτ333 = O ( h). αβ 3 = hτ3 αβ = hτ αβ 3 = O ( h ) , and the symmetric couple stresses are as S 2 mα 3 = mα(0)3 + O ( h 2 ) = 2µl22 χ(0) α3 + O ( h ) ,

(1) (1) mαβ = hmαβ = O ( h ) , m33 = hm33 = O ( h).

(30)

(31)

(32)

Out-of-plane traction boundary conditions

The traction-prescribed boundary conditions and the equilibrium equations are derived as follows: the leading terms in the expansion of the out-of-plane traction boundary conditions Eq. (19)1 are (0)(1)  ∂τ33 ∂m3(0) 11 γ δ (33) at s = ± 12 , −  2 e3 γδ +  = 0  h ∂s ∂s  and the second order terms are (2)(1)   ∂τ33  (1) ∂m3(2) γ (1)  (1)(1) δ h σ3(1)γ − 12 e3 γδ  mαδ + − m − 2 τ − + p3,(1)γ  = 0 at s = ± 12 . (34) ,α 33,δ  3αγ ,α ∂s ∂s     Expanding the out-of-plane traction boundary condition Eq. (19)2, we have the leading term as (1)(1) ∂p (1) ∂τ333 (0) σ33 − pα(0),α − 3 − 3τ(0)(1) − = 0 at s = ± 12 . (35) α 33,α ∂s ∂s An expansion of the out-of-plane traction boundary conditions Eq. (19)3 gives (0)(1) 1 m3(0) (36) γ + 2eγ 3α τ33α = 0 at s = ± 2 . The leading term of the out-of-traction boundary condition Eq. 4 gives (1)(1) h ( p3(1) + τ 333 (37) ) = 0 at s = ± 12 . The leading term of the out-of-plane edge boundary condition, Eq. (17)2 gives (0) 3nα τα(0)(1) 33 + pα nα = 0. The in-plane equilibrium equations

Expanding the in-plane equations Eq. (9), we have the leading term as (0)(1)  ∂ 2 m3(0) ∂ 2 τ33 1 1 β γ − 2  2 eβ3 γ + =0  h  ∂s 2 ∂s 2 

(38)

and the second order term as (0) (0) (0)(1) (0) 1 σαγ ,α − 2 e3 δγ mα 3,αδ − ταβγ ,αβ + pα , αγ (1) (1) 1 σ3(1)γ − 12 e3δγ m33,  δ − 2 eβ 3 γ mαβ ,α . ∂   (2) (2)(1) +  = 0 ∂ ∂τ m 3β 33 γ (1)(1) (1)  ∂s − 1 e  2 β3 γ ∂s − 2τ3αγ ,α − ∂s − p3, γ  Integrating Eq. (38) from − 12 to s twice and with Eq. (33) and Eq. (36), we obtain 1 2

(0)(1) (0)(1) eβ3 γ m3(0) = 0 or m3(0) β + τ33 γ γ + 2eγ 3α τ33α = 0 .

(39)

(40)

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179

2 (0) ∂ 2uα2 ∂ 2uα(0) ∂ uβ ∂ 2u3(1) From Eq. (40), we can obtained in terms of , and . ∂s 2 ∂xβ ∂xβ ∂xα ∂xβ ∂xα ∂s

We have the following result by integrating the Eq. (39) from s = − 12 to s = + 12 (i.e., over the thickness) and using Eq. (34), we have 1

∫ (σ 2

−1

2

(0) αγ ,α

(0)(1) (0) − 12 e3δγ mα(0)3,αδ − ταβγ , αβ − pα ,αγ ) ds = 0 .

(41)

From the results obtained in Eq. (26) and Eq. (40), all the terms of the integrand are independent of s. Therefore, the integration is zero and gives (0) (0) (0)(1) (0) 1 σαγ (42) ,α − 2 e3 δγ mα 3,αδ − ταβγ , αβ − pα ,αγ = 0 . Equation (42) is the in-plane equilibrium equations in terms of the in-plane stresses, symmetric couple stresses, higher-order stresses. The out-of-plane equilibrium equations

Now, we consider the out-of-plane equilibrium equations Eq. (10). The leading is (1)(1)  ∂p3(1) ∂τ333 1 ∂  (0) 1 (0) (0) (0)(1) − 2τα 33,α −  σ33 − 2 eβδ 3 m3β,δ − pα ,α −  = 0. h ∂s  ∂s ∂s 

(43)

Integrating Eq. (43) from − 12 to s and substituting of Eq. (35) and Eq. (36), we have (0) (0) σ33 − 12 eβδ3 m3(0) β , δ − pα ,α −

Integrating Eq. (44) from s = − 12

∫

1

2

−1

2

(1)(1) ∂p3(1) ∂τ333 − 2τ(0)(1) − = 0. (44) α 33,α ∂s ∂s to s = 12 (i.e., over the thickness) and using Eq. (37), we have

(0) (0) (0)(1) (σ33 − 12 eβδ 3m3(0) β , δ − pα , α − 2 τα 33, α ) ds = 0

(45)

Since all the terms in the integrand in Eq. (45) are independent of s, therefore, we have (0) (0) (0)(1) σ33 − 12 eβδ3m3(0) (46) β , δ − pα , α − 2τ α 33, α = 0 From Eq. (44) and Eq. (46), we have (1)(1) ∂p3(1) ∂τ333 + =0 (47) ∂s ∂s Eq. (46) can be simplified by using Eq. (40) as (0) σ33 − pα(0), α − 3τ(0)(1) (48) α 33, α = 0 . With constitutive equations, Eqs (29) to (31), we have equation 32 2   ∂2M  l12 ( 15 l1 + 14 l22 )  (0) l12l22 v 2 (0) 2 ε −  l0 + 8 2 1 2  +M + ε αα −  l0 + = 0, (49) 8 2 1 2  αα , ββ    ∂ x ∂ x − v 1 2 l + l l + l 5 5 ( ) ( ) α α 1 2 1 2 15 4 15 4     (1) ∂u where 3 = M . ∂s Equation (49) is a second-order differential equation with small coefficient characteristics and which can be solved by perturbation method to get the inner and outer for M. The boundary conditions is the edge boundary condition (0) 3nα τα(0)(1) (50) 33 + pα nα = 0 which can be written as       3l12l22 9 22       l1 l2 2 2 l1 l2 4 4 ε  = 0  ∂M + n  nα l02 + εαβ,β +  l02 − α 2  2 2  ββ , α     8 8 2 l  8 l  ∂x l  5  l12 + 2   α 5  l12 + 2    5  l1 + 2     4   4 4    15  15    15 

(51)

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Macro-, Meso-, Micro- and Nano-Mechanics of Materials

After solving Eq. (49), the terms

∂ 2uα2 ∂ 2u3(1) and are expressed according to the leading terms of ∂s 2 ∂xα ∂s

(0) (0) (0)(1) (0)(1) (0) in-plane displacements uα(0) . All the zeroth-order terms σ(0) αβ , σ 33 , pα , τ αβγ , τ α 33 and mα 3 are (1) (1)(1) (1)(1) (1) (1) independent of s (or x3), while the first-order terms σ(1) α 3 , p3 , ταβ 3 , τ 333 , mαβ and m33 are linearly proportional to s (or x3).

