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Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved. Composite Laminates: Properties, Performance and Applications : Properties, Performance and Applications, Nova Science Publishers, Incorporated,

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved. Composite Laminates: Properties, Performance and Applications : Properties, Performance and Applications, Nova Science Publishers, Incorporated,

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COMPOSITE LAMINATES: PROPERTIES, PERFORMANCE AND APPLICATIONS

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MATERIALS SCIENCE AND TECHNOLOGIES SERIES

COMPOSITE LAMINATES: PROPERTIES, PERFORMANCE AND APPLICATIONS

ANDERS DOUGHETT AND Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

PEDER ASNAREZ EDITORS

Nova Science Publishers, Inc. New York

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Copyright © 2010 by Nova Science Publishers, Inc.

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Published by Nova Science Publishers, Inc. Ô New York

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CONTENTS

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Preface

ix

Chapter 1

Post –Impact-fatigue Behaviour of Composite Laminates: Current and Novel Technologies For Enhanced Damage Tolerance Alkis Paipetis and Dionysios T.G. Katerelos

Chapter 2

Composite Multilayer Coatings for Improved Barrier Properties of Packaging Board Caisa Andersson

83

Chapter 3

Simulation of Ultimate Strength of Fiber-Reinforced Composites by Means of Bridging Micromechanics Model Zheng-Ming Huang and Ye-Xin Zhou

121

Chapter 4

Micromechanical Analysis for Laminated Composite Materials B.R. Kim and H.K. Lee

201

Chapter 5

Smart Structures for Wireless Communications Seong-Ho Son and Woongbong Hwang

233

Chapter 6

Comparison between Low Velocity Impact and Quasi-Static Indentation Tests on CFRP Composite Laminates Daniele Ghelli and Giangiacomo Minak

271

Chapter 7

Micro-Nano Engineered Composites – New Directions in Biosensing Technologies Maria Marti Villalba and James Davis

305

Chapter 8

Review on Methodologies of Progressive Failure Analysis of Composite Laminates P.F. Liu and J.Y. Zheng

327

Chapter 9

Optic Fibre System for Damage Monitoring in Composite Materials R. de Oliveira and O. Frazão

347

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1

viii

Contents

Chapter 10

Formulation of a Macro-Element to Analyze the Mechanical Behavior of General Composite Laminated Plates Liz G. Nallim and Sergio Oller

365

Chapter 11

Delamination of Composite Structure under Various Types of Loading by Hybrid Mongrel Elements Nguyen Tien Duong and Nguyen Dang Hung

403

Chapter 12

The Surface Integrity of Composite Laminates Subjected to Drilling A.M. Abrão, J. Paulo Davim, J.C. Campos Rubio and P.E. Faria

439

Chapter 13

Failure Process of Carbon Fiber Composites Alexander Tesar

453

Chapter 14

Towards Diffuse Interface Models with a Nonlinear Polycrystalline Elastic Energy Thomas Blesgen and Anja Schlömerkemper

465

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Index

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491

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PREFACE In materials science, composite laminates are assemblages of layers of fibrous composite materials which can be joined to provide required engineering properties, including inplane stiffness, bending stiffness, strength, and thermal expansion. The individual layer consists of high-modulus, high-strength fibers in a polymeric, metallic, or ceramic matrix material. Most fibers in use include graphite, glass, boron, and silicon carbide, and most of the matrix materials are epoxies, polyimides, aluminum, titanium, and alumina. With the ongoing development of the high-tech industry, demand for advanced materials has led to the development of substitutes for traditional engineering materials (i.e., wood, aluminum, steel, concrete). Composite materials have emerged as superior engineering materials due to attributes that are not attainable with existing engineering materials. Due to these advantages, composite materials provide an opportunity for cost-efficient high performance in many weight-critical applications in spite of a product cost impediment compared with traditional materials. This new book gathers the latest research from around the globe in this dynamic field in relation to the properties, performance and application of composite materials. During their service life, composite laminates are subjected to various transient loadings which often lead to the deterioration of their structural integrity. This deterioration may be manifested in various forms; with increasing impact energy these forms range from delamination to flexural damage to penetration. Among the aforementioned damage modes, delamination is of primary importance as (i) it is invisible and consequently requires non destructive evaluation and (ii) it comprises a considerable damage volume in the affected composite laminate compared to other impact induced damage. The residual properties of laminates deteriorate largely for energies just sufficient to cause penetration; at higher energies, the penetrating impactor passes through the laminate cleanly causing minimum damage away from the penetration area. Furthermore, the impact induced damage is prone to further propagation as the composite is subjected to cyclic loading during its service life. As the propagation of delamination damage is of critical importance for the structural integrity of the composite, it is imperative to predict the delamination growth once present and, at the same time, to develop technologies that minimize the damage initiation and propagation. The scope of Chapter 1 is to present a thorough overview of the aspects related to post impact damage in composite laminates and its subsequent development under fatigue loading. An introduction on the performance of impacted composite laminates and their constituent phases under cyclic loading will be presented, including special loading cases such as environmental or impact-fatigue. Theoretical semi empirical and numerical models that relate delamination

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Anders Doughett and Peder Asnarez

propagation to fatigue loading are also thoroughly reviewed. Then, an overview of the existing technologies that minimize impact damage such as stitching or interleaving will be presented and critically evaluated. Special focus is given in non destructive technologies to detect damage and its propagation. The state of the art technologies to minimise impact damage are presented together with the novel concept of hybrid composite systems that employ nano scale phases in order to enhance specific energy absorption and minimize damage propagation. Finally, the applications of damage tolerance technologies in composite structures are discussed. Replacement of petroleum-based plastic films with more environmentally friendly, waterbased coatings that provide sufficient barrier protection to paper or paperboard is a major future challenge for the packaging industry. Natural polymers, such as starch or cellulose polysaccharides, exert many interesting properties for packaging applications besides originating from renewable sources. They are abundant in nature and can be extracted and processed at a reasonable cost. They also provide good film forming barrier properties for oxygen and grease. However, the moisture sensitivity of biobased polymers makes them inappropriate as protective films for use in food packaging. This chapter aims to present new research in the field of composite barrier coatings by incorporation of reinforcing fillers to enhance the moisture barrier properties. Synthetic as well as biobased polymer coatings have been investigated. Composite polymer-filler formulations were prepared by blending nanosized clay in polymer dispersions. A composite material was also prepared by blending synthetic polyester into a biobased polymer dispersion. The composite formulations were applied on top of a three-ply packaging board to form various single- and multilayer laminates. Barrier properties with respect to water vapour permeability, oxygen permeability and water absorption are qualitatively discussed. Both commercial talc-filled barrier dispersion coatings and plastic packaging films were used as references. The properties of single layer and combinations of layers in terms of adhesive properties and synergistic effects were studied by investigation of the surface properties. Chapter 2 shows that a biopolymer based coating can be reinforced with nanosized clay to give barrier properties competitive to commercial, synthetic coating formulations. A dramatic improvement in barrier properties upon application of a thin top nanocomposite coating on various pre-coated structures could also be observed. Bridging model is a well-developed micromechanical theory, which can be used to predict mechanical behaviors particularly strengths of laminated composites based on the properties of constituent fiber and resin materials. Internal stresses generated in the fibers and resin of a laminate are explicitly correlated with externally applied stresses on the laminate. Thermal residual stresses due to un-match between thermal expansion coefficients of the fibers and the resin are obtained rigorously. In Chapter 3, the bridging model development incorporated with updated theory is presented. Computer routine together with input data examples for the First Would Wide Failure Exercise (WWFE-I) problems is given in the appendixes of the chapter. Laminated composite materials are stacked with unidirectional layers at various orientations of the fiber directions to obtain the desired stiffness and strength properties required an acceptable design (Herakovich, 1998). The synergism between constituents of laminated composite materials gives the superior material properties (e.g., low weight, high

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stiffness, and high strength) unavailable from the individual constituent (Herakovich, 1998; Walker and Smith, 2003; Wimmer et al., 2006). With rapid growth of the use of laminated composite materials, the need for the performance prediction and estimation of these materials has been increased. The physical properties of the laminated composite materials are generally not isotropic in nature, but rather are typically orthotropic or anisotropic in which the material properties are dependent on the reinforced fibers and material used in the laminated composites, type of the layer, orientation of fiber axis to the applied forces, etc. (Herakovich, 1998; Kaw, 1997). The micromechanical analysis has been extensively used to solve the problems on a finer scale and to relate the mechanics of materials to their microstructure which cannot directly solve by using the traditional continuum analysis based on the continuity, isotropy and homogeneity of materials (Lee and Simunovic, 2000, 2001). This micromechanical analysis for laminated composite materials is reviewed in Chapter 4. Traditional practice in antenna design for wireless communications has been to design the mechanical structure and antenna as separate entities. Thus, most of the conventional antennas are attached to the surface of a structure, which causes structural instability. The structural stability and light-weight aspect are essential demands on load-carrying structures such as a vehicular body. Recently there has been interest in designing a structurally integrated antenna, the so-called smart-skin antenna. The smart structure is a composite sandwich structure into which microstrip antennas are embedded for radiating electromagnetic waves. The microstrip antenna is very compatible to the sandwich structure. However, it is important to satisfy both structural and electrical requirements that often conflict. Chapter 5 covers the design concept, material selection, fabrication, mechanical characteristics (bending, buckling, fatigue, and impact), and antenna performance (reflection coefficient, radiation pattern, and antenna gain). Furthermore, several experimental models using the smart-skin antenna technology are presented, from single element antenna to phased-array antenna. According to several studies, the damage produced in laminated composites during quasistatic indentation tests may be considered similar to the damage caused by foreign object impact. In Chapter 6, in order to verify this similarity rigorously, low velocity impact and quasi-static indentation tests were done on quasi-isotropic carbon fibre-epoxy resin matrix laminated plates; the elastic behaviour and material damage induced in the two cases were compared. Both dynamic and quasi-static tests were simulated by a two-dimensional finite element model including geometrical nonlinearity. The examination of failures by visual inspection and by optical and scanning electron microscope showed qualitatively similar damage types in impacted and indented specimens: surface indentation, back-face splitting, delamination, fibre fracture and fibre kinking. On the other hand, energy considerations, together with load-displacement relationship and numerical analysis, indicated that in impacted specimens damage is somewhat more severe, thus limiting the validity of the analogy between low velocity impact and quasi-static indentation. As discussed in Chapter 7, the detection of biological molecules has always been of great interest but the complexity of many biofluids and the possible interference of other matrix components can often make the detection of the target analyte difficult. The resolution capabilities inherent to chromatographic techniques has lead to the latter dominating much of the analytical research effort in recent years. There has, however, been an increasing interest in the development of decentralised testing whereby direct measurement and reporting of the analyte concentration is achieved at the site of the analysis – whether it be in the home, in the

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workplace or in the field. Electrochemical technologies have found a niche application in such technologies given their inherent suitability towards miniaturisation but their selectivity has always been open to question – particularly when compared with the more established laboratory based procedures. Composite systems based on the complex interplay of biological and synthetic recognition components with modern materials has led to a revolution in the applicability of such hybrid devices and has effectively opened up a new vista of analytical science. The transfer of laboratory based systems for use by the average person has, until recently, been the dream of Science Fiction writers but the technology has matured considerably in the past few decades and numerous commercial products are now widely available within the retail sector. The devices offer rapid response and hence proffer the possibility of immediate action (i.e. glucose measurements by diabetics) or, in some instances, an opportunity for taking more long term preventative measures (typified by weekly cholesterol measurements). They obviate the need for the transfer of the samples to the lab, the inevitable delays in processing and the possibility of sample degradation that can occur either in transport or storage. Stiffness degradation for laminated composites such as carbon fiber/epoxy composites is an important physical response to the damage and failure evolution under continuous or cyclic loads. The ability to predict the initial and subsequent evolution process of such damage phenomenon is essential to explore the mechanical properties of laminated composites. Chapter 8 gives a general review on the popular methodologies which deal with the damage initiation, stiffness degradation and final failure strength of composite laminates. These methodologies include the linear/nonlinear stress calculations, the failure criteria for initial microcracking, the stiffness degradation models and solution algorithms in the progressive failure analysis. It should be pointed out that the assumption of constant damage variable which is introduced into the constitutive equations of laminated composites to simulate the stiffness degradation properties is less effective and practical than that of changed damage variable with loads in the framework of continuum damage mechanics (CDM). Also, different damage evolution laws using CDM should be assumed to describe three failure modes: fiber breakage, matrix cracking and interfacial debonding, respectively. Composite structures integrity is sensible to service life. Their application in the aeronautical and space engineering implies the necessity to insure their integrity through nondestructive evaluations. On-line health monitoring procedure capable to detect, acquire, and identify damage in fibre reinforced plastic composite materials are necessary. Among the different non-destructive techniques, acoustic emission was chosen for its ability to detect evolutive defects during in-service life of structures. Traditionally, the AE waves are detected at the surface of the structure by piezoelectric transducers. Such transducers have some limitations (e.g. they can’t be used at low/high temperature, and are sensible to electromagnetic interferences). Optic fibre sensors have revealed to be a good alternative. Due to their low dimensions they can be easily embedded in fibre reinforced composite at manufacturing. Chapter 9 discusses the use of an optic fibre system developed for damage monitoring in composite materials from the rapid release of elastic strain energy they generate, detected in the form of elastic waves. Among the different optic fibre sensors, the Fabry-Pérot interferometer is chosen for its high sensitivity to transient phenomena. The propagating acoustic emission waves induce variations of the light in the interferometer. The difficulty

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Preface

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when using such sensor remains the phase recovery. In this study an original set-up is proposed for phase recovery based on the generation of two quadrature-shifted phase interferometric signals from two fibre Bragg gratings. The optic fibre sensor is embedded in a cross-ply carbon fibre/epoxy laminate. The optic fibre sensor system successfully detects periodic ultrasonic waves propagating into the material as well as simulated acoustic emission waves. These tests demonstrate that the optic fibre system is suitable for damage detection from acoustic emission waves. Such in-service health monitoring methodology can be used to locate damage and to determine its severity. A general analytical - numerical approach developed for the statical and dynamical analysis of unsymmetrically laminated plates of general quadrilateral shapes is presented in Chapter 10. An arbitrary quadrilateral flat laminate is mapped onto a square basic one, so that a unique macro-element is constructed for the whole plate. The Ritz method is applied to evaluate the governing equation in which all coupling effects, including those of bending and stretching, are contained. Kinematical assumptions of the Classical Lamination Plate Theory (CLPT) are considered in the equivalent single-layer laminate theories (ESL) context. The plate deflection is approximated by sets of beam characteristic orthogonal polynomials generated using the Gram-Schmidt procedure, always working in the generated macroelement. All possible transverse boundary conditions combining with the different in-plane constraints are considered in the analysis. The convergence studies and the comparisons with results available from the literature indicate that the approach presented is reliable and accurate. Sets of numerical results are given in tabular and graphical form illustrating the influence of different number of layers, fiber stacking sequences and edge conditions on the static deflection, natural frequencies and nodal patterns of a selection of laminated plates. The anisotropic elasticity theory and solution of Lekhnitskii for generalized plane strain state are presented. Wang’s singular solution is used to determine the stress singularity order and eigen coefficients. The stress intensity factors Ki (i = I, II and III), which characterize the toughness of the structure, are calculated. A solution is also considered by the calculation of the energy release rates (Gi and Gtotal) for delamination crack. The local and precise evaluation of the interlaminar stresses which those of Ki, Gi and Gtotal requires a very fine mesh of finite elements near the free edges and at the crack tip of delamination. In order to appreciably reduce the fine mesh in finite elements of pure displacement type, we undertook the development of a finite element in 2 dimensions of the mongrel type (metis element) in regular and singular stress fields. This special class of hybrid finite element assures the monotone convergence was proposed by Nguyen Dang Hung [1]. This element has advantages of the classical finite element and the hybrid element [2-4]. A method allowing the determination of the order of the stress singularity of the delamination crack was presented in Chapter 11. The determination of the stress singularity of the delamination is necessary to know the behavior of the stress in the vicinity of the crack tip. Results of three stress intensity factors and the energy release rates for delamination crack in composite laminates under various types of loading (axial extension, bending, twisting) are presented and compared with the literature to demonstrate the efficiency of the present method.

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In general, composite laminates are produced near net shaped, thus requiring machining operations to achieve the specified dimensions/tolerances and to allow assembly. Owing to the fact that drilling is the machining operation most frequently applied to composite laminates, Chapter 12 is focused on the influence of the machining parameters on the surface integrity of the finished component. More specifically, delamination, surface finish and the dimensional and geometric deviations induced after drilling fiber-reinforced polymeric laminates are discussed based on the cutting phenomena involved. The findings suggest that tool geometry plays a critical role on the surface integrity of machined composites and that standard drill geometries recommended for metal cutting are not suitable for this grade of materials. In addition to that, feed rate is the most relevant machining parameter affecting the integrity of fiber-reinforced polymeric composites, i.e., the higher the feed rate, the higher the damage induced. In Chapter 13, some research results of failure behaviour of carbon fiber composites are presented. The solution of material instability on the basis of fiber kinking theory is adopted for the treatment of the failure process. The micromechanical modeling adopting the FETMapproach is used for numerical analysis of the problem. Some numerical and experimental results with actual application are submitted in order to demonstrate the efficiency of the approaches suggested. Recently in [8], an extension of the Cahn-Hilliard model was derived that takes into account nonlinear elastic energies of the precipitates and includes composite laminates in the physical description. The aim of Chapter 14 is to provide a basis for the further generalization of isothermal diffuse interface models, which we do by developing our methods exemplary for the Allen-Cahn/Cahn-Hilliard equations. Since segregated phases in typical physical applications are polycrystalline, it is natural to incorporate also effects present in polycrystals rather than in single crystals, leading to a polycrystalline lamination theory. To this end we recall some models and methods used in the context of polycrystalline materials and composites. Finally, we outline how the Allen-Cahn/Cahn-Hilliard model can be extended to polycrystalline geometrically linear elasticity.

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Chapter 1

POST –IMPACT-FATIGUE BEHAVIOUR OF COMPOSITE LAMINATES: CURRENT AND NOVEL TECHNOLOGIES FOR ENHANCED DAMAGE TOLERANCE Alkis Paipetis1 and Dionysios T.G. Katerelos2 1

Department of Materials Science & Engineering, University of Ioannina, GR-45110 Ioannina, Greece 2 Department of Sound & Musical Instruments Technology, Technological Educational Institute of Ionian Islands, GR-28200, Lixouri, Greece

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Abstract During their service life, composite laminates are subjected to various transient loadings which often lead to the deterioration of their structural integrity. This deterioration may be manifested in various forms; with increasing impact energy these forms range from delamination to flexural damage to penetration. Among the aforementioned damage modes, delamination is of primary importance as (i) it is invisible and consequently requires non destructive evaluation and (ii) it comprises a considerable damage volume in the affected composite laminate compared to other impact induced damage. The residual properties of laminates deteriorate largely for energies just sufficient to cause penetration; at higher energies, the penetrating impactor passes through the laminate cleanly causing minimum damage away from the penetration area. Furthermore, the impact induced damage is prone to further propagation as the composite is subjected to cyclic loading during its service life. As the propagation of delamination damage is of critical importance for the structural integrity of the composite, it is imperative to predict the delamination growth once present and, at the same time, to develop technologies that minimize the damage initiation and propagation. The scope of this chapter is to present a thorough overview of the aspects related to post impact damage in composite laminates and its subsequent development under fatigue loading. An introduction on the performance of impacted composite laminates and their constituent phases under cyclic loading will be presented, including special loading cases such as environmental or impact-fatigue. Theoretical semi empirical and numerical models that relate delamination propagation to fatigue loading are also thoroughly reviewed. Then, an overview of the existing technologies that minimize impact damage such as stitching or interleaving will be presented and critically evaluated. Special focus is given in non destructive technologies to detect damage and its propagation. The state of the art technologies to minimise impact

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2

Alkis Paipetis and Dionysios T.G. Katerelos damage are presented together with the novel concept of hybrid composite systems that employ nano scale phases in order to enhance specific energy absorption and minimize damage propagation. Finally, the applications of damage tolerance technologies in composite structures are discussed.

Abbreviations AE – Acoustic Emission CAI – Compression After Impact CDM – Continuum Damage Mechanics CF – Carbon Fibre CFRP – Carbon Fibre Reinforced Plastic FBG – Fibre Bragg Grating FEA – Finite Elements Analysis GF – Glass Fibre GFRP – Glass Fibre Reinforced Plastic HVI – High Velocity Impact LVI – Low Velocity Impact NDI – Non-Destructive Inspection OM – Optical Microscopy SEM – Scanning Electron Microscopy SERR – Strain Energy Release Rate

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Introduction Impact as a cause of premature failure of air structures is of major importance. One of the most important air tragedies, the Concord accident that occurred in Paris in 2000 was attributed to impact damage induced by a tire fragment which hit the jet’s fuel tanks. Composite materials, and particularly polymer matrix laminates, are particularly vulnerable to foreign object damage; in many cases, impact results in invisible damage due to the inherent through thickness anisotropy of the laminated structure in conjunction with the brittleness of the matrix. In this case, impact may result to a series of failure modes, not all of which are straightforwardly detected. Low-velocity impact (LVI) in particular is characterized by non catastrophic failure of a specimen; specimens subjected to LVI do not suffer penetration and still exhibit a considerable amount of residual strength [1]. This is expected since LVI involves by definition relatively low impact energy. However, LVI is well known as a major cause of delamination onset. When a projectile or an impactor hits a laminate with low velocity, the localized contact stresses lead to large local out-of-plane stress components. At higher velocities, this phenomenon is driven by localized contact stresses and the propagation of a generated internal stress wave. This stress wave is the cause of delamination initiation at the ply interfaces, where there is a major change of ply stiffness due to the change in the ply orientation [2, 3]. The amount and the type of damage in the laminate depend upon the mass and geometry of the impactor [4, 5] as well as the configuration of the laminate [2].

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Post –Impact-fatigue Behaviour of Composite Laminates

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In general, composite damage is categorized into four main groups, matrix cracking, interlaminar failure (delamination), interfacial failure (fibre-matrix debonding) and fibre failure. While broken fibres are seldom observed after low or moderate velocity impact, matrix damage such as intralaminar cracks and delaminations are always present bellow the apparently intact surface layer of a composite structure (Figure 1). Both these damage modes could be catastrophic for the structure at a secondary level. During the service loading of the structure, local stress concentrations and strain magnifications are created at the vicinity if the impact induced defects, which in their turn give rise to secondary damage modes. These damage modes lead to the reduction of the load bearing and the load transfer ability of the structure and potentially include fibre failures [6], which accumulate to lead to premature failure.

Figure 1. Blind damage induced by Low Velocity Impact (LVI).

Delamination between the constituent laminae of a composite laminated structure is categorized into two groups, interior and exterior. Interior delamination appears far from the structure edges and is mainly caused by incomplete curing, the introduction of foreign particles during manufacturing, or matrix cracks in the longitudinal and the transverse directions that propagate during operation and bridge each other forming a delaminated area. Exterior delamination is due to edge effects, i.e. stress singularities of the out-of-plane stresses developing at the edges of a laminate in the arising stress fields. The ‘interlaminar stress field’ plays a dominant role as far as delamination onset and growth is concerned. This field refers to the stresses arising through the thickness of the laminate and transverse to the plane of the laminate. The onset and growth of delamination after impact is also of major importance as a damage tolerance design parameter. The aim of this study is to review the up-to-date work and present the recent developments in relation to the post-impact damage initiation and growth in polymeric composite laminates. More analytically, the following fields are considered in detail. As

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matrix properties are of primary importance to the performance of the composite, a review of the impact response of polymer matrix materials is performed. The impact behaviour of composite laminates is then examined, taking into consideration thermal effects, natural composites and impact-fatigue. Subsequently, theoretical models for the delamination growth under fatigue loading are presented. Special emphasis is given to the inspection techniques for delamination – and damage in general – developed within a structure, i.e. methodologies for measurement, characterisation and growth quantification. To this end, a detailed investigation of the non-destructive inspection (NDI) methods, such as ultrasonics, acoustography, electrical measurements, optical and imaging techniques is performed. Works that describe the post impact behaviour under static and cyclic loading are summarized and available delamination propagation models are reviewed. A special mention is made for the case of natural composites. An overview of the existing technologies that minimize impact damage, i.e. z-pinning, stitching, interleaving and matrix toughening are presented and critically evaluated. The concept of hybrid composite systems that employ nano scale phases in order to enhance specific energy absorption and minimize damage propagation is presented and elaborated. Finally applications of the post-impact/ fatigue behaviour of specific systems, such as composite joints and innovative materials, like GLARE, are detailed. Summarising, this work presents an integrated approach on the impact and post impact behaviour of composite structures, starting from the causes of the impact damage initiation and its propagation due to post impact thermo-mechanical loading, continuing with the damage identification and quantifying procedures, elaborating on the available studying and predicting methodologies and closing by critically evaluating the available remedying and preventing technologies.

Impact Response Polymers The polymeric matrix of a laminate is of primary importance to the impact response of the composite structure. During the last decade, a series of works have been published examining and explaining the parameters involved in the impact response of polymeric materials used as matrices in composite materials. Perkins [7] reviewed the factors that influence the impact resistance of polymers, emphasizing on crystalline polymers. The phenomenology of polymeric fracture was examined, including brittle and ductile failure and the ductile-brittle transition temperature (DBTT). Subsequently, an in-depth discussion of the effects of crystalline morphology was presented, with special attention paid to the influence of spherulite size, fillers, processing conditions, transitions/relaxations, and multilayer co-extrusion. Rubber phase addition was considered, including mechanisms, morphology, rubber type and particle size, and test conditions. Finally, common impact test methods were surveyed, including pendulum, falling weight, tensile impact, and tensile elongation tests. General guidelines were listed in the review conclusions for improving the impact resistance of a polymeric system: (i) microscopical examination of the fracture surfaces to ascertain the fracture mechanisms under conditions similar to its intended use, (ii) determination of the impact test correlated with the type of fracture, (iii) reviewing the processing procedures and determination whether the morphology imparted during fabrication can be improved, (iv) investigation the addition of

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various types of fillers, (v) determination if manufacturing procedures benefit or harm the structure, (vi) consideration of deformation mechanisms that may provide enhanced ductility and impact resistance. Arogn and Cohen [8] demonstrated that all solid polymers are intrinsically brittle and undergo a ductile to brittle fracture transition based on the nature of their bonding alone. The most effective way of avoiding a ductile to brittle transition is to reduce the plastic resistance to delay reaching the brittle strength which in unoriented polymers is governed by intrinsic cavitation. While a number of possibilities for this exist, the most widely used techniques involve incorporation of rubbery particles or rigid particles. In both approaches the continuous homo-polymer is transformed into a quasi-regular cellular solid that is much more capable of undergoing large local plastic flow by ligament stretching between cavitated particles and is less susceptible to the propagation of brittle cracks under the usual conditions of tensile straining. In many instances, as in epoxies with high flow stresses and strain hardening rates, such cavitation alone can result in attractive levels of toughening by crack tip shielding. In semi-crystalline polymers, however, dramatic effects can be achieved with either rubbery or rigid particles when the quasi-uniformly cavitated cellular polymer can undergo wide spread large strain plastic extension of inter-particle ligaments in slow tension resulting in impressive crack extension resistances J. Arogn and Cohen [8] proved that, under impact conditions, a notched sample rubber particle-modified polymer can still exhibit considerable toughening. However rigid-particle-modified polymers suffered severely from clustering of rigid particles into super critical flaws that trigger brittle response, much like what is encountered in structural steels. Based on their known mechanical response in neat form, six semi-crystalline polymers were analyzed in detail to evaluate their potential for toughening under impact conditions. The results correlated very well with experimental findings.

Impact Response of Fibrous Composites An experimental investigation was presented by Chotard and Benzeggagh [9]. They studied the dynamic behaviour of composite pultruded glass/polyester profiles subjected to low-velocity impact (LVI) with several incident kinetic energy ranges. The results of the drop-weight tests provided specific information about the effects of the impactor velocity, mass and nose radius on the impact response of the profiles. A big-impactor (BI) strike produced less delamination than a small-impactor (SI) one. However, the location of delaminated areas was different for BI and SI impacts. Thus, damage localisation had more influence on the mechanical response of these profiles than the damage extent itself. An experimental study of the sequence of damage mechanisms during impact loading was also carried out. The damage sequence over the impact duration was identified using a methodology based on combined accelerometer/strain-gauge signal analysis. This analysis allowed for the explanation of the impact event and the identification of damage chronology. The accelerometer provided displacements values used to determine the vibration energy. The attenuation (loss) factor calculated from the acceleration signal and its variation was correlated with damage extension. The chronology of separated damage mechanisms was determined with good accuracy. Two different types of failure -cracking and delaminationwere observed. Finally, an inverse variation of damage characteristics versus velocity and

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energy magnitude was emphasised, which demonstrated the separate effects observed between mass and velocity characteristics and the influence of experimental parameters on the impact response of pultruded beams. The effect of laminate structural geometry on impact performance of woven aramid fibre - polyethylene (PE) fibre/vinylester hybrid composites was examined by Park and Jang in [10]. The impact properties of one- and two-layer hybrid composites were compared with respect to impact mechanism and deformation extent. In one-layer composites, two intraply hybrids exhibited different deformation extent and load transfer at crossover points. Intraply fabric hybrid laminates with the same amount of aramid and PE fibres absorbed most of impact energy through large deformation of PE fibres with an elliptical damage shape. The bundle of PE fibres existed around the perforated point with a slippage of PE fibres toward the impact point. The laminate with a 3:1 ratio in the aramid/PE fibres volume fraction showed the higher impact energy because both aramid and PE fibres contributed to the absorption of impact energy with a round damage zone and the load transfer to the PE fibre direction was restricted by adjacent aramid fibres. The laminate exhibited aramid fibre breakages near the impact point with little slippage of PE fibres. In the case of two-layer composites, interply hybrid composites exhibited higher impact energy than intraply hybrid composites. The impact energy of interply hybrid composites was mainly absorbed through delamination, and thus impact energy was well correlated to delamination area. The impact energy of intraply hybrid composites was primarily dominated by full exertion of deformation in PE fibre rather than a delamination process. Dutta et al [11] observed the fracture morphology and texture of graphite/epoxy composite fragments after impact. They identified and explained the observed changes in the surface texture of the fragments for different impact velocities (122 to 610 m/s, 400 to 2000 ft/s) and over a wide range of specimen temperatures (-54 to 24 °C, -64 to 75 °F). The composite panels were impacted by spherical steel projectiles. The spall was differentiated according to different sizes and shapes. A few fragments representing each shape and size were selected to analyze the surface morphology using SEM. Changes in the surface texture were observed according to the different sizes and shapes, and the change in size and shape of the fragments was credited to the change in impact force. Unidirectional panels absorbed more energy at low velocity range. A series of interesting conclusions were drawn from the experimental data. Cross-ply laminates absorb more energy than 6061-T6 aluminium at room temperature, but not at low temperature. At low temperature, composites require much less energy to delaminate per unit delaminated area. The energy absorbed by quasiisotropic panels was higher than the energy absorbed by cross-ply panels indicating that toughness increases with multiple orientations of the fibres. Of the three different types of surface textures, i.e., gravel, hackles and matrix rich areas, gravel was the dominant morphology. At low temperature, because of clean debonding, the smooth matrix area was more common; while at room temperature the matrix-rich area was relatively rough (Figure 2).

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Figure 2. Texture changes with depth for impacted CFRPs at room temperature [11].

The fracture behaviour and failure mechanisms of composite laminates containing woven fabrics in mode I and mode II delamination and under impact loading were investigated in the overview performed by Kim and Sham [12].Potential advantages of using woven-fabric composites over those made from angle-ply unidirectional prepreg tapes were discussed in terms of the interlaminar fracture toughness, the impact damage resistance and damage tolerance (Figure 3). Angle-ply laminates possessed higher mode I interlaminar fracture toughness, GIC, than (0°/0°) unidirectional laminates as a result of the extra toughening mechanisms associated with crack branching into the adjacent layers, leading to multiple delamination. Woven-fabric laminates also showed much higher GIC values (often more than 4 – 5 times) than their unidirectional counterparts. Unique features and advantageous failure mechanisms were identified: inherent roughness of the fabric; the availability of resin-rich regions between the fabrics; crack propagation along the undulating pattern of the yarns creating a large fracture surface area; and multiple crack front formation. The mode I interlaminar fracture behaviour of glass-woven-fabric composites was shown to be controlled by the fibre-surface treatment. A weak interface bond offered by a low silane concentration promoted the stability of crack propagation, with high interlaminar fracture toughness. The major failure mechanisms taking place in mode II delamination were fibre/matrix interface debonding in shear, and fracture of matrix resin between the reinforcing fibres. There was little dependence of GIIC value on interply angle as this was dominated by the interfacial bond quality. The glass-woven-fabric laminates containing high silane concentrations had a strong interface bond with high GIIC values. Woven-fabric laminates in general exhibited a lower maximum load, a smaller damage area, a higher ductility index, and a higher residual compression-after-impact (CAI) strength than cross-ply laminates. Higher mode II interlaminar fracture toughness, lower thermal and elastic mismatches along with a more ductile and compliant nature of the fabrics were mainly responsible for these observations for

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woven-fabric laminates. The effects of fibre/matrix interface properties influenced by different silane coupling agents on interlaminar fracture and the impact performance of woven-glass-fabric composites were evaluated. A higher silane concentration results in improved impact performance of the composite through its ameliorating influence on GIIC values.

Figure 3. Schematic illustration of the sequence of delamination crack propagation in a woven fabric laminate [12].

A plane stress continuum damage mechanics (CDM) model for composite materials was implemented in the non-linear finite element analysis (FEA) code, LS-DYNA3D and its predictive capability was evaluated by carrying out a series of numerical simulations involving impact of laminated composite plates [13]. The performance of the model relative to that of an existing model in LS-DYNA3D was assessed by comparing the computed results with instrumented impact test results. It was demonstrated that a CDM based approach offered a versatile tool for predicting damage progression in composite structures. Although

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the parameters used in such models are often non-physical and difficult to characterize, the formulation presented a unique and interesting approach to the numerical simulation of damage in composite materials. According to the authors [13], the work reported represented the first step in an ongoing effort to develop a versatile, numerically robust, and physically meaningful CDM based constitutive model for composite materials. Morozov et al [14] studied the damage resistance of laminated composite helicopter blades subjected to impact loading using combined theoretical and experimental approaches. Dynamic stress intensity factors were determined for composite laminates and separate layers as functions of the size of the defects caused by the projectile impacts. The effect of the projectile size on the damage tolerance of composite blades was investigated and tolerable sizes of the defects were estimated. The methodology adopted for the study proved capable to produce results applicable to the analysis and damage tolerant design of composite structural components subjected to impact loading by projectiles. The comparison of the experimentally determined critical values of stress intensity factors with the computational results yielded maximum allowable sizes of the defects depending on the size of the projectiles. Corum et al [15] attempted to experimentally characterize the susceptibility of three candidate automotive structural composites to incidental low-energy impact damage. The composites, each of which had the same urethane matrix, were produced by a rapid moulding process suitable for high-volume automotive applications. The reinforcement for the first composite was a random chopped-GF, while the remaining two were reinforced with stitchbonded carbon-fibre (CF) mats, one in a cross-ply layup, and the other in a quasi-isotropic layup. A pendulum device, representative of events such as tool drops (LVI), and a gas-gun projectile, representative of events such as kick-ups of roadway debris (high velocity impact – HVI), were used to impact plate specimens. The impactor point in both cases was a 12.7-mmdiameter hardened-steel hemisphere. The effect of hydrothermal exposure on the impact performance of the specimens was investigated. Brick-drop tests were also performed to assess the applicability of the baseline pendulum and gas-gun data to other events. Following the impacts, the damage areas were measured and the plates were cut into either tensile, standard compressive, or CAI specimens for determining strength degradation. The GF composite was least susceptible to damage, followed by the cross-ply CF laminate with the same thickness. The quasi-isotropic CF composite, which was thinner than the other two, exhibited the maximum damage. While compressive strength was significantly degraded by moderate damage in the random-GF composite, tensile strength was not since fibre breakage rarely occurred in the damage region. On the other hand, both tensile and compressive strengths were degraded in the cross-ply CF laminate. The compressive strength degradation for a given damage area was similar in the two CF laminates. Both exhibited less degradation than the GF composite. For the quasi-isotropic CF laminate, it was shown that strength degradation produced by an open circular notch provided a reasonable lower bound to the degradation due to an impact damage area of the same size. This was in line with the common assumption often used in damage tolerance evaluations [16], that impact-induced defects introduce the same strength reduction as a hole of the same size (Figure 4). While damage in the moulded random GF composite correlated well with kinetic energy, the same was not true for the CF laminates. For these composites, the damage area produced by the pendulum was about twice that produced by the gas-gun projectile for the same kinetic energy. To correlate the aforementioned experimental findings, the parameter mav was employed, where m and v were the impactor mass and velocity, and the exponent a was a constant. A single baseline

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curve relating damage area to impactor mass and velocity was obtained for each composite. These curves were also shown to provide adequate approximations of the damage produced in pre-soaked specimens and specimens tested at -40 °C. With the significant exception of two tests, they gave generally conservative approximations for damage produced by brick drops.

Figure 4. Comparison of CAI strength degradation due to circular holes in quasi isotropic laminates to that due to impact damage [15].

The suitability of damage – force maps for characterising LVI damage in a glass/polyester composite was investigated at low and intermediate energies [17]. Tests were performed on circular plates with diameters ranging between 50 and 300 mm and on square plates with edge lengths of 75 and 200 mm. Damage appeared in the form of delamination under the point of impact and widespread matrix cracking. The translucent nature of the composites facilitated the determination and quantification of the extent of damage within each specimen. Plots of delaminated area against impact force yielded maps in which the experimental data laid within a narrow band over the range of conditions examined. For a given impact energy, the impact force was greater and damage was more severe in smaller coupons. The damage – force maps showed that the impact force to generate delamination lay between 600 and 800 N. An energy-balance model was used to successfully predict the impact response of circular structures and to predict the onset of damage within these composite plates. The model could predict the impact response of targets that incur reasonably high levels of damage and provide an initial estimate of the impact conditions required to initiate interlaminar failure in fibre-reinforced composites. Kang et al [18] dealt with the effect of constituent materials on impact damage and the strength reduction of a sandwich structure composed of laminated facesheets and Nomex®

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honeycomb core. LVI tests were performed and damage was inspected by scanning acoustic microscopy. In addition, static tests were conducted under flexural loading. The damage behaviour was characterized through the energy absorbed during impact. The strength reduction was evaluated via a residual strength prediction model and was found to depend both on the facesheet material and core thickness. A statistical model was developed to identify the variation of residual strength of the impacted structure. It showed that the residual strength was affected by both the facesheet material and core thickness. Quasi-isotropic and cross-ply glass/epoxy laminate plates were investigated in crush and LVI crush tests [19].The characterization of the damage was done in relation to the type of test, the ply stacking sequence, the plate dimensions and the maximum force reached during impact. There was a strong correlation between the force-displacement curves from the crush tests and impact tests. The maximum load caused the identical deflection in both tests corresponding to the same maximum force, therefore indicating that the deformation was not sensitive to the velocity of the load application. For a specific laminate sequence, damage parameters were similar for crush and impact tests when flexibility was less than a threshold value. From this and the previous statement, it was concluded that for laminate structural components with certain boundary conditions and whose dimensions and thickness give a flexibility factor under a certain value, crush tests are a viable alternative for evaluating low velocity object impacts. In the case of highly flexible components, although force displacements curves were similar, damage area and length were different for the same lamination sequence and so the aforementioned conclusion was not valid. The quasi-isotropic plates in general supported higher maximum loads than cross-ply specimens, although, in general, they exhibited greater damage. The increased non-linear force/deflection behaviour of the most flexible plates was verified. For a Delamination Threshold Load, the duration of the impact depended on the transverse flexibility of the plate. Contact time increased with flexibility. The shape of the damage depended on the fibre orientation, and could be approximated with a circle for low flexibility. The through thickness distribution of damage for the stiffest plates was of the typical trapezoid shape with the longest base in the face opposite to the load application. For the most flexible plates, damage was practically constant across the thickness. The indentation depth increased with decreasing flexibility of the plate. Finally, plates absorbed more energy in the crush tests than in the impact tests and that difference was more prominent with decreasing flexibility.

Effect of Temperature on Impact Response The effect of temperature variations for both low and high temperature ranges was studied experimentally in relation to impact damage to carbon fibre reinforced plastic (CFRP) laminates [20]. CF/epoxy orthotropic laminated plates with cross-ply lay-ups ([06/906]s and [04/904])s, and CF/PEEK orthotropic cross-ply laminated plates ([04/904]s) were used. A steel ball launched by an air gun inflicted the impact damage on the CFRP laminate. Nondestructive evaluation (NDE), such as scanning acoustic microscopy (SAM) was performed on the delamination-damaged samples to characterize damage growth at different temperatures. The increase in the temperature of a CFRP laminate resulted in decreasing impact induced delamination areas.

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Gómez-del Rio et al [21] examined the impact response of CFRP laminates at LVI under low temperature conditions. Square specimens of CF/epoxy laminates with different lay-ups (unidirectional, cross-ply, quasi-isotropic and woven laminates) were tested using a drop weight device. The test temperature ranged from 20 down to –150 °C. After the impact tests, the damage extension was measured by C-Scan ultrasonic inspection and the damage mechanisms were studied by optical and scanning electron microscopy (OM and SEM, respectively). The influence of temperature, ply reinforcement architecture and stacking sequence on the mechanical behaviour of the CFRP laminates subjected to low velocity impulsive loads was evaluated (Figure 5). The experimental results obtained showed that cooling the laminate before impact had an effect on damage similar to that of increasing the impact energy, i.e., larger matrix cracking and delamination extension, deeper indentation on the impacted side, and more severe fibre – matrix debonding and fibre fracture on the opposite side. However, an essential difference between both damage-increasing effects was observed; whereas the energy absorbed by the laminate was roughly proportional to the impact energy, larger damage extent favoured by low temperatures was not followed by a greater dissipation of energy due to the lower specific fracture energy of the epoxy matrix at cryogenic conditions. The threshold energy decreased up to 50% in quasi-isotropic laminate when temperature fell from 20 to -150 °C. This thermally induced effect was severe in tape laminates (cross-ply and quasi-isotropic), where orientation of the carbon fibres was different in each layer. In these materials, low temperatures produced interlaminar residual thermal stresses, high enough to accelerate matrix cracking and delamination during LVI. For multidirectional laminates manufactured with plain woven unidirectional fabric, the aforementioned phenomenon was not as prominent, and damage extension after impact did not increase substantially at low temperature.

Figure 5. Effect of temperature on impacted quasi isotropic plates 20 °C (left), -60 °C (center) and -150 °C (right) [21].

Natural Composites Sabeel Ahmed et al [22] explored experimentally the effects of hybridization of GF on LVI behaviour and damage tolerance capability of woven jute fabric reinforced isophthalic polyester composite. Laminates were fabricated by hand lay–up and cured under light pressure at room temperature. LVI tests were conducted on jute and hybrid samples (150×150mm) using an instrumented drop weight impact tower. Some of the samples were subjected to the NDI (C-scan) to study the nature and extent of damage and to measure the

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delamination area. Post-impact tension tests were conducted to assess the damage tolerance capability of the composites. The results of the study indicated that jute laminates had better impact energy absorption capacity than jute–glass hybrid laminates; however their damage tolerance capability was less than that of jute–glass hybrid laminates.

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Impact-Fatigue Impact-fatigue as a distinct case of fatigue is of major importance, since it can simulate realistic working conditions of composite components. As it is correlated to fatigue [23], it will occupy a part in the present review. The theoretical analysis of impact-fatigue phenomena was presented [24] and the resultant fatigue equation was tested by experimental impact-fatigue data. Fatigue life proved to be a function of fibre orientation with respect to the impact load direction. There was a minimum in impact energy for failure to occur, or an impact-fatigue limit. The lifetime equation of impact-fatigue was derived and then analysed statistically to predict the engineering lifetime for design purposes. As postulated, impactfatigue behaviour can be analysed by means of fracture mechanics concepts, because crack propagation is the governing mechanism [25]. One of the very first studies of impact-fatigue was performed on 63.5% GF/vinylester resin, notched composites [26]. A well-defined impact-fatigue (S–N) behaviour, having a progressive endurance below the threshold single cycle impact fracture strength and possessing a fatigue limit was experimentally defined. Fractographic analysis revealed fracture by primary debonding, with fibre breakage and pull-out in the tensile zone and a shear fracture of fibre bundles in the compressive zone of the specimen (Figure 6). The residual strength measured after impact-fatigue showed initially retention of the strength at high impact energy levels and then a gradual drop followed by a rapid drop. The residual modulus and toughness showed a gradual drop with increasing number of impacts. It was suggested that few large cracks for high impact endurance and an increased volume of microcracks in the matrix for low impact endurance account for failure of the composite under impact-fatigue.

Figure 6. SEM micrographs of composite plates submitted to impact-fatigue showing shear failure of the fibres [26]. Composite Laminates: Properties, Performance and Applications : Properties, Performance and Applications, Nova Science Publishers,

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Azouaoui et al carried out an investigation [27] on the impact-fatigue damage of glass/epoxy laminated composites. Damage accumulation, such as matrix cracking, delamination and fibre breakage was the immediate result of repeated impact of the composite material and led to the reduction of the overall stiffness. A model was proposed for characterizing damage as a function of the impact number. The macroscopic failure mode and the internal damage in glass/epoxy laminated specimens as a consequence of impact-fatigue were analysed for different levels of incident impact energy. In the work presented by Barkoula et al [28], the influence of stacking sequence, existence and position of interleaves on the solid particle erosion in CFRP composites was investigated. The erosive wear behaviour was studied in a modified sandblasting apparatus at 90° impact angle. The erosion behaviour was considered as a repeated impact process or impact-fatigue. A semi-empirical approach initially developed for the prediction of the residual strength after single impact was adopted and evaluated in the case of erosion conditions. The excellent agreement between theoretical predictions and experimental values proved the reliability of this model as a useful tool for the prediction of the post impact residual strength in the case of solid particle erosion (Figure 7).

Figure 7. Normalised residual tensile strength versus impact energy and model predictions: various laminate configurations [28]. Composite Laminates: Properties, Performance and Applications : Properties, Performance and Applications, Nova Science Publishers,

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Schrauwen and Peijs [29] described the results of falling weight impact tests on GFreinforced laminates. The test program consisted (i) of falling weight impact tests for the determination of the penetration energy and the influence of laminate construction on damage development and (ii) repeated falling weight impact tests for the determination of the impactfatigue life and damage development under repeated impact conditions at sub-penetration energy levels. The objective of the work was to compare the impact behaviour of cross-ply laminates based on a brittle unsaturated polyester resin and a more ductile vinylester resin system with two types of glass reinforcement, i.e. woven and multiaxial non-crimp fabric. The penetration energy of the various composite laminates depended on the type of reinforcement, whereas damage development during repeated impact was strongly influenced by both fibre architecture and resin. No significant effect of the different material parameters investigated on the number of impacts to penetration was observed. Especially when the repeated impact energy was normalised with respect to the penetration energy, all laminates exhibited similar behaviour. Williams et al [30] focussed on the development, implementation, and verification of a plane-stress continuum damage mechanics (CDM) model for composite materials. A physical treatment of damage growth based on the extensive experimental literature on the subject was combined with the mathematical rigour of a CDM description to form the CDM model. The model was implemented in the commercial FEA code LS-DYNA. Its prediction of impact damage growth and its representation on the impact force histories in carbon fibre reinforced plastic laminates were shown to be physically meaningful and accurate. Furthermore, it was emphasised that the material characterization parameters could be extracted from the results of standard test methodologies, for which a large body of published data is available in the literature. The impact-fatigue properties of three kinds of GF or glass-bead reinforced polymers were studied [23] via uniaxial and cyclic indentation tests with concurrent Acoustic Emission (AE) and visco-elastic properties measurements. The detailed failure mechanisms and damage development mechanisms were investigated with acoustic velocity measurements and OM observations. Prior to fatigue tests, the materials were characterized by basic mechanical tests such as tensile tests, indentation tests, and drop weight tests. When performing fatigue tests, special attention was paid to the effect of interval times between loading on the fatigue properties of the composites. It was found that the numbers of cycles to failure were strongly dependent on the duration of the interval time. In the case of uniaxial fatigue tests, longer interval times resulted in larger numbers of cycles to failure. On the other hand, in cyclic indentation tests, longer interval time resulted in smaller numbers of cycles to crack initiation and ultimate failure. The results of AE measurements were in good agreement with the results of cycles to failure in fatigue tests and the test conditions. Attention was also paid to the relationship between the size of the damage zones and interval time between loadings. The difference in the cycles to failure due to the different loading modes and interval time were understood by the differences in the mechanisms of damage zone development, where breakage of glass, micro-voiding, change in the orientation of glass fibres, and plastic deformation of matrix occurred. The degree of damage localization had a strong effect on the fatigue life. The differences in the damage development mechanisms were attributed to the difference in the elastic response of the specimens due to the different loading mode (uniaxial or indentation) and the interval time.

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Repeated LVI drop weight tests were carried out [31] on advanced composites (glass, carbon, Kevlar® in epoxy matrix) using an instrumented impact test machine. Number of drops to failure (Nf) data was obtained with simultaneous recording of the load–time and energy–time traces. The extent of impact damage for glass and Kevlar® composites was interpreted using final delamination area maps or tracings. Results for all three composites showed that with an increase in drop numbers, the peak load (PL) steadily decreased while the total energy (Et) to failure increased. Also, as the incident energy (Ein) was varied in arithmetic progression, the number of drops to failure (Nf) varied in harmonic progression. It was concluded that repeated drop tests with final delamination area maps assist in understanding the impact damage tolerance of polymer composites. Meo et al [32] described the results of an experimental investigation and a numerical simulation on the impact damage for a range of sandwich panels. The test panels were representative of the composite sandwich structure of the engine nacelle Fan Cowl Doors of a large commercial aircraft. The LVI response of the sandwich panels was studied at five energy levels, in order to assess damage initiation, damage propagation, and failure mechanisms. A numerical simulation was performed using LS-DYNA3D transient dynamic FEA code for calculating contact forces during impact along with a failure analysis for predicting the threshold of impact damage and initiation of delamination. Good agreement was obtained between numerical and experimental results (Figure 8). In particular, the numerical simulation was able to predict the extent of impact damage and impact energy absorbed by the structure. It was shown that a rigid numerical model can yield significant information towards the understanding of the mechanisms involved in LVI even prior to testing, and therefore aid to designing a more efficient impact-resistant aircraft structure.

Figure 8. Impact damage area (a) matrix cracking mode (0 failed-1 elastic) and experimental versus FE results [32].

The effect of repeated LVI on the performance of carbon/epoxy composites with three different stacking sequences was evaluated [33]. The evolution of the macroscopic damage was characterized by a phenomenological quadratic equation. The coefficients of the equation were parametrically adjusted to the experimental points and related to the degree of freedom of the composite to deform, the rate of change of the composite properties due to the impact events, and the change of the damage mode. These coefficients were shown to be dependent on the laminate stacking sequence and successfully described the macroscopic behaviour of

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the composites. The experimental results showed that the cross-ply and non- symmetric laminates had better endurance against LVI than unidirectional laminates. As was expected, this result was due to multidirectional reinforcement which does not allow for “easy paths” for cracks at matrix rich areas. As for unidirectional laminates, fast trough thickness cracks developed, rapidly leading to failure. For all studied configurations, the surface crack pattern followed the fibre orientation of the top laminae, indicating matrix and fibre-matrix interface rupture. Moreover, the cross-ply non-symmetric laminates developed huge surface indentations, easily observed by naked eye, well in advance of their failure. Therefore, these two configurations were safer with respect to the repeated LVI. The influence of laminate thickness on the resistance to repeated low energy impacts of glass, carbon and aramid fabric reinforced composites was evaluated for two levels of impact energy [34]. According to the study, below a certain energy level the cross section of the laminate was the most dominant parametre that determined impact resistance. For the same cross section, the experimental points for all tested laminates fell on a single curve, irrespective of the reinforcement. When the energy level of the impactor was increased, fibre characteristics became significant. The glass fabric reinforced composite showed the steepest increase on the impact resistance with increasing laminate thickness. This behaviour was attributed primarily to the higher areal coverage of the glass fabric used. The more isotropic behaviour of GF in relation to the anisotropic character of aramid and CF was also found to be of relevance. The experimental data were fitted against laminate thickness using a quadratic equation [33]. Studies were undertaken on the effects of different post-cure schedules on the impact resistance of room temperature-cured glass/epoxy laminates subjected to repeated drop tests, using an in-house built un-instrumented drop weight impact tester [35]. The impact resistance was assessed in terms of two parameters, viz., the number of drops to failure (Nf) and the delamination area growth (dA). The specimens were post-cured at different time–temperature combinations and the impact response was studied at three different incident energy levels. It was found that, the Nf and dA values increased to a maximum at an optimum post-cure schedule of 85 °C/4h. The extent of composite cure for the different post-cure schedules was ascertained by measuring the glass transition temperatures (Tg), in order to obtain a correlation between the impact resistance and the chemical cure status of the composite. In addition, single drop impact studies were made using an instrumented impact tester and the efficiency of the optimum post-cure schedule was confirmed. The objective of the experimental campaign presented by Baucom et al [36, 37, 38] was to obtain a detailed understanding of the effects of reinforcement geometry on damage progression in woven composite panels under repeated drop-weight impact loading conditions. The composite systems included a 2D plain-woven laminate, a 3D orthogonally woven monolith, and a biaxially reinforced warp-knit. The radial spread of damage was smallest for the 2D laminates and largest for the 3D woven composites. The 3D composites had the greatest resistance to penetration and dissipated more energy in total than the other systems. This was attributed to unique energy absorption mechanisms, which involved the crimped portion of z-tows in the 3D composites implying that failure was controlled by the configuration of the z-tows. The 3D systems exhibited both an inherent capability to dissipate energy over a larger area and greater perforation strength than other systems with comparable areal densities and fibre-volume-fractions.

