Composite Materials Research Progress

COMPOSITE MATERIALS RESEARCH PROGRESS

COMPOSITE MATERIALS RESEARCH PROGRESS

LUCAS P. DURAND EDITOR

Nova Science Publishers, Inc. New York

Copyright © 2008 by Nova Science Publishers, Inc.

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Published by Nova Science Publishers, Inc.

2008 2007034054

New York

CONTENTS Preface

vii

Chapter 1

Multi-scale Analysis of Fiber-Reinforced Composite Parts Submitted to Environmental and Mechanical Loads Jacquemin Frédéric and Fréour Sylvain

Chapter 2

Optimization of Laminated Composite Structures: Problems, Solution Procedures and Applications Michaël Bruyneel

Chapter 3

Major Trends in Polymeric Composites Technology W.H. Zhong, R.G. Maguire, S.S. Sangari and P.H. Wu

109

Chapter 4

An Experimental and Analytical Study of Unidirectional Carbon Fiber Reinforced Epoxy Modified by SiC Nanoparticle Yuanxin Zhou, Hassan Mahfuz, Vijaya Rangari and Shaik Jeelani

129

Chapter 5

Damage Evaluation and Residual Strength Prediction of CFRP Laminates by Means of Acoustic Emission Techniques Giangiacomo Minak and Andrea Zucchelli

165

Chapter 6

Research Directions in the Fatigue Testing of Polymer Composites W. Van Paepegem, I. De Baere, E. Lamkanfi, G. Luyckx and J. Degrieck

209

Chapter 7

Damage Variables in Impact Testing of Composite Laminates Maria Pia Cavatorta and Davide Salvatore Paolino

237

Chapter 8

Electromechanical Field Concentrations and Polarization Switching by Electrodes in Piezoelectric Composites Yasuhide Shindo and Fumio Narita

257

1

51

vi Chapter 9

Index

Contents Recent Advances in Discontinuously Reinforced Aluminum Based Metal Matrix Nanocomposites S.C. Tjong

275

297

PREFACE Composite materials are engineered materials made from two or more constituent materials with significantly different physical or chemical properties and which remain separate and distinct on a macroscopic level within the finished structure. Fiber Reinforced Polymers or FRPs include Wood comprising (cellulose fibers in a lignin and hemicellulose matrix), Carbon-fiber reinforced plastic or CFRP, Glass-fiber reinforced plastic or GFRP (also GRP). If classified by matrix then there are Thermoplastic Composites, short fiber thermoplastics, long fiber thermoplastics or long fiber reinforced thermoplastics There are numerous thermoset composites, but advanced systems usually incorporate aramid fibre and carbon fibre in an epoxy resin matrix. Composites can also utilise metal fibres reinforcing other metals, as in Metal matrix composites or MMC. Ceramic matrix composites include Bone (hydroxyapatite reinforced with collagen fibers), Cermet (ceramic and metal) and Concrete. Organic matrix/ceramic aggregate composites include Asphalt concrete, Mastic asphalt, Mastic roller hybrid, Dental composite, Syntactic foam and Mother of Pearl. Chobham armour is a special composite used in military applications. Engineered wood includes a wide variety of different products such as Plywood, Oriented strand board, Wood plastic composite (recycled wood fiber in polyethylene matrix), Pykrete (sawdust in ice matrix), Plastic-impregnated or laminated paper or textiles, Arborite, Formica (plastic) and Micarta. Composite materials have gained popularity (despite their generally high cost) in highperformance products such as aerospace components (tails, wings , fuselages, propellors), boat and scull hulls, and racing car bodies. More mundane uses include fishing rods and storage tanks. This new book presents the latest research from around the world. The purpose of Chapter 1 is to present various application of statistical scale transition models to the analysis of polymer-matrix composites submitted to thermo-hygro-mechanical loads. In order to achieve such a goal, two approaches, classically used in the field of modelling heterogeneous material are studied: Eshelby-Kröner self-consistent model on the one hand and Mori-Tanaka approximate, on the second hand. Both models manage to handle the question of the homogenization of the microscopic properties of the constituents (matrix and reinforcements) in order to express the effective macroscopic coefficients of moisture expansion, coefficients of thermal expansion and elastic stiffness of a uni-directionally reinforced single ply. Inversion scale transition relations are provided also, in order to identify the effective unknown behaviour of a constituent. The proposed method entails to inverse

viii

Lucas P. Durand

scale transition models usually employed in order to predict the homogenised macroscopic elastic/hygroscopic/thermal properties of the composite ply from those of the constituents. The identification procedure involves the coupling of the inverse scale transition models to macroscopic input data obtained through either experiments or in the already published literature. Applications of the proposed approach to practical cases are provided: in particular, a very satisfactory agreement between the fitted elastic constants and the corresponding properties expected in practice for the reinforcing fiber of typical composite plies is achieved. Another part of this work is devoted to the extensive analysis of macroscopic mechanical states concentration within the constituents of the plies of a composite structure submitted to thermo-hygro-elastic loads. Both numerical and a fully explicit version of Eshelby-Kröner model are detailed. The two approaches are applied in the viewpoint of predicting the mechanical states in both the fiber and the matrix of composites structures submitted to a transient hygro-elastic load. For this purpose, rigorous continuum mechanics formalisms are used for the determination of the required time and space dependent macroscopic stresses. The reliability of the new analytical approach is checked through a comparison between the local stress states calculated in both the resin and fiber according to the new closed form solutions and the equivalent numerical model: a very good agreement between the two models was obtained. The purpose of the final part of this work consists in the determination of microscopic (local) quadratic failure criterion (in stress space) in the matrix of a composite structure submitted to purely mechanical load. The local failure criterion of the pure matrix is deduced from the macroscopic strength of the composite ply (available from experiments), using an appropriate inverse model involving the explicit scale transition relations previously obtained for the macroscopic stress concentration at microscopic level. Convenient analytical forms are provided as often as possible, else procedures required to achieve numerical calculations are extensively explained. Applications of this model are achieved for two typical carbon-fiber reinforced epoxies: the previously unknown microscopic strength coefficients and ultimate strength of the considered epoxies are identified and compared to typical expected values published in the literature. In Chapter 2 the optimal design of laminated composite structures is considered. A review of the literature is proposed. It aims at giving a general overview of the problems that a designer must face when he works with laminated composite structures and the specific solutions that have been derived. Based on it and on the industrial needs an optimization method specially devoted to composite structures is developed and presented. The related solution procedure is general and reliable. It is based on fibers orientations and ply thicknesses as design variables. It is used daily in an (European) industrial context for the design of composite aircraft box structures located in the wings, the center wing box, and the vertical and horizontal tail plane. This approach is based on sequential convex programming and consists in replacing the original optimization problem by a sequence of approximated sub-problems. A very general and self adaptive approximation scheme is used. It can consider the particular structure of the mechanical responses of composites, which can be of a different nature when both fiber orientations and plies thickness are design variables. Several numerical applications illustrate the efficiency of the proposed approach. As explained in Chapter 3, composites have been growing exponentially in technology and applications for decades. The world of aerospace has been one of the earliest and strongest proponents of advanced composites and the culmination of the recent advances in

Preface

ix

composite technology are realized in the Boeing Model 787 with over 50% by weight of composites, bringing the application of composites in large structures into a new age. This mostly-composite Boeing 787 has been credited with putting an end to the era of the all-metal airplane on new designs, and it is perhaps the most visible manifestation of the fact that composites are having a profound and growing effect on all sectors of society. It is generally well-known that composite materials are made of reinforcement fibers and matrix materials, and light weight and high mechanical properties are the primary benefits of a composite structure. Accordingly, the development trends in composite technology lie in 1) new material technology specifically for developing novel fibers and matrices, enhancing interfacial adhesion between fiber and matrix, hybridization and multi-functionalization, and 2) more reliable, high quality, rapid and low cost manufacturing technology. New reinforcement fiber technology including next generation carbon fibers and organic fibers with improved mechanical and physical properties, such as Spectra®, Dyneema®, and Zylon®, have been developing continuously. More significantly, various nanotechnology based novel fiber reinforcements have conspicuously and rapidly appeared in recent years. Matrix materials have become as complex as the fibers, satisfying increasing demands for impact resistant and damage tolerant structure. Various means of accomplishing this have ranged from elastomeric/thermoplastic minor phases to discrete layers of toughened materials. Nano-modified polymeric matrices are mostly involved in the development trends for matrix polymer materials. Technology for enhancing the interfacial adhesion properties between the reinforcement and matrix for a composite to provide high stress-transfer ability is more critically demanded and the science of the interface is expanding. Fiber/matrix interfacial adhesion is vital for the application of the newly developed advanced reinforcement materials. Effective approaches to improving new and non-traditional treatment methods for better adhesion have just started to receive sufficient attention. Multifunctionality is also an important trend for advanced composites, in particular, utilizing nanotechnology developments in recent years to provide greater opportunities for forcing materials to play a more comprehensive role in the designs of the future. More reliable and low cost manufacturing technology has been pursued by industry and academic researchers and the traditional material forms are being replaced by those which support the growing need for high quality, rapid production rates and lower recurring costs. Major trends include the recognition of the value of resin infusion methods, automated thermoplastic processing which takes advantage of the unique advantages of that material class, and the value of moving away from dependence on the large and expensive autoclaves. In Chapter 4, an innovative manufacturing process was developed to fabricate nanophased carbon prepregs used in the manufacturing of unidirectional composite laminates. In this technique, prepregs were manufactured using solution impregnation and filament winding methods and subsequently consolidated into laminates. Spherical silicon carbide nanoparticles (β-SiC) were first infused in a high temperature epoxy through an ultrasonic cavitation process. The loading of nanoparticles was 1.5% by weight of the resin. After infusion, the nano-phased resin was used to impregnate a continuous strand of dry carbon fiber tows in a filament winding set-up. In the next step, these nanophased prepregs were wrapped over a cylindrical foam mandrel especially built for this purpose using a filament winder. Once the desired thickness was achieved, the stacked prepregs were cut along the length of the cylindrical mandrel, removed from the mandrel, and laid out open to form a rectangular panel. The panel was then consolidated in a regular compression molding

x

Lucas P. Durand

machine. In parallel, control panels were also fabricated following similar routes without any nanoparticle infusion. Extensive thermal and mechanical characterizations were performed to evaluate the performances of the neat and nano-phased systems. Thermo Gravimetric Analysis (TGA) results indicate that there is an increase in the degradation temperature (about 7 0C) of the nano-phased composites. Similar results from Differential Scanning Calorimetry (DSC) and Dynamic Mechanical Analysis (DMA) tests were obtained. An improvement of about 50C in glass transition temperature (Tg) of nano-phased systems were also seen. Mechanical tests on the laminates indicated improvement in flexural strength and stiffness by about 32% and 20% respectively whereas in tensile properties there was a nominal improvement between 7-10%. Finally, micro numerical constitutive model and damage constitutive equations were derived and an analytical approach combining the modified shearlag model and Monte Carlo simulation technique to simulate the tensile failure process of unidirectional layered composites were also established to describe stress-strain relationships. A new approach that integrates acoustic emission (AE) and the mechanical behaviour of composite materials is presented in Chapter 5. Usually AE information is used to evaluate qualitatively the damage progression in order to assess the structural integrity of a wide variety of mechanical elements such as pressure vessels. From the other side, the mechanical information, e.g. the stress-strain curve, is used to obtain a quantitative description of the material behaviour. In order to perform a deeper analysis, a function that combines AE and mechanical information is introduced. In particular, this function depends on the strain energy and on the AE events energy, and it was used to study the behaviour of CFRP composite laminates in different applications: (i) to describe the damage progression in tensile and transversal load testing; (ii) to predict residual tensile strength of transversally loaded laminates (condition that simulates a low velocity impact). For a long time, fatigue testing of composites was only focused on providing the S-N fatigue life data. No efforts were made to gather additional data from the same test by using more advanced instrumentation methods. The development of methods such as digital image correlation (strain mapping) and optical fibre sensing allows for much better instrumentation, combined with traditional equipment such as extensometers, thermocouples and resistance measurement. In addition, validation with finite element simulations of the realistic boundary conditions and loading conditions in the experimental set-up must maximize the generated data from one single fatigue test. This research paper presents a survey of the authors’ recent research activities on fatigue in polymer composites. For almost ten years now, combined fatigue testing and modelling has been done on glass and carbon polymer composites with different lay-ups and textile architectures. Chapter 6 wants to prove that a synergetic approach between instrumented testing, detailed damage inspection and advanced numerical modelling can provide an answer to the major challenges that are still present in the research on fatigue of composites. Chapter 7 presents an overview of the damage variables proposed in the literature over the years, including a new variable recently introduced by the Authors to specifically address the problem of thick laminates subject to repeated impacts. Numerous impact data are used as a basis to address and comment potentials and limitations of the different variables. Impact data refer to single impact events as well as repeated impact tests performed on laminates with different fiber and matrix combinations and various lay-ups. Laminates of different thickness are considered, ranging from tenths to tens of millimeters.

Preface

xi

The analysis shows that some of the variables can indeed be used for assessing the damage tolerance of the laminate. In single impact tests, results point out the existence of an energy threshold at about 40-50% of the penetration energy, below which the damage threat is quite negligible. Other variables are not directly related to the amount of damage induced in the laminate but rather give an indication of the laminate efficiency of energy absorption. The electromechanical field concentrations due to electrodes in piezoelectric composites are investigated through numerical and experimental characterization. Chapter 8 consists of two parts. In the first part, a nonlinear finite element analysis is carried out to discuss the electromechanical fields in rectangular piezoelectric composite actuators with partial electrodes, by introducing models for polarization switching in local areas of the field concentrations. Two criteria based on the work done by electromechanical loads and the internal energy density are used. Strain measurements are also presented for a four layered piezoelectric actuator, and a comparison of the predictions with experimental data is conducted. In the second part, the electromechanical fields in the neighborhood of circular electrodes in piezoelectric disk composites are reported. Nonlinear disk device behavior induced by localized polarization switching is discussed. Aluminum-based alloys reinforced with ceramic microparticles are attractive materials for many structural applications. However, large ceramic microparticles often act as stress concentrators in the composites during mechanical loading, giving rise to failure of materials via particle cracking. In recent years, increasing demand for high performance materials has led to the development of aluminum-based nanocomposites having functions and properties that are not achievable with monolithic materials and microcomposites. The incorporation of very low volume contents of ceramic reinforcements on a nanometer scale into aluminumbased alloys yields remarkable mechanical properties such as high tensile stiffness and strength as well as excellent creep resistance. However, agglomeration of nanoparticles occurs readily during the composite fabrication, leading to inferior mechanical performance of nanocomposites with higher filler content. Cryomilling and severe plastic deformation processes have emerged as the two important processes to form ultrafine grained composites with homogeneous dispersion of reinforcing particles. In Chapter 9, recent development in the processing, structure and mechanical properties of the aluminum-based nanocomposites are addressed and discussed.

In: Composite Materials Research Progress Editor: Lucas P. Durand, pp. 1-50

ISBN: 1-60021-994-2 © 2008 Nova Science Publishers, Inc.

Chapter 1

MULTI-SCALE ANALYSIS OF FIBER-REINFORCED COMPOSITE PARTS SUBMITTED TO ENVIRONMENTAL AND MECHANICAL LOADS Jacquemin Frédéric and Fréour Sylvain* GeM -Institut de Recherche en Génie Civil et Mécanique, Université de Nantes-Ecole Centrale de Nantes-CNRS UMR 6183, 37 Boulevard de l’Université, BP 406, 44 602 Saint-Nazaire, France

Abstract The purpose of this work is to present various application of statistical scale transition models to the analysis of polymer-matrix composites submitted to thermo-hygro-mechanical loads. In order to achieve such a goal, two approaches, classically used in the field of modelling heterogeneous material are studied: Eshelby-Kröner self-consistent model on the one hand and Mori-Tanaka approximate, on the second hand. Both models manage to handle the question of the homogenization of the microscopic properties of the constituents (matrix and reinforcements) in order to express the effective macroscopic coefficients of moisture expansion, coefficients of thermal expansion and elastic stiffness of a uni-directionally reinforced single ply. Inversion scale transition relations are provided also, in order to identify the effective unknown behaviour of a constituent. The proposed method entails to inverse scale transition models usually employed in order to predict the homogenised macroscopic elastic/hygroscopic/thermal properties of the composite ply from those of the constituents. The identification procedure involves the coupling of the inverse scale transition models to macroscopic input data obtained through either experiments or in the already published literature. Applications of the proposed approach to practical cases are provided: in particular, a very satisfactory agreement between the fitted elastic constants and the corresponding properties expected in practice for the reinforcing fiber of typical composite plies is achieved. Another part of this work is devoted to the extensive analysis of macroscopic mechanical states concentration within the constituents of the plies of a composite structure submitted to thermo-hygro-elastic loads. Both numerical and a fully explicit version of Eshelby-Kröner model are detailed. The two approaches are applied in the viewpoint of predicting the mechanical states in both the fiber and the matrix of composites structures submitted to a *

E-mail address: [email protected]. Fax number : +33240172618. (Corresponding author)

2

Jacquemin Frédéric and Fréour Sylvain transient hygro-elastic load. For this purpose, rigorous continuum mechanics formalisms are used for the determination of the required time and space dependent macroscopic stresses. The reliability of the new analytical approach is checked through a comparison between the local stress states calculated in both the resin and fiber according to the new closed form solutions and the equivalent numerical model: a very good agreement between the two models was obtained. The purpose of the final part of this work consists in the determination of microscopic (local) quadratic failure criterion (in stress space) in the matrix of a composite structure submitted to purely mechanical load. The local failure criterion of the pure matrix is deduced from the macroscopic strength of the composite ply (available from experiments), using an appropriate inverse model involving the explicit scale transition relations previously obtained for the macroscopic stress concentration at microscopic level. Convenient analytical forms are provided as often as possible, else procedures required to achieve numerical calculations are extensively explained. Applications of this model are achieved for two typical carbon-fiber reinforced epoxies: the previously unknown microscopic strength coefficients and ultimate strength of the considered epoxies are identified and compared to typical expected values published in the literature.

Keywords: scale transition modelling, homogenization, identification, polymer-matrix composites.

1. Introduction Carbon-reinforced epoxy based composites offer design, processing, performance and cost advantages compared to metals for manufacturing structural parts. Among the advantages, provided by carbon-reinforced epoxies over metals and ceramics, that have been recognised for years, improved fracture toughness, impact resistance, strength to weight ratio as well as high resistance to corrosion and enhanced fatigue properties have often been put in good use for practical applications (Karakuzu et al., 2001). Now, the accurate design and sizing of any structure requires the knowledge of the mechanical states experienced by the material for the possibly various loads, expected to occur during service life. Since high performance composites are being increasingly used in aerospace and marine structural applications, where they are exposed to severe environmental conditions, these composites experience hygrothermal loads as well as more classical mechanical loads. Now, unlike metallic or ceramic materials, composites are susceptible to both temperature and moisture when exposed to such working environments. These environmental conditions are known to possibly induce sometimes critical stresses distributions within the plies of the composite structures or even within their very constituents (i.e. the reinforcements on the one hand and the matrix on the second hand). Actually, carbon/epoxy composites can absorb significant amount of water and exhibit heterogeneous Coefficients of Moisture Expansion (CME) and Coefficients of Thermal Expansion (CTE) (i.e. the CME/CTE of the epoxy matrix are strongly different from the CME/CTE of the carbon fibers, as shown in: Tsai, 1987; Agbossou and Pastor, 1997; Soden et al., 1998), moreover, the diffusion of moisture in such materials is a rather slow process, resulting in the occurrence of moisture concentration gradients within their depth, during at least the transient stage (Crank, 1975). As a consequence, local stresses take place from hygro-thermal loading of composite structures which closely depends on the experienced environmental conditions, on the local intrinsic properties of the constituents and on its microstructure (the morphology

Multi-scale Analysis of Fiber-Reinforced Composite Parts…

3

of the constituents, the lay-up configuration, ... fall in this last category of factors). Now, the knowledge of internal stresses is necessary to predict a possible damage occurrence in the material during its manufacturing process or service life. Thus, the study of the development of internal stresses due to thermo-hygro-elastic loads in composites is very important in regard to any engineering application. Numerous papers, available in the literature, deal with this question, using Finite Element Analysis or Continuum Mechanics-based formalisms. These methods allow the calculation of the macroscopic stresses in each ply constituting the composite (Jacquemin and Vautrin, 2002). But, they do not provide information on the local mechanical states, in the fibers and matrix of a given ply, and, consequently, do not allow to explain the phenomenon of matrix cracking and damage development in composite structures, which originate at the microscopic level. The present work is precisely focused on the study of the internal stresses in the constituents of the ply. In order to reach this goal, scale transition models are required. The present work underlines the potential of scale-transition models, as predictive tools, complementary to continuum mechanics in order to address: i) the estimation of the effective hygro-thermo-elastic properties of a composite ply from those of its constituents (section 2), ii) the identification of the hygro-thermo-elastic properties of one constituent of a composite ply (section 3), iii) the estimation of the local mechanical states experienced in each constituent of a composite structure (section 4), iv) the identification of the local strength of the constitutive matrix (section 5). Section 6 of this paper is mainly dedicated to conclusions about the above listed sections the whereas section 7 is devoted to the introducing some scientifically appealing perspectives of research in the field of composites materials which are highly considered for further investigation in the forthcoming years.

2. Scale-Transition Model for Predicting the Macroscopic Thermo-Hygro-Elastic Properties of a Composite Ply 2.1. Introduction Scale transition models are based on a multi-scale representation of materials. In the case of composite materials, for instance, a two-scale model is sufficient: -

-

The properties and mechanical states of either the resin or its reinforcements are respectively indicated by the superscripts m and r. These constituents define the socalled “pseudo-macroscopic” scale of the material (Sprauel and Castex, 1991). Homogenisation operations performed over its aforementioned constituents are assumed to provide the effective behaviour of the composite ply, which defines the macroscopic scale of the model. It is denoted by the superscript I. This definition also enables to consider an uni-directional reinforcement at macroscopic scale, which is a satisfactorily realistic statement, compared to the present design of composite structures (except for the particular case of woven-composites that will be specifically discussed in section 7.1).

4

Jacquemin Frédéric and Fréour Sylvain

As for the composite structure, it is actually constituted by an assembly of the above described composite plies, each of them possibly having the principal axis of their reinforcements differently oriented from one to another. This approach enables to treat the case of multi-directional laminates, as shown, for example, in (Fréour et al., 2005a).

2.2. The Classical Practical Strategy for the Direct Application of Homogenisation Procedures Within scale transition modeling, the local properties of the i−superscripted constituents are usually considered to be known (i.e. the pseudo-macroscopic stiffnesses, Li, coefficients of thermal expansion Mi and coefficients of moisture expansion βi), whereas the corresponding effective macroscopic properties of the composite structure (respectively, LI, MI and βI) are a priori unknown and results from (often numerical) computations. Among the numerous, available in the literature scale transition models, able to handle such a problem, most involve rough-and-ready theoretical frameworks: Voigt (Voigt, 1928), Reuss, (Reuss, 1929), Neerfeld-Hill (Neerfeld, 1942; Hill, 1952), Tsai-Hahn (Tsai and Hahn, 1980) and Mori-Tanaka (Mori and Tanaka, 1973; Tanaka and Mori, 1970) approximates fall in this category. This is not satisfying, since such a model does not properly depict the real physical conditions experienced in practice by the material. In spite of this lack of physical realism, some of the aforementioned models do nevertheless provide a numerically satisfying estimation of the effective properties of a composite ply, by comparison with the experimental values or others, more rigorous models. Both Tsai-Hahn and Mori-Tanaka models fulfil this interesting condition (Jacquemin et al., 2005; Fréour et al., 2006a). Nevertheless, in the field of scale transition modelling, the best candidate remains KrönerEshelby self-consistent model, because only this model takes into account a rigorous treatment of the thermo-hygro-elastic interactions between the homogeneous macroscopic medium and its heterogeneous constituents, as well as this model enables handling the microstructure (i.e. the particular morphology of the constituents, especially that of the reinforcements).

2.3. Estimating the Effective Properties of a Composite Ply through EshelbyKröner Self-consistent Model Self-consistent models based on the mathematical formalism proposed by Kröner (Kröner, 1958) constitute a reliable method to predict the micromechanical behavior of heterogeneous materials. The method was initially introduced to treat the case of polycrystalline materials, i.e. duplex steels, aluminium alloys, etc., submitted to purely elastic loads. Estimations of homogenized elastic properties and related problems have been given in several works (François, 1991; Mabelly, 1996; Kocks et al., 1998). The model was thereafter extended to thermoelastic loads and gave satisfactory results on either single-phase (Turner and Tome, 1994; Gloaguen et al., 2002) or two-phases (Fréour et al., 2003a and 2003b) materials. More recently, this classical model was improved in order to take into account stresses and strains due to moisture in carbon fiber-reinforced polymer–matrix composites.

Multi-scale Analysis of Fiber-Reinforced Composite Parts…

5

Therefore, the formalism was extent so that homogenisation relations were established for estimating the macroscopic CME from those of the constituents (Jacquemin et al., 2005). Many previously published documents have been dedicated to the determination of (at least some of) the effective thermo-hygro-elastic properties of heterogeneous materials through Kröner-Eshelby self-consistent approach (Kocks et al., 1998; Gloaguen et al., 2002; Fréour et al., 2003a-b; Jacquemin et al., 2005). The main involved equations are:

[

(

])

−1 LI = Li : I + E I : Li − LI

I

β =

I

1 ΔC

(L + L : R ) i

I

[

I

I −1

I

−1

(

i

[

i

(1) i = r, m

I

: L + L :R

:L

)

: L : β ΔC

]

:L :M

I −1

i

i

I

M = L +L :R

]

I −1

I

−1

I

: L +L :R

:L

i = r, m

I −1

(2) i =r,m

i =r,m

i

i

i

i

(3) i = r, m

Where ΔCi is the moisture content of the studied i element of the composite structure. The superscripts r and m are considered as replacement rule for the general superscript i, in the cases that the properties of the reinforcements or those of the matrix have to be considered, respectively. Actually, the pseudo-macroscopic moisture contents ΔCr and ΔCm can be expressed as a function of the macroscopic hygroscopic load ΔCI (Loos and Springer, 1981), so that the hygro-mechanical states cancels in relation (2) that can finally be rewritten as a function of the materials properties only, but at the exclusion of the ΔCi that are unexpected to appear in such an expression (Jacquemin et al., 2005). Relation (2), that is provided in the present work for predicting the macroscopic CME, is given for its enhanced readability, compared to the more rigorous state exclusive relation. In relations (1-3), the brackets < > stand for volume weighted averages (that in fact replace volume integrals that would require Finite Elements Methods instead). Empirically, as stated by Hill (Hill, 1952), arithmetic or geometric averages suggest themselves as good approximations. On the one hand, the geometric mean of a set of positive data is defined as the nth root of the product of all the members of the set, where n is the number of members. On the other hand, in mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. For Young’s modulus, as an example, the Geometric Average YGA of the moduli GA = YR YV , according to the Reuss (YR) and Voigt (YV) models is defined as Y whereas the corresponding Arithmetic Average YAA is: Y

AA

=

YR + YV . 2

6

Jacquemin Frédéric and Fréour Sylvain

In statistics, given a set of data, X = { x1, x2, ..., xn} and corresponding weights, W = { GA i w1, w2, ..., wn}, the weighted geometric (respectively, arithmetic) mean X i =1,2,..., n (respectively, X

i

AA i =1,2,..., n

) is calculated as:

⎛ n ⎞ 1/⎜ ∑ wi ⎟ ⎟ GA ⎛ n ⎞ ⎜ = ⎜ ∏ x iw i ⎟ ⎝ i =1 ⎠ Xi ⎟ i =1,2,..., n ⎜⎝ i =1 ⎠

Xi

AA i =1,2,..., n

=

1 n

(4)

n

∑ xi wi

(5)

∑ w i i =1

i =1

Both averages have been extensively used in the field of materials science, in order to achieve various scale transition modelling over a wide range of materials. The interested reader can refer to: (Morawiec, 1989; Matthies and Humbert, 1993; Matthies et al., 1994) that can be considered as typical illustrations of works taking advantage of the geometric average for estimating the properties and mechanical states of polycrystals (nevertheless, EshelbyKröner self-consistent model was not involved in any of these articles), whereas the previously cited references (Kocks et al., 1998; Gloaguen et al., 2002; Fréour et al., 2003; Jacquemin et al., 2005) show applications of arithmetic averages for studying of polycrystals or composite structures. According to equations (4) and (5), the explicit writing of a volume weighted average directly depend on the averaging method chosen to perform this operation. Since the present work aims to express analytical forms involving such volume averages, it is necessary to select one average type in order to ensure a better understanding for the reader. Usually, in this field of research, the arithmetic and not the geometric volume weighted average is used. Moreover, in a recent work, the alternative geometric averages were also used for estimating the effective properties of carbon-epoxy composites (Fréour et al., to be published). Nevertheless, the obtained results were not found as satisfactory than in the previously studied cases of metallic polycrystals or metal ceramic assemblies. Actually, the very strongly heterogeneous properties presented by the constituents of carbon reinforced polymer matrix composites yields a strong underestimation of the effective properties of the composite ply predicted according to Eshelby-Kröner model involving the geometric average, by comparison to the expected (measured) properties. Thus, the geometric average should not be considered as a reliable alternate solution to the classical arithmetic average for achieving scale transition modelling of composite structures. Consequently, arithmetic averages satisfying to relation (4) only will be used in the following of this manuscript.