The independent in-plane traction boundary conditions

The leading term in the in-plane surface traction Eq. (11)1 is (1)  (0) 1  ∂τ3(1)(1)  (0) ∂m33   (0) ∂p3(1)  (0)(1) βγ tγ = nβ σβγ − 2 eβγ 3  mα 3,α +   − δβγ  pα ,α −  − ταβγ ,α − 2 ∂s  ∂s  ∂s     + ( Dζ nζ ) ( nα nγ p

(0) α

(0)(1) α β αβγ

+n n τ

(0)(1) α β δ γ αβδ

+n n n n τ

)−

1 2

eβ3 γ nα nψ nβ

(1) ∂mαψ

∂s

(52)

∂τ3(1)(1) αδ ∂s Differentiating Eq. (40) with respect to xα and substituting the result and the terms (0)(1) + nα nδ nγ τ(0)(1) + Dγ ( nα pα(0) ) − Dβ ( nα ταβγ αβδ ) − nα nδ nγ

(1)(1) (1) ∂m(1) ∂p (1) ∂τ ∂m33 ∂τ(1)(1) , 3 , 3βγ , αψ and 3αδ into Eq. (52) we have: ∂s ∂s ∂s ∂s ∂s (0) (0) tγ = nβ ( σβγ − 12 eβγ 3 mα 3,α − δβγ pα(0),α − τ(0)(1) αβγ ,α ) (0)(1) + ( Dζ nζ ) ( nα nγ pα(0) + nα nβ τ(0)(1) αβγ + nα nβ nδ nγ τ αβδ )

(53)

(0)(1) (0)(1) + Dγ ( nα pα(0) ) − Dβ ( nα ταβγ + nα nδ nγ ταβδ ) + nβ Aβγ(0)

Equation (53) is the leading term for the in-plane surface traction. The in-plane tangential torque traction Eq. (12)1 is zero. The leading term of the in-plane torque traction Eq. (12)2 is (0)(1) q3 = nα mα(0)3 + 2e3αβ nα nγ nλ τβγλ . (54) The double force is r = nα pα(0) + nα nβ nγ τ(0)(1) (55) αβγ The in-plane surface traction t3 in Eq. (11)2 is zero by Eq.(40). For non-smooth lateral surfaces, the line traction Eq.(18) becomes (0)(1) pα = ∆ ( kα nβ pβ(0) + k γ nβ τβγα + kδ nγ nβ nα τ(0)(1) (56) γβδ ) and p3 = O(h) . Therefore, line traction pα will exist along the non-smooth lateral surfaces. As the thickness h in the out-of-plane direction approaches zero, terms in the order of O(h) or higher are vanished. This is in fact the plane stress condition. Therefore, the governing equations and boundary conditions of plane stress condition are obtained with terms up to the order of O(1). We note that the in-plane equilibrium equation, Eq. (42) consists of two fourth-order differential equations in terms of the two in-plane displacement uα(0) . As a result, four independent boundary conditions should be prescribed on any boundaries in the x1-x2 plane. Now we have four prescribed boundary conditions, two force tractions, Eq.(53), one torque traction, Eq.(54), and one double force, Eq.(55). With these, the plane stress deformation problem can be completely determined. The governing equations and boundary conditions for plane strain and plane stress are summarized in Table 1 and in Appendix. For plane-strain problems, the displacement is uα = uα ( xβ ), u3 = 0 (57) It is a truly two dimensional problem via plane-stress is not [19] and the procedure to obtain the solution is more straight forward. The result for plane strain is given in Table 1.

Advanced Materials Research Vol. 9

εαβ

Plane-strain deformation Kinematic relations: 1 = 2 ( uα ,β + uβ,α )

εαβ

Plane-stress deformation = 12 ( uα ,β + uβ,α )

(*)

γ α = εββ ,α +

γ α = εββ, α 1 η(1) αβγ = 3 ( ηαβγ + ηβγα + ηγαβ )

(*)

∂M ∂xα

1 η(1) αβγ = 3 ( ηαβγ + ηβγα + ηγαβ )

− 151 ( δαβ ηλλγ + δβγ ηλλα + δ γα ηλλβ ) χαs 3 = χ3s α = 14 e3γλ ηαγλ

σαβ

181

− 151 ( δαβ Bγ + δβγ Bα + δ γα Bβ )

(58)1

Constitutive equations: ν   = 2µ  ε αβ + ε γγ δαβ  1 − 2ν   2 pα = 2µl0 γ α

(*)

χ

S α3

 ∂ 2uβ(0)   ∂M  = e3αδ  − Cδ  + e3δβ  ∂xα ∂xδ   ∂xδ   1 4

(59)1

ν   σαβ = 2µ  ε αβ + ε γγ δαβ  1 − 2ν   2 pα = 2µl0 γ α

2 (1) τ(1) αβγ = 2µl1 ηαβγ

(*)

2 (1) τ(1) αβγ = 2µl1 ηαβγ

m3α = 2µl22 χ3Sα Equilibrium equations: (1) 1 σαγ , α − 2 e3βγ mα 3, αβ − pα , αγ − ταβγ , αβ = 0

(58)2

m3α = 2µl22 χ3Sα

(1) (58)3 σαγ , α − 12 e3βγ mα 3, αβ − pα , αγ − ταβγ , αβ = 0

(59)2 (59)3*

Traction-prescribed boundary conditions: (1) (1) tγ = nβ (σβγ − 12 e3βγ mα 3, α − δβγ pα , α − ταβγ tγ = nβ ( σβγ − 12 eβγ 3 mα 3,α − δβγ pα ,α − ταβγ ,α ) ,α ) (1) + ( Dλ nλ )(nα nγ pα + nα nβ τ(1) αβγ + nα nβ nδ nγ ταβδ ) (*) (1) α αβγ

+ Dγ (nα pα ) − Dβ (n τ

(1) α δ γ αβδ

+n n n τ

)

(1) (1) + ( Dζ nζ ) ( nα nγ pα + nα nβ ταβγ + nα nβ nδ nγ ταβδ ) (1) (1) + Dγ ( nα pα ) − Dβ ( nα ταβγ + nα nδ nγ ταβδ ) + nβ Aβγ

(1) 3αβ α γ λ βγλ

q3 = nα mα 3 + 2e n n n τ

(1) q3 = nα mα 3 + 2e3αβ nα nγ nλ τβγλ

(1) α β γ αβγ

r = nα pα + n n n τ pα = ∆ ( nβ kα pβ + n k τ

(1) β γ βγα

(1) α β γ δ γβδ

+n n n k τ

)

(*)

(58)

(1) r = nα pα + nα nβ nγ ταβγ (1) (1) pα = ∆ ( kβnγ τ(1) γβα + kδ nγ nβ nα τγβδ − kα nγ nβ nδ τγβδ )

(59)4

Table 1. (where Bα and Cα are defined in Appendix) Comparison

We have derived the plane strain and stress strain gradient framework for linear elastic isotropic solids. The governing equations and boundary conditions are summarized in Table 1. The governing equations for conventional plane-strain and plane-stress deformations are similar, but the governing equations for the corresponding strain gradient versions are significantly different. For the plane strain case, the governing equation is a fourth-order differential equation. Since displacement is zero in one direction, four independent traction boundary conditions should be prescribed on any boundaries in the x1-x2 plane. These independent boundary conditions are equations (58)4 in Table 1. For the plane stress case, since the deformation measures involve the ∂ 2 uα(2) ∂ 2 u3(1) ∂ 3u3(3) , and , these terms should be first expressed according to the leading terms ∂s 2 ∂s ∂s 3 terms of the in-plane displacements. From the leading terms of in-plane and out-of-plane equilibrium equations, we can obtain the second-order differential equation, Eq.(49). The boundary condition for the differential equation is the in-plane line traction and is detailed in Eq.(51).

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Macro-, Meso-, Micro- and Nano-Mechanics of Materials

Conclusions

Plane-strain governing equations are derived from two-dimensional displacement equations, whereas plane-stress governing equations are derived from the leading terms of the power series equations expanded in terms of the thickness. Comparisons of the kinematic relations, the constitutive equations, the equilibrium equations and the traction-prescribed boundary conditions for both deformations indicate that the derivation of equations for plane-stress deformation was complex as second order differential equation was involved. Given that higher order materials length scale parameters ranges from microns to nanometers, applicability of conventional planar solutions to micron-scale and nanometer-scale structures is best assumed after due considerations for higher order effects. Appendix  16 2 l22   2l12 l22   2 2 l22  − l + 2 +      l1 −  1 15 4  ∂M 15 4  15 2 ∂ 2 uα(0)    Bα = (A.1) + + 2 ε αβ,β + 2 εββ,α 2 ∂xδ ∂xδ  l2 8 2  ∂xα  l2 8 2   l2 8 2   + l1   + l1   + l1   4 15   4 15   4 15  Cδ =

∂ 2uδ(2) ∂s 2

(A.2)