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Belingardi et al [39] investigated the response of a non-symmetric GFRP to repeated impacts. The laminate was intended for nautical applications, was manufactured using hand lay-up with subsequent vacuum bagging and possessed different matrix systems on each side, i.e. polyester and vinylester. With the aim of looking into the role of the matrix, two series of static indentation and repeated impact tests were performed, impacting the vinylester face and the polyester face of the laminate respectively. For the impact tests, four impact velocities were considered, and a minimum of four specimens for a given velocity were subjected to forty repeated impacts or up to perforation. The impact response was evaluated in terms of damage progression by visual observation of the impacted specimens, evolution of the peak force and of the bending stiffness with the number of impacts and by calculating the damage degree, defined as the ratio between the absorbed energy and the impact energy. For impact velocities for which no perforation occurred within the test duration, the experimental data for the two series essentially overlapped. On the contrary, for perforation tests, the vinylester resin exhibited higher peak forces but a lesser number of impacts to perforation. Visual observation of the impacted specimens revealed that, in both series of tests, lamina splitting occurred at the interface between the two resin systems. Moreover, the delamination area grew very rapidly during the first few impacts and then levelled off, following closely the behaviour of the bending stiffness against the number of impacts. As for fibre damage, the vinylester impacted face specimens exhibited little damage confined to the contact area with the impactor top, while the polyester impacted face specimens showed cross-like damage patterns along fibre directions. The impact-fatigue properties of unidirectional CF-reinforced polyetherimide (PEI) composites were evaluated [40] by subjecting standard izod impact samples to LVI at energy levels ranging from 0.16 to 1.08 J by using a pendulum impact tester. The effect of the previous LVIs on the residual impact properties of the laminates was investigated. Composite laminates were also subjected to repeated LVI tests up to fracture. Results of repeated impact study were reported in terms of peak load, absorbed energy and number of impacts. Fractographic analysis revealed fracture by primary debonding, with fibre breakage and pullout in the tensile zone, accompanied by a shear failure of fibre bundles in the compressive zone of the specimen. The effect of solid particle erosion on the strength and fatigue properties of E-glass/epoxy composite was investigated [41]. Solid particle erosion with SiC particles of 400–500 mm in diameter was simulated on 12-ply [45°/–45°/0°/45°/–45°/0°]S composites with a constant particle velocity of 42.5 m/s and a solid particle to air volume ratio of 6 kg/m3 at impact angles of 90°, 60°, and 30° for 30, 60, 90, and 120 s. Damaged and undamaged specimens were subjected to tensile tests while monitoring their AE activity. SEM images of the erosiondamaged specimens revealed that the same damage mechanism occurred at different impact angles. The AE stress delay parameter was used to predict the residual tensile strength of erosion-damaged composites. Tension-tension fatigue tests were performed on virgin specimens and specimens exposed to erosion damage for 60 s and 90 s at 90° particle impact angle to observe the effects of erosion damage on the fatigue life. A modified Basquin’s equation was employed to predict the fatigue life of the erosion-damaged specimens. Azouaoui et al in their work [42] reported and discussed an experimental method for characterising the damage behaviour of glass/polyester laminated composite plates at lowenergy impact-fatigue. Experimental tests were performed with increasing impact energy and increasing number of impact events. The effects of impact number, energy level and

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cumulative impact energy on delamination area evolution, and also on crater dimensions progress (diameter and depth) were graphically presented. The delamination behaviour and crater dimension in the laminated specimens depended largely upon the level of incident energy. Well-defined impact-fatigue (E–Nf) behaviour was obtained, showing an endurance limit at above 104 impact cycles. The response of adhesive joints to three fatigue regimes, namely constant amplitude sinusoidal loading (standard fatigue, SF), cyclic in-plane impacts (impact-fatigue, IF) and a combination of the two (CSIF), was investigated [43]. The samples used in the study were CFRP lap–strap joints (LSJs) bonded with a rubber modified epoxy adhesive. It was observed that fatigue fracture at very low load amplitudes occurred in IF. Two main patterns of failure were observed in SF; cohesive failure in the adhesive, which was linked to slow fatigue crack growth behaviour, and a mixed-mode failure, involving failure in both the adhesive and the CFRP. In addition, it was observed that the transition from cohesive to mixed-mode failure was accompanied by crack growth acceleration (Figure 9). In the case of IF, all failures were due to a mixed-mechanism. In the combined standard and impact-fatigue it was seen that the introduction of a relatively small number of impacts between SF blocks drastically changed the dynamics of fatigue crack propagation, increasing the crack growth rate. Furthermore, the cavitation of rubber particles in the adhesive was affected by the repetitive impact loading, which was seen as evidence of active toughening.

Figure 9. Impact-fatigue crack growth: failure modes [43].

The effects of thermal cycles on the impact-fatigue properties of unidirectional carbon fibre reinforced polyetherimide (PEI) matrix composites were investigated [44]. During the thermal cycles, the samples were immersed into boiling water (100 °C) and subsequently to ice water (0 °C), 50, 200 and 500 times. The changes in the viscoelastic properties of the composites were investigated by means of dynamic mechanical thermal analysis (DMTA). At the second step, thermally cycled composites were subjected to repeated impact loadings, with different impact energies. Instrumented impact test results were presented as a function of force, energy and deformation. SEM studies were performed in order to understand the morphology of fractured samples after impact-fatigue loading. As was expected, the number of thermal cycles and applied impact energy of the impactor were found to significantly influence the fracture morphology of repeatedly impacted laminates. A damage variable (damage degree DD) representing the ratio between the absorbed energy and the impact energy was introduced in 1998 by Belingardi and Vadori to assess damage accumulation caused by LVI. More recently, repeated impact tests carried out on thick laminates pointed out the significance and extent of the penetration process in thick

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laminates, which DD was unable to monitor; by definition, it reached the value of one at penetration to remain unchanged over the entire penetration process. In a recent work [45], a new damage variable (damage index DI) was proposed to overcome the shortcomings of DD with respect to thick laminates. By introducing the displacement of the impactor into the definition of the DI, the depth of the penetration process was taken into account. Normalization by the displacement of the quasi-static perforation test allowed for a nondimensional damage variable which reached the value of one at complete perforation. The validity of the approach was demonstrated against impact data obtained for different fibrematrix architectures and laminate thickness. The DI could effectively differentiate between penetration and perforation thresholds, increasing monotonically within the range of the penetration process. In particular, the DI increased linearly with the impact energy up to penetration and rose abruptly within the range of the penetration process. Results for repeated impact tests proved that initially DI increased almost linearly impact after impact, due to a steady accumulation of damage. A few impacts before perforation a sudden rise in the DI value pointed out a change in the rate of damage accumulation. The main aim of Casas-Rodriguez et al [46] was to investigate the behaviour of adhesively bonded CFRP joints subjected to cyclic LVI and compare it with specimens tested in standard constant amplitude sinusoidal fatigue. It was seen that the accumulated energy associated with damage in impact-fatigue was significantly lower than that associated with similar damage in standard fatigue. It was also found that the mechanisms of failure were very different for the two loading regimes. For both types of loading, fracture initiated in the adhesive layer and then propagated into the 0° ply of the composite adjacent to the adhesive layer. However, the fracture surfaces after impact-fatigue were generally less uniform and exhibited more signs of high rate or brittle fracture than those observed in fracture surfaces after standard fatigue testing. A parametric approach was proposed to characterise damage in standard and impact-fatigue and it was shown that crack velocity, accumulated absorbed energy and normalised maximum force were all useful parameters for characterising damage evolution. Crack propagation rates tended to be higher in impact-fatigue and damage occurred at significantly lower energy levels than that required for similar damage in standard fatigue. This led to the conclusion [47] that impact-fatigue is a potentially dangerous form of loading, both for adhesives and polymer composite materials. Impact-fatigue properties of unidirectional CF reinforced polyetherimide (PEI) composites were investigated [48]. Repeated LVI was performed by using a pendulum type instrumented impact tester at energy levels ranging from 0.54 J to 0.94 J. The results of the repeated impact study were reported in terms of peak load (Fmax), absorbed energy (Emax) and number of repeated impacts. The life time of the composite materials was correlated to repeated impact loadings using an analytical model. The response of natural fibre composites on the impact-fatigue loading has also been studied. An impact-fatigue study was made for the first time [49, 50, 51] on 35% jute/vinylester composites containing both untreated and alkali treated fibres. Longer alkali treatment removed the hemicellulose, improved the crystallinity and gave better fibre dispersion. The flexural strength of the composite made from treated fibres was superior. 4 h alkali treated jute fibres gave the optimum combination of improved interfacial bonding and fibre strength properties. However this was not reflected upon their impact-fatigue behaviour. On the contrary, the composites reinforced with 8h alkali treated fibres displayed superior impact-fatigue properties. SEM micrographs revealed that the fibres suffered catastrophic

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fracture with microfibrillar pull-out at some places, which improved the fatigue performance of the composites. Hemp fibre reinforced unsaturated polyester composites (HFRUPE) were subjected to LVI tests in order to study the effect of non-woven fibre reinforcement on their impact properties [52]. HFRUPE composites specimens of various fibre volume fractions (Vf) were prepared and their impact response was compared with samples containing an equivalent fibre Vf of chopped strand mat E-glass fibre reinforcement. Post-impact damage was assessed using SEM. A significant improvement in load bearing capability and impact energy absorption was found following the introduction hemp fibre as reinforcement. The results indicated a clear correlation between fibre volume fractions, stiffness of the composite laminate, impact load and total absorbed energy. Unreinforced unsaturated polyester control specimens exhibited brittle fracture behaviour with a lower peak load, lower impact energy and less time to fail than hemp reinforced unsaturated polyester composites. The impact test results showed that the total energy absorbed by 0.21 Vf (four layers) of hemp reinforced specimens was comparable to the energy absorbed by the equivalent Vf of chopped strand mat E-glass fibre reinforced unsaturated polyester composite specimens.

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Fatigue Modelling – Life Predictions A methodology which combines relatively short-term fracture mechanics data obtained from experimental measurements with a FEA of the component or structure was proposed for predicting the fatigue life of fibre-composite components and structures [53]. The approach was used to study the damage growth due to cyclic fatigue of I-beams. The beams were manufactured using CFRP and contained a 60 mm diameter notch in the web. Experimental work showed that the development of significant damage was limited to the region of the material in the web around the 60 mm diameter notch. A significant amount of matrix microcracking damage occurred within the first 0.5×106 fatigue cycles, mainly in the +45° plies and 0° plies, in which the fibres were oriented at 90° and at 45°, respectively, to the local tensile stress. This matrix cracking eventually led to limited delamination after approximately 0.5×106 cycles which took place mainly along the global +45°/–45° ply interfaces. Limited delamination was also observed along the +45°/0° ply interface. The growth of these two types of damage eventually led to fibre fracture which was the final cause of structural failure of the web material, and hence of the I-beam. The model concentrated upon modelling these types of damage mechanisms. The agreement between the results from the theoretical model and the experiments was good, especially when taking into account that there were no “adjustable factors” involved in the modelling. The effect of matrix toughness on the fatigue behaviour of cross-ply reinforced composites was discussed by Gassan and Dietz [54]. Brittle and impact-modified epoxy resins containing commercial E-GF with a good fibre matrix adhesion were used. Loadcontrolled tension – tension fatigue tests with different applied maximum loads between 150 and 330 MPa were performed. The damage, as measured by stiffness reduction, was more significant for composites with a brittle matrix. The energy loss was shown to be sensitive to the matrix toughness and was higher for impact-modified resins for a given strain amplitude. A non-linearity in the S/N curve was observed for the impact-modified composites, while the S/N curve for the brittle ones was linear when plotted on a semi log-scale. Furthermore, the

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brittle composites showed an extensive region of final damage propagation, whereas the impact modified composites failed in a sudden death manner. An artificial neural network (ANN) was shown to be an efficient tool for modelling fatigue life of multidirectional composite laminates made of GFRP composite materials and tested under constant amplitude loading patterns [55]. The modelling efficiency of the network was satisfactory for both on- and off-axis coupons’ life, irrespective of the test conditions. Tension–tension, compression–compression and even tension–compression loading patterns were investigated and the modelling accuracy of the proposed ANN model was validated. The main benefit of this modelling tool was that only a portion, in the range of 40–50%, of the experimental data was needed for the whole analysis. Thus, expensive and time consuming tests required for the compilation of S-N curves could be significantly reduced without significant loss of accuracy. The neural network method was applied using experimental data from two different material systems and proved that constant life diagrams (CLDs) are very useful for the design of structures loaded under variable amplitude loading spectra. CLDs can be efficiently constructed in conjunction with the proposed ANN model using a smaller set of experimental data than that needed for the construction of CLDs conventionally (Figure 10).

Figure 10. Constant Life Diagram (CLD) for on-axis multidirectional laminates: Artificial neural network predictions versus experimental data in the range 104-107 loading cycles [55].

Delamination Propagation under Fatigue – Models of Prediction A critical review of the published experimental research was presented [56] concerning delamination onset and growth in composite laminate interfaces of different lay-ups under single-mode loadings. It was found that, depending on the loading mode and interface lay-up, the traditional fracture toughness characterization by unidirectionally reinforced composite tests can lead to marked under- or overestimation of material resistance to crack growth. Empirical models of fracture toughness as a function of delamination front orientation with respect to reinforcement directions of the adjacent laminae were validated and their applicability range was established. The delamination onset and growth in composite laminates was also characterized in two phenomenological works by Ramkumar [57, 58].

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Fracture mechanics was used to model delamination growth during static loading [2, 59, 60, 61, 62]. Models for describing delamination growth under cyclic loading were proposed [63, 64]. Trethewey et al [65] studied the delamination propagation under pure Mode II conditions. Behesty et al [66] elaborated on earlier work on the development of a life-prediction method for CF/epoxy laminates. In order to compare the behaviour of a number of different CFRP laminates already studied, further constant-life fatigue data were obtained for a CFRP composite and a GFRP laminate of similar construction – a 16-ply [(±45,02)2]S lay-up. Fatigue tests were carried out in both virgin condition and after LVI damage. The analysis of the new data and the re-assessment of the older data base, led to the appropriate modification of the constant-life model. A prediction procedure for the fatigue response of composite materials in the virgin and impact-damaged conditions was presented, based only on the tensile and compressive strengths of the composite in question. The model was equally applicable to both GFRP and CFRP, despite the fact that the fatigue response of the GFRP laminate was different from that of the equivalent CFRP material. Shen et al [67] dealt with the computational modelling of delamination and the prediction of delamination growth in laminated composites. In their analysis of post-buckled delaminations, an important parameter was the distribution of the local strain-energy release rate (SERR) along the delamination front. A study using virtual crack closure technique was performed for three-dimensional FEA models of circular delaminations embedded in woven and non-woven composite laminates. The delamination was embedded at different depths through the thickness of the laminates. The issue of symmetry in boundary conditions was discussed. It was found that fibre orientation of the plies in the delaminated part played an important role in the distribution of the local SERR. This implied that the popular use of quarter models in order to save computational effort is unjustified and may lead to erroneous results. A comparison was made with experimental results and growth of the delamination front with fatigue cycling was predicted. A methodology for the prediction of delamination areas and directions using evolution criteria derived from mechanical tests was also described. It was found that evolution criteria based on components of the SERR predicted the rate of delamination growth much better than evolution criteria based on the total SERR. A thermodynamically consistent damage model was proposed for the simulation of progressive delamination in composite materials under variable-mode ratio [68]. The model was formulated in the context of Damage Mechanics. A novel constitutive equation was developed to model the initiation and propagation of delamination. A delamination initiation criterion that evolved from the Benzeggagh – Kenane propagation criterion was proposed to assure that the formulation can account for changes in the loading mode in a thermodynamically consistent way. The formulation accounted for crack closure effects to avoid interfacial penetration of two adjacent layers after complete decohesion. The model was implemented in a finite element formulation, and the numerical predictions were compared with experimental results obtained in both composite test specimens and structural components (Figure 11). Kim and Hwang [69] proposed an impact response model that can be used to compute the fatigue damage. The prediction of fatigue damage can be made through investigation of impact response. The stiffness reduction induced in composite laminates by matrix cracking or delamination is an important measure of fatigue damage. Additionally, stiffness reduction affected impact response. The authors identified a correlation between fatigue and impact

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damage manifested as stiffness degradation. Impact induced stiffness reduction was investigated using simplified methods and the predicted impact force histories were compared with numerically calculated ones using FEA.

Figure 11. Experimental and numerical load-displacement relations for fatigue after impact [68].

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Ndt Inspection of Initial Damage and Damage Propagation Acoustography Acoustography provides a very powerful means of monitoring damage in composites in real time during both static loading and long-term dynamic fatigue tests. The integration of an acoustography ultrasonic imaging system with a mechanical testing machine was described [70, 71, 72]. The apparatus was designed for real-time ultrasonic imaging of impact damage growth in composite material samples subjected to either static or fatigue loading (Figure 12). Experimental results were presented of the damage growth to failure of an aerospace standard CF composite subjected to compressive cyclic loading. Experimental results for static loading revealed a reversible increase and decrease in the damage area, indicating an opening and closing of delamination defects, which pointed to the possibility of an under-estimation of defect size by the conventional ultrasonic testing of unloaded components. Results presented [71] showed the damage-area growth during fatigue cycling under high compressive loads. After an initial small enlargement, damage grew at a constant rate until the third stage was reached when there was further growth at an increasing rate to final failure. A “fatigue limit” was also observed.

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Figure 12. Acoustography: Schematic diagram of the system [70].

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Acoustic and Lamb Waves’ Methods The work of Grondel et al [73] was devoted to the development of a health monitoring system designed for aerospace applications. These applications were concerned with the detection of damaging impacts and debonding between stiffeners and composite skins, since these are the major causes of in-service damage of aircraft structures. The chosen health monitoring system was first based on the excitation and reception of Lamb waves along the structure by using thin piezoelectric transducers (active mode) and secondly on a continuous monitoring taking the same transducers used as acoustic emission sensors (passive mode). The composite specimen used was consistent with an aircraft wing-box in terms of structure and loading. Several impacts with incrementing energy were applied on the composite specimen. In passive mode, the study showed the ability of using the acoustic signature of an impact to detect possible damage. Moreover, the damage initiation in the case of damaging impact was confirmed in active mode. Further measurements during fatigue testing were performed. The ability of the system to monitor disbond between the stiffener and the composite skin was demonstrated and the sensitivity of the health monitoring system to the disbond growth was further evaluated. Toyama and Takatsubo [74] proposed an inspection technique using Lamb waves to detect impact-induced delamination in composite laminates. The technique consisted of two line scans. The first scan measured the arrival times of the transmitted S0 mode along the 0° direction to detect delamination and evaluate its size. The second scan measured the maximum amplitude of the earliest wave packet in a line, including the longest delamination, to locate its edge. The technique was performed on impacted CFRP cross-ply laminates. A remarkable decrease in the arrival times due to the delamination was detected, and the delamination length was calculated using a simple model for Lamb-wave propagation.

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Furthermore, the delamination edge was located as a sudden decrease in the amplitude. The technique enabled the detection of the delamination and the evaluation of its size and location using only two scans. The article of Monnier [75] used an in situ NDI method for the impact damage monitoring of carbon/epoxy laminated composites. This structural health monitoring (SHM) method aimed at improving the operational safety of the structure using integrated piezoelectric sensors. The propagation of Lamb waves inside a stiffened aeronautical structure, and the definition of associated data processing systems were described. The detection of realistic defects was made feasible by the definition of a damaging parameter, based on the real time analysis of the ultrasonic signatures transmitted inside the structure. An LVI simulation by loading at the centre of quasi-isotropic CF reinforced epoxy resin laminate was presented in order to characterize damage development [76]. Acoustic emission (AE) was used to detect damage accumulation. It was found that the drops in the load – displacement curve were due to delamination and fibre failure. The damage induced by the 7 mm radius indenter was much higher than the one induced by the 4 mm radius as a higher fraction of fibres failed, further reducing the residual strength. The tensile strength after indentation was also investigated and correlated with AE parameters. In the tensile test the residual strength of the lay-up [0/90/+45/–45]s was higher as the delamination and splitting of the external lamina prevented fibre failure during transverse loading. A FEA model for the prediction of both the first ply failure (FPF) and the ultimate ply failure (UPF) of the laminate was built using the ANSYS® software and correlated well with experimental findings.

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Electrical Measurements Self-sensing of damage by measurement of the DC electrical resistance or potential away from the damaged region was demonstrated [77] in quasi-isotropic continuous CF epoxymatrix composite laminates under impact at energy up to 5 J. The through-thickness potential drop was substantial up to 240– 480 mm (at 0.25–99 mA correspondingly) in the longitudinal direction from the position of through-thickness current application, due to current spreading in the longitudinal direction. A model for the current spreading was also provided. The fractional change in resistance resulting from damage decreased with increasing distance from the point of impact (diameter of indentation up to 3.5 mm and depth of indentation up to 0.16 mm), but was detectable even at a distance of 150 mm from the point of impact. Both the through-thickness resistance and the oblique resistance were effective indicators. The ability for the resistance measured away from the damaged region to indicate damage in the damaged region was attributed to the considerably lower electrical resistivity in the longitudinal than in the through-thickness or oblique direction in the composite. Drop impact damage of continuous carbon fibre epoxy–matrix composite laminates, was studied [78] by electrical resistance measurement, and was proved to be more sensitive than the ultrasonic method. The oblique resistance at an angle between the longitudinal and through-thickness directions was more effective than the surface longitudinal resistance in indicating damage, particularly in the interior of the laminate (Figure 13). The oblique resistance values from longitudinal segments of a specimen could not be integrated, but the surface resistance values were. In the case of a unidirectional composite, electrical contacts at 45° from the longitudinal direction in the plane of the laminate were more effective than those

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at 90°. Even minor damage associated with negligible indentation was sensed. The spatial distribution of damage was also studied.

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Figure 13. Fractional change in oblique resistance upon impact at 0.73J (thin curve and 3.63 J (thick curve) [78].

The method of sensing impact damage in carbon fibre polymer-matrix structural composite by DC electrical resistance measurement was evaluated by measuring the resistance of the impacted surface [79]. The resistance was obtained by using the four-probe method as a more sensitive, more precise (less data scatter) and more accurate indicator of composite damage than that obtained by using the two-probe method. The data scatter was low for both four-probe and two-probe resistance for impact energy up to 5 J, but it was lower for the four-probe resistance than the two-probe resistance and increased with damage. The four-probe resistance, of the 8-lamina composite, increased upon impact, and the fractional increase diminished as the distance from the point of impact increased. The four-probe resistance of the 24-lamina composite increased upon impact for the specimen segment containing the point of impact, but decreased slightly upon impact for the segments within 20 mm from the point of impact. The two-probe resistance decreased less upon impact than the four-probe resistance. Self-sensing damage, as obtained by electrical resistance measurements, was found to be effective in a carbon fibre polymer–matrix composite cylinder made by filament winding [80]. Resistance was measured in the axial, radial, oblique, and circumferential directions by using circumferential or axial electrical contacts on the outer and/or inner surfaces of the cylinder. Minor damage upon drop impact at 10 J or below caused the radial resistance to decrease irreversibly, whereas major damage upon drop impact above ≈10 J caused the radial, oblique, and axial resistances to increase irreversibly. The circumferential resistance ratio i.e. the ratio of the circumferential resistance of a damaged area to that away from the damaged area was most sensitive and it increased monotonically with impact energy over 1.4 J. The electrical resistance method and the electrical potential method for composite damage self-sensing were compared for the case where the direction of the applied current and the direction of the measured potential gradient were parallel [81, 82]. The electrical potential method involved measuring the potential at a distance from the line of current application and was effective for damage sensing of carbon fibre polymer–matrix composites

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when the distance in the through-thickness direction was sufficiently small or else for thin laminates up to eight-plies. However it was ineffective for 16 ply laminates i.e. for a 2.1 mm thickness. On the other hand, the electrical resistance method, which involved measuring the potential on the line of current application, was not limited by the thickness of the laminate. However, it suffered from current path distortion upon damage and the consequent reduced sensitivity for damage sensing. Interestingly enough, the measured resistance was reported to decrease with increasing damage for the region of the composite not containing the point of impact. However, this effect was counterbalanced by the resistance increase for the regions containing the point of impact. The self-sensing of flexural strain and damage was demonstrated [83] in carbon fibre polymer-matrix composite by measuring the DC electrical resistance. Upon strain in the elastic regime, the compression surface resistance decreased reversibly (due to increase in the current penetration), while the tension surface resistance increased reversibly (due to decrease in the current penetration), and the oblique resistance increased reversibly. Upon minor damage the oblique resistance after unloading decreased, the oblique resistance decreased with load application near the onset of loading and the curve of the oblique resistance or the resistance of the tension or compression surface vs. deflection became nonlinear. Upon major damage, all resistances abruptly and irreversibly increased. The onset of damage as manifested by the increase of surface and oblique resistance occurred earlier for the compression surface than for the tension surface. The surface resistance was a superior indicator for strain, whereas the oblique resistance was a superior indicator for damage. Shen et al [84] used beam-type specimens with and without delamination damage to carry out numerical analyses for the application of the four-probe electrical resistance measurement. The validity range and the applicability of the method were studied. It was found that the four-probe resistance method was valid only when the through-thickness conductivity was comparable to or larger than the longitudinal conductivity. For the potential method, which measured directly the voltage values between the voltage contacts, the results showed that the percentage change in damage-induced voltage drop between a pair of voltage contacts was not consistent with the percentage change in resistance. The underlying reason was that the damage-induced voltage change depended on the location of the applied current, while the resistance change did not. Todoroki et al [85] employed an electrical resistance change method for monitoring delamination. They found both experimentally and analytically that the electrical resistance change method using response surfaces was very effective in identifying delamination in CFRP laminates. The effect of the electrode spacing was investigated using Finite Element Analysis (FEA). Five types of spacing were analyzed for two fibre volume fractions. Crossply beam type specimens were adopted for the analyses. It was shown that the effect of the spacing depended on the fibre volume fraction. For laminates of high fibre volume fraction, dense spacing was required to obtain an accurate estimation of delamination location and length. Matsuzaki and Todoroki [86] proposed a wireless system for detecting delamination of carbon/epoxy composites to overcome the difficulties of in-service delamination detection for rotating composite components like helicopter and wind turbine blades. In this system, a tiny oscillation circuit was attached to the composite component. When delamination of the component occurred, the electrical resistance changed causing a change in the oscillating frequency of the circuit (Figure 14). Since this system used the composite structure itself as a

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sensor and the oscillating circuit was very small, it was applicable to rotating components. The electrical resistance change and oscillating frequency change due to delamination was experimentally measured for carbon/epoxy specimens. The effect of temperature variation was also measured. The wireless method was found to successfully detect delamination, and to estimate its size. The effect of temperature change was minimized by means of a temperature compensation circuit.

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Figure 14. Wireless delamination detection system using electrical resistance and oscillating frequency change [85].

Ueda and Todoroki [87] employed a two-stage electric potential change method (EPCM) to identify delamination. Delamination is usually estimated using response surfaces, which requires numerous experiments. An equivalent electrical conductivity method was introduced to reduce this number. Delaminations were successfully estimated using response surfaces based on FEA results for equivalent electrical conductivity. The electrical potential technique was applied [88] to detect and locate impact damage in CFRP plates. The potential field across the surface of the laminates was measured using arrays of electrical contacts. A constant current of 100 mA was applied to the plate, and the changes in the potential distribution resulting from impact damage were measured. These results were compared to those calculated using 3D FEA simulation of current flow in CFRP laminates. The results showed good agreement between simulation and experimental results and could allow for detailed analytical studies of optimum network and current input configurations for the practical realization of a damage sensing system. A comparison of the potential distribution on the top surface for damaged and undamaged laminates showed a substantial difference in the potential field around the impact damaged area. Quasi-isotropic samples of CFRP composite 2 mm thickness were manufactured and instrumented with an array of electrodes [89]. The electrodes were used to introduce a DC current and to measure the resultant potential distribution on the top and bottom surfaces of

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the laminate. Instrumented samples were impacted at energies between 2 and 12 J, and the potential fields were measured before and after impact. The impact damage caused significant changes in potential values. Indications of the extent and severity of the impact damage could be best obtained by representing the data as contours of equi-potential change. There was excellent correlation between regions of potential change and impact damage as measured by ultrasonic C scan. The results of FEA calculations of potential change distribution on the surface of current carrying quasi-isotropic laminates containing delaminations of a range of sizes and locations were qualitatively similar to those obtained experimentally, but differed in detail. The differences between real and simulated impact damage were discussed to assess the performance of a damage detection system based on equi-potential contour mapping.

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Embedding Optical Fibres Kuang and Cantwell [90] presented the results of a study involving the use of conventional glass optical fibres for detecting impact damage in a thermoplastic-based FibreMetal Laminate (FML) system. Although a number of researchers have investigated the use of intensity-based optical fibre sensors for detecting impact damage in composites, almost all of the published literature has focused on detecting damage in plain thermosetting polymer composites. This study investigated the possibility of using intensity-based optical fibres for detecting impact damage in thermoplastic-based FML systems. The results suggested that for higher fracture detection sensitivity, optical fibres should be embedded close to the impact surface in order to be sufficiently sensitive for energies of 1 to 2 Joules in these hybrid structures. Although the optimum choice of optical fibre and its location should be reestablished for other types of material system, this study highlighted some important observations for the future development of damage detection systems for use in tough composite materials and fibre metal laminates. Step-index multi-mode optical fibres were embedded in a thermoplastic glass fibre polypropylene (GF/PP) [91]. Three types of commercially available optical fibres were investigated to evaluate their potential for detecting impact-induced damage in thermoplasticbased composite structures. Preliminary findings confirmed the feasibility of using these inexpensive multi-mode optical fibres as damage sensors in high-performance composite materials. When embedded between the uppermost ply of a cross-ply unidirectional [0/90]s laminate, impact energies as low as 0.8 J could be detected. The optical fibres were embedded at different locations within the composite structure to investigate the effect of embedment locations on their sensitivity for detecting impact damage. Kister et al [92] reported the use of conventional reinforcing E-Glass Fibres that were employed as light guides. These reinforcing fibre light guides were used to detect damage induced in the composite by impact, indentation and flexure. The glass fibres were converted into light guides by applying an appropriate cladding material. The coating resins used in this study consisted of an epoxy- and a polyurethane-based resin system. These self-sensing fibres or reinforcing fibre light guides were either surface mounted or embedded at two specified locations within 16-ply glass fibre-reinforced epoxy prepreg composites. The data generated in this study demonstrated that the self-sensing concept could be used to study in situ and in real time the failure processes in Glass Fibre reinforced composites. A detailed study was also undertaken to characterize the various failure modes observed when the composites with the

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self-sensing light guides were subjected to impact, indentation and flexural loading. The damaged areas in the composite were easily located by means of the “bleeding” light emanating from the broken self-sensing E-GFs (Figure 15).

Figure 15. Delamination of a composite subjected to 2 J impact, were the glass fiber reinforcement is used as a light guide (a) front face, (b) back face [92].

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Small-diameter fibre Bragg grating (FBG) sensors were employed [93] for the monitoring of delamination induced by LVI. The FBG sensors were embedded into CFRP laminates (Figure 16). Using a drop-weight impact tester, impact loading was applied to the laminates at four impact energy levels. After the impact tests, the internal damage including delaminations was observed by ultrasonic C-scan, and the reflection spectra from the embedded FBG sensors were measured. The form of the spectrum changed significantly depending on the delamination size. Furthermore, the spectra were calculated theoretically for confirmation of the measured spectra. Since the change in the measured spectrum was consistent with that for the calculated spectrum, the relationship between the delamination size and the form of the spectrum could be clarified. From the results, the present method using small-diameter FBG sensors was found to be effective for the monitoring of the delamination.

Figure 16. Embedding a small diameter FBG sensor into the test specimen to detect delamination [93].

The purpose of Silva et al [94] was to quantitatively evaluate the effect of embedding optical fibres on the mechanical behaviour of a CF/epoxy composite in order to assess whether their presence could potentially degrade the mechanical performance of the host material. The existing literature on this subject is not conclusive about the nature and intensity of this effect. Three kinds of mechanical tests were performed in this work: impact tests, static flexural tests, and fatigue tests. The results pointed to a possible detrimental effect related to the presence of optical fibres which was different in nature and intensity for each of these tests. The mechanical behaviour in static loading conditions was not significantly affected by the presence of the optical fibres. However, impact and fatigue performance was strongly affected albeit in different ways. Based on these results, the possible failure mechanisms that can explain the detected differences were discussed. Damage in unidirectional CF composite resulting from both LVI and HVI was evaluated using embedded FBG sensors, C-scan and microscopic analysis [95]. It was found that the FBG sensors located 10 mm from the impact site could detect residual strains from a 0.33 J (1.3 m s-1) impact which was not detectable by C-scan or visual inspection. The measured

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residual strain increased with impact energy and damage changed from matrix cracking to severe delaminations. HVI (225 m s-1, 11 J) resulted in test panel perforation and delaminations. FBG sensors located within a distance of 2–3 the damage radius detected residual strain from the impact. An array of embedded sensors could possibly be used to identify the site of both LVI and HVI and predict the damage from the response of the adjacent sensors providing the sensors are located sufficiently close to the impact site. FBG sensors were applied to durability tests of a composite wing structure in order to verify its health monitoring capabilities for long-term use [96]. The durability tests included drop-weight impact tests and two periodic fatigue tests based on the design service life of the aircraft. The seven FBG sensors were installed to the surface of the test panel for the monitoring of impact and fatigue damage. The impact damage was detected by using the spectrum change of the sensor output. The strain changes during the fatigue tests were measured by the wavelength shift of the sensor output. The damage evolutions in the test panel were also evaluated by using various NDI technologies, i.e. AE, ultrasonic C-scan and pulse thermography. Compared with results of the other non destructive techniques, the FBG sensors could provide valuable information for monitoring the structural integrity of the composite wing structure. As a result, it was confirmed that FBG sensors were capable for the long-term health monitoring of large-scale composite structures. A study of the response to LVI and perforation of two-dimensional and three-dimensional woven composites was undertaken utilizing local measurements of post impact residual strain measured by surface mounted and embedded optical fibre Bragg grating (FBG) strain sensors [97]. Concurrent global measurements of contact force, dissipated energy, and point of impact displacements were undertaken. The study delineated five distinct regimes ranging from initial impact to complete penetration. Sensor and host damage were separated by signal intensity and the evolution of Bragg peaks due to repeated impact loads. The results indicated that a local-global framework could be used to monitor damage progression in different host materials, and hence it could be potentially used to mitigate damage effects. The impact strain relaxation occurred in two forms: a global relaxation of the average strain over the grating seen by uniform Bragg peak shifts, and a reduction in the strain fields’ non-uniformity over the grating length as indicated by the non-uniform reductions in the peak bandwidth around the Bragg wavelength. The measured residual impact strain relaxation reached a maximum of one-third of the peak residual impact strains. Due to strain relaxation, the amount of time between impact events provided enough time for sample equilibration, which had an effect on the composites’ response to additional impacts.

Imaging Techniques LVI tests were performed to investigate the impact behaviour of CF/epoxy composite laminates reinforced by short fibres and other interleaving materials. Characterisation techniques, such as cross-sectional fractography and scanning acoustic microscopy (SAM), were employed [98] to assess quantitatively the internal damage of composite laminates at the sub-surface under impact (Figure 17). SEM was used to observe impact fractures and damage modes at the fracture surfaces of the composites. The laminates experienced various types of fracture; delamination, intra-ply cracking, matrix cracking, fibre breakage and damage depending on the interlayer materials. The trade-off between impact resistance and residual

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strength was minimised for composites reinforced by Zylon fibres. For composites interleaved by poly(ethylene-co-acrylic acid) (PEEA) film, the deterioration of residual strength was substantial, although the damaged area was significantly reduced. Damage induced on the front and back surfaces of impact was also observed and compared for different laminates.

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Figure 17. Total damage versus impact energy estimated by Scanning Acoustic Microscopy [98].

The rapid estimation of the damage of a composite plate was possible with thermal methods. The use of high performance cameras based on focal plane array of quantum detectors and fast post processing led to the substantial improvement of thermal methods. Post processing consisted in associating the experimental advantage of the flash method and some characteristics of periodic methods with the processing of a big quantity of data. The main idea was to use as a reference signal the spatially averaged image evolution. Some practical aspects of the method applied to the estimation of delamination characteristics were presented by Guillaumat et al [99]. The experiments were obtained from damaged composite plates (transparent fibre glass epoxy sample covered with a black painting). Mian et al described the application of a short ultrasonic pulse in combination with infrared surface imaging to assess fatigue damage in an unnotched [100] and notched [101] graphite/epoxy composites. The technique used a short pulse of sound wave that infused into the material and resulted in frictional heating at the crack surfaces. The heating at the crack defuses into the surface of the sample, and the temperature rise at the crack location was sufficient to be captured by an infrared (IR) camera. The ability of the sonic infrared imaging technique in detecting fatigue damage in composite materials was demonstrated. FEA modelling was performed to better understand the effect of the applied sound pulse at the damage locations and to interpret the vibrometer data. The same samples were also tested using a thermal wave imaging technique. In the case of notched composites the technique revealed the presence of semi-elliptical delaminations near the circular notch as modelled in the FEA. A separate FEA model was developed for transient heat transfer analysis by utilizing the calculated frictional energy as a heat source. Variation in surface temperature within the delamination regions was plotted with time, and was compared with the temperature–time plot obtained from the sonic IR technique.

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The study by Meola et al [102] focused on the use of lock-in thermography (LT) for NDI evaluation of aerospace materials and structures. The experimental analysis was performed by testing several specimens, which were made of different materials employed in the fabrication of aircraft (composites, hybrid composites, sandwiches, metals) and which included the most commonly encountered types of damage (delamination, impact damage, fatigue failure). The data presented showed that LT provided useful information, which could be exploited for detection of defects and evaluation of their size, position and nature (Figure 18).

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Figure 18. Lock-in thermography: Phase images and micrographs of Glare with (a) and without (b) lateral restraints [102].

A digital imaging technique for characterizing dynamic delamination in laminated composites was presented by Wosu and Hoy [103]. Perforation-induced delamination surfaces were generated using a conical and protruding hemispherical penetrator applied in the thickness direction using the penetrating split Hopkinson pressure bar (P-SHPB). The studies revealed that damage layers on perforated specimens increased linearly below the perforation threshold energy and sharply after perforation.

Ultrasonics Delamination extent is usually measured by ultrasonic C-scan, which provides an in plane view of the damaged zone. By this technique, some information on the actual damage state is lost, as it is typically made of several interlaminar cracks located at different heights along the material thickness [104]. Nevertheless, a C-scan procedure is easy to perform and relatively inexpensive and experimental evidence given that the in-plane dimensions of delamination can be correlated with the material residual strength. Furthermore, the delamination area as revealed by ultrasonic C-scan is frequently used to rank different laminates on the basis of their impact resistance. Unidirectional graphite/epoxy composites used in the aerospace industry, were impacted by a free falling weight [105]. The subsequent changes/ degradation in elastic moduli, strength, toughness, and fatigue properties were measured after different number of impacts.

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It was found that for all energy levels these properties varied linearly with the number of impacts. Attenuation changes were not a good parameter for degradation estimation, since they did not incorporate the micro- and macro-cracks beyond the impact point. However, these micro- and macro-cracks had significant effect on the mechanical properties. In contrast to the attenuation, the stress wave factor, which indicated the efficiency of wave propagation along the specimen, correlated very well with degradation, and could be used effectively to measure the residual strength after impact. The ultrasonic characteristics of specimens subjected to combined fatigue and impact were also studied. Based on these experiments, it was concluded that the loss in fatigue residual life due to impact could be predicted by measuring the effects of the impact load on attenuation and the stress wave factor. It was found that the reduction in fatigue life was proportional to sudden changes in attenuation and stress wave factor. Damage accumulation models based on the Coffin-Manson equation were suggested for impact and combined fatigue and impact. It was found that residual properties and fatigue life could be estimated using these models. The aim of Adden et al [106] was to demonstrate the possibility of fatigue damage characterization in GFRP-tube-like components by using circumferential plate waves (Figure 19). For that purpose, fatigue tests with different loading directions were conducted and the stiffness degradation was monitored. After a preset number of loading cycles, non-destructive ultrasonic tests using circumferential plate waves were performed. The correlation between damage induced amplitude changes of the plate waves and stiffness degradation in noncrimped fabric composites was discussed. The results indicated that the technique was applicable to fatigue damage assessment in complex-shaped components of composite materials and was applicable for a wide range of applications.

Figure 19. Schematic representation of the fatigue damage measuring device using circumferential plate acoustic waves [106].

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Other Methods

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Optical coherence tomography (OCT) is an emerging technique for imaging of synthetic materials. OCT is attractive because it combines high sensitivity (> 90 dB), high resolution (5 μm to 20μm), and low cost. The value of any new technology is evaluated by how well it compares with existing methods. In the work by Dunkers et al [107], impact damage of an epoxy/E-glass composite was imaged using OCT, and the results were compared with microfocus X-ray computed tomography. This technique was regarded as a good benchmarking method for OCT because both techniques have the ability to locate features precisely and have comparable resolutions. OCT was considered to be a confocal technique so it was also compared to laser scanning confocal microscopy (LSCM). Contrast mechanisms, sensitivity, resolution, depth of penetration, and artefacts of the techniques were compared. The impact damage features revealed using OCT was also briefly discussed.

Figure 20. Non-contact detection of interfacial fracture: (a) center crack (primarily mode I) and (b) shear crack (primarily mode II) configuration [108].

A novel experimental technique was developed for time-resolved detection and tracking of damage in the forms of delamination and matrix cracking in layered materials such as composite laminates [108]. The technique was non-contact in nature and used dual or quadruple laser interferometers for high temporal resolution (Figure 20). Simultaneous measurements of differential displacement and velocity at individual locations were obtained to analyze the initiation and progression of interfacial fracture and/or matrix cracking/delamination in a polymer matrix composite laminate system reinforced by graphite fibres. The measurements at multiple locations allowed the determination of the speeds at which the interfacial crack front (mode-I) or the matrix cracking/delamination front (mode-II) dominated. Experiments were carried out using three-point bending configurations. Impact loading was achieved using a modified Kolsky bar apparatus with a complete set of diagnostics for load, deformation, deformation rate, and input energy measurement. This technique was used to characterize the full process of damage initiation and growth. The experiments also focused on the quantification of the speed at which delamination or damage

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propagated under primarily mode-I and mode-II conditions. The speed of delamination (mode-I) or the speed of matrix cracking/delamination (primarily mode-II) increased linearly with impact velocity. Furthermore, speeds of matrix failure/delamination under primarily mode-II conditions were much higher than the speeds of induced delamination under mode-I conditions. Růžek et al [109] compared the results of visual, ultrasonic C-Scan and laser shearography for impact damage assessment of sandwich panels, which were cut out from sandwich skins. Considering the reliability, simplicity and rapidity of each technique and with the digital indicator measurement as a base, the laser shearography was evaluated as the most suitable method for that purpose. The objective of the work presented by Amaro et al [110] was to evaluate the features and capabilities of four different methods when utilized to detect and quantify impact damages on CF-reinforced epoxy composite. Experimental tests were performed on [0/90/0/90]2s and [904/04]S laminates, using a drop weight-testing machine with impact energy of 3 J, which corresponded to a maximum load of approximately 3000 N. The damage in the laminates was analysed by electronic speckle pattern interferometry (ESPI), Shearography, Ultrasonic testing and X-radiography. All techniques were able to detect almost all the defects, however, the interferometric methods showed some limitations. X-radiography was presented as a promising alternative technique, but was not able to localise delaminations in the through thickness direction. The ultrasonic methods, A-scan and C-scan showed to be the best solutions to inspect the samples. According to the experimental results, these techniques were able to detect and measure defect extension on whole samples with good precision. The main purpose of the Corigliano et al study [111] was to assess the performance of the unscented Kalman filter for real-time identification of unknown properties of the debonding surface(s) on the basis of free-surface measurements only. The problem of impact induced delamination in layered composites was numerically modelled on the basis of the following assumptions: the layers are elastic; softening interface laws model the progressive interlaminar de-cohesion due to delamination. An explicit dynamic finite element procedure was used for the step by step analysis. Results concerning impact on a GFRP composite were discussed. Hatsukabe et al [112] presented their latest progress in the studies on SQUID-NDE for CF composites (CFCs), including CFRPs and CF reinforced carbon matrix composites (C/Cs). A new SQUID-NDE system for the CFCs was developed, utilizing a pulse tube cryocooler for more practical and easy-handling use. Electromagnets using U-shaped ferrite cores have been introduced to the system in order to induce enough eddy current density in the relatively low-electric-conductivity CFCs. By using this system, the detection of deeplying defects in thick and/ or multi-layer CFCs specimens was demonstrated by partly including a metal layer upon or below the CFCs layer. CFRP plates and multi-layer C/C plates with total thickness of 20 mm having deep-lying slots at various depths were prepared. The slots at 15 mm below the surface in the CFRP plate and the multi-layer C/C plate were successfully detected. Defect detection on more ‘realistic’ multi-layer structures, including aluminium plates upon/below C/C plates, was also attempted.

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Post – Impact Behaviour under Static and Cyclic Loading In the following paragraphs, a detailed presentation of research works conducted to investigate the post-impact behaviour of composite laminates under static or cyclic loading will be given. The experimental studies and their conclusions are summarized. The effect of prior damage on the tensile strength and toughness of a number of CFRPs and GFRPs angle-ply laminates was investigated [113]. The methods of introducing this prior damage were tension-tension fatigue, transverse compression, repeated impact, and stress corrosion. The effects of damage were assessed by measuring the un-notched and notched tensile strengths and, in some instances, the work of fracture of notched samples. AE monitoring and microstructural studies were carried out in support of the mechanical tests. Under various circumstances, the effects of microstructural damage resulted in independent changes in the notched and un-notched strengths of a composite. The toughness to strength ratio, KQ/σf, either rose or fell, depending on the material and the damaging conditions. However, the results for all of the tests presented still fell within the 90% confidence limits for the KQ/σf relationship previously identified. Ma et al [114] examined the hydrothermal effects on the fatigue behaviour of the CF/PEEK laminates before and after impact damage. [0/45/90/-45]2s AS-4/PEEK laminated composites were immersed in 80 °C hot water for 45, 90 and 200 days, subjected to falling weight impact with an energy of 8.58 J and then immersed in 80 °C hot water for 45 days. It was found that the tensile strength of AS-4/PEEK laminated composites decreased with the increase of exposure period. The injured AS-4/PEEK composites were subjected to a static load and tension-tension fatigue at various stress amplitudes. The effect of stress amplitude on the fatigue life was studied. The experimental fatigue life for different stress amplitudes were estimated by the median rank order statistic cumulative distribution function. Subsequently, the fitting curves for estimated data were analyzed by the Weibull distribution function. The S-N curves for a series of cyclic loads at various survival probabilities were presented. The damage behaviour of composites subjected to fatigue was also studied by SEM. The fatigue life of immersed specimens was shorter than that of the pristine specimens; overall, the impact damage affected the fatigue life of the composites significantly. The effects of LVI and cyclic thermal loading on the fatigue behaviour of CF/PEEK quasi-isotropic laminates were examined [115]. The fatigue behaviour of the pristine composites, LVI composites, and LVI and thermally exposed composites were investigated. Cyclic thermal loading was performed in the temperature range between 60 and 260 °C. The residual tensile strength was measured to understand the influence of LVI on the retention of tensile strength. Fatigue testing was performed at a stress ratio of 0.1, with a frequency of 3 Hz. The Weibull distribution function was employed to model the ultimate tensile strength and fatigue life. S–N curves were plotted and the influence of thermal cycling and LVI on the fatigue performance of the CF/PEEK laminates was investigated. The stiffness variation during fatigue testing was monitored and the differences in stiffness reduction for three test conditions were compared. C-scan was used to investigate the damage zone under different LVIs and to understand damage propagation during fatigue testing. Moreover, SEM was used to examine the fracture morphologies of CF/PEEK composites for both tensile and fatigue failure.

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The effects of laminate thickness on the tension-compression fatigue behaviour of a quasi-isotropic CF/epoxy composites subjected to LVI were investigated [116]. The penetration impact properties and the residual tensile strength under various LVIs were experimentally defined. The initial slope during penetration impact revealed that the thicker laminate had a higher stiffness than the thinner one. Moreover, the results showed that the total energy required to penetrate the composite was not proportional to the thickness. Static tensile strength measurements and tension-compression fatigue tests at various stress levels were also carried out on the impacted and un-impacted laminates. The median-rank method and Weibull distribution function were applied to predict the failure probability of the composites under given applied loading and fatigue cycles. The relationship between stress levels and fatigue life (S-N curve) was established and reductions in fatigue life of the impacted laminates at various stress levels were compared. The different slopes of the S-N curves implied that different fracture mechanisms were activated, depending on the laminate configuration. The S-N curves of the composites with different thicknesses were also compared. Ultrasonic C-scanning was used to examine the damage zones in composites subjected to various impact energies and fatigue loading. Plain-weave CFRPs were subjected to drop-weight impact at energies of up to six joules [117]. Tensile and compressive residual-strength tests were performed over the range of impact energies and two energy levels were selected for evaluation of the fatigue performance. Fatigue load amplitudes were chosen to be fractions of the static strengths. The behaviour of impact-damaged specimens was investigated under tension-tension and tensioncompression fatigue loads. The sequence of loading (impact/cycle or cycle/impact) was reversed to study its effect on the strength or the life of the composite materials. The experimental data on impact tests were compared with the corresponding results obtained by using a theoretical model. Cyclic loading followed by impact gave better performance than impact damage followed by cyclic loading, and tension-tension performance was superior to tension-compression cyclic loading. An investigation was undertaken to determine the effects of hail damage on the fatigue strength of a graphite/epoxy composite laminate [118]. Sixteen-ply coupons were subjected to simulated hail impact by ice-balls of 25.4 or 38.1 mm in diameter. Upon impact, the 25.4 mm diameter ice-ball possessed 7.1 J of kinetic energy, while the 38.1 mm diameter ice-ball possessed a kinetic energy of 27.4 J. For further comparison, additional coupons were impacted with a 12.7 mm diameter aluminium sphere with either 7.1 or 27.4 J of kinetic energy. Inspection revealed that the ice-ball impact events did not cause internal damage, while each aluminium sphere impact caused delaminations within each coupon. After being impacted and inspected, each coupon was subjected to constant amplitude tension-tension fatigue. It was determined that neither the 25.4 mm diameter ice-ball impact nor the 38.1 mm diameter ice-ball impact affected the fatigue performance of this particular laminate. The 7.1 J aluminium sphere impact did not affect the laminate performance either, while the 27.4 J aluminium impact did deteriorate the fatigue properties of the material. Chotard et al [119] reported detailed results on the impact damage and residual mechanical behaviour of pultruded GF/polyester structures. The impact aspects studied included damage analysis and the impact behaviour of distinct geometries. The influence of test parameters such as impact velocity, impactor mass and impactor size on the type of damage observed were emphasised. Four-point flexural tests were performed on box beams and U sections in order to determine the post-impact residual performance of these beams

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under static and fatigue loading. Several types of failure mode were observed on damaged and undamaged specimens. The residual performance strongly correlated with the impact testing configuration. Moreover, the residual performance was influenced by the profile geometry (in fatigue bending tests) and especially by the geometrical characteristic of the structures (opened for U sections and closed for the box beams). The main objective of Margueres et al [120] was to monitor the stiffness reduction and damage evolution of a composite GFRP obtained by resin transfer moulding (RTM). The study was based on mechanical tests such as tension, torsion, impact, fatigue and post-impactfatigue. Ultrasonic tests, the analysis of the amplitude of AE signals, SEM observations and a numerical analysis completed their study. The material translucence allowed for a direct visual analysis. The results highlighted the reliability of ultrasonic measurements to follow the damage evolution in a composite. This approach was proposed for the definition of residual properties of composite materials damaged during service. Chotard et al [121] presented an experimental investigation on the residual mechanical behaviour of patch-repaired composite pultruded structures initially submitted to LVI loading. They reported detailed results about static, fatigue and impact tests performed on different GF/polyester impact-damaged structures repaired by low-cost manual techniques. All the tests were conducted at room temperature. For all the studied structures, the initial static strength was completely recovered via the repair process. On the other hand, for two types of pultruded structures, the fatigue crack-growth life did not recover its initial values but when compared with the damaged specimen, the lifetime of the repaired structure was significantly enhanced. Residual performances of both undamaged and repaired specimens were influenced by the profile geometry (in fatigue bending tests) and especially by the geometry of the structures. Carefully designed, external scarf patch repairs can recover more than 85% of the undamaged mechanical behaviour, depending on the type of residual applied loading (Figure 21). Fatigue tests were conducted on impact damaged coupons of T300/914 CFRPs [122]. Two damage levels were used to represent barely visible impact damage (BVID) and visible impact damage (VID). Fatigue tests were conducted at a frequency of 5 Hz and a load ratio of R= –1. Progression of fatigue damage was monitored by ultrasonic C-scan, measurement of changes in the coupon modulus and measurement of hysteresis in the impact damaged coupons. Horn et al [123] proposed a new experimental method for quantifying impact damage and estimating the remaining fatigue lifetime of impact damaged polymer matrix composites. The procedure was demonstrated using GF reinforced polyurethane produced by injection moulding and structural reaction injection moulding. Thermoelastic stress analysis (TSA) was used to quantify the stress concentration associated with impact-damage in test samples for each composite. Following impact and TSA imaging, the samples were fatigued to failure over a range of stress amplitudes. The TSA-derived stress concentration factors were used to determine a modified stress amplitude that integrated the impact-fatigue data onto a master stress-life curve. This approach provided a quantitative measure of impact damage and a practical methodology for estimating the residual fatigue lifetime of impact damaged composites.

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Figure 21. Residual fatigue damage for undamaged, impacted and repaired composites: (a) 80 J impacted specimens and (b) 110 J impacted specimens [121].