Multi-scale Analysis of Fiber-Reinforced Composite Parts…

7

Now, in the present case, where the macroscopic behaviour is described by two separate heterogeneous inclusions only (i.e. one for the matrix and one for the reinforcements), convenient simplifications of equation (5) do occur. Actually, introducing vr and vm as the volume fractions of the ply constituents, and taking into account the classical relation on the summation over the volume fractions (i.e. vr + vm=1), equation (5) applied to the volume average of any tensor A writes:

Ai

AA i = r, m

= Ai

i = r, m

= v r Ar + v m Am

(6)

In the following of the present work, the superscript AA denoting the selected volume average type will be omitted. According to equations (1-3), the effective properties expressed within Eshelby-Kröner self-consistent model involve a still undefined tensor, RI. This term is the so-called “reaction tensor” (Kocks et al., 1998). It satisfies:

(

)

I I R I = I − S esh : S esh

−1

= ⎛⎜ LI ⎝

−1

− E I ⎞⎟ : E I ⎠

−1

(7)

In the very preceding equation, I stands for the fourth order identity tensor. Hill’s tensor E , also known as Morris tensor (Morris, 1970), expresses the dependence of the reaction tensor on the morphology assumed for the matrix and its reinforcements (Hill, 1965). It can I I I I −1 . It has to be be expressed as a function of Eshelby’s tensor Sesh , through E = Sesh : L underlined that both Hill’s and Eshelby’s tensor components are functions of the macroscopic stiffness LI (some examples are given in Kocks et al., 1998; Mura, 1982). In the case, when ellipsoidal-shaped inclusions have to be taken into account, the following general form enables the calculation of the components of this tensor (see the works of Asaro and Barnett, 1975 or Kocks et al. 1998): I

2π ⎧ I 1 π sinθ dθ ∫ γ ikjl dφ ⎪E ijkl = ⎪ 4π 0∫ 0 ⎨ ⎪ − 1 I ⎪⎩γ ikjl = K ik (ξ ) ξ j ξ l

[

(8)

]

In the case of an orthotropic macroscopic symmetry, the components Kjp(ξ) were given in (Kröner, 1953): ⎡LI ξ 2 + LI ξ 2 + LI ξ 2 66 2 55 3 ⎢ 11 1 I KI = ⎢ L12 + LI66 ξ1ξ 2 ⎢ LI13 + LI55 ξ1ξ 2 ⎢⎣

( (

) )

(L + L )ξ ξ ξ +L ξ +L (L + L )ξ ξ

I 12 LI66 12 I 23

I 66 1 2 I 2 I 2 22 2 44 ξ 3 I 44 2 3

(L (L

) )

I I ⎤ 13 + L 55 ξ1ξ 2 ⎥ I I ⎥ 23 + L 44 ξ 2 ξ 3 I 2 I 2 I 2⎥ L 55 ξ1 + L 44 ξ 2 + L 33ξ 3 ⎥ ⎦

(9)

8

Jacquemin Frédéric and Fréour Sylvain

with

ξ1 =

sinθ cosφ sinθ sinφ cosθ , ξ2 = , ξ3 = a1 a2 a3

(10)

where 2 a1, 2 a2, 2 a3 are the lengths of the principal axes of the ellipsoid (representing the considered inclusion) assumed to be respectively parallel to the longitudinal, transverse and normal directions of the sample reference frame. According to equations (2-3, 7), the determination of both the macroscopic coefficients of thermal and moisture expansion are somewhat straightforward, while the effective stiffness is known, because the involved expressions are explicit. On the contrary, the estimation of the macroscopic stiffness of the composite ply through (1) cannot be as easily handled. Expression (1) is implicit because it involves LI tensor in both its right and left members. Moreover, calculating the right member of equation (1) entails evaluating the reaction tensor (7) which also depends on the researched elastic stiffness, at least because of the occurrence of Hill’s tensor (or Eshelby’s tensor, if that notation is preferred) in relation (1). As a consequence, the effective elastic properties of a composite ply satisfying to Eshelby-Kröner self-consistent model constitutive relations are estimated at the end of an iterative numerical procedure. This is the main drawback of the self-consistent procedure preventing from achieving an analytical determination of the effective macroscopic thermo-hygro-elastic properties of a composite ply, in the case where this scale transition model is employed. Therefore, managing to express explicit solutions for estimating the macroscopic properties (or at least the macroscopic stiffness) requires focusing on a less intricate, less rigorous model but still providing realistic numerical values. Mori-Tanaka approach suggest itself as an appropriate candidate, for reasons that will be comprehensively explained in the next subsection.

2.4. Introducing Mori-Tanaka Model as a Possible Alternate Solution to Eshelby-Kröner Model As Eshelby-Kröner self-consistent approach, Mori Tanaka estimate is a scale transition model derived from the pioneering mathematical work of Eshelby (Eshelby, 1957). Mori and Tanaka actually investigated the opportunity of extending Eshelby’s single-inclusion model (which is sometimes presented as an “infinitely dilute solution model”) to the case where the volume fraction of the ellipsoidal heterogeneous inclusion embedded in the matrix is not tending towards zero anymore, but admits a finite numerical value (Mori and Tanaka, 1973; Tanaka and Mori, 1970). Calculations show that, in many cases, the effective homogenised macroscopic properties deduced from Mori-Tanaka approximate are close to their counterparts, estimated from the previously described Eshelby-Kröner self-consistent procedure (Baptiste, 1996, Fréour et al., 2006a). Exceptions to this statement occur nevertheless in the cases where extreme heterogeneities in the constituents properties have to be accounted for. For example, handling a significant porosities volume fraction yields MoriTanaka estimations deviating considerably from the self-consistent corresponding

Multi-scale Analysis of Fiber-Reinforced Composite Parts…

9

calculations, according to (Benveniste, 1987). However, Mori-Tanaka approach is reported to remain reliable for treating cases similar to those aimed by the present work. It has previously been demonstrated that the effective macroscopic homogenised thermohygro-elastic properties exhibited by a composite ply, according to Mori and Tanaka approximation satisfy the following relations (Baptiste, 1996; Fréour et al., 2006a):

I

i

i

L = T : L :T

i

−1

−1

−1

= Li : T i i = r,m : T i i =r,m i = r,m i = r,m

−1 ⎛ i i ⎜ T : L : Li : T i β = I i = r, m ΔC ⎜⎝

1

I

−1 ⎛ M = ⎜ T i : Li : Li : T i ⎜ i = r, m ⎝ I

(11)

T

⎞ ⎟ : β i ΔC i ⎟ ⎠

(12) i = r, m

T

⎞ ⎟ : Mi ⎟ ⎠

(13) i = r, m

The superscript T appearing in relations (12-13) denotes transposition operation. The same remarks as indicated in the preceding subsection holds for the determination of the effective macroscopic CME using relation (12). This equation can be rewritten as a function of the materials properties only, thus excluding the moisture contents. In equations (11-13), Ti is the elastic strain localisation tensor, expressed for the isuperscripted phase that is considered to interact with the embedding phase (denoted by the superscript e). Actually, Mori-Tanaka model is based on a two-step scale-transition procedure. In this theory, contrary to the case of Eshelby-Kröner self-consistent model, the inclusions are not considered to be directly embedded in the effective material having the behaviour of the composite structure (and thus interacting with it). In Mori and Tanaka approximation, the n constituents of a n-phase composite ply are separated in two subclasses: one of them is designed as the embedding constituent, whereas the n-1 others are considered as inclusions of the first one. The inclusion particles are embedded in the matrix phase, itself being loaded at the infinite by the hygro-mechanical conditions applied on the composite structure. In consequence, the inclusion phase does not experience any interaction with the macroscopic scale, but with the matrix only. In consequence, Mori and Tanaka model corresponds to the direct extension of Eshelby’s single inclusion model (Eshelby, 1957) to the case that the volume fraction of inclusions does not remain infinitesimal anymore. Within Mori and Tanaka approach, this localisation tensor Ti writes as follows:

[

(

)]

−1 Ti = I + Ei : Li − Le

(14)

Contrary to the case of Eshelby-Kröner scale-transition model (refer to subsection 2.3. above), the localisation involved within Mori-Tanaka approximate does not explicitly involve

10

Jacquemin Frédéric and Fréour Sylvain

the macroscopic stiffness. Nevertheless, according to the already cited same subsection, the reaction tensor involved in Eshelby-Kröner model was also implicitely depending on the macroscopic stiffness through the calculation procedure entailed for estimating Hill’s tensor. Within Mori-Tanaka procedure (Benveniste, 1987; Baptiste 1996; Fréour et al., 2006a), Hill’s tensor Ei expresses the dependence of the strain localization tensor on the morphology assumed for the embedding phase and the particulates it surrounds (Hill, 1965). It can be i expressed as a function of Eshelby’s tensor Sesh , through: i Ei = Sesh : Le

−1

(15)

In practice, the calculation of Hill’s tensor for the embedded inclusions phase only would e be necessary, since obvious simplifications of (14), leading to T = I , occur in the case that the embedding constituent localisation tensor is considered. According to relations (14-15), the strain localization tensor Ti does not involve the macroscopic stiffness tensor (or any other macroscopic property). As a consequence, contrary to Eshelby-Kröner self-consistent procedure, Mori-Tanaka approximation provides explicit relations (actually, the homogenization equations (11-13)) for estimating the researched macroscopic effective properties of a composite ply.

2.5. Example of Homogenization: The Case of T300-N5208 Composites The present subsection is focused on the application of the theoretical frameworks described in the above 2.3 and 2.4 sections to the numerical simulation of the effective properties of a typical, high-strength, fiber-reinforced composite made up of T300 carbon fibers and N5208 epoxy resin. The choice of such a material is justified because of the strong heterogeneities of the hygro-thermo-elastic properties of its constituents (actually, the numerical deviation occurring among the macroscopic properties of composites determined through various scale transition relations rises with this factor, see Jacquemin et al, 2005; Herakovich, 1998). Table 1 accounts for the pseudo-macroscopic properties reported in the literature for these constituents. The comparison between the results obtained through the two, considered in the present work alternate scale transition framework of Mori-Tanaka model are displayed on figure 1, for: -

I I the longitudinal and transverse Young’s moduli Y11 , Y22 , I I Coulomb’s moduli G12 , G 23 ,

I the coefficients of thermal expansion M11 , I the coefficients of moisture expansion β11 ,

M I22 β I22 .

Multi-scale Analysis of Fiber-Reinforced Composite Parts…

11

Table 1. Hygro-thermo-mechanical properties of T300/5208 constituents.

ν12 G 23 G12 ν13 [GPa] [GPa]

ρ [g/cm3]

Y1

Y2 , Y3

[GPa]

[GPa]

T300 fibers (Soden et al., 1998; Agbossou and Pastor, 1997)

1200

230

15

0.2

7

N5208 epoxy matrix (Tsai, 1987; Agbossou and Pastor, 1997)

1867

4.5

4.5

0.4

6.4

β11, β 22

M11 [10-6/K]

M22, M33 [10-6/K]

15

-1.5

27

0

6.4

60

60

0.6

β33

The calculations were achieved assuming that the reinforcements exhibit fiber-like morphology with an infinite length axis parallel to the longitudinal direction of the ply. For the determination of the CME, a perfect adhesion between the carbon fibers and the resin was assumed. Moreover, it also was assumed that the fibers do not absorb any moisture. Thus, the ratio between the pseudo-macroscopic and the macroscopic moisture contents is deduced from the expression given in (Loos and Springer, 1981):

ΔCm ΔCI

=

ρI vm ρm

(16)

where ρ stands for the densities. The macroscopic density can be deduced form the classical rule of mixture:

ρI = vm ρm + vr ρr

(17)

The equations required for achieving Mori-Tanaka estimations involve relations (8-17). For the purpose of the strain localization, the embedding constituent was considered to be the epoxy matrix, whatever the considered volume fraction of reinforcements (thus, the e m transformation rule L = L was considered to be valid in any case). Figure 1 also reports the numerical results obtained through Kröner-Eshelby Self-Consistent model (1-3, 6-10, 1617), in the same conditions (identical inclusion morphology and constituents properties as for Mori-Tanaka computations).

Jacquemin Frédéric and Fréour Sylvain Macroscopic stiffness component [GPa]

12

Macroscopic stiffness component [GPa]

250 200

I Y11

150 100 I Y22

50 0 0

0,25

0,5

0,75

16 14 12

I G12

10 8 6 4

G I23

2 0 0

1

0,25

0,75

1

80

1,4

-6

-1

CTE component [10 K ]

β I22

1,2 CME component

0,5

epoxy volume fraction

matrix volume fraction

1 0,8 0,6 0,4

I β11

0,2

60

I M11

40 20

M I22

0 0

0 0

0,25

0,5

0,75

1

0,25

0,5

0,75

1

-20

epoxy volume fraction

epoxy volume fraction

0, 2 5

Longitudinal (KESC)

0, 2

Transverse (KESC)

0, 1 5 0, 1 0, 0 5 0 0

Longitudinal (Mori-Tanaka) 0 , 25

0, 5

Transverse (Mori-Tanaka) 0 ,7 5

1

matrix volum e fraction

Figure 1. Macroscopic effective hygro-thermo-mechanical properties of T300/N5208 plies, estimated as a function of the epoxy volume fraction, through scale transition homogenisation procedures. Comparison between Mori-Tanaka approximate and Kröner-Eshelby self-consistent model.

Figure 1 shows the following interesting results: 1) In pure elasticity, both the investigated scale transition methods manage to reproduce the expected mechanical behaviour of the composite ply: the material is stiffer in the longitudinal direction than in the transverse direction. Moreover, the bounds are satisfying: the properties of the single constituents are correctly obtained for those of the composite ply in the cases where the epoxy volume fraction is either taken equal to vm=0 (transversely isotropic elastic properties of T300 fibers) or vm=1 (isotropic elastic properties of N5208 resin). 2) The curves drawn for each checked elastic constant are almost superposed, except for I Coulomb’s modulus G12 . Thus Mori-Tanaka model constitutes a rather reliable alternate homogenization procedure to Eshelby-Kröner rigorous solution for estimating the macroscopic elastic properties of typical carbon-epoxies.

Multi-scale Analysis of Fiber-Reinforced Composite Parts…

13

3) Kröner-Eshelby self-consistent model and Mori-Tanaka approach both also do manage to achieve a realistic prediction of the macroscopic coefficients of thermal expansion. Especially, the expected boundary values are attained when the conditions vm=1 (isotropic CTE of N5208 resin) or vm=0 (transversely isotropic thermal properties of T300 fibers) are taken into account. 4) Mori-Tanaka approximate correctly reproduces the expected macroscopic coefficients of moisture expansion in the longitudinal direction. In the transverse direction, however, Mori-Tanaka model properly follows Eshelby Kröner model estimates while the epoxy m volume fraction is higher than 0.5. In the range 0 ≤ v ≤ 0.5 , discrepancies occur between two considered scale transition models. In the case that the considered strain localization assumes the epoxy as the embedding constituent within Mori-Tanaka approximate, the I relative error on β 22 induced by this localization procedure, compared to Kröner-Eshelby reference values remains weaker than 9%, and falls below 6% in the range of epoxy volume fraction that is typical for designing composites structures for engineering applications (0.3 ≤ vm ≤ 0.7) . 5) In the range of the epoxy volume fraction, that is typical for designing composites m structures for engineering applications i.e. 0.3 ≤ v ≤ 0.7 , according to the above

(

)

discussed results 3) and 4), Mori-Tanaka model can be employed as an alternative to EshelbyKröner self-consistent model for estimating the effective macroscopic hygro-thermomechanical properties of composite plies. The above listed elements 1) to 5) finally indicate that the effective macroscopic thermohygro-elastic properties of composite plies can be estimated in a reliable fashion using MoriTanaka approximate, assuming the epoxy as the embedding constituent, instead of the more rigorous Kröner-Eshelby model. This statement is true while the epoxy volume fraction remains higher than 40%. Beyond this boundary value, some significant relative error (less than 10%) may be expected to occur in the estimated transverse CME. The results, obtained in the present section, will be used in the following as input parameters for estimating the mechanical states experienced at macroscopic but at microscopic scale also in composite structures submitted to various loads (the interested reader should refer to section 4 for details).

3. Inverse Scale Transition Modelling for the Identification of the Hygro-Thermo-Elastic Properties of One Constituent of a Composite Ply 3.1. Introduction The precise knowledge of the pseudo-macroscopic properties of each constituent of a composite structure is required in order to achieve the prediction of its behavior (and especially its mechanical states) through scale transition models. Nevertheless, the pseudomacroscopic stiffness, coefficients of thermal expansion and moisture expansion of the matrix

14

Jacquemin Frédéric and Fréour Sylvain

and its reinforcements are not always fully available in the already published literature. The practical determination of the hygro-thermo-mechanical properties of composite materials are most of the time achieved on unidirectionnaly reinforced composites and unreinforced matrices (Bowles et al., 1981; Dyer et al., 1992; Ferreira et al., 2006a; Ferreira et al., 2006b; Herakovich, 1998; Sims et al., 1977). In spite of the existence of several articles dedicated to the characterization of the properties of the isolated reinforcements (Tsai and Daniel, 1994; DiCarlo, 1986; Tsai and Chiang, 2000), the practical achieving of this task remains difficult to handle, and the available published data for typical reinforcing particulates employed in composite design are still very limited. As a consequence, the properties of the single reinforcements exhibiting extreme morphologies (such as fibers), are not often known from direct experiment, but more usually they are deduced from the knowledge of the properties of the pure matrices and those of the composite ply (which both are easier to determine), through appropriate calculation procedures. The question of determining the properties of some constituents of heterogeneous materials has been extensively addressed in the field of materials science, especially for studying complex polycrystalline metallic alloys (like titanium alloys, cf. Fréour et al., 2002 ; 2005b ; 2006b) or metal matrix composites (typically Aluminum-Silicon Carbide composites cf. Fréour et al., 2003a ; 2003b or iron oxides from the inner core of the Earth, cf. Matthies et al., 2001, for instance). The required calculation methods involved in order to achieve such a goal are either based on Finite Element Analysis (Han et al., 1995) or on the inversion of scale transition homogenization procedures similar to those already presented in section 2 of the present paper. It was shown in previous works that it was actually possible to identify the properties of one constituent of a heterogeneous material from available (measured) macroscopic quantities through inverse scale transition models. Such identification methods were successfully used in the field of metal-matrix composites for the determination of the average elastic (Freour et al., 2002) and thermal (Freour et al., 2006b) properties of the β-phase of (α+β) titanium alloys. The procedure was recently extended to the study of the anisotropic elastic properties of the single-crystal of the β-phase of (α+β) titanium alloys on the basis of the interpretation of X-Ray Diffraction strain measurements performed on heterogeneous polycrystalline samples in (Freour et al., 2005b). The question of determining the temperature dependent coefficients of thermal expansion of silicon carbide was handled using a similar approach from measurements performed on aluminum – silicon carbide metal matrix composites in (Freour et al., 2003a; Freour et al., 2003b).Numerical inversion of both Mori-Tanaka and Eshelby-Kröner self-consistent models will be developed and discussed here.

3.2. Estimating Constituents Properties from Eshelby-Kröner Self-consistent or Mori-Tanaka Inverse Scale Transition Models 3.2.1. Application of Eshelby-Kröner Self-consistent Framework to the Identification of the Pseudo-macroscopic Properties of one Constituent Embedded in a Two-Constituents Composite Material The pseudomacroscopic stiffness tensor of the reinforcements can be deduced from the inversion of the Eshelby-Kröner main homogenization form over the constituents elastic properties (1) as follows :

Multi-scale Analysis of Fiber-Reinforced Composite Parts…

Lr =

[ (

) ]

[ (

m

) ] [ (

15

) ]

−1 v LI : E I : Lr − LI + I − Lm : E I : Lm − LI + I : E I : Lr − LI + I (18) vr vr

1

The application of this equation implies that both the macroscopic stiffness and the pseudomacroscopic mechanical behaviour of the matrix is perfectly determined. The elastic stiffness of the matrix constituting the composite ply will be assumed to be identical to the elastic stiffness of the pure single matrix, deduced in practice from measurements performed on bulk samples made up of pure matrix. It was demonstrated in (Fréour et al., 2002) that this assumption was not leading to significant errors in the case that polycrystalline multi-phase samples were considered. The similarities existing between multi-phase polycrystals and polymer based composites suggest that this assumption should be suitable in the present context, at least when scale factors do not occur. Nevertheless, in the case that significant edge effects, due for instance to a reduced thickness of the matrix layer constituting the composite ply, might be expected to occur, the identification of the ply embedded matrix elastic properties to those of the corresponding bulk material would not systematically be appropriate. Consequently, the application of inverse form (18) given above could lead to an erroneous estimation of the reinforcements elastic stiffness. Moreover, identification based on such inverse homogenization methods are sensitive to both the precise knowledge of the constituents volume fractions (i.e. vm and vr) and to the presence of porosities (which lowers the effective stiffness LI of the composite ply). An expression, analogous to above-relation (18) can be found for the elastic stiffness of the matrix, through the following replacement rules over the superscripts/subscripts: m → r, r → m . Nevertheless, the situation, where the properties of the reinforcements are known, when those of the matrix are unknown is highly improbable. The pseudomacroscopic coefficients of moisture expansion of the matrix can be deduced from the inversion of the homogenization form (2) as follows :

βm =

1 v

m

ΔC

m

Lm

−1

)

(

: Lm + LI : R I : G m

(19)

where Gm writes :

G

m

= ΔC

I

(L + L : R ) i

I

I −1

I

I

r

(

r

I

:β − v L + L : R

:L

)

I −1

r r r : L : β ΔC (20)

i =r,m

An expression, analogous to above-relation (19) can also be found for the coefficients of moisture expansion of a permeable reinforcement type, through the following replacement rules over the superscripts/subscripts: m → r, r → m . In the particular case, where impermeable reinforcements are present in the composite structure, the coefficients of moisture expansion of the matrix simplifies as follows (an extensive study of this very question was achieved in Jacquemin et al., 2005):

16

Jacquemin Frédéric and Fréour Sylvain

β

m

=

ΔC I v

m

ΔC

m −1

m

L

(

m

I

I

) (

i

I

: L + L :R : L + L :R

)

I −1

I

:β

:L

I

(21)

i =r,m

The pseudomacroscopic coefficients of thermal expansion of the matrix can be deduced from the inversion of the homogenization form (3) as follows:

M

m

m −1

=L

(

m

:L

⎡ ⎤ I I i I I −1 I I r r I I −1 r r (22) :L :M − v L + L :R :L :M ⎥ + L :R :⎢ L + L :R ⎢ ⎥ i =r,m ⎣ ⎦

) (

)

)

(

Form (22) can be easily rewritten for expressing the coefficients of thermal expansion of the reinforcements, using the same replacement rules over the superscripts/subscripts: m → r, r → m , than for the previous cases.

3.2.2. Application of Mori-Tanaka Estimates to the Identification of the Pseudomacroscopic Properties of one Constituent Embedded in a Two-Constituents Composite Material 3.2.2.1. Inverse Mori-Tanaka Elastic Model In the present work, it is be considered, that the reinforcements are surrounded by the matrix, thus, Tm=I and (11) develops as follows:

(

)(

)

−1 LI = v m Lm + v r Lr : T r : v m I + v r : T r

(23)

Thus, from (11) two alternate equations are obtained for identifying the pseudomacroscopic stiffness of the composite ply constituents: •

On the first hand, the elastic properties of the matrix satisfies m

L

I

=L +

1- vm v

m

(LI - Lr ): Tr

(25)

Equation (25) is an implicit equation since both its left and right hand sides involve the researched stiffness tensor Lm. •

whereas, on the second hand, the elastic stiffness of the reinforcements respects

Lr = LI +

−1 ( LI - Lm ) : T r 1- vm

vm

(26)

Multi-scale Analysis of Fiber-Reinforced Composite Parts…

17

For the same reasons as above (i.e. comments about equation (25)), expression (26) is an implicit relation. As a consequence, the need of an inverse modelling for achieving the identification of the elastic properties exhibited by any one constituent of a composite ply through Mori-Tanaka scale-transition approximate yields the loss of the main advantage of this very model over the more rigorous Eshelby-Kröner self-consistent approach: the opportunity to express analytical explicit relations instead of having to perform successive numerical calculations for solving implicit equations. Moreover, the general remarks about the sensitivity of identification methods to certain factors, expressed in subsection 3.2.1 are valid in the present context also. 3.2.2.2. Inverse Mori-Tanaka Model for Identifying Coefficients of Moisture of Thermal Expansion Following the same line of reasoning as above, in the purely elastic case, one can inverse relation (12) in order to express the coefficients of moisture expansion of a constituent embedded in a composite ply according to Mori-Tanaka estimates, or its coefficients of thermal expansion, from the homogenization relation (13). In the case of the pure matrix, one gets:

Mm =

βm =

1 vm

1 m

v ΔC

m

Lm

Lm

−1

−1

⎡ ⎤ : ⎢ Li : T i : M I − v r Lr : T r : M r ⎥ ⎢⎣ ⎥⎦ i = r, m

⎡ ⎤ : ⎢ Li : T i : β I ΔC I − v r Lr : T r : β r ΔC r ⎥ i = r, m ⎣⎢ ⎦⎥

(27)

(28)

This last relation (valid for the general case of a possibly permeable reinforcement type) yields to the following simplified form if impermeable reinforcements are considered:

β

m

=

m −1

1 m

v ΔC

m

L

i

: L :T

i

I

i = r, m

: β ΔC

I

(29)

Due to the localization procedure which does not treat in an equivalent way the embedding matrix and the embedded inclusions (reinforcements) in the point of view of Mori-Tanaka scale-transition approach, the inverse forms satisfied by the coefficients of thermal expansion and coefficients of moisture expansion of the reinforcements are not anymore deduced from the above-relations established for the matrix through simple replacement rules. Actually, unlike the inverse forms obtained according to Eshelby-Kröner self-consistent model, Mori-Tanaka model yields non-equivalent inverse forms for the matrix one the one hand and for the reinforcements, on the second hand. The expressions, required for identifying the thermal or hygroscopic properties of reinforcements within Mori-Tanaka model are:

18

Jacquemin Frédéric and Fréour Sylvain

Mr =

βr =

1 vr

1 v r ΔC r

Tr

Tr

−1

−1

: Lr

: Lr

−1

−1

⎡ ⎤ : ⎢ Li : T i : M I − v m Lm : M m ⎥ i = r, m ⎣⎢ ⎦⎥

(30)

⎡ ⎤ : ⎢ Li : T i : β I ΔC I − v m Lm : β m ΔC m ⎥ (31) ⎢⎣ ⎥⎦ i = r, m

3.3. Examples of Properties Identification in Composite Structures Using Inverse Scale Transition Methods 3.3.1. Determination of Reinforcing Fibers Elastic Properties The literature often provides elastic properties of carbon-fiber reinforced epoxies (see for instance Sai Ram and Sinha, 1991), that can be used in order to apply inverse scale transition model and thus identify the properties of the reinforcing fibers, as an example. Table 2 of the present work summarizes the previously published data for an unidirectional composite designed for aeronautic applications, containing a volume fraction vr=0.60 of reinforcing fibers. In order to achieve the calculations, according to relations (18) or (26) depending on whether Eshelby-Kröner model or Mori-Tanaka approximation, input values are required for the pseudo-macroscopic properties of the epoxy matrix constituting the composite ply. The elastic constants considered for this purpose are listed in Table 3 (from Herakovich, 1998). Both the above-cited inverse scale transition methods have been applied. The obtained results are provided in Table 4, where they are compared to typical values, reported in the literature, for high-strength reinforcing fibers (Herakovich, 1998). It is shown that a very good agreement between the two inverse models is obtained. Moreover, the calculated values are similar to those expected for typical reinforcements according to the literature. Nevertheless, r some discrepancies between the identified moduli do exist, especially for G12 (that r corresponds to L 55 stiffness component). Actually, the value deduced for this component through Mori-Tanaka inverse model deviates from both the expected properties and the estimations of Eshelby-Kröner model. This deviation, occurring for this very component, is Table 2. Macroscopic elastic moduli (from the literature) and stiffness tensor I components (calculated) considered for the composite ply at ΔC = 0 % and TI = 300 K, according to (Sai Ram and Sinha, 1991). Elastic moduli Stiffness tensor components

Y1I [GPa]

Y2I [GPa]

I ν12 [1]

I G12 [GPa]

G I23 [GPa]

130

9.5

0.3

6.0

3.0

LI11 [GPa]

LI22 [GPa]

LI12 [GPa]

LI44 [GPa]

LI55 [GPa]

134.2

14.8

7.1

6.0

3.0

Multi-scale Analysis of Fiber-Reinforced Composite Parts…

19

obviously directly related to the discrepancies previously underlined in subsection 2.5 where the question of comparing the homogenization relations of the two scale transition methods presented in this paper, was investigated. Table 3. Pseudomacroscopic elastic moduli and stiffness tensor components assumed for I the epoxy matrix of the composite plies at ΔC = 0 % and TI = 300 K, according to (Herakovich, 1998). Elastic moduli Stiffness tensor components

Y1m [GPa]

Y2m [GPa]

m ν12 [1]

m G12 [GPa]

Gm 23 [GPa]

5.35

5.35

0.350

1.98

1.98

Lm 11 [GPa]

Lm 22 [GPa]

Lm 12 [GPa]

Lm 44 [GPa]

Lm 55 [GPa]

8.62

8.62

4.66

1.98

1.98

Table 4. Pseudomacroscopic elastic moduli and stiffness tensor components identified I for the carbon fiber reinforcing the composite plies at ΔC = 0 % and TI = 300 K, according to either Mori-Tanaka estimates, or Eshelby-Kröner self-consistent model. Comparison with the corresponding properties exhibited in practice by typical highstrength carbon fibers, according to (Herakovich, 1998).