References

1. N.A. Fleck, G.M. Muller, M.F. Ashby and J.W. Hutchinson, Acta Metall. Mater. Vol. 42 (1994) 475-487. 2. J.S. Stölken and A.G. Evans, Acta Metall. Mater. Vol. 46 (1998) 5109-5115. 3. W.D. Nix, Metall. Trans. A20 (1989) 2217-2245. 4. N.A. Stelmashenko, M.G. Walls, L.M. Brown and Y.V. Milman, Acta Metall. Mater. Vol. 41(1993) 2855-2865. 5. Q. Ma and D.R. Clarke, J. Mater. Res. Vol. 10 (1995) 853-863. 6. W.J. Poole, M.F. Ashby and N.A. Fleck, Scripta Metall. Mater. Vol. 34(1996) 559-564. 7. E. Wong, P.E. Sheehan and C.M. Lieber, Science, Vol. 277 (1997) 1971-1975. 8. R. Gao, Z.L. Wang, Z. Bai, W.A. de Heer, L. Dai, L and M. Gao, Phy. Review Lett. Vol. 85 (2000) 622-625. 9. D.C.C. Lam and A.C.M. Chong, J. Mater. Res. Vol. 14 (1999) 3784-3788. 10. A.C.M. Chong and D.C.C. Lam, J. Mater. Res. Vol. 14 (1999) 4103-4110. 11. D.C.C. Lam, F. Yang, A.C.M. Chong, J. Wang and P. Tong, J. Mech. Phys. Solids. Vol. 51 (2003) 1477-1508. 12. R.A. Toupin, Arch. Ration. Mech. Anal. Vol. 11 (1962) 385-414. 13. R.D. Mindlin and H.F. Tiersten, Arch. Ration. Mech. Anal. Vol. 11 (1962) 415-448. 14. W.T. Koiter, Proc. K. Ned. Akad. Wet. (B) Vol. 67 (1964) 17-44. 15. R.D. Mindlin, Arch. Ration. Mech. Anal. Vol. 16 (1964) 51-78. 16. R.D. Mindlin, R. D. Int. J. Solids Struct. Vol. 1 (1965) 417-438. 17. N.A. Fleck and J.W. Hutchinson, Strain gradient plasticity. In: Hutchinson, J. W., Wu, T. Y. (Eds.) Advances in Applied Mechanics. Vol. 33. Academic Press, New York, (1997) 295-361. 18. J.Y. Chen, Y. Huang, K.C. Hwang and Z.C. Xia, J. App. Mech. Vol. 67 (2000) 105-111. 19. Y.C. Fung and P. Tong, Classical and computational solid mechanics, World Scientific (2001).

Advanced Materials Research Vol. 9 (2005) pp 183-190 © (2005) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMR.9.183

A piezoelectric screw dislocation interacting with a dielectric crack in a hexagonal piezoelectric material J. X. Liua and X. L. Liu Department of Engineering Mechanics, Shijiazhuang Railway Institute Shijiazhuang 050043, P.R. China

a

[email protected]

Key words: Piezoelectric material, Screw dislocation, Intensity factors, Shielding effect.

Abstract. This paper is concerned with the interaction of a piezoelectric screw dislocation with a semi-infinite dielectric crack in a piezoelectric medium with hexagonal symmetry. The solution of the considered problem is obtained from the dislocation solution of a piezoelectric half-plane adjoining a gas medium of dielectric constant ε0 by using the conformal mapping method. The intensity factors of stress, electric displacement and electric field and the image force on the dislocation are given explicitly. The effect of electric boundary conditions on the dislocation-crack interaction is analyzed and discussed in detail. The results show that ε0 only influences the electric displacement and electric field intensity factors and the image force produced by the electric potential jump. Introduction The study of the interaction between a dislocation and a crack plays an important role in understanding the fracture behavior of many traditional or composite materials. This is mainly due to the fact that dislocations near a crack can retard or enhance crack propagation [1]. In the past decades, much effort has been devoted to this subject for materials without piezoelectric effect. With wide applications of piezoelectric materials such as piezoceramics, III-V and II-VI compounds, wurtzite GaN in engineering, such topic also attracts increasing attention. Recently there were a few investigations that took into account interaction of a screw dislocation with various kinds of mode III cracks in hexagonal piezoelectric medium. It should be pointed out that dislocations in piezoelectric materials not only suffer finite displacement jumps bi (i = 1, 2, 3) but also may have the electric potential jump across the slip plane. The latter corresponds to the electric dipole layer [2]. Hereafter a piezoelectric dislocation will refer to dislocation with electric potential jump. Lee et al. [3] and Kwon and Lee [4] studied the interaction between singularities (a line force, a line charge and a piezoelectric screw dislocation) and an electrically impermeable semi-infinite and finite long crack respectively. They discussed the shielding effects of crack tip due to the said singularities. Chen et al. [5] analyzed a piezoelectric screw dislocation interacting an electrically impermeable wedge-shape crack. Zhang et al. [6] investigated the interaction between a piezoelectric screw dislocation and a dielectrically elliptical cavity, where the condition that the cavity degenerates into electrically permeable or impermeable crack is given. The above investigations on the crack-dislocation interaction, except for the work of Zhang et al. [6], adopted assumption that the crack surfaces are impermeable to electric fields. This assumption

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Macro-, Meso-, Micro- and Nano-Mechanics of Materials

can simplify some analysis, but it may give rise to non-physical or erroneous results for the problems related to cracks [7-10]. In reality, cracks in piezoelectric materials are filled with vacuum or air. Therefore, we reconsider the interaction between a piezoelectric screw dislocation and a semi-infinite mode III crack in a hexagonal piezoelectric medium. Herein the crack is regards as a thin layer filled with a gas of dielectric constant ε0. The special attention is paid to the influence of ε0 on the interaction. Basic Equations For a piezoelectric screw dislocation whose line is perpendicular to the isotropic basal plane (x1x2-plane) of hexagonal piezoelectric materials, non-zero field quantities associated with the screw dislocation are the out-of-plane elastic displacement u3, shear strainsγβ3, and shear stresses σβ3 as well as the electric potential ϕ, in-plane electric fields Eβ and electric displacements Dβ, β =1, 2. These quantities can be expressed by the two complex potentials and their derivatives with respect to the complex variable z = x1 + i x2 [6, 9, 11]:

u3 = Im Φ ( z )  , ϕ = Im Ψ ( z )  ,

γ 23 + i γ 13 =

d Φ/ ( z )

, E2 + i E1 = −

dz d Φ/ ( z )

σ 23 + i σ 13 = c44

+ e15

(1)

dΨ ( z )

dΨ ( z )

dz

,

(2)

, D2 + i D1 = e15

d Φ/ ( z )

− ε11

dΨ ( z )

dz dz dz dz where c44, e15 and ε11 are the elastic, piezoelectric and dielectric constants. Screw Dislocation in a Piezoelectric Half-plane Consider a half-plane piezoelectric

medium, its exterior being a gas with dielectric constant ε0, as shown in Fig.1. A piezoelectric

,

(3)

x2 − zd

zd

screw dislocation (b3, ∆ϕ) is located at point zd, where b3 and ∆ϕ represent the classical elastic displacement jump and the electric potential jump, respectively. The surface of the piezoelectric

half-plane is traction free, i.e.

σ 31 ( 0, x2 ) = 0 .