Impact-damaged CF/epoxy composite laminates were tested in tension-compression fatigue under constant-amplitude loading with emphasis on studying mechanisms leading to delamination growth and fatigue failure [124]. The shapes of the buckles were measured with an optical whole-field measurement technique, while the extension and depth of delaminations was imaged several times during testing by ultrasonic C-scanning. The delamination growth occurred mainly transversely to the load application direction and buckles observed on the backside usually had the same shape as some of the delaminations. This indicated that buckling which took place during the compressive part of the load cycle

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instigated delamination growth. In specimens run to the fatigue limit, buckling and delamination growth occurred only in the outer 2 or 3 layers on the backside while a specimen that failed before the fatigue limit buckled globally. Constant amplitude tension–compression fatigue was performed on two layups of impact damaged CF/epoxy composite laminates [125]. The shape and the amplitude of the buckles were measured using an optical whole field measurement technique. The experimental conditions were chosen to be representative of damage in aircraft structures. The R-value was found to have a small influence on the fatigue life indicating that the compressive part of the load cycle had more importance than the tensile part, as the compressive load caused local buckling around the damage zone. The buckling was inward on the impact side and outward on the backside with larger buckles on the backside. Also the backside buckles showed a larger growth. Fatigue lives were compared with a conservative analytical model. Puhui et al [126] presented a new method for the determination of the CAI strength of composite laminates. In this method, the impact damage zone was modelled as an equivalent hole. The most outstanding characteristic of the method was that the simplification of the impact damage was based on the compressive failure mechanisms of impacted laminates. Such a simplification was not discussed in the literature before. A technique was established for determining the shape and size of the equivalent hole. The stress distribution around damage was calculated using the complex potential method and the classical lamination theory. A lay-up independent failure criterion was used to predict CAI strength. The predictions of the present approach were compared to test results and were found to be in very good agreement over a wide variety of materials. The study indicated that the width of damage zone was a key factor governing CAI strength. The influence of damage area, impact energy, impactor shape and dimension, etc. on CAI strength was characterized by their effects on the damage width. The study provided a very simple and effective approach for CAI strength prediction compared with previous methods. Lee et al [127] investigated the relationship between LVI damage under static indentation loading and the residual compressive static and fatigue strength in hybrid composites with nonwoven carbon tissue (NWCT). The hybrid laminates were made by interleaving the NWCT layer in the CFRP layer interfaces. The indentation damage was associated to the delamination area and the indentation energies of angle-ply laminates. The delamination area of the hybrid laminates were reduced to about half compared to the area induced in plain CFRP by indentation loading. After the indentation damage, compressive static and fatigue tests were carried out using an anti-buckling guide for the angle-ply laminates. The failure modes of the angle-ply CFRP and hybrid specimens were caused by global buckling under the compressive static loading. However, the failure modes of the angle-ply CFRP and hybrid specimens at Nf < 106 cycles were caused by shearing of the damaged part under the compression–compression fatigue loading. Compared with CFRP laminates, compressive fatigue lives of the hybrid laminates were significantly extended in all the stress ranges. The NWCT interleaving effect was assessed based on the damage extent in the hybrid specimens after indentation. The compressive static and fatigue failure processes were discussed from the observation of the fracture surface of damaged parts and the appearance of failed specimens when subjected to the compressive loadings after indentation. The impact properties and the fatigue properties of sandwich composites after impact were addressed [128]. Specifically, the LVI resistance of CF faced sandwich composites and the effect on their fatigue life was studied. Two and four layer face sheet CF sandwich

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composite samples with foam filled honeycomb core were impacted in a low velocity drop tester with varying masses and velocities of the impactor. After impact, the area of damage was determined using ultrasonic techniques. The fatigue life was then determined and compared to the fatigue life of non-impacted samples in a four point bend test. For all impact energies the impactor completely penetrated the front face of the sample. At 20 and 30 J the impactor also penetrated the back face. The 30 J caused a larger penetration area whereas the 20J caused more delamination of the back face. Two different densities of foam were used for the two layer samples. The two layer high density sample failure mode changed from in shear for the virgin materials to bending for the impacted samples. The penetration from the top face deteriorated the fatigue properties of the two layer samples. The fatigue life of the four layer and two layer low density samples did not appear to be affected by the impact. The damage tolerance of various lay-ups of thin carbon/epoxy laminates was examined [129] by CAI tests, using a testing device which adapted to the thicknesses of the specimens and did not require tabs or any modification of the specimen geometry. The compression stress state was not modified by the device geometry, as was verified by numerical simulation. With this device, CAI tests were performed for different carbon/epoxy laminate lay-ups (quasi-isotropic, cross-ply and woven) and the values of the residual strength and the normalized residual strength of the laminates were obtained as a function of the impact energy. The woven laminate was found to possess the highest residual strength for all impact energies, and the quasi-isotropic laminate the least normalized strength loss with increasing impact energy. Petit et al [130] presented an experimental study of impact and CAI tests performed on composite laminate covered with a cork thermal shield (TS) intended for launchers’ fairings. Drop weight impact tests were performed on composite laminate sheets with and without TS in order to study its effect on the impact damage. The results showed the TS was a good mechanical protection towards impact as well as a good impact revealing material. Nevertheless, totally different damage morphology was obtained during the impact test with or without TS, and in particular at high impact energy, the delaminated area was larger with TS. CAI tests were also performed in order to evaluate the TS effect on the residual strength. The TS increased the residual strength for the same impact energy, but at the same time, it caused a decrease in the residual strength for energies where delamination was not visible. As was postulated, during the impact tests with TS, invisible fibres’ breakages took place before delamination was visible, which was not the case for the unshielded sheets. Xiong et al [131] studied the static CAI and fatigue properties and the failure mechanisms of CF reinforced composite laminates with different lay-ups in order to optimize the stacking sequence. Ten different lay-ups were studied and the results for static strength, CAI and fatigue residual strength under different fatigue stress amplitudes were evaluated and compared to each other. The results of this study provided an insight into fatigue damage development in composites and constituted a fundamental basis for the development of a strain-based residual strength model. A new formulation was presented for a fatigue-driven residual strength surface based on controlling fatigue strain. The determination of the parametric formulae of the residual strength surface was established effectively and easily for the experimental data. The application of the model to test data, demonstrated its practical and effective use. The fatigue-driven residual strength could be obtained realistically from experimental data using the new formulae.

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The LVI and tension–tension fatigue behaviour of glass fibre reinforced polyester resin composites were investigated by Yuanjian and Isaak [132]. Two fibre geometries, namely [±45°]4 and [0/90°]2s stitch bonded GF were studied. The results revealed that LVI could seriously impair the tensile properties of [±45°]4 composites. For the [0/90°]2s GF composites, a critical impact energy was found. Below this energy level the tensile properties were hardly affected by the impact but above it the tensile properties deteriorated with increasing energy. LVI also reduced the fatigue lives of the composites and this reduction was related to the degradation in tensile strength. By normalizing the fatigue stress against the post-impact residual tensile strength of the composite, it was found that for each fibre geometry a single S–N curve could be drawn, which encompassed both undamaged and impacted samples. This implied that fatigue lifetimes of impact damaged composites could be predicted from measurement of the residual tensile strength of impacted specimens and the S–N curve of undamaged samples. Saito and Kimpara [133] evaluated the damage evolution behaviour considering the effect of the textile structure and water absorption. Damage observation was conducted using both non-destructive and direct observation methods. Candidate textile reinforcements were T300-3k plain woven fabric (PW) and T700S-12k multi-axial knitted fabric (MA) (Figure 22). The effect of water absorption on the performances of CAI and Post-Impact-fatigue (PIF) were small in PW CFRP laminates. Conversely, PIF properties of water-absorbed MA CFRP laminates drastically decreased. CAI strength was not affected by water absorption. PIF performance of dry MA CFRP was fairly higher than that of the other configurations. Using microscopic observation, some evidence of interfacial deterioration caused by water absorption was confirmed for both PW and MA CFRP laminates.

Figure 22. Compressive strength after impact for plain woven and multi axial knitted CFRP under dry and wet condition [133].

Minak et al [134] dealt with the relation between damage and tension–tension fatigue residual strength (FRS) in a quasi-isotropic CF reinforced epoxy resin laminate. The work Composite Laminates: Properties, Performance and Applications : Properties, Performance and Applications, Nova Science Publishers,

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was organized in two phases: during the first one, composite laminates were damaged by means of an out-of-plane quasi-static load that was supposed to simulate LVI; in the second phase, fatigue tests were performed on damaged and undamaged specimens obtained from the original composite laminates. During the quasi-static transverse loading phase, damage progression was monitored by AE. The measurement of the strain energy accumulated in the specimens and of the acoustic energy released by fracture events made it possible to estimate the amount of induced damage and evaluate the quasi-static residual tensile strength of the specimens. A probabilistic failure analysis of the fatigue data, reduced by the relative residual strength values, led to the correlation of the FRS of damaged specimens with the fatigue strength of undamaged ones.

Natural Composites

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Jute/polyester woven laminates have been impacted to energies up to 20 Joules with an impact velocity of 2 m/s using a drop-weight impact tower [135]. The diameter of the hemispherical impactor was 12.7 mm. Some of the laminates were further subjected to postimpact tensile tests to failure. The aim of these tests was to measure the degradation of their mechanical performances with increasing impact energy. Tensile failure occurred by macroscopic matrix cracking under the point of impact. The interlaminar fracture toughness of this material was sufficiently high and failure did not involve delamination. Fatigue loading with a maximum applied stress of 13 MPa i.e. around 25% of the ultimate tensile stress of the laminate was also performed. The fatigue ratio was 0.5 and the loading frequency used was 10 Hz. The damage produced by impact and the subsequent fatigue loading was observed using an optical microscope.

Post – Impact Static Behaviour Modelling A semi-empirical analysis on residual compressive strength (RCS) of CF/epoxy woven composite laminate was developed [136] which included the damage effects caused by impact and hydrothermal cycling. Impact damage was modelled as a soft inclusion with an exponentially decreasing stiffness which was further reduced due to hydrothermal cycling. A complex variable method was used to determine the in-plane stress distribution near the impact induced damage and a point stress failure criterion was then used to predict the failure load. Based on the semi-empirical model, the RCS could be related to damage width, damage intensity, undamaged strength and a degradation factor due to hydrothermal cycling. The results from the analysis coincided reasonably well with the experimental data for plainwoven fabric laminates. Nilsson et al [137] presented a combined numerical and experimental study of slender composite panels loaded in compression with artificial delaminations at two different depths. The study was motivated by FEA where this change in delamination depth induced a transition in the direction of delamination growth along with a change in the basic fracture models and stability. Tests were then carried out to verify the transition in delamination growth. The predicted transitions were to a large extent verified experimentally. The paper gave an outline of the computational model, which included contact between delaminated

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members, calculation of energy release rate with fracture mode separation by an approximate as well as a reliable method for general lay-ups, and a moving mesh scheme to account for delamination growth. The experimental work included monitoring of delamination growth by C-scan and acoustic emission along with a detailed fractographic study. The correlation between experimental observations and computer results were discussed in detail. Previous solutions to the low mass impact problem were based on laborious numerical approaches. Olsson in his paper [138] presented a closed form solution based on three asymptotic cases with deformation purely by indentation, bending or shear. Solutions were presented for a Hertzian contact law, suitable for monolithic plates, and for a linear contact law, suitable for sandwich panels. A good agreement was shown between “exact” numerical solutions and experimental results. The solution was also combined with an existing quasistatic delamination threshold load criterion. The comparison with data published in the literature indicated that the criterion also may be applicable to small mass impacts. The effect of delaminations in a post-buckling stiffened structure manufactured from laminated composite materials was modelled [139]. Driven by aircraft certification requirements, the emphasis of the technique was towards establishing whether delamination growth would initiate under given loading conditions. A geometric non-linear FEA was used to calculate the SERR around the circumference of a circular delamination using the virtual crack closure technique. In order to deal with the complex structural response in a computationally efficient manner, the structure was modelled using plate elements with two layers of plate elements used in the delaminated region. The effect of delamination size on the post-buckling strength panels was shown to be a complex phenomenon in which trends were difficult to predict. Large delaminations could significantly affect the global and sub-laminate buckling modes and therefore be less critical than smaller delaminations. It was concluded that the method could accurately predict the load and location at which delamination growth would initiate with the provision of reliable critical Strain Energy release Rate (SERR) data. Tay and Shen [140] presented the results of a buckling and post-buckling analysis and modelling of embedded delaminations, and the prediction of delamination growth in laminated composites with consideration for residual thermal stresses. The distribution of the local SERR along the delamination front was obtained via the virtual crack closure technique applied to three-dimensional (3D) FEA models of circular delaminations embedded in woven and nonwoven composite laminates. In each case, the delamination was embedded at a different depth through the thickness of the laminates. The fibre orientation of the plies bounding the delamination influenced significantly the distribution of the local SERR. There was qualitative agreement between the predicted directions of delamination growth and Cscan results of the embedded delaminations. It was found that residual thermal stresses affected significantly the onset of buckling of the delaminated sub-laminate, but had negligible influence on the distribution of the local SERR in the post-buckled regime. Furthermore, the effect of residual thermal stresses was more pronounced for delaminations that were closer to the surface. A method for the prediction of delamination areas and directions using fatigue growth criteria derived from test coupon data was also presented. It was found that growth criteria based on components of the SERR predicted the rate of delamination growth much better than those based on the total SERR. Riccio et al [141] performed a crack growth analysis on composite panels containing embedded delaminations using a geometrically non linear FEA code based on the total Langragian formulation. The code was improved with an effective virtual crack closure

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technique to evaluate SERR and with a penalty method for contact forces. The validation of the proposed tool was performed with experimental and numerical data available in the literature for double cantilever beam (DCB) specimens. Finally, the influence of the geometrical parameters of the delamination (size and location along the thickness) on the SERR distribution and delamination growth stability in composite panels under compression was analyzed. In a following article [142], Riccio and Gigliotti proposed a novel numerical approach for the delamination growth simulation in composite panels under compressive load. This approach, suitable for preliminary design and optimization purposes, was able to simulate the delamination propagation by means of a limited number of linear analyses. It was based on the determination of delamination buckling and on the evaluation of the energy released during the delamination propagation by means of eigenvalue and linear static analyses. The proposed approach was implemented into the FEA code ANSYS and applied to composite panels with circular embedded and rectangular through-the-width delaminations under compressive load. A first validation was carried out by comparing the results of delamination growth load and SERR distributions along the delamination front to two-dimensional and three-dimensional nonlinear results taken from literature. A numerical simulation of the structural behaviour of delaminated composite plates under compressive load was developed and presented by Riccio and Pietropaoli [143]. In this article the effect of fibre – matrix failure on the buckling behaviour of a delaminated structure up to final failure was investigated in detail. A geometrically non linear FEA approach was adopted for elastic instability simulation and the modified virtual crack closure technique (MVCCT) was used for the SERR evaluation at the delamination front. A fracture mechanics progressive damage approach was introduced for the simulation of fibre – matrix damage onset and evolution. Comparisons of numerical results with existing experimental data on composite panels with an embedded circular delamination under compressive load were presented for preliminary validation purposes. Furthermore, the influence of the different failure mechanisms on the compressive behaviour of delaminated composite plates was assessed, by comparing numerical results obtained with models of different degrees of complexity. Finally, results of a limited sensitivity analysis were presented, pointing out the influence of some key parameters of the adopted numerical approach on the accuracy of numerical simulations. Surface free energy has been often treated as a scalar constant without considering its dependence on propagation direction. It is desirable, however, to investigate how surface free energy or fracture toughness of delamination in a single interface varies with both the local mismatch angle of fibre directions and the direction of crack propagation in polymeric laminate composites. As a materials constant, fracture toughness is effectively used for various mechanical analyses of fibre-reinforced composites as well as conventional materials. Kim and Mayer [144] investigated quantitatively and qualitatively the dependence of delamination fracture toughness on mismatch angle and crack propagation direction in laminated structures. AS4-Carbon/Epoxy prepregs were used for fabricating test specimens, and 50 different mismatch angles of fibre direction were applied on the delaminated interface of laminates. Fracture toughness was measured using the Mixed-mode Bending (MMB) test. This test method imposed both mode-I and mode-II fracture, and the mixed-mode ratio (GII/G) can be controlled. The mixed-mode ratios used here were 20, 35, 50, 65, and 80%. The crack path and the delamination fracture toughness were observed and calculated for

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specimens, and the dependence on fracture toughness was shown to be related to the mismatch angle of ply fibres at the delaminated interface. The relationship between mismatch angle and delamination fracture toughness was newly revealed and discussed for various angles (Figure 23). These results could be usefully applied to various fracture mechanics analyses in fibre-reinforced laminated composites.

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Figure 23. Fracture toughness (Gc) as functions of incline angle and mismatch angle at mixed mode ration GII/G =80% [144].

Attia and Kinloch [145] attempted to predict growth of impact damage in CFRP skin/stringer structures subjected to cyclic-fatigue loading. They proposed a method that combined an experimental relationship between the SERR, G, obtained from direct measurements and the number of fatigue cycles Ng with a linear FEA of the composite structure under cyclic-fatigue loading. Kang and Kim investigated the fatigue behaviour and variability of fatigue life in carbon/epoxy laminates with impact-induced damage under tensile [146] and 2-stage tensile [147] fatigue loading. To describe the fatigue behaviour of the impacted laminates, the strength reduction concept was introduced, based on the Broutman’s model for 2- and 3stage loading, respectively. By using this concept combined with the expected fatigue behaviour of the un-impacted laminates, a fatigue life prediction model for the impacted laminates was derived and then verified through the fatigue tests on the impacted laminates under constant amplitude loading. In order to account for the probabilistic nature of the phenomenon, a statistical model was developed by introducing the random variable Z: The model described well the distribution of fatigue life in laminates. As was reported, the variation of fatigue life decreased rapidly as the applied impact energy increased. A test method, with specimen design similar to that proposed by O’Brien et al. [63, 148, 149 150] but in 3-point bending, was proposed [151] to measure the critical SERR (Gc) for delamination in fibre-reinforced polymers (FRP) under out-of-plane or transverse loading. Unlike end-notch-flexure (ENF) or double-cantilever-beam (DCB) tests that are currently used, the proposed method did not require an insert film or a pre-crack to initiate delamination. Instead, the delamination was initiated from a FRP layer of the same composition but with different fibre orientation. For this study, the FRP layer orientation was

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90°, i.e. one 90° layer in the mid-thickness and the rest 0° layers. Under 3-point bending, matrix shear cracking was firstly initiated in the 90° layer, mimicking the mechanism for delamination initiation in the FRP when subjected to transverse loading. The matrix shear cracking led to delamination in the adjacent interlaminar regions and the delamination area was measured after the test. Calculation of Gc was based on “the area method”, that is, dividing the total energy loss by the delamination area. The Gc was compared to that from plate specimens of the same composition and fibre lay-up but subjected to transverse point loading. The comparison indicated that very similar Gc values were acquired from the two types of testing, even though unstable crack growth occurred under 3-point bending and stable crack growth under transverse point loading. Therefore, the proposed method could potentially quantify delamination resistance of FRP under the transverse loading conditions. Andrews and Massabò [152] studied the static and dynamic response of singly and multiply delaminated plates subjected to cylindrical bending using models based on Timoshenko‘s shear deformable beam theory. The static interaction of multiple delaminations and its effects on fracture parameters and macro-structural behaviour was investigated. Interaction effects led to phenomena of amplification and shielding of the SERR and modifications of the mode ratio when compared to systems with a single delamination. The behaviour was controlled by the through thickness spacing of the delaminations. The dynamic delamination fracture of a double cantilever beam was studied and the capability of the proposed model to follow the different phases of delamination growth, including arrest, was highlighted [153]. The effect of loading pulse duration on stationary delamination systems was investigated and the response was described by SERR shock spectra to highlight phenomena of dynamic amplification. The results of the proposed models were compared with two-dimensional FEA solutions and showed excellent agreement. A computational model for investigating the local extent and the through-the-thickness position of the damage induced by LVIs on laminated composites was presented [154]. The model was based on a refined 3D zig-zag approach. It fulfilled the interlaminar transverse shear and normal stress contact conditions at the layer interfaces, those on the transverse normal stress gradient and the boundary conditions at the upper and lower bounding faces, as required by the theory of elasticity. Its functional degrees of freedom are the three components of the elastic displacement and the two shear rotations, like in widespread smeared laminate models. Several classical models were particularized from it and confronted together to assess whether a sophisticated structural model could have practical advantages for the study of impacts, since this is a still open question. In order to improve the structural modelling with an affordable computational effort, the contributions of the present model were incorporated updating the strain energy of a C0 parent eight-node plate element based on the First Order Shear Deformation Plate theory and through a post-processing procedure based on spline interpolation of the involved quantities. The impact load was computed by the Hertz’s contact law, integrating the equilibrium equations by the Newmark’s algorithm. The Galerkin’s method was used to obtain these equations, instead of using FEA discretization, because accurate results were obtained at lower costs. Fibre, matrix and delamination failures were predicted using two different strength-based criteria for each single mode. Since their predictions could greatly diverge, a comparison provided an estimation of the variation range so as to choose the appropriate one for each failure mode. According to the ply-discount theory, the degradation of properties after failure was simulated by reducing the elastic properties of the failed plies, in conformity with the failure mode. Different models with a

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different representation of displacements and interlaminar stresses were shown to provide different results in terms of displacements, contact force and damage. The application to a laminated stiffened panel indicated that the discrete-layer effects played an important role in the simulation of impacts, even when the laminates were not thick, since high-order dynamic effects were involved. The structural model, the contact force simulation and the failure criteria simulated the impact induced damage, extent and position across the thickness with a satisfactory accuracy. This was verified by comparison with ultrasonic detection.

Figure 24. Shearography images of impacted plated (a) 5.9 J, (b) 7.8 J and (c) 12.9 J. The damaged areas are within the marked ellipses [155].

Impact-induced damage in fibre-reinforced laminated composite plates was characterized by Watkins et al [155]. An instrumented impact tower was used to carry out LVIs on thirteen Composite Laminates: Properties, Performance and Applications : Properties, Performance and Applications, Nova Science Publishers,

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clamped glass/epoxy composite plates. A range of impact energies was experimentally investigated by progressively varying impactor masses (holding the impact height constant) and varying impact heights (holding the impactor mass constant). The in-plane strain profiles as measured by poly-vinylidene fluoride (PVDF) piezoelectric sensors were shown to indicate damage initiation and to correlate to impact energy. Plate damage included matrix cracking, fibre breakage, and delamination. Electronic shearography validated the existence of the impact damage and demonstrated an actual damage area larger than visible indications (Figure 24). The strain profiles that were associated with damage were replicated using an inhouse built FEA code. Using these simulated strain signatures and the shearography data, a back-propagation artificial neural network (ANN) was employed to detect and classify the type and severity of damage. This result was an extension of a prior work that showed that a strain-based neural network could characterize impact energy and contact forces for nondamaging impacts [156]. An analytical model for predicting the compressive fatigue limit strain of composite laminates which contain barely visible impact damage was presented [157]. The model represented the complex damage morphology as a single, circular delamination, and calculated the strain at which thin-film buckling of the circular region of delaminated plies occurred. The fatigue limit strain was defined as the strain at which the SERR for a thin postbuckled strip of the delaminated plies was equal to the critical Mode I value (GIC) for the resin. The model predicted a “critical” depth at which propagation of damage during fatigue was likely to occur (Figure 25). Results obtained using the model were compared with two sets of experimental results, and showed agreement of fatigue limit strain to within 4% of the experimental value.

Figure 25. Strain Energy release rate for delamination at 1, 2, and 3 ply depths in [45, -45, 0, 90]2s CFRP laminates [157].

The influence of compressive stress on mode II damage evolution was investigated [158] based on the cut-ply and dropped-ply experiments. An interfacial failure model with modified failure initiation and propagation criteria was proposed in order to take into account the effect of compression on matrix shear strength and mode II critical fracture energy, GIIC,. The model

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used one independently determined parameter to relate the compression to the increase in interlaminar shear strength and GIIC. Using this failure criterion, two types of cut-ply models and two types of dropped-ply models, using the same input parameter produced excellent correlation with experimental delamination stresses. As a validation case, the single-lap model was run with the above criterion and the same input parameters. The model correlated well with the experimental failure stress. The dynamic compressive strength of quasi-isotropic CF/epoxy composite was investigated experimentally and simulated numerically [159]. In-plane compression tests at strain rates around 400/s of quasi-isotropic laminates were performed using the Split Hopkinson Pressure Bar (SHPB). The dynamic strength of quasi-isotropic laminates exhibited a considerable increase when compared to the static values. The FEA model used ABAQUSTM three-dimensional solid elements C3D8I with 8 nodes and a user-defined interface with 8 nodes finite elements. These interface elements interrelated the threedimensional solid elements which modelled the composite layers and included a cohesive damage model which allowed for the simulation of delamination initiation and propagation. The proposed model assumed that the phenomenon of failure under these conditions was mainly dictated by interface delamination. This was supported by experimental tests which showed that all quasi-isotropic laminates split into several almost intact sub-laminates. The model compared very well with experimental results, confirming the formulated hypothesis that the internal layer damage did not markedly contribute to the quasi-isotropic laminate failure. Craven et al [160] examine the effect of impact damage on local tensile stiffness of multidirectional tape laminates. A FEA model was used for parametric studies on the effect of various patterns of delaminations and cracks across the fibres. Regular and random crack patterns were considered for 8 and 16 ply quasi-isotropic laminates. The effect on local stiffness was evaluated by a recently developed inverse numerical approach. It was found that although fibre cracks control the loss of stiffness, delaminations were instrumental in extending the zone influenced by these cracks. Fractographic observations were used to generate a detailed model of the damage in a real laminate. A close agreement between the predicted and measured strain fields and the corresponding stiffness distributions calculated using the inverse method was found.

Delamination Growth Prediction (Cyclic) The most up-to-date research, in the mechanics of delaminations and related crack-like defects in laminate and fibre composites, was discussed in [161] and was updated in [162]. Both internal and near-surface delaminations were considered. In the latter case, local buckling of delaminations and the interaction between buckling, damage accumulation, crack growth and global buckling were considered. The problem of the evaluation of the residual load-carrying capacity of delaminated structural components was discussed, including the assessment of the fracture toughness with respect to impact loading. Quasi-isotropic carbon/epoxy laminates with holes, polymer plugs and cut fibres were studied experimentally and analytically [163]. Strain fields were measured using digital speckle photography. The results were used to validate an inverse method, where elastic properties of inclusions were determined by matching computed and measured displacements.

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The tensile and compressive strength were measured and the applicability of notch failure criteria to soft inclusions was examined. Available closed form solutions agreed well with measured strains. The elastic properties predicted by the inverse method were in fairly good agreement with data from coupon tests although predictions were sensitive to measuring errors. The laminate toughness in compression was higher than in tension, as expected from the different failure mechanisms such as fibre kinking. Laminates with inclusions were tougher than laminates with holes, which suggested that inclusions restrain in-plane fibre kinking. Higher toughness was reflected in larger characteristic lengths. Fatigue tests of a hat-shape stringer stiffened panel were conducted [164]. The panel was a typical part of upper skin of a lightweight composite wing using a new production technology of stitching, co-bonding and RTM method. Impact damage was induced on the skin/stringer co-bonded part and on the typical skin part of the test panel by a drop-weight impact machine. The test protocol consisted of two phases. The first phase involved fatigue tests to verify durability of the structure with barely visible impact damage. The second phase was flaw growth tests for evaluation of visible impact damage growth to estimate inspection intervals. The Mini-TWIST (shortened version of The Transport WIng STandard load program) spectrum loading was used for both tests. NDI was carried out by pulsed thermography during the test to observe damage propagation. Finally, static load was applied up to the design limit load to verify the residual strength after all the spectrum loading tests. A composite laminated plate containing an internal delamination and subject to compressive loads was considered [165]. The plate was modelled as two sub-laminates partly bonded together by an elastic interface. The interface in its turn was represented by a continuous distribution of linear elastic springs acting in both the normal and tangential directions with respect to the interface plane. Based on an explicit solution, already presented in previous works by the authors, the expressions for the I and II (opening and sliding) modal contributions of the potential SERR, G, and the mode-mixity angle, ψ, were determined. In the reviewed paper, the model was further developed to describe the mechanical response of delaminated plates under fatigue loads. A mode-dependent fatigue growth law was applied, which accounted for the simultaneous presence of both opening and sliding crack propagation modes. The results could be used to easily predict the number of cycles needed for a delamination to extend to a given length or for complete failure, for any value of the minimum and maximum applied load.

Natural Composites A series of experiments was carried out to characterize the residual tensile and fatigue after impact properties of non-woven hemp fibre mat reinforced polyester [166]. Additionally, the degradation of tensile modulus during fatigue cycling was studied and related to damage accumulation. For comparison purposes, ±45° GF reinforced polyester samples were also subjected to similar tests. It was necessary to apply a relatively high pressure to the hemp composite during the curing stage in order to ensure a high enough fibre fraction and provide a significant reinforcing effect. With similar fibre weight fractions, the hemp and glass reinforced materials exhibited similar static tensile properties and fatigue lifetimes. Although the slightly steeper S–N curve of the hemp based material indicated a higher rate of reduction in fatigue strength with increasing cycles, it remained above the S–N

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curve for the glass based material. It was thus concluded that the hemp-based material was able to withstand slightly higher cyclic stress levels for the same number of cycles. The major difference in mechanical performance was the poorer resistance of the hemp based composite to impact. Additionally, the hemp based material failed in a much more brittle manner, without any visible signs of damage, such as the matrix cracking that was observed in the GF based composite. It was found that, if the fatigue lifetime data of impact damaged samples were normalized against the post-impact residual tensile strength, all data points lay close to a common S–N curve. This implied that residual fatigue lifetimes of damaged samples could be predicted from their residual strength and the S–N curve of the undamaged material.

Models of Prediction

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Katerelos et al [16] studied the fatigue behaviour of composite panels subjected to LVI. Impacted specimens were tested under compression–compression fatigue. A delamination propagation model based on the derivation of the SERR was used. The shape of the delamination was experimentally monitored by c-scan imaging and it was transferred to a polar diagram, where the load direction was at 90°. The stress distribution around the initially induced delamination was derived analytically, based on the assumption that both the initially delaminated area and the delamination growth after certain amount of fatigue cycles was simulated by an ellipse with semi-axes defined from the original delamination shape. The definition of the ellipse simulating a C-scan detected delaminated area was presented (Figure 26).

Figure 26. Definition of an ellipse to simulate a C-scan detected delamination.

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Stress distribution functions formulated according to the Lekhnitskii – Savin theory were derived. The orientation and aspect ratio of the ellipse were used to calculate the corresponding strain energy-release rates, which were subsequently used to predict the direction of delamination propagation. A flowchart representation of the algorithm for the calculation of the SERR around the ellipse was outlined (Figure 27).

Initial Delamination

Calculation of the stress components, σij Material Properties

σθ εθ ELAM

CLT

Model

Ei , t i , m

G(θ) E*

Figure 27. Calculation of the SERR distribution. Flow-chart representation of the algorithm.

The reduced stiffness properties (E*) of the delaminated plate were derived using the model proposed by O’Brien [63]. Thus the direction of delamination growth could be easily predicted. 0.01

IMS/SXB 9J Prediction (9 J) IMS/SXB 16J Prediction (16 J) 1E-3

(dA/dN)/A0

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1E-4

1E-5

1

10

G (kN / m) Figure 28. Prediction of the normalised delamination growth vs. the maximum SERR.

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In order to quantify the delamination area growth, a power law similar to the Griffith’s law was used. The constants of the power law were derived from experimental data. By the application of this law to impacted and subsequently fatigued laminates, with different initial delamination areas resulting from different impact energies, it was proved that the delamination growth rate with respect to the fatigue cycles was independent from the impact energy and, therefore, from the initial delaminated area. The delamination growth rate was predicted as a function of SERR (Figure 28). Models to predict the split growth in notched AS4/3501-6 graphite/epoxy quasi-isotropic laminates under tension-dominated fatigue were presented [167]. First, a power law model and an artificial neural network (ANN) model were developed to describe the split growth under constant-amplitude fatigue. They were then applied in conjunction with a linear damage growth model to predict the split growth under spectrum fatigue. The ANN model was found to work better than the power law model as a predictive tool for split growth. A multi-layer perception (MLP) network using error back propagation algorithm was employed by Garg et al [168] in order to estimate the damage parameters from broad-band spectral data as a diagnostic signal. Various existing models of damage in laminated composite and the resulting stiffness degradation were discussed. Degradation of ply properties as a function of transverse matrix cracks in cross-ply, splitting in longitudinal ply, and evolution of consecutive stages of damage, such as delaminations and fibre fracture was considered to be one of the damage model parameters. The stiffness degradation factor and the location and size of the damaged zone in a laminated composite beam were considered as damage model parameters. Fourier spectral data, which were typical for most of the diagnostic wave measurements, were used as input to the neural network. Training of the neural network involved many data sets difficult to generate using experiments, and therefore a spectral finite element model (SFEM) with embedded degraded zone for the laminated composite beam was developed. A numerical simulation using this model was carried out, in order to estimate the temporal signals that were likely to be measured. Analytical studies on the performance of the neural network were presented based on numerically simulated data. The effect of measurement noise on the network performance was also reported. Mixed-mode open-notch flexure (MONF), anti-symmetric loaded end-notched flexure (MENF) and centre-notched flexure (MCNF) specimens were used to investigate dynamic mixed I/II mode delamination fracture using a fracturing split Hopkinson pressure bar (FSHPB) [169]. An expression for dynamic SERR Gd was formulated and evaluated. The experimental results showed that dynamic delamination increased linearly with mode mixing. At low input energy Ei ≤ 4.0 J, the dynamic (Gd) and total (GT) SERRs were independent of mixed-mode ratio. At higher impact energy of 4.0 ≤ Ei ≤ 9.3 J, Gd decreased slowly with mixed I/II mode ratio while GT was observed to increase more rapidly. In general, Gd increased more rapidly with increasing delamination than with increasing energy absorbed. For impact energy of 9.3 J before fragmentation of the plate, the effect of kinetic energy was not significant and could be neglected. For the same energy-absorption level, the delamination was greatest at low mixed-mode ratios corresponding to highest Mode II contribution (Figure 29). The results of SERRs from MONF were compared with a mixedmode bending (MMB) formulation and were in agreement for Mode II but not for Mode I. Hackle (Mode II) features on SEM photographs decreased as the impact energy increased but increased as the Mode I/II ratio decreased. For the same loading conditions, more pure Mode

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II features were generated on the MCNF specimen fractured surfaces than the MENF and MONF specimens.

Figure 29. (a) Delamination crack growth as a function of peak energy absorbed at 9.3 J impact Energy for varying mode ratios and (b) regions of crack growth and area plot of energy absorbed-delamination curve for varying impact energy [169].

Another approach to calculate the stiffness reduction due to the delamination presence was proposed by Kim et al [170]. A damage detection problem was formulated as the identification of the spatial stiffness distribution in a damaged composite plate. Full-field heterogeneous curvature fields obtained using an optical deflectometry technique were processed by the virtual fields method adapted to retrieve the 2D stiffness distribution map of a damaged carbon-epoxy plate (Figure 30). The method located the damage and provided a fairly good estimate of the stiffness reduction in the damaged area. In this paper, the procedure was described, validated on simulated measurements and some initial experimental results were given.

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Figure 30. Optical deflectometry curvature fields processed with the virtual field method to retrieve the stiffness distribution [170].

An analytical model for the prediction of compressive fatigue threshold strains in composite plates with barely visible impact damage (BVID) was presented [171]. The model represented the complex damage morphology as a single circular delamination at a critical level and calculated the strain at which thin-film buckling of the circular delaminated region occurred. The threshold strain was defined as the strain at which the SERR for the fracture of post-buckled delaminated plies along the delamination was equal to the critical Mode I value (G1C) for the resin. The model predicted the critical through-thickness level for delamination, the stability of delamination growth and the sensitivity to experimental error in geometric measurements of the damage area. Results obtained using the model were in agreement to within 4% for fatigue strain experimental values for four sets of data reported in the literature.

Technologies for the Reduction or Elimination of Damage Propagation In the past various methodologies have been proposed, studied and used to hinder interlaminar damage and delamination. These include matrix toughening procedures and through thickness laminate reinforcement. The later could be divided into two categories, Zpinning and stitching.

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z-Pinning

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Mouritz [172] reviewed published research into polymer composite laminates reinforced in the through-thickness direction with z-pins (Figure 31). The current research into the manufacture, microstructure, delamination resistance, damage tolerance, joint strength and mechanical properties of z-pinned composites was outlined. The benefits of reinforcing composites with z-pins were assessed, including improvements to the delamination toughness, impact damage resistance, post-impact damage tolerance and through-thickness properties. The enhancement of the failure strength of bonded and bearing joints due to zpinning was also examined (Figure 32). The paper reviewed research into the adverse effects of z-pins on the in-plane mechanical properties, such as reduced elastic modulus, strength and fatigue performance. Mechanisms responsible for the reduction of the in-plane properties were discussed, and techniques to minimise the adverse effect of z-pins were described. The benefits and drawbacks of z-pinning on interlaminar toughness, damage tolerance and inplane mechanical properties were compared against other common types of through-thickness reinforcement for composites, such as 3D weaving and stitching. Gaps in our understanding and unresolved research problems with z-pinned composites were identified to provide a road map for future research into these materials.

Figure 31. (a) Typical size of a z-pin and (b) z-pins inside a composite [172].

Partridge and Cartié [173] provided an introduction into the technology of through-thethickness reinforcement of thermosetting composites by the insertion of z-fibre pins. The manufacture of the raw materials was described, as was the method of pin insertion. Delamination tests samples were prepared from unidirectional continuous CF/epoxy prepreg (IMS/924), made into 3 mm thick unidirectional laminates with and without a block of z-pins in the crack path. Fracture testing was carried out under Mode I (standard DCB test) and Mode II (3-point-ENF) loading conditions. In the Mode II loading case, the z-pins underwent a significant bending deformation prior to failure by internal shear [174]. The data presented were obtained by using the commonly used data reduction schemes. The applicability of such data treatment was also critically assessed.

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Figure 32. Plots of the effects of z-pins on the apparent modes I and II delamination toughness for CFRPs [172].

The effectiveness of z-pins in co-cured joints was illustrated on the model of a composite double cantilever beam (DCB) subject to a standard fracture toughness test [175]. A comprehensive solution was presented accounting for a broad spectrum of issues that affect the problem. The accurate evaluation of the rotational constraint provided by the intact section of DCB, possible transverse shear deformation in the delaminated section, and effects of uniform and non-uniform temperature were considered. A simple criterion for the effectiveness of z-pins in co-cured joints was introduced and its application was illustrated on numerous examples. As indicated by the analysis, z-pinning was an effective method of enhancing delamination resistance of composite joints. Even a very small volume fraction of z-pins (less than 1.5%) arrested delamination in co-cured composite joints. The mode I interlaminar fracture in z-pin reinforced composite laminates was modelled [176] using a cohesive volumetric finite element (CVFE) scheme. The test configuration used in this study was a z-pin reinforced double cantilever beam specimen. A bilinear rateindependent but damage-dependent cohesive traction – separation law was adopted to model the fracture of the unreinforced composite and discrete nonlinear spring elements to represent the effect of the z-pins were employed. The delamination toughness and failure strength of the z-pin reinforced composites were determined by a detailed comparative study of the numerical modelling results with experimental data. To further reduce the computational effort, an equivalent distributed cohesive model was introduced as a substitute for the discrete nonlinear spring representation of the z-pins. The cohesive model was implemented on various test problems with varying failure parameters and for varying spatial z-pin reinforcement configurations and was in good agreement with experimental results.

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Chang et al [177] examined the effect of pinning on the flexural properties, fatigue life and failure mechanisms of carbon/epoxy laminates. Five-harness satin weave carbon/epoxy laminates were reinforced in the through-thickness direction with different volume fractions and sizes of fibrous composite pins. Microscopic examination of the laminates before flexural testing revealed that the pins caused considerable damage to the microstructure, including out-of-plane crimping, in-plane distortion and breakage of the fibres and the formation of resin-rich zones around each pin. The pins also caused swelling of the laminate that reduced the fibre volume content. Despite the damage, the pins did not affect the flexural modulus of the laminate as was previously reported by Chang et al [178]. However, increasing the volume content or the diameter of the pins caused a deterioration in the flexural strength and fatigue life, which appeared to be governed by fibre rupture on the tensile side of the laminate. Property changes were discussed in terms of transitions in the dominant failure mechanisms due to the presence of pins. The effect of z-pinning on the in-plane compression properties and failure mechanisms of polymer laminates was experimentally studied by Mouritz [179]. The reduction of the compression modulus, strength and fatigue performance of carbon/epoxy laminates with increasing volume content and diameter of pins was determined. The elastic modulus decreased at a quasi-linear rate with increasing pin content and pin diameter. Softening was caused by fibre waviness around the pins and reduced fibre volume content due to volumetric swelling of the laminate from the pins. A simple model was presented for calculating the compression modulus of pinned laminates, which took into account the softening effects of fibre waviness and fibre dilution. The compression strength and fatigue life also decreased with increasing volume content and pin diameter. The strength and fatigue properties deteriorated due to fibre kinking caused by fibre waviness around the pins and the reduced fibre content caused by swelling. The deterioration of the compression properties was also dependent on the laminate lay-up. The decrease in properties was dependent on the percentage of 0° or load bearing fibres in the laminate. The paper provided suggestions for minimizing the loss of compression properties to laminates due to pinning.

Stitching Carpino et al [104] claimed that stitching technologies were particularly effective in improving damage tolerance. Stitching is expected to play an important role in primary structures of next generation commercial airplanes. However, they postulated that stitching could be detrimental to some mechanical properties, such as quasi-static in-plane stiffness and strength. Perforation energy was also adversely affected by stitching. Carpino et al [104] explained the detrimental effects of stitching as an effect of the synergies of stitching in combination with sample-bullet contact and performed the required numerical analyses for their description and modelling. In their review [104], dynamic tests were performed on stitched CFRP panels of different thicknesses. The panels were struck by two spherical projectiles of different diameter. The experimental results were in agreement with two closedform models for the prediction of the perforation energy, one of which was suitably modified. The damaged area of the impacted samples was recorded by ultrasonic C-scan, in order to study the correlation between delamination extent and the maximum energy absorbed by the target during impact.

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Experimentally measured perforation data were given for woven, z-stitched and throughthickness z-stitched glass polyester laminates for a number of laminate thicknesses, a number of geometries of impactor (cone, flat and hemisphere), and two missile masses [180]. Impact perforation velocities ranged up to 571 m s-1 on 200 by 200 mm laminates with fully clamped boundary conditions (Figure 33). Results were expressed in terms of static and impact perforation energies. The discussion included a study of energy absorption mechanisms during perforation, with a view to identifying improved combinations of materials. It was concluded that all configurations behave in a similar manner and that the flat ended missile had the largest dynamic enhancement factor, i.e. ratio of impact perforation energy to static perforation energy.

Figure 33. Static energy to perforation versus thickness and number of plies for (a) Woven roving laminates and (b) z-stitched laminates [180].

Aymerich et al [181] studied the effect of edge stitching on tensile static and fatigue properties of graphite fibre reinforced laminates. In a following work [182], they examined the effect of stitching on the impact performance of a class of graphite/epoxy cross-ply laminates with the aim of investigating the ability of through-thickness reinforcement to

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improve delamination resistance. The influence of Kevlar stitching on edge delaminations, ultimate tensile strength and tension–tension fatigue life was examined on laminates of two stacking sequences ([±45/0/90]s and [±30/90]s), as representative of two classes of fibredominated and matrix-dominated laminates. Through-thickness stitching offered a significant improvement in the static delamination resistance of laminated composites, but had adverse effects on the ultimate tensile stress of the laminates under study, with an increase in strength of [±30/90]s laminates and a decrease in strength of [±45/0/90]s laminates. Similarly the fatigue life of matrix dominated [±30/90]s specimens was considerably extended by stitching, while the fatigue resistance of fibre-dominated [±45/0/90]s specimens was reduced, particularly at high fatigue stresses. Stitching appeared very efficient in arresting delamination in [±30/90]s specimens for both static and fatigue loads, but was not as efficient in stopping fatigue delaminations in [±45/0/90]s specimens, since Kevlar threads were found to be prone to breakage under cyclic loading. While the static delamination resistance of laminated composites was improved by stitching, this did not automatically result in better fatigue performance for all lamination sequences. In the work by Hosur et al [183], damage resistance of stitched/unstitched S2-glass/epoxy composites was studied. Five layer stitched/unstitched plain weave S2-glass woven fabric composite laminates were manufactured using a toughened SC15 epoxy resin system. For stitching, two configurations: one with 25.4 mm grid and other with 12.7 mm grid, were used with 6 mm pitch. Damage resistance was evaluated by subjecting 100×100 mm2 samples to LVI loading at energy levels ranging from 10 J to 80 J using an instrumented impact tester. Three samples were tested at each energy level. The extent of damage was evaluated using ultrasonic C-scan. The outcome of the study indicated that stitching confined the damage size. As damage was little over most of the energy range, the study was further extended to determine the effect of repeated impact loading. Under this study, laminates were subjected to repeat impact loading up to a maximum of 40 impacts at energy levels ranging from 10 to 50 J. Results of the repeated impact study were reported in terms of peak load, absorbed energy and projected damage area. All the laminates, sustained repeated impact loading up to 30 J. Beyond 30 J, laminates failed to carry repeated loading and failure depended on the laminate configuration. Saito and Kimpara [184] focused on a multi-axial stitched fabric, which was a thick, high performance reinforcement for large-scale composite structures. The effects of impact damage on multi-axial stitched CFRP laminates moulded by vacuum-assisted resin transfer moulding (VARTM) method were evaluated. Impact damage within the material was evaluated by an ultrasonic scanning device and optical cross-sectional observations. Probed images obtained by both non-destructive and destructive methods were compared, and internal damage distributions of multi-axial stitched CFRP laminates were compiled. In addition, the residual compressive strength and the fatigue properties of impact-damaged CFRP laminates were evaluated by in situ damage growth monitoring using the thermoelastic stress analyzer (TESA). Three-dimensional damage distributions of impacted CFRP laminate were obtained from ultrasonic C-scan images and cross-sectional photographs. The damage progress behaviour was observed on a destructive and non-destructive basis by postimpact-fatigue (PIF) tests. LVI tests were carried out [185] on stitched CF-reinforced plastic laminates of various thicknesses, with reference to the overall force – displacement curve, first failure load, penetration, indentation and damage extent. The results obtained were compared to similar

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data available for 2D laminates. Apparently, the presence of stitches did not affect substantially the material behaviour in terms of force – displacement curve, first failure load, and indentation. However, the stitched laminates exhibited penetration energy about 30% lower than their 2D equivalent. The advantage of stitching in terms of impact damage resistance was evident only for high thickness composites. The data generated suggested that the use of stitches could be unnecessary to hinder delamination in thin 2D laminates. Velmurugan and Solaimurugan [186] examined the effect of plain stitching by untwisted fibre rovings in the in-plane mechanical properties and Mode I interlaminar fracture toughness of glass/polyester composites. Through-the-thickness reinforcement was performed by stitching with twisted fibre yarns in a sewing machine to increase the delamination strength. However, stitching degraded the in-plane mechanical properties. In this work glass woven roving mats (WRM) were stitched in the thickness direction with untwisted Kevlar®, glass and carbon fibre rovings by plain stitch. It was observed that in plain stitch, no thread cross was formed and thus no resin rich pockets were seen. The uniform distribution of fibres in the stitch roving, the absence of resin rich region and reduced fibre damage resulted in increased in-plane tensile, lap shear, flexural, transverse shear and impact strengths. The effect of stitching on Mode I delamination toughness (GIc) of glass/polyester laminates was investigated by performing double cantilever beam (DCB) test. Stitching increased the Mode I delamination toughness up to 20 times compared to that of unstitched specimens. The influence of pure assembly seams using a thin polyester yarn in a zigzag geometry on the resulting mechanical performance of a non-crimped fabric (NCF) CF-reinforced epoxy composite manufactured by vacuum-assisted resin transfer moulding was presented by Beier et al [187]. This study aimed at generating a solid foundation regarding the overall performance level of stitched NCF composites and at identifying critical property changes. The comprehensive evaluation of the mechanical composite properties included static as well as dynamic tests of the in-plane properties as well as a characterisation of the interlaminar properties, such as apparent interlaminar shear strength (ILSS) and CAI. It was demonstrated that mechanical properties such as the tensile and compression stiffness and CAI strength were not degraded by the chosen stitching parameters, whereas the tensile and compression strength, ILSS as well as the tensile fatigue behaviour were reduced as a result of pronounced localised fibre undulations. A direct comparison with the properties of a commonly used 5H satin woven fabric composite verified that the overall performance of the stitched NCF composites were superior with regard to the key criteria for aircraft applications and maintained the performance advantage of NCF composites as compared to standard woven fabrics. Promising approaches included the use of different yarn materials based on soluble thermoplastics and/or modified stitching parameters. The tensile fatigue properties of specific types of 3D woven, stitched and z-pinned composites with through-thickness reinforcement were compared by Mouritz [188]. Tensile tests under monotonic and cyclic loading were performed on the 3D composite materials to determine the influence of the z-reinforcement type – woven z-binder, stitch or z-pin – on the tensile modulus, strength and fatigue life. The in-plane Young’s modulus of the composites was not affected by the type or volume content of the z-reinforcement. The tensile strength of the 3D woven and stitched composites was also not affected by the z-reinforcement. However the strength of the z-pinned composite dropped steadily with increasing volume content of zreinforcement. The fatigue life of the 3D composites was reduced by the z-reinforcement, regardless of the type of z-reinforcement, i.e. woven z-binders, stitches or z-pins. Z-

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reinforcement damaged the microstructure of the 3D composites and resulted in the degradation of the tensile fatigue properties. The fatigue damage mechanisms caused by the different types of z-reinforcement were described. The results indicated that through-thethickness reinforcement was detrimental to the tensile fatigue life. However, the study was restricted to specific types of materials and further research into a wider variety of 3D woven, stitched and z-pinned composites was required for a general assessment of their fatigue performance. Tan et al [189] investigated the effect of selective stitching on impact damage tolerance and fatigue durability of stiffened composite panels and compared it with unstitched stiffened composite panels. Two-blade stiffened composite panels were fabricated by the resin film infiltration technique. Impact damage was inflicted on the stiffened panels using a dropweight with impact energy of 30 J from the skin-side over the stiffener and the flange. The experimental results indicated that selective stitching was an effective way to improve the compressive failure strength and fatigue strength of stiffened panels with both clearly visible flange damage (CVFD) and clearly visible stiffener damage (CVSD). The buckling and postbuckling FEA were performed to predict the static compressive strength of the selectively stitched stiffened panels. A good agreement was obtained between analytical and experimental results. The failure behaviour of the undamaged, CVFD and CVSD panels was also investigated.

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Matrix Toughening – Interleaving Matrix toughening procedures can be categorized into traditional interleaving reinforcement and the incorporation of nano-phases to produce hybrid composites. Kishore et al [190] examined the dynamic behaviour of laminated carbon/epoxy composites with inserted polytetrafluoroethylene (PTFE)-coated interleaves. Instrumented impact tests performed on the interleaved test samples showed significant differences in the energy absorption characteristics that could be correlated with the failure mode. It was inferred that with the introduction of tougher layers of material at specific locations, the loadcarrying ability decreased while the energy-absorbing capability improved considerably. The microstructure and fracture behaviour of epoxy mixtures containing two monomers of different molecular weights were studied by Zubeldia et al [191]. The variation of fracture toughness by the addition of other modifiers was also investigated. Several amounts of highmolecular-weight diglycidyl ether of bisphenol A (DGEBA) oligomer were added to a nearly pure DGEBA monomer. The mixtures were cured with an aromatic amine and exhibited phase separation after curing. The curing behaviour of the epoxy mixtures was investigated with thermal measurements. A significant enhancement of the fracture toughness was accompanied by a slight increase in both the rigidity and strength of the mixtures that corresponded to the content of the high-molecular-weight epoxy resin. Dynamic mechanical analysis and atomic force microscopy studies indicated that the generated two-phase morphology was a function of the content of the added epoxy resin. The influence of the addition of an oligomer or a thermoplastic on the morphologies and mechanical properties of both epoxy-containing mixtures was also investigated. Hybrid composite materials with non-woven tissue (NWT) were developed to improve the mechanical properties of conventional FRP composite materials [192]. As was shown, the

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interleaving of the non-woven carbon tissue (NWCT) layer largely increased the Mode II interlaminar fracture toughness while it did not significantly change the Mode I interlaminar fracture toughness. In addition, compared with conventional CFRP composites, the hybrid composites with NWCT exhibited high reliability and improved strength in their response to quasi static tension and fatigue. The low-energy impact damage mechanism for the NWCT hybrid composites was also investigated under quasi-static indentation loading. The toughening mechanism of the NWCT interleaving was described in relation to observed damage. Indentation caused shear failure due to the compressive fatigue loading of the CFRP and hybrid composites, but in the case of hybrid composites, the propagation of shear cracks was prevented by NWCT located between the CFRP layers. It was found that the Mode II interlaminar fracture was predominant in the delamination by the indentation loads.

Figure 34. Static sequence for non-interleaved and PET interleaved laminates [193].

Pegoretti et al [193] studied the correlation between interlaminar fracture toughness and impact energy absorption for the fracture of carbon/epoxy laminates. CF/epoxy cross-ply prepreg layers were interleaved with thin poly(ethylene-terephthalate) (PET) films. Before the lamination process, 1 mm diameter circular holes were drilled in the PET films at several surface densities (from 0 up to 44 holes/cm2) in order to selectively increase the interlaminar contact area between the carbon/epoxy laminae (Figure 34). In this way, the interlaminar

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contact area gradually varied from 0%, corresponding to the case in which non-perforated PET films were used, up to 100% in the case of non-interleaved laminates. The Mode I interlaminar fracture toughness of the resulting laminates was determined according to the ASTM D-5528-01 standard test method on double cantilever beam (DCB) specimens. The critical values of the SERR determined at the point at which the load versus opening displacement curve became non-linear (GIC,NL) varied from 40 up to 260 J/m2, depending on the interlaminar contact area. All the laminates were subsequently tested in three point flexure performed both under quasi-static (5 mm/min) and impact (2 m/s) loading conditions. The elastic modulus of the laminates was found to be practically independent of the level of interlaminar adhesion, while the bending strength decreased as the interlaminar fracture toughness decreased. The total energy to fracture evaluated under impact conditions showed a non-monotonic correlation with interlaminar fracture toughness, reaching a maximum level for a GIC,NL value of about 60 J/m2. At the same time, the ductility index, i.e. the ratio between the propagation and the initiation energies, evaluated by instrumented Charpy impact tests, markedly increased as the interlaminar fracture toughness decreased (Figure 35).

Figure 35. Ductility index values as a function of the interlaminar fracture toughness [193].