Y1r [GPa]

Y2r [GPa]

r ν12 [1]

G r23 [GPa]

r G12 [GPa]

Mori-Tanaka estimate

213.1

13.7

0.27

4.1

22.7

Eshelby-Kröner model

213.2

13.3

0.27

4.0

12.1

232

15

0.279

5.0

15

Lr44 [GPa]

Lr55 [GPa]

Elastic moduli

Typical expected properties Stiffness tensor components

Lr11 [GPa] Lr22 [GPa] Lr12 [GPa]

Mori-Tanaka estimate

219.2

24.9

11.2

4.1

22.7

Eshelby-Kröner model

219.2

23.9

10.8

4.0

12.1

Typical expected properties

236.7

20.1

8.4

5.02

15

3.3.2. Determination of AS4/3501-6 Matrix Coefficients of Moisture Expansion Macroscopic values of the Coefficients of Moisture Expansion are sometimes available, contrary to the corresponding pure epoxy resin CME. Simulations were performed in the cas e of an AS4/3501-6 composite, with a reinforcing fiber volume fraction vr=0.60. The calculations were achieved using the elastic properties given in Table 5, and the macroscopic coefficients of moisture expansion listed in Table 6. The same table summarizes the results obtained with both inverse Eshelby-Kröner self-consistent model (21) and Mori-Tanaka

20

Jacquemin Frédéric and Fréour Sylvain

m estimates (29) assuming a moisture content ΔC = 3.125 (the ratio between composite and ΔC I resin densities being 1.25 in this material, the moisture content ratio assumed in the present study corresponds to the maximum expected value), in the case that impermeable reinforcements are considered. According to Table 6, a very good agreement is obtained between the two inverse models. This result is compatible with the homogenisation calculation previously achieved in subsection 2.5: for such a volume fraction of reinforcements, Eshelby-Kröner and Mori-Tanaka models provide identical macroscopic coefficients of moisture expansion from the pseudomacroscopic data. As a consequence, the corresponding inverse forms (21) and (29) yields the same estimation for the pseudomacroscopic CME of the matrix constituting the composite ply.

Table 5. Macroscopic and pseudo-macroscopic mechanical elastic properties of AS4/3501-6 constituents.

E 1 [GPa]

E 2 , E 3 [GPa]

ν12 , ν13

AS4 fibers (Soden et al., 1998)

225

15

0.2

0.40

15

3501-6 epoxy matrix (Soden et al., 1998)

4.2

4.2

0.34

0.34

1.567

135.2

9.2

0.25

0.36

5.2

AS4/3501-6 (KESC homogenisation)

ν23

G 12

[GPa]

Table 6. Macroscopic and pseudomacroscopic (3501-6 matrix only) coefficients of moisture expansion of AS4/3501-6 composite. The pseudomacroscopic values results from the two inverse scale transition models described in the present work.

β11

β 22 , β 33

AS4/3501-6 (Daniel and Ishai, 1994)

0.01

0.2

3501-6 epoxy from Eshelby-Kröner self-consistent inverse model

0.148

0.148

3501-6 epoxy from Mori-Tanaka inverse model

0.148

0.148

Moisture expansion coefficient

4. From the Numerical Model to Analytical Solutions for Estimating the Pseudo-macroscopic Mechanical States 4.1. Introduction It was extensively discussed in previously published works (the interested reader can, for instance refer to Benveniste, 1987 and Fréour et al., 2006a, where the question is addressed), that Mori and Tanaka constitutive assumptions were not suitable for a reliable estimation of the localization of the macroscopic mechanical states within the constituents of typical

Multi-scale Analysis of Fiber-Reinforced Composite Parts…

21

composites conceived for engineering applications, which often present a significant volume fraction of reinforcements. As a consequence, only Eshelby-Kröner approach will be considered in the present section.

4.2. Numerical SC Model Extended to a Thermo-Hygro-Elastic Load Within Kröner and Eshelby self-consistent framework, the hygrothermal dilatation generated by a moisture content increment ΔCi is treated as a transformation strain exactly like the thermal dilatation occurring after a temperature increment ΔTi (that last case was extensively discussed in the literature, see for example Kocks et al., 1998). Thus, the pseudo-macroscopic stresses σi in the considered constituent (i.e. i=r or i=m) are given by:

(

σ i = Li : ε i − M i ΔT i − β i ΔC i

)

(32)

Where, ε stands for the strain tensor. In general case, the moisture content differs at macroscopic scale and pseudo-macroscopic scale, contrary to the temperature. Actually, the reinforcements generally do not absorb moisture. In consequence, the mass of water contained by the composite is: either found in the matrix, locally trapped in porosities (Mensitieri et al., 1995) or located where fiber debonding occurs. Replacing the superscripts i by I in (32) leads to the stress-strain relation that holds at macroscopic scale.

(

σ I = LI : ε I − M I ΔT I − β I ΔC I

)

(33)

The so-called “scale-transition relation” enabling to determine the local stresses and strains from the macroscopic mechanical states was demonstrated in a fundamental work, starting from the assumption that the elementary inclusions (here the matrix and the fiber) have ellipsoidal shapes (Eshelby, 1957):

(

σ i − σ I = −LI : R I : ε i − ε I

)

(34)

Actually, (34) is not very useful, because both the unknown pseudo-macroscopic stresses and strains appear. Nevertheless, combining (32-34) enables to find the following expression for the pseudo-macroscopic strain (the demonstration is available in Jacquemin et al., 2005 and Fréour et al., 2003b):

(

ε i = Li + LI : R I

) : [(L + L : R )..ε + (L : M −1

I

I

I

I

i

i

)

]

− LI : M I ΔT + Li : β i ΔC i − LI : β I ΔC I (35)

In relation (35), the classical replacement rule ΔTi=ΔTΙ=ΔT was introduced (i.e. the temperature field is considered to be uniform within the considered ply).

22

Jacquemin Frédéric and Fréour Sylvain

Moreover, it was established in (Hill, 1967), that the self-consistent model was compatible with the following volume averages on both pseudo-macroscopic stresses and strains:

σ ε

i i = r, m

i i = r, m

=σ =ε

I

I

(36)

For a given applied macroscopic thermo-hygro-elastic load {σI, ΔCI, ΔT} one can easily determine εI through (33), provided that the effective elastic behaviour LI of the ply has been calculated using either the homogenization procedure corresponding to Eshelby-Kröner model or the corresponding Mori-Tanaka alternate solution (see previous developments provided in section 2 above). Then, the pseudo-macroscopic strains are determined through (35).

4.3. Analytical Expression for Calculating the Mechanical States Experienced by the Constituents of Fiber-Reinforced Composites According to Eshelby-Kröner Model The main impediment requiring to be overcome in order to achieve closed-forms from relation (35) is the determination of Morris’ tensor EI. Actually, according to the integrals appearing in relation (8), this tensor will admit only numerical solutions in most cases. However, some analytical forms for Morris’ tensor are actually available in the literature; the interested reader can for instance refer to (of Mura, 1982; Kocks et al., 1998; or Qiu and Weng 1991). Nevertheless, these forms were established considering either spherical, discshaped of fiber-shaped inclusions embedded in an ideally isotropic macroscopic medium, that is incompatible with the strong elastic anisotropy exhibited by fiber-reinforced composites at macroscopic scale (Tsai and Hahn, 1987). In the case of carbon-epoxy composites, a transversely isotropic macroscopic behaviour being coherent with fiber shape is actually expected (and predicted by the numerical computations). Assuming that the longitudinal (subscripted 1) axis is parallel to fiber axis, one obtains the following conditions for the semi-lengths of the microstructure representative ellipsoid: a1→∞, a2=a3. Moreover, the macroscopic elastic stiffness should satisfy :

LI11 ≠ LI12 ≠ LI22 ≠ LI23 ≠ LI44 ≠ LI55 . Now, it is obvious, that these additional hypotheses lead to drastic simplifications of Morris’ tensor (8), in the case that fiber morphology is considered for the reinforcements. The line of reasoning required to achieve the writing of analytical expressions for Morris’ tensor is extensively presented in (Welzel et I

al., 2005; Fréour et al., 2005). Actually, one obtains (in contracted notation i.e, E ij components are given here):

Multi-scale Analysis of Fiber-Reinforced Composite Parts… 0 ⎡0 ⎢ 3 1 ⎢0 + ⎢ 8LI22 4LI22 − 4LI23 ⎢ ⎢ LI22 + LI23 ⎢0 2 ⎢ 8LI22 LI23 − 8LI22 I ⎢ E = ⎢0 0 ⎢ ⎢ ⎢0 0 ⎢ ⎢ ⎢0 0 ⎢ ⎣

0 I I L 22 + L 23 2

0

0

0

0

0

0

8LI22 LI23 − 8LI22 3 8LI22

+

1 4LI22 − 4LI23 1

0

8LI22

+

1

0

4LI22 − 4LI23

0

0

0

0

1 I

8L 55 0

23 0 ⎤ ⎥ 0 ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ 1 ⎥ I 8L 55 ⎥⎦

(37)

In fact, the epoxy matrix is usually isotropic, so that three components only have to be m m considered for its elastic constants: Lm 11 , L12 and L 44 . One moisture expansion coefficient is m. sufficient to describe the hygroscopic behaviour of the matrix: β11

In the case of the carbon fibers, a transverse isotropy is generally observed. Thus, the corresponding elasticity constants depend on the following components: Lr11 , Lr12 , Lr22 , Lr23 , Lr44 , and Lr55 . Moreover, since the carbon fiber does not absorb water, r r its CME β11 and β 22 will not be involved in the mechanical states determination. Introducing these additional assumptions in (35), and taking into account the form (37) obtained for Morris’ tensor, one can deduce the following strain tensors for both the matrix and the fibers: i i ⎤ ⎡ε i 11 ε12 ε13 ⎥ ⎢ i i i i (38) ε = ⎢ε12 ε 22 ε 23 ⎥ ⎢ i i i ⎥ ε ε 23 ε 33 ⎥ ⎦ ⎣⎢ 13

where, in the case of the matrix, ⎧ε m = ε I 11 ⎪ 11 I I ⎪ m 2 L 55 ε12 ⎪ε12 = I m ⎪ L 55 + L 44 ⎪ I ⎪ m 2 LI55 ε13 ⎪ε13 = I m L 55 + L 44 ⎪ ⎪⎪ m m m m m ⎨ε m = N1 + N 2 + N 3 + N 4 + N 5 22 m ⎪ D1 ⎪ I I I I ⎪ 2 L 22 L 22 − L 23 ε 23 ⎪ε m = 23 2 ⎪ I I I m I m I m 2 L 22 + L 23 L 44 − L 44 + L 22 3 L 44 − 2 L 23 − 3 L 44 ⎪ ⎪ I I I I L 22 − L 23 ε 22 − ε 33 ⎪ε m = ε m − 4 LI 33 22 22 2 ⎪ I I m m I m I m ⎪⎩ L 22 + 3 L 22 L11 − L12 - L 23 L11 + L 23 - L12

(

(

) ( (

(

)

)( ) ) (

(39)

)

)

24

Jacquemin Frédéric and Fréour Sylvain

⎧N m ⎪ 1 ⎪N m ⎪ 2 ⎪N m ⎪ 3 ⎪ ⎪N m ⎨ 4 ⎪ ⎪ ⎪ ⎪ N 5m ⎪ ⎪ ⎪ m ⎩D1

(

m

)(

m

m

= L11 + 2L12 β11 ΔC =

(

=

I L 22

=

I L 22

I

(

)

m

m

+ M11 ΔT

I I I I −L12 β11ΔC + M11ΔT I m I L12 − L12 ε11

=

=

(

m

)− (

I L 22

m

I

)

)(

I

I

I

I

+ L 23 β 22 ΔC + M 33ΔT

) ( )( ) )+ L (L + 3 L )(L − L )+ L I

m

m

I

)

) I2

L 22 5 L11 − L12 + 3 L 22 − L 23 3 L11 + L12 + 4 L 22 + L 23 I ε 22 2 2 I I m m I I 3 L 22 − L 23 L11 − L12 + L 22 − L 23

(

(

m I LI22 Lm 11 − 5 L12 − L 22

(3 L

I 23

I I 22 − L 23 m I I Lm 11 + L12 + L 22 − L 23

m 11

m 11

m I 12 + 4 L 22 2 2 I I 22 − L 23

m 12

(40)

)− 3 L

2 I 23 I ε 33

The pseudo-macroscopic stress tensors are deduced from the strains using (32). Thus, in the matrix, one will have:

⎡ σm 11 m m m ⎢ σ = ⎢2 L 44 ε12 ⎢ m m ⎢⎣ 2 L 44 ε13

m

m

2 L 44 ε12 σm 22

m 2 Lm 44 ε 23

2 L 44 ε13 ⎤ m⎥ 2 Lm ε 44 23 ⎥ ⎥ m σ 33 ⎥⎦ m

m

(41)

with

( ( (

) ( ) ( ) (

) ( ) ( ) (

) ) )

⎧ σ m = Lm ε m − M m ΔT + Lm ε m + ε m − 2 M m ΔT − β m Lm + 2 Lm ΔC m 11 11 11 12 22 33 11 11 11 12 ⎪⎪ 11 m m m m m m m m m m m m = − + + − − + σ L ε M ΔT L ε ε 2 M ΔT β L 2 L ⎨ 22 11 22 11 12 11 33 11 11 11 12 ΔC ⎪σ m = Lm ε m − M m ΔT + Lm ε m + ε m − 2 M m ΔT − β m Lm + 2 Lm ΔC m 11 33 11 12 11 22 11 11 11 12 ⎪⎩ 33

(42)

The local mechanical states in the fiber are provided by Hill’s strains and stresses average laws (36):

εr =

r

σ =

1 I vm m ε − ε vr vr 1 v

r

I

σ −

vm v

r

σm

(43)

(44)

Multi-scale Analysis of Fiber-Reinforced Composite Parts…

25

4.4. Examples of Multi-scale Stresses Estimations in Composite Structures: T300-N5208 Composite Pipe Submitted to Environmental Conditions 4.4.1. Macroscopic Analysis 4.4.1.1. Moisture Concentration

Consider an initially dry, thin uni-directionally reinforced composite pipe, whose inner and outer radii are a and b respectively, and let the laminate be exposed to an ambient fluid with boundary concentration c0. The macroscopic moisture concentration, cI(r,t), is solution of the following system with Fick's equation (45), where DI is the transverse diffusion coefficient of the composite. Boundary and initial conditions are described in (46): 2 I ∂c I 1 ∂c I ⎤ I ⎡∂ c =D ⎢ 2 + ⎥ , a
(45)

⎧⎪c I (a , t ) = c 0 and c I (b, t ) = c 0 ⎨ I ⎪⎩c (r,0) = 0

(46)

Applying the Laplace transform to the latter system and using the residue theory to express the solution in time space (Crank, 1975), we finally obtain the macroscopic moisture concentration:

2 exp(−ω 2 m τ) { A m J 0 (ωm r ) + B m Y0 (ωm r ) } m =1 ω m Δ ′u (ω m ) ∞

c I ( r , τ) = c 0 + ∑

where J 0 and Y0 are Bessel’s functions of order zero,

[a ] .

(47)

Δu is the determinant of 2*2 matrix

Am and Bm are determinants of matrices deduced from [a ] by substituting

respectively column 1 and 2 by the constant vector {g } . with respect to

Δu′ ( ω m ) is the derivative of Δu

ω calculated for ω m the mth positive root of Δu . r and τ are defined by the

relations r = r/b and τ = (D t)/b . I

2

Furthermore, the elements of

[a ]

and

{g}

are: a 11 = J 0 (ωa ) , a 12 = Y0 (ωa ) ,

a 21 = J 0 (ω) , a 22 = Y0 (ω) , g1 = c 0 , g 2 = c 0 . 4.4.1.2. Macroscopic Stresses

At the initial time, let us assume that the pipe is stress free. Therefore, the hygro-elastic orthotropic behaviour writes as follows in (48-49), where βI and LI are respectively the in-

26

Jacquemin Frédéric and Fréour Sylvain

plane tensors of hygroscopic expansion coefficients and moduli. Those tensors are assumed to be material constants.

⎡ LI I ⎫ ⎧σ11 11 ⎪ I ⎪ ⎢ I ⎪σ 22 ⎪ ⎢ L12 ⎨ I ⎬=⎢ I ⎪σ 33 ⎪ ⎢L12 ⎪ I ⎪ ⎢ ⎩τ12 ⎭ ⎢⎣0

I I I I ⎫ L12 0 ⎤ ⎧ε11 ⎥ ⎪ − β11ΔC ⎪ I I L 22 L 23 0 ⎥ ⎪ε I22 − β I22 ΔC I ⎪ ⎥ ⎨ ⎬ I I I I I L 23 L 22 0 ⎥ ⎪ε 33 − β 22 ΔC ⎪ ⎥ ⎪ I I I ⎪ I 0 0 L 55 ⎥⎦ ⎩γ 12 − β12 ΔC ⎭ I

L12

I ⎫ I ⎧⎪τ 32 ⎪ ⎡L 44 = ⎨ I ⎬ ⎢ ⎪⎩τ13 ⎪⎭ ⎣⎢0

I

with, ΔC =

I ⎫ 0 ⎤ ⎧⎪γ 32 ⎪ ⎥ ⎨ I ⎬ I L 55 ⎦⎥ ⎪⎩γ 13 ⎪⎭

(48)

(49)

cI I . c and ρI are respectively the macroscopic moisture concentration and the I ρ

mass density of the dry material. To solve the hygromechanical problem, it is necessary to express the strains versus the displacements along with the compatibility and equilibrium equations. Introducing a characteristic modulus L0 , we introduce the following dimensionless variables:

σ Ι = σ Ι / L 0 , LI = LI /L 0 , ( w I , u I , v I ) = ( w I , u I , v I ) / b. Displacements with respect to longitudinal and circumferencial directions, respectively I

u (x , r ) and v I (x , r ) are then deduced: ⎧ u I ( x , r) = R 1 x ⎪⎪ I ⎨ v ( x , r) = R 2 x r ⎪R , R are constants. ⎪⎩ 1 2 I

(50)

I

It is worth noticing that the displacements u (x , r ) and v (x , r ) do not depend on the moisture concentration field. Finally, to obtain the through-thickness or radial component of I

the displacement w , we shall consider in the following the analytical transient concentration (47). The radial component of the displacement field w

I

satisfies the following equation:

Multi-scale Analysis of Fiber-Reinforced Composite Parts…

r2

∂w I ∂2w I r − wI = + 2 ∂r ∂r

K1r 2

27

∂ΔC I ∂r

(51)

LI22

I with, K1= LI12 β11 + LI23 β I22 + LI22 β I22 It is shown that the general solution of equation (51) writes as the sum of a solution of the homogeneous equation and of a particular solution (Jacquemin et Vautrin, 2002).

w I (r ) = R 3 r +

R4 − r

∞ 2exp(-ω 2m τ) K 1 [A i ∑ ∑ I k =0 m =1 ω m Δ ′u (ω m ) L 22 ∞

1 ( −1) k ( ) 2 k +1 (ω m ) 2 k + 2 B r 2 k +3 2 + i k!(k + 1)! ((2k + 3) 2 − 1) π

{

1 (-1) k ( ) 2 k +1 (ω m ) 2 k + 2 1 r ( 2 k + 3) 2 [2 ln( ω m ) − ψ( k + 1) − ψ(k + 2)] + ∑ k =0 k!(k + 1)! 2 ((2k + 3) 2 − 1) ∞

1 (-1) k ( ) 2 k +1 (ω m ) 2 k + 2 ln(r ) r 2 k +3 2(2k + 3) r 2 k +3 2 2∑ [ − ] − r ln(r ) k =0 k!(k + 1)! ((2k + 3) 2 − 1) ((2k + 3) 2 − 1) 2 ∞

}]

Finally, the displacement field depends on four constants to be determined : Ri for i=1..4. These four constants result from the following conditions :

• •

global force balance of the cylinder; nullity of the normal stress on the two lateral surfaces.

4.4.2. Numerical Simulations of Internal Stresses in T300/5208 Composite Laminated Pipes 4.4.2.1. Introduction

Thin laminated composite pipes, with thickness 4 mm, initially dry then exposed to an ambient fluid, made up of T300/5208 carbon-epoxy plies, with a fiber volume fraction vr=0.6, were considered for the determination of both macroscopic stresses and moisture content as a function of time and space. The closed-form formalism used in order to determine the mechanical stresses and strains in each ply of the structure is described in subsection 4.4.1. This model ensures the calculation of the macroscopic moisture content, too. When the equilibrium state is reached, the maximum moisture content of the neat resin may be estimated from the maximum moisture content of the composite. By assuming that the fibers do not absorb any moisture, ΔCI and ΔCm are related by expression (16) given by (Loos and Springer, 1981). In the case of T300/5208, since the ratio between composite and resin densities is 1.33 (due to the constituents properties listed in table 1), the maximum moisture content ratio given by (16) is about 3.33.

28

Jacquemin Frédéric and Fréour Sylvain

c (%)

Figure 2 shows the time-dependent concentration profiles, resulting from the application of a boundary concentration c0, as a function of the normalized radial distance from the inner radius rdim. At the beginning of the diffusion process important concentration gradients occur near the external surfaces. The permanent concentration (noticed perm in the caption) holds with a constant value because of the symmetrical hygroscopic loading. The macroscopic mechanical states were calculated for two types of composites structures: a) a unidirectionnaly reinforced cylinder, and b) a [55°/-55°]S laminated cylinder.

1,5 1,4 1,3 1,2 1,1 1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0

0.5 month 1 month 1.5 months 2 months 2.5 months 3 months 6 months perm

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

r dim

Figure 2. Time dependent concentration profiles in T300/5208 as a function of the normalised radial distance from the inner radius rdim.

Starting with the macroscopic stresses deduced from continuum mechanics, the local stresses in both the fiber and matrix were calculated either with the new analytical forms or the fully numerical model. The comparison between the two approaches is plotted on figures 3 and 4. These figures show the very good agreement between the numerical approach and the corresponding closed-forms solutions. The slight differences appearing are due to the small deviations on the components of Morris’ tensor calculated using the two approaches. Actually, it is not possible to assume the quasi-infinite length of the fiber along the longitudinal axis in the case of the numerical approach, because the numerical computation of Morris’ tensor is highly time-consuming. Thus, the numerical version of Eshelby-Kröner self-consistent model constitutes only an approximation of the real microstructure of the composite. In consequence, it seems that the new analytical forms, that are able to take into account the proper microstructure for the fibers, are not only more convenient, but also more reliable than the initially proposed numerical approach. 4.4.2.2. Interpretation of the Simulations

The highest level of macroscopic tensile stress is reached for the uni-directional composite, in the transverse direction and in the central ply of the structure (figure 3). The transverse stresses exceed probably the macroscopic tensile strength in this direction. The choice of a

Multi-scale Analysis of Fiber-Reinforced Composite Parts…

29

[+55°/-55°]S laminated allows to reduce the macroscopic stress in the transverse direction. Nevertheless, a high shear stress rises along the time in the fibers of the central ply of such a structure (figure 3).

50

50

0 0,5

1

1,5

2

2,5

3

6

perm

[MPa]

100

[MPa]

100

-50

0 0,5

1

1,5

2

2,5

3

6

perm

6

perm

-50

a)

-100

a)

-100

-150

-150

month

month

50

40 35

0 1

1,5

2

2,5

3

6

perm [MPa]

[MPa]

b)

30

0,5 -50

25 20 15

-100

10

-150

b)

5 0 0,5

-200

1

1,5

month

2 2,5 month

3

A S 4 5 0 _ 1_ 1 Δ C m / Δ C I = 2

composite (CMF)

200

matrix (numerical)

fiber (numerical)

σ 11

0 -200 -400

1

2

3

4

5

matrix (analytical)

6

7

8

fiber (analytical) cas

Figure 3. Local stresses in T300/5208 composite for the central ply, in the case of a) the unidirectionaly reinforced composite and b) the [+55°/-55°]S symmetric laminate. CMF stands for Continuum Mechanics Formalisms.

Moreover, the figure 4 shows that the micro-mechanical model always predict a very high compressive stress in the matrix of the inner ply whatever the laminate studied (the macroscopic stress is negligible in the radial direction because thin structures are considered). These local stresses could help to explain damage occurrence in the surface of composite structures in fatigue.

30

Jacquemin Frédéric and Fréour Sylvain 150

150

100

a)

100

50

1,5

2

2,5

3

6

perm

-100

0 -50

0,5

,

,

,

1

[MPa]

[MPa]

0,5

-50

,

50

0

1

1,5

2,5

3

6

perm

3

6

perm

a) -100

-150

-150

-200 -250

-200 month

month 50

0 -5

0 0,5

1

1,5

2

2,5

3

6

perm

1

1,5

2

2,5

[MPa]

-15 -20

,

,

,

,

-100

0,5

-10

-50 [MPa]

2

-150

b)

-25 -30

-200

b)

-35

-250

-40 month

month A S 4 5 0 _ 1_ 1 Δ C m / Δ C I = 2

composite (CMF)

200

matrix (numerical)

fiber (numerical)

σ 11

0 -200 -400

1

2

3

4

5

matrix (analytical)

6

7

8

fiber (analytical) cas

Figure 4. Local stresses in T300/5208 composite for the inner ply, in the case of a) the uni-directionaly reinforced composite and b) the [+55°/-55°]S symmetric laminate. CMF stands for Continuum Mechanics Formalisms.

This work demonstrates the complementarities of continuum mechanics and micromechanical models for the prediction of a possible damage in composite structures submitted to hygro-elastic loads. In the following section, the analytical expressions presented here for the localization of the macroscopic mechanical states within the plies constituents, will be inversed in order to achieve the identification of the strength of the constitutive matrix of a composite ply.

5. Identification of the Local Strength of the Constitutive Matrix of a Composite Ply 5.1. Introduction Damage predictions are important for design and for guiding materials improvement for engineering applications. Composite structures encountered in engineering applications

Multi-scale Analysis of Fiber-Reinforced Composite Parts…

31

are designed to endure combined mechanical, thermal and hygroscopic loads during their service life. Besides, composite structures usually benefit from improved properties granted by a multidirectional arrangement of their plies. The multiplicity of both possible loads and ply arrangements is not compatible with an extensive experimental investigation of composite structures damage. As a result, only uniaxial and pure shear test data of unidirectional composites are usually available in the literature. By consequence, the estimation of damage occurrence in composite structures requires introducing adapted failure criteria extending the available data to the combined loads and composite laminates considered for one particular application. Many published papers have dealt with this problem: see for instance (Tsai, 1987; Cuntze, 2003). Nevertheless, it is established for a long time, that in composite structures the damage initiates at microscopic scale, either (and most of time) in the matrix or (sometimes) in the fibers. The failure of a ply is thus closely related and explained by the failure of its microscopic constituents (Tsai, 1987; Cuntze, 2003; Fleck and Jelf, 1995; Kaddour et al., 2003; Khashaba, 2004). As a consequence, the reliable prediction of a possible damage occurrence of multi-directionnal laminates submitted to complex loading requires the knowledge of the microscopic failure criteria of the epoxy matrix and carbon fibers constituting the plies. Nevertheless, previous published works have emphasised the following remarkable result: the strength of the pure constituents (i.e. pure epoxy resin) strongly depends on the size of the sample, and especially on its thickness (Fiedler et al., 2001). Besides, the thickness of a ply in thin laminates has the magnitude of 150 microns, that is generally strongly weaker than the thickness of the samples tested for the experimental determination of the strength of the pure constituents. As a consequence, the experimental strengths of pure carbon fibers and epoxy matrices, determined on bulk specimen can hardly be directly used to properly estimate microscopic failure criteria in real structures. In particular, as shown for instance in (Garett and Bailey, 1977; Christensen and Rinde, 1979), the effect of the matrix on transverse failure of composite structures is of interest. The strain to failure of the pure matrix in uniaxial tension varied from 1.5 to 70 % whereas transverse strains to failure of corresponding fiber reinforced composites were dramatically smaller and varied only in the range 0.2 to 0.9%. In the present study, an innovative method, dedicated to the determination of the microscopic stress/strain failure criteria of the epoxy matrix embedded in a composite structure is described. This method is based on the inversion of the analytical expressions presented in section 4.3. The present work describes developments relating the macroscopic failure envelopes to the microscopic ones. The conditions, indicated in already published literature, when the macroscopic failure can exclusively be attributed to matrix failure modes are taken into account as fundamental hypotheses of the present approach. The model enables the identification of both the strength coefficients and ultimate strength, so that the microscopic stress/strain failure envelopes can also be drawn. Applications to the case of two typical carbon/epoxy composites (T300/5208 and AS4/3501) are achieved: the failure conditions of the N5208 and 3501-6 epoxy resins will be determined and compared.