(4)

ε0

x1 c44, e15, ε11

Fig.1. A half-plane piezoelectric medium subjected to a piezoelectric screw dislocation

Since electric field can exist in a gas, the normal components of the electric displacement and the tangential component of electric field on the boundary of the piezoelectric half-plane should satisfy D1 ( 0, x2 ) = D10 ( 0, x2 ) , E2 ( 0, x2 ) = E20 ( 0, x2 ) ,

(4)

where the subscript “0” refers to the electric quantities in the gas. Eβ0 and Dβ0 are expressed in

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185

terms of the complex potential Ψ0(z): dΨ 0 ( z ) dΨ 0 ( z ) , D20 + i D10 = −ε 0 , (5) E20 + i E10 = − dz dz where ε0 stands for the dielectric constant of the gas. The complex potentials that satisfy Eqs. 4 and 5 are assumed to have the following forms:

Φ ( z) =

1 1 b3 ln ( z − zd ) + A ln ( z + zd ) , 2π 2π

(6)

Ψ ( z) =

1 1 ∆ϕ ln ( z − zd ) + B ln ( z + zd ) , 2π 2π

(7)

Ψ 0 (z) =

1 B0 ln ( z − zd ) , 2π

(8)

where the first term on the right hand side of Eq. 7 are the complex potential associated with b3 in an infinite piezoelectric medium, while the second denotes the complex potential corresponding to the image elastic displacement jump with strength A at point − zd . The two terms on the right hand side of Eq. 8 have the same physical meanings for ∆ϕ and B. Eq. 9 is the complex potential corresponding to the image electric potential jump with strength B0 at zd . Using Eqs. 7-9 and the conditions given in Eqs. 4 and 5, one obtains A = −b3 −

2e15ε 0

(

c44 ε eff + ε 0

)

∆ϕ , B = −

ε − ε0 2ε ∆ϕ , B0 = ∆ϕ , ε + ε0 ε + ε0 eff

eff

eff

(9)

eff

where ε eff = ε11 + e152 / c44 is the effective dielectric constant of the hexagonal piezoelectric materials [9]. As a result, the complex potentials are determined as Φ (z) =

 2e15ε 0 1  b3 ln ( z − zd ) − b3 ln ( z + zd ) − ∆ϕ ln ( z + zd )  , 2π  c44 ε eff + ε 0  

Ψ ( z) =

  ε eff − ε 0 1 ∆ϕ ln ( z − zd ) − ln ( z + zd )  , ε eff + ε 0 2π  

(11)

Ψ 0 (z) =

1 2ε eff ∆ϕ ln ( z − zd ) 2π ε eff + ε 0

(12)

(

)

(10)

It is seen from Eqs. 11 and12 that the gas outside a piezoelectric half-plane only influences the electroelastic fields induced by the electric potential jump ∆ϕ. Screw Dislocation-crack Interaction In this section we will analyze the interaction of a piezoelectric screw dislocation with a semi-infinite dielectric crack, as shown in Fig. 2.

It is well-known that the mapping function, ω = z , transforms the right half-plane in the λ-plane into the z-plane with a semi-infinite long slit, as indicated in Fig. 3. Therefore, by replacing z,

zd and

zd by z , zd and zd in Eqs. 11 and 12 respectively, solution to the problem

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Macro-, Meso-, Micro- and Nano-Mechanics of Materials

mentioned above can be expressed as x2

zd

x1 c44, e15, ε11

Fig.2. A piezoelectric screw dislocation in a piezoelectric solid with a semi-infinite crack

ζ2

x2

ω= z

ζ1

x1 z-plan

ω-plan Fig.3. A schematic of the conformal mapping Φ ( z) =

1  b3 ln 2π  

Ψ (z) =

 1 ∆ϕ ln 2π 

(

)

z − zd − b3 ln

(

)

z − zd −

(

)

z + zd −

ε − ε0 ln ε + ε0 eff

2e15ε 0

(

c44 ε eff + ε 0

)

∆ϕ ln

(

 z + zd  , 

(

)

eff

 z + zd  , 

)

(13)

(14)

which are the two complex potentials corresponding to the problem in Fig. 2. By substituting Eqs. 14 and 15 into Eqs.2 and 3, the stress, electric displacement and electric field are obtained as follows: σ 23 + i σ 13 =

D2 + i D1 =



1 4π z

 1 1 + , z z z z − +  d d  

( c44b3 + e15 ∆ϕ ) 

 1 ( e15b3 − ε11∆ϕ ) −  4π z  z − zd

(15)

1

 2e ε 1 e15b3 + z + zd  c44 ε eff + ε 0

(

2 15 0

)

  ε eff − ε 0 ε11∆ϕ   ∆ϕ − ε eff + ε 0   

,

(16)

Advanced Materials Research Vol. 9

E2 + i E1 = −

 ε eff − ε 0 1 ∆ϕ  − 4π z  z − zd ε eff + ε 0 1

187

 1 . z + zd 

(17)

In analyzing the interaction of a crack with a dislocation, two quantities of interest are the intensity factors induced by a dislocation near a crack as well as the image force on the dislocation due to the crack. The former can be used to evaluate the effect of a dislocation on crack propagation. The latter can provide information about the motion of a dislocation. Making use of the definition of the stress, electric displacement and electric field intensity factors kIII = lim 2π zσ 32 , k D = lim 2π zD2 , k E = lim 2π zE2 , z →0

z →0

(18)

z →0

and Eqs. 16-18, one obtains θ 1 kIII = − ( c44b3 + e15∆ϕ ) cos d , 2 2π rd kD = −

kE =

1 2π rd

(19)

  c ε ε − e2 ε θ  e15b3 − 44 11 eff 15 0 ∆ϕ  cos d ,   2 c44 ( ε eff + ε 0 )  

(20)

ε eff θ 1 ∆ϕ cos d , 2 2π rd ε eff + ε 0

(21)

where zd = rd ( cos θ d + i sin θ d ) has been used. The above equations show that ε0 only affects the electric displacement and electric field intensity factors due to the electric potential jump ∆ϕ. The image force acting on a dislocation is a configuration force. It is used to judge equilibrium positions and motion of a dislocation. According to the generalized Peach-Koehler formula derived by Pak [11] for piezoelectric materials, we have ∗ F1 = σ 23 b3 + D2∗ ∆ϕ , F2 = −σ 13∗ b3 − D1∗ ∆ϕ ,

(22)

where σ β∗ 3 ( zd ) and Dβ∗ ( zd ) , β =1, 2, are the perturbed stresses and electric displacement at zd due to the crack. They are obtained by subtracting the stress and electric displacement due to a screw dislocation in an infinite piezoelectric medium without a crack. Omitting some details, the two components of the image force are: Fx = −

1 8π rd

{( c

)

b + 2e15b3∆ϕ − ε11∆ϕ 2 cos θ d +

2 44 3

   ε eff − ε 0 2e152 ε 0 c44b32 + 2e15b3 ∆ϕ + ε11∆ϕ 2   ∆ϕ 2 − ε eff + ε 0 c44 ( ε eff + ε 0 )   

,

(23)

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Macro-, Meso-, Micro- and Nano-Mechanics of Materials

Fy = −

1 8π rd

{( c

)

b + 2e15b3∆ϕ − ε11∆ϕ 2 sin θ d +

2 44 3

  ε eff − ε 0 θ  2e152 ε 0 c44b32 + 2e15b3∆ϕ + ∆ϕ 2 − ε11∆ϕ 2  tan d  . ε eff + ε 0 2 c44 ( ε eff + ε 0 )   

(24)

From Eqs. 24 and 25, it can be found that ε0 only has influence upon the contribution of ∆ϕ2 to the image force. When ε0 vanishes, the intensity factors and image force reduce to θ 1 kIII = − ( c44b3 + e15∆ϕ ) cos d , 2 2π rd

θ 1 ( e15b3 − ε11∆ϕ ) cos d , 2 2π rd

kD = −

(25) (26)

kE =

θ 1 ∆ϕ cos d , 2 2π rd

Fx = −

θ 1 c44b32 + 2e15b3 ∆ϕ − ε11∆ϕ 2 cos 2 d , 4π rd 2

(27)

(

)

(28)

θ  1  c44b32 + 2e15b3∆ϕ − ε11∆ϕ 2  sin θ d + tan d  . (29) 8π rd 2  The above are the results for the interaction of a piezoelectric screw dislocation and an electrically impermeable semi-infinite crack [3].

(

Fy = −

)

If letting ε0 approaches infinity, the electric displacement intensity factor and the image force on the dislocation become kD = − Fx = −

 e152 θ 1  ∆ϕ  cos d  e15b3 + c44 2 2π rd  

1 8π rd

{( c

)

b + 2e15b3∆ϕ − ε11∆ϕ 2 cos θ d +

2 44 3

    2 2  2 e15 + ε11  ∆ϕ 2   c44b3 + 2e15b3∆ϕ +   c44     Fy = −

1 8π rd

{( c

(30)

,

(31)

)

b + 2e15b3∆ϕ − ε11∆ϕ 2 sin θ d +

2 44 3

  2 2  2 θ d  , 2 c44b3 + 2e15b3∆ϕ +  e15 + ε11  ∆ϕ  tan  2   c44   

(32)

which correspond to the case of the interaction between a conductive semi-infinite crack and a piezoelectric screw dislocation. From Eqs. 27 and 31, it is found that the electric displacement intensity factors of the two limited cases due to ∆ϕ have opposite sign.