Hybrid Composite Systems Incorporating Nano- phases Carbon nanotubes (CNTs) are considered to be highly promising as fillers for improving the properties of polymers. However, questions concerning the appropriate type of CNTs, e.g., single-wall CNTs (SWCNT), double-wall CNTs (DWCNT) or multi-wall CNTs (MWCNT), and the relevance of a surface functionalisation are still to be answered. In their study [194] Gojny et al focused on the evaluation of the influence of different types of nanofillers on the mechanical properties of epoxy-based nano-composites and the relevance of

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surface functionalisation. The produced nano-composites exhibited enhanced strength and stiffness and more importantly, a significant increase in fracture toughness (43% at 0.5 wt% amino-functionalised DWCNT). The influence of filler content, the dispersion, the aspect ratio, the specific surface area and amino-functionalisation on the nano-composite properties were discussed and correlated to the identified micro-mechanical mechanisms (Figure 36).

Figure 36. Experimentally obtained fracture toughness of epoxy-based composites containing (a) nonfunctionalised and (b) amino-functionalised nanoparticles. [194]. Composite Laminates: Properties, Performance and Applications : Properties, Performance and Applications, Nova Science Publishers,

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Sadeghian et al [195] presented an experimental study of using carbon nanofibres (CNF) to improve the interlaminar fracture properties of GF/polyester composites. Surfactant-treated CNF were dispersed in polyester resin and then the CNF-resin suspension was used to impregnate a GF preform using VARTM. The possibility to use VARTM for thick and large CNF toughened composite parts was experimentally investigated. The influence of CNF concentration on the CNF filtration in the GF preform, the resin viscosity, and the micro-void formation were examined (Figure 37). By choosing appropriate manufacturing parameters, they were able to use the VARTM process to infuse the surfactant-treated CNF/resin matrix into the GF preform and successfully manufacture the CNF toughened GF/polyester specimens for mode-I delamination tests. The critical SERR of mode-I delamination (GIC) was estimated for several composite specimens with 1 wt% CNF concentrations. Significant improvement in the GIC was consistently observed when 1 wt% CNF were added to toughen the polyester resin. Microscopic examination showed that the fracture surface of the 1 wt% CNF toughened GF/polyester samples was more complex than the fracture surface of regular GF/polyester composites.

Figure 37. Filtration in flow direction with inflow with 1.0 % CNF concentration (a) perform and setup with two layers of random glass fiber mats with thickness 1.2 mm (side view) and (b) contrast difference due to CNF filtration [195].

Wetzel et al [196] focused on the effect of nano-particles on the mechanical and fracture mechanical properties of epoxy resins, particularly with regard to fracture and toughening mechanisms. A comprehensive study was carried out on a series of nano-composites containing varying amounts of nano-particles, either titanium dioxide (TiO2) or aluminium oxide (Al2O3). Nano-composites were systematically produced by applying high shear energy during a controlled dispersion process, in order to reduce the size of agglomerates and produce a homogeneous distribution of individual nano-particles within the epoxy resin. The mechanical performance of the nano-composites was subsequently characterized in flexure, dynamic mechanical analysis (DMA) and fatigue crack growth testing (FCP), using a linear fracture mechanics approaches (LEFM). The microstructure of specimens and the corresponding fracture surfaces were examined by TEM, SEM and AFM techniques in order to identify the relevant fracture mechanisms, and to gain information about the dispersion quality of nano-particles within the polymer. It was found that the presence of nano-particles

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in epoxy triggered various fracture mechanisms, e.g. crack deflection, plastic deformation, and crack pinning. It was observed that nano-particles could overcome the drawbacks of traditional tougheners (e.g. glass beads or rubber particles) by simultaneously improving stiffness, strength and toughness of epoxy, without sacrificing thermo-mechanical properties (Figure 38).

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Figure 38. Increase of the fracture toughness with increasing diameter to distance ratio (2r0/2c) of the fillers. Experimental data compared with theoretical models for crack pinning [196].

Figure 39. In plane rigidity of quasi isotropic CFRP laminate and Mode II interlaminar fracture toughness versus CNF interlayer thickness [197].

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Mode I and II interlaminar fracture toughness were investigated [197] for CF/epoxy laminates toughened by a carbon nanofibre/epoxy interlayer. Vapour grown carbon fibre (VGCF) and vapour grown carbon nanofibre (VGNF) were chosen as the reinforcement for the interlayer. To illustrate the effect of the interlayer on the fracture toughness of the laminates, several types of CF reinforced plastics/carbon nanofibre (CFRP/CNF) hybrid laminates were fabricated. The laminates were manufactured using unidirectional carbon/epoxy prepregs and a carbon nanofibre interlayer with varying thickness. The mode I interlaminar fracture toughness was evaluated by a standard double cantilever beam (DCB) test. The mode II interlaminar fracture toughness was evaluated by the end notched flexure (ENF) test. The DCB tests confirmed that the mode I interlaminar fracture toughness for hybrid laminates was about 50% greater than that of the plain CFRP laminates. Furthermore, the mode II fracture toughness tests confirmed that the interlaminar fracture toughness for hybrid laminates was 2–3 times greater than that of plain CFRP laminates. The recommended range of CNF interlayer thickness was between 100 and 150 μm (approximately 20 g/m2 carbon nanofibre area density) (Figure 39). The difference in the effect of the interlayer fracture properties under mode I and II fracture was discussed on the basis of SEM fractographic observations.

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Natural Composites The influence of the fibre-matrix adhesion on the fatigue and impact performance of jute fibre-reinforced polypropylene was investigated [198]. It was shown that a strong interface was connected with a higher dynamic modulus and stiffness degradation with increasing load cycles and maximum applied stress. The specific damping capacity was higher for composites with poorly bonded fibres. Furthermore, stronger fibre-matrix adhesion reduced the loss energy for non-penetration impact tests by approximately 30%. Tests performed at different temperatures, exhibited higher loss energies for cold and warm test conditions compared to room temperature tests. The post-impact dynamic modulus (after 5 impact events) was approximately 40% and 30% lower for composites with poor and good fibre-matrix adhesion, respectively.

Applications A new processing method was developed for spreading fibre tows to make thin-ply laminated composites. The proposed method employed a constant airflow through sagged fibre filaments to efficiently spread the thick tows without damaging any fibres. The method was robust and provided comparable outcome with other available thin-ply methods. The thin plies of thickness less than one-third of the conventional plies could easily be made with the tow-spreading technology. Experiments were performed [199] to evaluate the performance of tow-spread, thin-ply laminated composites. To study the thickness effect of the laminated composites, the test specimens were made with the same material and the same spread tows, but with dispersed and grouped laminations of the plies. Uniaxial tension static and fatigue loadings were applied on both unnotched and open-hole specimens. Impact and compressionafter-impact tests were also conducted. From stress–strain curves, acoustic emission counts,

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X-ray photos, c-scan images and observation of damage modes of failed specimens, it was observed that the thin-ply laminated composites can suppress the micro-cracking, delamination and splitting damage for static, fatigue and impact loadings without special resin and/or 3-D reinforcements. Therefore, the laminate design could be simplified by using higher strain allowables without a progressive failure analysis.

Composite Joints

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Lanciotti et al [200] described the results of a research activity, carried out at the Department of Aerospace Engineering at the University of Pisa in conjunction with AGUSTA, aimed at the characterisation of the fatigue behaviour of composite joints. The activity was performed as a support to the certification of the tail of the new helicopter EH 101. Two types of specimens were defined as representative of the structure; the specimens differed in the thickness of the laminates, namely 2 and 4 mm, and were representative of critical areas. The specimens were impact damaged with an energy level capable of inducing barely visible indentation, as required by the Certification Authority. Fatigue tests, under sinusoidal loading, R=O-2, were carried out in both dry and wet conditions; the stiffness of the specimens was monitored during the fatigue tests. The comparison between the results obtained showed the considerable effect of impact damage and of humidity absorption on the fatigue resistance of composite joints.

Figure 40. C-cured z-pinned joint between the skin and the stiffener [201]. Composite Laminates: Properties, Performance and Applications : Properties, Performance and Applications, Nova Science Publishers,

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Alkis Paipetis and Dionysios T.G. Katerelos

Although z-pins were effective in preventing delaminations in adhesively bonded and cocured joints, their applicability depends on the reliable assessment of the strength of a z-pincomposite assembly (Figure 40). In particular, high residual thermal stresses experimentally observed dictated the necessity for local stress analysis. Elevated temperature experienced by the joint during its lifetime may also affect its resistance in delamination. Byrd and Birman [201] illustrated an approach for determining local residual stresses and confirmed the possibility of delamination and cracking in the composite structure due to high postprocessing transverse stresses. The analysis of the effect of elevated temperature applied at one of the surfaces on the response of a z-pinned joint was conducted using the concept of a double cantilever beam with an “insulated” crack. In addition, it was shown that elevated temperature could actually enhance the integrity of the joint provided that the z-pin-composite interfacial strength increased.

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Wind Turbines The size of wind turbine rotors has increased in the past decade from 40 m to more than 120 m diameter. The resulting mass of about 18 tons per rotor blade causes high bending moments at the inner part of a blade due to the gravitational loads. More than 108 load cycles will take place in the expected lifetime of 20 years of a turbine. During this time the rotor blades are exposed to various hostile conditions such as extreme temperatures, humidity, rain, hail impact, snow, ice, solar radiation, lightning and salinity. In order to withstand these external conditions without diminishing the safety a sound knowledge of the fatigue behaviour of the material and structural properties is needed. To meet the upcoming requirements Kensche [202] highlighted some fatigue and lifetime aspects on wind turbine rotor blades made of composite materials. These included a historical part in connection with glider technology, the presentation of relevant S – N curves not only for the 0°-orientated fibres representing the spar cap but also for ±45°-lay-ups in shear web and shell, the influence of fibre content and architecture, of environmental effects, with a view to lifetime prediction of structural elements.

Fibre Metal Laminates Fibre metal laminates offer significant improvements over current available materials for aircraft structures. While weight reduction and improved damage tolerance characteristics were the prime drivers to develop this new family of materials, they also possess additional advantages, critical for today's designers, such as cost reduction and improved safety. Originally developed for their outstanding fatigue resistance, other advantages of fibre metal laminates include high specific static properties, ease of manufacture, excellent impact resistance, burn through capabilities rivalling titanium alloys, and good corrosion resistance. The combination of these properties in one material was an extraordinary achievement, rendering Glare [203] particularly inviting for aircraft applications. The very nature of Glare is its crack bridging mechanism, which provides superior damage tolerance properties. Depending on the property, Glare shows either monolithic metal or composite behaviour, which challenges the definition of strength justification and certification procedures. Through

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its unique combination of properties, Glare was chosen as the material for pressurized fuselage skin structures of the new aircraft generation, such as the A380. Glare exhibited excellent resistance to crack growth [203]. This was attributed to the combination of delamination and fibre bridging mechanisms. The fatigue insensitive fibres restrained the crack opening and transferred load over the crack in the metal layers. During the initiation phase fibre bridging did not occur and the behaviour was dominated by the metal properties. Mechanically fastened joints introduced additional effects such as secondary bending, load transfer and effects related to the fastener installation. The residual strength of Glare depended on the amount of broken fibres and the delamination size and was described with the R-curve approach. The impact resistance of Glare was related to the aluminium and glass/epoxy properties and was significantly higher than the impact resistance of monolithic aluminium [204]. The same was proven for fire resistance. Depending on the Glare grade and thickness, the outer aluminium layer melted away, whereas the other layers remained intact due to carbonisation of the glass/epoxy layers and delamination of the laminate. The air in the delaminations acted as insulation, keeping the temperatures at the non-exposed side relatively low. Some aspects of the detailed design of aircraft structures in Glare, as well as the design of splices and riveted joints were discussed [205]. In order to apply Glare in very large fuselage panels, a splice concept was developed, which allowed for a number of longitudinal splices to be cured in the same curing cycle as the basic material. Through the introduction of this splicing concept, the width of a panel was no longer limited to the maximum width of the aluminium sheet. Internal local reinforcements (doublers) may be integrated into the panel during lay-up.

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Summary An overview of the post-impact-fatigue behaviour of composites was performed. The response of impacted composite structural composites to service loading is critical as impact events may significantly degrade the material properties. This work initially focuses on the impact response of composite materials and their constituent phases as they are documented in the literature. The effect of polymer matrix materials, laminate geometry reinforcement geometry are reported, including sandwich structures, and a thorough account of the experimental, and semi-empirical approaches is provided for the quantification of damage. The effect of temperature and thermal stresses is also assessed. A thorough investigation of the effect of impact-fatigue on the composite properties is performed together with the experimental and modelling approaches that appear in the literature. Delamination growth under fatigue loading for impacted composite panels is also examined. Delamination growth is assessed in the literature both under mode I and II conditions both numerically and experimentally. Finally, the models for fatigue life prediction are presented, accounting for the presence of notches, matrix toughness, loading modes and stiffness reduction. The non destructive evaluation of initial damage and its propagation is of primary importance for composite structures. To this end, the up-to-date methods are presented. These methods are generally based on wave propagation, electric conductivity or via the

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incorporation of sensors in the parent material, such as Fibre Bragg gratings. Full field techniques, such as optical and lock-in thermography, speckle pattern interferometry and xradiography are also widely used in the literature. The post impact response both under static and cyclic loading is also examined. The extended literature on the subject includes among others the effect of notches, the effect of hydrothermal exposure, hail damage as well as the effect of different geometries and fabrication processes with a view service life prediction and/ or optimisation of the composite. The post-impact behaviour of composites is extensively modelled in the literature. The associated models include semi empirical analyses, numerical studies in conjunction with experimental findings and analytical solutions. Special attention was given to buckling and post-buckling analyses of plates in the presence of delamination. The existing models focus on the different fracture modes as well as the prediction of the critical strain energy release rate. A thorough investigation of the technologies that inhibit damage initiation and propagation is also presented. These technologies include z-pinning and stitching, interleaving, and matrix toughening. Special focus is given to novel technologies where the incorporation of phases at the nanoscale is employed to increase the toughness of the composite and subsequently its damage tolerance. Throughout this work, detailed reference to the impact and post-impact behaviour of natural fibre composite is made. Finally, an overview of the structural applications where increased damage tolerance is a prerequisite is presented with a special mention to fibre metal laminates.

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[81] Wang, D.; Wang, S.; Chung, D.D.L.; Chung, J.H. Jrnl Intel Mater Syst Struct. 2006, 17, 853-861. [82] Wang, D.; Chung, D.D.L. Smart Mater Struct. 2006, 15, 1332-1344. [83] Wang, S.; Chung, D.D.L. Carbon 2006, 44, 2739-2751. . [84] 84 Shen, L.; Li, J.; Liaw, B. M.; Delale, F.; Chung, J.H. Comp Sci Tech. 2007, 67, 2513-2520. [85] Todoroki, A.; Tanaka, M.; Shimamura, Y. Comp Sci Tech. 2005, 65, 37-46 [86] Matsuzaki, R.; Todoroki, A. Comp Sci Tech. 2006, 66, 407-416 [87] Ueda, M.; Todoroki, A. Eng Fract Mech. 2008, 75, 2737-2750 [88] Angelidis, N.; Khemiri, N.; Irving, P.E. Smart Mater Struct. 2005, 14, 147-154 [89] Angelidis, N.; Irving, P.E. Comp Sci Tech. 2007, 67, 594-604 [90] Kuang, K.S.C.; Cantwell, W.J. Jrnl Mater Sci Lett. 2002, 21, 1351-1354 [91] Kuang, K.S.C.; Cantwell, W.J. Jrnl Thermopl Comp Mater. 2003, 16, 213-229 [92] Kister, G.; Ralph, B.; Fernando, G.F. Smart Mater Struct. 2004, 13, 1166-1175 [93] Takeda, S.; Minakuchi, S.; Okabe, W.; Takeda, N. Comp A 2005, 36, 903-908 [94] Silva, J.M.A.; Devezas, T.C.; Silva, A.P.; Ferreira, J.A.M. Jrnl Comp Mater. 2005, 39, 1261-1281 [95] Chambers, A.R.; Mowlem, M.C.; Dokos, L. Comp Sci Tech. 2007, 67, 1235-1242 [96] Takeda, S.; Aoki, Y.; Ishikawa, T.; Takeda, N.; Kikukawa, H. Comp Struct. 2007, 79, 133-139. [97] Pearson, J.D.; Zikry, M.A.; Prabhugoud, M.; Peters, K. Jrnl Comp Mater. 2007, 41, 2759-2783. [98] Sohn, M.S.; Hu, X.Z.; Kim, J.K.; Walker, L. Comp B 2000, 31, 681-691. [99] Guillaumat, L.; Batsale, J.C.; Mourand, D. Comp A 2004, 35, 939-944. [100] Mian, A.; Newaz, G.; Han, X.; Mahmood, T.; Saha, C. Comp Sci Tech. 2004, 64, 11151122. [101] Mian, A.; Han, X.; Islam, S.; Newaz, G. Comp Sci Tech. 2004, 64, 657-666. [102] Meola, C.; Carlomagno, G.M.; Sqillace, A.; Vitiello, A. Eng Fail Anal. 2006, 13, 380388. [103] Wosu, S.N.; Hoy, D. Jrnl Comp Mater. 2006, 40, 1577-1602. [104] Carpino, G.; Lopresto, V.; Santoro, D. Comp Sci Tech. 2007, 67, 325-335. [105] Nayeb-Hashemi, H.; Cohen, M.D.; Zotos, J.; Poormand, R. Jrnl Non-Destr Eval. 1986, 5, 119-131. [106] Adden, S.; Pfleiderer, K.; Solodov, I.; Horst, P.; Busse, G. Comp Sci Tech. 2008, 68, 1616-1623. [107] Dunkers, J.P.; Sanders, D.P.; Hunston, D.L.; Everett, M.J.; Green, W.H. The Jrnl Adh. 2002, 78, 129-154. [108] Minnaar, K.; Zhou, M. Jrnl Mech Phys Sol. 2004, 52, 2771-2799. [109] Růžek, R.; Lohonka, R.; Jironč, J. NDT&E Inter. 2006, 39, 132-142. [110] Amaro, A.M.; de Moura, M.F.S.F.; Reis, P.N.B. (2006). Detection of Load Velocity Impact Damage in Carbon-Epoxy Plates using NDT. Fracture of Nano and Engineering Materials and Structures. Proceedings of the 16th European Conference of Fracture, Alexandroupolis, Greece, July 3-7, 2006. Proceedings in CD-ROM. [111] Corigliano, A.; Ghisi, A.; Mariani, S. (2006). Detection of Load Velocity Impact Damage in Carbon-Epoxy Plates using NDT. Fracture of Nano and Engineering

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In: Composite Laminates Editors: A. Doughett and P. Asnarez, pp. 83-119

ISBN: 978-1-60741-620-3 © 2010 Nova Science Publishers, Inc.

Chapter 2

COMPOSITE MULTILAYER COATINGS FOR IMPROVED BARRIER PROPERTIES OF PACKAGING BOARD Caisa Andersson Karlstad University, Faculty of Technology and Science, Department of Chemical Engineering, SE-651 88 Karlstad, SWEDEN

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Abstract Replacement of petroleum-based plastic films with more environmentally friendly, waterbased coatings that provide sufficient barrier protection to paper or paperboard is a major future challenge for the packaging industry. Natural polymers, such as starch or cellulose polysaccharides, exert many interesting properties for packaging applications besides originating from renewable sources. They are abundant in nature and can be extracted and processed at a reasonable cost. They also provide good film forming barrier properties for oxygen and grease. However, the moisture sensitivity of biobased polymers makes them inappropriate as protective films for use in food packaging. This chapter aims to present new research in the field of composite barrier coatings by incorporation of reinforcing fillers to enhance the moisture barrier properties. Synthetic as well as biobased polymer coatings have been investigated. Composite polymer-filler formulations were prepared by blending nanosized clay in polymer dispersions. A composite material was also prepared by blending synthetic polyester into a biobased polymer dispersion. The composite formulations were applied on top of a three-ply packaging board to form various single- and multilayer laminates. Barrier properties with respect to water vapour permeability, oxygen permeability and water absorption are qualitatively discussed. Both commercial talc-filled barrier dispersion coatings and plastic packaging films were used as references. The properties of single layer and combinations of layers in terms of adhesive properties and synergistic effects were studied by investigation of the surface properties. This chapter shows that a biopolymer based coating can be reinforced with nanosized clay to give barrier properties competitive to commercial, synthetic coating formulations. A dramatic improvement in barrier properties upon application of a thin top nanocomposite coating on various pre-coated structures could also be observed.

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Introduction Cellulose fiber based materials are widely used for packaging. Paper or paperboard packages are used for dry food such as sugar, flour and biscuits as well as for moisturecontaining or wet foodstuff like cheese, butter, frozen food, milk and juice. Other applications are electronics, consumer goods, luxury products, detergents and a wide variety of industrial products. Packages are used to preserve the integrity of the packaged product but also for sales and marketing purposes. All of these applications have their own requirements on the packaging material regarding mechanical, forming, printing, sealing, adhesion and barrier properties, to mention a few, but of utmost importance especially for food and medical packages is the protection against chemical contamination and/or physical damage of the packaged product caused by interaction with the environment or even counterfeit. Uptake or loss of moisture, chemicals and gases, i.e. transportation of these molecules in or out of the package, may be deleterious for both the goods and the environment. Paperboard is usually a laminate made up of a series of cellulose fiber plies all of which exert their own properties and contribute to the overall mechanical and physical properties of the final structure, depending on the fiber composition, pulping and papermaking processes. Paperboard further constitutes a supporting basis for the coating or lamination of protective layers, which can be applied in one or more layers, on one or both sides of the board. The protective layer can be a pure polymer as well as a composite polymer-filler coating. Each coating layer provides some specific property for protection, and multilayer coating is often necessary to achieve adequate properties of the laminate to fulfill all requirements set by the end use of the product. Figure 1 shows a schematic sketch of a paperboard laminate. The purpose of the front side coating might be enhancement of printability, i.e. on the surface that will be the outside of the packaging. The backside of the board, i.e. the inside of the packaging, may have a pigment coating layer (denoted pre-coating) that is applied with the aim to reduce surface porosity to facilitate application of the protective coatings (denoted middle- and top coating layers) which may provide grease, oxygen and moisture barrier properties. Often it is necessary to coat both the front side and the backside with barrier layers.

Inside

Top coating Middle coating Pre-coating Three-ply paperboard

Front side coating Outside

Figure 1. Schematic sketch of the laminate structure of coated paperboard. Composite Laminates: Properties, Performance and Applications : Properties, Performance and Applications, Nova Science Publishers,

Composite Multilayer Coatings for Improved Barrier Properties of Packaging Board 85

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Paperboard Laminates The paperboard multi-ply structure typically includes mechanical or chemical pulp and contains recycled fibers in one or more of the plies. Furthermore, each single ply can be made of bleached or unbleached pulp and the board can be un-sized, medium- or hard sized. The most usual structure is outer smooth, strong plies and inner bulky layers to achieve maximum bending stiffness. The outer plies often contain a mix of short and long fiber kraft pulps with or without fillers while the center plies usually contain a mix of chemical pulp, high yield pulp or recycled fibers together with broke. Paperboard is commonly defined as having a basis weight of 225 g/m2 or higher, while products with a basis weight below 225 g/m2 is denoted paper, even though no exact distinction can be made since an overlapping region exists [1]. Paper substrates can be surface-treated to improve barrier properties for further utilization as inner pouches in direct contact with the packaged goods (primary packaging), or as wrapping paper, sacks or bags. Paperboard is widely used for production of packages and can be used in the form of folding cartons for physical protection of products (secondary packaging). However, a great deal of the paperboard produced is used for primary packaging, e.g. for milk, juice, frozen fish and vegetables, pizza, pet food and candies. Carton board can be divided into different categories based on the raw materials and enduse purpose. Solid bleached sulfate (SBS) is made up of 1–3 plies of hardwood or softwood bleached pulp [2]. Typical applications are packaging of cosmetics, frozen food, cheese, tea, coffee and chocolate [3, 4]. Folding boxboard (FBB) generally contains a middle ply including mechanical pulp laminated between two outer plies of bleached chemical pulp [2, 4]. The middle ply or plies are often made up of ground wood pulp (GW), thermo-mechanical pulp (TMP) or chemical-thermo mechanical pulp (CTMP) mixed with chemical pulp for increased strength and high yield pulp to give highest possible bulk. FBB is used for a wide range of food packaging, pharmaceutical products etc. White lined chipboard (WLC) usually contains 3 to 4 plies with recycled fibres in the middle plies. Typical basis weight is 200–450 g/m2 [2]. Coated WLC is used for packaging of non-food products such as detergents, pet food, toys or tools but have also found use for cigarettes and pharmaceutical products. Liquid packaging board (LPB) and carrier boards are typically made of unbleached kraft and high yield pulp in the middle plies, unbleached kraft pulp in the bottom ply and bleached kraft pulp in the top ply [4]. Liquid packages for milk and juice, drinking cups, plates and fast-food cartons are typical end-use purposes. Only virgin fibres are used because of the high requirements on cleanliness and purity. LPB is usually laminated with polyethylene plastic film (LDPE) on both sides for packaging of dairy products. An aluminum foil can be laminated to the sandwiched structure for decorative properties or when high barrier against flavors, oxygen or UV light is required as in the case of fruit juices which are sensitive to Cvitamin loss through photo oxidation [4-6]. Carton board is in general pigment coated in single- double- or triple layers on the outer (front) side for appearance and print quality purposes [5], often with a total coat weight of 20–25 g/m2 [3]. The backside is sometimes single-coated to improve quality or water-treated for curl control purposes. Typical coat weight is then 8–10 g/m2 [3]. Coating or lamination of the inner (back) side is carried out to meet the demands for different barrier properties against liquid water, moisture, oxygen and grease [7]. Figure 1 shows a typical barrier-coated carton board structure.

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Containerboard consists of liner (115-440 g/m2) and a corrugating medium called fluting of 80–200 g/m2 [1]. The liner and the fluting are glued together to form several multilayer structures [8]. Corrugated board is characterized by high compressive strength at relatively low weight [5]. Liner can be divided into kraftliner made of virgin, unbleached chemical pulp and testliner made with recycled fibres. Both grades typically consist of two plies [3]. Liner is often surface treated by sizing to improve the water resistance [2, 9]. The fluting is a singleply product, which can be made of semichemical pulp or from recycled fibres. Use of recycled fibres usually require surface sizing to reach sufficient strength of the product [2]. Containerboard is the largest paper and paperboard sector worldwide in terms of tonnage [4]. Boxes and trays of containerboard are used as secondary packaging or for packaging of multiple packs for storage and distribution.

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Barrier Coating of Paperboard Sizing is carried out to improve the internal and surface strength of paperboard [9-11]. Paperboard for packaging of liquids, frozen or chilled food is hard-sized [4]. Common chemicals for surface sizing are chemically modified starches, carboxymethyl cellulose, poly(vinyl alcohol), polyurethanes and fluorochemical emulsion sizing agents [5]. All these sizes provide oil and grease repellent properties to various extents, besides enhancing the water resistance. It has been demonstrated that the penetration of sizing components into the substrate can be adjusted by control of the colloidal stability in starch-hydrophobic sizing agent formulations [12], thereby providing wide possibilities to reach the desired surface functionality of paper or paperboard. A package should provide protection of the content as well as the environment. It should prevent mechanical damage and spoilage of the goods packed. For food applications, this includes both hygienic and health aspects covering preservation of nutritional value and avoidance of toxicological substances coming in contact with the food. The functions of a package also includes communication of important information about content and quality such as freshness and expiry date, proper design for marketing and reduction of the risk of tampering or adulteration, reduction of waste by prolongation of shelf life and convenience throughout the packaging chain [13, 14]. A functional package should also provide low production cost and reduce the usage of energy by eliminating the need of refrigeration or freezing of the product [14]. Paperboard offers both mechanical strength and flexibility for production of packages but lack barrier properties and thus need to be surface-treated for improvement of functionality and protection. The packaged goods (food) need protection against physical, biochemical and microbiological deterioration as well as protection against organoleptic changes, including any changes in flavour, odour, texture, colour or taste of the food by interaction with the surrounding environment. Time, temperature, moisture, light, gases and pressure are factors that indirectly affect the shelf-life and quality of foods [13, 15], while direct spoilage of food can be caused by mechanical damage during transport or by attack of microorganisms [16]. Permeation of small molecules through a polymeric coating will mainly take place in amorphous regions since the high molecular packing in crystalline structures effectively hinders the movement of permeating species [13, 17, 18]. However, of higher relevance is the permeation through defects like cracks, voids and pinholes in the polymer coating.

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Composite Multilayer Coatings for Improved Barrier Properties of Packaging Board 87 Permeation of a small molecule through a polymer has since long time ago been described as occurring in a stepwise process [13]: 1) 2) 3) 4)

absorption of the permeating species onto the surface of the polymer solution of the gas or vapour into the polymer matrix diffusion through the wall along a concentration gradient desorption from the other surface

Water Vapour Barrier The water vapour transmission rate, WVTR, is defined as the amount of water vapour transmitted through a unit area in a unit time under specified conditions of temperature and humidity. WVTR is typically expressed in grams per square meter and day (g/m2·d). The WVTR decreases exponentially with increased homogeneous coating layer or film thickness [19]. When the thickness and difference in vapour pressure across the testing material are taken into account, the property water vapour permeability, WVP, can be calculated. The unit for WVP is typically g·m/m2·Pa·s.

Oxygen Barrier

Low

OTR, cm3/m2·d

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The oxygen transmission rate, OTR, is defined as the quantity of oxygen gas passing through a unit area per unit time under specified conditions of temperature, humidity and pressure (ASTM D3985-05). Oxygen is strongly and irreversibly absorbed by polymers naturally present in different foodstuffs [13], thereby causing permanent changes in the food quality. Exposure to oxygen can cause oxidation followed by rancidity of fats or enzymatic browning of fruits and vegetables [15]. Intermediate

High

750

10

0

1

50 WVTR, g/m2·d

Figure 2. Low, intermediate and high transmission rates of permeable materials with respect to OTR and WVTR, at 23°C and 50% RH [20]. Composite Laminates: Properties, Performance and Applications : Properties, Performance and Applications, Nova Science Publishers,

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OTR is commonly given in cm3/m2·d. The oxygen permeability, OP, can be calculated for a homogeneous material as the ratio of the OTR value to the difference between the partial pressure of O2 on the two sides of the film times the thickness of the film (ASTM D 398505). The unit for OP is typically cm3·µm/(m2·d·kPa). The pressure difference is usually 1 atm, i.e. 101.3 kPa. Figure 2 displays the range of fields generally considered as low, intermediate and high transmission rates of permeable materials with respect to OTR and WVTR. Measurements of OTR are commonly carried out at 23°C and 50% relative humidity, RH, at atmospheric pressure, but testing at other relative humidities are reported in the literature as well. For hygroscopic materials the OTR will increase with increasing RH due to absorption of water and subsequent swelling of the polymer, resulting in a more open structure [21]. The oxygen permeability of hydrophilic biobased polymers increase exponentially with increased RH [22].

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Water Absorption and Surface Hydrophobicity Water absorption of paperboard surfaces is generally recorded as the Cobb value. The Cobb value is defined as the amount of water in g/m2 that is absorbed over the test area under a water pressure of 10 mm during a specified time, usually 60 seconds (ISO 535). Packaging materials are commonly sized to reach Cobb60 values of about 20 g/m2. Hard-sized substrates are defined as having a Cobb60 < 10 g/m2 which requires a combination of internal and surface sizing with hydrophobic sizing agents such as paraffin wax emulsion, polyurethanes and styrene-based copolymers [23]. The spreading and absorption of liquid water at short time intervals can be recorded by measurement of the contact angle in the air-water-paper interface by placing a water drop on the paper surface. The contact angle is defined as the angle formed between the plane of the solid surface and the tangent of the fluid surface, where they met at the three-phase line [24]. The contact angle is a measure of the wettability of the paper and is strongly related to the surface hydrophobicity. To produce a food package fulfilling all requirements on grease, moisture and gas barrier while providing sufficient mechanical integrity, it is often necessary to create a multilayer coating or laminate structure in which each layer contributes with specific protection to the overall properties. Coated paperboard may be considered as a series of membranes [5]. In a very simplified model, the total permeability of the coated paperboard is a function of thickness and permeability for the paperboard and coating layers respectively [13]. In reality, even though the porous paper or paperboard material usually does not express any significant contribution to the gas barrier, the combination of multiple plies and multiple coating layers may lead to combined effects.

Converting of Coated Paperboard To form a package, the coated paperboard must be closed by either heat sealing or gluing. Thermoplastic polymer coatings can be melted and joined by application of heat and pressure. LDPE films are readily heat sealed whereas moderate to poor heat sealing can be found for several other plastic films [17, 25]. Heat sealing with no need for additional glue is thus another argument for use of dispersion coatings or thermoplastic, bio-based polymers as

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Composite Multilayer Coatings for Improved Barrier Properties of Packaging Board 89 alternatives for plastic laminates based on mineral oils. Acrylic coatings are easily heat sealed over a wide range of temperatures [17]. Factors that may have an unfavorable effect on heat sealing of dispersion coatings are presence of fillers, additives or waxes [19]. Enrichment of surfactant by migration to the surfaces to be sealed was shown to result in weaker seals [26]. Packages for dry products like flours, cereals and detergents are often closed directly on the filling line by straight side seam gluing [7]. Final closing of the package can be done by flap tucking of the top while the bottom is either flap tucked or glued (Figure 3). Other types of packages involve gluing of the paperboard at multiple points to form a tray with a lid. Water-based dispersions including acrylates or acetates are often used as glues. Efficient gluing then requires a well defined porosity of the paperboard surface for rapid penetration of water into the substrate, which promotes good film formation and adhesion between the glued parts [2]. Dispersion coatings are generally compatible in gluing applications with starch or vinyl acetate based glues [19]. To solve problems with wetting of water-based glues a hot melt adhesive is sometimes required for gluing of board surfaces coated with plastic films or dispersions [7, 27]. Packaging of cocoa powder, cereals or biscuits is often done by use of an inner pouch of flexible paper or plastic as a primary package. The inner pouch may provide a denser packaging for fine-grain powders than does the porous paperboard. Migration of chemical substances can sometimes occur when the packaged product, e.g. chocolate, comes in direct contact with the cartonboard. Thus, to prevent undesired change in taste, the chocolate must be protected by a primary packaging. The flexible material is usually formed into a tube, which is then glued to the carton. The ends of the tube can either be heat sealed or glued. Gluing can also be necessary to fasten a transparent window to a carton filled with e.g. pasta or candies. Some conventional board packaging designs are shown in Figure 3. Sealing can be done through a foldover seal where two identical surfaces are sealed together or through lap seal where the inner side is sealed against the outer side [17]. The latter case requires that both sides of the coated paperboard are heat sealable or gluable.

Folding carton

Side seam glued folding carton

Gable-top liquid packaging

Foldover seal

Lap seal

Figure 3. Packaging designs.

Gluing with water-based glues may have deteriorating effects on the barrier properties of water-sensitive bio-based coatings since interaction with water will cause the polymer to swell [21].

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Barrier Materials for Paperboard Some characteristic properties of selected barrier materials used in the research that is presented in this chapter are presented below. For a deeper description on the advantages and disadvantages of each material, see [28].

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Synthetic Materials The most widely used barrier materials for paper or paperboard packaging until today have been plastic films such as high density (HDPE) or low density (LDPE) polyethylene; polypropylene (PP); polyethylene terephthalate (PET), or ethylene vinyl alcohol (EVOH) [5, 25, 29]. These plastic films differ in barrier and mechanical properties and are hence often used in combination to offer the appropriate protection for different types of packaging. Polymer dispersions or latexes have gained attraction during recent years as barrier materials for paper and paperboard products [30-33]. These water-based dispersions consist of polymers or co-polymers of styrene, butadiene, acrylates, vinyl acetate and polyolefins and can be applied by conventional on-line or off-line coating techniques [27]. Typical packaging applications are corrugated board, disposables and cartons for frozen or chilled food or ready meals [19]. Additives are used to set the desired level of consistency, durability and runnability, e.g. colloidal stabilizers, thickeners, waxes, anti-foaming agents, biocides and pesticides. Fillers can be added to improve the barrier or optical properties, runnability and cost competitiveness or to reduce the blocking tendency [27]. To achieve an appropriate level of water vapour barrier and water resistance, a coat weight of more than 10 g/m2 is required [19]. Dispersion coatings also provide grease and oil resistance and are hence suitable also for production of packages for confectionery or bakery products. Acrylates provide hydrophobicity and good grease resistance, while styrenebutadiene copolymers show moderate barrier properties [34]. Poly(ethylene terephthalate), PET, plastic films offer good resistance to oils and different solvents [29]. Waste products of PET like bottles and sheet material can be recycled via consumer collecting systems and used for production of new barrier material. Water-soluble hydrophobic polyester resins formed from such waste materials that can be applied on paper or paperboard substrates by conventional coating techniques have recently been developed. They are interesting for enhancement of packaging barriers not the least because their origin is recycled, inexpensive and non-toxic materials. The polyester backbone polymerized from recycled PET plastics is further modified to give the product water and oil repellent properties by reaction with a hydroxyl functional compound. Water dispersibility is achieved by incorporation of ionic groups. The hydrophobicity of the resin is enhanced by introduction of aliphatic groups consisting of straight chain or branched fatty acids or triglycerides. When films of these types of polymers are dried, the hydrophobic parts orient themselves so that the surface becomes water-repellent at much lower material requirements than with traditional resins [35].

Biobased Materials Several studies have been undertaken in recent years to investigate the potential of biobased materials for packaging applications, e.g. based on starch [21, 36, 37]. Polysaccharides

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Composite Multilayer Coatings for Improved Barrier Properties of Packaging Board 91 have the ability of forming crystalline films with low permeability to oxygen, thus making them interesting to use in packaging applications. Starch has found wide use in paper, paperboard and packaging manufacture. Different types of modified starches are used in the sizing and coating of paper and paperboard, and can act either as a thickener or an adhesive. The considerations regarding use of biobased polymers with respect to food packaging requirements is discussed by Krochta and de Mulder-Johnston [22] and by Petersen et al. [6]. Starch-based products that can replace conventional plastics are today commercially available on the market. Bio-based film-forming materials have the potential for use in surface treatment of cellulose-based substrates, either by coating or by extrusion/lamination. Biobased materials have also been used in composite formulations, e.g. starch-LDPE blends [38, 39]. Starch-PVOH blends have shown competitive properties comparable to e.g. conventional LDPE and PS films [22]. An overview of different biobased polymers for enhancement of paperboard functionality is given in [28]. Starch is a naturally abundant plant polymer whose function is to serve the plant with carbohydrates. Starches used for coating and surface sizing purposes mainly come from potato, maize, wheat, or corn. In European paper production, starch originating from potato is the most widely used [10]. Starch production starts with normal cultivation and harvesting of potatoes. The potatoes are then washed and grated into smaller pieces at the starch production unit. The grated material is sieved in centri-sieves and is then passed through a series of hydrocyclones to separate water from the solid material. The powder is then filtered by means of belt- and vacuum filters and is finally dried [40]. The final starch powder shipped from the production mill will have an equilibrium moisture content of ca ~20% by weight, thus mirroring the highly hygroscopic character of the material. The production of potato starch is a closed eco-cycle. Potatoes normally contain about 76% water and the starch fraction makes up about 19% of the total weight. Besides these materials, amino acids, various sugars, salts, proteins and fibers are also present. After separation of the starch fraction in the production line, the portion of pulp can be isolated for further use as fibers in food production or for pet feed. The remaining fruit juice is brought back to the field and is used for fertilization of a new cultivation cycle [40]. The hydroxyl groups on starch glucose units are possible sites for substitution reactions to create different types of starch ethers or esters. The hydroxypropylation substitution involves reaction with starch and propylene oxide under alkaline conditions. Incorporation of hydrophilic hydroxypropyl groups prevents association of the amylose chains after cooking (retrogradation). Hydroxypropylation improves the low-temperature stability, the viscosity stability of the dispersion and the flexibility of films. Hydroxypropylated starch has found use both in food and non-food applications and shows excellent film forming properties along with high binding power and gluing potential, high adhesive strength, and beneficial rheological properties. Non-ionic starch ethers are furthermore not sensitive to changes in electrolyte concentration and pH in the production process.

Reinforcement of Polymer Coatings Nanosized materials have gained interest in recent years to enhance the functionality of paper and paperboard by improvement barrier and mechanical properties [41]. Much of the characteristic properties of nanosized materials are controlled by their small size, which give

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a very large total surface area. Nanoparticles dispersed in a polymer matrix give a composite material. Nanoparticles are generally defined as having a size below 100 nm and can be either inorganic (silicates, metal oxides) or organic (polymers, dyes). Inorganic silicates, e.g. montmorillonite, are naturally abundant clay minerals that are often used in nanocomposite technology due to their high surface area and large aspect ratio, which makes them effective for barrier improvement already at very low (≤ 5 weight-%) concentrations [42-44]. Improved barrier efficiency leads to less material requirements, which in turn leads to reduced costs and reduced amounts of waste [45]. The mechanism of barrier improvement by incorporation of plate-like nanoparticles is by increasing the path length of the molecules diffusing through the film, i.e. forcing them to take a tortuous path, which leads to significantly prolonged transmission rates [46, 47], Figure 4. This prolonged pathway can be described by a tortuosity factor, τ, which is the ratio between the tortuous path length d´ and the permeation length of an unhindered molecule, d, which simply represents the thickness of the coating layer [48]. Tortuous pathways for permeating molecules Filler particles

Polymer matrix

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Barrier coating

Cellose fiber substrate

Figure 4. Extended transport ways for permeation of molecules through a filled polymer matrix.

The barrier properties of a nanocomposite material depend on the relative orientation of the silicate layers and the state of aggregation and dispersion. Dispersion of layered silicates in a polymer matrix can result in a number of different states, leading to microcomposites, intercalated- or exfoliated nanocomposites. It is believed that complete and homogeneous dispersion with the clay platelets arranged in a non-parallel manner (exfoliation) will give the highest performance improvements in coatings [44]. Improvement of barrier properties up to four orders of magnitude has been reported by incorporation of nanoclays in a polymer coating.

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Composite Multilayer Coatings for Improved Barrier Properties of Packaging Board 93

Experimental Substrate A three-ply commercial packaging board coated with a double clay coating on the front side (Performa Natura, Stora Enso Imatra Mill, Finland) was used as a substrate to study the effect of multilayer coatings on the barrier properties of the paperboard. The board is built up of a middle ply mixed of bleached sulphate pulp and bleached CTMP sandwiched between the top and back plies consisting of bleached sulphate pulp. The pigment coated side constitutes a basis for further printing of packages for e.g. detergent and household cleaners, chocolate and confectionery and various food packages. The backside is normally polymer coated or laminated with plastic film to meet the barrier requirements for the target product. The basis weight of the substrate is 255 g/m2 with an average thickness of 350 µm.

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Coating Materials The biopolymer used was a commercial oxidized, hydroxypropylated potato starch supplied by Lyckeby Industrial AB, Kristianstad, Sweden. This starch ether was selected due to its excellent film-forming properties [49]. The granular starch was dissolved in hot water at 95°C under vigorous stirring for 30 minutes. The resulting starch solution had a concentration of 20% by weight, corresponding to the maximum concentration for complete granule swelling of this starch grade. A synthetic anionic polyester resin, EvCote® PWRH-100, supplied by EvCo Research LLC, Atlanta, GA, USA was also used. The polyester resin is proven to form flexible, amorphous films with excellent grease resistance and high water repellency. The resin is FDA approved and might thus be used for food packaging. Furthermore, the coatings are claimed to be heat sealable and coated products are both repulpable and recyclable. Its composition is 40 weight-% recycled PET while the rest of the polyester is made up of saturated behenic acid (a natural constituent in vegetable and animal fats) and soybean oil. Maximum concentration for water dispersion is ~35% by weight. The polyester was supplied as ground powder and was dispersed in a hot water/NH3 solution and held at 90°C for 5 minutes while keeping the pH between 6.8 and 7.2. Target pH in the dispersions should be in the range 7.5-7.9 at room temperature for maximum stability. The average particle size determined by dynamic light scattering is 60 nm. The Tg was determined to 35°C by differential scanning calorimetry (DSC 2920 CE, TA Instruments, New Castle, DE, USA). As reference materials for barrier properties a medium-carboxylated styrene-acrylate (SA) latex (Dow xz 95085.01, Dow Europe GmbH) with a Tg of 22°C and a commercially ready-to-use barrier dispersion (Rebarco RB 736, Ciba Specialty Chemicals, Basel, Switzerland) were used. The latter is a styrene-butadiene (SB) latex comprising talc particles as filler materials. Commercial packaging plastic films were also used for reference purpose: a standard low density polyethylene (LDPE) film (defa-Folien/Heikoflex, Lohmar, Germany; thickness 30 µm) and a universal polyethylene terephthalate (PET) film (Hostaphan®, Mitsubishi Polyester film, Wiesbaden, Germany; thickness 12 µm). The composite coating formulations were a mixture of water dispersions of starch or polyester and a nanosized clay (Southern Clay Products Inc., Gonzales, TX, USA). Three

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different classes of composites were formed: starch-nanoclay; starch-polyester resin; polyester resin-nanoclay (Table 1). The relative proportions between continuous and dispersed phase was adjusted to reach an appropriate viscosity for application of the coatings on the paperboard substrate. The final solids content in all composite formulations was kept in the range 17-24 weight-%. Table 1. Overview of investigated composite formulations Composite Formulation Starch-nanoclay Starch-polyester Polyester-nanoclay

Continuous Phase Starch Starch Polyester resin

Dispersed Phase Nanoclay Polyester resin Nanoclay

Characterization of Composite Formulations Viscosity The viscosity of the polymer matrices and the composite formulations were measured by a controlled shear stress rheometer (Physica MCR 300, Physica Messtechink GmbH, Ostfildern, Germany) with shear rates from 1 to 4000 s-1 in concentric cylinder geometry (CC 17).

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Charge Density The charge density of the outgoing materials as well as the composite formulations was determined by polyelectrolyte titration using a Particle Charge Detector (Mütek PCD 03, Herrsching, Germany). Measurements were carried out at room temperature and at the pH values prevailing in the formulations after dilution with deionized water to a concentration of 0.25% by weight. All dispersions were found to be anionic, therefore titration was carried out using a cationic polyelectrolyte titrant, poly(diallyl dimethyl ammonium chloride), polyDADMAC, until the isoelectric point was reached (0 mV). Anionic sodium polyethylene sulphonate, 0.001 eq/l, was used for calibration of the instrument. The specific charge quantity in µeq/g was calculated as the titrant consumption in liters times the concentration of titrant (0.001 eq/l) divided by the active substance in grams. The amount of active substance was set to the total amount of solid material in each sample. Since the colloidal stability of the water-dispersed polyester resin is strongly dependent on pH, the zeta potential in the pH-range from 3 to 9 was measured using a Zetasizer 2000 (Malvern Instruments Ltd., Malvern, UK) calibrated with a latex standard sample (Zeta Potential Transfer Standard) with surface charge 50±5 mV. Five runs were automatically performed on each sample. The electrolyte concentration was set to 1 mM with NaCl and the resin was diluted 1:500, which gave output concentration signals in a range similar to the standard sample (300-500 kCts/s). The potential was measured at pH 3.0, 5.0 and 7.0 after adjustment with 0.01 M HCl and at 8.0 and 9.0 after adjustment with 0.01 M NaOH.

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Composite Multilayer Coatings for Improved Barrier Properties of Packaging Board 95

Surface Tension of Coating Formulations The surface tension of the coating formulations (at concentrations used during coating application) was measured by the Wilhelmy plate technique (Thermo Cahn Radian Model 322, Thermo Haake, Karlsruhe, Germany) using a standard glass plate as the solid probe. The glass plate was immersed in the liquids down to a depth of 4 mm and the surface tension was calculated from the receding step values after constant weight reading. The effect of small incremental additions of a non-ionic surfactant (Surfynol 402, Air Products, Utrecht, The Netherlands) on the surface tension of a 1% starch solution (by weight) was also evaluated.

Laboratory Coating

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A laboratory rod coater (K Control Coater, RK Print Coat Instruments Ltd.) was used to apply the composite formulations in single- and multiple layers using various primary coatings to form a series of different laminate structures. A wire-wound rod (no. 5) designed to give a wet film deposit of 50 µm was used. All coated substrates were dried at 105°C for 90 seconds. The average coat weights were determined by weighing the coated samples of known areas and subtracting the determined basis weight of the uncoated substrate. The coating structural thickness was measured by STFI Thickness Tester M201 (TJT-Teknik AB, Järfälla, Sweden). Five replicates with a measurement length of 45 mm were analyzed. The thickness of the primary and secondary coating layers were determined by subtracting the average thickness of the uncoated substrate and the primary coated substrate respectively, from the measured total sample thickness. The coated paperboard laminate structure is shown in Figure 5. Secondary coating Primary coating Top ply Center ply Bottom ply Pigment coating Figure 5. Coating laminate structures.

The first primary coating to be investigated was a conventional pigment coating with the primary intention to give a better print quality by providing a whiter and more closed paper structure with lower porosity. The aim was to achieve a more surface-located secondary barrier layer. In this case, the secondary layer was applied on the double clay coated front side of the commercial substrate (denoted pigment coating in Figure 5).

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Both the unfilled starch and the polyester matrices were applied as primary coatings, the former also acting as a base for further application of a secondary protective layer. SA-latex and SB-latex-talc were used as primary coatings to provide water barrier properties of laminates having moisture sensitive starch-based secondary layers. The starch-based composites were also applied in double layers, i.e. both in primary and secondary layers. Table 2 summarizes the combinations of primary coatings (vertical columns) and secondary coatings (horizontal lines). Table 2. Overview of investigated primary and secondary coatings (marked with +). A minus sign (-) means that the combination was not tested and nw means non-wetting

Secondary Layer None Starch Polyester Starchnanoclay Starchpolyester Polyesternanoclay

Pigment Coating + + nw

Starch + + nw

Primary Layer SA- SB-Latex- StarchStarch- PolyesterPolyester Latex Talc Nanoclay Polyester Nanoclay + + + + + + + + -

+

+

-

+

+

+

-

-

+

-

-

-

-

-

+

-

nw

+

-

-

-

-

-

-

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Water Vapour Transmission Rate The water vapour transmission rate, WVTR, was measured by the gravimetric cup method (ASTM E96) at 23°C and 50% RH using silica gel as desiccant and the coated side exposed to the humid air.

Oxygen Transmission Rate The oxygen barrier properties of coated substrates was measured by a Mocon® OxTran® oxygen transmission rate tester, Model 2/21 (Mocon Inc., Minneapolis, MA, USA) at 23°C and 50% RH (ASTM D3985-05) under atmospheric pressure. The coated substrates were enveloped in an adhesive aluminum foil mask leaving an exposed surface of 5 cm2. The oxygen concentration in the test gas was 21%. The reported OTR results were compensated to 100% O2.

Interaction with Liquid Water The absorption of water of coated substrates was measured according to ISO 535 at 23°C and 50% RH. The absorption time was 60 s (Cobb60) and the substrate area exposed to the test liquid was 25 cm2.

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Composite Multilayer Coatings for Improved Barrier Properties of Packaging Board 97 The static contact angles for a drop of de-ionized water applied on the surface of coated substrates were measured using a FTA 200 Dynamic Contact Angle Analyzer (First Ten Angstrom, Portsmouth, NH, USA) under the conditions of 23°C and 50% RH. The drop size was 10 µl. The drop spreading over the surface was recorded by means of a video camera. Contact angles were calculated by the software and the average of at least three measurements was calculated. Angles after contact times of 1.0 and 10.0 s were selected for comparison of the surface hydrophobic character.

Surface Energy

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A quick analysis of the total surface energy for uncoated and primary coated substrates was undertaken by application of a series of dyne test liquids (Vetaphone A/S, Kolding, Denmark). The method implies application of a test liquid with known surface tension, followed by visual inspection of the surface-test liquid interaction within 2 seconds (ASTM D2578). The surface energy of the polyester resin coating as a function of curing temperature was determined from contact angle measurements by the automatic DAT drop spreading technique (DAT 1100 Dynamic Contact Angle Tester, FIBRO System AB, Stockholm, Sweden) using water, ethylene glycol and methylene iodide as test liquids. The coatings were dried in 23°C for 24 hours, and in 80°C, 105°C or 140°C for 90 seconds respectively. Measurements were carried out at 23°C and 50% RH. The drop size was 4 µl and the contact angles, θ, given below were the average from 9 drops. The total surface energy for the coated substrates was calculated from the contact angle at 5 s and split in its polar and dispersive components through a least squares fit for the three test liquids. The total, dispersive and polar surface tension components of the test liquids are shown in Table 3. Table 3. Surface tensions of test liquids. Values from [24]

Total Dispersive Polar

Water 72.8 21.8 51.0

Surface Tension, mN/m Ethylene Glycol 48.0 29.0 19.0

Methylene Iodide 50.8 50.8 0

Surface Gloss The gloss of a coated surface is defined as the percentage of reflected light relative the incident intensity at a given incidence and viewing angle. The gloss of coated paperboard is affected by the substrate smoothness and the smoothness of the coating layer(s). The coating thickness, the pore size distribution and pore volume are also important parameters. A glossy surface is usually desirable from packaging aesthetic purposes. In case of non-calendered paper, the gloss value can also be taken as a measure of the coating coverage. The gloss of primary and secondary coatings was measured by a Zehntner ZGM 1022 glossmeter at a viewing angle of 75° (Zehntner GmbH Testing Instruments, Hoelstein, Switzerland) according to the TAPPI T480 standard.

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Results and Discussion Both the biopolymer and the synthetic resin provide low maximum concentration for complete dispersion and stability of water suspensions (20 and 35 per cent solids respectively). This is indeed a limiting factor for industrial processing since low solids content of the coating formulations implies that a large amount of water is transferred to the paper substrate upon coating application, thereby involving the risk of causing swelling of the cellulose fibres as well as other interactions that will negatively affect the substrate properties. A low solids content also mean that extensive drying is necessary for complete removal of water, and this is a very energy consuming procedure. Furthermore, it is hard to achieve coating layers thick enough to provide sufficient barrier properties with a low overall solids content of the formulation. A solution to this latter problem can be a stepwise increase of the coat weight by application in multiple layers.