32

Jacquemin Frédéric and Fréour Sylvain

5.2. Determination of the Local Failure Criterion of the Matrix from the Macroscopic Strength Data of the Composite Ply 5.2.1. Introduction – Choice of a Failure Criterion In this paper, failure is taken in the general sense previously defined in the literature, including fracture, but also yield, etc. Since this works aims applications to multidirectional structures submitted to triaxial stresses, general failure criteria are necessary to the description of the strength in both stress and strain spaces. Failure criteria serve important functions in the design and sizing of composite laminates. They should provide a convenient framework or model for mathematical operations. The framework should be the same for different definitions of failures, such as the ultimate strength, endurance limit, or a working stress based on design or reliability considerations. However, the criteria are not intended to explain the mechanisms of failure, that can occur concurrently or sequentially. The quadratic criterion will be used in the present study: it includes interactions among the stress or strain components analogous to the Von Mises criterion for isotropic materials, and is compatible with the existence of strength having the properties, often met in the case that composite structures are considered, to be anisotropic and also possibly different in tension or compression. The criterion, expressed in stress space writes as follows : i i Fmnop σ imn σ iop + Fmn σ imn = 1

(52)

where F stands for the strength parameters respectively expressed in stress space. The superscript i represents the scale considered for failure prediction (macroscopic: i=Ι or pseudomacroscopic: i=m or i=r). In order to use the failure criteria (52) presented above, it is necessary to identify the i i quadratic ( Fmnop ) and linear ( Fmn ) strength parameters involved in the equation. In the present work, for helping fixing the ideas, the simplified case of three-dimensional stresses and strains (for both macroscopic and microscopic scales), with a single shear component, usually met in multi-directional composite laminates submitted to mechanical i i loads (see examples given in Tsai, 1987) will be assumed to hold (i.e. σ13 = σ 23 = 0 MPa , i ε13 = ε i23 = 0 , where the subscripts 1, 2 and 3 respectively denotes the directions parallel to the fiber axis, the transverse direction and the normal direction, in the orthogonal frame of reference of the considered ply). Besides, the strength should be unaffected by the direction or i sign of the shear stress component σ12 : if shear stress is reversed, the strength should be kept i i constant. However, sign reversal for the longitudinal ( σ11 ) and transverse ( σ 22 ) stresses components from tension to compression is expected to have a significant effect on both the macroscopic and microscopic strength of the composite. As a consequence, terms of equation (52) containing first-degree shear stress should be null. Finally, taking into account the definition chosen for the reference frame, and the properties of (at least) transverse isotropy

Multi-scale Analysis of Fiber-Reinforced Composite Parts…

33

exhibited at any (i.e. macroscopic or microscopic) scale in one ply, the strength parameters have to satisfy the following relations: i ⎧F i = F3333 , ⎪ 2222 ⎪ i i ⎨F1122 = F1133 , ⎪ i i ⎪⎩F22 = F33 .

(53)

Taking into account the above listed simplifications, equation (52) can be rewritten:

)

(

⎧ ⎛ i2 i i2 i i2 i i2 ⎞ i i i i i i i ⎪⎪1 = F1111σ11 + 2 F1212 σ12 + F2222 ⎜⎜ σ 22 + σ 33 ⎟⎟ + 2 F1122 σ11 σ 22 + σ 33 + 2 F2233σ 22 σ 33 + (54) ⎝ ⎠ ⎨ ⎪ i i i i i ⎪⎩ F11σ11 + F22 σ 22 + σ 33

(

)

5.2.2. Direct Identification of the Macroscopic Strength Parameters Most of the unknown macroscopic strength parameters in stress space, appearing in equation (54) can be identified using information deduced from simple mechanical tests (uniaxial tension, compression or longitudinal shear tests Tsai, 1987): I F1111 =

1 XI X

I/

I , F2222 =

1 YI Y

I/

I , F11 =

1 X

I

-

1 X

I/

I , F22 =

1 Y

I

-

1 Y

I/

I , F1212 =

1 2 SI

2

(55)

Where XI and YI are respectively the longitudinal and transverse tensile stress strength, /

/

X I and Y I the longitudinal and transverse compressive stress strength, whereas SI is the longitudinal shear stress. I

I

The two unknown remaining terms, F1122 and F2233 are related to the interaction between two orthogonal stress components. The practical determination of these interaction terms requires performing biaxial tests, which are not as easy to achieve than uniaxial tests. As a consequence, the required data are often not available in the literature. There are, however, geometric and physical conditions fixing the mathematical form of the failure criterion (54): for instance, the failure envelope has to be closed so that the material cannot present infinite strength when submitted to any load. Let us introduce a dimensionless interaction term: i* Fmmnn =

i Fmmnn i i Fmmmm Fnnnn

(56)

34

Jacquemin Frédéric and Fréour Sylvain i* For closed envelopes, the condition − 1 ≤ Fmmnn ≤ 1 has to be satisfied. But a more

detailed theoretical study (see Liu and Tsai, 1998) reduces the admissible range to the domain

1 I* [-1,0]. The same reference (Liu and Tsai, 1998) advises the choice of Fmmnn = - for the 2

macroscopic interaction term (which corresponds to the generalised Von Mises model), since this value is reasonable for a wide range of laminates. Taking into account this additional I

I

I

assumption in equation (56), the knowledge of F1111 and F2222 = F3333 ensures the I

I

determination of the last two missing interaction terms F1122 and F2233 , in stress space. One similar method could be applied in order to determine the macroscopic strength parameters expressed in strain space from the ultimate strains. Nevertheless, this method is not useful in practice since uniaxial strains are difficult to apply to a sample. Thus, the ultimate strains are generally deduced from the ultimate stresses: to reach this goal, one has to introduce the macroscopic properties, i.e. the stiffness tensor LI, in order to relate both failure criteria through Hooke’s law (33) expressed at macroscopic scale assuming a purely elastic load.

5.2.3. Identification of the Microscopic Strength Parameter (of the Matrix Only) Using an Inverse Method From the standpoint of the structural designer, it is desirable to have failure criteria which are applicable at the level of the lamina, the laminate, and the structural component. Nevertheless, failure at macroscopic scale is often the consequence of an accumulation of micro-level failure events (Tsai, 1987; Liu and Tsai, 1998). Laminated materials typically exhibit many local failures prior to rupture. Thus, it is important to build up tools enabling to enhance the understanding of micro-level failure mechanisms in order to develop higherstrength materials. The ultimate goal is to have a failure theory that the designer can use with confidence under the most general structural configuration and loading conditions and that the developer of materials can use to design and fabricate new products to meet specific needs. In order to reach this goal, the estimation of microscopic strength criteria would be of a valuable help. Since the epoxy resins involved in composite structures generally exhibit an isotropic hygro-mechanical behaviour, the microscopic strength criterion expressed in terms of stresses (54) simplifies as follows:

[ (

)

]

⎧ m ⎛ m2 m2 m2 ⎞ m m2 m m m m m m ⎪1 = F1111 ⎜⎝ σ11 + σ 22 + σ 33 ⎟⎠ + 2F1212 σ12 + 2 F1122 σ11 σ 22 + σ 33 + σ 22 σ 33 (57) ⎨ ⎪ + Fm σ m + σ m + σ m 11 11 22 33 ⎩

(

)

Thus, only four strength parameters have to be determined in order to enable failure m

m

m

m

predictions at microscopic scale: F1111 , F1212 , F1122 , F11 . Hypotheses being compatible with the experimental observations are necessary to build an inverse model enabling the

Multi-scale Analysis of Fiber-Reinforced Composite Parts…

35

determination of these four parameters from the corresponding, available from practical mechanical tests, macroscopic strength stress failure criterion. The present work is focused on the development of modelling tools for the prediction of a possible damage occurrence in fiber-reinforced epoxy laminates submitted to mechanical loads. Actually, fibrous composite materials fail in a variety of mechanisms at the fiber/matrix microscopic scale. Besides, according to the literature, i) fiber-dominated failures usually occur when the plies are loaded in planes perpendicular to the fibers axis (longitudinal tension and compression), whereas ii) matrix-dominated failures often occur in the cases that the plies are loaded along the transverse and normal directions in tension and compression or when shear stresses are applied to the considered ply (Tsai, 1987; Liu and Tsai, 1998). Thus, matrix-dominated failure modes often occur in practice. As a consequence, the above listed i) and ii) statements will be used in order to identify microscopic strength parameters in stress and strain spaces for the matrix. According to the developments of section 4, it is possible to derive the pioneering numerical self-consistent model of Kröner and Eshelby in order to find the relation between the macroscopic mechanical states and the researched corresponding microscopic stresses and strains existing in the matrix of a composite material. In the present work, the strength parameters in either the matrix or the ply will be considered to remain independent from the magnitude of the applied mechanical load. Since the damage envelope has been defined as the strain or stress threshold beyond which nonlinearity occurs in the behaviour of the material at the scale concerned by damage, and in the case that a purely mechanical load is taken into account, the material is assumed to behave elastically until failure occurs. Now, in these conditions, both stress and strain ultimate strength are simultaneously reached, and satisfy either macroscopic elastic Hooke’s law (33) or the corresponding microscopic relations that are deduced from (38-42), assuming ΔC I = ΔC m = 0 and ΔT I = ΔT m = 0 K . It will be assumed that macroscopic failure occurring in the transverse and normal I

directions, for a longitudinal stress σ11 = 0 MPa , is governed by local failure of the matrix. Various macroscopic stress states, compatible with that last hypothesis, are taken on the macroscopic strength envelope (54), expressed in stress space and, finally implemented in the scale transition relations (38-42). This leads to the determination of microscopic mechanical stresses and strains states in the matrix, that are, according to our hypotheses, responsible for macroscopic damage governed by matrix failure. As a consequence, these local mechanical states should be compatible with the microscopic failure envelopes of the matrix as written in equations (57). According to this relation, four, non equivalent, macroscopic stress states suffice to find m

m

m

m

the eight researched coefficients involved in (57): F1111 , F1212 , F1122 , F11 . The whole method required to perform such estimation is described on table 7. Actually, four macroscopic loading states taken on the stress failure envelope (defined on table 7) σ a , σ b , σ c and σ d I

I

I

I

are required for the determination of the four coefficients of the failure envelopes since numerical tests shows that equation (56) rewritten at microscopic scale for the epoxy matrix m

m

does not provide an additional relation between F1111 and F1122 :

36

Jacquemin Frédéric and Fréour Sylvain m* = F1122

m F1122 m F1111

≠−

1 2

(58)

Moreover, according to (38-42) an uniaxial macroscopic tension or compression along the transverse (or normal) direction induces local mechanical states in the matrix generally exhibiting no zero strain and stress on-diagonal components (see for instance the cases of the macroscopic loads σ a and σ b on Table 7). As a consequence, only the strength coefficient I

I

m F1212 can be determined independently from the three others, from the single macroscopic m

m

m

load σ d . Concerning the calculation of F1111 , F1122 , F11 , one has to solve numerically the I

system (60) (cf. Table 7). Finally, the uniaxial microscopic ultimate stresses of the epoxy matrix embedded in the composite structure can be deduced from the set of equations (55) expressed at microscopic scale (i.e. replacing the subscripts I by the subscript m), provided that the coefficients of the local failure envelope are already known:

⎧ ⎪X m = Y m = Z m = 1 ⎛⎜⎜ m ⎪ 2 F1111 ⎝ ⎪ 1 ⎪ m/ m/ m/ ⎨X = Y = Z = m 2 F1111 ⎪ ⎪ 1 ⎪S m = m ⎪ 2 F1212 ⎩

2 m m⎞ − F11 F11m + 4 F1111 ⎟⎟ ⎠

⎛ m2 m m⎞ + F11 ⎟⎟ ⎜⎜ F11 + 4 F1111 ⎠ ⎝

(62)

The method, developed in the present paragraph, enables the determination of a) the coefficients of the microscopic failure envelope of the epoxy matrix in stress and/or strain space from the macroscopic failure envelope of the ply and scale transition relations linking macroscopic loads to the corresponding local microscopic mechanical states experienced by the matrix, only thereafter, b) the local maximum strength of the matrix embedding the carbon fibers which can be evaluated from the classical formalism relating the strength to the coefficients of the failure envelope. This inverse method provides an alternative to the classical direct approach leading to the determination of the failure envelope from the maximum strength measured on pure epoxies, in the cases that the required data is not available or when the behaviour of the matrix embedded in the composite structure is expected to be significantly different from the behaviour of the pure matrix, as shown for example, in references (Garett and Bailey, 1977; Christensen and Rinde, 1979).

Table 7. One possible set of trials enabling the determination of the microscopic strength coefficients of the matrix expressed in stress space.

σ cI

Applied macroscopic load

Corresponding macroscopic strain

σ Ia

ε

I i

⎡0 0 0⎤ / ⎥, I ⎢ 0 ⎥ σ b = ⎢0 Y I ⎢0 0 0⎥⎦ ⎣

⎡0 0 ⎢ = ⎢0 Y I ⎢⎣0 0

⎡ I ⎢ε 11i ⎢ =⎢ 0 ⎢ ⎢ 0 ⎣

0 ε

I 22i 0

0⎤ ⎥ 0⎥ 0⎥⎦

0 ⎡0 ⎢ I = ⎢0 σ 22c ⎢⎣0 0

0 ⎤ ⎥ 0 ⎥, I ⎥ σ33c ⎦

σ dI

⎧ I ⎛ I2 I2 ⎞ ⎪F2222 ⎜⎜ σ 22c + σ33c ⎟⎟ + ⎝ ⎠ ⎪ ⎨ ⎪2 FI σ I σ I + FI ⎛⎜ σ I 2 + σ I 2 ⎞⎟ − 1 = 0 22 ⎜ 22c 33c ⎟ ⎪ 2233 22c 33c ⎝ ⎠ ⎩

⎤ I I σ I22i , ⎥ ε11i = S12 ⎥ I I I 0 ⎥ ε 22i = S22 σ 22i , I = SI23 σ I22i ⎥ ε 33i I ε ⎥ 33i ⎦ 0

⎡ε I ⎢ 11c ε Ic = ⎢ 0 ⎢ ⎢ 0 ⎣⎢

0 ε I22c 0

0 ⎤ ⎥ 0 ⎥ ⎥ ε I33c ⎥ ⎦⎥

I ε11c ε I22c I ε33c

(

)

I I σ I22c + σ33c , = S12 I I I I = S22 σ 22c + S23 σ33c , I = SI23 σ I22c + SI22 σ33c

Corresponding conditions for finding the microscopic strength coefficients in stress space from (10, 19)

⎡σ m ⎢ 11i ⎢ 0 = σm i ⎢ ⎢ 0 ⎣⎢

0 m

σ 22i 0

0 ⎤ ⎥ 0 ⎥ , i = a, b, c ⎥ ⎥ σm 33i ⎦⎥

⎧Fm A + Fm B + F m C − 1 = 0 ⎪ 1111 i 1122 i 11 i ⎪ 2 2 2 ⎪A = σ m + σ m + σ m 11i 22i 33i ⎪⎪ i , i = a, b, c ⎨ ⎡ ⎤ ⎞ ⎛ ⎪ B = 2 ⎢σ m ⎜ σ m + σ m ⎟ + σ m σ m ⎥ 33i ⎟ 22i 33i ⎥ ⎪ i ⎢ 11i ⎜⎝ 22i ⎠ ⎣ ⎦ ⎪ ⎪ m m m ⎩⎪Ci = σ11i + σ 22i + σ33i

⎡ 0 ⎢ I εdI = ⎢ε 12d ⎢ ⎢ 0 ⎣

0⎤ ⎥ 0⎥ 0⎥⎥ ⎦

I ε 12d

0 0

0⎤ ⎥ 0⎥ ⎥ 0⎥ ⎦

I I I ε12d = S66 S

i = a, b Corresponding microscopic stress according to (15-17)

⎡ 0 SI ⎢ = ⎢SI 0 ⎢0 0 ⎢⎣

⎡ 0 Sm ⎢ = ⎢Sm 0 ⎢0 0 ⎢⎣

0⎤ ⎥ 0⎥ 0⎥⎥ ⎦

(59)

σm d

(60)

m S m 2 - 1 = 0 (61) 2 F1212

38

Jacquemin Frédéric and Fréour Sylvain

5.3. Numerical Applications and Examples 5.3.1. Identification of the Microscopic Failure Criteria of Two Typical Epoxies from the Knowledge of the Macroscopic Failure Envelope of AS4/3501-6 and T300/N5208 Composite Plies In the present paper, two types of high strength carbon fiber reinforced epoxies are considered: a) AS4/3501-6 and b) T300/N5208 composites having identical fiber volume fraction: vf=0.6. These two materials constitute good candidates for the present work, since the microscopic strength of their respective matrix is not yet available (at our knowledge) in the already published literature, in spite of they are quite often considered for illustrating scientific works in this field of research (Tsai, 1987). Table 8. Macroscopic strength data. XI

X I´

YI, ZI

YI´, ZI´

SI

T300/5208 (Tsai, 1987)

1500

1500

40

246

68

AS4/3501-6 (Liu and Tsai, 1998)

1950

1480

48

200

79

Strengths [MPa]

Table 9. Quadratic macroscopic stress failure criteria deduced from the strength data. Quadratic ijkl subscripted coefficients [MPa-2] and linear ij subscripted coefficients [MPa-1]. Strength parameters

I F1111

I F2222 , I F3333

I F1122 ,

I F1212

I F1133

I F2233

F11I

I , F22 I F33

T300/5208

4.44 10-7

1.02 10-4

1.08 10-4

-3.36 10-6

-5.08 10-5

0

0.0209

AS4/3501-6

3.46 10-7

1.04 10-4

8.01 10-5

-3.00 10-6

-5.02 10-5

-0.0002

0.0158

The macroscopic strength of single plies are given in Table 8. The coefficients of the corresponding quadratic macroscopic stress failure criteria, deduced from the classical direct method, through equation (55) are listed in Table 9. Table 10. Macroscopic stiffness components [GPa] of 60% volume uni-directionally fiber reinforced plies. Fiber axis is parallel to longitudinal direction. Stiffness components

LI11

LI22 , LI33

LI12 , LI13

LI23

LI44

LI55 , LI66

T300/5208

142.72

13.92

5.79

7.19

3.34

7.00

AS4/3501-6

137.27

11.60

4.20

5.22

3.68

6.45

Multi-scale Analysis of Fiber-Reinforced Composite Parts…

39

Table 11. Quadratic local stress failure criteria in N5208 and 3501-6 epoxy matrices respectively deduced from the macroscopic failure envelopes of T300/5208 and AS4/3501-6 plies, taking into account the microscopic elastic properties given on tables 1 and 5. Quadratic ijkl subscripted coefficients [MPa-2] and linear ij subscripted coefficients [MPa-1]. Strength parameters

m m m F1111 , F2222 , F 3333

m F1212

m m m F1122 , F 1133 , F2233

m, m F11m , F22 F33

N5208

2.18 10-4

7.82 10-4

-8.77 10-5

0.0162

3501-6

2.15 10-4

5.04 10-4

-8.07 10-5

0.0143

Table 12. Local (matrix embedded in a composite ply) strength data deduced from the local quadratic stress failure criteria of a N5208 and 3501-6 epoxy matrices respectively. Strengths [MPa]

Xm, Ym, Zm

Xm´, Ym´, Zm´

Sm

N5208

40.1

114.7

25.3

3501-6

42.6

108.9

30.9

In order to achieve the identification of the coefficients of the quadratic microscopic failure criteria of the pure epoxies (3501-6 and N5208, respectively), the method previously explained in subsection 5.2.3 was applied. The macroscopic stiffnesses considered for the simulation are provided in Table 10, whereas the elastic constants of the elastically isotropic resins, required for localising the macroscopic stress/strain states at the microscopic scale in the matrices, according to equations (38-42), were previously given in tables 1 and 5. In order to find the microscopic strength coefficients, four

σI

σI

σI

σI

independent macroscopic stress states a , b , c , d located on the macroscopic failure envelope according to the conditions described on the first raw of Table 7. Table 11 shows the strength coefficients found for the quadratic microscopic failure criterion in stress space of both epoxies by solving equations (60-61). Besides, the microscopic ultimate uniaxial stresses of the two studied epoxies have been determined by introducing in equation (62) the results of the previous identification of the strength coefficients of their respective quadratic failure criterion in stress space (still Table 11). The corresponding results have been listed in Table 12. Finally, instances of the microscopic failure envelopes have been drawn and superimposed to the corresponding macroscopic failure envelopes. Pictures of Figure 5 compare the results obtained in stress space for each couple epoxy/composite.

40

Jacquemin Frédéric and Fréour Sylvain 50 50 1000

2000

&& &[MPa]

&22 [MPa]

-4000 -3000 -2000 -1000-50 0

-150

-4000 -3000 -2000 -1000-50 0

-250

-350

-350 AS4/3501

&11 [MPa]

N5208

2000

-150

-250

T300/5208

1000

σ 11 [MPa]

3501-6

200

200 100

-400

-300

-200

-100 0 -100

100

-200

σ 33 [MPa]

&& [MPa]

0 -500

0 -500

-400

-300

-200

-100

0

100

0

50

-200

-300 -400

-400

-500 T300/5208

σ 22 [MPa] 100

100

50

50

0 -250

-200

-150

-100

σ 22 [MPa]

3501-6

&& [MPa]

N5208

&& [MPa]

AS4/3501

-50

0

50

0 -250

-200

-150

-100

-50

-50

-100 T300/5208

-100 AS4/3501

&22 [MPa]

N5208

-50

3501-6

σ 22 [MPa]

Figure 5. Examples of macroscopic and local (matrix only) stress failure envelopes of T300/5208 and AS4/3501-6 plies.

5.3.2. Observations on Predicted Results and Discussion According to the identification procedure described in subsection 5.2.3, an infinite number of macroscopic stress states sets { σ a , σ b , σ c , σ d } can be considered for the determination I

I

I

I

of the researched microscopic failure envelope strength coefficients. Actually, σ c only may I

I

I/

vary whereas σ a , σ b and σ d are fixed by the macroscopic ultimate stresses Y , Y , S I

I

I

I

of the considered composite structure (see the first raw of Table 7). Several tests were

Multi-scale Analysis of Fiber-Reinforced Composite Parts…

41

performed, introducing various numerical stress states (compatible with the constitutive hypotheses of the present work) for σ c . The tests showed that the microscopic strength I

coefficients are, as expected, independent from the choice of the initial macroscopic stress state σ c : one set of coefficients only is found as the unique solution of system (60). This I

demonstrates that the inverse model presented here is reliable from a numerical point of view. The obtained results for the ultimate uniaxial stresses of 3501-6 and N5208 epoxies are close together (Table 12), whereas the macroscopic strength present significant discrepancies (Table 8). As an example, the relative deviation between the macroscopic longitudinal tensile ultimate stress of the two composites reaches around 25% when the relative deviation between the longitudinal tensile ultimate stress of the two epoxies is limited to 6%. Moreover, the representation of the microscopic failure envelopes are rather similar for the two considered resins, (Figure 5), whereas the macroscopic failure envelopes differ from one composite to the other (Figure 5, also). This could be interpreted as follows: for the considered composites, the observed deviation in the macroscopic failure envelopes comes from the choice of the reinforcing fibers and not from the choice of the resin. This is remarkable, since the considered epoxies exhibit a very different elastic mechanical behaviour (see Tables 1 and 5). Moreover, the predicted microscopic ultimate uniaxial stresses are coherent with experimental results measured on plain resins. For instance, reference (Fiedler et al., 2001) reports a strength value of 117 MPa in compression, and elastic limits reaching respectively 29 MPa in tension and 31 MPa in torsion for small specimen of plain unreinforced BisphenolA type resin (i.e. “small” denotes a significantly reduced sized in normal and transverse directions compared to “bulk” specimen). These measured strength are of the same order of magnitude than the strength, calculated in the present work, for 3501-6 and N5208 epoxies. At the opposite, the strengths determined on bulk specimens of 5208 and 3501-6 plain epoxies are approximately two times higher than the values obtained in the present work, for the strength of the corresponding epoxies embedded in thin composite plies. This last result is also compatible with both the experimental comparison achieved in reference (Fiedler at al., 2001) on various sized pure epoxies and the practical comparisons of the failure mechanisms exhibited by composites structures and their constitutive epoxy resin (see Garett and Bailey, 1977; Christensen and Rinde, 1979). The present work allows to represent the scale effects observed in practice on the composite constituents strengths, because the composite ply strengths involved in the calculations do actually depend on both the constituents properties and microstructure.

6. Conclusions The present work dealt with the question of scale transition modelling of polymer matrix composites and its application to several fields of investigation. Therefore, Mori-Tanaka and Eshelby-Kröner self-consistent models, taking advantage of arithmetic averages, were both considered for achieving the determinaiton of the homogenized properties of composite ply as a function of the properties of its constituents (on the one hand, the matrix , and on the second hand, the reinforcements).

42

Jacquemin Frédéric and Fréour Sylvain

The theoretical models properly take into account the specific microstructure of such materials. Especially the extreme morphology of the reinforcements can be considered, while the morphology and orientation of the reinforcing inclusions are kept constant in a single ply. As a consequence, the models manage to reproduce realistically the strong macroscopic anisotropy observed in practice on uni-directionally fiber-reinforced epoxies. The obtained results have shown that the two approaches, presented here, yield close together estimations of the macroscopic coefficients of thermal expansion, coefficients of moisture expansion and elastic moduli, in the range of the epoxy volume fraction, that is typical for designing m composites structures for engineering applications i.e. 0.3 ≤ v ≤ 0.7 . Nevertheless, an

(

)

I exception to this statement occurs for Coulomb modulus G12 , that is strongly underestimated in the case that the calculations are performed according to Mori-Tanaka approximation, in the same range of epoxy volume fractions. Moreover, realistic inverse scale transition procedures based on Kröner-Eshelby selfconsistent model and Mori-Tanaka estimates were also provided for achieving the numerical determination of the mechanical, hygroscopic or thermal properties of one constituent of an uni-directionally reinforced composite ply. Both models were used in order to estimate the elastic stiffness of reinforcing fibers embedded in a composite ply, from the knowledge of the macroscopic properties and those of the matrix. The obtained numerical results were successfully compared with expected practical results. A similar study was achieved in the standpoint of estimating the coefficients of moisture expansion of the matrix constituting a composite ply. In both cases the proposed theoretical approaches led to similar results, which is satisfying. Thus, the two inverse models described in the present work can be equally used in order to achieve such an identification. Another section of this article was devoted to the analysis of the macroscopic mechanical states localization within the constituents of a composite ply. Since it was previously demonstrated in the literature, that Mori-Tanaka approximation was not reliable for handling such a task, only Eshelby-Kröner model was considered. A numerical model, valid for any morphology of the reinforcing inclusions, was provided. Moreover a rigorous fully analytical treatment of the classical Kröner and Eshelby Self-Consistent model including morphology effects was achieved also. Especially, the determination of Morris’ tensor was performed in a satisfactory agreement with the transverse macroscopic elastic anisotropy expected for the fiber shape that should be taken into account in order to satisfactory represent the specific microstructure of carbon-fiber reinforced composites. The new closed-form solutions obtained for the components of Morris’ tensor were introduced in the classical hygro-thermoelastic scale transition relation in order to express analytically the internal strains and stresses in both the fiber and the resin of a ply submitted to a hygro-thermo-elastic load. The closedform solution demonstrated in the present work was compared to the fully numerical selfconsistent model for various geometrical arrangements of the fibers: uni-directional or laminated composites. A very good agreement was obtained between the two models for any component of the local stress tensors. It was also demonstrated that continuum mechanics and micro-mechanical models give complementary information about the occurrence of a possible damage during the loading of the structure. In a last part, the present study explained a procedure enabling to achieve the identification of one single set of strength parameters defining completely the microscopic

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43

failure envelope of the matrix entering in the composition of a composite structure, in the cases that a pure mechanical load is applied. The identification method was built around an inverse scale transition method which requires the knowledge of the macroscopic strengths, and both the macroscopic and microscopic elastic stiffnesses. Besides, it was necessary to consider some hypotheses in order to proceed to the identification of the coefficients of the microscopic quadratic failure criteria. In the present work, it was assumed that the macroscopic failure of a uni-directionally reinforced ply is dominated by the local failure of the matrix when the external load is applied in planes perpendicular to the fiber axis. Numerical applications of the proposed inverse method were made considering the cases of two high-strength composites structures: AS4/3501-6 and T300/N5208. The determination of the microscopic quadratic failure criterion of the pure epoxies (3501-6 and N5208, respectively) was achieved. The obtained results are close together and present a good agreement with ultimate strengths measured on reduced sized plain resins (available from already published literature). This demonstrates the reliability of the present predictive method for estimating the local failure behaviour of epoxies whose experimental failure criterion has not yet been determined. In further works, the proposed approach will be extended to the more general case of hygro-thermo-mechanical loads. This will imply to take into account the stress free strains in order to keep consistency between the failure envelopes expressed in stress and strain spaces. Besides, the rigorous treatment of the hygro-thermo-mechanical load requires to consider the dependence on the temperature and moisture content of a) the elastic stiffness, coefficients of thermal expansion and coefficients of moisture expansion and b) the ultimate strength (and in general, the coefficients of the considered failure criterion), at both macroscopic and microscopic scales. Others perspectives of research are proposed in the following section below.

7. Perspectives Scale transition modelling based theoretical analysis of composite structures constitutes an overexpanding field of research, due to multiple factors. Among them, the emergence of new materials exhibiting a specific, more advanced microstructure, the ambition to account for additional, sometimes only recently discovered, physical phenomena and the relentless research for building faster, more convenient but still reliable models stand for the three essential motivations for achieving further developments in the incoming years.

7.1. Emergence of New Materials The present development stage of Eshelby’s single inclusion theory involved in the mechanical modeling of composites is not intended for a rigorous treatment of the morphology presented by the reinforcements used for manufacturing woven-composites. As a consequence, answering to the question of a theoretical study, through scale transition models, of mechanical parts made of such composites will require a specific and still missing solution.