Advanced Materials Research Vol. 9

189

In order to have a better understanding of the influence of ε0 on the interaction, the electric displacement intensity factor and angular component Fθ of the image force are calculated for b3 = 0 . By virtue of Eqs.32 and 33, the radial and angular components, Fr and Fθ, of the image force are obtained as follows: Fr =

2 1 c44ε11ε eff − e15ε 0 2 ( ∆ϕ ) , 4π rd c44 ( ε eff + ε 0 )

(33)

2 θ 1 c44ε eff ( ε11 − ε 0 ) − e15ε 0 2 Fθ = ( ∆ϕ ) tan d . 8π rd 2 c44 ( ε eff + ε 0 )

(34)

Eq.34 indicates that the Fr is not dependent on the angular position θd of the dislocation. Illustrated in Figs.4 and 5 are the variations of the normalized electric displacement intensity factor and the angular component of the image force with the angular position θd of the dislocation for differential values of ε0. They are normalized by K D = 8π rd k D ×108 / ∆ϕ , fθ = 8π rd Fθ × 107 / ( ∆ϕ ) .

Normalized electric displacement intensity factor KD

2

1.6

ε 0 = α c44ε11ε eff / e152

1.4

α=0

1.2

α = 0.05

(35)

1.0 0.8 0.6 0.4

α = 0.5

0.2 0.0

α = 1.0

-0.2 -0.4

α = 5.0 α = 10

-0.6

α →∞

-0.8 -180

-135

-90

-45

0

45

90

135

180

θd (degree)

Fig. 4 Variation of the normalized electric displacement intensity factor KD versus the angular position θd of the dislocation for different values of ε0 The material employed in the calculation is PZT-5H piezoelectric ceramic that possesses hexagonal symmetry. Its material constants are [11]: c44 = 3.53 ×1010 N/m2, e15 = 17 C/m2, ε11= 151 ×10−10 C/Vm, where, N is the force in Newtons, C is the charge in coulombs, V is the electric potential in volts, and m is the length in meters. It is observed from Fig. 4 that in the case of ∆ϕ being positive, kD is large than zero when 0 ≤ ε 0 < c44ε11ε eff / e152 , and smaller than zero when

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Macro-, Meso-, Micro- and Nano-Mechanics of Materials

c44ε11ε eff / e152 < ε 0 < ∞ for θ d ≠ 0o . From Fig.5, It is seen that fθ is positive when 0o < θ d < 180o , but

negative when −180o < θ d < 0o for 0 ≤ ε 0 < c44ε11ε eff / c44ε eff + e152  . However, fθ exhibits the opposite feature for c44ε11ε eff /  c44ε eff + e152  < ε 0 < ∞ . When ε 0 = c44ε11ε eff / c44ε eff + e152  , fθ is equal to zero for arbitrary θd. These results show that ε0, i.e. the electric boundary condition on the crack face, has significant influence on the crack-dislocation interaction in piezoelectric media. 4

β →∞

3

Normalized image force fθ

ε0 = βc44ε11εeff / c44εeff + e152 

2

β =5

β =1

1

0

-1

-2

-3 -180

β = 0.5

β=0

-135

-90

-45

0

45

90

135

180

θd (degree)

Fig. 5 Variation of the normalized image force fθ versus the angular position θd of the dislocation for different values of ε0

Conclusions The interaction between a piezoelectric screw dislocation and a semi-infinite crack in hexagonal piezoelectric materials has been analyzed, where the crack is regards as a thin layer filled with a gas of dielectric constant ε0. The main results can be summarized as follows: (1) The electric potential jump ∆ϕ can induced the strain owing to the presence of a gas in the crack; (2) ε0 has no influence on the stress produced by ∆ϕ but is able to affect the electric displacement and electric field due to ∆ϕ; (3) ε0 only alters the value of the image force associated with ∆ϕ2.

References 1. J. Weertman: Dislocation Based Fracture Mechanics, World Scientific, New Jersey (1996). 2. D.M. Barnett and J. Lothe: Phys. Stat. Sol. (b) Vol. 67 (1975), p. 105 3. K.Y. Lee, W.G. Lee and Y.E. Pak: J. Appl. Mech. Vol. 67 (2000), p. 165 4. J.H. Kwon and K.Y. Lee: J. Appl. Mech. Vol. 69 (2002), p. 55

Advanced Materials Research Vol. 9 (2005) pp 191-198 © (2005) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMR.9.191

Surface electrode problems in piezoelectric materials Kun-Li Zhou, Zhi-Dong Zhou and Zhen-Bang Kuang1 Department of Engineering Mechanics, Shanghai Jiaotong University, Shanghai 1 200240. P.R. China [email protected] Keywords: Surface electrode, Stroh Formulation, analytical continuation method

Abstract In this paper, the surface electrode problems in a piezoelectric half-plane with surface electrode are presented. By the analytical continuation method these problems can be reduced to Reimann-Hilbert problems and the closed-form expressions of the mechanical and electric fields are obtained. This research is useful for the piezoelectric apparatuses. Introduction The electroelastic interactions of piezoelectric ceramics with surface electrodes are very important in piezoelectric transducers and various surface wave devices. It is of great importance to analyze the electric fields and stress distributions at the edge of electrodes and in order to understand the function of the devices and the mechanical and electric failure phenomena. Shindo et al. [1], Deng and Meguid [2] and Ru [3] and others conducted the studies of rigid and soft electrodes at different cases. In this paper attentions are focused on the piezoelectric half-plane with a rigid or a soft surface electrode. Using the standard analytical continuation method and the Reimann-Hilbert theory, the closed-form expressions of the displacements and the electric potentials of these mixed boundary problems are obtained. Some numerical calculations are also given. Basic Equations In a fixed rectangular coordinate system

σ ij = cijkl ε kl − elij E l 1 2

ε kl = (u k ,l + u l ,k )

( x1, x2 , x3 ) , the constitutive equations is: Di = eikl ε kl + κ il E l E i = −φ,i

(i, j, k , l = 1,2,3)

u k ,l = ∂u k / ∂xl ,...

(1)

where cijkl , eikl , κ il are the elastic stiffnesses, piezoelectric stress constants, and permittivities, respectively. σ ij , ε ij u j , Di , Ei , and φ are stress, strain, displacement, electric displacement, electric field and electric potential respectively. Here the general two-dimensional problems in

( x1 = x, x2 = y ) -plane are studied. Following Suo [4]、Chung and Ting [5] and Kuang and Ma [6] the generalized displacement u and generalized stress function Ψ each can be obtained by considering a linear combination of four complex analytical functions, 4

4

α =1

α =1

u = 2 Re ∑ aα fα ( zα ) = 2 Re Af ( z ), u J = 2 Re ∑ AJα fα ( zα )

(2)

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Macro-, Meso-, Micro- and Nano-Mechanics of Materials

4

4

α =1

α =1

Ψ ( z ) = 2 Re ∑ bα fα ( zα ) = 2 Re Bf ( z ), Ψ J ( z ) = 2 Re ∑ BJα fα ( zα )

(3)

where aα J = AJ α , bα J = BJα , zα = x + pα y , ui (i = 1, 2,3) are displacements and u4 is electric potential φ . Ψ j ( j = 1, 2, 3) are resultant forces and Ψ4 is the electric displacement flux on an arc. The uppercase Latin subscript and the Greek subscript all range from 1 to 4, the lowercase Latin subscript from 1 to 3. In this paper that the implicit summation convention is used only for Latin indices, while for Greek indices we write the summation symbol explicitly. The eigenvalues Pα and the eigenvectors a α , b α are determined by

(

)

[Q + R + R T p + Tp 2 ]a = 0 where

Q E Q= T  e11

R E e11  , R =   T − κ 11   e 31

T E e 31  , T =   T − κ 13   e 33

(4)

e 33   − κ 33 

(5)

(e ij )s = eijs QikE = ci1k1 , RikE = ci1k 3 , TikE = ci 3k 3， bα = ( R T + pα T ) aα = − ( Q + pα R ) aα pα

(6)

Let Fα ( zα ) = df α ( zα ) / dzα , the generalized stresses vectors t1 , t 2 are as follows t 2 ( z ) = ΨJ ,1 = 2 Re{BF ( z )},

where

t 1 ( z ) = − ΨJ , 2 = −2 Re{B < p > F ( z )}

(7)

denotes diagonal matrix, t2 j = σ 2 j , t24 = D2 , t1 j = σ 1 j , t14 = D1 .