Viscosity All polymer dispersions and composite formulations showed shear thinning, i.e. the viscosity decreased continuously with increased shear rate. The viscosity of the starch and polyester matrices are strongly dependent on polymer concentration (Figure 6). Starch (10%)

Starch (20%)

Polyester (25%)

Polyester (35%)

Viscosity, mPas

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100000 10000 1000 100 10 1 1

10

100

1000

Shear rate, s-1 Figure 6. Viscosity as a function of shear rate for starch and polyester dispersions at intermediate and maximum concentrations. Dispersion solids content within brackets.

The starch-polyester composite formulation viscosity as a function of polyester resin concentration was studied for polyester resin added stepwise to a starch matrix at 20, 40, 60 and 80 weight-% on dry basis. Pure starch and pure polyester resin were used as reference materials. The blends showed a steep increase in the viscosity at low additions of polyester, which could not solely be explained by the simultaneous small increase in solids content. Adjusting the measured viscosities at a fix shear rate (126 s-1) to the same nominal solids content as in the pure polyester resin resulted in the curve shown in Figure 7. As can be seen

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Composite Multilayer Coatings for Improved Barrier Properties of Packaging Board 99 from the curve, a maximum viscosity was found in a blend of 20% polyester resin and 80% starch by weight. A further increase in polyester addition led to a continuous decrease in viscosity until the final value of around 10 mPas for the pure polyester-water dispersion. The stable shear thinning observed in all starch-polyester composite formulations however indicated that the starch and polyester are readily compatible with each other. The steep increase in viscosity at low additions of polyester suggests a destabilizing mechanism. Adsorption of starch onto the polyester particles might lead to flocculation, thereby dramatically increasing the viscosity. Increasing the polyester concentration above 40 weight% however enhances the stability in the dispersion, probably due to increased repulsion between the anionic starch and the anionic polyester when the amount of the latter increases. It is also possible that the polyester hydrophobic fatty acid side groups might contribute to the repulsive forces by steric effects. Incorporation of the polyester into a starch suspension however facilitates to keep the overall viscosity at a more appropriate level even at higher shear rates since the very low viscosity of a pure water dispersion of the polyester sometimes lead to unwanted splashing and uneven spreading over the substrate when applied by the laboratory coater. The effect of polyester concentration on barrier properties is further discussed below.

Adjusted viscosity, mPas

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700 600 500 400 300 200 100 0 0

20

40

60

80

100

Concentration polyester, weight-% Figure 7. Viscosity as a function of polyester concentration in starch-polyester composite formulations at a fix shear rate. The data is adjusted to a nominal solids content of 25% as in the 100wt% polyester formulation.

The rapid increase in the formulation viscosity by incorporation of nanoclay in a starch matrix is shown in Figure 8. The shape of the starch-polyester curve was similar even though the viscosity effects were not as dramatic. At high shear rates the viscosities however converged and fell within the critical viscosity range of conventional coating applicators, which are typically 500-1500 mPas (blade coater), 80-1000 mPas (film applicators) and 100300 mPas for curtain coaters [28]. The viscosity of the polyester-nanoclay composite was just slightly higher than that of the pure polyester (cf. Figure 6).

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Starch

Starch-polyester

Starch-nanoclay

Polyester-nanoclay

Viscosity, mPas

100000 10000 1000 100 10 1 1

10

100

1000

10000

Shear rate, s-1

Figure 8. Viscosity as a function of shear rate for composite formulations. The starch flow curve is shown for comparison.

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Charge Density and Zeta Potential A weak specific charge quantity of 44 µeq/g was found for the starch ether in a water suspension at pH 6.1 (Figure 9). This anionic charge originates from the –COOH groups introduced by oxidation with sodium hypochlorite. The charge quantity of the nanoclay was 359 µeq/g in a water dispersion at pH 8.9. The highly anionic character of the polyester is illustrated by the value close to 1000 µeq/g at a pH of 7.8. The measured charge quantity in the starch-nanoclay and the starch-polyester composites was between the values for the pure components, suggesting that the almost uncharged starch chain screens some of the negative charges, presumably by adsorption onto the anionic clay particles or by association with the highly anionic polyester. The observed tendency for an increase in the overall anionic charge found in the polyester-nanoclay composite suggests that the materials repel each other, i.e. no screening occur. All dispersions showed stable potential values once the point of zero charge were reached, thus indicating colloidal stable and homogeneously dispersed formulations. The pH of water dispersions of the polyester resin should be in the range 7.5-7.9 at room temperature for maximum stability, according to manufacturer recommendations. The strong effect of pH on the dispersion stability was visualized by a stepwise change in turbidity of the water dispersion of the polyester resin from pH 3.0 (opaque) to pH 9.0 (clear). In line with expectations, the Zeta potential of the diluted polyester resin was low (-18.9±0.5 mV) at the lowest pH at which measurement was carried out. At pH 5.0 the –OH and –COOH groups are partially protonized and the Zeta potential reached a value of -46.6±3.7 mV. Above this pH level, the potential remained stable around -45±4 mV, indicating that the dispersion is stable with regard to small fluctuations in pH. Mixing the resin with starch or nanoclay only caused slight changes in pH, and the overall pH in composite formulations prevailed within this range of constant particle potential.

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Composite Multilayer Coatings for Improved Barrier Properties of Packaging Board 101

Specific charge quantity, µeq/g

1200

1000

800

600

400

200

0 Clay

Starch

Starchpolyester

Starchnanoclay

Polyester

Polyesternanoclay

Figure 9. Specific charge quantity in water dispersions of the original components and of the corresponding composite formulations.

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Wettability of Primary and Secondary Layers Appropriate wetting is a pre-requisite to form a continuous, evenly distributed barrier film on top of an uncoated or a coated surface. A liquid will spread spontaneously on a surface only if its surface tension is lower than the surface energy of the surface in question [50]. This is of considerable importance with water-based coating formulations which may have surface tensions well above that of paper based substrates. The surface tension of water is 72.8 mN/m and water-borne latex dispersions often have surface tensions between 30-38 mN/m [51]. The surface energy of cellulose is 54.5 mN/m [24]. Values for uncoated paper is typically 35-40 mN/m while coated paper can have surface energies of 45-50 mN/m. Besides the surface tension, the liquid viscosity is also a factor that affects the wetting and spreading of a coating over a surface [52]. In a coating process, the liquid is however forced to wet the substrate by the external hydrodynamic or mechanical forces induced by the constantly moving applicator or substrate [53]. That means, the dynamic contact angle may be well above that of the static angle measured under equilibrium conditions. The measured surface tension of the liquids and the surface energies of primary coated paperboard are shown in Figure 10. The polyester has the lowest surface tension. Incorporation of nanoclay in the polyester resin did not significantly affect the surface tension. However, the pure starch solution had a surface tension ~64 mN/m and this was further increased by addition of either nanoclay or polyester. From Figure 10 b it is evident that the pure polyester coating and the polyester composite coatings have very low surface energy (30 mN/m), i.e. lower than the surface energy of 43 mN/m specified for the standard PET plastic film. The pigment- and latex primary coatings all show surface energy values around 36 mN/m whereas the starch and starch-nanoclay

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Caisa Andersson

composite show the highest values. Application of the starch, the polyester or the starchnanoclay in double layers did not change the measured surface energy from the primary coated surfaces. The surface energy of starch-nanoclay secondary coatings on top of primary coatings of different nature was slightly lower than that of the pure starch-nanoclay, presumably due to interactions with the low-energy surface beneath. 100 90

Surface tension, mN/m

80 70 60 50 40 30 20 10 0 Starch

Starch-polyester

Starch-nanoclay

Polyester

Polyester-nanoclay

(A)

50

Surface energy, mN/m

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60

40

30

20

10

0 Pigment coating

Starch

Starchpolyester

Starchnanoclay

Polyester

Polyesternanoclay

SA-latex

SB-latextalc

(B) Figure 10. a) surface tension of coating formulations and b) coated substrate surface energy.

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Composite Multilayer Coatings for Improved Barrier Properties of Packaging Board 103 Surfactants are often added to coating formulations to enhance their colloidal stability. Surfactant addition also aids in wetting of the substrate by reducing the surface tension of the liquid [52]. Addition of 0.1 weight-% non-ionic surfactant to the diluted starch solution reduced the surface tension from 69.8 mN/m to 43.1 mN/m wheras a surfactant concentration of 0.5 weight-% resulted in a surface tension of 31.6 mN/m. Further increase of the surfactant concentration up to 2 weight-% did not give any further reduction in the surface tension. The measured reference values for deionized water used for dilution of the starch solutions were 73.1 mN/m and 31.4 mN/m with 0.01 g surfactant/g water. The polyester resin showed complete non-wetting when applied on top of a pre-dried coating of the same polyester or on any other pre-coated surface. Barrier properties are governed by cross-linking which may take place in polyester resin films when cured, i.e. dried at high temperature. High temperature drying however increases the surface energy of the dried film. The non-polar moiety of the polyester is claimed to be oriented away from most substrates after drying, thus making the films highly hydrophobic [35]. To further investigate the reasons behind the observed wetting problems with the polyester resin, contact angles for test liquids with varying polar character were measured. Contact angles (5 s) for the different curing temperatures for all three test liquids on the polyester coated substrate are given in Figure 11. Water

Ethylene glycol

Methylene iodide

120

Contact angle, °

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100

80

60

40

20

0

23°C

80°C

105°C

140°C

Figure 11. Contact angles at 5 s for polyester-coated substrates as a function of drying temperature.

The contact angle θ is related to the surface energy through Young’s equation (1) where γSL, γSV and γLV refers to the solid-liquid, solid-vapour and liquid-vapour interfacial free energies respectively [24].

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(1)

104

Caisa Andersson

The total solid-liquid surface energy can be split in terms counting for its non-polar, dispersive forces, γ d, and its polar (acid-base), γ p, forces, which can be expressed by equation (2) where subscript L and S refer to the liquid and surface respectively. When using a set of test liquids with known surface tensions (Table 3) and measuring their contact angles, a set of equations are obtained that can be solved to give the unknown parameters, hence the surface energy of the substrate can be calculated [52].

γ L (1 + cos θ ) = 2 γ Sd + γ Ld + 2 γ Sp + γ Lp

(2)

The contact angle at 5 s can be regarded as an equilibrium contact angle for cured samples and thus suits well as standard for calculations of surface energy. The contact angle for each test liquid decreased with time for the substrate dried at 23°C, presumably due to penetration of the test liquid drops into the more open (incompletely film formed) substrate, thus lowering their contact angles and raising their apparent energy components. The energy values obtained at this lowest drying temperature are thus a function of both true surface energy and film permeability. For substrates dried at higher temperature, the contact angle was almost constant over all times of interest. The calculated total surface energy and its polar component are shown in Figure 12.

Polar

Total

50

Surface energy, mN/m

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45 40 35 30 25 20 15 10 5 0 23°C

80°C

105°C 140°C

Figure 12. Total surface energy and its polar component as a function of drying temperature.

The high presence of fatty acids in the polyester resin provides highly hydrophobic surfaces. Drying the coated substrates at enhanced temperature had a strong effect on the polar surface energy of the polyester coatings. Increasing the temperature from 80°C to 105°C to 140°C did however not significantly affect the surface energy. The very low

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Composite Multilayer Coatings for Improved Barrier Properties of Packaging Board 105 contribution from the polar components to the total surface energy of cured substrates could explain the wetting problems observed with multilayer barrier coatings. To override the problems with non-wetting of the polyester coatings, the secondary polyester containing layer was simply added on a wet primary layer (i.e. wet-on-wet application). This application method was also tested in case of the starch-nanoclay composite to increase the total coat weight. The method showed perfect wetting and appropriate adhesion between the layers. The technique of wet-on-wet application however resulted in poor leveling of the secondary layer. This was seen as regular stripes over the surface, caused by the wire-wound rod (Figure 13). This should be explained by a strong increase in viscosity of the wet primary layer due to rapid absorption of water into the porous substrate. The observed effects on barrier properties are further discussed below.

Figure 13. Structure of single-and double layer coating in wet-on-wet application.

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However, forming a composite coating of hydrophilic, highly polar starch and hydrophobic polyester resin facilitated wetting and application also on a primary coated surface. Incorporation of the synthetic polyester into a biopolymer matrix not only provides advantageous rheological properties, it also facilitated the spreading of the coating on a variety of substrates.

Coat Weight and Thickness Forming of a paperboard on a machine normally leads to variations in average grammage between different cut sheets but the local board grammage on a small scale also show variations within an A4 sheet. Reference data given by the manufacturer is 255 g/m2 with a tolerance level of ±4%, i.e. the basis weight is expected to vary between 245 and 265 grams per square meter. The calculated coat weights are therefore subjected to uncertainty factors. Average values for both the SA-latex and the SB-latex-talc were 28±3 g/m2, while the starch primary coating was about 13±2 g/m2, due to the low solids content of the starch suspension. Values for the polyester and starch-polyester primary coatings were about 21±2 g/m2 and 15±2 g/m2 respectively. The average paperboard thickness of 350 µm is given a tolerance level of ±4%, i.e. variations between 336-364 µm can be expected. The thickness measurements showed that a single starch-nanoclay coating was 15-20 µm thick and that a double coated barrier structure had the expected thickness of 30-40 µm in total. The primary SA-latex and the primary SBlatex-talc coatings roughly made up about 30 µm whereas the polyester coating was about 24 µm thick. The secondary starch-nanoclay coatings increased the total coat weight with 15-20

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µm, i.e. the same thickness as in case of primary coatings. The starch-polyester and polyesternanoclay coatings were measured to be within 18 and 26 µm.

Barrier Properties Water Vapour Barrier The WVTR for starch-polyester composites as a function of polyester concentration is shown in Figure 14. The variation in coat weight due to small variations in solids content of the formulations was accounted for by recalculating the measured data to water vapour permeability, WVP.

WVP, g/(Pa·s·m) x 10 -6

12.0 10.0 8.0 6.0 4.0 2.0 0.0

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0

20

40

60

80

100

Concentration polyester, weight-%

Figure 14. WVP of starch-polyester composites as a function of polyester concentration.

The data strongly indicates that incorporation of polyester resin in a starch formulation reduces the permeability of water vapour through the coating layer. Adding 40 weight-% polyester to a starch solution leads to a 50 % decrease in WVP. Based on the outcome of viscosity and water vapour barrier measurements, a starch-polyester composite with 40-50 weight-% polyester was selected for further studies. Figure 15 shows WVTR values in g/m2·d for the primary coatings. Reference value for the uncoated paperboard is 373±4 g/m2·d. On the market foodstuff with intermediate requirements on moisture protection have a critical level of WVTR 0, INDEX IN MODIFIED MAXIMUM NORMAL STRESS CRITERION') WRITE(20,260) BETA,ALFA 260 FORMAT(10X,'Beta(in defining bridging elements a2)=',F5.3, & 2X,'Alfa(in defining a3)=',F5.3) C*** NQ = -1, CLASSICAL MAXIMUM NORMAL STRESS CRITERION IS USED C*** > 0, INDEX IN MODIFIED MAXIMUM NORMAL STRESS CRITERION C*** BETA & ALFA = COEFFICIENTS IN DEFINEING a2 & a3 READ(10,*) NL,NSTP,JPRIT,IDSOL,IDRES,IDPLY,JSTP C*** NL = LAYERS IN THE LAMINATE; NSTP = INCREAMENTAL STEPS C*** JPRIT =-2, PRINT in-plane STRESS/STRAIN CURVE PLUS FIRST & LAST STEP MODULI C*** =-3, PRINT bending/curvature CURVE PLUS FIRST & LAST STEP MODULI C*** = 4, ONLY PRINT INTERNAL STRESSES AT THE LAST STEP C*** = 0, NO PRINT C*** IDSOL = 1, FIRSTLY APPLY THERMAL, THEN ISOTHERMAL MECHANICAL LOADS; C*** = 2, COUPLED THERMO-MECHANICAL ANALYSIS (e.g. TMF problem) C*** (THERMAL RESIDUAL STRESSES INCLUDED IF "TL" NOT EQUAL TO "TU") C*** IDRES = 0, NO ADDITIONAL RESIDUAL STRESS C*** = 1, ADDITIONAL RESIDUAL STRESSES TO BE INPUTED C*** IDPLY = 0, EACH PLY HAS SAME GEOMETRIC PARAMETERS C*** = 1, EACH PLY HAS DIFFERENT GEOMETRIC PARAMETERS C*** JSTP > 0, PRINT STRESS(idss)/STRAIN(idss) CURVE, C*** ONCE AFTER "JSTP" STEPS C*** = 0, NOT PRINT C (SSMAX(1)-SSMIN(1)),...,(SSMAX(3)-SSMIN(3))=Stresses:Sxx,Syy,Sxy,if IDSEC=0 C =Forces:Nxx,Nyy,Nxy,if IDSEC=1 or 2 C (SSMAX(4)-SSMIN(4)),...,(SSMAX(6)-SSMIN(6))=Mxx,Myy,Mxy(per unit length),if 0 C =Total moments(N-mm),if IDSEC=1 or 2 IF(NL.EQ.1) NL2=NL IF(NL.GE.2) THEN NL2=NL NL=2*NL-1 ENDIF READ(10,*) (Z2(I),I=1,NL2+1)

1

DO 1 I=1,NL2+1 Z(I)=Z2(I) IF(NL.GE.2) THEN Z(1)=Z2(1) Z(2*NL2)=Z2(NL2+1)

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150

Zheng-Ming Huang and Ye-Xin Zhou DO 5 I=2,NL2 Z(2*I-2)=Z2(I)-(Z2(I)-Z2(I-1))/40. Z(2*I-1)=Z2(I)+(Z2(I+1)-Z2(I))/40.

5 ENDIF

IF(NL.GT.ML) THEN WRITE(20,2000) NL,ML 2000 FORMAT(15X,'ACTUAL ALLOWABLE',I3) STOP ENDIF

LAYERS',I3,2X,'MORE

THAN

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READ(10,*) (ANGLE(2*I-1),I=1,NL2) IF(NL.GE.2) THEN DO 15 I=1,NL2-1 15 ANGLE(2*I)=0. ENDIF READ(10,*) TL,TU,T0,T1 READ(10,*) NTSUB READ(10,*) (SSMIN(I),I=1,6),(SSMAX(I),I=1,6) DO 190 IL=1,NL LAYER(I)=1 DO 190 J=1,3 SSP(IL,J)=0. 190 continue C CALL GLOBAL(ML,NL,ANGLE,TS,TC) C** Z(i) -Z-COORDINATE OF THE i-th LAYER, Z=0 IS THE MIDDLE SURFACE OF THE LAMINATEC* C*** ANGLE(i) - INCLINED ANGLE OF i-th PLY x-AXIS WITH THE GLOBAL X-AXIS C*** TS(i,.,.),TC(i,.,.) - TRANSFORMATION MATRICES OF i-th LAYER C*** LAYER(i) = 1, i-th LAYER DOES NOT FAIL; C*** =-k, i-th LAYER HAS FAILED AT k-th ORDER (k=1, INITIAL)C* C*** TL - STRESS FREE TEMPERATURE; TU - PRESENT TEMPERATURE (TO APPLY LOAD) C*** T0 - INITIAL WORKING TEMP. (USUALLY=TU); T1 - FINIAL WORKING TEMP. C*** NTSUB - SUBINTERVALS TO COVER [TL,TU] C*** SSMIN,SSMAX - MINIMUM & MAXIMUM APPLED STRESSES & MOMENTS ON THE LAMINATE CALL PARAM(ML,NL,MUM,MUF,MEM,MEF,NEM,NEF,MSEG,RESIN,FIBER) IF(IDPLY.NE.0) THEN READ(10,*) (VF(2*IL-1),IL=1,NL2) ELSE READ(10,*) VF1 DO 150 IL=1,NL 150 VF(IL)=VF1 ENDIF IF(NL.GE.2) THEN

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14

151

VF(1)=VF1*100./97.5 VF(2*NL2-1)=VF1*100./97.5 IF(NL2.GE.3) THEN DO 13 I=2,NL2-1 VF(2*I-1)=VF1*100./95. ENDIF DO 14 I=1,NL2-1 VF(2*I)=0.000001

ENDIF WRITE(20,70) 70 FORMAT(/5X,'PLY',3X,'PLY-ANGLE',5X,'Vf',8X,'Z(i)',8X,'Z(i+1)') DO 80 IL=1,NL ALFF(IL,3)=0. ALFM(IL,3)=0. 80 WRITE(20,90) IL,ANGLE(IL),VF(IL),Z(IL),Z(IL+1) 90 FORMAT(4X,I3,4X,F6.2,7X,F5.3,5X,F6.3,7X,F6.3) CALL INITIL0(ML,NL,SSF,SSM,SA,SS) IF(IDRES.GT.0) CALL RESID(ML,NL,SSF,SSM) IF(TL.NE.TU) THEN WRITE(20,10) TL,TU 10 FORMAT(/10X,'THERMAL RESIDUAL STRESSES COOLING FROM',1X,F8.2, & 3X,'TO',1X,F8.2) CALL PROSS(ML,MUM,MUF,MEM,MEF,NL,NEM,NEF,MSEG,ALFA,BETA, & 4,LFAIL,NTSUB,JSTP,NQ,TL,TU,A,B,BM,SS,SA,DSS, & RESIN,FIBER,SSF,SSM,SSP,EUF,EUM,SF,SM,SUF,SUUM, & ALFF,ALFM,VF,LAYER,Z,SP,CG,TS,TC,BTE,ALF,FAIL,NL2) ENDIF IF(IDSOL.EQ.1) THEN IF(ABS(T1-T0).GT.0.) THEN WRITE(20,20) T0,T1 20 FORMAT(/10X,'DECOUPLED SOLUTION BY FIRST APPLYING', & ' TEMPERATURE FROM',F6.1,2X,'TO',2X,F6.1) CALL PROSS(ML,MUM,MUF,MEM,MEF,NL,NEM,NEF,MSEG,ALFA,BETA, & 4,LFAIL,NTSUB,JSTP,NQ,T0,T1,A,B,BM,SS,SA,DSS, & RESIN,FIBER,SSF,SSM,SSP,EUF,EUM,SF,SM,SUF,SUUM, & ALFF,ALFM,VF,LAYER,Z,SP,CG,TS,TC,BTE,ALF,FAIL,NL2) WRITE(20,40) 40 FORMAT(3X,'ABOVE STRESSES ARE TOTAL RESIDUAL STRESSES', & ' BEFORE APPLYING ANY MECHANICAL LOAD'/) ENDIF DO 30 I=1,6 SA(I)=0.

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DSS(I)=(SSMAX(I)-SSMIN(I))/REAL(NSTP) continue CALL PROSS(ML,MUM,MUF,MEM,MEF,NL,NEM,NEF,MSEG,ALFA,BETA, & JPRIT,LFAIL,NSTP,JSTP,NQ,T1,T1,A,B,BM,SS,SA,DSS, & RESIN,FIBER,SSF,SSM,SSP,EUF,EUM,SF,SM,SUF,SUUM, & ALFF,ALFM,VF,LAYER,Z,SP,CG,TS,TC,BTE,ALF,FAIL,NL2) ELSE WRITE(20,50) T0,T1 50 FORMAT(/'COUPLED THERMO-MECHANICAL SOLUTION FROM', & ' INITIAL TEMPERATURE',F6.1,2X,'TO FINAL', & ' TEMPERATURE',F6.1) DO 60 I=1,6 SA(I)=0. DSS(I)=(SSMAX(I)-SSMIN(I))/REAL(NSTP) CALL PROSS(ML,MUM,MUF,MEM,MEF,NL,NEM,NEF,MSEG,ALFA,BETA, & JPRIT,LFAIL,NSTP,JSTP,NQ,T0,T1,A,B,BM,SS,SA,DSS, & RESIN,FIBER,SSF,SSM,SSP,EUF,EUM,SF,SM,SUF,SUUM, & ALFF,ALFM,VF,LAYER,Z,SP,CG,TS,TC,BTE,ALF,FAIL,NL2) 60 continue ENDIF RETURN END C************************************************************************** ***** SUBROUTINE PARAM(ML,NL,MUM,MUF,MEM,MEF,NEM,NEF,MSEG,RESIN,FIBER) C*** TO DETERMINE FIBER & MATRIX EFFECTIVE MODULI OF EACH PLY, FIBER C*** & MATRIX COMPLIANCES, ULTIMATE STRENGTHS, AND THERMAL EXPANSION C*** COEFFICIENTS USING CURRENT STRESS STATES, TEMPERATURE, STRAIN RATES C*** SUPPOSE THE LAMINATE CONSISTS OF TWO PHASE MATERIALS C*** MUM (MUF) - MAXIMUM SETS OF MATRIX (FIBER) PARAMETERS C*** MEM (MEF) - MAXIMUM NUMBER OF ONE PARAMETER SET FOR MATRIX (FIBER) C*** NEM (NEF) - ACTUAL NUMBER OF ONE PARAMETER SET FOR MATRIX (FIBER) C*** MSEG - NUMBER OF SEGMENTS FOR MATRIX STRESS/STRAIN CURVE C IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION RESIN(2,MUM,MEM),FIBER(MUF,MEF),ID(5) ID(1)=1 ID(2)=2 READ(10,*) NEF

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Simulation of Ultimate Strength of Fiber-Reinforced Composites… IF(NEF.GT.MEF.OR.MUF.LT.7) THEN NUF=7 GOTO 2000 ENDIF READ(10,*) (FIBER(1,I),I=1,5) WRITE(20,200) (FIBER(1,I),I=1,5),NEF 200 FORMAT('FIBER MODULI(E1,U12,E2,U23,G12)=',E10.4,1X,F5.3,1X, & E10.4,1X,F5.3,1X,E10.4/1X,'TEMPERATURE POINTS (Nef)=', & I4,2X,'ON WHICH FIBER PROPERITES ARE CHANGED') DO 20 J=2,7 READ(10,*) (FIBER(J,I),I=1,NEF) 20 CONTINUE IF(NEF.LE.8) THEN WRITE(20,110) (FIBER(2,I),I=1,NEF) 110 FORMAT(2X,'T(1-Nef)=',8(F7.1,1X)) WRITE(20,120) (FIBER(3,I),I=1,NEF) 120 FORMAT(1X,'E1(1-Nef)=',8(F7.0,1X)) WRITE(20,130) (FIBER(4,I),I=1,NEF) 130 FORMAT(1X,'Su(1-Nef)=',8(F7.1,1X)) WRITE(20,140) (FIBER(5,I),I=1,NEF) 140 FORMAT('Suc(1-Nef)=',8(F7.1,1X)) WRITE(20,150) (FIBER(6,I),I=1,NEF) WRITE(20,160) (FIBER(7,I),I=1,NEF) 150 FORMAT('Af1(1-Nef)=',8(F7.3,1X)) ELSE WRITE(20,210) (FIBER(2,I),I=1,NEF) 210 FORMAT(2X,'T(1-Nef)=',8(F7.1,1X)/11X,8(F7.1,1X)) WRITE(20,220) (FIBER(3,I),I=1,NEF) 220 FORMAT(1X,'E1(1-Nef)=',8(F7.0,1X)/11X,8(F7.0,1X)) WRITE(20,230) (FIBER(4,I),I=1,NEF) 230 FORMAT(1X,'Su(1-Nef)=',8(F7.1,1X)/11X,8(F7.1,1X)) WRITE(20,240) (FIBER(5,I),I=1,NEF) 240 FORMAT('Suc(1-Nef)=',8(F7.1,1X)/11X,8(F7.1,1X)) WRITE(20,250) (FIBER(6,I),I=1,NEF) 250 FORMAT('Af1(1-Nef)=',8(F7.3,1X)/11X,8(F7.3,1X)) WRITE(20,260) (FIBER(7,I),I=1,NEF) 260 FORMAT('Af2(1-Nef)=',8(F7.3,1X)/11X,8(F7.3,1X)/) ENDIF C READ(10,*) MSEG,NEM WRITE(20,1010) MSEG 1010 FORMAT(/'SEGMENTS (Mseg) OF THE MATRIX STRESS-STRAIN', & 1X,'CURVE =',I4) C NUM=5+2*MSEG IF(NEM.GT.MEM.OR.NUM.GT.MUM) GOTO 2000 ID(3)=2*MSEG+5

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Zheng-Ming Huang and Ye-Xin Zhou

500

470 TENSION:')

ID(4)=2*MSEG+3 ID(5)=2*MSEG+4 DO 500 J=1,5 READ(10,*) (RESIN(1,ID(J),I),I=1,NEM) DO 500 I=1,NEM RESIN(2,ID(J),I)=RESIN(1,ID(J),I) DO 490 K=1,2 IF(K.EQ.1) THEN WRITE(20,470) FORMAT(/20X,'MATRIX PROPERTIES UNDER ELSE

480 COMPRESSION:')

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50

WRITE(20,480) FORMAT(/18X,'MATRIX PROPERTIES UNDER ENDIF DO 50 J=3,2*MSEG+2 READ(10,*) (RESIN(K,J,I),I=1,NEM) CONTINUE WRITE(20,410) NEM FORMAT(5X,'TEMPERATURE POINTS (ON WHICH MATRIX

410 PROPERTIES', & 1X,'ARE VARIED) =',I4/2X,'TEMP.',4X,'Alfm',4X, & 'Poissons Ratio',2x,'Tensile Strength',2x,'Com.', & 1x,'Strength') DO 55 I=1,NEM WRITE(20,420) (RESIN(K,J,I),J=1,2),RESIN(K,2*MSEG+5,I), & RESIN(K,2*MSEG+3,I),RESIN(K,2*MSEG+4,I) 420 FORMAT(F7.1,2X,F7.2,6X,F5.3,10X,F7.1,9X,F7.1) 55 continue WRITE(20,415) 415 FORMAT(2X,'TEMP.', 19X,'YILED STRENGTH, Ys(1--Nem)') DO 56 I=1,NEM WRITE(20,430) RESIN(K,1,I),(RESIN(K,J,I),J=3,2+MSEG) 56 continue WRITE(20,460) 460 FORMAT(2X,'TEMP.',17X,'TANGENTENT MODULI, ET(1--Nem)') DO 57 I=1,NEM WRITE(20,430) RESIN(K,1,I),(RESIN(K,J,I),J=3+MSEG,2*MSEG+2) 430 FORMAT(F7.1,2X,9(F8.1,1X)/9X,9(F8.1,1X)) 57 continue 490 CONTINUE C RETURN C 160 FORMAT('Af2(1-Nef)=',8(F7.3,1X)/)

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1000 FORMAT(/'CONSTITUTIVE MODEL FOR FIBERS =',I3/ & 30X,'=0..FIBER PROPERTIES NEVER CHANGED'/ & 30X,'=-1.PROPERTIES VARY WITH TEMPERATURE (isotropically),'/ & 34X,'BUT CAN BE LINEARLY INTERPOLATED'/ & 30X,'=1..ONLY E11 VARIES WITH TEMPERATURE'/ & 30X,'=2 OR OTHER VALUE..NOT DEFINED') 2000 WRITE(20,2100) MUM,MEM,MUF,MEF,NUM,NEM,NUF,NEF 2100 FORMAT(/'ALLOWABLE (MUM,MEM,MUF,MEF)=',4(I4,1X), & 'ACTUAL (NUM,NEM,NUF,NEF)=',4(I4,1X)) STOP END C************************************************************************** **** SUBROUTINE RESID(ML,NL,SSF,SSM) IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION SSF(ML,3),SSM(ML,3),SF(3),SM(3) DO 10 IL=1,NL READ(10,*) (SF(I),I=1,3),(SM(I),I=1,3) DO 10 I=1,3 SSF(IL,I)=SSF(IL,I)+SF(I) SSM(IL,I)=SSM(IL,I)+SM(I) 10 CONTINUE RETURN END C********************************************************************** SUBROUTINE FIBERS(ML,NL,MUF,MEF,NEF,TEMP, & FIBER,EGF,SF,SUF,ALFF,LAYER) IMPLICIT DOUBLE PRECISION(A-H,O-Z) DIMENSION FIBER(MUF,MEF),EGF(ML,5),SF(ML,3,3),SUF(ML,2), & ALFF(ML,3),LAYER(ML) C DO 40 I=1,NEF-1 IF(TEMP.GE.FIBER(2,I).AND.TEMP.LE.FIBER(2,I+1)) THEN I0=I GOTO 50 ENDIF 40 CONTINUE WRITE(20,25) TEMP,FIBER(2,1),FIBER(2,NEF) STOP 50 CONTINUE DO 60 IL=1,NL IF(LAYER(IL).LT.0) GOTO 60 DO 70 I=1,5 70 EGF(IL,I)=FIBER(1,I) DO 80 J=6,7 ALFF(IL,J-5)=FIBER(J,I0)+(TEMP-FIBER(2,I0))*

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(FIBER(J,I0+1)-FIBER(J,I0))/(FIBER(2,I0+1)-FIBER(2,I0)) SUF(IL,J-5)=FIBER(J-2,I0)+(TEMP-FIBER(2,I0))* (FIBER(J-2,I0+1)-FIBER(J-2,I0))/ (FIBER(2,I0+1)-FIBER(2,I0)) continue EGF(IL,1)=FIBER(3,I0)+(TEMP-FIBER(2,I0))* (FIBER(3,I0+1)-FIBER(3,I0))/(FIBER(2,I0+1)-FIBER(2,I0)) CONTINUE GOTO 100

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C 100 DO 110 IL=1,NL IF(LAYER(IL).LT.0) GOTO 110 CALL ELASC(ML,IL,EGF(IL,1),EGF(IL,3),EGF(IL,2),EGF(IL,5),SF) 110 CONTINUE RETURN 25 FORMAT(/'GIVEN TEMPERATURE T=',E11.4,1X,'IS OUT OF RANGE(', & E11.4,1X,E11.4,') OF THE FIBERS'/) END C************************************************************************** SUBROUTINE ELASC(ML,IL,E1,E2,U,G,SE) IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION SE(ML,3,3) C***TO CALCULATE PLANE ELASTIC COMPLIANCE MATRIX SE DO 10 I=1,3 DO 10 J=1,3 SE(IL,I,J)=0. 10 continue SE(IL,1,1)=1./E1 SE(IL,1,2)=-U/E1 SE(IL,2,1)=SE(IL,1,2) SE(IL,2,2)=1./E2 SE(IL,3,3)=1./G RETURN END C************************************************************************** *** SUBROUTINE PLASC(ML,IL,MSEG,ETM,SSM,EGM,SP,ID,LAYER) IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION ETM(2,20),SSM(ML,3),EGM(ML,5),SP(3,3),SA(3),LAYER(ML) ID=0 S1=SQRT(SSM(IL,1)**2+SSM(IL,2)**2-SSM(IL,1)*SSM(IL,2) & +3.*SSM(IL,3)**2) IF(S1.LE.ETM(1,1)) RETURN ID=1 DO 10 I=1,MSEG-1 IF(S1.GT.ETM(1,I).AND.S1.LE.ETM(1,I+1)) THEN ET1=ETM(2,I+1)

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Simulation of Ultimate Strength of Fiber-Reinforced Composites… GOTO 50 ENDIF

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10 CONTINUE ET1=ETM(2,MSEG) 50 IF(ET1.LE.0.) ET=0.01 ET=ET1 E=ETM(2,1) IF(LAYER(IL).LT.0) ET=0.01*ET1 SA(1)=SSM(IL,1)-(SSM(IL,1)+SSM(IL,2))/3. SA(2)=SSM(IL,2)-(SSM(IL,1)+SSM(IL,2))/3. SA(3)=SSM(IL,3) C=9.*(E-ET)/(4.*E*ET*S1*S1) SP(1,1)=C*SA(1)*SA(1) SP(1,2)=C*SA(1)*SA(2) SP(1,3)=2.*C*SA(1)*SA(3) SP(2,2)=C*SA(2)*SA(2) SP(2,3)=2.*C*SA(2)*SA(3) SP(3,3)=4.*C*SA(3)*SA(3) DO 20 I=1,3 DO 20 J=I,3 20 SP(J,I)=SP(I,J) C*** CHANGE MODULUS TO DEFINE BRIDGING MATRIX EGM(IL,1)=ET1 EGM(IL,2)=0.5 EGM(IL,3)=ET1 EGM(IL,4)=0.5 EGM(IL,5)=ET1/3. RETURN END C********************************************************************** SUBROUTINE STATUS(ML,IL,SSM,L) IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION SSM(ML,3) IF((SSM(IL,1)+SSM(IL,2)).GE.0.) THEN L=1 ELSE L=2 ENDIF RETURN END C********************************************************************** SUBROUTINE MATRIX(ML,NL,MUM,MEM,NEM,MSEG,LAYER, & TEMP,RESIN,SSM,EGM,SM,SUUM,ALFM) IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION RESIN(2,MUM,MEM),SSM(ML,3),EGM(ML,5),SM(ML,3,3), Composite Laminates: Properties, Performance and Applications : Properties, Performance and Applications, Nova Science Publishers,

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Zheng-Ming Huang and Ye-Xin Zhou &

SUUM(ML,2),ALFM(ML,3),ETM(2,20),LAYER(ML),EGM1(ML,5)

IF(MSEG.GT.20) STOP DO 40 I=1,NEM-1 IF(TEMP.GE.RESIN(1,1,I).AND.TEMP.LE.RESIN(1,1,I+1)) THEN I0=I GOTO 50 40 25 RANGE(', &

E11.4,1X,E11.4,') OF THE MATRIX'/) STOP CONTINUE

50

& &

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ENDIF CONTINUE WRITE(20,25) TEMP,RESIN(1,1,1),RESIN(1,1,NEM) FORMAT(/'GIVEN TEMPERATURE T=',E11.4,1X,'IS OUT OF

DO 60 IL=1,NL CALL STATUS(ML,IL,SSM,L) ALFM(IL,1)=RESIN(L,2,I0)+(TEMP-RESIN(L,1,I0))* (RESIN(L,2,I0+1)-RESIN(L,2,I0))/(RESIN(L,1,I0+1) -RESIN(L,1,I0)) ALFM(IL,2)=ALFM(IL,1) DO 80 J=1,2 J1=2*MSEG+2+J SUUM(IL,J)=RESIN(L,J1,I0)+(TEMP-

RESIN(L,1,I0))* & (RESIN(L,J1,I0+1)-RESIN(L,J1,I0))/ & (RESIN(L,1,I0+1)-RESIN(L,1,I0)) 80 continue EGM(IL,1)=RESIN(L,MSEG+3,I0)+(TEMPRESIN(L,1,I0))* & (RESIN(L,MSEG+3,I0+1)-RESIN(L,MSEG+3,I0))/ & (RESIN(L,1,I0+1)-RESIN(L,1,I0)) EGM(IL,2)=RESIN(L,2*MSEG+5,I0)+(TEMPRESIN(L,1,I0))* & (RESIN(L,2*MSEG+5,I0+1)-RESIN(L,2*MSEG+5,I0))/ & (RESIN(L,1,I0+1)-RESIN(L,1,I0)) EGM(IL,3)=EGM(IL,1) EGM(IL,4)=EGM(IL,2) EGM(IL,5)=0.5*EGM(IL,1)/(1.+EGM(IL,2)) DO 90 K=1,MSEG ETM(1,K)=RESIN(L,K+2,I0)+(TEMPRESIN(L,1,I0))* & (RESIN(L,K+2,I0+1)-RESIN(L,K+2,I0))/ & (RESIN(L,1,I0+1)-RESIN(L,1,I0))

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Simulation of Ultimate Strength of Fiber-Reinforced Composites…

159

90 ETM(2,K)=RESIN(L,K+2+MSEG,I0)+(TEMPRESIN(L,1,I0))* & (RESIN(L,K+2+MSEG,I0+1)-RESIN(L,MSEG+K+2,I0))/ & (RESIN(L,1,I0+1)-RESIN(L,1,I0)) IF(LAYER(IL).LT.0) THEN EGM1(IL,1)=0.01*EGM(IL,1) EGM1(IL,3)=0.01*EGM(IL,3) EGM1(IL,5)=0.01*EGM(IL,5) ELSE EGM1(IL,1)=EGM(IL,1) EGM1(IL,3)=EGM(IL,3) EGM1(IL,5)=EGM(IL,5) ENDIF EGM1(IL,2)=EGM(IL,2) EGM1(IL,4)=EGM(IL,4) CALL ELAPS(ML,IL,MSEG,ETM,SSM,EGM1,SM,LAYER,ID) EGM(IL,1)=EGM1(IL,1) EGM(IL,3)=EGM1(IL,3) EGM(IL,5)=EGM1(IL,5) EGM(IL,2)=EGM1(IL,2) EGM(IL,4)=EGM1(IL,4) 60 CONTINUE RETURN END C************************************************************************** * SUBROUTINE ELAPS(ML,IL,MSEG,ETM,SSM,EGM,SM,LAYER,ID) IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION ETM(2,20),SSM(ML,3),EGM(ML,5),SM(ML,3,3),SP(3,3), & LAYER(ML) CALL ELASC(ML,IL,EGM(IL,1),EGM(IL,3),EGM(IL,2),EGM(IL,5),SM) CALL PLASC(ML,IL,MSEG,ETM,SSM,EGM,SP,ID,LAYER) IF(ID.EQ.0) RETURN DO 10 I=1,3 DO 10 J=1,3 SM(IL,I,J)=SM(IL,I,J)+SP(I,J) 10 continue RETURN END C************************************************************************** * SUBROUTINE GLOBAL(ML,NL,ANGLE,TS,TC) IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION ANGLE(ML),TS(ML,3,3),TC(ML,3,3) PAI=3.14159265359/180. DO 10 IL=1,NL RL1=COS(ANGLE(IL)*PAI)

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20

RM2=RL1 RL2=-SIN(ANGLE(IL)*PAI) RM1=-RL2 TC(IL,1,1)=RL1*RL1 TC(IL,1,2)=RL2*RL2 TC(IL,1,3)=2.*RL1*RL2 TC(IL,2,1)=RM1*RM1 TC(IL,2,2)=RM2*RM2 TC(IL,2,3)=2.*RM1*RM2 TC(IL,3,1)=RL1*RM1 TC(IL,3,2)=RL2*RM2 TC(IL,3,3)=RL1*RM2+RL2*RM1 DO 20 I=1,2 DO 20 J=1,2 TS(IL,I,J)=TC(IL,I,J) continue TS(IL,1,3)=RL1*RL2 TS(IL,2,3)=RM1*RM2 TS(IL,3,1)=2.*RL1*RM1 TS(IL,3,2)=2.*RL2*RM2 TS(IL,3,3)=TC(IL,3,3)

10 CONTINUE RETURN END C********************************************************************** SUBROUTINE INVER(D,A,N,L) C*** SUBROUTINE TO FIND THE INVERSION S OF THE POSITIVE DEFINITE MATRIX C*** N 0.6. The maximum value of total strain energy release rate G in present method and in the method of Wang [7] (figure 15) occurs at a delamination length approximately equal to one or two ply thickness that depends width of composite laminate. In the results of Kim and Hong [15], the maximum value of G is almost independent of the laminate width. In this example, the maximum value of G in present solution and the solution of Wang [7] occurs at a/b = 0.125 ÷ 0.25. In the solution of Kim and Hong [15], it always occurs at a/b ≈ 0.28. 0.7

G/106e2 (psi in)

0.6 0.5 0.4 0.3 0.2

Present Kim & Hong [15] Wang [7]

0.1

a/b

0. 01 0. 03 0. 05 0. 10 0. 15 0. 20 0. 25 0. 30 0. 35 0. 40 0. 45 0. 50 0. 55 0. 60 0. 65 0. 70 0. 75 0. 80 0. 85 0. 90 0. 95 0. 99

0

G/106e2 (psi in)

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Figure 15. G with various crack lengths, b=8 in, h=1 in, θ = ± 45o. 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

Present Wang [7]

h1/h2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

2

3

4

5

6

7

8

9

10

Figure 16. G with various ratio h1/h2.

The influence of laminate geometric variables on the delamination behavior is illustrated by changing the total strain energy release rate G with the relative thickness of the upper and lower plies h1/h2. In the graphite-epoxy composite [45o/-45o]s with: h1 + h2 = w = 2 in; Crack length: a = 1 in; Laminate width: 2b = 4 in,

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Nguyen Tien Duong and Nguyen Dang Hung

The material properties and the loading are same in (71) and (73). The result of this influence is presented in the figure 16 and compared with that of Wang [7]. The agreement is very good found between the two methods. The difference of the values of total strain energy release rate G in present method and those of Wang is less than 2%.

2). Delamination of Composite Laminates under Bending The composite laminate with the material properties as in (71) and the dimensions of the plate as in (72) is subjected to a bending K3 = 1/h (see loading condition in (74)). The seconddegree triangular elements and a mesh of 56 elements for one half of the cross section of laminate are considered (figure 17). The stress intensity factors Ki (i = I, II, III) for the two cracks at various fiber orientations of ±θ degrees are presented in figures 18 and 19. As in the case of delamination under extension, in this case the stress intensity factor KIII is dominant; KI is significant; KII is very weak. When θ is ±60 degrees, the three modes are simultaneously cancelled. For the crack 1, KI is positive when |θ| < 60o, so the delamination propagation in the mode I can happen. For the crack 2, KI is negative when |θ| < 60o, so the delamination propagation in the mode I doesn’t happen. It if found that in the case of bending, the stress intensity factors Ki of the crack 2 in the part of the laminate subjected to axial extension are similar to those for the axial extension (see figures 8 and 19).

Singular metis element of the crack 2

y

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h

h

h

x Δ

h

Δ a

Singular metis element of the crack 1 b

Figure 17. Mesh of one half of the cross section.

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Delamination of Composite Structure under Various Types of Loading…

431

Ki/106 K3 (psi in1/2)

1 0,5 0 -0,5 0

10

20

30

-1

±θ

40

50

60

70

80

-1,5

90

KI KII KIII

-2 -2,5 -3 -3,5 -4

Figure 18. Stress intensity factor Ki of the crack 1 under bending.

Ki/106 K3 (psi in1/2)

0,5 0 -0,5 0

10

20

30

±θ

40

50

60

70

80

90

-1 -1,5

KI KII KIII

-2 -2,5 -3 -3,5

Figure 19. Stress intensity factor Ki of the crack 2 under bending. 7

Gi/106 K32 (psi in)

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-4

6 5

Crack 1 Crack 2

4 3 2 ±θ

1 0 -1 0

10

20

30

40

50

60

70

80

90

Figure 20. Total strain energy release rate G under bending.

The total strain energy release rate G for various fiber orientations of ±θ degrees of composite laminates under bending is shown in figure 20. The energy release rate is highest for a composite laminate [±θ]s with θ = 16o and has the small value with |θ| ≥ 45o. This is Composite Laminates: Properties, Performance and Applications : Properties, Performance and Applications, Nova Science Publishers,

432

Nguyen Tien Duong and Nguyen Dang Hung

similar with the case of laminates under extension. The total strain energy release rate of two cracks (crack 1 and crack 2) is similar.

3). Delamination of Composite Laminates under Twisting The composite laminate with the material properties as in (71) and the dimensions of the plate as in (72) is subjected to a warping angle K5 = 1/h (see loading condition in (75)). In this case, the entire cross section of laminate with 224 elements and 509 nodes is examined (figure 21).

Crack 3

Crack 4

y

h

h x h

Δ

Crack 1

Δ

Crack 2

a

2b

Figure 21. Mesh of entire cross section of laminate (224 elements).

Ki/106 K5 (psi in1/2)

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h

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 -0.05 0

KI KII KIII

10

20

30

40

±θ 50

60

70

Figure 22. Stress intensity factor Ki of the crack 1 under twisting. Composite Laminates: Properties, Performance and Applications : Properties, Performance and Applications, Nova Science Publishers,

80

90

Delamination of Composite Structure under Various Types of Loading…

433

Ki/106 K5 (psi in1/2)

0.3 0.25

KI

-KI

KII

0.2

KIII

0.15 0.1 0.05 0 0

10

20

30

40

±θ 50

60

70

80

90

Figure 23. Stress intensity factor Ki of the crack 2 under twisting.

Ki/106 K5 (psi in1/2)

0.4 0.35

KI KII KIII

0.3 0.25 0.2

-KI

0.15 0.1 0.05 0 -0.05 0

10

20

30

40

±θ

50

60

70

80

90

Figure 24. Stress intensity factor Ki of the crack 3 under twisting.

Ki/106 K5 (psi in1/2)

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0.3 KI

0.25

KII

0.2

KIII

0.15 0.1 0.05 0 -0.05

0

10

20

30

40

±θ

50

60

70

80

90

Figure 25. Stress intensity factor Ki of the crack 4 under twisting.

The stress intensity factors Ki (i = I, II, III) for cracks 1, 2, 3 and 4 at various fiber orientations of ±θ degrees are presented in figures 22, 23, 24 and 25. In the cases of uniform extension and of uniform bending, the stress intensity factor KI is very weak but in the case of uniform twisting, the stress intensity factor KI is very important. The KI is even more important than KIII in some cases (for example θ = ± 60o for cracks 2 and 4). In the cases of uniform extension and of uniform bending, the stress intensity factors Ki (i = I, II or III) is very weak when θ ≥ 60 o whereas in the case of uniform twisting they are still enough high when θ = 75o.

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434

Nguyen Tien Duong and Nguyen Dang Hung The comparisons of Ki (i = I, II or III) for 4 cracks are given in figure 26, 27 and 28.

KI/106 K3 (psi in1/2)

0.25

-KI

KI of the crack 1 KI of the crack 2

0.2

KI of the crack 3 0.15

KI of the crack 4

0.1

KI

0.05

±θ

0 0

10

20

30

40

50

60

70

80

90

Figure 26. Comparison of KI between 4 cracks under twisting.

KII/106 K5 (psi in1/2)

0.012

KII

-KII

0.008 0.006

1 2 3 4

0.004 0.002 0 -0.002

0

10

20

30

40

±θ 50

60

70

80

90

Figure 27. Comparison of KII between 4 cracks under twisting. 0.4

KIII/106 K3 (psi in1/2)

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KII of the crack KII of the crack KII of the crack KII of the crack

0.01

0.35

Crack 1

0.3

Crack 2

0.25

Crack 3 Crack 4

0.2 0.15 0.1

±θ

0.05 0 0

10

20

30

40

50

60

70

Figure 28. Comparison of KIII between 4 cracks under twisting.

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80

90

Delamination of Composite Structure under Various Types of Loading…

435

The sign of the KIII of 4 cracks is identical. But the sign of the KI and KII of cracks 2 and 3 is the opposite of that of cracks 1 and 4. The KI of cracks 2 and 3 is negative; it means that the crack propagation does not take place for cracks 2 and 3. The absolute values of Ki of crack 1 are equal to those of crack 3. The absolute values of Ki of crack 2 are equal to those of crack 4. They can be expressed by the following formulae:

K I1 = − K I3 ;

− K I2 = K I4

− K II1 = K II3 ;

K II2 = − K II4

1 3 K III = K III ;

(76)

2 4 K III = K III

The absolute value of KI of cracks 2 and 4 is more important than that of cracks 1 and 3. The variation of KII of 4 cracks is very complex (figure 27). The absolute value of KIII of cracks 1 and 3 is more important than that of cracks 2 and 4. In the case of the axial extension and uniform bending: KI and KII reach the maximum value at θ = ± 30o; KIII takes the maximum value at θ = ± 15o. In the case of uniform twisting: the maximum value of KI corresponds to the interval θ = ± 45o ÷ ± 60o; the maximum values of the KII and KIII are located at ± 15o and ± 30o, respectively. The strain energy release rates Gi and Gtotal for various fiber orientations of ±θ degrees of composite laminates under twisting are shown in figures 29 to 33. It is found that the variations of Gi and Gtotal of 4 cracks are similar (see figures 30 to 33), so only strain energy release rates (Gi and Gtotal) of crack 1 for various orientations of the fibers ±θ degrees of the structure are presented in figure 29.

Gi/106 K52 (psi in)

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0.06

GI

0.05

GII

0.04

GIII

0.03

Gtotal

0.02 0.01 0 -0.01

0

10

20

30

40

±θ

50

60

70

80

90

Figure 29. Strain energy release rate Gi of the crack 1 under twisting.

It is noted that the strain energy release rates of the three modes are always positive. Thus, the three modes of crack are presented simultaneously. The strain energy release rate of mode III (GIII) is always most important. The GII is always weakest. It means that:

G III > G I > G II

(77)

The comparisons of the strain energy release rate of 4 cracks are presented in figures 30 to 33. The strain energy release rates Gi and Gtotal of 4 cracks are similar (the difference of Gi Composite Laminates: Properties, Performance and Applications : Properties, Performance and Applications, Nova Science Publishers,

436

Nguyen Tien Duong and Nguyen Dang Hung

and Gtotal between 4 cracks is small). In 4 cracks, the total train energy release rate is largest for the composite laminate ±θ = 45o.

GI/106 K52 (psi in)

0.016 0.014

Crack Crack Crack Crack

0.012 0.01 0.008

1 2 3 4

0.006 0.004 0.002

±θ

0 -0.002 0

10

20

30

40

50

60

70

80

90

Figure 30. Comparison of GI between 4 cracks.

GII/106 K52 (psi in)

0.006 0.005

Crack Crack Crack Crack

0.004 0.003

1 2 3 4

0.002 0.001

±θ

-0.001

0

10

20

30

40

50

60

70

80

90

Figure 31. Comparison of GII between 4 cracks. 0.04

GIII/106 K52 (psi in)

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0

0.035 Crack Crack Crack Crack

0.03 0.025 0.02

1 2 3 4

0.015 0.01 0.005

±θ

0 0

10

20

30

40

50

60

70

Figure 32. Comparison of GIII between 4 cracks.

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80

90

Gtotal/106 K52 (psi in)

Delamination of Composite Structure under Various Types of Loading…

437

0.06 Crack Crack Crack Crack

0.05 0.04

1 2 3 4

0.03 0.02 0.01

±θ

0 0

10

20

30

40

50

60

70

80

90

Figure 33. Comparison of Gtotal between 4 cracks.

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5. Conclusions The results obtained in section 4.1 show that there are the stress singularities for an interlaminar crack between two dissimilar materials, the anisotropic laminates contain a pair of complex conjugation, δ1,2 = -0,5 ± iγ, and a constant, δ3 = -0,5. This situation is unique and different from that of interface crack between two isotropic or orthotropic materials [16-18] in the sense that δ1, δ2 and δ3 present always simultaneously in the current delamination problem. In the cases of degradation such as ±θ = 0o and 90o, composite laminates become unidirectional. After some numerical tests in section 4.2 for composite laminates subjected to uniform extension, bending, and twisting, it appears that the metis element method constitutes a very effective tool to calculate the stresses, the stress intensity factors and the strain energy release rates for the delamination problem. The metis finite element gives new method to calculate and analyze structures in general and composite structures in particular. This element is very useful for studying the crack or delamination where there is a high stress concentration that demands a fine mesh when it is used by the classical finite element methods (pure or displacement or equilibrium finite element methods) and the hybrid elements (displacement or stress hybrid element methods).