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Since the recent discovery of carbon nanotubes in the 90’s, researchers worldwide have engaged in fundamental studies of this novel material (Treacy et al., 1996). The pioneering works have underlined the characteristics of carbon nanotubes such as an extraordinarily high stiffness (Salvetat et al., 1999) coupled to a high tensile strength (Demczyk et al., 2002)., high aspect ratio and an especially low density. Actually, for instance, the experimental direct mechanical measurement of the elastic properties of carbon nanotubes provided Young’s moduli in the range of 1 TPa, which considerably exceeds the corresponding modulus of any currently available fiber material (Salvetat et al., 1999; Demczyk et al., 2002). In consequence, the technological applications of carbon nanotubes as reinforcements for elastomers (Frogley et al., 2003) or polymer-based composites (Liu and Wagner, 2005; Breton et al., 2004; Xiao et al., 2006) was very recently investigated. Furthermore, multimaterials made up of polymer matrix, carbon fibers and carbon nanotubes are considered also for achieving a new generation of engineering composites.

7.2. Accounting for Additional Physical Factors The present work is focused on the theoretical prediction of the mechanical behaviour of composite structures submitted also to environmental conditions. However, every aspect of the consequences of environmental loading on the constituents of composite materials have not always been considered in this paper, for the sake of simplicity. Nevertheless, accounting for some additional physical factors would improve the realism and the reliability of the predictions obtained through the scale-transition models. For instance, the moisture diffusion process was assumed, in the present work, to follow the linear, classical, established for a long time, Fickian model. Nevertheless, some valuable experimental results, already reported in (Gillat and Broutman, 1978), have shown that certain anomalies in the moisture sorption process, (i.e. discrepancies from the expected Fickian behaviour) could be explained from basic principles of irreversible thermodynamics, by a strong coupling between the moisture transport in polymers and the local stress state (Weitsman, 1990a, Weitsman, 1990b). The present work yields several perspectives of research concerning the application of scale transition model to the identification of composite materials properties. Moisture and temperature are not the only parameters leading to an evolution of the mechanical properties of epoxies. According to the literature, thermo-oxidation is reported to enhance the stiffness of the epoxies (Decelle et al., 2003 ; Ho and al., 2006). The inverse methods presented here could for instance be directly applied to the estimation of the epoxy stiffening from the knowledge of the macroscopic elastic properties evolution as a function of the mass loss during the thermo-oxidation process. Furthermore, extensions of the inverse models could be achieved in order to account for the variation of the coefficients of thermal and/or moisture expansion of the constituents of a composite ply, enabling to identify them and their evolutions as a function of the environmental conditions. Finally, a similar approach could be developed in order to identify the damage induced evolution of the mechanical behaviour of the constituents of composite plies from the inelastic part of macroscopic stress/strain curves. The experimental data required for achieving such analysis is already available in the literature (Soden et al., 1998). Nevertheless, local and macroscopic damage have still to be

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implemented in the theoretical laws. The above-listed perspectives of research will be successively considered in further works.

7.3. Improving the Calculation Time While Ensuring the Most Reliable Predictions The present work underlines the sometime existing opportunity to replace purely numerical mathematical solutions by analytical forms enabling to significantly reduce both the time required for designing the software and the time necessary for achieving one simulation. It was demonstrated in this paper that Eshelby-Kröner could be, at least partially, presented as an analytical model, while it was used for predicting mechanical states. Nevertheless, the estimation of the macroscopic properties (elastic stiffness, coefficients of thermal expansion and coefficients of moisture expansion) through the homogenization relations deduced from this very model do still involve an implicit iterative procedure. It was already shown in the literature by Welzel and his co-authors, that under specific conditions, it was possible to build a model, numerically equivalent to Eshelby-Kröner model, from the combination of two (separately less successful) other models (Welzel, 2002 ; Welzel et al. 2003). The concept is similar to the idea based on empirical comparisons, historically proposed by Neerfeld (Neerfeld, 1942) and Hill (Hill, 1952) to average Reuss and Voigt rough hypotheses in order to get a numerically acceptable theoretical solution. In the field of micro-mechanical modelling of composite materials, a combination of the two possible localization procedures considered for Mori-Tanaka model in the present work would enable to numerically reproduce the homogenized properties obtained from Eshelby-Kröner model. Building an effective model from the two main ways of writing Mori-Tanaka model would mainly enable to obtain closed-form solutions for the elastic stiffness tensor, instead of having to numerically solve the iterative procedure involved in Eshelby-Kröner self-consistent model. Thus, a coupling of this numerically effective solution for predicting realistic hygrothermomechanical macroscopic properties to the already proposed in this very article analytical forms for the local mechanical states would yield to a faster but still extremely reliable innovative scale-transition approach for studying composite materials. The analytical forms required for achieving the effective Mori-Tanaka model should be derived and published in the near future.

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Benveniste, Y. (1987). A New Approach to the Application of Mori-Tanaka’s Theory in Composite Materials, Mechanics of Materials, 6: 147-157. Bowles, D.E., Post, D., Herakovich, C.T., and Tenney, D.R. (1981). Moiré Interferometry for Thermal Expansion of Composites, Exp. Mech., 21: 441-447. Breton, Y., Désarmot, G., Salvetat, J.P., Delpeux, S., Sinturel, C., Béguin, F. et Bonnamy, S. (2004). Mechanical properties of multiwall carbon nanotubes/epoxy composites : influence of network morphology, Carbon, 42: 1027-1030. Christensen, R. M. and Rinde, J. A. (1979). Transverse tensile characteristics of fiber composites with flexible resins : theory and test results, Pol. Eng. Sci., 19: 506-511. Crank, J. (1975). The mathematics of diffusion, Clarendon Press, Oxford. Cuntze, R. G. (2003). The predictive capability of failure mode concept-based strength criteria for multi-directional laminates – part B, Composite Science and Technology, 64: 487-516. Daniel, I. M., and Ishai, O. (1994). In: “Engineering Mechanics of Composite Materials”, Oxford University Press. Decelle, J., Huet, N. and Bellenger, V. (2003). Oxidation induced shrinkage for thermally aged epoxy networks, Polymer Degradation and Stability, 81: 239-248. Demczyk, B.G., Wang, Y.M., Cumings, J., Hetman, M., Han, W., Zettl, A., and Ritchie, R. O. (2002). « Direct mechanical measurement of the tensile strength and elastic modulus of multiwalled carbon nanotubes », Mat. Sci. Eng., A334: 173-178. DiCarlo, J.A. (1986). Creep of chemically vapor deposited SiC fiber, J. Mater. Sci., 21: 217-224. Dyer, S.R.A., Lord, D., Hutchinson, I.J., Ward, I.M. and Duckett, R.A. (1992). Elastic anisotropy in unidirectional fibre reinforced composites, J. Phys. D: Appl. Phys., 25: 66-73. Eshelby, J. D. (1957). The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems, In: Proceedings of the Royal Society London, A241: 376–396. Ferreira, C., Casari, P., Bouzidi, R. and Jacquemin, F. (2006a). Identification of Young Modulus Profile in PVC Foam Core thickness using speckle interferometry and Inverse Method, Proceedings of SPIE - The International Society for Optical Engineering. Ferreira, C., Jacquemin, F. and Casari, P. (2006b). Measurement of the nonuniform thermal expansion coefficient of a PVC foam core by speckle interferometry - Influence on the mechanical behavior of sandwich structures, Journal of Cellular Plastics, 42 (5): 393-404. Fiedler, B., Hojo, M., Ochiai, S., Schulte, K. and Ando, M. (2001), Failure behaviour of an epoxy matrix under different kinds of static loading, Composite Science and Technology, 61: 1615-1624. Fleck, N. A. and Jelf, P. M. (1995). Deformation and failure of a carbon fibre composite under combined shear and transverse loading, Acta metall. mater., 43: 3001-3007. François, M. (1991). Détermination de contraintes résiduelles sur des fils d’acier eutectoïde de faible diamètre par diffraction des rayons X, Doctoral Thesis, ENSAM, Paris. Fréour, S., Gloaguen, D., François, M., Guillén, R., Girard, E. and Bouillo, J. (2002) Determination of the macroscopic elastic constants of a phase embedded in a multiphase polycrystal – application to the beta-phase of Ti17 titanium based alloy, Materials Science Forum, 404-407: 723-728.

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Fréour, S., Gloaguen, D., François, M. and Guillén, R. (2003a). Study of the Coefficients of Thermal Expansion of Phases Embedded in Multiphase Materials, Material Science Forum, 426–432: 2083–2088. Fréour, S., Gloaguen, D., François, M. and Guillén, R. (2003b). Thermal properties of polycrystals - X-ray diffraction and scale transition modelling, Physica Status Solidi a, 201: 59-71. Fréour, S., Jacquemin, F. and Guillén, R. (2005a). On an analytical Self-Consistent model for internal stress prediction in fiber-reinforced composites submitted to hygro-elastic load, Journal of Reinforced Plastics and Composites, 24: 1365-1377. Fréour, S., Gloaguen, D., François, M., Perronnet, A. and Guillén, R. (2005b). Estimation of Ti-17 β−phase Single-Crystal Elasticity Constants using X-Ray Diffraction measurements and inverse scale transition modelling, Journal of Applied Crystallography, 38: 30-37. Fréour, S., Jacquemin, F. and Guillén, R. (2006a). Extension of Mori-Tanaka Approach to Hygroelastic Loading of Fiber-Reinforced Composites – Comparison with EshelbyKröner Self-consistent Model, Journal of Reinforced Plastics and Composites, 25: 10391052. Fréour, S., Gloaguen, D., François, M. and Guillén, R. (2006b). Application of inverse models and XRD analysis to the determination of Ti-17 β−phase Coefficients of Thermal Expansion, Scripta Materialia, 54: 1475-1478. Fréour, S., Jacquemin, F. and Guillén, R. (to be published). On the use of the geometric mean approximation in estimating the effective hygro-elastic behaviour of fiber-reinforced composites, Journal of Materials Science. Frogley, M.D., D. Ravich, D., and Wagner, H.D. (2003). “Mechanical properties of carbon nanoparticle-reinforced elastomers”, Comp. Sci. Tech., 63: 1647-1654. Garett, K. W. and Bailey, J. E. (1977). The effect of resin failure strain on the tensile properties of glass fiber-reinforced polyester cross-ply laminates, J. Mater. Sci., 12: 2189-2194. Gillat, O. and Broutman, L.J. (1978). “Effect of External Stress on Moisture Diffusion and Degradation in a Graphite Reinforced Epoxy Laminate”, ASTM STP, 658: 61-83. Gloaguen, D., François, M., Guillén, R. and Royer, J. (2002). Evolution of Internal Stresses in Rolled Zr702, Acta Materialia, 50: 871–880. Han, J., Bertram, A., Olschewski, J., Hermann, W. and Sockel, H.G. (1995). Identification of elastic constants of alloys with sheet and fibre textures based on resonance measurements and finite element analysis. Materials Science and Engineering, A191: 105-111. Herakovitch, C. T. (1998). Mechanics of Fibrous Composites, John Wiley and Sons Inc., New York. Hill, R. (1952). The elastic behaviour of a crystalline aggregate. Proc. Phys. Soc., 65: 349-354. Hill, R., (1965). Continuum micro-mechanics of elastoplastic polycrystals, J. Mech. Phys. Solids, 13: 89-101. Hill, R. (1967). The essential structure of constitutive laws for metals composites and polycrystals, Journal of the Mechanics and Physics of Solids, 15: 79-95.

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Ho, N.Q., Pons, F. and Lafarie-Frenot, M.C. (2006). Characterization of an Oxidized Layer in Epoxy Resin and in Carbon Epoxy Composite for Aeronautic Applications, Proceedings of ECCM 12. Jacquemin, F. and Vautrin, A. (2002). A Closed-form Solution for the Internal Stresses in Thick Composite Cylinders Induced by Cyclical Environmental Conditions, Composite Structures, 58: 1–9. Jacquemin, F., Fréour, S., and Guillén, R. (2005). A self-consistent approach for transient hygroscopic stresses and moisture expansion coefficients of fiber-reinforced composites, Journal of Reinforced Plastics and Composites, 24: 485-502. Kaddour, A. S., Hinton, M. J. and Soden, P. D. (2003). Behaviour of ± 45° glass/epoxy filament wound composite tubes under quasi-static equal biaxial tension-compression loading: experimental results, Composites : part B, 34: 689-704. Karakazu, R., Atas, C. and Akbulut, H. (2001). Elastic-plastic Behaviour of Woven-steelfiber-reinforced Thermoplastic Laminated Plates under In-plane Loading, Composites Science and Technology, 61: 1475-1483. Khashaba, U. A., (2004). In-plane shear properties of cross-ply composite laminates with different off-axis angles, Composite Structures, 65: 167-177. Kocks, U. F., Tomé , C. N. and Wenk, H. R. (1998). Texture and Anisotropy, Cambridge University Press. Kröner E. (1953). Dissertation, Technischen Hochschule Stuttgart. Kröner, E. (1958). “Berechnung der elastischen Konstanten des Vielkristalls aus des Konstanten des Einkristalls”, Zeitschrift für Physik, 151: 504–518. Liu, K-S and Tsai, S. W. (1998). A progressive quadratic failure criterion for a laminate, Composite Science and Technology, 58: 1023-1032. Liu, L. et Wagner, H.D. (2005). Rubery and gassy epoxy resins reinforced with carbon nanotubes, Comp. Sci. Tech., 65: 1865-1868. Loos, A. C. and Springer, G. S. (1981). Environmental Effects on Composite Materials, Moisture Absorption of Graphite – Epoxy Composition Immersed in Liquids and in Humid Air, pp. 34–55, Technomic Publishing. Mabelly, P. (1996). Contribution à l’étude des pics de diffraction – Approche expérimentale et modélisation micromécanique, Doctoral Thesis, ENSAM, Aix en Provence. Matthies, S., and Humbert, M. (1993). The realization of the concept of a geometric mean for calculating physical constants of polycrystalline materials, Phys. Stat. Sol. b, 177: K47K50. Matthies, S., Humbert, M., and Schuman, Ch. (1994). On the use of the geometric mean approximation in residual stress analysis”, Phys. Stat. Sol. b, 186 : K41-K44. Matthies, S. Merkel, S., Wenk, H.R., Hemley, R.J. and Mao, H. (2001). Effects of texture on the determination of elasticity of polycrystalline ε-iron from diffraction measurements, Earth and Planetary Science Letters, 194: 201-212. Mensitieri, G. M., Del Nobile, M. A., Apicella, A. and Nicolais, L. (1995). Moisture-matrix interactions in polymer based composite materials, Revue de l’Institut Français du Pétrole, 50: 551-571. Morawiec, A. (1989). Calculation of polycrystal elastic constants from single-crystal data, Phys. Stat. Sol. b, 154 : 535-541.

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Mori, T. and Tanaka, K. (1973). Average Stress in Matrix and Average Elastic Energy of Materials with Misfitting Inclusions, Acta Metallurgica, 21: 571-574. Morris, R. (1970). Elastic constants of polycrystals, Int. J. Eng. Sci., 8: 49. Mura, T. (1982). Micromechanics of Defects in Solids, Martinus Nijhoff Publishers, The Hague, Netherlands. Neerfeld, H. (1942). Zur Spannungsberechnung aus röntgenographischen Dehnungsmessungen, Mitt. Kaiser-Wilhelm-Inst. Eisenforschung Düsseldorf 24: 61-70. Qiu, Y. P. and Weng, G. J. (1991). The influence of inclusion shape on the overall elastoplastic behavior of a two-phase isotropic composite, Int. J. Solids Structures, 27 (12): 1537-1550. Reuss, A. (1929). Berechnung der Fliessgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle, Zeitschrift für Angewandte und Mathematik und Mechanik, 9: 49–58. Sai Ram K.S., Sinha, P.K. (1991). Hygrothermal effects on the bending charactericstics of laminated composite plates, Computational Structure, 40 (4): 1009-1015. Salvetat, J.-P., Andrews, G., Briggs, D., Bonard, J.-M., Bacsa, R.R., Kulik, A.J., Stöckli, T., Burnham, N.A., and Forro, L. (1999). “Elastic and Shear Moduli of Single-Walled Carbon Nanotube Ropes”, Phys. Rev. Lett., 82: 944-947. Sims, G.D., Dean, G.D., Read, B.E. and Western B.C. (1977). Assessment of Damage in GRP Laminates by Stress Wave Emission and Dynamic Mechanical Measurements, Journal of Materials Science, 12 (11): 2329-2342. Soden, P. D., Hinton M. J. and A. S. Kaddour, A. S. (1998). Lamina properties lay-up configurations and loading conditions for a range of fiber-reinforced composite laminates, Composites Science and Technology, 58: 1011-1022. Sprauel, J. M., and Castex, L. (1991). First European Powder Diffraction International Conference on X-Ray stress analysis, Munich. Tanaka, K. and Mori, T. (1970). The Hardening of Crystals by Non-deforming Particules and Fibers, Acta Metallurgica, 18: 931-941. Treacy, M.M.J., Ebbesen, T.W. and Gibson, T.M. (1996). “Exceptionally high Young’s modulus observed for individual carbon nanotubes”, Nature, 381: 678-680. Tsai, C.L. and Daniel I.M. (1993). Measurement of longitudinal shear modulus of single fibers by means of a torsional pendulum. 38th International SAMPE Symposium 1993:1861-1868. Tsai C.L. and Daniel I.M. (1994). Method for thermo-mechanical characterization of single fibers, Composites Science and Technology, 50: 7-12. Tsai, C.L. and Chiang, C.H. (2000). Characterization of the hygric behavior of single fibers. Composites Science and Technology, 60: 2725-2729 Tsai, S. W. and Hahn, H. T. (1980). Introduction to composite materials, Technomic Publishing Co., Inc., Lancaster, Pennsylvania. Tsai, S. W. (1987). Composite Design, 3rd edn, Think Composites. Turner, P. A. and Tome, C. N. (1994). “A Study of Residual Stresses in Zircaloy-2 with Rod Texture”, Acta Metallurgica and Materialia, 42: 4143–4153. Voigt, W. (1928). Lehrbuch der Kristallphysik, Teubner, Leipzig/Berlin. Weitsman, Y. (1990a). “A Continuum Diffusion Model for Viscoelastic Materials”, Journal of Physical Chemistry, 94: 961-968.

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Weitsman, Y. (1990b). “Moisture in Composites: Sorption and Damage”, in: Fatigue of Composite Materials. Elsevier Science Publisher, K.L. Reifsnider (editor), 385-429. Welzel, U. (2002). “Diffraction Analysis of Residual Stress; Modelling Elastic Grain Interaction.”, PhD thesis, University of Stuttgart, Germany. Welzel, U., Leoni, M. and Mittemeijer, E. J. (2003). « The determination of stresses in thin films; modelling elastic grain interaction », Philosophical Magazine, 83: 603-630. Welzel, U., Fréour, S. and Mittemeijer, E. J. (2005). « Direction-dependent elastic graininteraction models – a comparative study », Philosophical Magazine, 85: 2391-2414. Xiao, K.Q., Zhang, L.C. et Zarudi, I. (2006). « Mechanical and rheological properties of carbon nanotube-reinforced polyethylene composites », Comp. Sci. Tech., 67: 177-182.

In: Composite Materials Research Progress Editor: Lucas P. Durand, pp. 51-107

ISBN: 1-60021-994-2 © 2008 Nova Science Publishers, Inc.

Chapter 2

OPTIMIZATION OF LAMINATED COMPOSITE STRUCTURES: PROBLEMS, SOLUTION PROCEDURES AND APPLICATIONS Michaël Bruyneel SAMTECH s.a., Liège Science Park Rue des Chasseurs-ardennais 8, 4031 Angleur, Belgium

Abstract In this chapter the optimal design of laminated composite structures is considered. A review of the literature is proposed. It aims at giving a general overview of the problems that a designer must face when he works with laminated composite structures and the specific solutions that have been derived. Based on it and on the industrial needs an optimization method specially devoted to composite structures is developed and presented. The related solution procedure is general and reliable. It is based on fibers orientations and ply thicknesses as design variables. It is used daily in an (European) industrial context for the design of composite aircraft box structures located in the wings, the center wing box, and the vertical and horizontal tail plane. This approach is based on sequential convex programming and consists in replacing the original optimization problem by a sequence of approximated subproblems. A very general and self adaptive approximation scheme is used. It can consider the particular structure of the mechanical responses of composites, which can be of a different nature when both fiber orientations and plies thickness are design variables. Several numerical applications illustrate the efficiency of the proposed approach.

1. Introduction According to their high stiffness and strength-to-weight ratios, composite materials are well suited for high-tech aeronautics applications. A large amount of parameters is needed to qualify a composite construction, e.g. the stacking sequence, the plies thickness and the fibers orientations. It results that the use of optimization techniques is necessary, especially to tailor the material to specific structural needs. The chapter will cover this subject and is divided in three main parts.

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After recalling the goal of optimization, the different laminates parameterizations will be presented with their limitations (the pros and the cons) in the frame of the optimal design of composite structures. The issues linked to the modeling of structures made of such materials and the problems solved in the literature will be reviewed. The key role of fibers orientations in the resulting laminate properties will be discussed. Finally the outlines of a pragmatic solution procedure for industrial applications will be drawn. Throughout this section, a profuse and state-of-the-art review of the literature will be provided. Secondly, a general solution procedure used daily in industrial problems including fibers reinforced composite materials will be described. The related optimization algorithm is based on sequential convex programming and has proven to be very reliable. This algorithm is presented in detail and validated by comparing its performances to other optimization methods of the literature. Finally, it will be shown how this optimization algorithm can efficiently solve several kinds of composite structure design problems: amongst others, solutions for topology optimization with orthotropic materials will be presented, important considerations about the optimal design of composites including buckling criteria will be discussed, optimization with respect to damage tolerance will be considered (crack delamination in a laminated structure). On top of that, some key points of the solution procedure based on this optimization algorithm applied to the pre-sizing of (European) industrial composite aircraft box structures will be presented.

2. The Optimal Design Problem and Available Optimization Methods The goal of optimization is to reach the best solution of a problem under some restrictions. Its mathematical formulation is given in (2.1), where g0(x) is the objective function to be minimized, gj(x) are the constraints to be satisfied at the solution, and x={xi, i=1,…,n} is the set of design variables. The value of those design variables change during the optimization process but are limited by an upper and a lower bound when they are continuous, which will be the case in the sequel.

min g 0 ( x ) g j ( x ) ≤ g max j

j = 1,..., m

x i ≤ xi ≤ x i

i = 1,..., n

(2.1)

The problem (2.1) is illustrated in Figure 2.1, where 2 design variables x1 and x2 are considered. The isovalues of the objective function are drawn, as well as the limiting values of the constraints. The solution is found via an iterative process. xk is the vector of design variables at the current iteration k, and xk+1 is the estimation of the solution at the iteration k+1. Typically a local solution xlocal will be reached when a gradient based optimization method is used. The best solution xglobal can only be found when all the design space is looked over: this last can be accessed with specific optimization methods that include a non deterministic procedure, as the genetic algorithms.

Optimization of Laminated Composite Structures… x2

53

g j (X)

S (k )

X

X(k )

( k +1)

α

a b

X*local

X*global

x1

Initial design Structural analysis

Optimization

New design

no

Optimal design ? yes

End

Figure 2.1. Illustration of an optimization problem and its solution.

In structural optimization, the design functions can be global as the weight, the stiffness, the vibration frequencies, the buckling loads, or local as strength constraints, strains and failure criteria. When the design variables are linked to the transverse properties of the structural members (e.g. the cross-section area of a bar in a truss), the related optimization problem is called optimal sizing (Figure 2.2a). The value of some geometric items (e.g. a radius of an ellipse) can also be variable: in this case, we are talking about shape optimization (Figure 2.2b). Topology optimization aims at spreading a given amount of material in the structure for a maximum stiffness. Here, holes can be automatically created during the optimization process (Figure 2.2c). Finally, the optimization of the material can be addressed, e.g. the local design of laminated composite structure with respect to fibers orientations, ply thickness and stacking sequence (Figure 2.2d).

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Initial designs

Final designs

a) Optimal sizing

b) Shape optimization

c) Topology optimization

d) Material optimization

Figure 2.2. The structural optimization problems.

The structural optimization problems are non linear and non convex, and several local minimum exist. It is usually accepted that a local solution xlocal gives satisfaction. The global solution xglobal can only be determined with very large computational resources. In some cases when the problem includes a very large amount of constraints, a feasible solution is acceptable. A lot of methods exist to solve the problem (2.1). Morris (1982), Vanderplaats (1984), and Haftka and Gurdal (1992) present techniques based on the mathematical programming approach used in structural optimization. Most of them are compared by Barthelemy and Haftka (1993), and Schittkowski et al. (1994). Non deterministic methods, such as the genetic algorithm (Goldberg, 1989), are studied by Potgieter and Stander (1998), and Arora et al. (1995). Those authors also present a review of the methods used in global optimization. Optimality criteria for the specific solution of fibers optimal orientations in membrane (Pedersen 1989) and in plates (Krog 1996) must be mentioned as well. Finally the response surfaces methods are also used for optimizing laminated structures (Harrison et al. 1995, Liu et al. 2000, Rikards et al. 2006, Lanzi and Giavotto 2006). The approximation concepts approach, also called Sequential Convex Programming, developed in the seventies by Fleury (1973), Schmit and Farschi (1974), and Schmit and Fleury (1980) has allowed to efficiently solve several structural optimization problems: the optimal sizing of trusses, shape optimization (Braibant and Fleury, 1985), topology optimization (Duysinx, 1996, 1997, and Duysinx and Bendsøe, 1998), composite structures optimization (Bruyneel and Fleury 2002, Bruyneel 2006), as well as multidisciplinary optimization problems (Zhang et al., 1995 and Sigmund, 2001). In sizing and shape optimization the solution is usually reached within 10 iterations. For topology optimization, since a very large number of design variables are included in the problem, a larger number of design cycles is needed for converging with respect to stabilized design variables values over 2 iterations. Those approximation methods consist in replacing the solution of the initial optimization problem (2.1) by the solution of a sequence of approximated optimization problems, as illustrated in Figure 2.3.

Optimization of Laminated Composite Structures… x2

55

g j ( X)

X(k )

g~ (jk ) ( X)

X(k )* X*local

X*global

x1

Initial design

Approximated optimization problem

Solution of the approximated problem

no

Optimal design ? yes End

Figure 2.3. Definition of an approximated optimization problem based on the information at the current design point x(k). The corresponding feasible domain is defined by the constraints of (2.2).

Each function entering the problem (2.1) is replaced by a convex approximation (k ) ~ g j ( X) based on a Taylor series expansion in terms of the direct design variables xi or intermediate ones as for example the inverse design variables 1 xi . For a current design x(k) at iteration k, the approximated optimization problem writes:

min g~0( k ) (x) g~ (jk ) ( x ) ≤ g max j (k )

x i( k ) ≤ xi ≤ x i

j = 1,..., m

(2.2)

i = 1,..., n

where the symbol ~ is related to an approximated function. The explicit and convex optimization problem (2.2) is itself solved by dedicated methods of mathematical

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Michaël Bruyneel

programming (see Section 7). Building an approximated problem requires to carry out a structural and a sensitivity analyses (via the finite elements method). Solving the related explicit problem does no longer necessitate a finite element analysis (expensive in CPU for large scale problems). The solution obtained with this approach doesn’t correspond to the global optimum, but to a local one, since gradients and deterministic information are used. Nevertheless this local solution is found very quickly and several initial designs could be used to try to find a better solution, as proposed by Cheng (1986). Finally it must be noted that when a very large number of constraints is considered in the optimal design problem (say more than 105) the user is often satisfied with a feasible solution.

3. Parameterizations of Laminated Composite Structures Before presenting the several possible parameterizations of laminates, with their advantages and their disadvantages, the classical lamination theory is briefly recalled in order to introduce the notation that will be used throughout the chapter. See Tsai and Hahn (1980), Gay (1991) and Berthelot (1992) for details.

3.1. The Classical Lamination Theory 3.1.1. Constitutive Relations for a Ply Fibers reinforced composite materials are orthotropic along the fibers direction, that is in the local material axes (x,y,z) illustrated in Figure 3.1. Homogeneous macroscopic properties are assumed at the ply and at the laminate levels. Material axes (orthotropy)

x

y

2 z,3

θ 1

Figure 3.1. The unidirectional ply with its material and structural axes.