Soft surface electrode on a piezoelectric half-plane Consider a piezoelectric half-plane with a thin electrode layer of length 2a attached at the surface as shown in Fig.1. Let the electric charge of the electrode layer be q0 and the electric potential be V0. The corresponding mixed boundary conditions are of the form, x ∈ L1 ∪ L3

x ∈ L2 | z |→ ∞

t 2 = 0 or σ 2 j ( x ) = 0, ( j = 1, 2,3) D2 ( x ) = 0

t 2i = 0 (i = 1, 2,3),

φ = V0 ,

∫

a

−a

D2 dx = −q0

σ ij → 0, D j → 0 i = 1, 2,3 j = 1, 2

Using Eqs.(1), (3) and (7) the boundary conditions (8a) and (8b) are reduced to

(8a) (8b) (8c)

Advanced Materials Research Vol. 9

y

S+

L1

L3

L2

−a

193

x

a

0 S−

Fig.1 A half-plane piezoelectric ceramics with a surface electrode x ∈ L1 ∪ L3

BF + ( x) + BF − ( x) = 0

x ∈ L2

BF + ( x ) + BF − ( x) = (0, 0, 0, D2 ( x ))T Af − ( x) + Af − ( x) = (∗, ∗, ∗, V0 )T ,

(9)

∫

a

−a

D2 ( x )dx = − q0

in which ‘*’ denotes some unconcerned quantities and D2 ( x) is unknown. Using the relation +

F − ( x) = F ( x) . We define a new function Φ( z ) F ( z ) Φ( z ) =  −1 −B BF ( z )

z ∈S−

(10)

z ∈S+

Φ( z ) is analytic in whole plane except on L2 . Then Eq.(9 ) 2 reduces to

Φ + ( x ) − Φ − ( x ) = −B −1 (0, 0, 0, D2 ( x )) T = −B −41 D2 ( x )

(11)

where B 4 = [ B14 B24 B34 B44 ]T .The solution can be obtained by Cauchy formula

Φ( z ) = −B 4−1

1 D2− ( x) D2− ( x)dx −1 1 dx + C , Φ ( z ) = − B α α α4 2π i ∫L x − z 2π i ∫ x − zα

(12)

According to Eq. (12) and the zero remote loadings, we get C = 0 . Let D3 ( z ) = X a ( z )iγ 1

X a ( z ) = 1/ z 2 − a 2

(13)

where γ 1 is a real constant. According to Eq.(10) 1 , Eq.(12) is reduced to [6][7] F ( z ) = −B 4−1 Fα ( zα ) =

1 iγ 1 iγ 1 dx = B 4−1 , ∫ 2 2 2 2 L 2π i x − a ( x − z) 2 z −a iγ 1

2 zα − a 2

2

z ∈S−

Bα−14

According to (8b), γ 1 is obtained as γ 1 = q 0 / π . Through integration of Eq.(14) we get

(14)

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Macro-, Meso-, Micro- and Nano-Mechanics of Materials

f ( z) =

iq0 lnβ ( z + z 2 − a 2 ) B −4 1 2π

(15)

where β is a real constant. Let H = iAB −1 which is Hermit matrix and H 44 = ia4α bα−14 is real [3,4]. Using boundary condition Eq.(9), β can be obtained, i.e. q0

π

H 44 ln β a = V0 or β =

 π V0  1 Exp   a  q0 H 44 

(16)

Thus we get f ( z) =

iq0 iV z z ln( + ( )2 − 1) + 0 B −41 2π a a 2 H 44

(17)

  iq0 t 2 ( z ) = 2 Re[BF( z )] = Re B B −41   π z 2 − a2   

(18)

  iq0 iV z z2 ln[( ) + 2 − 1] + 0 B −4 1  u( z ) = 2 Re Af ( z ) = Re  A a a H 44 π  

(19)

It is obvious that near the electrode the general stresses have singularity with the order of 1/√r. For general dielectric without piezoelectric effect, we only need to consider the electricity effect. In this case Q, R , T, A, B, H each has one component,

B44 = −(ε12 + p4ε 33 ),

A44 = −i ε 11ε 33 − ε132 a44

Q44 = −ε11 ， R44 = −ε13 ， T44 = −ε 33 . H 44 = −(ε11ε 22 − ε122 ) −1 2

,

and

p4 = p = [−ε13 + i ε11ε 33 − ε132 ]/ ε 33 . From Eq.(19) the electric potential is obtained φ = V0 −

q0

z z Re ln[ + ( ) 2 − 1] a a π ε11ε 33 − ε

(20)

2 13

For isotropic medium, ε ij = εδ ij and φ = V0 −

q0

z z Re ln[ + ( ) 2 − 1] , which is identified with πε a a

the result in usual dielectric theory. Here we take PZT-5H piezoelectric ceramics to carry out numerical calculation. The engineering material constants are listed in Table 1. The polarization direction of the material is along the 0y axis. We assume that the length of a is 10-2 m. Table 1 Engineering material constants of PZT-5H Elastic stiffnesses Piezoelectric coefficients Dielectric constants (×1010 N / m2 ) C11

C12

12.6 5.3

C13 C33

(×10−10 C / Vm)

(C / m 2 ) C44

5.3 11.7 3.53

e31 -6.5

e33 23.3

e15 17.0

κ11

κ 33 151

130

Advanced Materials Research Vol. 9

195

Fig2 (a) (b) show the distributions of the D2 q0 and σ 22 q0 at x / a = 0,0.5,1.0,1.5 in the depth direction, respectively. With the increasing of the y value, the distribution curves become smooth. We can see in the depth direction that the nearer the point to the electrode edge, the steeper the distribution curve of this point is. On the figure the distributions along positive axis of x are only drawn. The results also show that D2 and σ11, σ22 are of symmetric distributions and D1 and σ22 are of anti-symmetric distributions about the y axis. 2 σ22/ q0 ( x10 Nm C )

0

-1 -2

-4

x/a=0 x/a=0.5 x/a=1 x/a=1.5

-6 -8 0.0

-0.5

-1.0 y/a

0 -2

2

D2/ q0 (x10-6 m-2)

-2

-1.5

x/a=0 x/a=0.5 x/a=1 x/a=1.5

-4 -6 -8 0.0

-2.0

-0.5

-1.0 y/a

(a)

-1.5

-2.0 (b)

Fig 2 Distribution of D2 / q0 (a) and σ 22 / q0 (b) along y axis for soft surface electrode Rigid surface electrode on a piezoelectric half-planes In this case the boundary conditions (8a) and (8c) are still held, but Eq. (8b) changes to x ∈ L2 u = u 0 or u j = u0 j , ( j = 1, 2,3), φ = V0

∫

a

−a

D2 ( x)dx = − q0

∫

a

−a

σ i 2 ( x) dx = 0

(21)

Since translation of general displacements has no effect on the general stresses, so we can let u0 j = V0 = 0 . Substituting Eq.(1), (3) and (7) into the boundary condition (8a) and (21) , we get

x ∈ L1 ∪ L3 BF ( x) + BF ( x) = (0, 0, 0, 0)T (22) x ∈ L2 Af ( x) + Af ( x) = (u01 , u02 , u03 , V0 ) T or AF ( x) + AF ( x) = (0, 0, 0, 0) T Define a new function of complex variable z as z ∈S−

ΒF (z) Φ(z) =   −BF (z)