References [1]

[2]

Nguyen Dang Hung, “On the monotony and the convergence of a special class of hybrid finite element: the mongrel element”, Variational methods in mechanics of solids (Ed. by S. Nemat-Nasser), Pergamon (1978). Nguyen Dang Hung, De Saxcé G. and Kang C. H., “The computation of 2-D stress intensity factors using a hibrid mongrel displacement finite elements”, Engineering Fracture Mechanics, 1991, 38, pp 197-205

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[7] [8]

[9]

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[11] [12] [13]

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Nguyen Tien Duong and Nguyen Dang Hung Nguyen Tien Duong, Nguyen Dang Hung, “Regular and singular metis finite element model for delamination in composite laminates”, Finite Elements in Analysis and Design, pp. 650-659, Vol. 42, 2006. Nguyen Tien Duong and Nguyen Dang Hung, “Interlaminar stresses and delamination of composite laminates under extension and bending”, Structural Engineering and Mechanics, An International Journal, Vol. 25, No 6, April 2007, pp. 733-751. Lekhnitskii, S.G., “Theory of elasticity of an anisotropic body”, Holden-day, San/Fransco, California, USA, 1963. Saxcé G. D., “Conception d’un élément fini hybride métis monocouche pour la modélisation du delaminage dans les matériaux composites”, Rapport interne du service Mécanique des Matériaux et des structures, Faculté Polytechnique de Mons, Programme Mobilisateur Multimatériaux, octobre 1992. Wang S. S., “Edge delamination in angle-ply composite laminates”, A.I.A.A. Journal, Vol. 22, 1984, pp 256-264 Fleury C., Nguyen D.H., Guerlement G., Fryns G., “Développement et validation de modèles pour la caractérisation des propriétés mécaniques et la description des mécanismes d’endommagement de structures en matériaux composites”, Rapport SF193, Université de Liège, 1994. De Saxcé G. and Kang C.H., “Application of the hybrid mongrel displacement finite element method to computation of stress intensity factors in anisotropic material”, Volume 41, Issue 1, January 1992, Pages 71-83. Raju I.S., Crews J.H. and Aminpour M.A., “Convergence of strain energy release rate components for edge-delaminated composite laminates”, Eng. Fract. Mech., Vol. 30, No3, pp. 383-396, 1988. Byron R. and Pagano N. J., “Interlaminar stresses in composite laminates under uniform axial extension”, J. Comp. Materials, Vol. 4 (1970), pp 538-548. Wang S. S. and Choi I., “Boundary-layer effects in composite laminate”, Journal of Applied Mechanics, September 1982, Vol. 49 / 549. Wen-Hwa Chen and Tain-Fu Huang, “Stress singularity of edge delamination in angleply and cross-ply laminates”, Journal of Applied Mechanics, September 1997, Vol. 64, Page 525. Qian W., Sun C.T., “Calculation of stress intensity factors for interlaminar cracks in composite laminates”, Compostes Science and Technology, Vol. 57, pp. 637-650, 1997. Kim K.S. and Hong C.S., “Characteristics of free edge delamination in angle-ply laminate”, Fifth International Conference on Composite Materials ICCM-V, July 29 – August 1, 1985, pp. 347-361. Erdogan F., “Stress distributions in bonded dissimilar materials with cracks”, Journal of Applied Mechanics, Vol. 32, 1965, pp. 403-410. Rice J.R., “Plane problems of cracks in dissimilar media”, Journal of Applied Mechanics, Vol. 32, 1965, pp. 418-423. Sih G.C. and Rice J.R., “The bending of plates of dissimilar materials with cracks”, Journal of Applied Mechanics, Vol. 31, 1964, pp. 477-483. Chi-Hang Kang, “Une famille d’éléments hybrides singuliers pour l’étude des plaques fissurées métalliques et composites”, Thèse de doctorat, Université de Technologie de Compiègne, 1991.

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Chapter 12

THE SURFACE INTEGRITY OF COMPOSITE LAMINATES SUBJECTED TO DRILLING A.M. Abrãoa, J. Paulo Davimb,*, J.C. Campos Rubioa and P.E. Fariaa a

Department of Mechanical Engineering, University of Minas Gerais, Av. Antônio Carlos, 6627 – Pampulha, Belo Horizonte MG, CEP: 31.270-901, Brazil b Department of Mechanical Engineering, University of Aveiro, Campus Santiago, 3810-193, Aveiro, Portugal

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Abstract In general, composite laminates are produced near net shaped, thus requiring machining operations to achieve the specified dimensions/tolerances and to allow assembly. Owing to the fact that drilling is the machining operation most frequently applied to composite laminates, this chapter is focused on the influence of the machining parameters on the surface integrity of the finished component. More specifically, delamination, surface finish and the dimensional and geometric deviations induced after drilling fiber-reinforced polymeric laminates are discussed based on the cutting phenomena involved. The findings suggest that tool geometry plays a critical role on the surface integrity of machined composites and that standard drill geometries recommended for metal cutting are not suitable for this grade of materials. In addition to that, feed rate is the most relevant machining parameter affecting the integrity of fiber-reinforced polymeric composites, i.e., the higher the feed rate, the higher the damage induced.

Keywords: fiber-reinforced composites, delamination, surface finish, dimensional and geometric deviations, drilling.

*

E-mail address: [email protected]. (Corresponding author)

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Introduction The manufacture of components with combined properties as distinct as high specific strength, creep resistance, toughness, resistance to corrosion, wear and crack propagation has led to the increasing use of fiber-reinforced polymeric composites over the last decades. Furthermore, due to their anisotropy, different values for stiffness and damping can be achieved (Adams and Maheri, 2003), which are critical to specific applications, such as aerospace structures. Fiber-reinforced composites are widely recognized owing to their superior mechanical properties and advantages for applications in aerospace, defense and transportation sectors (Hocheng and Tsao, 2006). They also possess lower radar and thermal signatures compared with metals and resist degradation caused by chemicals, biological agents and even nuclear radiation (Velayudhama et al., 2005). In general, these components are produced near net shape and drilling, milling and trimming are the principal machining operations required for assembly of these parts. Drilling is probably the machining process most widely applied to composite materials, especially in the automotive and aircraft industries, where structural components present holes for various purposes, such as bolted and riveted joints; therefore, the quality and accuracy of the hole drastically affects the strength of the joint (Persson et al., 1997). El-Sonbaty et al. (2004) report that over 100,000 holes are required in a small engine aircraft, mostly for fasteners. The cutting mechanisms involved when machining composite materials differ considerably from metals (Santhanakrishanan et al., 1993). Fibers’ orientation, for instance, affects the cutting mechanism and, consequently, chip formation, cutting forces and temperature, tool life and integrity of the finished component. When subjected to machining operations, a variety of damages may be induced in the work material. In the particular case of fiber-reinforced composites, delamination is probably the most critical defect owing to the fact that it can severely impair the performance of the component. Additionally, as the cutting edge shears the work material producing the swarf, the imposed plastic strain promotes temperature elevation which, in association with the machining forces, also affects the quality of the machined part. In contrast to metallic alloys, the machining of composite materials offers a number of challenges mainly due to the quite distinctive properties of the matrix and reinforcing materials. Consequently, special attention must be paid to aspects such as tool material and geometry, machining parameters, tool wear, cutting forces and temperature. The principal aim of this chapter is to investigate the quality of composite laminates subjected to the drilling operation with emphasis on delamination, machined surface finish and the dimensional and geometric deviations of the machined holes.

Delamination Delamination is undoubtedly the principal damage induced when machining fiberreinforced polymeric composites and, for this reason, this subject matter has been addressed by a number of research works: Khashaba (2004), Stone and Krishnamurthy (1996), Hocheng and Tsao (2005), Zhang et al. (2001), Ghidossi et al. (2006) Tsao and Hocheng (2007), Campos Rubio et al. (2008a).

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Two mechanisms of delamination, known as peel-up at the drill entrance and push-out at the drill exit, are reported by El-Sonbaty et al. (2004). According to these authors, the key for solving the problem lies in reducing the thrust force during the machining operation. In contrast, Lachaud et al. (2001) detail the damages induced when drilling fiber-reinforced polymeric composites into four categories: delamination at drill entry, geometric defects, temperature damages and delamination at drill exit. The delamination at drill entry is not always present. The tool geometry related damages are associated to the angle between the fibers’ orientation and the cutting edge. This occurs due to the fact that before shearing takes place, the fibers are subject to alternate torsion and compression, resulting in an elliptical hole. In general, temperature-related damages appear as a result of friction between the drill and the wall of the hole. Delamination at the drill exit probably happens owing to the fact that under this circumstance not all fibers are cut, thus resulting in a normal stress which opens the matrix/fiber interface. In addition to reducing the structural integrity of the material, delamination leads to poor assembly tolerances and has the potential for deteriorating the component’s long-term performance. Therefore, delamination can often be a limiting factor for the usage of composite materials in structural components. According to Khashaba (2004), delamination is responsible for the rejection of approximately 60% of the components produced in the aircraft industry. The economic impact of drilling-related damages is significant for this sector, especially when considering the added value associated with the component when it reaches the assembly stage (Stone and Krishnamurthy, 1996). As far as the cutting parameters are concerned, feed rate is recognized as the most significant factor affecting delamination, followed by cutting speed and depth of cut (drill radius). A number of approaches are available to assess the damage and the influence of the cutting parameters and drill geometry on delamination. Due to its simplicity, visual inspection using an optical microscope associated to software for image analyzing is the most employed technique; however, this method cannot evaluate internal damages. In order to overcome this difficulty, alternative methods have been proposed: Hocheng and Tsao (2006) used ultrasonic scanning to evaluate delamination and Seif et al. (2007) proposed a method to measure delamination based on the shadow moiré laser technique. Gao and Kim (1999) present a comparative study on destructive and non-destructive evaluation techniques for characterizing the damage in carbon fiber-reinforced composites. Caprino and Tagliaferri (1995) measured the damage extent by cross sectioning and polishing laminate samples before observation under the optical microscope. The results indicated that when drilling at high feed rate values step-like delamination, intralaminar cracks and high density microfailure zones are the typical damages observed, whereas delamination at the intersection of the primary and secondary cutting edges is the most frequent type of damage observed at low feed rates. Generally, the extent of the damage is measured using the delamination factor, calculated as the ratio of the maximum diameter (Dmax) of the delamination zone to the drill diameter (D0). Alternatively, the ratio of the delaminated area to the hole area has also been used. The influence of the cutting parameters and tool material on delamination after drilling woven glass fiber-reinforced epoxy resin laminates can be qualitatively assessed in Figure 1, where it can be seen that delamination induced by the high speed steel tool was considerably larger compared with the tungsten carbide tool. Attention must be paid to the fact that these photographs were taken after 1,000 holes were produced by each drill. These photographs

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suggest that when drilling using the worn high speed steel drill, the lower the feed rate, the larger the delaminated area. The reason for that resides in the fact that using low feed rate values, the contact length between the reinforcing fibers and the cutting edge increases, thus accelerating the tool wear rate and increasing the extension of the damage area. In the case of the tungsten carbide drill, the influence of cutting speed and feed rate on delamination was negligible. Figure 2 presents the values for delamination areas obtained when drilling glass fiberreinforced epoxy resin laminates using worn high speed steel drills. It can be noticed that an increase in feed rate and a reduction in cutting speed leads to smaller delamination areas owing to the reason previously explained, albeit feed rate promotes the opposite effect when a fresh tool is employed. Davim et al. (2007) assert that the ratio of the maximum diameter (Dmax) of the delamination zone to the drill diameter (D0), see Equation (1), can satisfactory represent delamination when a regular damage pattern is obtained.

Fd =

Dmax D0

(1)

Nevertheless, when delamination presents an irregular form, such as that observed when drilling carbon fiber-reinforced composites, this parameter is not suitable to represent the damage. Therefore, an adjusted delamination factor (Fda) is proposed and calculated using Equation (2), where the first part represents the size of the crack contribution (conventional delamination factor, Fd) and the second part represents the damage area contribution.

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Fda = α

Dmax A + β max D0 A0

(2)

Figure 1. Delamination observed after drilling 1000 holes under various cutting conditions using high speed steel and tungsten carbide drills (Ø5 mm). Composite Laminates: Properties, Performance and Applications : Properties, Performance and Applications, Nova Science Publishers,

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10 vc=55 m/min vc=86 m/min

Delaminated area (mm2)

8

6

4

2

0 0,04

0,2 Feed rate (mm/rev)

Figure 2. Delamination areas induced by worn high speed steel drills under various cutting conditions.

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Where Amax is the area related to the maximum diameter of the delamination zone (Dmax) and A0 is the area of the nominal hole (D0). β is considered as the ratio of the damage area (Ad) to the area corresponding to Dmax (Amax) minus the nominal hole area (A0). The parameter α is the complement of β. Therefore, if the trend is a delamination area equal to the maximum diameter (Dmax) area of the delamination zone, the adjusted delamination factor (Fda) presents a value equal to the square of the conventional delamination factor (uniform behaviour). However, if the delamination area is minimal, the adjusted delamination factor presents a value tending to the conventional delamination factor. Drilling tests using tungsten carbide helical drills with 5mm diameter were performed on carbon fiber-reinforced plastic laminates and the damages around the holes were analyzed and measured by means of an image acquisition system connected to a 600 dpi resolution scanner. The influence of cutting speed and feed rate on both delamination factors can be seen in Figure 3. Not surprisingly, higher values are obtained for the adjusted factor values than the conventional parameter and the difference between both tend to increase slightly as cutting speed and feed rate are elevated. In addition to that, it seems that there is a critical feed rate of approximately 0.30 mm/rev above which severe delamination is observed. The advances in high speed machining with regard to both machine tools and tool materials have allowed the widespread use of this technology beyond the mold and die industry. Kassapoglou (1999) assert that an appreciable reduction in costs of helicopter fuselage can be achieved through high speed cutting. Work by Lin and Chen (1996) on high speed drilling carbon fiber-reinforced composite laminates suggests that the reduction in cutting forces due to higher cutting speeds may become delamination more controllable. Nevertheless, tool wear is the major limiting factor when high speed drilling carbon fiberreinforced composites.

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Delamination factor

2.25

2

1.75 Fd (f=0,25 mm/rot) Fd (f=0,30 mm/rot) Fd (f=0,35 mm/rot) Fda (f=0,25 mm/rot) Fda (f=0,30 mm/rot) Fda (f=0,35 mm/rot)

1.5

1.25

1 50

60

70

Cutting speed (m/min)

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Figure 3. Effect of cutting speed and feed rate on conventional (Fd) and adjusted (Fda) delamination factors.

Campos Rubio et al. (2008b) investigated high speed drilling of glass fiber reinforce epoxy composite material aiming minimal damage. High speed drilling experiments were conducted on a machining centre equipped with an aerostatic headstock with 40,000 rpm maximum rotational speed. Three tungsten carbide drills (grade K20) with 5mm diameter and 25o helical angle were used as cutting tools: two twist drills with 85o and 115o point angles and a “Brad & Spur” special geometry drill (candle stick drill), see Figure 4.

Campos Rubio et al. 2008b.

Figure 4. Tool geometries tested when high speed drilling composite laminates.

The results indicated that the delamination factor increased with feed speed but decreased as the spindle speed was elevated. As far as the tool geometries are concerned, the twist drill with 115o point angle promoted higher delamination. The drill with a point angle of 85o gave best results at low and intermediate spindle speeds and when high speed drilling at 40,000 rpm the Brad & Spur drill was responsible for the lowest delamination factors.

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Surface Finish Standard high speed steel and tungsten carbide drills are generally employed to produce holes in composite laminates, in spite of the fact that a considerable number of authors have asserted that conventional twist drills are unsuitable to machine these materials, the principal drawback being related to the drill geometry. The inadequate selection of the tool geometry leads due to excessive friction between the drill and the machined hole, promoting higher temperatures and increasing tool wear rates, drilling force and torque. As a consequence, unacceptable quality and surface damage are obtained. Ogawa et al. (1997) reported that feed rate is the most significant factor affecting the surface roughness of holes (an increase in feed rate from 5μm/rev to 50µm/rev promoted the elevation of the parameter Rmax from 10μm to 30µm) and that an increase in thrust force results in inferior surface finish on the hole wall. In order to reduce the drilling force and, consequently, to improve the surface roughness, the above mentioned authors carried out machining trials using drills with 1 mm diameter on holes previously drilled with 0.2, 0.4 and 0.8 mm diameter. The results indicated that surface roughness can be drastically reduced by pre-drilling, irrespectively of the diameter of the drill employed in the first operation. The influence of the drilling parameters (cutting speed and feed rate) on surface roughness when machining glass fiber-reinforced epoxy laminates was investigated by Davim et al. (2004a). The findings indicated that surface roughness decreases as cutting speed is elevated and feed rate is reduced. Drilling tests on carbon fiber-reinforced epoxy laminates using high speed steel drills with negative point angle (candle stick) were performed by Tsao and Hocheng (2008), who found that feed rate and spindle speed statistically affect the surface roughness of the machined wall, whereas the influence of the drill diameter was found to be negligible. El-Sonbaty et al. (2004) also investigate the effect of spindle speed, feed rate and drill diameter on the surface finish of holes produced on glass fiber-reinforced epoxy composites with four different values for fiber volume fraction. The findings indicated that better hole quality was obtained by increasing spindle speed and fiber volume fraction. Moreover, increasing drill diameter up to 10 mm resulted in the elevation of the surface roughness. A further increase in drill diameter led to the reduction of the surface roughness. According to the authors, the reduction in surface roughness may be due to the increase in cutting temperature, which may reach the glass-rubber transition temperature, causing a smoother surface finish on the hole wall. Davim et al. (2004b) studied the surface roughness obtained after drilling two distinct polyester matrix reinforced with glass fibers with tungsten carbide tools. The results indicated that spindle speed possesses the largest percentage of contribution to surface roughness for both composite materials, followed by feed rate and that lowest surface roughness values are obtained when drilling at highest cutting speed and lowest feed rate. In order to assess the influence of drill geometry on surface roughness, these authors compared the performance of a helical drill (118° point angle) with negative point angle drill (candle stick drill) when cutting glass fiber-reinforced polyester composite and found that the latter produced better surface finish. Ogawa et al. (1997) investigated the effect of the cutting edge position angle on the hole surface roughness when drilling plain woven glass fiber-reinforced epoxy laminates using

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drills with 1 mm diameter and found maximum surface roughness values at position angles of 30° and 120° due to differences in the thrust force for fibers and resin. Furthermore, feed rate was found to be more influential on surface roughness than cutting speed. Information concerned with drilling of aramid fiber-reinforced polymeric composites is not easily available in the published literature. Bhattacharyya and Horrigan (1998) report drilling tests of aramid fiber-reinforced epoxy composite using high speed steel tools with standard and modified (candle stick) geometries under dry cutting and using liquid nitrogen as cutting fluid. The use of liquid nitrogen promoted a drastic reduction in the surface roughness values, which could be further reduced by using negative drill point and large helix angles. The influence of the machining parameters on the surface texture obtained after grinding multi and unidirectional carbon fiber-reinforced laminates with an aluminum oxide wheel was investigated by Hu and Zhang (2003). The results suggest the both materials present the same behavior with regard to the grinding depth, however, the mutual constraints between the plies result in better surface finish on the multidirectional composite. Furthermore, the local surface roughness of the multidirectional laminate is drastically affected by the fiber orientation angle, with best roughness results for 90° and 45° ply bands and worst results for the plies with an orientation of -45°. Due to the abrasive nature of reinforcing materials employed in composite laminates, the use of non-traditional machining processes has increased in last years. Abrasive water jet is among the principal non-traditional machining operations indicated to cut polymeric composite laminates due to the low mechanical and thermal stresses imposed. Azmir and Ahsan (2008) employed this method to trim woven and chopped glass fiber-reinforced epoxy laminates and found that abrasive material, hydraulic pressure and traverse speed are the most significant factors affecting the machined surface roughness. Optimum machining performance was obtained using aluminum oxide as abrasive, highest pressure and lowest traverse speed. As far as the composite material is concerned, best surface roughness was obtained when cutting the thinnest (5 mm) woven reinforced laminate with a fiber volume fraction of 50%. Hocheng and Hsu (1995) studied the influence of the ultrasonic parameters when drilling carbon fiber-reinforced epoxy and PEEK composites and found that surface roughness increased with abrasive grain size and concentration and input current of the ultrasonic oscillations. Figure 5 gives an overview of typical surface roughness values obtained after cutting fiber-reinforced composite laminates using conventional and non-conventional machining operations. Owing to variations on the work and tool materials and on the machining parameters, straightforward relationships between machining operations and surface roughness cannot be easily drawn. In addition to that, variations in feed rate and tool geometry are responsible for the wide roughness range observed for conventional drilling. . Nevertheless, Figure 5 suggests that polymeric laminates reinforced with carbon fibers present lower roughness values compared with composites reinforced with glass and aramid fibers. In addition to drilling, only ultrasonic machining was used to produce holes. In the former, feed rate is the principal factor affecting surface roughness, whereas in the latter the size of the abrasive particles is critical to the surface finish. The influence of the abrasive grain size is overwhelming in the case of the operations used to produce flat surfaces (grinding and abrasive water jet machining).

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Drilling (GFRE) Drilling (GFRP) Drilling (CFRE) Drilling (AFRE) Grinding (C prepeg) AWJM (GFRE) USM (CFRE) 0

1

2

3

4

5

6

7

8

Average surface roughness (um)

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GFRE: glass fiber-reinforced epoxy resinAWJM: abrasive water jet machining GFRP: glass fiber-reinforced polyester resinUSM: ultrasonic machining CFRE: carbon fiber-reinforced epoxy resin AFRE: aramid fiber-reinforced epoxy resin Azmir and Ahsan, 2008; Bhattacharyya and Horrigan, 1998; Davim et al., 2004b; El-Sonbaty et al., 2004; Hocheng and Hsu, 1995; Hu and Zhang, 2003; Tsao and Hocheng, 2008.

Figure 5. Typical surface roughness values (Ra) obtained after drilling fiber-reinforced composite laminates using traditional and non-traditional machining operations.

Dimensional and Geometric Deviations In contrast to the analysis of delamination and surface finish of laminate composites, which have been investigated in depth by a number of authors, dimensional and geometric deviations produced after drilling composites have not received as much attention, in spite of their relevance for the performance of the component. According to Everson and Cheraghi (1999) the tolerances required for an aircraft assembly may range from 0.25 mm (for riveting) to ± 13 µm (straight shank fasteners). Brinksmeier and Janssen (2002) add that diameter tolerances of 30 µm or less are required for bolt holes produced in an aeroplane wing and tail composite structures. The abrasive nature of reinforcing materials leads to accelerated tool wear rates, thus making rather difficult to maintain tight tolerances when producing an elevated number of holes. After drilling carbon fiber-reinforced epoxy laminates, these authors reported that tolerances within the range of 80 µm were obtained using a standard tungsten carbide steel drill and dry cutting. However, using a step drill and minimal quantity lubrication resulted in considerably

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lower dimensional deviations. Surprisingly, drilling with a diamond coated step drill generated holes with largest dimensional deviation. The use of drills especially designed is probably the preferred alternative in order to produce holes with tight dimensional and geometric tolerances when drilling fiber-reinforced composite laminates (Abrão et al., 2007). Alterations on the helix and point angles, changes on the chisel and lip length together with the use of step drills, trepanning tools and drills with four flutes are typically reported. Arul et al. (2007) reported that the reduction in the hole diameter when drilling glass fiber-reinforced epoxy laminates with high speed steel drills should be attributed, in addition to tool wear, to the stressing of the composite followed by relaxation. Moreover, the higher the feed rate, the larger the reduction in the hole diameter. A straightforward relationship between cutting speed and hole diameter, however, was not obtained. Finally, the authors proposed the use of acoustic emission for monitoring the dimensional tolerances of the holes produced, additionally to monitoring tool wear and thrust force. Kim and Ramulu (2004) investigated the dimensional and geometric quality of holes drilled on graphite reinforced bismaleimide composite stacked on a titanium alloy. The findings indicated that when drilling with a standard high speed steel tool, undersized holes are generated, whereas oversized holes are produced using a tungsten carbide drill. Furthermore, the amount of oversize recorded when drilling with the carbide tool increased with spindle speed and feed rate. As far as the geometric deviations are concerned, higher circularity deviation values were obtained using the high speed steel drill compared with the carbide tool. The former tool material was more sensitive to the machining parameters and circularity increased with spindle speed and decreased as feed rate was elevated. Similarly to the surface roughness results obtained when ultrasonically drilling carbon fiber-reinforced epoxy composite, Hocheng and Hsu (1995) noticed that the hole oversize increases with abrasive grain size and input current. Feed rate and abrasive concentration did not affect hole dimension significantly. The dimensional deviations observed on holes drilled in aramid fiber-reinforced epoxy resin was studied by Bhattacharyya and Horrigan (1998), who found that closer tolerances are obtained when using drills with high helix angle and negative point angle, in addition to liquid nitrogen as cutting fluid. The recorded dimensional errors ranged from approximately 20 µm to 110 µm. Drilling tests on glass fiber-reinforced epoxy resin composite laminates were conducted in order to investigate the influence of tool material and machining conditions on dimensional and geometric deviations. Two cutting tool grades were tested: high speed steel and tungsten carbide (drill diameter of 5 mm). Figure 6 indicates that similar results were obtained for the first hole, however, the tungsten carbide drill outperformed the high speed steel after 1,000 holes. The reason for that resides on the capacity of the tungsten carbide drill to withstand abrasion against the glass fibers. Furthermore, the increase in the diameter of the hole 1,000 produced by the carbide drill is probably due to the run out of the spindle. Similar results were obtained when considering the geometric deviation of the holes: the tungsten carbide drill induced a roundness deviation inferior to 20 μm after producing 24,000 holes. Cross sections of the thousandth holes produced in glass fiber-reinforced epoxy laminates using tungsten carbide and high speed steel drills are given in Figure 7, where it can be seen that the carbide drill produced a sharp wall, contrasting with the poor quality of the hole generated by the steel tool.

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5.1 High speed steel 5.05

Tungsten carbide

Hole diameter (mm)

5 4.95 4.9 4.85 4.8 4.75 4.7 1

1000

10000 Hole number

20000

24000

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Figure 6. Effect of tool wear on the diameter of holes produced on glass fiber-reinforced laminates using high speed steel and cemented carbide drills.

Figure 7. Cross sections of the thousandth holes on composite laminates produced using tungsten carbide (left) and high speed steel (right) drills with ∅5 mm.

Conclusion The increasing use of polymeric composites in a variety of industrial sectors allied to the distinctive behavior of this range of materials when subjected to cutting in comparison with metals has led to intense research on this subject over the last decade, albeit further work is still necessary in order to allow the thorough understanding of the phenomena involved when cutting polymeric reinforced composite laminates. Delamination is the principal damage induced by drilling of reinforced polymeric laminates and is drastically affected by feed rate and tool geometry. Tool wear can also dramatically affect delamination; therefore, the selection of the most adequate tool grade for a

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given operation and work material is critical in order to minimize the damage induced in the finished component. Among the techniques used to assess delamination, the delamination factor measured by employing optical microscopy is the most widely used method owing to its simplicity, in spite of the fact that it cannot identify internal damages. The quality of the machined surfaces, evaluated in terms of microgeometric deviations (surface roughness) and dimensional and geometric deviations relies on the development of cutting tools with geometry especially designed for this purpose. In addition to that, predrilling and the proper selection of the cutting parameters, especially feed rate, is crucial when tighter tolerances are required.

Acknowledgements The authors would like to express their gratitude to CAPES (Brazil) and GRICES (Portugal) for supporting this research project.

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References [1] Abrão, A. M.; Faria, P. E.; Campos Rubio, J. C.; Reis, P.; Davim, J. P. J. Mater. Process. Technol. 2007, 186, 1–7. [2] Adams, R. D.; Maheri, M. R. J. Alloys Compounds 2003, 355, 126–130. [3] Arul, S.; Vijayaraghavan, L.; Malhotra, S. K. J. Mater. Process. Technol. 2007, 185, 184–190. [4] Azmir, M. A.; Ahsan, A. K. J. Mater. Process. Technol. 2008, 198, 122-128. [5] Bhattacharyya, D.; Horrigan, D. P. W. Compos. Sci. Technol. 1998, 58, 267-283. [6] Brinksmeier, E.; Janssen, R. CIRP Annals – Manufact. Technol. 2002, 51, 87-90. [7] Campos Rubio, J. C.; Abrão, A.M.; Faria, P. E., Correia, A. E.; Davim, J. P. J. Compos. Mater. 2008a, 42, 1523-1532. [8] Campos Rubio, J.; Abrão, A. M.; Faria, P. E.; Correia, A. E.; Davim, J. P. Int. J. Mach. Tools Manufact., 2008b, 48, 715–720. [9] Caprino, G.; Tagliaferri, V. Int. J. Mach. Tools Manufact. 1995, 35, 817-829. [10] Davim, J. P.; Reis, P.; António, C. C. J. Mater. Process. Technol. 2004a, 155–156, 1828–1833. [11] Davim, J. P.; Reis, P.; António, C. C. Compos. Sci. Technol. 2004b, 64, 289–297. [12] Davim, J.P.; Campos Rubio J. C.; Abrão, A. Compos. Sci. Technol. 2007, 67, 19391945. [13] El-Sonbaty, I.; Khashaba, U. A.; Machaly, T. Compos. Struct. 2004, 63, 329–338. [14] Everson, C. E.; Cheraghi, S. H. Int. J. Mach. Tools Manufact. 1999, 39, 371–387. [15] Gao, S. L.; Kim, J. K. Compos. Sci. Technol. 1999, 59, 345-354. [16] Ghidossi, P.; El Mansori, M.; Pierron, F. Compos. Sci. Technol. 2006, 66, 1857–1872. [17] Hocheng, H.; Hsu, C. C. J. Mater. Process. Technol. 1995, 48, 255-266. [18] Hocheng H.; Tsao, C. C. J. Mater. Process. Technol. 2005, 167, 251–264. [19] Hocheng, H.; Tsao, C. C. Int. J. Mach. Tools Manufact., 2006, 46, 1403–1416. [20] Hu, N. S.; Zhang, L. C. J. Mater. Process. Technol. 2003, 140, 152–156.

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The Surface Integrity of Composite Laminates Subjected to Drilling [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33]

Kassapoglou, C. Compos. Part A, 1999, 30, 895–904. Khashaba, U. A. Compos. Struct. 2004, 63, 213-327. Kim, D.; Ramulu, M. Compos. Struct. 2004, 63, 101–114. Lachaud, F.; Piquet, R.; Collombet, F.; Surcin, L. Compos. Struct. 2001, 52, 511–516. Lin, S. C.; Chen, I. K. Wear, 1996, 194, 156–162. Ogawa, K.; Aoyama, E.; Inoue, H.; Hirogaki, T.; Nobet, H.; Kitahara, Y.; Katayama, T.; Gunjima, M. Compos. Struct. 1997, 38, 343-350. Persson, E.; Eriksson, I.; Zackrisson, L. Compos. Part A, 1997, 28, 141–151. Santhanakrishnan, G.; Krishanamuthy, R.; Malhotra, S. K. Proc. Machining of Compos. Mater. Symposium 1993, 139-148. Seif, M. A., Khashaba, U. A., Rojas-Oviedo, R. Compos. Struct. 2007, 79, 113–118. Stone, R.; Krishnamurthy, K. Int. J. Mach. Tools Manufact. 1996, 36, 985–1003. Tsao, C. C.; Hocheng. H. Int. J. Mech. Sci. 2007, 49, 983–988. Tsao, C. C.; Hocheng, H. J. Mater. Process. Technol. 2008, 203: 342–348. Velayudhama, A.; Krishnamurthya, R.; Soundarapandian, T. Int. J. Mach. Tools Manufact. 2005, 45, 399–406. Zhang, H.; Chen, W.; Chen, D.; Zhang, L. Precis. Mach. Adv. Mater. 2001, 196, 43–52.

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Chapter 13

FAILURE PROCESS OF CARBON FIBER COMPOSITES Alexander Tesar* Institute of Construction and Architecture, Slovak Academy of Sciences, Dubravska cesta 9, 842 20 Bratislava, Slovak Republic

Abstract

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Some research results of failure behaviour of carbon fiber composites are presented. The solution of material instability on the basis of fiber kinking theory is adopted for the treatment of the failure process. The micromechanical modeling adopting the FETM-approach is used for numerical analysis of the problem. Some numerical and experimental results with actual application are submitted in order to demonstrate the efficiency of the approaches suggested.

Key words: FETM- approach, fiber kinking, carbon fiber composite, failure process, material instability, transformaton strain, ultimate behaviour.

1. Introduction A long-standing difficulty in designing of carbon fiber composites is the formulation of a consistent theory that describes their failure behaviour under nonuniform stress fields. As problem appears there the discrepancy between the four-point bend and simple tensile test data. The bend specimens fail at higher strain compared with the tensile specimens. When the bend and tensile data are analysed using classical linear elastic theory the bend stress at any strain prior to failure of a tensile test specimen is 20 - 35 % higher than corresponding uniaxial tensile stress. Such strength discrepancy remains unresolved even when corrections are made for the nonlinearity of the stress-strain curves. By attempts to explain such discrepancy only very limited success has been achieved with failure theories, including the Weibull`s statistical model and the fracture mechanics approach. Similar experiences also appeared by the application of linear fracture mechanics or couple-stress theory.

*

E-mail address: [email protected]

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The carbon fiber composites adopted in present structural engineering are made of typical components listed as: 1. carbon fibers, with strength and elasticity moduli in scope 2. 5.7 GPa and 300 – 700 GPa, respectively, 3. aramide fibers, with strength 3.5 GPa and elasticity moduli in scope 80 – 185 GPa. The carbon fiber composites consist of micromechanical fibers and surface resin skin. The calculation on the micromechanical level takes into account the behaviour of single fiber in interaction with another fibers and surface skin. In present time is made the development of new types of fiber composites equipped with surface skin on the basis of advanced ceramics or metals having high strength and load-bearing capacity as well as increased temperature resistance and fatigue reliability. The material instability appearing in the failure process of carbon fiber composites is treated below adopting the fiber kinking theory and using the analysis on the micromechanical level. In this paper the following is submitted: 1. fiber kinking approach for the failure analysis of carbon fiber composites, 2. mathematical formulation of governing equations for modeling and numerical treatment of the problem, 3. numerical and experimental assessment with actual structural application.

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2. Analysis In carbon fiber composites the transformation strains and other field quantities appearing in elastic moduli are periodic functions of space, time and temperature. The periodicity is exploited in an effort to obtain accurate estimates for the transformation strains used to approximate mechanical properties of such composites. The Washizu´s variational principle is adopted in order to include initial stress and strain components into analysis. The stress in the carbon fiber microelement at the beginning of time and temperature increments studied is considered as initial stress with thermal strain increments. The variational principle under consideration is then written in the terms of time rate quantities given by I = {∫V [Sij εij + 0.5 Wij uki ukj – (εoij + 0.5 ε′ij) Sij] dV - ∫A1 ri ui dA1 - ∫A2 si (ui – wi) dA2}(dt)2 + {∫V Wij εij dV - ∫A1 ri ui dA1 - ∫A2 pi (ui – wi) dA2}dt

(1)

where Wij and Sij are the Piola-Kirchhoff stress tensors for initial stress and strain rate states, respectively, pi and si are the Lagrangian surface traction and its time rate quantity, respectively, ri and ri(.) are prescribed on surface area A1 and wi on area A2 and V is the volume bounded by the surface area A=A1+A2. The total strain rate εij is composed of the initial strain rate εoij and ε´ij, corresponding to the instantaneous stress rate Sij . To evaluate the thermal strain rate the thermal expansion coefficient at temperature T is α(T) and

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at temperature T+dT is α(T+dT). By expanding α(T+dT) into Taylor series the average thermal strain rate is obtained. The governing equation is given by μ η(wt) + (λ + μ) grad(div wt) + f = ρ ∂2wt/∂t2

(2)

where λ and μ are Lame´s constants, mass density is ρ, corresponding Laplace operator is η, body force vector is f and vector of displacements is wt (Tesar and Svolik 1993). In the terms of derivatives of displacements wt the governing equation is given by c2 wt + (c12 - c22) wt + fi/ρ = at

(3)

with propagation velocities for dilatational displacements c1 = √[(λ + 2μ)/ρ]

(4)

c2 = √(μ/ρ)

(5)

and shear displacements

Strain and stress components are given by εij = (wi,j + wj,i)/2

(6)

σij = λ εkk δij + 2 μ εij , i,j = 1, 2, 3,

(7)

and

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with Kronecker delta function δij .

3. Failure Process The kinking of microscopic fiber bundles focuses the attention on this type of material instability when dealing with failure process of carbon fiber composites. Material buckling of carbon fibers under static or dynamic compression and flexure is assumed as possible mode of failure as well as the item influencing the above bend/tensile discrepancy. The local shortwave imperfections as well as general long-wave imperfections along the length of fibers (Budiansky, B. 1983) are to be taken into account. Elastic plane strain inextensional deformation in compressed part of the carbon fiber composite gives displacements w(x,y) in normal direction to the fibers, governed by (1 - σ/G) ∂2w/∂x2 + ET/G ∂2w/∂y2 = (σ/G) ∂2wo/∂x2

(8)

where wo(x,y) is an initial displacement pattern. Taking into account the half-plane y ≥ 0 the effect of short-wave imperfection is given by

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Alexander Tesar wo = δD(x) δD(y)

(9)

where δD is the symbol for the Dirac delta function. A long-wave imperfection concentrated at the edge y = 0 is given by wo = - ⎜x ⎜ δ(y)

(10)

The deviations from ideal fiber alignment due to fiber spacing irregularities induce the patterns of angular misalignment (elastic distortion) that arrange themselves into inclined domains. Such rotations induce the plastic kinking into similarly inclined kink bands. The failure follows there after the start of plastic deformation with kinking failure stress σs. The consequent correlations between σs and kink angle β for long wave imperfections (see Figure 1) are given by tan β = ± √ [(1 - σs/G)/(ET/G)]

(11)

and for short wave imperfections by

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tan β = ± (√(2) – 1) √ [(1-σs/G)/(ET/G)]

(12)

Figure 1. Kinking of fiber bundles.

To find rational estimates for the kink width W (see Figure 1) the single fiber dimensions are to be taken into account. The carbon fiber diameter d is the meaningful size in the problem. The kink width is clearly delineated by bending in the fibers so that the local fiber bending resistance must be considered explicitly. The fibers are assumed to undergo inextensional bending until they break. At the same time, the elastic strain in the matrix can be neglected with respect to its plastic strains, i.e., the matrix is assumed to be rigid-plastic. Smoothing of the fibers then states a simple couple-stress formulation which gives no fiber rotations outside a band. The rotation φ(x + y tan β) within the band is governed by d2φ/dX2 + σ/τr = 1

(13)

with τr = (τy2 + σTy2 tan2 β] and

X = (4x/d) [τr /(c E)]1/2

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(14)

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and with c as volume concentration of carbon fibers. By solving Eq. (13) with the condition that the rotations vanish at the ends of the kinking domain, the evolution of the plastic kinking is studied. The carbon fiber breaking is presumed to occur when a critical tensible strain εF is reached in combined compression and flexural action at the points of the maximum curvature. For the case of perfectly brittle fibers then pays the formula for the kink width given by W/d = π/4 [(2 τry)/(c E)]-1/3

(15)

The above analysis holds for perfectly straight fibers. However, the studies that take initial misalignment into account have shown that W can substantially differ from the perfect fiber case. An idea appears there to adopt the micromechanical simulation of single fibers by micro strings and to study the problem. The resistance of such single string fibers contains elastic, plastic, visco-elastic and visco-plastic parameters possibly appearing (see Simo, J.C. 1990). One approach for such model is submitted below. For physical interpretation of above definitions internal and left-hand external displacements of one string microelement are denoted by wa and wb. The internal displacement vector wi is eliminated beforehand, giving the stiffness matrix by K(ω) = ⎡Kaa Kab⎤ ⎣Kba Kbb⎦

(16)

w = ⎡wa ⎤ ⎣ wb⎦

(17)

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and the deformation vector by

Corresponding force vectors are given by na = Kaa wa + Kab wb

(18)

nb = -Kba wa - Kbb wb

(19)

The state vector v is defined as combination of displacements and internal forces given by v = [w, n]T

(20)

The state vector at boundaries a and b is given by vb = S va

(21)

with corresponding transfer matrix S. It holds S = ⎡Saa Sab⎤ ⎣Sba Sbb⎦

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(22)

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with Saa = -Kab-1 Kaa , Sab = Kab-1 , Sba = -Kba + Kbb Kab-1 Kaa , Sbb = -Kbb Kab-1

(23)

The damping parameters are partially contained in the isothermal bulk modulus KI given by KI = Ko(1 + i ηB)

(24)

and appearing in the stiffness terms of the transfer matrix S. In Eq. (24) ηB is the damping factor. The heterogeneity is an essential ingredient of the thermoelastic dissipation in the body of the carbon fiber composites studied. The adiabatic bulk modulus KA is related to the isothermal modulus KI by KA = KI [ 1 + KI γ2 T/CV ]

(25)

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where γ is the volumetric thermal expansion coefficient, CV is the heat capacity at the constant volume and T is the absolute temperature. The increased stiffness under adiabatic conditions is due to the fact that compression produces heating and therefore more pressure is needed to produce a given volumetric strain compared with isothermal conditions. In carbon fiber composite the adiabatic heating induce nonuniform temperatures under oscillatory loading and the heat will then flow among the constituents. The consequent phase difference between stress and strain leads to energy dissipation in each stress cycle. The velocity v versus stress σ is given by v = ρ/G + σ/η

(26)

with ρ as time differentiation of σ. Considering the substitution υ = G/η and taking into account initial conditions v(0) = 0 and σ(0) = 0 for t = 0, the average stress is given by σa = G ∫ e-υ(t) a(t) dt

(27)

with acceleration a(t). With the spectral function N(υ), specifying the density of υ and G in the microelement carbon fiber spring studied, the resulting stress is given by σR = Go v + ∫ N(υ)dυ ∫ e-υ t a(t) dt

(28)

Gs = ∫ N(υ) dυ

(29)

Ψ(t) = Gs(1 – Gs-1 ∫ N(υ) e-υt dυ)/(Go+Gs)

(30)

Adopting the value

as well as the function of relaxation

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the resulting stress is modified into σR(t) = (Go+Gs) [v(t) - ∫ a(t) Ψ(t) dt]

(31)

For the approach of N(υ) the Dirac function given by δ(x) = 0 for x ≠ 0 and by ∫ δ(x) dx = 1 is adopted. Because for each function f(x) there holds ∫ f(x) δ(x – xo) dx = f(xo)

(32)

the approach required may be done by replacing the function υ e-υt by substitution t-2 δ(υ - t-1). The approximation for N(υ) is then given by N(υ) ≈ υ-1 Ψ(1/υ)

(33)

Adopting the above analysis the complex modulus of elasticity appearing as the function of the frequency ω is given by E = Go + iω ∫ N(υ)/(υ + iω) dυ

(34)

The division into real and imaginary components then yields E1 = Go + ∫ ω2/(ω2 + υ2) N(υ) dυ

(35)

E2 = ∫ ωυ/(ϖ2 + υ2) N(υ) dυ

(36)

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and

with Go valid for ω = 0 . Instead of infinite number of microelements the above approach allows the modeling of the carbon fiber material by single elements with modulus G(ω) and stiffness η(ω), both appearing as functions of the frequency ω. The complex modulus of elasticity is then given by E = G(ω) + η(ω) ωI (37) and is implemented into the stiffness terms of corresponding complex transfer matrix S. The calculation run of the FETM-wave approach (Finite Element versus Transfer Matrix Methods), using the above matrix S, is adopted with updated variability of micromechanical mesh size in space, time and temperature. The details of such approach are summed up, for example, in (Tesar and Fillo 1988) or in (Tesar and Svolik 1993). Adopting the above approach the ultimate analysis of carbon fiber elements is given by 1. Micromechanical modeling of the material and macromechanical modeling of the structural configuration in space, time and temperature. 2. Updated calculation of stress and strain in space, time and temperature. 3. Automatic comparison with ultimate strength of the elements adopted.

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Alexander Tesar 4. Initiation of cracks in micromechanical elements trespassing the ultimate strength. 5. Updated calculation of the crack distribution in space, time and temperature until total failure of structure.

The regime of the crack initiation and distribution is rather complex. One or several cracks develop and propagate slowly along the critical regions of the structure studied. In the case of the shear the cracks turn inside of the body in a direction being quasi-perpendicular to the tension.

4. Numerical and Experimental Verification Numerical and experimental assessment of the standard carbon fiber specimen as shown in Figure 2 was made first of all. The specimen was subjected to tension until the failure and the results were compared with failure strength of the same specimen made of steel.

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Figure 2. Standard carbon fiber specimen with thickness 1.4 mm.

Figure 3. Experimental facility for tensile testing of standard carbon fiber specimen studied. Composite Laminates: Properties, Performance and Applications : Properties, Performance and Applications, Nova Science Publishers,

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Figure 4. Explosive failure of the standard carbon fiber specimen tested.

Figure 5. Stress-stran curves of standard specimens studied.

The experimental facility adopted is shown in Figure 3. The carbon fiber failure is shown in Figure 4. The results obtained have stated that carbon fiber specimen has almost eight times higher tensile strength compared with the steel equivalent. The comparison of numerical and experimental results obtained shows good correspondency of both approaches (see Figure 5).

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5. Application The spider-web-like roof made of carbon fiber composites combined with glass elements (see Figure 6) is studied as actual application of above topic. Such light-weight structure is prone to ultimate wind induced vibrations mentioned as follows:

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1. The wind turbulences force the structure with a considerable power. The forced movements owing to turbulence and associated mechanisms are stochastic in nature. 2. Carbon fiber members can produce a strong vortex wake associated with aerodynamic drag forces appearing. Depending on the wind speed and the crosssection’s shape, the shedding of vortices is regular with the periods inversely proportional to the wind speed. In resonant conditions the structure controls the rhythm of the vortex shedding and limited amplitude vibrations appear there. Aside the well known vortex trail type excitation the more general types of aerodynamic excitation mechanism appear there. Possible re-attachment of separated flow, the vortices generated by local shape geometry and by the movement of the structure contribute to periodic aerodynamic forcing. 3. Aerodynamic forces proportional to the movement of the structure can produce divergent vibrations. In theoretical treatment of the phenomenon the concepts of aerodynamic damping and aerodynamic stiffness are to be applied. 4. In the design of carbon fiber structures is to be avoided that absolute value of negative aerodynamic damping exceeds the positive mechanical damping producing torsional or flexural mode aerodynamic instability. 5. At the onset of divergence the critical wind velocity and aerodynamic instability of the structure is initiated.

Figure 6. Geometry of slender spider-web-like roof studied.

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The time and frequency domain are two frequently used basic approaches for the analysis of aerodynamic response of the structure studied. For the loads changing arbitrarily in time, the first approach aims the solution of convolution type integrals, while the second one involves the Fourier-transformed equations of motion with the frequency as fundamental parameter. For specification of aeroelastic loads in time domain the indicial aerodynamic functions are to be applied. The advantages of the frequency-domain method for modeling of aeroelastic behaviour of the structure studied are obvious. The flutter derivatives are functions of frequency of vibration and can directly be applied to Fourier-transformed equations of motion given by (-ω2 M + i ω C + K) X(ω) = [i ω CAe(ω) + KAe(ω)] X(ω) + Fb(ω)

(38)

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where ω is the circular frequency, X and Fb are the vectors of the Fourier transform of nodal degrees of freedom and of nodal dynamic loads, respectively, and i is the imaginary unit. Here, M, C and K are the mass, damping and stiffness matrices, respectively, related to the mechanical properties of the structure studied, CAe(ω) is the aeroelastic damping matrix and KAe is the aeroelastic stiffness matrix defined in the terms of flutter derivatives appearing. In the structure shown in Figure 6 the carbon fiber elements were designed as tubes 60/5 mm. The filling members were made as glass plates with thickness 10 mm. As ultimate behaviour was tested the development of antimetric flexural displacements into aeroelastic response of the structure subjected to the laminar wind action with speed 30 m/sec and with pressure 1.0 MN/m2. The structural response obtained is plotted in Figure 7.

Figuer 7. Development of ultimate aeroelastic response of the spider-web-like roof studied – antimetric flexural displacements (in m) in nodes 1 - 100 along the length of structure from 1 sec (minimal displacement) until 40 sec (maximal displacement).

Another results obtained are submitted, for example, in references [7] and [8].

Conclusion Analysis and numerical as well as experimental results sampled up submit some image on the ultimate response of structures made of carbon fiber composites. The analysis of material instability with adoption of the fiber kinking theory for study of the failure process

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resulted in approximations for the treatment of the problem. Micromechanical analysis performed is based on the discrete simulation of the problem in space, time and temperature. Some results with actual application are submitted.

Acknowledgements The author is indebted to Slovak Grant Agencies VEGA and APVV for supporting above research.

References

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[1] Tesar, A., Fillo, L.: Transfer Matrix Method, KLUWER Academic Publishers, Dordrecht/Boston/London, 1988. [2] Tesar, A. and Svolik, J,: Wave distribution in fibre members subjected to kinematic forcing. Int. Journal for Communication in Numerical Mechanics, 9, 1993. [3] Simo, J.C.: On a fully three-dimensional finite strain viscoelastic damage model formulation and computational aspects. Comput. Meth. Engng. 29, 1990, [4] Budiansky, B.: Micromechanics. Computers & Structures, Vol. 16, No. 1-4, 1983, [5] Duk-Hyun Kim: Composite Structures for Civil and Architectural Engineering, South Korea, 1995. [6] Extren Design Manual, Strongwell Corporation, Bristol, Virginia, 1998. [7] Tesar, A. and Simoncic, M.: Fatigue analysis of fiber glass composites. Building Research Journal, 2, 2002. [8] Tesar, A., Sotakova, D. and Minar, M.: Micromechanics of fiber glass composites at elevated temperatures. Int. Journal for Numerical Methods in Engineering, 4, 2002.

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In: Composite Laminates ISBN 978-1-60741-620-3 c 2010 Nova Science Publishers, Inc. Editors: A. Doughett and P. Asnarez, pp. 465-489

Chapter 14

T OWARDS D IFFUSE I NTERFACE M ODELS WITH A N ONLINEAR P OLYCRYSTALLINE E LASTIC E NERGY Thomas Blesgen∗ and Anja Schl¨omerkemper† Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, D-04103 Leipzig, Germany

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Abstract Recently in [8], an extension of the Cahn-Hilliard model was derived that takes into account nonlinear elastic energies of the precipitates and includes composite laminates in the physical description. The aim of this work is to provide a basis for the further generalization of isothermal diffuse interface models, which we do by developing our methods exemplary for the Allen-Cahn/Cahn-Hilliard equations. Since segregated phases in typical physical applications are polycrystalline, it is natural to incorporate also effects present in polycrystals rather than in single crystals, leading to a polycrystalline lamination theory. To this end we recall some models and methods used in the context of polycrystalline materials and composites. Finally, we outline how the Allen-Cahn/Cahn-Hilliard model can be extended to polycrystalline geometrically linear elasticity.

1.

Introduction

Diffuse interface models have been successfully applied to model segregation and precipitation phenomena in alloys and liquid mixtures. However, so far, elastic effects due to composite structures of the considered materials as well as effects due to polycrystalline structures of the considered materials have mostly been neglected. In this article we shall consider these effects and provide a basis for a generalization of the existing diffuse interface models. We focus on three cases: (i) single crystalline materials which follow the linear elastic theory developed by Eshelby, [17], in the context of elastic inclusions and inhomogeneities, ∗ †

E-mail address: [email protected] E-mail address: [email protected]

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(ii) single crystalline materials which are described well by a geometrically linear theory of elasticity that takes phase fractions on the microscale into account and was developed in [14], and (iii) polycrystalline materials that are described well by a geometrically linear theory of elasticity that takes phase fractions on the microscale as well as the underlying texture of the polycrystal on a mesoscopic scale into account. This is very important for many applications where the classical single crystal theory is not general enough. The first two cases are treated in Section 2.1. We develop the third case in Section 2.2., where we also provide a general introduction to the methods of composite materials and polycrystals and repeat concepts like texture of a material and homogenization as well as established bounds on the effective elastic energy of polycrystals. Furthermore we mention recent results for stress-induced phase transformations in polycrystalline materials, [7]. Starting from elasticity models for the three cases we then generalize diffuse interface models for precipitation and segregation phenomena, which is the topic of Section 3. Our approach is quite general and can be applied to any of the established models, provided the temperature is conserved (for non-isothermal settings, the validity of the second law of thermodynamics requires further correction terms which are not studied here). For practical reasons and in order to have a concise presentation, we will discuss in this article the coupling of the afore-mentioned elastic lamination theories to the Allen-Cahn/CahnHilliard equations (AC-CH equations for short). This model, first introduced in [13], contains both the Allen-Cahn equation and the Cahn-Hilliard equation as special cases, which are the two most-frequently used models to investigate segregation, precipitation, and phase change problems in materials science, engineering, theoretical physics, and biology, among others. The Allen-Cahn system with linear elasticity was studied before in [10], the Cahn-Hilliard system with linear elasticity in [19], [25] and [12]. An extension of the Cahn-Hilliard system with geometrically linear elasticity valid for single crystals was recently found in [8]. The coupling to elasticity changes significantly the morphology of the precipitates and the coarsening patterns, see, e.g., the classification in [18], which opens interesting research topics for the future. Next to the generalization of the AC-CH equations we are interested in proving existence and uniqueness of the new systems of differential equations. In Section 3.1. we prove existence and uniqueness of weak solutions for a class of functionals inspired by Eshelby’s linear theory of elasticity with lamination. In Section 3.2. we obtain similar results for materials falling in the second category, i.e., for single crystalline materials described by a geometrically linear elastic lamination theory. Section 3.3. is devoted to the generalization of the AC-CH equations to polycrystalline materials and provides the basis of further analytical and numerical research. We end this work with an outlook and a discussion of our results.