For a linear elastic behaviour, the stress-strain relations in the material axes are given by the Hook’s law σ = Qε where ε and σ are the strain and stress tensors, respectively, while Q is the matrix collecting the stiffness coefficients in the orthotropic axes. For a plane stress assumption, it comes that

Optimization of Laminated Composite Structures… ⎧ σ x ⎫ ⎡ mE x ⎪ ⎪ ⎢ ⎨ σ y ⎬ = ⎢mν xy E y ⎪σ ⎪ ⎢ 0 ⎩ xy ⎭ ⎣

mν yx E x mE y 0

0 ⎤ ⎧ ε x ⎫ ⎡Q xx ⎥⎪ ⎪ 0 ⎥ ⎨ ε y ⎬ = ⎢⎢Q yx G xy ⎥⎦ ⎪⎩γ xy ⎪⎭ ⎣⎢ 0

Q xy Q yy 0

0 ⎤⎧ ε x ⎫ ⎪ ⎪ 1 0 ⎥⎥ ⎨ ε y ⎬ , m = 1 − ν xyν yx Qss ⎦⎥ ⎪⎩γ xy ⎪⎭

57

(3.1)

The stresses and strains can be written in the structural coordinates (1,2,3) as in (3.2) and (3.3) where θ is the angle between the local and structural axes, defined in Figure 3.1. 2 ⎧ ε 1 ⎫ ⎡ cos θ ⎢ ⎪ ⎪ 2 ⎨ε 2 ⎬ = ⎢ sin θ ⎪ε ⎪ ⎢2 cos θ sin θ ⎩ 6 ⎭ ⎣⎢ 2 ⎧σ 1 ⎫ ⎡ cos θ ⎢ ⎪ ⎪ 2 ⎨σ 2 ⎬ = ⎢ sin θ ⎪σ ⎪ ⎢cos θ sin θ ⎩ 6 ⎭ ⎣⎢

sin 2 θ cos θ − 2 cos θ sin θ 2

sin 2 θ cos θ − cos θ sin θ 2

− cos θ sin θ ⎤ ⎧ ε x ⎫ ⎥⎪ ⎪ cos θ sin θ ⎥ ⎨ ε y ⎬ cos 2 θ − sin 2 θ ⎥⎥ ⎪⎩γ xy ⎪⎭ ⎦ − 2 cos θ sin θ ⎤ ⎧ σ x ⎫ ⎥⎪ ⎪ 2 cos θ sin θ ⎥ ⎨ σ y ⎬ cos 2 θ − sin 2 θ ⎥⎥ ⎪⎩σ xy ⎪⎭ ⎦

(3.2)

(3.3)

For a ply with an orientation θ with respect to the structural axes, the constitutive relations write:

⎧σ 1 ⎫ ⎡Q11 ⎪ ⎪ ⎢ ⎨σ 2 ⎬ = ⎢Q12 ⎪σ ⎪ ⎢Q ⎩ 6 ⎭ ⎣ 16

Q16 ⎤ ⎧ ε 1 ⎫ ⎪ ⎪ Q26 ⎥⎥ ⎨ε 2 ⎬ Q66 ⎥⎦ ⎪⎩ε 6 ⎪⎭

Q12 Q22 Q26

(3.4)

where the matrix of the stiffness coefficients in the structural axes takes the form:

⎡ c4 ⎧ Q11 ⎫ ⎢ 4 ⎪Q ⎪ ⎢ s ⎪ 22 ⎪ ⎢c 2 s 2 ⎪⎪Q12 ⎪⎪ =⎢ 2 2 Q (1, 2,3) = ⎨ ⎬ ⎢c s ⎪Q66 ⎪ ⎢ 3 ⎪Q16 ⎪ ⎢c s ⎪ ⎪ ⎪⎩Q26 ⎪⎭ (1, 2,3) ⎢⎣ cs 3

⎤ ⎥ c4 2c 2 s 2 4c 2 s 2 ⎥ ⎧Q xx ⎫ ⎪ ⎪ c 2s 2 c4 + s4 − 4c 2 s 2 ⎥ ⎪Q yy ⎪ ⎥⎨ ⎬ c2s 2 − 2c 2 s 2 (c 2 − s 2 ) 2 ⎥ ⎪Q xy ⎪ ⎥ − cs 3 cs 3 − c 3 s 2(cs 3 − c 3 s )⎥ ⎪⎩ Q ss ⎪⎭ ( x, y , z ) − c 3 s (c 3 s − cs 3 ) 2(c 3 s − cs 3 )⎥⎦ s4

2c 2 s 2

4c 2 s 2

(3.5)

with

c = cosθ

s = sin θ

The variation of the Q’s with respect to the angle θ is plotted in Figure 3.2. It is observed that the stiffness coefficients are highly non linear in terms of the fibers orientation.

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Figure 3.2. Stiffness coefficients in N/mm² in the structural axes for several values of the fibers orientation in a carbon/epoxy material T300/5208 (after Tsai and Hahn, 1980).

Based on the fact that the trigonometric functions entering the matrix in (3.5) can be written in the following way:

1 cos 4 θ = (3 + 4 cos 2θ + cos 4θ ) 8 1 3 cos θ sin θ = (2 sin 2θ + sin 4θ ) 8 1 2 2 cos θ sin θ = (1 − cos 4θ ) 8

1 cos θ sin 3 θ = (2 sin 2θ − sin 4θ ) 8 1 4 sin θ = (3 − 4 cos 2θ + cos 4θ ) 8

(3.6)

Tsai and Pagano (1968) derived an alternative expression for the Q’s coefficients in the structural axes given in (3.7):

⎡ Q11 Q12 Q16 ⎤ Q22 Q26 ⎥ = γ 0 + γ1 cos 2θ + γ 2 cos 4θ + γ 3 sin 2θ + γ 4 sin 4θ Q(1,2,3) = ⎢ ⎢ ⎥ ⎢⎣ sym Q66 ⎥⎦ where the parameters γ are functions of the lamina invariants U1-U5:

(3.7)

Optimization of Laminated Composite Structures…

⎡ U1 U 4 0 ⎤ U1 0 ⎥ γ0 = ⎢ ⎢ ⎥ ⎢⎣ sym U 5 ⎥⎦

⎡ U3 − U3 U3 γ2 = ⎢ ⎢ ⎢⎣ sym

0 ⎤ 0 ⎥ ⎥ − U 3 ⎥⎦

0 ⎡ U2 ⎢ γ1 = − U2 ⎢ ⎢⎣ sym U2 ⎤ ⎡ ⎢ 0 0 2 ⎥ ⎢ U2 ⎥ 0 γ3 = ⎢ ⎥ 2 ⎥ ⎢ 0 ⎥ ⎢ sym ⎢⎣ ⎥⎦

59

0⎤ 0⎥ ⎥ 0⎥⎦

(3.8)

⎡ 0 0 U3 ⎤ γ4 = ⎢ 0 − U3 ⎥ ⎥ ⎢ ⎢⎣ sym 0 ⎥⎦

and

1 U1 = (3Q xx + 3Q yy + 2Q xy + 4Qss ) 8 1 U 2 = (Q xx − Q yy ) 2 1 U 3 = (Q xx + Q yy − 2Q xy − 4Qss ) 8

1 U 4 = (Q xx + Q yy + 6Q xy − 4Qss ) 8 1 U 5 = (Q xx + Q yy − 2Q xy + 4Qss ) 8

3.1.2. Constitutive Relations for a Laminate Composite structures are thin membranes, plates or shells made of n unidirectional orthotropic plies stacked on the top of each other. Such structures can support in and out-of plane loadings. In the following the constitutive relations for a laminate made of several individual plies are derived. The notations are defined in Figure 3.3. In the case of plane stress, i.e. the effects of transverse shear is neglected, in-plane normal and shear loads N, as well as the flexural and torsional moments M are applied to the laminate. Those loadings are computed by considering the stress state in each ply with the relations (3.9):

⎧ N1 ⎫ h / 2 ⎧σ 1 ⎫ ⎪ ⎪ ⎪ ⎪ N = ⎨ N 2 ⎬ = ∫ ⎨σ 2 ⎬dz ⎪ N ⎪ − h / 2 ⎪σ ⎪ ⎩ 6⎭ ⎩ 6⎭

⎧ M 1 ⎫ h / 2 ⎧σ 1 ⎫ ⎪ ⎪ ⎪ ⎪ M = ⎨ M 2 ⎬ = ∫ ⎨σ 2 ⎬zdz ⎪ M ⎪ − h / 2 ⎪σ ⎪ ⎩ 6⎭ ⎩ 6⎭

(3.9)

For a first order cinematic theory, where the displacement through the laminate’s thickness is linear in the z coordinate measured with respect to the mid-plane of the plate/shell (Figure 3.3), the vector of laminate’s strains εl is linked to the in-plane strains and the curvatures via the relation ε l = ε 0 + zκ . With this definition it turns that the constitutive relations for a laminate are given by (3.10) where A, B and D are the in-plane, coupling and bending stiffness matrices of the laminate.

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⎧ N1 ⎫ ⎡ A11 ⎪ N ⎪ ⎢A ⎪ 2 ⎪ ⎢ 12 0 ⎫ ⎧ ⎪ N 6 ⎪ ⎢ A16 ⎧ N ⎫ ⎡ A B ⎤ ⎪ε ⎪ ⎬=⎢ ⎨ ⎬=⎢ ⎨ ⎬⇔ ⎨ ⎥ ⎩M ⎭ ⎣ B D⎦ ⎪⎩ κ ⎪⎭ ⎪ M 1 ⎪ ⎢ B11 ⎪ M 2 ⎪ ⎢ B12 ⎪ ⎪ ⎢ ⎩ M 6 ⎭ ⎣ B16

A12 A22

A16 A26

B11 B12

B12 B22

A26

A66

B16

B26

B12 B22

B16 B26

D11 D12

D12 D22

B26

B66

D16

D26

B16 ⎤ ⎧ε10 ⎫ ⎪ ⎪ B26 ⎥ ⎪ε 20 ⎪ ⎥ B66 ⎥ ⎪⎪ε 60 ⎪⎪ ⎥⎨ ⎬ D16 ⎥ ⎪κ1 ⎪ D26 ⎥ ⎪κ 2 ⎪ ⎥⎪ ⎪ D66 ⎦ ⎪⎩κ 6 ⎪⎭

(3.10)

(a) A laminate with its structural axes. h is the total thickness

(b) Several unidirectional plies stacked on top of each other. Material axes related to the kth ply . 3

n tk k zk

hk

hk-1

1

h2 2 1

h1 h0

(c). Definition of the plies location through the laminate’s thickness. hk and hk-1 are used to locate the kth ply of the stacking sequence Figure 3.3. A laminate with n layers (a) Structural axes (b) Material axes of ply k (c) Position of each ply in the stacking sequence.

Optimization of Laminated Composite Structures…

61

3.2. The Possible Parameterizations of Laminates There exist several parameterizations for the laminates depending on the way the coefficients of the stiffness matrices in (3.10) are computed and depending on the definition of the design variables. The advantages and disadvantages of those different parameterizations are compared in the perspective of the optimal design of the laminated composite structures.

3.2.1. Parameterization with Respect to Thickness and Orientation When the ply thickness and the related fibers orientation are chosen to describe the laminate, the coefficients of the stiffness matrices can be written as follows: n

n

k =1 n

k =1 n

Aij = ∑ [Qij (θ k )](hk − hk −1 ) ⇔ Aij = ∑ [Qij (θ k )]t k Bij = Dij =

1 2 2 ∑ [Qij (θ k )](hk − hk −1 ) ⇔ Bij = ∑ [Qij (θ k )]t k z k 2 k =1 k =1

(3.11)

3 n 1 n 2 tk 3 3 [ Q ( θ )]( h − h ) ⇔ = [ ( θ )]( + ) , i, j = 1,2,6 D Q t z ∑ ij k ∑ ij k k k ij k k −1 3 k =1 12 k =1

where zk and hk define the position of the kth ply in the stacking sequence. tk and θ k are the ply thickness and the fibers orientation, respectively (Figure 3.3). With such a parameterization the local values (e.g. the stresses in each ply of the laminate) are available via the relations (3.1) and (3.4). On top of that the design problem is written in terms of the physical parameters used for the manufacturing of the laminated structures. Finally several different materials can be considered in the laminate when the parameterization (3.11) is used. However when fibers orientations are allowed to change during the structural design process the resulting mechanical properties are generally strongly non linear (see Figure 3.2) and non convex, and local minima appear in the optimization problem. This is also illustrated in Figure 3.4 that draws the variation of the strain energy density in a laminate over 2 fibers orientations. In Figure 3.5 it is shown that the structural responses entirely differ when either ply thickness or ply orientation is considered in the design, resulting in mixed monotonousnon monotonous structural behaviors. It turns that the optimal design task is more complicated since the optimization method should be able to efficiently take into account simultaneously both different behaviors. Additionally using such a parameterization increases the number of design variables that may appear in the optimal design problem since the thickness and fibers orientation of each ply are possible variables. Finally optimizing with respect to the fibers orientations is known to be very difficult and few publications are available on the subject. For a sake of completion, the sensitivity analysis of the structural responses of composites with respect to those variables can be found in Mateus et al. (1991), Geier and Zimmerman (1994), and Dems (1996).

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Michaël Bruyneel Strain energy density (N/mm)

θ2

θ1

Figure 3.4. Variation of the strain energy density in a [θ1/θ2]S laminate with respect to the fibers orientations θ1 and θ2. Strain nenergy density (N/mm)

1.2 1.4 1.6 1.8

θ

t

2

Figure 3.5. Variation of the strain energy density in an unidirectional ply with respect to its thickness t and its fibers orientation θ.

3.2.2. Parameterization with Sub-laminates The design parameters are no longer defined based on single unidirectional plies but instead on predefined sub-laminates. Each sub-laminate is itself made of several single unidirectional plies. The design parameters are assigned to the sub-laminates and no longer to each individual ply. Examples of sub-laminates may be [0/45/-45/90], [0/60/-60] or [0/90]. This parameterization allows to decrease the number of design variables. However the control at the ply level is lost. The previously presented parameterization in terms of ply thickness and orientation is a limiting case.

Optimization of Laminated Composite Structures…

2

3

63

S u b -la m in a te 1 [3 0 /-3 0 ]

1

S u b -la m in a te 2 [0 /4 5 /-4 5 /9 0 ]

Figure 3.6. Parameterization with sub-laminates. Here the symmetric laminate is made of 2 sublaminates.

3.2.3. The Lamination Parameters The stiffness matrix in (3.10) can be expressed with the lamina invariants defined in (3.8) together with the lamination parameters. For a given base material identical for each ply of the laminate the lamination parameters are given by (3.12) in the structural axes: h/2

ξ [1A,2,B,3,,D4] = ∫ z 0,1,2 [cos 2θ ( z ), cos 4θ ( z ), sin 2θ ( z ), sin 4θ ( z )]dz

(3.12)

−h / 2

The lamination parameters are the zero, first and second order moments relative to the plate mid-plane of the trigonometric functions (3.6) entering the rotation formulae for the ply stiffness coefficients (3.5). With this definition the stiffness matrices A, B and D in (3.10) write:

A = hγ 0 + γ 1ξ1A + γ 2ξ 2A + γ 3ξ 3A + γ 4ξ 4A B = γ 1ξ1B + γ 2ξ 2B + γ 3ξ 3B + γ 4ξ 4B D=

(3.13)

3

h γ 0 + γ1ξ1D + γ 2ξ 2D + γ 3ξ 3D + γ 4ξ 4D 12

Twelve lamination parameters exist in total and characterize the global stiffness of the laminate. This number is independent of the number of plies that contains the laminate. In most applications the lamination parameters are normalized with respect to the total thickness of the laminate (Grenestedt, 1992, and Hammer, 1997). In the case of symmetric laminates the 4 lamination parameters ξ B defining the coupling stiffness B vanish. Moreover when the structure is either subjected to in-plane loads or to out-of-plane loads only the 4 lamination parameters related to the in-plane stiffness ξ A or the out-of-plane stiffness ξ D must be considered, respectively. In the case of composite membrane or plates presenting orthotropic material properties 2 lamination parameters are sufficient to characterize the problem. Lamination parameters are not independent variables. Feasible regions of the lamination parameters exist which provide realizable laminates. Grenestedt and Gudmundson (1993)

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demontrated that the set of the 12 lamination parameters is convex. It is also observed from (3.13) that the constitutive matrices A, B and D are linear with respect to the lamination parameters. This means that the optimization problem is convex if it includes functions related to the global stiffness of the laminate, as for example the structural stiffness, vibration frequencies and buckling loads (Foldager, 1999). Feasible regions were determined for specific laminate configurations (e.g. Miki, 1982 and Grenestedt, 1992), but the region for the 12 lamination parameters has not yet been determined. Recently the relations between the lamination parameters were derived for ply angles restricted to 0, 90, 45 and -45 degrees by Liu et al. (2004) for membrane and bending effects, and by Diaconu and Sekine (2004) for membrane, coupling and bending effects. One of the feasible regions of lamination parameters is illustrated in Figure 3.7 in the case of a symmetric and orthotropic laminated plate subjected to bending. As the plate is assumed orthotropic in bending ξ1D and ξ 2D are enough to identify the stiffness of such a problem. Those two lamination parameters take their values on the outline delimited by the points A, B, C, and in the dashed zone. Any combination of the lamination parameters that is outside of this region will produce a laminate which is not realizable. When this plate is simply supported and subjected to a uniform pressure, the vertical displacement is a function of ξ1D and ξ 2D . The iso-values of this structural response are the parallel lines illustrated in Figure 3.7. According to Grenestedt (1990), the plate stiffness increases in the direction of the arrow. The stiffest plate is then characterized by the point D in Figure 3.7, which corresponds to a [(±θ)n]S laminate, defined by a single parameter θ. A

C 1 0.8 0.6

ξ2D

0.4 0.2

ξ1D

0 -0.2

E

-0.4 -0.6 -0.8 -1 -1

D -0.5

B

0.5

1

Figure 3.7. Feasible domain (outline plus dashed zone) of the lamination parameters for a symmetric and orthotropic laminated plate subjected to a uniform pressure (after Grenestedt, 1990). The points A, B, C correspond to [0], [(±45)n]S and [90] laminates, respectively. The point D defines a [(±θ)n]S laminate. The point E is a combination of laminates defined on the outline. The laminate of maximum stiffness is located on the outline (point D)

Optimization of Laminated Composite Structures…

65

This kind of parameterization has allowed to show that optimal solutions – in terms of the stiffness – are often related to simple laminates with few different ply orientations. For example only one orientation is necessary for characterizing the optimal laminate in a flexural problem (Figure 3.7), and at most 3 different ply orientations are sufficient to define the optimal stacking sequence in the case of a membrane of maximum stiffness (Lipton, 1994). Table 3.1 summarizes some of those important results. When using such a parameterization the number of design variables is very small (12 in the most general case) irrespective to the number of plies that contains the laminate. As seen in Figure 3.7 the design space is convex, and only one set of lamination parameters characterizes the optimal solution. However acording to the relations (3.8) and (3.13) only one kind of material can be used in the laminate: defining a different material for the core of a sandwich panel is for example not allowed (Tsai and Hahn, 1980). Additionally the local structural responses (e.g. the stresses in each ply) can not be expressed in terms of the lamination parameters since those last are defined at the global (laminate) level and are linked to the structural stiffness. However the global strains of the laminate (but not in each ply) can be computed with relation (3.10) and used in the optimization, as is done by Herencia et al. (2006). The feasible regions of the 12 lamination parameters is not yet determined. As said before those regions are only known for specific laminate configurations. This strongly limit their use in the frame of the optimal design of composite structures. Finally when the optimal values of the lamination parameters are known, coming back to corresponding thicknesses and orientations is a difficult problem and the solution is non unique (Hammer, 1997). Foldager et al. (1998) proposed a technique based on a mathematical programming approach while Autio (2000) used a genetic algorithm to find this solution when the number of layers is limited or for prescribed standardized ply angles. Table 3.1. Summary of some important results obtained with the lamination parameters Kind of structure Plate

Laminate configuration

Symmetric/orthotropic

Criteria

Stiffness Vibration Buckling

Symmetric

Buckling

Membrane Symmetric

Stiffness

General Cylindrical Symmetric/orthotropic shell

Stiffness Buckling

Optimal sequence

[(±θ ) n ]S [(±θ ) n ]S [(±θ ) n ]S [θ ]S [(α / 90 + α ) n ]S [θ ] , [α / 90 + α ] [(±θ ) n ]S , [0 / 90]S , quasi-isotropic

Reference

Grenestedt (1990), Miki and Sugiyama (1993) Grenestedt (1991) Fukunaga and Sekine (1993) Hammer (1997) Fukunaga and Vanderplaats (1991b)

3.2.4. Combined Parameterization As shown by Foldager et al. (1998) and Foldager (1999), composite structures can be designed by combining two parameterizations: the lamination parameters on one hand, and the plies thickness and fibers orientations on the other hand. The benefit of the approach

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relies on using a convex design space with respect to the lamination parameters, while keeping in the problem’s definition the physical variables in terms of thickness and orientation. This iterative procedure – between both design spaces – consists in determining a first (local) solution in terms of thicknesses and orientations. A new search direction towards the global optimum is then computed by evaluating the first order derivative of the objective function at the local solution with respect to the lamination parameters. The global optimum is reached when this sensitivity is close to zero. Otherwise a new design point is calculated in the space of the fibers orientations, and the process continues, usually by adding new plies in the laminate. As seen in Figure 3.8, the structural response is not convex with respect to θ while it is convex in terms of the lamination parameter ξ. With this technique the knowledge of the feasible regions of the lamination parameters is not mandatory. Although efficient, this solution procedure can only be used for global structural responses like the stiffness, the vibration frequencies and the buckling load.

f(θ)

f 1

f(ξ)

3 2 4

θ, ξ Figure 3.8. Illustration of the optimization process after Foldager et al. (1998) in both spaces of the lamination parameters ξ and the fibers orientation θ.

3.2.5. Alternative Parameterization In order to decrease the non linearities introduced by the fibers orientation variables, Fukunaga and Vanderplaats (1991a) proposed to parameterize the laminated composite membranes with the following intermediate variables:

xi = sin 2θi or xi = cos 2θi based on the relation (3.12) and (3.13). This formulation was tested by Vermaut et al. (1998) for the optimal design of laminates with respect to strength and weight restrictions. As in the previous section, the main difficulty is to compute the orientations corresponding to the optimal intermediate variables values xi.

Optimization of Laminated Composite Structures…

67

4. Specific Problems in the Optimal Design of Composite Structures For designing laminated composite structures a very large number of data must be considered (material properties, plies thickness and fibers orientation, stacking sequence) and complex geometries must be modelled (aircraft wings, car bodies). Therefore the finite element method is used for the computation of the structural mechanical responses. Usually mass, structural stiffness, ply strength and strain, as well as buckling loads are the functions used in the optimization problem. The design variables are classically the parameters defining the laminate: fibers orientations, plies thickness, and indirectly the number of plies and the stacking sequence. Some specific problems appear in the formulation of the optimization problem for laminated structures. They are reported hereafter. Large number of design variables. Even for a parameterization in terms of the lamination parameters, the number of design variables can easily reach a large value when the plies thickness and fibers orientations are allowed to change over the structure, leading to non homogenenous plies (Figure 4.1) and curvilinear fibers formats (Hyer and Charette 1991, Hyer and Lee 1991, Duvaut et al. 2000). In industrial applications (Krog et al. 2007), thicknesses related to specific orientations (0°, ±45°, 90°) are used and several independent regions are defined throughout the composite structure, what increases the number of design variables. Large number of design functions. Not only global structural responses related to the stiffness are relevant in a composite structure optimization, but also the local strength of each ply. Damage tolerance and local buckling restrictions are important as well. For an aircraft wing, it is usual to include about 300000 constraints in the optimization problem (Krog et al. 2007).

Non homogeneous ply

Homogeneous ply

Figure 4.1. Homogeneous and non homogeneous ply in a laminate.

Problems related to the topology optimization of composite structures. In topology optimization one is looking for the optimal distribution of a given amount of material in a predefined design space that maximizes the structural stiffness (Figure 4.2).

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Solid

Domain where the material is distributed

Void

Figure 4.2. Illustration of a topology optimization problem (after Bruyneel, 2002)

For composite structures, and due to the stratification of the material, it results that 2 topology optimization problems must be defined and solved simultaneously: the optimal distribution of plies at a given altitude in the laminate (Figure 4.3) and the transverse topology optimization where the optimal local stacking sequence is looked for (Figure 4.4). Continuity conditions between adjacent laminates should also be imposed.

Figure 4.3. Topology optimization at a given altitude in the non homogeneous laminate.

Figure 4.4. Transverse topology optimization in a composite structure.

Optimization of Laminated Composite Structures…

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Specific non linear behaviors of laminated structures. In order to improve the accuracy in the model, non linear effects, and especially the design with respect to the limit load, should be considered in the formulation of the optimization of composite structures. This dramatically increases the computational time of the finite element analysis, and can only be used for studying small structural parts such as super-stringers, i.e. some stiffeners and the panel (Colson et al., 2007). Although simple fracture mechanics criteria have been considered (Papila et al. 2001), damage tolerance and propagation of the cracks (delamination) should be taken into account in the same way. Uncertainties on the mechanical properties of composites. There is a larger dispersion in the mechanical properties of the fibers reinforced composite materials than for metals. Moreover, some uncertainties concerning the orientations and the plies thickness exist. Robust optimization should be used in these cases (Mahadevan and Liu, 1998, Chao et al., 1993, Chao, 1996, and Kristindottir et al., 1996). Strong link with the manufacturing process. Contrary to the design with metals, there is a strong link between the material design, the structural design and the manufacturing process when dealing with composite materials. The constraints linked to manufacturing can strongly influence the design and the structural performances (Henderson et al., 1999, Fine and Springer, 1997, Manne and Tsai, 1998) and should be taken into account to formulate in a rational way the design problem (Karandikar and Mistree, 1992). Singular optima in laminates design problems. When strength constraints are considered in the design problem, and if the lower bounds on the plies thickness is set close to 0 (i.e. some plies can disappear at the solution from the initial stacking sequence), it can be seen (Schmit and Farschi, 1973, Bruyneel and Fleury, 2001) that the design space can become degenerated. In this case the optimal design can not be reached with gradient based optimization methods. Such a degenerated design space is illustrated in Figure 4.5. It is divided into a feasible and an infeasible region according to the limiting value of the TsaiWu criteria. In this example a [0/90]S laminate’s weight is to be minimized under an inplane load N1. The optimal solution is a [0] laminate. Unfortunately this optimal laminate configuration can not be reached with a gradient based method since the 90 degree plies are still present in the problem even if their thickness is close to zero, and the related Tsai-Wu criterion penalizes the optimization process. A first solution consists in using the ε-relaxed approach (Cheng and Guo 1997), which slightly modifies the design space in the neighborhood of the solution and allows the optimization method to reach the true optimum [0]*. Alternatively (Bruyneel and Fleury, 2001, and Bruyneel and Duysinx, 2006) when fibers orientations are design variables the shape of the design space changes, the gap between the true optimal solution and the one constrained by plies with a vanishing thickness [0/x]* decreases and the real optimal solution becomes attainable (Figure 4.5). Optimizing over the fibers orientations allows to circumvent the singularity of the design space.

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* represents the obtained solutions, optimum or not

Figure 4.5. Design space for [0/90]S and [0/10]S laminates.

Importance of the fibers orientations in the laminate design. Besides their efficiency in avoiding the singularity in the optimization process as just explained before fibers orientations play a key role in the design of composite structures. Modifying their value allows for great weight savings, as illustrated in Figure 4.6. Let’s consider that the initial laminate design corresponds to fibers orientation and ply thickness at point A. A first way to obtain a feasible design with respect to strength restrictions is to increase the ply thickness and go to B, which penalizes the structural weight. Another solution consists in modifying the fibers orientation, here at constant thickness (point C). A better solution is to simultaneously optimize with respect to both kinds of design variables (point D). However taking into account such variables in the optimization problem is a real issue, and providing a reliable solution procedure is a challenge.

Figure 4.6. Design space for an unidirectional laminate subjected to either N1 or N6. Iso-values of the Tsai-Wu criterion. The ply thickness and fibers orientations are the design variables.

The optimal stacking sequence. A large part of the research effort on composites has been dedicated to the solution of the optimal stacking sequence problem. As it is a combinatorial

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problem including integer variables, genetic algorithms have been used (Haftka and Gurdal, 1992, Le Riche and Haftka, 1993). The topology optimization formulation of Figure 4.4 was used by Beckers (1999) and (Stegmann and Lund, 2005) to solve this problem with discrete and continuous design variables, respectively. Another approach, still based on the discrete character of the problem, is proposed by Carpentier et al. (2006). It consists in using a lay-up table defined based on buckling, geometric and industrial rules considerations. This table, which satisfies the ply drop-off continuity restrictions is determined numerically. Once it is obtained a given laminate total thickness corresponds to a stacking sequence (via a column of the table). The optimization process then consists in optimizing the local thickness of a set of contiguous laminates defining the structure. Each laminate has equivalent homogenized properties with 0, ±45 and 90° plies. Based on the lay-up table, the stacking sequence is therefore known everywhere in the structure for different local optimal thicknesses and the composite material can be drapped.

Figure 4.7. Illustration of a lay-up table for 0, ±45 and 90° plies.

5. Problems Solved in the Literature 5.1. Structural Responses When designing laminated composite structures the functions entering the optimization problem (2.1) are classically the stiffness, the vibration frequencies, the structural stability and the plies’ strength. (see Abrate, 1994, for a detailed review of the literature). It is interesting to note that for orthotropic laminates maximizing the stiffness, the frequency or the first buckling load will provide the same solution (Pedersen, 1987 and Grenestedt, 1990). On top of that, it should be noted that optimizing a laminated structure against plies strength or stiffness will result in different designs. It results that the local (stress) effects are very important in the optimal design of composite structures (Tauchert and Adibhatla, 1985, Fukunaga and Sekine, 1993, and Hammer, 1997).