(23)

z ∈S+

Φ( z ) is analytic in the whole plane except on the segment L2 ，Eq.(22) 2 is reduced to −1

−

Φ + (x)+ H HΦ (x) = 0

(24)

which is the standard Hilbert problem，the solution is Φ( z ) = G ( z + a )

1 − + iε 2

( z − a)

1 − −iε 2

+ D1 z + D2

(25)

where D1 ， D2 are complex constant vectors， ε is the eigenvalue of the following equations,

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Macro-, Meso-, Micro- and Nano-Mechanics of Materials

−1

−1

e 2πεα I − H H = 0,

e 2πεα I − H H G = 0

(26)

From Eq.(26), we can get four values of ε a (α = 1, 2,3, 4) , and ε = ε1 = −ε 2 , iκ = ε 4 = −ε 3 , where

ε and κ are real constants. Because the remote electro-mechanical loading is zero ， so D1 = D2 = 0 . From Eqs.(25) and (26) the general solution of Φ( z ) is, 4

Φ( z ) = ∑ G α Xα ( z ),

Xα (z) =

α =1

 z+a 2 2  z−a  z −a  1

iεα

, α = 1, 2, 3, 4

(27)

Thus, in the S − domain there is F( z ) = B −1Φ( z ) 4

4

4

Fk ( zk ) = ∑ Bkj−1Φ j ( zk ) = ∑∑ Bkj−1Gα j ( zk + a) j =1

(28) 1 − + iεα 2

( zk − a )

1 − −iεα 2

(no summation on k )

j =1 a =1

(1) For general piezoelectric medium ε ≠ 0 , according to the boundary condition when

x > a , t 2 = 2 Re BF( x) = 0 . It is obtained that G 2 = −G1 and G 3′＝iG 3 , G 4′ = iG 4 are pure imaginary constant vectors, where G j

are the eigenvectors of the Eq.(26). Let

−1

G i = [Gi1 , Gi 2 , Gi 3 , Gi 4 ]T , H αβ H βγ = Jαγ ， e 2πεα = g a . From Eq.(26) 2 the relationships between Gβ 1 , Gβ 2 , Gβ 3 and Gβ 4 are

Gβ k = g k( β )Gβ 4 (k = 1, 2,3)

(29)

Using the residue theory, we get the integral

∫

a

∫

a

−a

−a

iε

a 1 a+ x a+ x dx = ∫     2 2 2 2 − a a −x a−x a −x a−x

1

κ

dx =

2π e + e − επ επ

κ

(30)

2π a+x   dx = iπκ a − x e + e − iπκ  

1 a+ x   dx = ∫− a 2 a − x  a −x  a − x2 1

2

− iε

a

2

Using four boundary conditions contained in Eq.(21) and Eq.(30) we get G44 = −

2π (2 m24

q0 , n24 + m34 n34 − 1)

m m G14 = −G 24 = −i ( 14 + 24 )G44 , n14 n24

(31)

m G34 = − 34 G44 n34

The coefficients g k( β ) , mi 4 , ni 4 are the functions of g β , Jαγ and given in appendix. Now the closed-solutions of the normal stresses and normal electric potential are obtained. (2) For a transverse isotropic medium, ε = 0 , the displacement w and the components u , v, φ in xy plane are decoupling. So the characteristic equations are reduced to three-order

Advanced Materials Research Vol. 9

197

matrix. G2 can be deleted and G1 becomes pure imaginary constant vector. Here, the example discussed in section 3 is also taken as an example to carry out numerical calculation, but the rigid electrode substitutes for soft one. In this case we have

ε 1 = −ε 2 = ε = 0, − ε 3 = ε 4 = −0.0388901i = iκ

(32)

It is obvious that general stresses have singularity with ( z ± a) −1 2 , ( z ± a) −1 2+κ , ( z ± a) −1 2 −κ near the electrode ends. The constant vectors Gj are G1 = (0, − 0.126882, − 1.10917)T , G 3 =( − 0.0308026, 0.0634408, 0.634163)T G 4 =(0.0308026, 0.0634408, 0.634163)T

(33)

Numerical results also show that D2, σ11 and σ22 are of symmetrical distributions and D1, σ12 are of anti-symmetrical distributions about 0y axis. Boundary conditions are all satisfied. Fig.3 (a) (b) show the distributions of D2/q0 and σ22/q0 in the depth direction at x/a = 0.0, 0.5, 1.0, 1.5 respectively. Comparing Figs (2) and (3), it is seen that the distributions of electric displacement are similar in the soft and rigid electrode cases, but the stresses are different.

2 -1

σ22/ q0 ( x10 Nm C )

x/a=0 x/a=0.5 x/a=1 x/a=1.5

-4 -6 -8 0.0

-0.5

-1.0 y/a

Fig 3

0

-2

-2

-2

x/a=0 x/a=0.5 x/a=1 x/a=1.5

3

D2/ q0 (x10-6 m-2)

0

-1.5

-2.0 (a)

-4 -6 -8 0.0

-0.5

-1.0 y/a

-1.5

-2.0 (b)

Distributions of D2 / q0 (a) and σ 22 / q0 (b) along y axis for rigid surface electrode

Acknowledgements This work was supported by the National Natural Science Foundation through Grant No. 10132010. References 1. Y. Shindo, Int. J. Eng. Sci. Vol. 36 (1998), 1001-1009. 2. W. Deng, S.A. Meguid, J. Appl. Mech. Vol. 65 (1998), 76-84. 3. C.Q. Ru, J. Mech. Phy. Solids, Vol. 48 (1999) 693-708. 4. Z. Suo, C.M. Kuo, D.M. Barnett and J.R. Willis, J. Mech. Phy. Solids, Vol. 40 (1992), 739-765. 5. M. Chung and T.C.T. Ting, Piezoelectric solid with an elliptic inclusion or hole. Int. J. Solids Struct. Vol. 33 (1996) 3343-3461. 6. Z.B. Kuang and F.S. Ma, Crack tip fields. Xi’an Jiaotong University Press, Xi’an, 2002 (In

198

Macro-, Meso-, Micro- and Nano-Mechanics of Materials

Chinese). 7. N.L. Muskhelishvili, Some basic problems of the mathematical theory of elasticity, Noordhoff, Leyden, (1953) Appendix The coefficients g kβ , mi 4 and ni 4 g1( β ) =

γ 11( β ) + γ 12( β ) ( β ) γ 21( β ) + γ 22( β ) ( β ) γ 31( β ) + γ 32( β ) , g2 = (β ) , g3 = ( β ) γ 13( β ) − γ 14( β ) γ 23 − γ 24( β ) γ 23 − γ 24( β )

γ 11( β ) = [ J14 ( g β − J 22 ) + J12 J 24 ][( g β − J 22 )( g β − J 33 ) − J 23 J 32 ], γ 12( β ) = [ J 34 ( g β − J 22 ) − J 32 J 24 ][ J13 ( g β − J 22 ) + J12 J 23 ], γ 13( β ) = [( g β − J11 )( g β − J 22 ) − J12 J 21 ][( g β − J 22 )( g β − J 33 ) − J 23 J 32 ], γ 14( β ) = [ J 31 ( g β − J 22 ) + J 32 J 21 ][ J13 ( g β − J 22 ) + J12 J 23 ] γ 21( β ) = [ J 24 ( g β − J11 ) + J14 J 21 ][( g β − J11 )( g β − J 33 ) − J13 J 21 ], γ 22( β ) = [ J 34 ( g β − J11 ) + J14 J 31 ][ J 23 ( g β − J11 ) + J13 J 21 ], γ 23( β ) = [( g β − J11 )( g β − J 22 ) − J12 J 21 ][( g β − J11 )( g β − J 33 ) − J13 J 31 ], γ 24( β ) = [ J 32 ( g β − J11 ) + J12 J 31 ][ J 23 ( g β − J11 ) + J13 J 21 ] γ 31( β ) = [ J 34 ( g β − J11 ) + J14 J 31 ][( g β − J11 )( g β − J 22 ) − J12 J 21 ], γ 32( β ) = [ J 24 ( g β − J11 ) + J14 J 21 ][ J 32 ( g β − J11 ) + J12 J 31 ] (1)