2. 2.1.

A Polycrystalline Lamination Theory The Elastic Energy in Single Crystalline Composites

Our main objective in this section is to study geometrically linear elasticity for composites in the context of isothermal phase transitions. For systematic reasons, we first recall the linear ansatz dating back to Eshelby, [17]. This allows us, as a byproduct of the existence Composite Laminates: Properties, Performance and Applications : Properties, Performance and Applications, Nova Science Publishers,

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467

theory proved in Section 3.1., to obtain a new existence result for the Allen-Cahn/CahnHilliard equations with linear elasticity. Throughout this paper let Ω ⊂ RD for D ≥ 1 be a bounded domain with Lipschitz boundary which serves as the reference configuration. By u : Ω → RD we describe the displacement field, such that a material point x in the undeformed body Ω is at x′ = x+u(x) after the deformation. Then the (linearized) strain tensor is defined by

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ε(u) :=


1 ∇u + ∇ut , 2

(1)

D×D . As usual, · stands for the inner where At denotes the transpose PDof a matrix A ∈ R D product in R , i.e., u · v = i=1 ui vi , and for A, B ∈ RD×D we denote by A : B := P D×D . tr(At B) = D i,j=1 Aij Bij the inner product in R The linear theory by Eshelby, [17], developed in the context of elastic inclusions and inhomogeneities, can be summarized in the following ansatz for the elastic energy of a composite 1 (2) Wlin (d, ε) := (ε − ε(d)) : C(d)(ε − ε(d)) 2 D×D for all ε ∈ RD×D sym , d ∈ R, and ε(d) := d ε with a constant ε ∈ Rsym . The notion of d will become clear in Section 3. Here we only mention that d ∈ [0, 1] is a conserved or unconserved order parameter of a diffuse interface model that describes, e.g., segregation in a solid with reference configuration Ω ⊂ RD with D ≥ 1. By C(d) we denote the symmetric, positive definite and concentration dependent elasticity tensor of the system that maps symmetric tensors in RD×D to themselves. For the rest of this section we discuss the geometrically linear elasticity theory that takes the laminates of the material into account. As is shown in [8, Remark 1], the abovementioned linear elasticity theory by Eshelby is a special case of this geometrically linear theory. In the following we assume that two phases are present in the considered material which may form microstructures as e.g. displayed in Figures 7 and 8. We refer to the energy of c (d, ε(u)) in (4), each of the phases as microscopic energy, cf. (3), and to the energy W which reflects the effective behavior of the system with microstructures, as the mesoscopic energy. When we include polycrystalline structures, we moreover consider a macroscopic scale, see Section 2.2. c (d, ε(u)) in the geometrically linear case we need to solve To determine the energy W a local minimization problem, see (4) below, which we shall explain now. Consider an open ball B := Br (x0 ) ⊂ Ω containing a two-phase microstructure. We assume that the volumes occupied by each of the two phases in B are measurable sets. In ˜ d˜2 = 1 − d˜ characterize the two phases on the microscale, we have particular, if d˜1 ≡ d, ˜ di ∈ BV (B; {0, 1}) and d˜1 + d˜2 = 1 a.e. in B. The symbol BV denotes the space of functions of bounded variation, see, e.g., [1, 27]. By Z Z 1 ˜ dx := hmi ˜ := − m(x) m(x) ˜ dx |B| B B

we denote the average of a function m ˜ in B, where |E| is the D-dimensional Lebesgue measure of a set E. Composite Laminates: Properties, Performance and Applications : Properties, Performance and Applications, Nova Science Publishers,

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Let εTi ∈ RD×D sym , i = 1, 2, be the stress-free strain of the i-th phase relative to the chosen reference configuration and αi be its elasticity tensor. Then the elastic energy density of phase i subject to a strain ε˜ is given by

1 Wi (˜ ε) := αi ε˜ − εTi : ε˜ − εTi + wi 2

(3)

for constants wi ≥ 0. Under the assumption that the elastic energy adapts infinitely fast and that the surface energy between laminates of the microstructure can be neglected, the effective elastic energy is, [14], Z ˜ 2 (˜ c ε) + (1 − d)W ε) dx, (4) W (d, ε) := inf inf − d˜W1 (˜ ˜ u ˜|∂B =εx =d

B

where we write for short ε˜ = ε˜(˜ u) = 12 (∇˜ u + ∇˜ ut ). The infimum over d˜ is the result of homogenization subject to the constraint that the volume fraction of the selected phase is preset by d, see [16]. The other infimum in (4) is the result of relaxation theory, see [15], [20], which is now outlined. If for prescribed d = a + b the microscopic elastic energy density is Wd (˜ ε), then Z c − Wd (˜ ε(˜ u)) dx (5) Wd (ε) := inf

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u ˜|∂B =εx

B

is the elastic energy density of the material with mesoscopic strain ε after microstructure cd does not depend on B. Likewise, has formed. As is shown in [15], this definition of W (4) is independent of B = Br (x0 ) as long as B ⊂ Ω. We mention that there exist explicit c if D = 2, 3, [14]. The representation for D = 2 will be recalled analytic formulas for W in Section 3.2. Next we discuss a polycrystalline lamination theory before we come to the formulation of extended Allen-Cahn/Cahn-Hilliard models.

2.2.

A Polycrystalline Lamination Theory

The Allen-Cahn/Cahn-Hilliard system is an established model for describing precipitation in solids, segregation phenomena, and more general phase change problems, among others. Very often, the actual physical phenomenon is very complicated, as it additionally depends on the morphology of the material on the small scale, or on plastic effects like the formation and movement of dislocations and hardening. In this article, we do not focus on the description of the latter, but focus on the morphology of the material. Besides the lamination microstructure, also the polycrystalline nature of the solid is of importance. A polycrystalline material is a solid which is composed of many grains with a lattice subsequently assumed to be identical, but with different orientations. Each grain behaves like a single crystal, at least this is what we shall assume in the following, where we neglect all effects resulting from grain boundaries. We are interested in materials that form microstructures within the grains. Such composites can for instance be laminates of order one and two as indicated in Figure 1. The texture of polycrystalline materials is described by a matrix-valued function R : Ω → SO(3) which is constant on each grain (we assume that every grain has full Lebesgue

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Figure 1. Part of a polycrystal showing laminates of order one and two in its grains.

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measure in RD ). Here, SO(D) denotes the set of all rotations about the origin of RD (characterized by Rt R = Id and det(R) = 1). Thus the function R describes the number and shapes of the grains as well as their orientations. In Figure 2 we give a simple mathemat-

Figure 2. Part of a polycrystal prototype forming a chessboard structure.

ical example of a polycrystal which shows the structure of a chessboard. This texture can for instance form if the lattice structure in the light squares is the reference configuration, i.e., R = Id, whereas the lattice structure in the dark squares is obtained by a rotation, R = R π4 . Another typical example is the isotropic or random texture in which all rotations R ∈ SO(3) occur in the polycrystal identically distributed. In the following we again consider a general polycrystal. Let us pick one grain with reference configuration G ⊂ RD and choose its orientation as the reference. Analogous to (5), the elastic energy of this grain obtained by relaxation is then Z c ε(˜ u)) dx, W (ε) = min − W (˜ u ˜|∂G =εx G

where ε denotes again the symmetrized strain gradient given by (1), i.e., as before we work in the framework of geometrically linearized elasticity and W is the microscopic elastic Composite Laminates: Properties, Performance and Applications : Properties, Performance and Applications, Nova Science Publishers,

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energy having multiple wells related to different compatible phases. The relaxed elastic energy of a grain rotated by R with respect to the reference grain is c (Rt εR). The macroscopic behavior of a polycrystal is obtained by nonlinear homogeW nization (see, e.g., [4]), Z c (Rt (x)ε(u(x))R(x)) dx. (6) W (ε) = min − W u|∂Ω =εx Ω

c and W looks similar, but they take into The mathematical structure of the definitions of W account different issues: while the relaxation of the multi-well energy W involves averages c to W involves averages over composites on a subgrain length scale, the passage from W over grains and thus depends on the texture of the material. In other words, here we consider composites on different length scales: there is the lamination on a microscopic scale, i.e., within the grains, and there is the polycrystalline structure on a mesoscopic scale, i.e., on the scale of the body. The analytical computation of W for a given material is a subtle issue. Bhattacharya and Kohn [4] discuss this for shape-memory alloys and study upper and lower bounds on W and in particular on the zero-set of W . An upper bound on W is obtained by choosing a constant test field u = εx on Ω. Then Z c (Rt εR) dx =: W T (ε). W (ε) ≤ − W Ω

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We shall call W T (ε) the Taylor bound on W . In analogy to [4, p. 125] we next derive a lower bound on W (ε). To this end we recall the definition of the Legendre-Fenchel transform of a function f : RD×D sym → R, f ∗ (σ) =

sup {ε : σ − f (ε)} ,

D×D ε∈Rsym

σ ∈ RD×D sym .

Thus for any R ∈ SO(3) and σ ∈ RD×D sym , n o c ∗ (RσRt ) = c (ε) W sup ε : RσRt − W ε∈RD×D sym

=

sup

ε′ ∈RD×D sym

n o c (Rt ε′ R) ε′ : σ − W

c (Rt ε′ R) ≥ ε′ : σ − W

Integration yields

for any ε′ ∈ RD×D sym .

Z Z t ′ c ∗ (RσRt ) dx. c − W (R ε R) dx ≥ − ε′ : σ − W Ω

Ω

Note that the inequality still holds true if we maximize over all σ : Ω → RD×D sym . Then we ′ ′ minimize over all u such that u = εx on ∂Ω and obtain by (6) Z c ∗ (RσRt ) dx. max − ε′ : σ − W W (ε) ≥ ′min u|∂Ω =εx σ:Ω→RD×D sym

Ω

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471 R ′ R = εx, we have −Ωε : σ dx = −Ωε : σ dx.

Now note that for div σ = 0 and u′ with u′|∂Ω Hence we finally obtain the lower bound Z c ∗ (RσRt ) dx. W (ε) ≥ max − ε : σ − W div σ=0 Ω

(7)

From (7) we can go even one step further and consider constant stress-fields σ as test functions, which yields the Sachs bound W S (ε). Explicitly,   Z t ∗ c W (ε) ≥ max ε : σ − − W (RσR ) dx (8) σ∈RD×D sym

Ω

Z ∗ ∗ t c − W (RσR ) dx (ε) =: W S (ε). =

(9)

Ω

We refer to [4] for a discussion of upper and lower bounds for special cases of elastic energies. In particular, the bounds for scalar materials are studied there, which we recall and slightly extend here. Scalar materials reduce the dimension of the problem: instead of considering a vectorvalued displacement field u : R3 → R3 , the displacement is assumed to be a scalar-valued function on R2 , i.e., η : R2 → R. This corresponds to anti-plane shear. The strains f = f (η) = ∇η are vectors in R2 as are the stresses, which leads to the advantage of having a convex relaxed energy, [15]. The transformation behavior is now described by Rt f with R ∈ SO(2), instead of Rt εR, R ∈ SO(3), required above. With this change, all the above formulas can be defined and derived correspondingly for scalar materials. For instance, the effective behavior of a polycrystalline scalar material reads Z c (Rt (x)f (η(x))) dx. W (f ) = inf − W (10)

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η|∂Ω =f ·x Ω

In the following we consider the example of a four-variant scalar material with quadratic energy wells minimized at (1, 1), (−1, 1), (−1, −1) and (1, −1), cf., e.g., [4]. For f = (f1 , f2 ) ∈ R2 let W four (f ) :=

 1 min (f1 − 1)2 + (f2 − 1)2 , (f1 + 1)2 + (f2 − 1)2 , 2 (f1 − 1)2 + (f2 + 1)2 , (f1 + 1)2 + (f2 + 1)2

be the corresponding microscopic energy, see Figure 3. The mesoscopic energy is the convexification of W four and thus reads
c four (f ) = 1 (|f1 | − 1)2+ + (|f2 | − 1)2+ , W 2

where (a)+ = max{a, 0}, cf. Figure 4. Its zero-set is {f ∈ R2 | |f1 | ≤ 1, |f2 | ≤ 1}. To illustrate the effect of texture on the macroscopic energy W , we consider the chessboard texture as well as the isotropic texture. Ifthe material has a chessboard texture as in  1 −1 Figure 2, the rotations R0 = Id and R π4 = √12 occur equally distributed. Hence 1 1  !  ! f1 f1 + f2 1 c four 1 1 c four four √ + W , W T (f ) = W 2 2 f2 2 −f 1 + f 2

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Thomas Blesgen and Anja Schl¨omerkemper 2 0 2 4

W four 3

2 1

f2

0 2

f1 0 2

Figure 3. Plot of the microscopic energy W four .

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plotted in Figure 5. Its zero-set is the intersection of √ {|f1 | ≤ 1, |f2 | ≤ 1} with this set rotated by R π4 , thus {|f1 | ≤ 1, |f2 | ≤ 1, |f1 ± f2 | ≤ 2}. In an isotropic texture all rotations in SO(2) occur equidistributed. Hence  ! Z 2π f cos ϑ + f sin ϑ 1 1 2 c W T (f ) = W dϑ, 2π 0 −f 1 sin ϑ + f 2 cos ϑ

which has {f ∈ R2 | |f | ≤ 1} as zero-set and deviates from this quadratically with rotational symmetry. So far we have assumed that the constant temperature in our system is such that the microscopic elastic energy has several global minima. For instance, for shape memory alloys this means that we are below the transformation temperature in the martensitic phase. This is a realistic assumption for real-life segregation processes. However, here we do not want to exclude another case which is interesting in particular if external forces are applied. Having shape-memory alloys and martensitic phase transformations in other materials such as steels in mind, we discuss in the following what happens if the material is above its phase transformation temperature. Then the microscopic elastic energy has one global minimum only that corresponds to the lattice structure of the so-called austenitic phase; and it has several local minima that correspond to the lattice structures of the martensitic variants. A phase transformation from austenite to martensite can be induced by an applied load and results in pseudo-elastic behavior. We allow the applied load to be not only uniaxial but multi-axial; for multi-axial loading experiments in shape-memory alloys see for instance [23]. For a comparison of the models cited in the following with other models related to multi-axial loading experiments in shapememory alloys we refer to [22]. In [5, 26], Bhattacharya and Schl¨omerkemper discuss polycrystalline vectorial materials under an applied stress with a special focus on the yield set, which is defined as the set of

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473

2 1 0 1 2 1.0

c four W

0.5

0.0

f2

2 1

f1

0 1 2

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c four . Figure 4. Plot of W

all stresses such that the material is in its austenitic state. The boundary of this set gives the yield stress, i.e., the stress at which the transformation from austenite to martensite starts. For the definition of the yield set, Sachs and Taylor bounds are taken into account and the effect of texture on the yield set is studied. This is made explicit for cubic-to-orthorhombic phase transformations in shape-memory alloys there. In [6, 7], the same authors study the scalar case, to which we shall come back below. Once again we begin by formulating the theory for the vector-valued case in the geometrically linear setting, i.e., stresses and strains are elements of RD×D sym . As outlined in (23), the energy due to a uniform external applied load is Wext (ε) = −σext : ε. Since in a single crystalline material the integrand does not depend on x, minimization of this energy corresponds to minimizing its integrand W σext (ε) := W (ε) − σext : ε

over all ε ∈ RD×D sym . Assume that the global minimum of W is 0. Then there is no phase transformation as long as σext is such that inf

ε∈RD×D sym

W σext (ε) ≥

inf

ε∈RD×D sym

{Waust (ε) − σext : ε} ,

where Waust denotes that part of the energy W which corresponds to austenite, i.e., it denotes the energy of the high-symmetry phase whose well is close to the global minimum. Equivalently we have W ∗ (σext ) =

sup {σext : ε−W (ε)} ≤

ε∈RD×D sym

∗ sup {σext : ε−Waust (ε)} = Waust (σext ).

ε∈RD×D sym

Hence, the yield set in a single crystal is naturally defined as  ∗ ∗ Y := σext ∈ RD×D sym | W (σext ) ≤ Waust (σext ) ,

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(11)

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Thomas Blesgen and Anja Schl¨omerkemper

1.0 0.5 0.0 0.5 1.0 0.08

four

WT

0.06 0.04 0.02 0.00

f2

1.0 0.5

f1

0.0 0.5 1.0

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Figure 5. The Taylor bound on W four with chessboard texture. where we neglect any fatigue of the material such as hardening. When we wish to work with the relaxed energy, we consider the mesoscopic energy under an applied load, namely Z σext c ε(˜ u)) − σext : ε˜(˜ u) dx (12) W (ε) = min − W (˜ u ˜|∂Ω =εx Ω  Z Z u) dx . = min − W (˜ ε(˜ u)) dx − σext : − ε˜(˜ u ˜|∂Ω =εx

Ω

Ω

R Due to (1) we obtain −Ωε˜(˜ u) dx = ε. Hence c σext (ε) = W

Z ε(˜ u)) dx − σext : ε min − W (˜

u ˜|∂Ω =εx Ω

c (ε) − σext : ε. = W

In analogy to (11), we set n o ∗ ∗ c c Yb := σext ∈ RD×D | W (σ ) ≤ W (σ ) . ext ext sym aust

Finally, the macroscopic energy of a polycrystal under uniform applied load reads Z σext c (Rt ε(u)R) − σext : ε(u) dx W (ε) = min − W u|∂Ω =εx Ω

= W (ε) − σext : ε.

Correspondingly, we set n o ∗ ∗ Y := σext ∈ RD×D | W (σ ) ≤ W (σ ) . ext sym aust ext

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475

Note that Y , Yb and Y can be defined correspondingly in the scalar setting. In the following we will elaborate on this further and consider the energy W : R2 → R defined by   C 2 |e − f | + w(f ) , (14) W (e) := min 2 f ∈R2 where   0 if e = 0, w(e) := ω if e = e(i) , i = 1, . . . , n,   ∞ else

with a constant ω > 0 and e(1) , . . . , e(n) being the local minima of W and w. In the case of shape-memory alloys these are the stress-free variants of the martensite and e = 0 corresponds to austenite. Note that here Waust (e) = C2 e2 for e close to 0, which leads to ∗ (s) = s2 . The applied load now yields the energy −s Waust ext · e. Thus, by (11), 2C   s2ext 2 ∗ Y = sext ∈ R W (sext ) ≤ . 2C

Furthermore, by [7] or elementary calculations, W ∗ (s) =  max 0, maxi s · e(i) − ω ≥ 0. Hence

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Y =

s2 2C

+ w∗ (s) and w∗ (s) =

  sext ∈ R2 max sext · e(i) ≤ ω . i

c = W ∗∗ and therefore, W c ∗ (s) = For the scalar case that we consider here we have W ∗ (s) = W caust (s), implying Yb = Y . caust W ∗ (s). Similarly, W In order to calculate also the yield set of a polycrystalline material Y , we apply a result from [7] for the energy in (14), which asserts that the Sachs bound YS on the yield set, which is obtained under the assumption of constant stress throughout the sample, equals Y and thus is sharp. In formulas, Y = YS , (15) S where YS = x∈Ω YR(x) with YR(x) = {s | Rt s ∈ Y }. To give a specific example, we consider a four-variant scalar material in the constrained model, i.e., for C → ∞. Then W in (14) reads   if e = 0, 0 W (e) = w(e) = ω if e ∈ {(1, 1); (−1, 1); (−1, −1); (1, −1)} ,   ∞ else and

W ∗ (s) = max{0, |s1 ± s2 | − ω}, whose zero-set is the square {s = (s1 , s2 ) ∈ R2 | |s1 ± s2 | ≤ ω} = Y , see Figure 6.

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Thomas Blesgen and Anja Schl¨omerkemper s2 ω

ω

s1

Figure 6. The yield set Y of a scalar four-variant single crystal.

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By (15) we thus obtain for a polycrystalline material with a chessboard structure as in Figure 2,   ω ω Y chess = {s | |s1 ± s2 | ≤ ω} ∩ s |s1 | ≤ √ , |s2 | ≤ √ 2 2   ω ω = s |s1 ± s2 | ≤ ω, |s1 | ≤ √ , |s2 | ≤ √ , 2 2

which is an octahedron. Similarly, for a polycrystal with isotropic texture, the macroscopic yield set Y is a disc with radius √ω2 . Our next goal is to generalize the above notions of energies for polycrystalline materials to energies that take into account also prescribed volume fractions of the phases. For this we return to the general case of vector-valued deformations. This generalization then allows us to develop an extension of the Cahn-Hilliard model for nonlinear elastic energies that takes into account effects of polycrystalline structures of the systems under consideration. c (d, ε) in the geometriTo this end we recall the definition of the mesoscopic energy W cally linear theory of elasticity which takes phase fractions into account, see Section 2.1. ˜ ˜ ˜ ˜ Let e(i) ∈ RD×D sym , i = 1, 2, be two stress-free strains and d1 ≡ d, d2 = 1 − d their corresponding phase fractions such that d˜i ∈ BV (Ω; {0, 1}) and d˜1 + d˜2 = 1 a.e. in Ω. Then Z ˜ 2 (˜ c (d, ε) = inf ε(˜ u)) + (1 − d)W ε(˜ u)) dx, W inf − d˜W1 (˜ ˜ u ˜|∂Ω =εx Ω =d

where Wi , i = 1, 2 are defined as in (3). Now let again Ω be the reference configuration of a polycrystalline material whose texture is described by some piecewise-constant map R : Ω → SO(3). For prescribed cd (ε) as in (5) and set in analogy phase fractions we proceed with the mesoscopic energy W to (6) for the macroscopic energy Z cd (ε(u)) dx. W d (ε) := inf − W u|∂Ω =εx Ω

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477

If the phase fractions are not prescribed, there is another step of homogenization to be done. Combining the earlier definitions, it is natural to define the macroscopic energy which takes volume fractions into account by Z c (d, ε(u)) dx. inf − W W (d, ε) := inf (16) =d u|∂Ω =εx Ω

In Subsection 3.3. we outline how this energy leads to an extension of the Allen-Cahn/CahnHilliard model to polycrystalline materials.

3.

The AC-CH Model and Extensions

Let as above Ω ⊂ RD , D ≥ 1, be a bounded domain with Lipschitz boundary. For a stop time T > 0, let ΩT := Ω × (0, T ) denote the space-time cylinder. To the AllenCahn/Cahn-Hilliard system, first derived in [13], we add elasticity, possibly respecting the lamination microstructure of the material, and introduce the system   ∂F , (17) ∂t a = λ div M (a, b)∇ ∂a ∂F ∂t b = −M (a, b) , (18) ∂b   c (a + b, ε(u)) , (19) 0 = div ∂ε W which has to be solved in ΩT subject to the initial conditions

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a(t = 0) = a0 , b(t = 0) = b0

in Ω

for given functions a0 , b0 : Ω → R subject to the Neumann boundary conditions for a, the no-flux boundary conditions, and the equilibrium condition for applied forces ∇a · ~n = 0,

J(a, b, u) = 0,

σ · ~n = σext · ~n

on ∂Ω, t > 0.

(20)

See also (17’)–(19’) below for an explicit formulation. In (17)–(19), the function a : ΩT → R+ 0 is a conserved order parameter, typically a is an unconserved order-parameter, specifying the reordering concentration, b : ΩT → R+ 0 of the underlying lattice, M (a, b) ≥ 0 denotes the mobility tensor, u : Ω → RD describes as before the displacement field, ε(u) is the (linearized) strain tensor defined in (1), and c (a + b, ε(u)) is λ > 0 is a small constant determining the interfacial thickness. Finally, W the stored elastic energy density as defined in (4) for composites. In (20), ~n is the unit outer normal to ∂Ω. For simplicity, body forces are neglected and it is assumed that the boundary tractions are dead loads given by a constant symmetric tensor σext as in Section 2. By J we denote the mass flux, given by J(a, b, u) := −M (a, b)∇µ = −M (a, b)∇ with µ :=

∂F ∂a

∂F (a, b, u), ∂a

the chemical potential.

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The system (17)–(19) is completed with the definition of the free energy Z
λ c (a + b, ε(u)) + Wext (ε(u)) dx, (21) F (a, b, u) := ψ(a, b) + |∇a|2 + |∇b|2 + W 2 Ω

where ψ(a, b) is the free energy density  ϑ g(a + b) + g(a − b) + κ1 a(1 − a) − κ2 b2 , ψ(a, b) := 2 g(s) := s ln s + (1 − s) ln(1 − s)

(22)

for scalars κ1 , κ2 > 0. The term 12 [g(a + b) + g(a − b)] in (22) defines the entropic part of the free energy, given in the canonical Bernoulli form for perfect mixing, and ϑ > 0 is the constant temperature. The functional Wext (ε) in (21) represents energy effects due to applied forces. In the absence of body forces, the work necessary to transform the undeformed body Ω into a state with displacement u is then Z Z Z u · σext~n = − ∇u : σext = − ε(u) : σext , − ∂Ω

Ω

Ω

where we use the symmetry of σext . Consequently, Wext (ε) = −σext : ε

(23)

is the energy density of the applied outer forces. The valid parameter range for a and b is, see Theorem 1,

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0 ≤ a + b ≤ 1,

0 ≤ a − b ≤ 1.

(24)

The inequalities are strict unless (a, b) = (0, 0) or (a, b) = (1, 0). The system (17)–(19) includes as special cases the elastic Cahn-Hilliard system (setting b ≡ 0, [19]) and the elastic Allen-Cahn equations (setting a ≡ 12 , [10]). The system studied here is exemplary for an isothermal model that exhibits simultaneously ordering and phase transitions. Equation (17) is a diffusion law for a governed by the flux J and states the conservation of mass in Ω. Equation (18) is a simple gradient flow in the direction ∂F ∂b . Equation (19) is a consequence of Newton’s second law under the additional assumption that the acceleration ∂tt u originally appearing on the left hand side can be neglected (this can be proved formally c (a+b, ε(u)) by a scaling argument and formally matched asymptotics). The term σ := ∂ε W defines the stress. Equation (19) serves to determine the unknown displacement u. Remark 1. Equations (17)–(19) can be generalized to vector-valued mappings a and b. This allows to study situations with more than two phases present. To fix ideas and for the sake of a clear presentation, we restrict ourselves throughout this paper to scalar quantities a and b.

Remark 2. Equations (17)–(19) with boundary conditions (20) comply with the second law of thermodynamics, which in case of isothermal conditions reads for a closed system ∂t F (a(t), b(t), u(t)) ≤ 0. This inequality can be verified by direct inspection similar to the calculations in [8].

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Next we discuss existence and uniqueness results for the Allen-Cahn/Cahn-Hilliard model extended to linear elasticity and geometrically linear elasticity, respectively. In Section 3.3. we show how the above model can be extended to polycrystalline materials exhibiting ordering and phase transition simultaneously.

3.1.

Existence and Uniqueness Results of the AC-CH Model with Linear Elasticity

The existence of solutions to the Allen-Cahn/Cahn-Hilliard equation without elasticity was studied in [11] with the help of a semigroup calculus. Existence and uniqueness of weak solutions to the Cahn-Hilliard equation with linear elasticity is proved in [19], with geometrically linear elasticity in [8]. Existence and uniqueness of weak solutions to the Allen-Cahn equation with linear elasticity is shown in [10]. Subsequently we provide existence and uniqueness results for (17)–(19). First we require some mathematical tools. We introduce the operator M associated to w 7→ −M △w as a mapping from H 1 (Ω) to its dual by Z M(w)η := M ∇w · ∇η, (25) Ω

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where M is the mobility tensor which we assume in the following to be positive definite. From the Poincar´e inequality and the Lax-Milgram theorem (which can be applied now thanks to the assumption that M is positive definite) we know that M is invertible and we denote its inverse by G, the Green function. We have (M ∇Gf, ∇η)L2 = hη, f i

for all η ∈ H 1 (Ω), f ∈ (H 1 (Ω))′ .

For f1 , f2 ∈ (H 1 (Ω))′ , we define the inner product (f1 , f2 )M := (M ∇Gf1 , ∇Gf2 )L2 with the corresponding norm kf kM :=

p

(f, f )M

for f ∈ (H 1 (Ω))′

which is applied in (27) in the proof of Theorem 1. For M ≡ 1, the explicit formulation of (17)–(19), used in the proofs below, is " # c
ϑ ′ ∂W ′ ∂t a = λ△ g (a+b) + g (a−b) + κ1 (1−2a) + (a+b, ε(u)) − △a ,(17’) 2 ∂d c  ∂W ϑ ′ g (a − b) − g ′ (a + b) + 2κ2 b − (a + b, ε(u)), ∂t b = λ△b + ∂d  2  c (a + b, ε(u)) . 0 = div ∂ε W

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(18’) (19’)

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c being the linear energy Now we prove the existence of solutions to (17)–(19) with W Wlin as given by (2). For the existence proof below, the energy does not have to have exactly the form of Wlin . We highlight the required conditions by introducing the general class of c = Wlin is elastic energies which satisfy the following assumption (A1). The functional W then one particular example. c ∈ C 1 (R × RD ; R) satisfies: (A1) The elastic energy density W c (d, ε) only depends on the symmetric part of ε ∈ RD×D , i.e., (A1.1) W c (d, ε) = W c (d, εt ) for all d ∈ R and all ε ∈ RD×D . W

c (d, ·) is strongly monotone uniformly in d, i.e., there exists a constant (A1.2) ∂ε W c1 > 0 such that for all ε1 , ε2 ∈ RD×D sym   c (d, ε2 ) − ∂ε W c (d, ε1 ) : (ε2 − ε1 ) ≥ c1 |ε2 − ε1 |2 . ∂ε W

(A1.3) There exists a constant C1 > 0 such that for all d ∈ R and all ε ∈ RD×D sym

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c (d, ε)| ≤ C1 (|ε|2 + |d|2 + 1), |W c (d, ε)| ≤ C1 (|ε|2 + |d|2 + 1), |∂d W c (d, ε)| ≤ C1 (|ε| + |d| + 1). |∂ε W

(26)

Theorem 1 (Existence of solutions for linear elasticity). Let the mobility tensor M in (25) c fulfill (A1) and let ψ be given by (22). In addition, let the initial be positive definite, let W data satisfy ψ(a0 , b0 ) < ∞. Then, there exists a solution (a, b, u) to (17)–(19) that satisfies
1 (i) a, b ∈ C 0, 4 [0, T ]; L2 (Ω) . (ii) ∂t a, ∂t b ∈ L2 (ΩT ).
(iii) u ∈ L∞ 0, T ; H 1 (Ω; RD ) . (iv) The feasible parameter range for (a, b) is given by (24). In particular, Theorem 1 confirms the existence of solutions to (17)–(19) with linear elasticity as defined in (2). Proof: The statements of the theorem can be proved with the methods developed in [10]. Here we only sketch the main steps. For a small discrete step size h > 0, chosen such that T h−1 ∈ N, for time steps m ∈ N with 0 < m < T hm−1 , and given values am−1 , bm−1 ∈ R, we introduce the discrete free energy functional F m,h (a, b, u) := F (a, b, u) +

1 1 ka − am−1 k2M + kb − bm−1 k2L2 , 2h 2h

(27)

where (in case of m = 1) it holds a0 = a0 , b0 = b0 , the initial values for a and b. By the direct method in the calculus of variations and Assumption (A.1), it is possible to Composite Laminates: Properties, Performance and Applications : Properties, Performance and Applications, Nova Science Publishers,

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show that for h sufficiently small, F m,h possesses a minimizer (am , bm , um ) ∈ H 1 (Ω) × H 1 (Ω) × H 1 (Ω; RD ). This minimizer solves the fully implicit time discretisation of (17’)– (19’). Next the discrete solution is extended affine linearly to (a, b, u) by setting for t = (τ m + (1 − τ )(m − 1))h with suitable τ ∈ [0, 1] (a, b, u)(t) := τ (am , bm , um ) + (1 − τ )(am−1 , bm−1 , um−1 ).

The validity of the second law of thermodynamics (cf. Remark 2) implies that F is nonincreasing in time. This allows to derive uniform estimates for (a, b, u). Compactness arguments then allow to pass to the limit h ց 0 and the limit solves (17)–(19). c = Wlin be given by (2), Theorem 2 (Uniqueness of solutions for linear elasticity). Let W let the material be homogeneous, that is the elasticity tensor C is independent of d, and let M ≡ 1. Then the solution (a, b, u) of Theorem 1 is unique in the spaces stated in the theorem. Proof: Fix t0 ∈ (0, T ). Let (ai , bi , ui ), i = 1, 2 be two pairs of solutions to (17’)–(19’) and (2). The differences a := a1 − a2 , b := b1 − b2 , u := u1 − u2 with corresponding ∂F difference of the chemical potentials µ := µ1 − µ2 := ∂F ∂a (a1 , b1 ) − ∂a (a2 , b2 ) solve the weak equations Z [−a∂t ξ + λ∇µ · ∇ξ] = 0, (28) ΩT

Z

[∂t bη + λ∇b · ∇η − ε : C (ε(u) − ε(a + b)) η]

ΩT

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= Z

Z 

ΩT


ϑ ′ ′ ′ ′ g (a2 +b2 ) − g (a1 +b1 ) + g (a1 −b1 ) − g (a2 −b2 ) η + 2κ2 bη , (29) 2

C (ε(u) − ε(a + b)) : ε(u) = 0

(30)

Ω t0

for every ξ, η ∈ L2 (0, T ; H01 (Ω)) ∩ L∞ (ΩT ) with ∂t ξ, ∂t η ∈ L2 (ΩT ), ξ(T ) = 0, where in order to get (30) we plugged in (u2 − u1 )X(0,t0 ) as a test function and integrated by parts. As a test function in (28) we pick  R t0 if t ≤ t0 , t µ(x, s) ds, ξ(x, t) := 0, if t > t0 . This shows

Z

aµ + λ∇(Ga) · ∇(∂t Ga) = 0.

Ω t0

The difference of the chemical potentials fulfills, with the help of (21), Z Z h   ϑ ′ µζ = g (a1 + b1 ) − g ′ (a2 + b2 ) + g ′ (a2 − b2 ) − g ′ (a1 − b1 ) ζ 2 ΩT ΩT i −2κ1 aζ + λ∇a · ∇ζ − ε : C(ε(u) − ε(a + b))ζ .

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(31)

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We pick ζ := (a1 − a2 )X(0,t0 ) . With (31) we obtain Z λ kak2M (t0 ) + λ|∇a|2 − aε : C(ε(u) − ε(a + b)) ≤ 2 Z

Ω t0

2κ1 a2 +

Ω t0

i ϑ h ′ g (a1 +b1 ) − g ′ (a2 +b2 ) + g ′ (a1 −b1 ) − g ′ (a2 −b2 ) |a|. (32) 2

In (29) we choose η := (b2 − b1 )X(0,t0 ) as a test function and add the resulting equation to (32) and use (30). We end up with Z h i
λ 1 2 c (a + b, ε(u)) kakM (t0 ) + kb(t0 )kL2 + λ |∇a|2 + |∇b|2 + W 2 2 ≤

Z

Ω t0

2(κ1 |a|2 + κ2 |b|2 )

Ω t0

Z h i ϑ ′ + g (a1 +b1 ) − g ′ (a2 +b2 ) + g ′ (a1 −b1 ) − g ′ (a2 −b2 ) (|a|+|b|). 2 Ω t0

From Theorem 1 we know that the terms g ′ (ai ± bi ), i = 1, 2 are finite, and g ′ is Lipschitz continuous (even real analytic). Applying first Young’s inequality, then Gronwall’s inequality, as t0 ∈ (0, T ) was arbitrary, we find a = b = 0 in ΩT . This finally yields Z ε(u) : Cε(u) = 0. Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

ΩT

With Korn’s inequality this proves u = 0 in ΩT .

3.2.

Existence and Uniqueness of the AC-CH model with Geometrically Linear Elasticity

For the existence theory in the case of the Allen-Cahn/Cahn-Hilliard model extended c is not practical since it is to geometrically linear elasticity, the above definition (4) of W based on a local minimization. To this end we collect here explicit analytic formulas for the c and its derivatives which are valid for D = 2. relaxed energy W c (d, ε) := d1 W1 (ε∗1 ) + d2 W2 (ε∗2 ) + β ∗ d1 d2 det(ε∗2 − ε∗1 ). W

(33)

Formula (33) is derived in [14], where also a corresponding formula in three dimensions can be found. To complete the definition, we have to introduce the quantities β ∗ , ε∗1 and ε∗2 . First we need some notations. Let γ ∗ > 0 be given by γ ∗ := min{γ1 , γ2 }, (34) −1/2

−1/2

where γi is the reciprocal of the largest eigenvalue of αi Qαi , αi is the elastic mod2×2 → R2×2 is given by ulus of laminate i, and the operator Q : Rsym sym Qε = ε − tr(ε)Id.

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In [8] a recipe is given for the practical computation of γ ∗ . Here, we only remark that if the space groups of the two existing laminates are cubic, it holds γ ∗ = min{C1,11 − C1,12 , C2,11 − C2,12 , 2C1,44 , 2C2,44 }. The first subscript of C denotes here the phase, the other two indices are the coefficients of the reduced elasticity tensor in Voigt notation, [24]. As shown in [14], the scalar β ∗ ∈ [0, γ ∗ ] determines the amount of translation of the laminates and is given by  0    0 β ∗ = β ∗ (d, ε) := β    II γ∗

if ϕ ≡ 0 if ϕ(0) > 0 if ϕ(0) ≤ 0 and ϕ(γ ∗ ) ≥ 0 if ϕ(γ ∗ ) < 0

(Regime 0), (Regime I), (Regime II), (Regime III).

(35)

In this definition, βII = βII (d, ε) is the unique solution of ϕ(·, d, ε) = 0 with ϕ defined by h i ϕ(β ∗ , d, ε) = −det(△ε∗ (β ∗ , d, ε)) = −det α(β ∗ , d)−1 e(ε) , △ε∗

(36)

= △ε∗ (β ∗ , d, ε) := ε∗2 (β ∗ , d, ε) − ε∗1 (β ∗ , d, ε),

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where the so-far undefined functions are specified below. The four regimes have the following crystallographic interpretation. Regime 0: The material is homogeneous and the energy does not depend on the microstructure. Regime I: There exist two optimal rank-I laminates. Regime II: The unique optimal microstructure is a rank-I laminate. Regime III: There exist two optimal rank-II laminates. For illustration, we visualize prototypes of rank-I and rank-II laminates in Figures 7 and 8, respectively.

-

Figure 7. A two-phase rank-I laminate in two space dimensions with corresponding normal vector. The strains are constant in the shaded and in the unshaded regions. The volume fraction of both phases, 0.5 in the picture, is prescribed by the macroscopic parameter d.

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h1









h2

Figure 8. A two-phase rank-II laminate in two space dimensions. The widths h1 and h2 of the slabs should be much larger than the thickness of the layers between the slab. To complete the definition (36) and for later use, set α(β ∗ , d) := d2 α1 + d1 α2 − β ∗ Q,

e(ε) := α2 (εT2 − ε) − α1 (εT1 − ε),

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ε∗i ≡ ε∗i (β ∗ , d, ε) := α−1 (β ∗ , d)ei (β ∗ , d, ε),

e1 (β ∗ , d, ε) := (α2 − β ∗ Q)ε − d2 (α2 εT2 − α1 εT1 ),

e2 (β ∗ , d, ε) := (α1 − β ∗ Q)ε + d1 (α2 εT2 − α1 εT1 ).

Hence ε∗2 − ε∗1 = [α(β ∗ , d)]−1 e(ε). We end this section by two explicit formulas, (37) and (38), for the first partial derivac . Their derivation is lengthy and can be found in [8]. tives of W

c
∂W (d, ε) = d1 α1 (ε∗1 − εT1 ) + d2 α2 (ε∗2 − εT2 ) : △ε∗ + W1 (ε∗1 ) − W2 (ε∗2 ) + V, (37) ∂d

where V depending on the different regimes is given by

and

  0 ∗ ∗ 2 V = β ∗ d(1 − d) ∂β ∂d kQ△ε k  ∗ ∗ (2d − 1)γ ϕ(△ε ) ∂β ∗ = ∂d

(

in Regimes 0, I, in Regime II, in Regime III

(Q(d2 α1 +d1 α2 −βQ)−1 (α2 −α1 )△ε∗ ):△ε∗ 0

((d2 α1 +d1 α2 −βQ)−1 (Q△ε∗ )):(Q△ε∗ )

in Regime II, otherwise.

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c with respect to the second variable yields The computation of the partial derivative of W i ∂h d1 W1 (ε∗1 ) + d2 W2 (ε∗2 ) + β ∗ d1 d2 det(ε∗2 − ε∗1 ) ∂ε i ∂β ∗ h

= d1 d2 α2 (ε∗2 −εT2 )−α1 (ε∗1 −εT1 ) : α−1 Q△ε∗ +det(△ε∗ ) ∂ε ∗ −1 ∗ T +d1 (α2 − β Q)α α1 (ε1 − ε1 )

c ∂W (d, ε) = ∂ε

+d2 (α1 − β ∗ Q)α−1 α2 (ε∗2 − εT2 ),

completed with

(38)

1 ∂β ∗ = − −1 ∗ Qα−1 (α1 − α2 )△ε∗ . ∂ε (α △ε ) : △ε∗

c and its partial derivatives is now With this result the collection of analytic formulas of W complete. The given representations are essential for a numerical implementation and are required for the existence proof in Theorem 3. c Theorem 3 (Existence of solutions for geometrically linear elasticity). Let D = 2 and W the energy defined in (33). Moreover, let the mobility tensor M be positive definite, let ψ be given by (22) and let the initial data satisfy

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ψ(a0 , b0 ) < ∞. Then, there exists a solution (a, b, u) to (17)–(19) that satisfies:
1 (i) a, b ∈ C 0, 4 [0, T ]; L2 (Ω) . (ii) ∂t a, ∂t b ∈ L2 (ΩT ).
(iii) u ∈ L∞ 0, T ; H 1 (Ω; RD ) . (iv) The feasible parameter range for (a, b) is given by (24). Proof: We can adapt the proof of Theorem 1. We can check with the formulas given c given by (33) and extended suitably to d ∈ R satisfies (A1) except for (26) above that W which needs to be modified to c (d, ε)| ≤ C1 (|ε| + |d|2 + 1), |∂ε W

(39)

see the explicit computation in (38). Condition (26) enters in the proof of the lower semicontinuity estimates and weak convergence estimates like Z Z c c (ak + bk , ε(uk )) W (a + b, ε(u)) ≤ lim inf W k→∞

Ω

Ω

for sequences (ak )k∈N , (bk )k∈N that converge to a and b, respectively, weakly in H 1 (Ω) and strongly in L2 (Ω) and for a sequence (uk )k∈N that converges weakly to u in H 1 (Ω; RD ). Since ak , bk converge strongly in L2 (Ω), the altered power |d|2 instead of |d| in (39) does not change the estimates. The proof now follows as in Theorem 1.

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c be the energy defined in (33). Assume further that the Theorem 4. Let D = 2 and W elastic moduli of the two phases are equal, i.e., α1 = α2 . Then the solution (a, b, u) in Theorem 3 is unique in the spaces stated in the theorem. Proof: Let again (ai , bi , ui ), i = 1, 2 be two solutions to (17)–(19). Under the assumption α1 = α2 the function ϕ, cf. (36), only depends on its first argument β, which implies that β ∗ is identical for any solution. Thus α is a constant matrix and we find that εi is identical for any solution. From this we learn that c (d1 , ε1 ) = ∂d W c (d2 , ε2 ). ∂d W

The theorem now follows from the Lipschitz continuity of g ′ analogous to Theorem 2. c (d1 , ε1 ) − Remark 3. For α1 6= α2 , the given proof fails. The critical term is ∂d W c c ∂d W (d1 , ε2 ). Even though ∂d W can be proved to be analytic, it is not possible to absorb powers of ε := ε(u1 ) − ε(u2 ) on the left.

3.3. The Extension of the AC-CH Model to Polycrystalline Elastic Materials In this section we outline how the Allen-Cahn/Cahn-Hilliard model can be extended to polycrystalline materials with laminates based on geometrically linear elasticity and homogenization. To this end we consider the first two equations of the system, (17) and (18), as before, where a and b now denote the corresponding macroscopic physical quantities of the polycrystalline material. Instead of the continuity equation (19), we now choose

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0 = div(∂ε W (a + b, ε)).

(40)

c is The boundary conditions are as in (20), while in the definition of the free energy (21), W replaced by W and ε is replaced by ε, i.e., Z
λ F (a, b, u) := ψ(a, b) + |∇a|2 + |∇b|2 + W (a + b, ε) + Wext (ε) dx, 2 Ω

where Wext is defined as in (23). This completes our extension of the Allen-Cahn/CahnHilliard model to polycrystalline elastic materials with prescribed volume fractions. To complete the theory, it remains to settle the existence of weak solutions also for this extended model. This question is currently open and a topic of ongoing research.

4.

Conclusion

In this article we derived extensions of the Allen-Cahn/Cahn-Hilliard system to elastic materials showing laminational structures. In particular we included (i) the linear elastic energy derived by Eshelby, (ii) a geometrically linear theory of elasticity for single crystals that takes phase fractions on the microscale into account, and (iii) a polycrystalline theory based on geometrically linear elasticity taking laminational structures into account, respectively. All generalized AC-CH models contain as special cases both the Allen-Cahn [10]

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and the Cahn-Hilliard equation [19], which are the two most-frequently used models in practical simulations and applications, e.g., in materials science, engineering, and biology. As a particular property, the considered elastic energy functionals incorporate the contributions of laminates on the microscale, which opens the floor for more advanced studies of segregation and precipitation phenomena in composite materials. We underline that we always assume that the temperature is kept constant. For nonisothermal settings, the thermodynamic analysis implies certain modifications to the model, which are especially challenging for the theory on the microscale. These extensions have to be done in such a way that the second law of thermodynamics continues to hold. It is currently open how this can be achieved. Besides the limiting assumption of constant temperature, the most important pending restriction is the postulation of small strain, included in (1). The technical problems of a large strain theory are striking and unsolved, and we refer to [3, 21] for further discussions and open questions. ˆ considered Next we come back to case (ii) and in particular to the specific energy W in Section 3.2. We point out that the formulas collected in Section 3.2. are essential for any numerics on the AC-CH model extended to microstructure. For further investigations and numerical studies we refer the interested reader to [8] and [9]. For the cases (i) and (ii) we showed in a mathematically rigorous way the existence and (in certain cases) the uniqueness of weak solutions, asserting the correctness of our approach. Analogous existence and uniqueness results for case (iii), i.e., for polycrystalline materials with laminational structures, remain open and are a topic of current research.

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[17] Eshelby, J. D. Elastic inclusions and inhomogeneities. Prog. Solid Mech. 1961, 2, 89–140. [18] Fratzl, P.; Penrose, O.; Lebowitz, J. L. Modelling of phase separation in alloys with coherent elastic misfit. J. Stat. Phys. 1999, 95, 1429–1503. [19] Garcke, H. On Cahn-Hilliard systems with elasticity. Proc. Roy. Soc. Edinburgh Sec. A 2003, 133, 307–331. [20] Kohn, R. V.; Vogelius, M. Relaxation of a variational method for impedance computed tomography. Comm. Pure Appl. Math. 1987, 60, 745–777. [21] Kohn, R. V.; Niethammer, B. Geometrically nonlinear shape-memory polycrystals made from a two-variant material. Math. Mod. Num. Anal. 2000, 34, 377-398. [22] Lexcellent, C.; Schl¨omerkemper, A. Comparison of several models of the determination of the phase transformation yield surface in shape-memory alloys with experimental data. Acta Mat. 2007, 55, 2995–3006. [23] Lexcellent, C.; Vivet, A.; Bouvet, C.; Calloch, S.; Blanc, P. Experimental and numerical determinations of the initial surface of phase transformation under biaxial loading in some polycrystalline shape-memory alloys. J. Mech. Phys. Solids 2002, 50, 2717– 2735. Composite Laminates: Properties, Performance and Applications : Properties, Performance and Applications, Nova Science Publishers,

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[24] Nye, J. F. Physical properties of crystals: their representation by tensors and matrices, Clarendon Press; New York: Oxford University Press, 1984. [25] Onuki, A. Ginzburg-Landau approach to elastic effects in the phase separation of solids. J. Phys. Soc. Japan 1989, 58, 3065–3068. [26] Schl¨omerkemper, A. On a Sachs bound for stress-induced phase transformations in polycrystalline shape memory alloys. PAMM Proc. Appl. Math. Mech. 2006, 6, 507– 508.

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[27] Ziemer, W. Weakly differentiable functions: Sobolev Spaces and Functions of Bounded Variation, Graduate Texts in Mathematics, Springer, New York, 1989.