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5.2. Optimal Design with Respect to Fibers Orientations Determining the optimal fibers orientation is a very difficult problem since the structural responses in terms of such variables are highly non linear, non monotonous and non convex. However it has just been show in the previous section that the design of laminated composite structures is very sensitive with respect to those variables. As explained by the editors of commercial optimization software (Thomas et al., 2000) there is a need for an efficient treatment of such parameters. A small amount of work has been dedicated to the optimal design of laminated structures with respect to the fibers orientations. Several kinds of approaches have been investigated and are reported in the literature: •

Approach by optimality criteria

Optimal orientations of orthotropic materials that maximize the stiffness in membrane structures were obtained by Pedersen (1989, 1990 and 1991), and by Diaz and Bendsøe (1992) for multiple load cases. When the unidirectional ply is only subjected to in-plane loads, Pedersen (1989) proposed to place the fibers in the direction of the principal stresses. The resulting optimality criterion was used in topology optimization including rank-2 materials (Bendsøe, 1995). This technique was used by Thomsen (1991) in the optimal design of non homogeneous composite disks. This criterion was extended by Krog (1996) to Mindlin plates and shells. •

Approach based on the mathematical programming

As soon as 1971, Kicher and Chao solved the problem with a gradients based method. Hirano (1979a and 1979b) used the zero order method of Powell (conjugate directions) for buckling optimization of laminated structures. Tauchert and Adibhatla (1984 and 1985) used a quasi-Newton technique (DFP) able to take into account linear constraints for minimizing the strain energy of a laminate for a given weight. Cheng (1986) minimized the compliance of plates in bending and determined the optimal orientations with an approach based on the steepest descent method. Martin (1987) found the minimum weight of a sandwich panel subjected to stiffness and strength restrictions with a method based on the Sequential Convex Programming (Vanderplaats, 1984). Watkins and Morris (1987) used a similar procedure with a robust move-limits strategy (see also Hammer 1997). In Foldager (1999), the method used for determining the optimal fibers orientations is not cited but belongs according to the author to the family of mathematical programming methods. SQP, the feasible directions method and the quasi-Newton BFGS were used by Mahadevan and Liu (1998), Fukunaga and Vanderplaats (1991a), and Mota Soares et al. (1993, 1995 and 1997), respectively. Those mathematical programming methods are reported and explained in Bonnans et al. (2003). •

Approach with non deterministic methods

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Genetic algorithms have been employed by several authors for determining the optimal stacking sequence of laminated structures (Le Riche and Haftka, 1993, Kogiso et al., 1994 and Potgieter and Stander, 1998) or in the treatment of fibers orientations (Upadhyay and Kalyanarama, 2000).

5.3. Formulations of the Optimization Problem Thickness and orientation variables were treated in several ways in the literature. They have been considered either simultaneously as in Pedersen (1991), and Fukunaga and Vanderplaats (1991a), or separately (Mota Soares et al. 1993, 1995 and 1997, and Franco Correia et al. 1997). Weight, stiffness and strength criteria have been separately introduced in the design problem and taken into account in a bi-level approach by (Mota Soares et al., 1993, 1995, 1997 and Franco Correia et al., 1997): at the first level the weight is kept constant and the stiffness is optimized over fibers orientations ; at the second level the ply thicknesses are the only variables in an optimization problem that aims at minimizing the weight with respect to strength and/or displacements restrictions. A similar approach can be found in Kam and Lai (1989), and Soeiro et al. (1994). Fukunaga and Sekine (1993) also used a bi-level approach for determining laminates with maximal stiffness and strength in non homogeneous composite structures (Figure 4.2) subjected to in-plane loads. In Hammer (1997), both problems are separately solved and the initial configuration for optimizing with respect to strength is the laminate previously obtained with a maximal stiffness consideration.

6. Optimal Design of Composites for Industrial Applications Based on the several possible laminate parameterizations and on the previous discussion it was concluded in Bruyneel (2002, 2006) that an industrial solution procedure for the design of laminated composite structures should preferably be based on fibers orientations and ply thicknesses, instead of intermediate non physical design variables such as the lamination parameters. Using those variables allows optimizing very general structures (membranes, shells, volumes, subjected to in- and out-of-plane loads, symmetric or not) and provides a solution that is directly interpretable by the user. On the other hand, an optimization procedure used for industrial applications should be able to consider a large number of design variables and constraints, and find the solution (or at least a feasible design) in a small number of design cycles. Additionally, the optimization formulation should be as much general as possible, and not only limited to specific cases (e.g. not only thicknesses, not only membrane structures, not only orthotropic configurations,…). For those reasons, a solution procedure based on the approximation concepts approach seems to be inevitable. Interesting local solutions can be found by resorting to other optimization methods (e.g. response surfaces coupled with a genetic algorithm) but on structures of limited size. For the pre-design of large composite structures like a full wing or a fuselage, or when non linear responses are defined in the analysis (post-buckling, non linear material behavior), the approximation concepts approach proved to be a fast method not expensive in CPU time for solving industrial problems (Krog and al, 2007, Colson et al., 2007).

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It results that robust approximation schemes must be available to efficiently optimize laminated structures. The characteristics of such a reliable approximation are explained in the following, and tests are carried out to show the efficiency and the applicability of the method.

7. Optimization Algorithm for Industrial Applications 7.1. The Approximation Concepts Approach In the approximation concepts approach, the solution of the primary optimization problem (2.1) is replaced with a sequence of explicit approximated problems generated through first order Taylor series expansion of the structural functions in terms of specific intermediate variables (e.g. direct xi or inverse 1/xi variables). The generated structural approximations built from the information known at least at the current design point (via a finite element analysis), are convex and separable. As will be explained latter a dual formulation can then be used in a very efficient way for solving each explicit approximated problem. According to section 2, it is apparent that the approximation concepts approach is well adapted to structural optimization including sizing, shape and topology optimization problems. However, the use of the existing schemes (section 7.2) can sometimes lead to bad approximations of the structural responses and slow convergence (or no convergence at all) can occur (Figure 7.1). x2

x2

X(k )

X (k )*

X (k ) X*local

X*global

X*local

X*global

x1

X ( k )*

x1

x2

X ( k )*

X*global

X*local

x1

a. A too conservative approximation b. A too few conservative approximation and unfeasible intermediate solutions c. An approximation not adapted to the problem, leading to zigzagging

Figure 7.1. Difficulties appearing in the approximation of highly non linear structural responses.

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Such difficulties are met for laminates optimization: their structural responses are mixed, i.e. monotonous with regard to plies thickness and non monotonous when fibers orientations are considered (Figure 3.5). Additionally, the non monotonous structural behaviors in terms of orientations are difficult to manage (Figure 3.4). It results that the selection of a right approximation scheme is a real challenge. In the next section a generalized approximation scheme is presented that is able to effectively treat those kinds of problems. This optimization algorithm will identify the structural behavior (monotonous or not) according to the involved design variable (orientation or thickness), and will automatically generate the most reliable approximation for each structural function included in the optimization problem. In section 8 numerical tests will compare the efficiency of the proposed approximation scheme and the existing ones for laminates optimization including both thickness and orientation variables.

7.2. Selection of an Accurate Approximation Scheme 7.2.1. Monotonous Approximations Based on the first order derivatives of the structural responses included in the optimization problem, linear approximations can be built at the current design point xk. It is a first order Taylor series expansion in terms of the direct design variables xi (7.1).

∂g j (x g~ (jk ) (x)=g j (x ( k ) ) + ∑ ∂xi i

(k )

)

( xi − xi( k ) )

(7.1)

As it is very simple this approximation is most of the time not efficient for structural optimization but can anyway be used with some specific move-limits rules (Watkins and Morris, 1987) that prevent the intermediate design point to go too far from the current one and to generate large oscillations during the optimization process (Figures 7.1b and 7.1c). Since the stresses vary as 1/xi in isostatic trusses where xi is the cross section area of the bars, a linear approximation in terms of the inverse design variables is more reliable for the optimal sizing of thin structures. The resulting reciprocal approximation is given in (7.2). (k ) ⎛ )⎜ 1 1 (k ) ( k ) 2 ∂g j ( x (k ) ~ g j (x)=g j (x ) − ∑ (x i ) − ⎜ k) ( x ∂xi i ⎝ i xi

⎞ ⎟ ⎟ ⎠

(7.2)

The Conlin scheme developed by Fleury and Braibant (1986) is a convex approximation based on (7.1) and (7.2). It is reported in (7.3) and illustrated in Figure 7.2.

∂g j (x g~ (jk ) (x)=g j (x ( k ) ) + ∑ ∂xi +

(k )

)

( xi − xi( k ) ) − ∑ ( xi( k ) ) 2 −

∂g j (x ( k ) ) ⎛⎜ 1 1 − ⎜ xi ∂xi xi( k ) ⎝

⎞ ⎟ ⎟ ⎠

(7.3)

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The symbols ∑ (+) and ∑ (−) in (7.3) denote the summations over terms having positive and negative first order derivatives. When the first order derivative of the considered structural response is positive a linear approximation in terms of the direct variables is built, while a reciprocal approximation is used on the contrary. Strain energy density (N/mm)

145 140 135 130 125

g~r( k ) ( x )

g (x )

120 115 110

g~l( k ) ( x )

105 100 90

45

x l(k )

xr(k )

180

Figure 7.2. The Conlin approximation.

Conlin can only work with positive design variables since an asymptote is imposed at xi=0. On top of that, the curvature of this approximation is imposed by the derivative at the current design point and can not be adapted to better fit the problem. The Method of Moving Asymptotes or MMA (Svanberg 1987) generalizes Conlin by introducing two sets of new parameters, the lower and upper asymptotes, Li and Ui, that can take positive or negative values, in order to adjust the convexity of the approximation in accordance with the problem under consideration. The asymptotes are updated following some rules provided by Svanberg (1987). The parameters pij and qij are built with the first order derivatives.

145

Strain energy density (N/mm)

140

145

g~ ( k ) ( x )

135

130

130

125

125

g (x )

g (x )

120

115

115

110

110

105

105

100 45

g~ ( k ) ( x )

140

135

120

Strain energy density (N/mm)

100

L(k ) 90

x

(k ) 135

x

(k )* 180

45

U (k ) 90

Figure 7.3. The MMA approximation.

x (k )*

x (k )

180

Optimization of Laminated Composite Structures… ⎛ 1 1 g~ (jk ) (x)=g j (x ( k ) ) + ∑ pij( k ) ⎜ − ⎜ (k ) (k ) (k ) + ⎝ U i − xi U i − xi

⎞ ⎛ 1 1 ⎟ + q (k ) ⎜ − ∑ ij ⎟ − ⎜ (k ) (k ) xi − L(i k ) ⎠ ⎝ xi − Li

77 ⎞ ⎟ ⎟ ⎠

(7.4)

As it will be seen later those monotonous schemes are not efficient for optimizing structural functions presenting non monotonous behaviors, as in Figure 3.4.

7.2.2. Non Monotonous Approximations Based on MMA, Svanberg (1995) developed the Globally Convergent MMA approximation (GCMMA). As illustrated in Figure 7.4 it is non monotonous and still only based on the information at the current design point (functions values, first order derivatives, asymptotes values). Here both Ui and Li are used simultaneously. It was not the case in (7.4). ⎛ 1 1 g~ (jk ) (x)=g j (x ( k ) ) + ∑ pij(k ) ⎜ − (k ) (k ) (k ) ⎜ i ⎝ U i − xi U i − xi

⎞ ⎛ 1 1 ⎟ + ∑ q (k ) ⎜ − ⎟ i ij ⎜ x − L( k ) x ( k ) − L( k ) i i i ⎠ ⎝ i

⎞ ⎟ ⎟ ⎠

(7.5)

Using this method can lead to slow convergence given that it can generated too conservative approximations of the design functions (Figure 7.1a).

145

Strain energy density (N/mm)

g~ ( k ) ( x )

140 135 130

g (x )

125 120 115 110 105 100 45

U (k )

L(k ) 90

x (k )

135

x (k )*

180

Figure 7.4. The GCMMA approximation.

In order to improve the quality of this approximation it was proposed in Bruyneel and Fleury (2002) and Bruyneel et al. (2002) to use the gradients at the previous iteration to improve the quality of the approximation, leading to the definition of the Gradient Based MMA approximations (GBMMA). In those methods the pij and qij parameters of (7.5) are computed based on the function value and gradient at the current design point and on the gradient at the previous iteration. The rules defined by Svanberg (1995) for updating the asymptotes are used.

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7.2.3. Mixed Approximation of the MMA Family When dealing with structural optimization problems including design variables of two different natures, for example in problems mixing ply thickness and orientation variables, one is faced to a difficult task because of the simultaneous presence of monotonous and nonmonotonous behaviors with respect to the set of design variables. In these conditions, most of the usual approximation schemes presented before have poor convergence properties or even fail to solve these kinds of problems. Knowing that the MMA approximation is very reliable for approximating monotonous design functions and based on the GBMMA approximations, a mixed monotonous – non monotonous scheme is presented in Bruyneel and Fleury (2002) and Bruyneel et al. (2002), which will automatically adapt itself to the problem to be approximated (7.6). ⎛ 1 1 − g~ (jk ) (x) = g j ( x ( k ) ) + ∑ pij( k ) ⎜ (k ) (k ) (k ) ⎜ i∈ A ⎝ U i − x i U i − xi ⎛ 1 1 + ∑ pij( k ) ⎜ − (k ) (k ) (k ) ⎜ + , i∈ B ⎝ U i − xi U i − x i

⎞ ⎛ 1 1 ⎟ + ∑ q (k ) ⎜ − ij ⎟ i∈ A ⎜ x − L( k ) x ( k ) − L( k ) i i i ⎠ ⎝ i

⎞ ⎛ 1 1 ⎟ + ∑ q (k ) ⎜ − ⎟ −, i∈B ij ⎜ x − L( k ) x ( k ) − L( k ) i i i ⎠ ⎝ i

⎞ ⎟ ⎟ ⎠

(7.6)

⎞ ⎟ ⎟ ⎠

In (7.6) the symbols ∑ (+, i ) and ∑ (−, i ) designate the summations over terms having positive and negative first order derivatives, respectively. A and B are the sets of design variables leading to a non monotonous and a monotonous behavior respectively, in the considered structural response. At a given stage k of the iterative optimization process, a monotonous, non monotonous or linear approximation is automatically selected, based on the tests (7.7), (7.8) and (7.9) computed for given structural response g j (X) and design variable xi .

∂g j (x k ) ∂xi ∂g j (x k ) ∂xi

×

×

∂g j (x k −1 )

∂g j (x k −1 )

∂g j (x k ) ∂xi

> 0 ⇒ MMA (monotonous)

(7.7)

< 0 ⇒ GBMMA (non monotonous)

(7.8)

∂xi ∂xi −

∂g j ( x k −1 ) ∂xi

= 0 ⇒ linear expansion

(7.9)

The selection of a right approximation is illustrated in Figure 7.5: when a monotonous approximation is used for approximating a non monotonous function, oscillations can appear, while a non monotonous approximation is too conservative when the function is monotnous. The best approximation is therefore selected based on tests (7.7) to (7.9). This strategy proved to be reliable for simple laminates design (Bruyneel and Fleury 2002) and for general laminated composite structures design problems (Bruyneel 2006, Bruyneel et al. 2007, Krog et al. 2007), for truss sizing and configuration (Bruyneel et al. 2002), for topology

Optimization of Laminated Composite Structures…

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optimization which includes a large amount of design variables (Bruyneel and Duysinx 2005). It has been made available in the BOSS Quattro optimization toolbox (Radovcic and Remouchamps, 2002). In the following this solution procedure based on a mixed approximation scheme is called Self Adaptive Method (SAM). Based on this approximation scheme, it is possible to resort to the other ones (GBMMA, MMA, Conlin and the linear approximation) by setting specific values to the asymptotes and by limiting the approximations to the sets A or B in (7.6).

145 140

Strain energy density (N/mm)

g~ MMA

g~MMA

g~GCMMA

Strain energy density (N/mm)

400

g~GCMMA

350

135 130

300

g (θ )

125 120

250

g (t )

115 200

110 105 100 45

(k )

(k )* θ MMA

150

L

U 90

θ

(k )

135

(k )

(k )* θ GCMMA

1.2

180

1.3

1.4

1.5

t (k )

1.7

(k )* tGCMMA

(k )* t MMA

Figure 7.5. The mixed SAM approximation.

A summary of the approximations that will be compared in the following is presented in Table 7.1. Table 7.1. Summary of the approximations that will be compared in the numerical tests Approximation Author MMA Svanberg (1987) GCMMA Svanberg (1995) SAM Bruyneel (2006)

Behavior Monotonous Non monotonous Mixed monotonous/non monotonous

7.3. Solution Procedure for Mono and Multi-objective Optimizations Since the approximations are convex and separable the solution of each optimization subproblem (Figure 2.3) is achieved by using a dual approach. Based on the theory of the duality, solving the problem (2.2) in the space of the primal variables xi is equivalent to maximize a function (7.10) that depends on the Lagrangian multipliers λ j , also called dual variables:

max min L(x, λ ) λ

λj ≥ 0

x

j = 0,..., m (λ0 = 1)

(7.10)

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Solving the primal problem (2.2) requires the manipulation of one design function, m structural restrictions and 2 × n side constraints (for mono-objective problems). When the dual formulation is used, the resulting quasi-unconstrained problem (7.10) includes one design function and m side constraints, if the side constraints in the primal problem are treated separately. In relation (7.10), L(x, λ ) is the Lagrangian function of the optimization problem, which can be written

L(x, λ ) = ∑ λ j (c j + ∑ j

i

pijk U ik − xi

+∑ i

qijk xi − Lki

)

(7.11)

according to the general definition of the involved approximations g~ j ( X ) of the functions. The parameter λj is the dual variable associated to each approximated function g~ j ( X ) . Given that the approximations are separable, the Lagrangian function is separable too. It turns that:

L( x, λ ) = ∑ L i ( xi , λ ) i

and the Lagrangian problem of (7.10)

min L(x, λ ) x

can be split in n one dimensional problems

min L i ( xi , λ )

(7.12)

xi

The primal-dual relations are obtained by solving (7.12) for each primal variable xi:

∂L i ( xi , λ ) =0 ∂xi

⇒

xi = xi ( λ )

(7.13)

Relation (7.13) asserts the stationnarity conditions of the Lagrangian function over the primal variables xi. Once the primal-dual relations (7.13) are known, (7.10) can be replaced by

max l (λ ) ⇔ max L(x(λ), λ ) λ

λ

λj ≥ 0

(7.14)

j = 1,..., m

Solving problem (2.2) is then equivalent to maximize the dual function l (λ ) with non negativity constraints on the dual variables (7.14). As it is explained by Fleury (1993), the

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maximization (7.14) is replaced by a sequence of quadratic sub-problems. Each sub-problem is itself partially solved by a first order maximization algorithm in the dual space. In the case of a multi-objective formulation the optimization problem writes :

min max g l 0 (x) X l =1,..., nc

g j (x) ≤ g j

j = 1,..., m

(7.15)

where nc is the number of load cases. Using the bound formulation (Olhoff, 1989) the problem (7.15) can be written as:

g l 0 ( x) ≤ β

1 2 β 2 l = 1,..., nc

g j ( x) ≤ g j

j = 1,..., m

min

(7.16)

where β is the multiobjective factor, that is an additional design variable in the optimization problem. Instead of solving (7.16) problem (7.17) is considered where a new variable δ is introduced for the possible relaxation of the set of constraints.

(

)

2 1 1 C min β 2 + (δ + p )2 + ∑ xi − xi( k −1) 2 2 2 i

g j 0 ( x) ≤ β j g j 0

j = 1,..., nobj

g j (x) ≤ g j (1 + δ )

j = 1,..., m

(7.17)

g j 0 are target values on the objective functions. The dual approach described for monoobjective optimisation problems is then applied to (7.17).

8. Applications of the Optimization Solution Procedure In the following examples (except the simple laminate designs and the topology optimization problem), the structural and semi-analytical sensitivity analyses are carried out with SAMCEF (http://www.samcef.com). The Boss Quattro optimisation tool box (http://www.samcef.com) is used for defining and solving the optimisation problem (Radovcic and Remouchamps 2002).

8.1. Laminate Subjected to in- and out-of-plane Loadings A symmetric 4 plies laminate made of carbon/epoxy is considered. The load case and the initial configuration are provided in Table 8.1. The fibers orientations of each ply are the

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design variables, while plies thicknesses are kept constant. The optimization consists in minimizing the laminate’s strain energy density, i.e. maximizing its stiffness. The evolution of this objective function with respect to the 2 angles θ1 and θ2 is reported in Figure 8.1, with the initial and optimal design points. A restriction is imposed on the relative variation of the 2 design variables. The optimization problem writes :

1 1 min ε T0 Aε T0 + κ T Dκ 2 θ 2

θ 2 − θ1 ≤ 45 0.001 ≤ θ i ≤ 180

(8.1)

i = 1,2

where the stiffness matrices A, B and D, and the laminate’s strain and curvature were previously defined in Section 3. Strain energy density (N/mm)

Initial design Optimal design

23.3°

22.3°

Solution

θ2 θ1

Figure 8.1. Variation of the strain energy density in the symmetric laminate subjected to the load case of Table 8.1.

Table 8.1. Problem’s definition: load case and initial design In-plane load case Out-of-plane load case ( N1 , N 2 , N 6 ) (M 1 , M 2 , M 6 )

Initial orientations θ = (θ 1 , θ 2 )

Initial thicknesses t = (t1 , t 2 )

in N/mm

in N

in degrees

in mm

(2000,0,1000)

(0,500,0)

(45,135)

(1,2)

In this application the laminate is subjected not only to in-plane but also to out-of-plane loadings. Since the plies thicknesses are not identical (Table 8.1) the objective function is not symmetric with regards to the axis θ1 = θ 2 (Figure 8.2).

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83

Strain energy density (N/mm) 180 160

θinit

140 120

θ2

100 80 60

θopt unconstrained

40

θopt

20 0

0

20

40

60

80

100

120

140

160

180

θ1 Figure 8.2. Illustration of the design space. Staring point, unconstrained and constrained optimum.

The iteration histories for the 3 approximation schemes are illustrated in the Figure 8.3. The convergence of the optimization process is controlled by the relative variation of the design variables at 2 successive iterations. The MMA approximation converges in 41 iterations. 29 iterations are enough for GCMMA. When the SAM approximation is used the solution is reached in a very small number of iterations. Objective function (N/mm)

Objective function (N/mm)

25 20

Objective function (N/mm)

20

20

15

15

10

10

5

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GCMMA Figure 8.3. Iteration history for the 3 approximation methods.

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8.2. Non Homogeneous Laminate In this application a non homogeneous composite membrane divided in regions of constant thickness and fibre orientations is studied. Each region is defined with an unidirectional laminate made of a glass/epoxy material. The design over stiffness is only considered here. The solution with respect to strength and stiffness is provided in Bruyneel (2006). 2

2

1

P

1

1

P

P

2

2

2

1

1

P

P

Figure 8.4. Initial configurations with 45 and -45 degrees plies orientations.

The quasi-unconstrained optimization problem (8.2) consists in finding the optimal values of the plies thickness and fibers orientations in each region of the laminated composite structure that maximize the overall stiffness (i.e. that minimize the compliance – the potential energy of the applied loads). The vectors of the design variables are given by θ = {θ i , i = 1,..., n} and t = {ti , i = 1,..., n} where n is the number of regions according to Figure 8.4. The initial thicknesses are of 1 mm.

min Compliance θ,t

0° ≤ θ i ≤ 180°

i = 1,..., n

(8.2)

0.01mm ≤ t i ≤ 5mm In this problem the optimal values of the thickness is 5 mm, that is their upper bound. Anyway this application illustrates the difficulties encountered when both kinds of design variables appear in the design problem. The optimal values of the compliances are reported in Figure 8.5 as a function of the number of regions. As already noticed by Foldager (1999) an increase of the number of regions of different orientations improves the overall optimal structural stiffness (i.e. it decreases the compliance). The optimal fibers orientations are illustrated in Figure 8.6, for the several membrane configurations of Figure 8.4. The iteration histories are reported in Figure 8.7. When the SAM method is used, about 10 iterations are enough for reaching a stationary solution with respect to a small relative variation of the objective at 2 successive iterations. The GCMMA approximation finds this solution in a larger number of design cycles. It is observed that when

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the SAM method is used, the structural responses in terms of both the fibers orientations and the thicknesses are well approximated, while using GCMMA, the approximation in terms of the thicknesses is too conservative, what slows down the overall convergence speed of the optimization process. Relative compliances 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5

1

4

8

12

20

Number of regions : n Figure 8.5. Evolution of the compliances in the problem (8.2) for the structures illustrated in Figure 8.4. The compliance of the one region structure is the reference (n = 1)

1 region

4 regions Figure 8.6. Continued on next page.

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8 regions

12 regions

20 regions Figure 8.6. Illustration of the optimal fibers orientations for the different composite membranes illustrated in Figure 7.9.

Pli 19

Pli 5

Figure 8.7. Continued on next page.

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GCMMA

5

x 10

4

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Mass (kg) and thickness of ply19 (mm) 8 7

4

Orientation of ply 5 (deg.) 140 120

Total mass 6

3

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80

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4

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120

6 3

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2

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100 80

Thickness of ply 19

3 1 0

60

2 0

5

10

1

15

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Figure 8.7. Convergence history for GCMMA and SAM for the membrane divided in 20 regions. Evolution of the thickness and the orientations of the plies number 5 and 19. Vertical displacement δmax under the load (mm)

2

O 320 finite elements + 80 finite elements * 20 finite elements

1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1

0

20

40

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100

120

140

160

180

Fibers orientation (deg.)

Figure 8.8. Evolution of the vertical displacement under the applied load for several discretizations of the homogeneous composite membrane (Figure 8.4, n=1).

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In Figure 8.8 the evolution of the vertical displacement under the load is drawn with respect to the fibers orientation in the case of the homogeneous membrane (Figure 8.4, n=1). The global minimum displacement is obtained for a value of 170°. When the starting point of the optimization process of the problem (8.2) is close to 45°, 0° fibers orientation is found as a local optimum. As -45° is chosen here for the initial design (i.e. 135°), the global optimum can be reached. This illustrates the fact that a gradient based method is not able to reach the global optimum, unless the starting point is in its vicinity. In Figure 8.8, the influence of the mesh refinement on the solution is presented, as well.

8.3. Multi-objective Optimization A symmetric laminate made of 4 plies and subjected to 2 in-plane load cases is considered. 1

N1

θ x N2

N2 2 N6 3

N1

Figure 8.9. Laminate subjected to in-plane loads.

The applied loads and the initial configuration are reported in Table 8.2. The load case (2) is variable : the factor k takes the values 0,1,2,…,8. The extreme load cases are, on one hand (1000,0,0) and on the other hand the combination of (1000,0,0) and (0,2000,0) N/mm. Table 8.2. Definition of the problem: load case and starting point Load case (1) ( N1 , N 2 , N 6 )

Load case (2) ( N1 , N 2 , N 6 )

Initial orientations θ = (θ 1 , θ 2 )

Initial thickness t = (t1 , t 2 )

in N/mm (1000,0,0)

in N/mm (0, k × 250 ,0)

in degrés (30,120)

en mm (1,2)

The performance of three approximation schemes are compared : GCMMA, MMA and SAM. The optimization problem writes :

1 T ε ( j ) Aε ( j ) θ,t j =1,2 2

min max

TW( j ) (θ i , t i ) ≤ 1

i, j = 1,2

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4

∑ ti ≤ 4

(8.3)

i =1

0.001 ≤ θ i ≤ 180

i = 1,2

0.001 ≤ t i ≤ 10

i = 1,2

where j is the number of the load case. This problem is solved by resorting the its bound formulation (Olhoff, 1989) including here 5 design variables (2 orientations, 2 thicknesses and the multi-objective factor β) and 7 constraints:

1 min β 2 2 1 T ε ( j ) Aε ( j ) ≤ β 2 TW( j ) (θ i , t i ) ≤ 1

j = 1,2 i, j = 1,2

(8.4)

4

∑ ti ≤ 4

i =1

0.001 ≤ θ i ≤ 180

i = 1,2

0.001 ≤ t i ≤ 10

i = 1,2

The results are reported in Figure 8.10 for the different values of k. The solution is obtained when the relative variation of the design variables at 2 successive iterations is lower than 0.01. It is seen that a large number of iterations is needed to reach the optimum when MMA is used. GCMMA converges in a lower number of iterations. As for mono-objective problems, SAM is the most effective optimization method. Maximum strain energy density (N/mm) 4

80

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6

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8

Number of iterations

0

0

2

4

6

8

Load parameter k

+ MMA o GCMMA Δ SAM

Figure 8.10. Variation of the strain energy density and number of iterations needed to reach the solution as a function of the parameter k.

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Objective functions (N/mm)

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Maximum constraints violations

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Figure 8.11. Convergence history for MMA. k is equal to 3. Objective functions (N/mm)

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Maximum constraints violations

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-6

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8

10

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Figure 8.12. Convergence history for SAM. k is equal to 3.

Figure 8.13 illustrates the optimum stacking sequence for the different values of the load parameter k. The solution corresponds to a [0/90]S with a variable proportion of 90° plies (depending on k). Figure 8.14 describes the design space for k = 4. The iso-values of both objective functions are drawn. The arrow indicates the direction for an increase of the stiffness. The optimal solution is characterized here by identical values of both objective functions.

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4

Evolution of the strain energy density

3.5

3

2.5

2

1.5

Laminate configuration for the several load cases

1

0.5

0

1

2

3

4

5

6

7

8

Load parameter k

Figure 8.13. Variation of the strain energy density and configuration of the corresponding optimal laminate.

Figure 8.14. Evolution of the strain energy densities for the [0/90]S laminate Subjected to ( N1 , N 2 , N 6 ) = (0,1000,0) N / mm and ( N1 , N 2 , N 6 ) = (1000,0,0) N / mm . t0° and t90° are the plies thickness.

The variation of the strain energy density for each single load case is illustrated in Figures 8.15 and 8.16. In those particular cases, the optimal solutions are given by only 90° or 0° orientations. This illustrates the need for a multi-objective formulation when several functions are considered as objective.

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Figure 8.15. Evolution of the strain energy density in the [0/90]S laminate subjected to N 2 = 1000 N / mm .