(1)

(1)

(1)

m14 = [ g1(4) ( g 2(1) + g 2 ) − g 2(4) ( g1(1) + g 1 )][ g1(1) ( g3(1) + g 3 ) − g 3(3) ( g1(1) + g 1 )] (1)

(1)

(1)

(1)

− [ g1(4) ( g3(1) + g 3 ) − g 3(4) ( g1(1) + g 1 )][ g1(3) ( g 2(1) + g 2 ) − g 2(3) ( g1(1) + g 1 )] (1)

(1)

(1)

(1)

(1)

(1)

n14 = [( g1(1) − g 1 )( g 2(1) + g 2 ) − ( g 2(1) − g 2 )( g1(1) + g 1 )][ g1(3) ( g3(1) + g 3 ) − g3(3) ( g1(1) + g 1 )] (1)

(1)

(1)

(1)

(1)

(1)

− [( g1(1) − g 1 )( g 3(1) + g 3 ) − ( g 3(1) − g 3 )( g1(1) + g 1 )][ g1(3) ( g 2(1) + g 2 ) − g 2(3) ( g1(1) + g 1 )] (1)

(1)

(1)

(1)

m24 = [ g1(4) ( g 2(1) − g 2 ) − g 2(4) ( g1(1) − g 1 )][ g1(3) ( g 2(1) − g 2 ) − g3(3) ( g1(1) − g 1 )] (1)

(1)

(1)

(1)

− [ g1(4) ( g3(1) − g 3 ) − g3(4) ( g1(1) − g 1 )][ g1(3) ( g 2(1) − g 2 ) − g 2(1) ( g1(1) − g 1 )] (1)

(1)

(1)

(1)

(1)

(1)

n24 = [( g1(1) + g 1 )( g 2(1) − g 2 ) − ( g 2(1) + g 2 )( g1(1) − g 1 )][ g1(3) ( g3(1) − g 3 ) − g3(3) ( g1(1) − g 1 )] (1)

(1)

(1)

(1)

(1)

(1)

− [( g1(1) + g 1 )( g3(1) − g 3 ) − ( g 3(1) + g 3 )( g1(1) − g 1 )][ g1(3) ( g 2(1) − g 2 ) − g 2(3) ( g1(1) − g 1 )] (1)

(1)

(1)

(1)

(1)

(1)

m34 = [ g1(4) ( g 2(1) − g 2 ) − g 2(4) ( g1(1) − g 1 )][( g1(1) + g 1 )( g3(1) − g 3 ) − ( g3(1) + g 3 )( g1(1) − g1 )] (1)

(1)

(1)

(1)

(1)

(1)

− [ g1(4) ( g3(1) − g 3 ) − g3(4) ( g1(1) − g 1 )][( g1(1) + g 1 )( g 2(1) − g 2 ) − ( g 2(1) + g 2 )( g1(1) − g1 )] (1)

(1)

(1)

(1)

(1)

(1)

n34 = [ g1(3) ( g 2(1) − g 2 ) − g 2(3) ( g1(1) − g 1 )][( g1(1) + g 1 )( g3(1) − g 3 ) − ( g3(1) + g 3 )( g1(1) − g1 )] (1)

(1)

(1)

(1)

(1)

(1)

− [ g1(3) ( g3(1) − g 3 ) − g3(3) ( g1(1) − g 1 )][( g1(1) + g1 )( g 2(1) − g 2 ) − ( g 2(1) + g 2 )( g1(1) − g 1 )]

Keywords Index A Active Finite Element Analytical Continuation Method Aspect Ratio

9 191 109

B Buckling Failure

31

C Clearance Coating Crack Crack Opening Displacement Cracking Patterns Cyclic Stress

109 21 69 101 117 41

D Damage Rate Duality Solution System Ductility Dynamic Finite Element Analysis

79 143 93 109

E Eigen-Solution Expansion Method Elastic Damage Mechanics Elastic Property

143 117 1

F Failure Process Ferroelectric Degradation Fiber Volume Fraction Finite Element Analysis (FEA) Finite Element Computation Finite Element Model (FEM) Fracture Behaviour Fracture Process of Masonry Functionally Graded Material (FGM) Furcated Fiber

163 69 41 51, 101 153 1 101 117 51 15

G Grain Effect Gun Tube Profile

69 109

H HAp/PLLA Composite Helicoidal Layup Heterogeneous Heterogeneous Material High Density Polyethylene (HDPE) Hydraulic Fracturing

1 15 117, 127 163 101 127, 163

I Insect Cuticle Intensity Factor Isometric Contraction

15 183 9

L Laminated Microstructure Life Local Approach Long Rod Penetrator

15 79 101 109

M Magnesium Alloy Material Constant Mechanical Alloying (MA) Mechanical Properties Mescoscopic Mesoscopic Damage Model Metal Matrix Composite (MMC) Micro-Electromechanical System (MEMS) Microstructure Molecular Orientation Multi-Scale

93 79 93 21 127 117 41 173 93 137 153

N Numerical Analysis Numerical Simulation

137 117, 163

P Phase Piezoelectric Material Piezoelectric Thin Films Plane Couple Stress

93 183 69 143

200

Macro-, Meso-, Micro- and Nano-Mechanics of Materials

Plane Strain Plane Stress Probability Fracture Mechanics Pullout Force Pulsed Laser Irradiation

173 173 101 15 69

79 41, 137 31

S Sabot Screw Dislocation Seepage Mechanics Shielding Effect Short Fiber Composite Shortening Contraction Silicon Carbide (SiC) Skeletal Muscle Small Crack Strain Gradient Elasticity Strain Relationship Strength Stress Concentration Factor (SCF) Stress Relationship Stroh Formulation Surface Electrode

109 183 163 183 41 9 51 9 79 173 21 93 1 21 191 191

T Thermal Barrier Ceramic Coatings Thermal Cycle Thermal Fatigue Thermal Shock Thermo-Elasto-Plastic Finite element Analysis FEA Thermoelasticity Thin Film Analysis Thin Wall Injection Molding Three Dimensional Dynamic Finite Element Analysis

31 31 69 69 41 153 173 137 109

U Unit Cell Model

1

V Von Mises Stress

Weibull Stress

109

101

X X-Ray Diffraction (XRD)

R Ratio Residual Stress Residual Thermal Stress

W

69

Authors Index A Ashihara, F.

59

C Cao, W.B. Chen, B. Chong, A.C.M. Chouanine, L. Chow, C.L. Cui, J.

51 15 173 59 1, 9 153

69

F Fan, J.H. Fan, J.P. Fang, G. Feng, X. Feng, Y.

15 1 143 15 153

51

H Hong, Y.L. Hui, S.C. Huo, L.X.

9 9 101

21 101

K Kamiya, O. Kim, H.G. Kim, J.B. Kuang, Z.B.

59 41, 137 109 191

L Lai, M.O. Lam, D.C.C.

Mai, Y.W. Mao, W.G.

93 31

Nishida, M. Noda , N.

59 69

P Pan, Y.

21

Q Qi, F.J.

101

Takano, M. Tang, C.A. Tang, C.Y. Tham, L.G. Tong, P. Tsui, C.P.

59 117, 127, 163 1, 9 127, 163 173 1, 9

W

J Jiang, Y.P. Jing, H.Y.

M

T

G Ge, C.C.

137 117, 127, 163 183 183 87 93

N

D Deng, S.F.

Lee, H.K. Li, L.C. Liu, J.X. Liu, X.L. Lu, G. Lu, L.

93 173

Wang, B. Wang, S.H. Wang, S.Y. Wu, A.H. Wu, X.L.

87 117, 163 127 51 15

Y Yan, C. Yan, L.L. Yang, F. Yang, G.E. Yang, T.H. Yao, W.

93 93 173 137 127, 163 143

202 Ye, L. Yu, Y.G.

Macro-, Meso-, Micro- and Nano-Mechanics of Materials 93 79

Z Zhang, Y.B. Zheng, X.J. Zhong, W.X. Zhou, K.L. Zhou, Y.C. Zhou, Z.D. Zhu, W.C.

117 69 143 191 21, 31, 69 191 127