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INDEX

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A absorption, viii, 2, 4, 6, 13, 17, 21, 45, 63, 66, 67, 73, 83, 87, 88, 96, 105, 109, 280, 281, 299, 300, 301 accuracy, 5, 22, 48, 51, 122, 133, 140, 273, 278, 299, 337, 367, 380, 383, 384, 386, 440 acetate, 89, 90, 321 ACI, 228 acid, 34, 93, 99, 117, 310, 315, 321 acidic, 317 acoustic, x, xi, 15, 25, 33, 36, 46, 47, 72, 342, 347, 348, 349, 350, 351, 352, 353, 354, 359, 361, 448 acoustic emission, x, xi, 25, 47, 72, 342, 347, 348, 349, 350, 351, 352, 353, 354, 359, 361, 448 acoustic microscopy, 11, 33 acoustic waves, 36 acrylate, 119 activation, 348 actuators, 348 additives, 89 adenine, 319, 321 ADH, 312, 313, 321 adhesion, 21, 68, 72, 84, 89, 105, 316 adhesive joints, 19 adhesive properties, viii, 83 adhesive strength, 91 adhesives, 20, 115, 118 adiabatic, 458 adjustment, 94, 335 adsorption, 100, 308, 311, 315 adulteration, 86 aeronautical, x, 26, 272, 347 aerospace, 24, 25, 35, 202, 258, 327, 348, 440 age, 16, 45, 54, 60 agent, 86, 116, 117, 307, 317, 358 agents, 8, 86, 88, 90, 311, 314, 440 aggregation, 92 aging, 118 aid, 16, 300, 309 air, 2, 11, 18, 75, 96, 114, 239, 259, 306, 352 air quality, 306 alcohol, 86, 90, 317, 321

algorithm, 50, 56, 57, 277, 334, 336, 337, 338, 345, 366, 367, 379, 395 alkali, 20 alkaline, 91, 115, 117 alloys, 74, 440, 465, 470, 472, 473, 475, 487, 488, 489 alternative, x, 11, 38, 311, 347, 349, 441, 448 alternatives, 89 aluminum, vii, 6, 38, 40, 70, 75, 85, 96, 113, 202, 249, 254, 258, 260, 261, 270, 317, 321, 330, 446 aluminum oxide, 321, 446 Alzheimer disease, 321 amine, 66 amino, 91, 314 amino acid, 91 amino acids, 91 ammonium, 94 ammonium chloride, 94 amorphous, 86, 93, 109, 310 amplitude, 19, 20, 21, 22, 25, 26, 36, 39, 40, 41, 43, 49, 298, 348, 356, 462 Amsterdam, 226, 230, 303 amylopectin, 117, 118 aniline, 314 anisotropy, 2, 203, 348, 440 ANN, 22, 52, 57 antenna, ix, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 248, 249, 250, 251, 252, 253, 254, 258, 259, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270 antibody, 307 antigen, 307, 318 APP, 168 aqueous solution, 320 Argentina, 365 argument, 88, 478, 486 arithmetic, 16 arrest, 50 aspect ratio, 56, 92, 209, 367, 384, 386, 393 assessment, 36, 38, 53, 66, 74, 199, 224, 268, 454, 460 assignment, 113 assumptions, xi, 38, 129, 279, 365, 366, 369

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Index

ASTM, 68, 76, 78, 80, 87, 88, 96, 97, 255, 270, 276, 303, 364 asymmetry, 255, 263 asymptotic, 47, 225, 246, 419, 478 atmospheric pressure, 88, 96 atomic force microscopy, 66, 70 ATP, 321 automation, 367 automotive application, 9 automotive applications, 9

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B background noise, 351 bandwidth, 33, 236, 238, 240, 241, 242, 243, 247, 248, 253, 254, 262, 351, 357 barrier, viii, 83, 84, 85, 86, 89, 90, 91, 92, 93, 95, 96, 98, 99, 101, 105, 106, 108, 109, 112, 113, 114, 115, 117, 118 barriers, 90, 108, 109 base catalysis, 310 beams, 6, 21, 40, 41, 240, 270 behavior, xi, 122, 139, 142, 200, 203, 204, 215, 216, 219, 220, 223, 224, 225, 226, 227, 229, 231, 253, 256, 257, 334, 336, 337, 342, 366, 379, 395, 403, 429, 446, 449, 467, 470, 471, 472 Beijing, 199, 362 benchmark, 367, 380 benchmarking, 37 biaxial, 199, 200, 332, 342, 343, 488 binding, 91, 307, 316, 317 biocompatibility, 310, 311, 314, 320 biocompatible, 320 biodegradable, 115, 117 biological stability, 315 biomarkers, 306 biomolecule, 308, 311, 313, 315, 319 biomolecules, 311, 315, 316, 317, 319 biopolymer, viii, 83, 93, 98, 105, 115 Biosensor, 306 biosensors, 312, 313, 314, 317 blends, 91, 98 bonding, 5, 112, 249, 308, 338 bonus, 320 Bose, 77, 78 Boston, 270, 464 boundary conditions, xi, 23, 50, 63, 216, 217, 219, 272, 276, 299, 303, 335, 365, 366, 367, 374, 375, 376, 380, 384, 386, 393, 395, 412, 477, 478, 486 bounds, 213, 214, 215, 231, 232, 466, 470, 471, 473 Bragg grating, xi, 32, 33, 76, 347, 348, 356, 357, 358 branching, 7, 310 broad spectrum, 61 broadband, 270, 353, 357 bubbles, 114 buffer, 313 burn, 74 butadiene, 90, 110

C calcium, 310 calcium carbonate, 310 calculus, 479, 480, 488 calibration, 94 capacity, 13, 53, 72, 139, 320, 328, 340, 356, 448, 454, 458 capillary, 109, 352, 355 capsule, 308 carbide, vii, 441, 442, 443, 444, 445, 447, 448, 449 carbohydrate, 118 carbohydrates, 91 carbon materials, 320 Carbon nanotubes (CNTs), 68, 315, 316, 317, 319, 321 carbonyl groups, 113 carboxyl, 319 carboxyl groups, 319 carboxylic, 315 carboxylic groups, 315 carboxymethyl cellulose, 86 carrier, 85 catalysis, 310 catalytic properties, 311 cavitation, 5, 19 CCC, 334 cell, 250, 258, 275, 276, 328, 338 cellulose, viii, 83, 84, 98, 101 cellulose fibre, 98 ceramic, vii, 113, 258, 265, 320 ceramics, 454 cereals, 89 certification, 47, 73, 74 Cetyltrimethylammonium bromide, 321 CFCs, 38 charge density, 94 chemical reactions, 112 chemical stability, 317 chemicals, 84, 86, 116, 117, 119, 440 chiral, 268 chitosan, 118, 313, 314, 317, 319, 320, 321 chloride, 321 chocolate, 85, 89, 93 chromatographic technique, ix, 305 cigarettes, 85 cladding, 30 classes, 64, 94, 202, 272, 309, 314 classical, xi, 43, 50, 133, 350, 366, 395, 403, 405, 437, 453, 466 classification, 202, 306, 466 clay, viii, 83, 92, 93, 95, 100, 115, 119, 318 closure, 23, 47, 48, 421 clustering, 5 coatings, viii, 83, 88, 89, 90, 92, 93, 94, 95, 96, 97, 101, 102, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 117, 118, 119 cocoa, 89

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Index coffee, 85 coherence, 37 collisions, 272 combined effect, 88 communication, 86, 234, 239, 253, 254, 268, 269 community, 306, 309 compatibility, 407, 410 compensation, 29 competitiveness, 90 compilation, 22 complement, 443 complexity, ix, 48, 234, 305 compliance, 123, 124, 125, 126, 127, 128, 129, 130, 137, 138, 139, 177, 406 composition, 49, 50, 84, 93, 309 compost, 115 compounds, 308 compressive strength, 9, 23, 46, 53, 54, 64, 66, 86, 139, 141, 179, 180, 331 computation, 437, 438, 470, 483, 485 computed tomography, 37 concentration, ix, 7, 8, 41, 70, 87, 91, 93, 94, 96, 98, 99, 103, 106, 132, 220, 305, 307, 309, 437, 446, 448, 457, 467, 477 conceptual model, 234 concrete, 202, 228, 229 condensation, 310 conductive, 306, 311, 319 conductivity, 28, 29, 75, 231, 313, 314 confidence, 39 configuration, 2, 17, 37, 40, 41, 61, 64, 235, 236, 264, 276, 281, 286, 294, 297, 459, 467, 468, 469, 476 conflict, ix, 233, 234 conformity, 50 Congress, 119, 303 conjugation, 437 conservation, 478 constant rate, 24 constituent materials, 10, 122, 125, 126, 129, 132, 133, 138, 142, 177, 180, 238 constraints, xi, 236, 338, 365, 374, 375, 380, 386, 446 construction, 15, 22, 23, 236, 238, 269, 308, 309, 314, 315, 317, 319, 320, 375 consumer goods, 84 consumption, 94, 307 contact time, 97, 109 contaminants, 115 contamination, 84 continuity, ix, 201, 203, 412, 415, 423, 486 control, 21, 53, 85, 86, 250, 266, 267, 276, 320, 348 convergence, xi, 207, 277, 278, 329, 334, 336, 337, 365, 367, 375, 380, 382, 383, 386, 403, 437, 485 conversion, 307, 309, 361 convex, 471 cooking, 91 cooling, 12 copolymers, 88, 90

core-shell, 321 corona, 112, 113 correlation, 11, 17, 21, 23, 30, 36, 46, 47, 53, 62, 67, 68, 140, 142, 286 correlations, 142, 456 corrosion, 39, 74, 327, 440 cosmetics, 85 costs, 50, 92, 202, 348, 443 counterfeit, 84 coupling, xi, 8, 238, 244, 248, 264, 278, 333, 358, 365, 367, 371, 373, 386, 405, 466 covalent, 307, 308, 315, 320 covalent bond, 308 covalent bonding, 308 coverage, 17, 97, 113 covering, 86 cracking, x, 3, 10, 12, 14, 16, 21, 23, 33, 37, 38, 46, 50, 52, 55, 74, 230, 279, 280, 301, 327, 328, 329, 331, 332, 334, 340, 342, 343, 349 CRC, 116, 117, 225, 401 creep, 328, 342, 405, 440 critical value, 9, 68 cross-linking, 103, 308, 317, 320 cross-sectional, 33, 64, 127 cryogenic, 12 crystalline, 4, 86, 91, 465, 466, 473 crystallinity, 20 crystals, 226, 489 CTAB, 321 cultivation, 91 cumulative distribution function, 39 curing, 3, 54, 66, 75, 97, 103, 241, 274 cutting force, 440, 443 cutting tools, 444, 450 cycles, 15, 19, 21, 22, 36, 40, 43, 49, 54, 55, 57, 72, 74, 255, 256, 257, 258 cycling, 23, 24, 39, 46, 54 cytosolic, 314

D dairy products, 85 damping, 72, 277, 336, 337, 440, 458, 462, 463 danger, 307 data analysis, 352 data processing, 26, 351 data set, 57 defects, x, 3, 9, 24, 26, 35, 38, 53, 86, 109, 113, 114, 141, 230, 347, 348, 350, 361, 441 defense, 440 definition, 2, 20, 26, 41, 55, 74, 122, 140, 207, 272, 273, 297, 299, 306, 309, 348, 468, 470, 473, 476, 478, 482, 483, 484, 486 deformation, 5, 6, 11, 15, 19, 37, 47, 60, 61, 71, 126, 129, 132, 137, 139, 140, 222, 228, 269, 272, 340, 350, 351, 395, 455, 456, 457, 467 degenerate, 423 degradation, x, 9, 10, 24, 35, 36, 45, 46, 50, 54, 57, 66, 72, 115, 122, 135, 139, 200, 243, 256, 299,

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494

Index

300, 306, 327, 328, 329, 332, 333, 334, 335, 336, 338, 340, 342, 348, 437, 440 degrees of freedom, 50, 366, 425, 463 dehydrogenase, 321 Delphi, 234, 235, 269 density, 38, 44, 72, 90, 112, 117, 217, 219, 222, 258, 274, 332, 370, 441, 455, 458, 468, 478 deposition, 319 derivatives, 115, 314, 372, 373, 396, 455, 463, 482, 484, 485 designers, 74, 199, 233 desorption, 87 detection, ix, xi, 25, 26, 28, 29, 30, 35, 37, 38, 51, 58, 300, 305, 306, 308, 310, 312, 313, 317, 319, 320, 347, 349, 355, 361 detergents, 84, 85, 89 deviation, 112, 292, 448 DFT, 163, 164, 165 DGEBA, 66 dialysis, 313 diamond, 260, 448 dielectric constant, 238, 248, 254, 258 dielectrics, 239 dietary, 306 differential equations, 466 differential scanning, 93 differential scanning calorimetry, 93 differentiation, 337, 372, 458 diffusion, 87, 351, 478, 488 diglycidyl ether of bisphenol, 66 diglycidyl ether of bisphenol A, 66 dimethylformamide, 315, 317 dipole, 243, 244, 245, 246 Dirac delta function, 456 direct measure, ix, 305, 361 direct observation, 45 disabled, 108 discontinuity, 348, 487 discretization, 50, 219, 221, 278, 366, 367, 419 discrimination, 349, 350, 356 dislocation, 351 dislocations, 468 dispersion, viii, 20, 69, 70, 83, 88, 89, 91, 92, 93, 98, 99, 100, 111, 112, 114, 115, 117 disposables, 90 distribution, 11, 23, 27, 29, 30, 39, 40, 43, 46, 47, 48, 49, 54, 55, 56, 58, 59, 65, 70, 86, 97, 217, 226, 406, 460, 464 distribution function, 39, 40, 56 division, 234, 459 DMF, 321 DNA, 321 dopamine, 314, 316 drop test, 16, 17 drying, 98, 103, 104, 112, 114 DSC, 93 DSM, 321 ductility, 5, 7, 68, 223 durability, 33, 54, 66, 90, 258, 327

duration, 5, 11, 15, 18, 50, 272, 273, 297, 300 dyes, 92 dynamic loads, 463 dynamic mechanical analysis, 70

E EDOT, 321 eigenvector, 413 elaboration, 349, 361 elastic constants, 225, 232, 328, 331, 335, 336, 340 elastic deformation, 126 elasticity, xi, xii, 50, 225, 226, 230, 231, 314, 403, 438, 454, 459, 465, 466, 467, 468, 469, 476, 477, 479, 480, 481, 482, 483, 485, 486, 488 electric conductivity, 75 electric potential, 29 electrical conductivity, 29, 314 electrical properties, 238, 239 electrical resistance, 26, 27, 28, 29 electrocatalysis, 317 electrocatalyst, 316 electrochemical deposition, 319 electrochemistry, 319 electrodes, 29, 311, 312, 313, 317, 319, 320, 321 electrolyte, 91, 94 electromagnetic, ix, x, 233, 239, 248, 252, 268, 276, 347, 348, 361 electromagnetic waves, ix, 233, 248, 252, 268 electron, ix, 271, 301, 307, 308, 309, 310, 311, 313, 314, 315, 316, 317, 319, 320 electron microscopy, 12, 301 electrostatic interactions, 317 ELISA, 307, 308 elongation, 4 emission, x, xi, 25, 26, 47, 72, 342, 347, 348, 349, 350, 351, 352, 353, 359, 361, 448 encapsulation, 320 endurance, 13, 17, 19 energy density, 468, 477, 478, 480 energy transfer, 300 entrapment, 308, 313, 314, 315 environment, 84, 86, 306, 310, 317, 319, 328 environmental advantage, 115 environmental effects, 74 enzymatic, 87, 115, 307 enzyme immobilization, 308, 317 enzymes, 307, 308, 313, 314, 315, 316, 319 epoxy resins, 70, 311 equilibrium, 50, 91, 101, 104, 134, 135, 335, 336, 404, 409, 415, 416, 417, 420, 437, 477 equilibrium state, 335 erosion, 14, 18 ESL, xi, 365 esterase, 312, 321 esters, 91 ethanol, 313, 319 ethers, 91 ethylene, 90, 97

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Index ethylene glycol, 97 ethylene vinyl alcohol (EVOH), 90 Europe, 93 evolution, x, 16, 18, 19, 20, 23, 33, 34, 41, 45, 48, 52, 57, 205, 209, 215, 216, 218, 219, 221, 222, 223, 228, 229, 234, 307, 327, 328, 329, 333, 334, 338, 340, 341, 342, 343, 348, 351, 457 excitation, 25, 264, 462 exercise, 122, 145, 146, 199, 342 exfoliation, 92 expansions, 135 exploitation, 311 exposure, 9, 39, 76, 87, 112, 114 extrusion, 91, 117

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F fabric, 6, 7, 8, 12, 15, 17, 36, 45, 46, 64, 65, 122 fabricate, 352 fabrication, ix, 4, 35, 76, 121, 233, 234, 248, 249, 269, 317 faith, 122 family, 74 fats, 87, 93 fatty acid, 90, 99, 104 FBG, 2, 32, 33, 358 FDA, 93 feeding, 236, 248 FEM, 200 ferrite, 38 fertility, 306 fertilization, 91 fiber bundles, 455, 456 fibers, vii, viii, ix, 85, 91, 121, 123, 126, 127, 129, 130, 132, 133, 137, 138, 139, 140, 141, 142, 201, 203, 209, 210, 215, 216, 217, 219, 222, 226, 227, 424, 435, 441, 442, 445, 446, 448, 454, 455, 456, 457 filament, 27, 232 filler particles, 111 fillers, viii, 4, 5, 68, 71, 83, 85, 89, 231 film, viii, 34, 49, 66, 83, 85, 87, 88, 89, 91, 92, 93, 95, 99, 101, 103, 104, 106, 109, 112, 115, 118, 119, 241, 249, 308, 310, 314, 321 film formation, 89, 119, 314 film thickness, 87 films, viii, 67, 68, 83, 88, 89, 90, 91, 93, 103, 108, 111, 112, 115, 117, 118, 119, 309, 310 filters, 91, 361 filtration, 70 financial support, 147 finite element method, 328, 334, 335, 337, 341, 400, 437 Finland, 93, 116, 117 fire, 75 fire resistance, 75 first generation, 314 fish, 85 flavors, 85

flexibility, 11, 86, 91, 314, 355, 380 flexural strength, 20, 62 flocculation, 99, 112 flow, 5, 29, 70, 100, 123, 333, 335, 458, 462, 478 fluctuations, 100, 353 fluid, 88, 446, 448 fluoride, 52 FMC, 321 folding, 85, 89 food, viii, 83, 84, 85, 86, 87, 88, 90, 91, 93, 108, 115, 116, 117, 118, 119, 306 food production, 91 Fortran, 277 Fourier, 57, 402, 463 fracture processes, 224 fractures, 33, 261 fragmentation, 57, 230 France, 302, 364, 487 FRC, 328 free energy, 48, 332, 478, 486 freedom, 16, 50, 319, 366, 425, 463 freezing, 86 friction, 351, 441, 445 Friedmann, 363 fruit juice, 85, 91 FTA, 97 fuel, 2, 328 functionalization, 315, 317 fusion, 320

G GAO, 302, 303 gas, 87, 88, 96 gas barrier, 88 gases, 84, 86 gel, 96, 308, 310, 311, 313, 314, 319, 321 gel formation, 310 generalization, xii, 465, 466, 476 generation, xi, 75, 313, 320, 347, 358, 361, 366 geometrical parameters, 48 Germany, 93, 94, 95, 116, 118, 465 GGT, 135 glass, vii, 5, 10, 11, 14, 15, 16, 17, 18, 30, 31, 34, 45, 52, 54, 55, 63, 65, 70, 71, 75, 95, 213, 214, 225, 238, 248, 254, 318, 342, 441, 442, 444, 445, 446, 447, 448, 449, 462, 463, 464 glass transition, 17 glass transition temperature, 17 glasses, 310 glucose, x, 91, 305, 310 glycol, 103 gold, 319, 321 gold nanoparticles, 319, 321 GPA, 339 GPS, 235 grades, 86, 116, 448 grain, 446, 448, 468, 469, 470 grain boundaries, 468

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Index

grains, 468, 469, 470 graph, 300 graphite, vii, 6, 34, 35, 37, 40, 57, 63, 142, 217, 218, 311, 317, 319, 381, 382, 384, 385, 386, 387, 388, 389, 390, 391, 392, 393, 448 gratings, xi, 76, 347, 356, 357, 358 Greece, 1, 77, 79, 80, 81 groups, 3, 90, 91, 99, 100, 112, 113, 258, 259, 281, 315, 319, 483 growth, vii, ix, 1, 3, 4, 11, 15, 17, 19, 21, 22, 23, 24, 25, 37, 42, 43, 46, 47, 48, 49, 50, 53, 54, 55, 56, 57, 58, 59, 64, 70, 75, 201, 202, 232, 315, 351 growth rate, 19, 57, 202 guanine, 319

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H haemoglobin, 315, 316 handling, 291, 306 harvesting, 91 HDPE, 90 health, x, xi, 25, 26, 33, 86, 306, 347, 348, 349, 361 healthcare, 306 heart, 306 heart disease, 306 heat, 34, 88, 89, 93, 117, 458 heat capacity, 458 heat transfer, 34 heating, 34, 458 height, 52, 259, 276, 295, 296, 298, 301 helix, 446, 448 hemicellulose, 20 hemisphere, 9, 63 hemp, 21, 54, 55 heterogeneity, 203, 348, 458 heterogeneous, 58, 203, 204, 206, 208, 225, 226, 228, 231, 307 hexafluorophosphate, 321 high fat, 64 high pressure, 338 high resolution, 37 high temperature, x, 11, 103, 347 high-speed, 361 high-tech, vii, 202 homogeneity, ix, 123, 201, 203 homogenized, 207 homogenous, 204, 208, 328 Horseradish peroxidase, 321 host, 32, 33, 306, 353, 356, 361 hot water, 39, 93 House, 270 HRP, 312, 313, 315, 318, 321 humidity, 73, 74, 87, 88 hybrid, viii, x, xi, 2, 4, 6, 12, 13, 30, 35, 43, 66, 67, 72, 231, 305, 310, 319, 320, 403, 415, 437, 438 Hybrid systems, 313, 318 hybridization, 12 hybrids, 6, 309, 319 hydro, 88, 91, 105, 109, 111, 314

hydrocarbon, 248, 254, 258, 265 hydrodynamic, 101 hydrogels, 314 hydrogen, 312, 319, 328, 329, 330, 338, 339, 340, 341 hydrogen peroxide, 312, 319 hydrophilic, 88, 91, 105, 109, 111, 314 hydrophobic, 88, 90, 97, 99, 103, 104, 105, 108, 109, 110, 111, 116, 118, 317 hydrophobicity, 88, 90, 109, 111, 115 hydrostatic tensile stress, 217 hydrothermal, 9, 39, 46, 76 hydroxides, 310 hydroxyl, 90, 91, 319 hydroxyl groups, 91, 319 hydroxypropyl, 91 hygienic, 86 hypothesis, 53, 273, 300, 406 hysteresis, 41, 257 hysteresis loop, 257

I IFT, 345 IgG, 315, 321 images, 18, 35, 51, 64, 73, 296 imaging, 4, 24, 34, 35, 37, 41, 55 imaging techniques, 4 immobilization, 308, 315, 317, 320 immunoglobulin, 321 immunological, 315 impact energy, vii, 1, 2, 6, 10, 12, 13, 14, 15, 16, 18, 19, 20, 21, 27, 32, 33, 34, 38, 44, 45, 46, 49, 52, 57, 58, 66, 67, 259, 260, 261, 262, 263, 273, 296 impact strength, 65 implementation, 15, 211, 234, 269, 329, 334, 343, 485 impulsive, 12 IMS, 56, 60 in situ, 26, 30, 64 in vitro, 310 in vivo, 310, 314 incidence, 97, 244 inclusion, 46, 206, 207, 208, 209, 211, 225, 226, 229, 231, 232, 314 independent variable, 299 India, 232 indicators, 26 indices, 356, 483 industrial, 84, 98, 230, 449 industrial processing, 98 industrial sectors, 449 industry, vii, viii, 35, 83, 121, 202, 258, 348, 351, 441, 443 inefficiency, 235 inelastic, 122, 147, 200 inequality, 333, 470, 478, 479, 482 inertia, 336, 337, 404, 408 inertness, 320

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Index infinite, 206, 209, 211, 225, 231, 367, 459 infrared, 34 infrastructure, 202 ingestion, 115 inhalation, 115 inhomogeneities, 206, 226, 227, 231, 465, 467, 488 inhomogeneity, 200 initiation, vii, x, 1, 2, 3, 4, 15, 16, 23, 25, 37, 50, 52, 53, 68, 75, 76, 280, 294, 327, 328, 329, 334, 340, 351, 460 injection, 41, 118, 321 injection moulding, 41 inorganic, 92, 309, 310, 311, 317, 318, 320 insertion, 60, 265 insight, 44, 216, 273, 367, 386 inspection, ix, 2, 4, 12, 24, 25, 32, 40, 54, 97, 258, 260, 271, 272, 277, 279, 301, 348, 350, 351, 352, 441, 478 instabilities, 352 instability, ix, xii, 48, 233, 336, 337, 453, 454, 455, 462, 463 Instron, 276 insulation, 75 integration, 24, 234, 235, 243, 268, 269, 276, 277, 278, 306, 320, 407 integrity, vii, x, xii, 1, 33, 74, 84, 88, 347, 351, 355, 439, 440, 441 Intel, 78, 79, 147 intensity, 32, 33, 46, 97, 348, 352, 353, 356, 403, 404, 425, 426, 427, 431, 432, 433, 437 interaction, 50, 53, 84, 86, 89, 97, 209, 215, 307, 315, 316, 320, 332, 344, 454 interaction effect, 209, 215 interaction effects, 209, 215 interactions, 98, 102, 119, 215, 309, 317, 329, 332, 369 interface, xii, 7, 8, 17, 18, 21, 22, 38, 48, 49, 53, 54, 72, 88, 112, 126, 129, 136, 204, 229, 306, 319, 334, 352, 356, 359, 412, 415, 423, 426, 437, 441, 465, 466, 467 interfacial bonding, 20 interference, ix, 305, 309, 348, 352, 356, 358, 361 interpretation, 351, 352, 457, 483 interval, 15, 109, 300, 435 intrinsic, 5, 356 iodide, 97 ionic, 90, 308, 313, 319, 321 ionic liquids, 313, 319 ions, viii, 22, 83, 98, 372, 383, 412 Islam, 79 ISO, 88, 96 isoelectric point, 94 isoparametric, 303 isothermal, xii, 177, 332, 333, 458, 465, 466, 478, 487 isotropic, ix, 6, 10, 12, 17, 71, 123, 126, 127, 128, 129, 147, 199, 200, 201, 203, 210, 211, 217, 218, 226, 230, 232, 244, 278, 330, 332, 333, 366, 372, 401, 437, 469, 471, 472, 476

isotropy, ix, 201, 203, 210 ITO, 311, 312, 318, 321

J Jacobian matrix, 372 joints, 4, 19, 20, 60, 61, 73, 74, 75, 440

K Kalman filter, 38 kinetic energy, 5, 9, 40, 57, 273, 300, 370, 371, 372 kinetics, 308, 317 Kirchhoff, 366, 367 Korea, 201, 233, 239, 248, 464 Korean, 226, 227, 323

L LAC, 321 Lagrange multipliers, 415 Lagrangian, 415, 416, 420, 454 lamina, 18, 26, 122, 125, 126, 132, 133, 135, 136, 137, 138, 139, 141, 142, 143, 144, 147, 148, 202, 204, 368, 369, 384, 422, 463 laminar, 231, 422, 463 lamination, xii, 11, 43, 64, 67, 84, 85, 91, 135, 386, 387, 388, 389, 390, 391, 392, 465, 466, 468, 470, 477 LAN, 248, 249, 251, 252 language, 140, 147, 277 large-scale, 33, 64, 366 laser, 37, 38, 275, 276, 356, 441 laser radiation, 356 lasers, 349 latex, 93, 94, 101, 109, 110, 111, 112, 113, 119 latexes, 90, 110, 115, 117, 119 lattice, 468, 469, 472, 477 law, 47, 50, 54, 57, 61, 204, 212, 228, 276, 277, 278, 303, 333, 406, 407, 411, 466, 478, 481, 487 laws, x, 38, 327, 328, 333, 334, 338, 341 LDH, 312, 321 leaching, 308, 314 lead, vii, ix, 1, 2, 3, 22, 23, 88, 99, 114, 272, 305, 309, 310, 317, 337, 359, 423 leaks, 351 life-cycle, 202 lifetime, 13, 41, 55, 74 ligament, 5 limitation, 128, 310 limitations, x, 38, 128, 313, 317, 337, 347 linear, x, xii, 21, 47, 48, 49, 54, 57, 70, 140, 141, 179, 180, 208, 211, 217, 219, 225, 251, 278, 291, 310, 327, 332, 353, 368, 406, 453, 465, 466, 467, 473, 476, 479, 480, 481, 482, 485, 486, 487, 488 lipoproteins, 306 liquid nitrogen, 446, 448 liquid water, 85, 88, 109 liquids, 86, 95, 97, 101, 103, 104, 313, 319, 320, 321 LOAD, 150, 151

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Index

location, 5, 26, 28, 30, 34, 47, 48, 57, 206, 350, 351 LOD, 312, 321 London, 116, 117, 119, 226, 228, 270, 464 long-term, 24, 33, 441 Los Angeles, 228 losses, 265, 348 low density polyethylene, 93 low temperatures, 12, 348, 361 low-density, 118, 236 low-temperature, 91 lubrication, 447 lying, 38, 423 lysis, 26

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M maize, 91 manipulation, 306, 329 manufacturer, 100, 105, 274 manufacturing, x, 3, 5, 70, 116, 234, 237, 241, 347 mapping, 30, 366, 371, 372, 479 market, 91, 106, 202 marketing, 84, 86 mask, 96 material degradation, 122, 200, 299, 300 material surface, 350, 351, 361 materials science, vii, 466, 487 Matrices, 396 MDB, 321 meals, 90 measurement, ix, 4, 26, 27, 28, 37, 38, 41, 42, 43, 45, 46, 57, 88, 95, 100, 113, 254, 276, 279, 291, 305, 306, 348, 356 measures, x, 305 mechanical behavior, viii, 121, 200, 203, 204, 215, 224, 342, 366, 395 mechanical performances, 46, 254, 350 mechanical properties, x, 36, 60, 62, 65, 66, 68, 70, 90, 91, 118, 119, 122, 126, 140, 200, 202, 204, 206, 226, 227, 230, 238, 248, 314, 327, 328, 329, 330, 335, 440, 454, 463 mechanical structure, ix, 233, 243, 269 mechanical testing, 24 media, 117, 226, 228, 231, 342, 438 median, 39, 275 MEF, 148, 150, 151, 152, 153, 155, 170, 171 melt, 89 membranes, 88, 313, 315, 317 memory, 472, 487, 489 mesoscopic, 206, 466, 467, 468, 470, 471, 474, 476 metal nanoparticles, 310, 317 metal oxide, 92, 317 metal oxides, 92, 317 metals, 35, 115, 328, 440, 449, 454 methanol, 321 methylene, 97 microcracking, x, 327, 328 microelectronics, 351 microencapsulation, 308

microenvironments, 320 microorganisms, 86 microscope, ix, 46, 271, 274, 277, 279, 280, 441 microscopy, 11, 33, 37, 66, 301, 351, 450 microspheres, 317 microstructure, ix, 60, 62, 66, 70, 201, 203, 227, 230, 467, 468, 477, 483, 487, 488 microstructures, 205, 224, 225, 228, 467, 468 microvoids, 332, 341 migration, 89, 112, 115 military, 202, 234, 268, 269 milk, 84, 85 mimicking, 50, 314 mineral oils, 89, 313 minerals, 92 mining, 230 Mitsubishi, 93 mixing, 57, 111, 478 mobility, 119, 477, 479, 480, 485 modeling, xii, 204, 205, 206, 210, 215, 216, 217, 224, 225, 227, 228, 229, 230, 231, 232, 244, 336, 344, 453, 454, 459, 463 models, vii, ix, x, xii, 1, 4, 9, 22, 23, 36, 46, 47, 48, 50, 53, 57, 62, 71, 75, 76, 203, 205, 216, 219, 224, 225, 227, 233, 327, 328, 329, 333, 465, 466, 468, 472, 486, 487, 488 modulation, 352, 356 modules, 265, 266 modulus, 13, 41, 54, 60, 62, 65, 68, 72, 124, 128, 129, 139, 140, 177, 179, 202, 207, 211, 213, 214, 215, 250, 277, 339, 340, 369, 381, 386, 458, 459, 482 moisture, viii, 83, 84, 85, 86, 88, 91, 96, 106 moisture content, 91 mold, 443 molecular weight, 66 molecules, ix, 84, 86, 92, 114, 305, 309, 316, 320 monolithic, 47, 74, 75, 141 monomer, 66 monomeric, 311, 313 monomers, 66, 313, 320 monotone, xi, 403, 480 montmorillonite, 92 Mori-Tanaka, 125, 126, 200, 206, 224, 230, 232 morphology, 4, 6, 19, 44, 52, 59, 66, 310, 466, 468 motion, 276, 463 moulding, 9, 41, 64, 65 movement, 86, 351, 462, 468 MPS, 313, 321 MTS, 255 MUA, 318, 321 multiphase materials, 226 multiplication, 209 multiplier, 416 muscle, 309 Myoglobin, 321

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N NaCl, 94 NADH, 313, 316, 321 Nafion, 311, 312, 313, 314, 317, 318 nanocomposites, 92, 118 nanofillers, 68 nanomaterials, 317, 319 nanoparticles, 69, 92, 310, 317, 319, 320, 321 nanosheets, 317 nanotubes, 311, 315, 316, 317, 319, 320, 321 NASA, 226, 302, 303, 344, 363, 400 natural, xi, xii, 4, 20, 76, 93, 365, 366, 368, 370, 372, 374, 375, 376, 379, 382, 386, 395, 396, 400, 465, 477 neglect, 468, 474 nerves, 309 Netherlands, 95, 230, 344 network, 22, 29, 52, 57, 242 neural network, 22, 52, 57 neurotransmitters, 314 New York, 117, 118, 119, 200, 224, 227, 228, 231, 232, 270, 344, 345, 400, 487, 489 Newton, 276, 478 next generation, 62 nitric oxide, 316 Nitrite, 313 nitrogen, 446, 448 no dimension, 382 nodal forces, 336 nodes, 53, 278, 418, 421, 425, 428, 432, 463 noise, 57, 266, 291, 348, 351, 361 non-destructive, x, 4, 36, 45, 64, 258, 347, 350, 351, 361, 441 nonlinear, x, xii, 8, 11, 28, 47, 48, 61, 68, 122, 129, 200, 223, 230, 232, 251, 277, 278, 291, 302, 327, 334, 335, 336, 337, 340, 345, 465, 470, 476, 487, 488 nonlinearities, 223 non-linearity, 21 non-uniform, 33, 61 non-uniformity, 33 normal, 50, 54, 91, 138, 177, 236, 244, 277, 344, 352, 367, 369, 375, 406, 426, 441, 455, 477, 483 novel materials, 306 nuclear, 440 nucleation, 216, 217, 219, 224 numerical analysis, ix, xii, 41, 271, 277, 278, 291, 303, 453

O observations, 7, 15, 30, 41, 47, 53, 64, 72, 273, 291, 355 OCT, 37 oil, 86, 90, 93, 311, 318 oils, 89, 90, 313 oligomer, 66

operator, 404, 409, 455, 479, 482 optical, ix, 4, 12, 30, 32, 33, 42, 43, 46, 58, 64, 76, 90, 271, 274, 277, 279, 280, 301, 320, 348, 349, 352, 353, 355, 356, 357, 358, 359, 441, 450 optical microscopy, 450 optical properties, 90 optimization, 48, 328 order statistic, 39 organic, 92, 235, 309, 311, 313, 314, 317, 318, 319, 320 organic polymers, 319 organic solvents, 314, 319 organoleptic, 86 orientation, ix, 2, 11, 12, 13, 15, 17, 22, 23, 47, 49, 56, 92, 201, 202, 203, 204, 232, 280, 319, 366, 367, 386, 387, 388, 389, 390, 391, 392, 405, 422, 428, 440, 441, 446, 469 oscillation, 28 oscillations, 296, 298, 300, 375, 446 oxidation, 85, 87, 100, 317 oxidative, 113, 316 oxidative damage, 113 oxide, 70, 91, 316, 317, 321, 446 oxide nanoparticles, 317 oxides, 310 oxygen, viii, 83, 84, 85, 87, 88, 91, 96, 108, 109, 112, 115 ozone, 112, 113

P PAA, 313, 321 packaging, viii, 83, 84, 85, 86, 89, 90, 91, 93, 97, 108, 109, 116, 117, 118, 119, 234 PANI, 318 paper, viii, 46, 47, 54, 58, 60, 62, 83, 85, 84, 86, 88, 89, 90, 91, 95, 97, 98, 101, 109, 114, 115, 116, 117, 118, 119, 302, 454, 467, 478 papermaking, 84 parameter, xii, 3, 9, 18, 23, 26, 36, 53, 125, 142, 177, 181, 273, 344, 386, 404, 414, 417, 419, 439, 442, 443, 445, 463, 467, 477, 478, 480, 483, 485 particles, 3, 5, 18, 19, 71, 92, 93, 99, 100, 111, 112, 114, 115, 231, 311, 446 partition, 415 passive, 25, 265, 350 pasta, 89 pathways, 92, 113, 114 PDC, 321 PEEK, 11, 39, 446 penalty, 48 pendulum, 4, 9, 18, 20 perception, 57 perforation, 17, 18, 20, 33, 35, 62, 63 periodic, xi, 33, 34, 227, 230, 231, 347, 349, 356, 359, 361, 370, 454, 462 periodicity, 454 permeability, viii, 83, 87, 88, 91, 104, 106, 108, 117, 243

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Index

permeation, 86, 92, 109 permittivity, 243 perturbation, 215, 356 pesticides, 90 PET, 67, 68, 90, 93, 101, 106, 107, 108, 109, 115 pharmaceutical, 85 phase shifts, 353 phase transformation, 207, 466, 472, 473, 487, 488, 489 phase transitions, 466, 478 phenol, 310 phenomenology, 4 phone, 97 photodetectors, 359, 361 photographs, 57, 64, 441 photosensitivity, 356 physical and mechanical properties, 329 physical properties, ix, 84, 201, 203, 232, 316 physics, 466 piezoelectric, x, 25, 26, 52, 275, 276, 347, 348, 350, 358, 361 pigments, 115 planar, 129, 236 plaques, 438 plastic, viii, x, 5, 11, 15, 64, 71, 83, 85, 88, 89, 90, 93, 101, 106, 108, 109, 112, 115, 118, 123, 124, 125, 129, 137, 140, 141, 142, 207, 302, 303, 306, 332, 333, 340, 345, 347, 351, 361, 440, 443, 456, 457, 468 plastic deformation, 71, 129, 142, 332, 333, 340, 456 plastic strain, 207, 440, 456 plasticity, 332, 333, 343 plastics, 72, 90, 91, 115, 116 platelets, 92 PLUS, 149 Poisson, 128, 179, 214, 277, 341, 369 Poisson ratio, 369 polar groups, 112 polarization, 348 polycrystalline, xii, 465, 466, 467, 468, 470, 471, 472, 475, 476, 477, 479, 486, 487, 488, 489 polyelectrolytes, 317 polyester, viii, 5, 10, 12, 15, 18, 21, 40, 41, 45, 46, 54, 63, 65, 70, 83, 90, 93, 94, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 113, 115, 118, 277, 445, 447 polyethylene, 6, 85, 90, 93, 94, 115, 118 polyethylene terephthalate, 90, 93 polyimides, vii polymer, viii, 2, 4, 5, 16, 20, 30, 37, 41, 53, 60, 62, 70, 75, 83, 84, 86, 87, 88, 89, 91, 92, 93, 94, 98, 111, 113, 117, 118, 225, 228, 229, 272, 311, 313, 314, 315, 316, 317, 319 polymer composite material, 20 polymer composites, 16 polymer film, 117 polymer films, 117 polymer matrix, 2, 4, 37, 41, 75, 87, 92, 225, 229, 272

polymer systems, 314 polymeric composites, xii, 439, 440, 441, 446, 449 polymers, viii, 4, 5, 15, 49, 68, 83, 87, 88, 90, 91, 92, 106, 111, 119, 309, 314, 317, 319, 320 polynomial, 329, 330, 342, 344, 375, 376 polynomials, xi, 365, 366, 375, 376, 382, 395, 401 polyolefins, 90 polypropylene, 30, 72, 90 polysaccharides, viii, 83, 90 polytetrafluoroethylene, 66 polyurethane, 41 polyurethanes, 86, 88 pores, 97, 109 porosity, 84, 89, 95, 116, 310, 317, 320 porous, 88, 89, 105, 109, 116, 228, 317, 320 porous materials, 116 Portugal, 347, 439, 450 potato, 91, 93, 115, 118 potential energy, 371, 377 pouches, 85 powder, 89, 91, 93 powders, 89 power, 57, 91, 113, 139, 141, 253, 265, 266, 333, 348, 353, 355, 356, 357, 358, 462, 485 PPD, 313, 321 PPI, 117 PPO, 311, 312, 318, 321 Prandtl, 405, 409 precipitation, 465, 466, 468, 487 prediction, ix, 11, 14, 15, 23, 26, 43, 47, 49, 59, 62, 74, 75, 76, 122, 142, 199, 200, 201, 203, 204, 205, 206, 209, 214, 215, 216, 219, 220, 221, 223, 224, 229, 303, 337 pre-existing, 276 preservative, 306 pressure, 12, 35, 54, 57, 86, 87, 88, 96, 241, 338, 339, 340, 341, 343, 345, 348, 351, 353, 446, 458, 463 printed-circuit, 236 printing, 84, 93, 112, 115 pristine, 39 probability, 40, 216, 217 probability distribution, 217 probe, 95, 264, 270, 313 production, 54, 85, 86, 90, 91, 115, 307, 317 program, 15, 54, 122, 147, 177, 216, 269, 277, 278, 339, 341 projectiles, 6, 9, 62 promote, 317 propagation, vii, viii, 1, 2, 4, 5, 7, 8, 13, 16, 19, 20, 22, 23, 25, 26, 39, 48, 52, 53, 54, 55, 56, 57, 67, 68, 75, 76, 244, 272, 286, 291, 296, 297, 300, 328, 352, 361, 426, 430, 435, 440, 455 property, 65, 74, 84, 87, 137, 205, 487 propylene, 91 protection, viii, 44, 83, 84, 85, 86, 88, 90, 106, 237 protective coating, 84, 109 protein, 91, 118, 307, 312, 314, 316, 319 protein films, 118

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Index protocol, 54, 119 prototype, 469 Pseudomonas, 321 PSS, 321 PTFE, 66, 238 pulps, 85 pulse, 33, 34, 38, 50 pure water, 99 PVA, 313, 317, 321 PVC, 321 pyrolytic graphite, 317 pyrrole, 314, 315

Q quantum, 34 quasi-linear, 62 quinone, 308

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R radar, 440 radiation, ix, 74, 233, 238, 240, 241, 242, 244, 250, 252, 253, 254, 263, 264, 356, 440 radical, 314 radiography, 76 radius, 5, 26, 33, 277, 337, 338, 441, 476 rain, 74 random, 9, 49, 53, 70, 225, 228, 351, 469 raw materials, 60, 85 Rayleigh, 401 reactivity, 315 reading, 95 real time, 24, 26, 30 reality, 88, 140 recall, xii, 465, 466, 470, 471, 476 reception, 25, 234, 235 receptors, 307 reciprocity, 244 recognition, x, 305, 306, 307, 308, 309 recovery, x, xi, 347, 355, 361 redistribution, 350 redox, 308, 314, 316, 319, 320 reduction, 3, 9, 10, 11, 14, 21, 23, 24, 33, 36, 39, 41, 45, 49, 54, 58, 60, 62, 74, 75, 86, 103, 107, 108, 109, 111, 139, 237, 342, 351, 386, 442, 443, 445, 446, 448 reflection, ix, 32, 233, 237, 240, 241, 242, 244, 246, 252, 254, 262, 263, 352, 358 refractive index, 354 refrigeration, 86 regular, xi, 70, 105, 278, 366, 403, 404, 405, 416, 418, 419, 424, 442, 462 reinforcement, 9, 12, 15, 17, 21, 22, 31, 59, 60, 61, 63, 64, 65, 66, 72, 75 reinforcing fibers, 442 rejection, 441 relaxation, 33, 277, 303, 448, 458, 468, 469, 470 relevance, 17, 68, 86, 447

reliability, 14, 38, 41, 67, 121 380, 383, 454 renormalization, 207, 208 repair, 41 research, vii, viii, ix, xii, 22, 39, 53, 60, 66, 73, 83, 90, 118, 272, 273, 288, 299, 305, 320, 334, 341, 380, 440, 449, 450, 453, 464, 466, 486, 487 research and development, 147, 341 researchers, 30, 199, 273, 366, 383 resins, 21, 30, 90, 115, 118, 130, 137 resistance, 4, 5, 7, 9, 17, 22, 26, 27, 28, 29, 33, 35, 43, 50, 55, 60, 61, 64, 65, 73, 74, 75, 86, 90, 93, 108, 115, 236, 237, 238, 268, 272, 273, 302, 327, 405, 440, 454, 456, 457 resistivity, 26 resolution, ix, 37, 305, 443 retail, x, 305 retention, 13, 39, 309 rheological properties, 91, 105 rhythm, 462 rice, 118 rigidity, 66, 71, 211, 237, 238, 258, 260, 404, 418, 420 rings, 275, 276, 286, 291 risk, 86, 98 risks, 115 Ritz method, xi, 365, 366, 372, 374, 376, 395, 401 robustness, 336 Rome, 363 room temperature, 6, 7, 12, 17, 41, 72, 94 root-mean-square, 351 rotations, 50, 212, 408, 456, 457, 469, 471, 472 roughness, 7, 111, 114, 116, 445, 446, 447, 448, 450 Royal Society, 226, 228 rubber, 4, 5, 19, 71

S safety, 26, 74, 234, 350 salinity, 74 salts, 91 sample, x, 5, 33, 34, 44, 94, 95, 206, 306, 314, 475 sampling, 250, 276, 352 SAR, 269 satellite, 236, 239, 253, 254, 269 satin weave carbon, 62 satisfaction, 374 scaffold, 310 scaffolding, 320 scaffolds, 309 scalar, 48, 217, 334, 471, 473, 475, 476, 478, 483, 487 scaling, 478 scanning electron microscopy, 2, 6, 12, 13, 18, 19, 20, 21, 33, 39, 41, 57, 70, 72, 114, 174, 274, 279 scatter, 27, 225, 273 scattering, 93 schema, 357 scientific knowledge, 306 seals, 89

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502

Index

searches, 345 segregation, 465, 466, 467, 468, 472, 487, 488 selecting, 366 selectivity, x, 305, 306, 307, 308, 320 SEM micrographs, 13, 20 semiconductor, 306 semiconductors, 317 semi-crystalline polymers, 5 semigroup, 479 sensing, 27, 28, 29, 306, 314, 315, 317, 320, 348, 350, 352, 356 sensitivity, viii, x, 25, 28, 30, 37, 48, 59, 83, 308, 319, 320, 347, 348, 353, 354, 361 sensors, x, 25, 26, 30, 32, 33, 52, 76, 275, 306, 313, 317, 320, 347, 348, 350, 356, 361 separation, 47, 61, 66, 91, 115, 354, 488, 489 series, 2, 4, 6, 8, 18, 39, 54, 70, 84, 88, 91, 95, 97, 203, 254, 359, 402, 455 serum, 321 serum albumin, 321 severity, xi, 30, 52, 347, 350, 351 sex, 369 Shanghai, 121 shape, 6, 11, 42, 43, 55, 99, 217, 220, 222, 259, 263, 276, 278, 279, 366, 372, 400, 401, 402, 404, 417, 440, 462, 472, 487, 489 shape memory alloys, 489 shape-memory, 470, 472, 473, 475, 487, 488 shear, 7, 13, 18, 37, 44, 47, 50, 52, 53, 60, 61, 65, 67, 70, 74, 94, 98, 99, 100, 141, 142, 199, 210, 211, 214, 225, 229, 236, 237, 250, 258, 277, 278, 331, 339, 369, 455, 460, 471 shear deformation, 61 shear rates, 94, 99 shear strength, 52, 53, 65, 142, 258, 331 shearography, 38, 52 shipping, 116 shock, 50 short-term, 21 shuttles, 307 sign, 96, 332, 426, 427, 435 signals, xi, 41, 57, 94, 237, 266, 276, 347, 348, 349, 351, 354, 359, 361 signal-to-noise ratio, 359 signs, 20, 55, 332 silane, 7, 8 silica, 96, 310, 319, 352, 356 silica glass, 356 silicate, 92, 118 similarity, ix, 271 simulation, 9, 16, 23, 26, 29, 44, 48, 51, 53, 57, 142, 147, 232, 239, 254, 274, 277, 278, 299, 344, 457, 464 simulations, 8, 48, 203, 216, 217, 218, 219, 223, 224, 242, 273, 300, 487 Singapore, 303, 363 single crystals, xii, 465, 466, 486, 488 singular, xi, 329, 336, 337, 403, 404, 405, 411, 414, 416, 418, 419, 421, 422, 424, 438

singularities, 3, 410, 437 sites, 91, 320 skin, 25, 49, 54, 73, 75, 233, 234, 264, 265, 268, 269, 309, 454 SME, 226 sodium, 94, 100 software, 26, 97, 122, 147, 254, 338, 352, 441 solar, 74 sol-gel, 310, 319, 320 solid polymers, 5 solubility, 315, 317 solutions, 38, 47, 50, 54, 76, 103, 215, 231, 335, 337, 366, 376, 380, 384, 386, 395, 402, 410, 466, 479, 480, 481, 485, 486, 487 solvents, 90, 313, 317 South Africa, 230 South Korea, 201 soybean, 93 space-time, 477 Spain, 119, 365 spatial, 27, 58, 61, 264, 370 spatial array, 264 species, 86, 87, 309, 314 specific surface, 69 specificity, 307 spectrum, 32, 33, 54, 57, 61, 248, 306, 352, 356, 357, 358, 359 speed, 37, 38, 113, 250, 255, 334, 441, 442, 443, 444, 445, 446, 448, 449, 462, 463 spheres, 232 spherulite, 4 spin, 308 spindle, 444, 445, 448 springs, 54 sputtering, 319 stability, ix, 7, 46, 48, 59, 86, 91, 93, 94, 98, 99, 100, 103, 233, 308, 315, 317, 320 stabilization, 112, 337, 349 stabilize, 314, 354 stabilizers, 90 stable crack, 50 stages, 57, 219, 223 standard deviation, 107, 108, 110, 113 standards, 303, 351 starch, viii, 83, 89, 90, 91, 93, 95, 96, 98, 99, 100, 101, 102, 103, 105, 106, 107, 108, 109, 110, 111, 112, 113, 115, 116, 118, 119 starches, 86, 91, 118 steel, vii, 6, 11, 202, 275, 276, 355, 441, 442, 443, 445, 446, 447, 448, 449, 460, 461 steric, 99 stiffeners, 25 stimulus, 348 stochastic, 462 Stora Enso, 93 storage, x, 86, 306, 328, 329, 330, 338, 339, 340, 341, 350, 351 strategic, 308 strategies, 228, 268, 307, 309

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Index stress fields, xi, 3, 207, 348, 403, 405, 409, 453 stress intensity factor, xi, 9, 403, 405, 421, 425, 426, 430, 433, 437, 438 stress level, 40, 55, 142 stress-strain curves, 142, 180, 453 stretching, xi, 5, 365, 371, 381, 393, 395 strikes, 276 structural health monitoring (SHM), 26 STRUCTURE, 403 styrene, 90, 110 substances, 86, 89, 115, 319 substitutes, vii, 202 substitution, 91, 111, 379, 396, 458, 459 substitution reaction, 91 substrates, 85, 88, 90, 91, 95, 96, 97, 101, 103, 104, 105, 111, 270 sugar, 84 sugars, 91 sulfate, 85 sulphate, 93 sulphur, 306 Sun, 202, 209, 211, 229, 231, 232, 277, 303, 322, 324, 344, 425, 438 superposition, 228, 356, 402 supply, 265, 266 supramolecular, 315 surface area, 7, 92, 311, 317, 454 surface energy, 97, 101, 102, 103, 104, 105, 112, 113 surface layer, 3, 111, 113 surface properties, 111 surface region, 109 surface roughness, 111, 114, 116, 445, 446, 447, 448, 450 surface tension, 95, 97, 101, 102, 103, 104, 112 surface wave, 350 surfactant, 89, 95, 103, 112, 113, 319 suspensions, 98 Sweden, 76, 93, 95, 97, 118 swelling, 62, 88, 93, 98, 320 switching, 266 Switzerland, 93, 97, 363 symbols, 282, 284, 285, 287, 289 symmetry, 23, 131, 217, 425, 472, 478 synergistic, viii, 83 synergistic effect, viii, 83 synthesis, 310 systems, viii, x, 2, 4, 17, 18, 22, 26, 30, 50, 90, 142, 202, 305, 306, 307, 310, 311, 312, 313, 314, 318, 320, 366, 419, 466, 476, 488

T talc, 93, 102, 107, 110, 111 tanks, 2, 351 targets, 10 taste, 86, 89 Taylor series, 455

503

technology, ix, x, 37, 54, 60, 72, 74, 92, 116, 117, 119, 121, 233, 234, 236, 268, 269, 305, 306, 348, 443 Teflon, 311, 319 television, 235 TEM, 70 temporal, 37, 57, 370 tensile, 4, 5, 9, 13, 14, 15, 18, 21, 23, 26, 39, 40, 43, 45, 46, 49, 53, 54, 55, 62, 63, 64, 65, 66, 124, 125, 138, 139, 140, 141, 142, 179, 180, 202, 210, 255, 331, 339, 453, 455, 460, 461 tensile strength, 9, 14, 18, 26, 39, 40, 45, 46, 55, 64, 65, 139, 179, 339, 461 tensile stress, 64, 139, 180, 453 tension, 5, 13, 21, 28, 40, 41, 54, 67, 72, 101, 102, 124, 139, 140, 142, 228, 293, 301, 343, 409, 460 TEOS, 318, 319, 321 test data, 44 textile, 45, 122, 200, 231 thermal analysis, 19, 132 thermal expansion, vii, viii, 121, 132, 135, 137, 138, 140, 179, 232, 454, 458 thermal load, 39, 135, 136, 342 thermodynamic, 328, 329, 332, 333, 487 thermodynamics, 333, 342, 466, 478, 481, 487 Thermoelastic, 41 thermo-mechanical, 4, 71, 85, 178 thermoplastic, 30, 66, 88 thermoplastics, 65 thermosetting, 30, 60 thermosetting polymer, 30 thin film, 310 thin films, 310 Thomson, 80, 343 three-dimensional, 23, 33, 47, 48, 53, 217, 272, 409, 464 threshold, 11, 12, 13, 16, 35, 47, 59, 217, 273, 290, 291, 292, 299 thresholds, 20, 274, 299, 301 thymine, 319 tin oxide, 321 tissue, 43, 66, 67 titania, 310 titanium, vii, 70, 74, 448 titanium dioxide, 70, 318 titration, 94 TLE, 369 TMP, 85 tolerance, viii, 2, 3, 7, 9, 12, 13, 16, 44, 60, 62, 66, 74, 76, 105, 225, 229, 328 torque, 445 total energy, 16, 21, 40, 50, 68, 280, 282, 284, 286, 287, 299, 300 toughness, xi, 6, 7, 13, 21, 22, 35, 39, 46, 48, 49, 53, 54, 60, 61, 65, 66, 67, 68, 69, 71, 72, 75, 76, 403, 440 toxicological, 86 traction, 61, 409, 412, 423, 454

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504

Index

transducer, 306, 307, 308, 309, 310, 311, 314, 317, 320, 350, 358 transduction, 306 transfer, x, 3, 6, 34, 41, 64, 65, 75, 147, 237, 245, 305, 307, 308, 309, 310, 311, 313, 314, 316, 317, 319, 320, 457, 458, 459 transformation, 134, 207, 212, 215, 216, 351, 369, 372, 454, 471, 472, 473, 487, 488 transformation matrix, 134 transformations, 487 transition, 4, 5, 19, 46, 272, 355, 445, 479 transition temperature, 4, 17, 445 transitions, 4, 46, 62, 117 translation, 483 translational, 367 transmission, 87, 88, 92, 96, 108, 109, 115, 243, 244, 245, 246, 266 transparency, 320 transparent, 34, 89, 310 transport, x, 86, 92, 108, 113, 306, 313, 315 transportation, 84, 440 transpose, 135, 467 trend, 290, 443 trial, 291, 383 trial and error, 291 triglycerides, 90 Tryptophan, 321 TSA, 41 tungsten, 441, 442, 443, 444, 445, 447, 448, 449 tungsten carbide, 441, 442, 443, 444, 445, 447, 448 two-dimensional, ix, 33, 48, 50, 129, 227, 271, 409 tyrosine, 319

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U ultimate analysis, 459 ultrasonic waves, xi, 347, 349, 358, 359, 361 uniform, 20, 33, 61, 65, 207, 208, 278, 367, 370, 383, 395, 424, 433, 435, 437, 438, 443, 473, 474, 481 urethane, 9 UV light, 85

V vacuum, 18, 91, 241 validation, 48, 53, 213, 224, 234, 343, 438 validity, ix, 20, 28, 215, 271, 279, 286, 466, 481 vapor, 117 variability, 49, 459 variable, x, 19, 20, 22, 46, 49, 299, 327, 328, 332, 333, 336, 341, 401, 404, 410, 485 variables, 204, 328, 332, 333, 334, 339, 341, 419, 429 variation, 5, 11, 29, 39, 49, 50, 66, 106, 132, 133, 135, 136, 147, 205, 245, 246, 247, 248, 349, 356, 360, 361, 386, 393, 405, 435, 467, 487 vector, 211, 334, 377, 404, 414, 417, 418, 419, 420, 455, 457, 471, 483

vegetables, 85, 87 vehicles, 235, 268, 328 velocity, ix, 2, 3, 5, 6, 9, 10, 11, 12, 15, 18, 20, 37, 38, 40, 44, 46, 237, 271, 272, 273, 274, 276, 286, 288, 294, 296, 297, 298, 299, 300, 301, 302, 303, 337, 338, 458, 462 vessels, 328, 339, 348, 351 vibration, 5, 300, 366, 367, 370, 374, 379, 383, 386, 387, 388, 389, 390, 391, 392, 393, 395, 400, 401, 402, 463 vinylester, 6, 13, 15, 18, 20 viscoelastic properties, 19 viscosity, 70, 91, 94, 98, 99, 101, 105, 106 visible, 41, 44, 52, 54, 55, 59, 66, 73, 275, 277, 279, 291, 294, 299, 358 voids, 86, 210, 230 vortex, 462 vortices, 462

W water, viii, 19, 45, 83, 86, 87, 88, 89, 90, 91, 93, 94, 96, 97, 98, 99, 100, 101, 103, 105, 106, 108, 109, 110, 113, 114, 115, 117, 314, 317, 446, 447 water absorption, viii, 45, 83, 109 water vapour, viii, 83, 87, 90, 96, 106, 113, 115, 117 wave packet, 25 wave propagation, 36, 75, 298 wavelengths, 354 waxes, 89, 90 wear, 14, 440, 442, 443, 445, 447, 448, 449 web, 21, 74 Weibull, 39, 40, 216, 217, 219, 453 Weibull distribution, 39, 40 weight reduction, 74 wells, 470, 471 wet coating, 114 wettability, 88, 112 wetting, 89, 101, 103, 105, 108, 112, 115, 119 wheat, 91, 118 whey, 118 wind, 28, 74, 462, 463 wireless, ix, 28, 29, 233, 248, 249, 251, 252, 268 Wireless LAN, 250 withdrawal, 113 wood, vii, 85, 202 workability, 348 workers, 317, 380 working conditions, 13 workplace, ix, 305

Z zeta potential, 94 ZnO, 317, 318

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