Figure 8.16. Evolution of the strain energy density in the [0/90]S laminate subjected to N1 = 1000 N / mm .

8.4. Optimal Design with Respect to Stiffness and Strength Restrictions In this application a stiffened laminated composite panel subjected to a uniform pressure is considered. The geometry, the boundary conditions and the stacking sequence of the different parts of the panel are illustrated in Figure 8.17. The plies thickness is equal to 0.125 mm and the base material is carbon/epoxy.

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laminate 1 :[(0/90/45/-45)2]S laminate 2 : [0/90/45/-45]4

Figure 8.17. Geometry and initial stacking sequence of the stiffened panel. GCMMA Relative compliance

Number of violations

Relative mass

3.5

2

3

1.8

50

2.5

1.6

40

2

1.4

30

1.5

1.2

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SAM Relative compliance 3 2.5

Number of violations

Relative mass

2

60

1.8

50

1.6

40

1.4

30

1.2

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1

10

2 1.5 1 0.5

0

5

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15

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15

0

0

Figure 8.18. Convergence history for GCMMA and SAM.

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The optimization problem consists in maximizing the structural stiffness for a given maximum weight, knowing that a safety margin of 0.15 on the Tsai-Hill criterion on the top and the bottom of each ply must be obtained at the solution. 64 strength restrictions are defined at the plies level. The design variables are the orientations of the plies initially oriented at 0, -45, +45 and 90 degrees and the related thicknesses. The problem includes 16 design variables. The convergence histories of GCMMA and SAM are compared in Figure 8.18. The SAM approximation succeeds in finding a solution in a very small number of iterations, with comparison to GCMMA. The optimal stacking sequence is illustrated in Figure 8.19. As already observed by Grenestedt (1990) and Foldager (1999), the optimal laminates include very few different orientations. laminate 1 :[90] 2.48 mm

laminate 2 : [0/93/96/93]4

Figure. 8.19. Optimal design of the stiffened panel.

8.5. Optimal Design under Buckling Considerations Anyone who has carried out optimal sizing with a buckling criterion has experienced an undesirable effect of very slow convergence speed and possibly large variations of the design functions during the iteration history. The reasons for the bad convergence of the buckling optimisation problem are multiple, and make it difficult to solve: discontinuous character of the problem due to the localized nature of local buckling, non differentiability of the eigenvalues and related problems in the sensitivity computation, modes crossing, selection of a right optimisation method, etc. A curved composite panel including 7 hat stiffeners is considered. The load case consists of a compression along the long curved sides, and in shear on the whole outline. The structure is simply supported on its edges. Bushing elements are used to fasten the stiffeners to the panel. In each super-stiffener (made of one stringer and the corresponding part of the whole panel), 3 design variables are used for defining the thickness of the 0°, 90° and ±45° plies in the panel and in the stiffener. 42 design variables are then defined. The goal is to find the structure of minimum weight with a minimum buckling load larger than 1.2. The results obtained in Bruyneel et al. (2007) are reported in Figures 8.20 for Conlin (Fleury and Braibant 1986) and SAM (Bruyneel 2006). The 12 first buckling loads are the design restrictions of the optimisation problem. In Figure 8.20, the evolutions of the weight and the

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first buckling load λ1 over the iterations are plotted, as well as some characteristic buckling modes.

Figure 8.20. Convergence history for the buckling optimisation with Conlin (left) and SAM (right) Bruyneel et al. (2007)

It is seen that when Conlin is used (Figure 8.20, left) a solution can not be reached. With SAM (Figure 8.20, right), the solution is obtained after an erratic convergence history. Those oscillations come from the fact that local buckling modes appear during the optimisation process, and some parts of the structures are no longer sensitive to this criterion. A small thickness is therefore assigned to those parts to decrease the weight, what makes them very sensitive to buckling at the next iteration, leading to oscillations of the design variables and functions values. It was observed in Bruyneel et al. (2007) that when a large number of buckling loads are used in the optimization problem (say 100 for the problem of Figure 8.20),

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a solution with SAM is reached in 6 iterations, while Conlin is still no longer able to converge.

8.6. Topology Optimization of Laminated Composite Structures The topology optimization problem of Figure 4.3 is here considered. In topology optimization of isotropic material (Bendsoe 1995), the design variable is a pseudo-density μi that varies between 0 and 1 in each finite element i (Figure 4.2). The so-called SIMP material law (Simply isotropic Material with Penalization) takes the following form:

Ei = μ ip E 0

ρi = μi ρ 0

(8.5)

where E0 and ρ0 are the Young modulus and the density of the base material (e.g. steel), E and ρ are the effective material properties, and p is the exponent of the SIMP law, chosen by the user (1
⎡Q11 p⎢ Q = μ i ⎢Q12 ⎢⎣Q16

Q12 Q22 Q26

Q16 ⎤ ⎡Q xx p⎢ ⎥ Q26 ⎥ = μ i ⎢Q yx ⎢⎣ 0 Q66 ⎥⎦

Q xy Q yy 0

0 ⎤ μ pE0 0 ⎥⎥ = i 1 −ν 2 Q ss ⎥⎦

⎡ ⎤ 0 ⎥ ⎢1 ν ⎢ν 1 0 ⎥ ⎢ 1 −ν ⎥ ⎢0 0 ⎥ 2 ⎦ ⎣

and the material stiffness matrix Q depends on the density design variable μ :

σ = Q(μ )ε

(8.6)

For orthotropic materials in a plane stress state, the stiffness in the material axes is given by the expression (3.1) where 4 material properties Ex, Ey, νxy and Gxy must be provided. For a material with orthotropic axes oriented at an angle θ with respect to the reference axes the material stiffness is given by (3.5). The SIMP parameterization (8.22 and 8.23) can be extended to a Simply Anisotropic Material with Penalization (Rion and Bruyneel 2006) , and the material law for topology optimization is now written as:

⎡Q11 p⎢ Q = μ i ⎢Q12 ⎢⎣Q16

Q12 Q22 Q26

σ = Q(μ , θ )ε

Q16 ⎤ Q26 ⎥⎥ Q66 ⎥⎦ (8.7)

The material stiffness now depends on both kinds of design variables, i.e. the material density and the fibers orientation.

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The problem in Figure 8.21 is solved with this parameterization. It includes 3750 design variables. The optimal topology and orientations obtained for an half of the structure are given in Figure 8.22. A comparison of the convergence speed for several approximations is provided in Figure 8.23.

? Figure 8.21. Definition of topology optimization problem. The initial structure is full of material.

Figure 8.22. Optimal topology with orthotropic material. Only one half of the structure is drawn. The fibers orientation is plotted in the few elements that contain full material at the solution

Figure 8.23. Convergence history for several approximation schemes for the topology optimization problem including orthotropic material.

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8.7. An Industrial Solution for the Pre-design of Composite Aircraft Boxes As reported in Krog et al. (2007), the pre-design of an aircraft wing is a large scale optimization problem including (up to now) about 1000 design variables and about 300000 constraints. Those variables are linked to the total thickness of the laminate made of 0, ±45 and 90° plies in the panel and to the dimensions of the cross section for the composite stiffener of each super-stringer defining the box structure (Figure 8.24). The constraints expressed as reserve factors (RF) are amongst others related to buckling and damage tolerance.

Figure 8.24. The principle of a composite wing made of super-stringers (from Krog et al., 2007).

Taking into account a so large number of design functions in the optimization problem will dramatically increase the CPU time spent in the optimizer. In order to decrease the size of the optimization problem, a technique for scanning the constraints (Figure 8.25) has been implemented in Boss Quattro (www.samcef.com). It consists in feeding the optimizer with the most critical constraints, based on their value at a given iteration. This leads to the definition of 2 sets of active and inactive constraints. The optimizer can only see the active restrictions. Those sets are not updated at each iteration but only when some inactive constraints tend to become violated after a given number of iterations (FREQ in Figure 8.25). When the SAM method (Bruyneel 2006) is used, the information at the previous design point is lost when the sets are updated, and the approximation is therefore only built based on the information at the current design point, that is with GCMMA (Svanberg 1995), for that specific iteration. The SAM approximation was found to be reliable in solving pre-design optimization problems of composite aircraft box structures in wings, center wing box, vertical and horizontal tail planes. Typically 30 iterations were enough to reach a stationary value of the weight and a nearly feasible design where very few constraints (less than 10) were still violated but of an amount of no more than 3 percents (RF larger than 0.97). Details of the results and of the implementation can be found in Krog et al. (2007).

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Figure 8.25. Strategy for scanning the constraints in large scale optimization problems (Krog et al. 2007).

8.8. Optimal Design with Respect to Damage Tolerance A simple DCB beam is considered (Figure 8.26). The energy release rates of modes I, II and III are computed at the straight crack front with a specific virtual crack extension method described by Bruyneel et al. (2006). The stacking sequence composed of 32 plies is given by: [θ/−θ/0/−θ/0/θ/θ/04/θ/0/−θ/0/−θ/θ/d/−θ/θ/0/θ/0/−θ/04/−θ/0/θ/0/θ/-θ] where d is the location of the interface where delamination will take place and θ is a variable. The goal is to find the optimal value of the orientation that will decrease the maximum value of GI along the crack front.

Figure 8.26. DCB beam and variation of GI along the crack front for the initial design. On the left the displacements of the lips are multiplied by 50.

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The solution is provided in Figure 8.27. The optimal value for the angle θ is zero. The convergence is achieved in 5 iterations with the SAM approximation and in 15 for MMA (Figure 8.28). Although the solution of this problem is trivial, the procedure could be used for more realistic structures subjected to several complex load cases.

Figure 8.27. DCB beam and variation of GI along the crack front for the optimal design. On the left the displacements of the lips are multiplied by 50

Figure 8.28. Convergence history for the optimization with respect to damage tolerance. SAM converges in 5 iterations while MMA needs 15 iterations to reach the solution

9. Conclusion In this chapter the optimal design of laminated composite structures was considered. After a review of the literature an optimization method specially devoted to composite structures was presented. This review helped us in selecting a formulation of the optimization problem that satisfies the industrial needs. In this context the fibers orientations and the ply thicknesses were selected as design variables. It was shown on the proposed applications that the developed solution procedure is general and reliable. It can be used for solving laminated composite problems including membrane, shells, solids, single and multiple load cases, in

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stiffness, buckling and strength based designs. It is routinely used in an (European) industrial context for the design of composite aircraft box structures located in the wings, the center wing box, and the vertical and horizontal tail plane. This approach is based on sequential convex programming and consists in replacing the original optimization problem by a sequence of approximated sub-problems. A very general and self adaptive approximation scheme is used. It can consider the particular structure of the mechanical responses of composites, which can be of a different nature when both fiber orientations and plies thickness are design variables.

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In: Composite Materials Research Progress Editor: Lucas P. Durand, pp. 109-128

ISBN: 1-60021-994-2 © 2008 Nova Science Publishers, Inc.

Chapter 3

MAJOR TRENDS IN POLYMERIC COMPOSITES TECHNOLOGY W.H. Zhong Department of Mechanical Engineering and Applied Mechanics North Dakota State University, Fargo, ND

R.G. Maguire Boeing 787 Program/Phantom Works, The Boeing Co. Seattle WA

S.S. Sangari Boeing Materials & Processes Technology, The Boeing Company, Seattle, WA

P.H. Wu Spirit Co., Wichita KS

Abstract Composites have been growing exponentially in technology and applications for decades. The world of aerospace has been one of the earliest and strongest proponents of advanced composites and the culmination of the recent advances in composite technology are realized in the Boeing Model 787 with over 50% by weight of composites, bringing the application of composites in large structures into a new age. This mostly-composite Boeing 787 has been credited with putting an end to the era of the all-metal airplane on new designs, and it is perhaps the most visible manifestation of the fact that composites are having a profound and growing effect on all sectors of society. It is generally well-known that composite materials are made of reinforcement fibers and matrix materials, and light weight and high mechanical properties are the primary benefits of a composite structure. Accordingly, the development trends in composite technology lie in 1) new material technology specifically for developing novel fibers and matrices, enhancing interfacial adhesion between fiber and matrix, hybridization and multi-functionalization, and 2) more reliable, high quality, rapid and low cost manufacturing technology. New reinforcement fiber technology including next generation carbon fibers and organic fibers with improved mechanical and physical properties, such as Spectra®, Dyneema®, and Zylon®, have been developing continuously. More significantly, various nanotechnology based novel fiber reinforcements have conspicuously and rapidly appeared in recent years. Matrix materials have become as complex as the fibers, satisfying increasing demands for

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W.H. Zhong, R.G. Maguire, S.S. Sangari et al. impact resistant and damage tolerant structure. Various means of accomplishing this have ranged from elastomeric/thermoplastic minor phases to discrete layers of toughened materials. Nano-modified polymeric matrices are mostly involved in the development trends for matrix polymer materials. Technology for enhancing the interfacial adhesion properties between the reinforcement and matrix for a composite to provide high stress-transfer ability is more critically demanded and the science of the interface is expanding. Fiber/matrix interfacial adhesion is vital for the application of the newly developed advanced reinforcement materials. Effective approaches to improving new and non-traditional treatment methods for better adhesion have just started to receive sufficient attention. Multi-functionality is also an important trend for advanced composites, in particular, utilizing nanotechnology developments in recent years to provide greater opportunities for forcing materials to play a more comprehensive role in the designs of the future. More reliable and low cost manufacturing technology has been pursued by industry and academic researchers and the traditional material forms are being replaced by those which support the growing need for high quality, rapid production rates and lower recurring costs. Major trends include the recognition of the value of resin infusion methods, automated thermoplastic processing which takes advantage of the unique advantages of that material class, and the value of moving away from dependence on the large and expensive autoclaves.

Introduction In sectors such as aerospace, wind energy, power transmission, marine, automotive and trucking, composites have been moving into the primary structure of wings, fuselages, chassis, hulls, and towers. In products such as sports goods and equipment, medical equipment, civil infrastructure, and dentistry composites are contributing to a market growth that will soon relegate homogeneous and isotropic materials to a niche category. The word “composite” is becoming synonymous with greater design flexibility and optimized materials utilization, leading to more opportunities for monolithic structural designs, less fasteners and holes, optimization of overall structural element architecture, improved fatigue and corrosion behavior, and high efficiency and maintainability. Composites are also particularly suitable for structural health monitoring systems with the associated advantages of reduced conservatism in designs. As they have evolved over the past several decades, composites now are spreading out and leaving their early material forms and traditional processes, and incorporating new constituents from nano particles to smart additives to hybridizing to capture the best of all technologies. This has led to lean and efficient automated processes that will enable these new developments to be cost-effective in production and performance-enhanced in products that serve us all. Over the past several decades polymeric composites have matured and evolved, sometimes fitfully, but much of the time in a steady development driven by the increasing awareness among industries of the values available from combinations of matrices and fibers. The world of aerospace has been one of the strongest proponents of advanced composites, eventually converging on a combination of carbon fibers and thermosetting materials as the preferred choice for the harsh environment and complex loading of aerostructures. Structural materials applied for airplane structures from metallic materials, to composites and then nanomodified composite materials are being developed, see Fig. 1.

Major Trends in Polymeric Composites Technology

ing as ls e r Inc ria rd Mate a g ow d T eerin n e Tr ngin ing of E u n nti lity Co pabi Ca Anisotropic Composite Materials

•Mechanical performance enhancements through alloying & heat treatments Isotropic Metal Materials

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Nano-Modified Composite Materials •Broad mechanical property improvements •Flammability and solvent resistance enhancements •Conductivity and CTE tailoring •Inherent color, optical qualities, etc. •Multi-functionality

•Fiber orientation optimizing effects on strength & stiffness •Laminate tailoring for coupled deformations •Independent toughness improvements through polymer alloying and controlling phase morphology Figure 1. Materials for airplane structures.

Composites have been applied in various areas including aerospace, automotives, renewable energy structures (e.g. windmill blades as shown in Fig. 2), marines, sports and construction. . The steps for pursuing composite materials with ultra-light weight, super mechanical properties and multi-functionalities have been developing fast, in particular, the nanotechnology speed up and provide revolutionary opportunities for the new trends of composite development, which can be summarized in the Fig. 3.

Figure 2. 38-meter European fiber glass windmill blade.

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Improve properties for FRP composites

Advanced materials technology

Reliable and low cost manufacturing technology

Reinforcement:

Interface:

Matrix:

a. New fibers: PBO, UHMWPE fibers, etc.

a. Nano-coating fibers (e.g., CNTs coated GF) Improve interfacial adhesion

Nano-filled Resin: (nanocomposites)

b. Nano-tech based: • 1D reinforcement: (i) Spun nanofibers (nanocomposites) (ii) Nano-scaled fibers (e.g. nano-PAN based CF) (iii) Ropes, yarns, bundles (e.g. CNT ropes, yarns)

b. Reactive nano-matrix (nanocomposites) (e.g. reactive nano-epoxy: improve wetting& adhesion)

a. Simple physical mixture of polymer and nano-fillers b. Integration of nano polymer systems (e.g. CNTs-epoxy, GNFsepoxy) c. multi-functionalization

• 2D reinforcement: Film, sheet, mat, etc. (e.g. Bucky paper, nano paper) c. Hybridization: e.g.ARALL

Figure 3. Routes of property enhancement for macro-composites.

With the success and proliferation of state-of-the-art composite materials in many areas including aerospace and renewable energy structures, there are new opportunities that are being considered now that the bar has been successfully raised with the 787. Some of these will be explored in this chapter. I.

Nanocomposites and Multifunctional Materials: carbon nanotubes (CNTs), nanofibers, and nanoplatelets have a natural affinity for polymeric materials and their inclusion in composites offers the promise of multi-functionality: electrical/thermal conductivity, acoustic damping and optical functionalities may be combined with load bearing capabilities. II. Hybridization: we see a new generation of metal/composite combinations deriving from early attempts such as GLARE®, ARALL®, titanium/graphite (TiGr), but utilizing new processes for the applications of monodisperse metallic coatings of high mechanical properties and durability, as well as multi-functionality. III. Alternatives to Carbon Fibers and Next Generation Carbon Fibers: as the understanding of the shortcomings of organic fibers such as UHMWPE increases, new approaches to interface enhancements are being developed which offer the

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benefit of a new supply chain for highly capable structural fibers. New generations of continuous carbon fibers are also being developed both on the micro and nano scales with improved performance and functionality. IV. Processing Technologies: the autoclave has been the mainstay for many years, but growing trends toward lean manufacturing make this a roadblock to improved efficiency. Continuous and inexpensive processing is making the curing step a part of the overall lean philosophy in composite manufacturing. A growing trend is the move away from prepregs to the family of materials and processing called resin infusion (RI). This includes Resin Transfer Molding (RTM), Vacuum-Assisted Resin Transfer Molding (VARTM), Resin Film Infusion (RFI), and the development of new continuous processes. All this is based on low-cost material forms of neat resin and fiber preforms that allow the manufacturer to put the two together in proprietary and efficient ways. V. Other trends include a new generation of thermoplastics being developed to serve the growing automation of TP parts, smart materials and structures which can de-couple requirements and reduce weight, low-cost carbon fibers from bio sources, and others.

1. Nanocomposites and Multifunctional Materials The definition of nanocomposites covers a variety of systems such as one-dimensional, twodimensional and, three-dimensional materials made of distinctly dissimilar components and mixed at the nanometer scale for achieving drastically enhanced properties. To obtain multifunctionality in nanocomposites, nanoparticles with high aspect ratio have been successfully employed. This denotes being functional in one property while either achieving new properties that are unknown in the individual components or improving and maintaining other intrinsic properties. Nanocomposites can be used as matrices for nano/macro-composites, or traditional composites when micro-scale fibers are included. Nano constituent composites have also been made into nanoscale fibers through spinning methods. To date, the creation of such new materials has resulted in: • • • • • • • • • •

Enhanced mechanical properties: strength, stiffness; toughness, impact resistance, structural durability, etc. Improved electrical conductivity Improved thermal conductivity and thermal management Improved flame resistance, thermal stability and increased service temperatures Enhanced acoustic damping Improved dimensional stability (low or tailored coefficient of thermal expansion) Enhanced tribological properties (wear, abrasion resistance, hardness) Improved barrier properties and environmental controls Decreased permeability Reduced shrinkage

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In many cases the product is a multifunctional material that has several of the above characteristics and properties in combination. Nanocomposites can improve many kinds of functionalities and typically can result in multi-functionalities. These functionalities include: This can be extremely attractive for engineered applications where costs of testing, evaluations, qualifications and certifications of each new material are high, and sometimes prohibitive. The prospect of a single material for multiple functions can therefore be very compelling, both for engineered performances and the non-recurring costs associated with bringing the material to readiness for the designer.

Nano-fillers: CNTs, CNFs, etc.

As loose fillers:

Combined further:

Nano-fillers + polymers --> Nanocomposites …

1D: ropes, yarns, bundles 2D: sheet/paper forms

+Matrix

Bulk form: Enhanced functionalities: - electrical - thermal - flame resistant - damping - mechanical …

Spinning fibers: 1D ? 2D sheet/paper Enhanced functionalities: - electrical - thermal - flame resistant - damping - mechanical…

FRP composites: (Hybrid composites)

+Matrix

Enhanced functionalities: - electrical - thermal - flame resistant - damping - mechanical… …

+ Fiber

Figure 4. Nano-scale materials in different forms for applications in composites.

Nano-sized materials as nano-fillers for nanocomposites can be classified into three categories according to the shape: particles (metals, metal compounds, organic and inorganic particles), fibrous materials (nanotubes, nanofibers, nanowires), and layered materials

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(graphene platelets, clay). These nano-fillers have exceptionally high specific surface areas so the overall amount of interfacial area is enormous if the nano-fillers are adequately dispersed within the matrix. This can result in creating various functionalities including mechanical, physical and other classes of properties. The large specific surface areas are highly desirable for stress transfer between the nano-fillers and the matrix, as well as providing increased chemical reactivity and energy levels compared to conventional bulk materials. Almost all nano-scale materials can be used as loose fillers for making nanocomposites. Fibrous nanofillers can be made into yarns/bundles, mats, braids, sheets/papers, which could be used in fiber reinforced polymer (FRP) composites. Nanocomposites can be applied in various forms such as coatings, films/sheets, spinning fibers, bulk materials as well as matrices for FRP composites due to the nano-fillers’ dramatic capability in enhancing functionalities for the polymer materials. When these nano-fillers or nanocomposites are used in fiber reinforced polymer (FRP) composites they become hybrid composite materials, which may exhibit multifunctional properties as illustrated in Fig. 4. Nanocomposites and hybrid composites with various functionalities have vast applications in structural applications in aircraft, space vehicles and renewable energy assemblies; impact protection systems; thermal management components; fuel cells; electronic devices, sensors, actuators, various functional coatings, electrostatic dissipation (ESD) and electromagnetic interference (EMI) radiation protection, lightning strike protection, etc. It has been established that improvements in the properties of nanocomposites are strongly affected by many factors including nano-filler size distribution, shape, aspect ratio, concentration, degree of dispersion, characteristics of the matrix, interactions between the filler and the matrix, and interfaces between the nano-particles themselves. According to the potential functionalities, nano-scale fillers can also be divided into (1) carbon types, such as CNT, carbon nanofibers (CNF, VGCF, or GNF), and graphite nanoplatelets (GNP), and (2) non-carbon types, such as nano-clay, POSS, nano-silica, metal nanoparticles and nano metal oxide particles. Nanocomposites with carbon type nano-fillers are mainly utilized for improvements of damping, mechanical, electrical and thermal properties. Nanocomposites with non-carbon type nano-fillers are predominantly used for flame retardency, improved barrier property, creep resistant, tribological properties and to some extent in early works, mechanical property enhancements. Nanocomposites have been undergoing rapid developments and significant progress has been made in the fields of nanocomposites and nanocomposite multi-functionalities over the past few decades. However, there are an abundant amount of questions and challenges left to be solved before taking full advantage of nano-scale fillers for development of stable, highquality nanocomposites. These include types, purity levels and polymer types (thermoset/thermoplastic), structure characteristics, viscosity at room and/or elevated temperature, appropriate treatment methods to be applied to the nano-fillers which will affect the interaction between the nano-fillers and polymer matrix, etc. In order to create a blend with controlled ratios of components and a well-dispersed nano-filler into the polymer matrix effective mixing methods and processing parameters should be understood and applied. Only when a complete understanding of these issues is established will the performance of nanocomposites with desired properties/functionalities be fully realized. Although many researchers have conducted remarkably successful experiments for achieving high performance nanocomposites, and obtained many encouraging empirical

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results, there is still a critical lack of comprehensive mathematical modeling that is needed to be used to make effective predictions for processing-structure-property relationship or when evaluating the multi-functionalities. An example is the wide array of electrical conductivity percolation threshold values for certain nano-fillers (e.g. CNT or CNF), when combined with the goal of improved strength and modulus. Modeling can increase the speed of selection and reduce the scale of actual testing, a huge advantage for bringing new products quickly to market. Mathematical modeling also provides practical benefits to industry in developing modeling capabilities for designing new materials. With the added dimensions of the nanocomposites options, conventional testing and down-selection for choices between the various and numerous materials can quickly become unfeasible. Nanostructures have unique physical and chemical properties different from bulk materials of the same chemical composition. The mechanical, electrical, thermal and magnetic properties of composites consisting of an insulating matrix and dispersed nanoparticles have been extensively studied over the past few decades. The significant progress in the understanding of nanocomposite systems within recent years has shown that multifunctional nanocomposites offer both great potential and great challenges, marking it as a highly active field of research. The research is continuing at an increasing pace, as the requirements for stronger and lighter materials are needed by a variety of industries. However, much research effort is continuing toward the development of new processing techniques that control the purity and dispersions of nanoparticles in the polymers.

2. Hybridization As composites develop and improve, and their applications grow, there will be inevitably, the realization that there are limitations and compromises in changing from metals to “nonmetals”. In many cases, the logical thought process is to consider how the best of both materials can be in included in hybrids. One of the major subsets of this segment of materials is based on the concept of going with one of the composites most valuable characteristics, that is, lamination of individual plies. In some cases the concept that comes most readily to mind is the replacement of one or more composite plies with metal foil or sheet and these are generally referred to as Fiber Metal laminates or FMLs. FMLs were first developed at Delft University in the Netherlands in the early 1980s, and marketed by Aluminum Company of America (Alcoa), combining sheets of aluminum in an alternating pattern with plies of traditional composite. As a result of this hybridization FMLs can theoretically combine the best of both the metal and the composite materials. The first FML was ARALL® (ARamid-ALuminum Laminate), a combination of aluminum and aramid/epoxy. This fiber-aluminum adhesive-bonded laminate is a super-hybrid composite material, which has many attractive properties such as good damage tolerance property, very high fatigue crack growth resistance, and high static strength along the fiber direction. The characteristics of ARALL® also include low density and resistance to the effects of temperature, humidity and acidity/alkalinity, etc. But greater applications became apparent if the aramid fiber composites were replaced by the ubiquitous glass fibers composites. In the 1980s, Delft developed a glass/epoxy FML called GLARE (GLAss-REinforced) composed of thin layers of aluminum sheet or foil interspersed with layers of fiberglass composite prepreg. The pre-preg layers may be aligned

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in various directions to accommodate the loading and because of this characteristic, it is truly a composite laminate with tailorable in-plane properties, and with the capability to add the aluminum plies in various locations and arrangements through the stack up, but with processing properties similar to bulk aluminum sheet metal. Its major advantages over conventional aluminum are lower density, better fatigue resistance (cracks are inhibited in growth due to the restraining effects of the adjacent composite plies) and better resistance to impact. An important consideration is the matching of the cure temperature of the composite with the thermal effects on the metal. For example, if the composite requires a 350°F degree cure, the aluminum alloy must be able to sustain its performace after exposure to that temperature because it is the total laminate that must go through the autoclave process. The original GLARE® used 250°F curing fiberglass composite, with an appropriate aluminum alloy, but in situations where a higher Tg is required for design thermal exposure, a 350F version was created and has been dubbed “New GLARE”. Another interesting version of FML is titanium-graphite (TiGr) laminates which consist of layers of titanium interleafed through the thickness of a Carbon Fiber Reinforced Plastic (CRFP) laminate. TiGr offers advantages over metallic structures in terms of weight, fatigue characteristics, damage tolerance, and design flexibility, and also advantages over traditional composite materials through higher bearing capabilities, greater toughness, and an expanded design space. There has been extensive testing in the industry to support development of mechanical properties for TiGr with various epoxy prepreg systems and success in optimization of surface preparation has resulted in extremely robust and environmentally durable TiGr. Production-related issues such as scale-up, compound contours, drilling and trimming, NDI, repair methods and even automation have been addressed for feasibility and optimization. These extreme hybrid composites have great potential for use in many applications including aircrafts. However, due to the large difference in thermal expansion coefficient of both the fiber and metal, and the anisotropy of the composites layers combined with isotropic metals, large residual stresses can be built up during the curing cycle which could cause an unsuitable residual stress system that may seriously hinder its outstanding performance. To regulate the residual stresses with controlled layups is necessary. In addition, there are many factors which influence the performance of these FMLs due to three kinds of material constituents involved, and the two interfaces: fiber/resin and metal/resin. To control the quality of the FML materials is critically important for the applications in practice. Other types of hybrids include co-mingling of different fibers for multifunctionality of the composite ply. Co-mingling of carbon fibers and glass fibers can add a softness to a laminate in places that need dimensional flexibility. Co-mingling of carbon and thermoplastic fibers can add toughness to strength and stiffness in a part. In some commercial products, such as the Cytec Priform technology, the thermoplastic actually melts during the cure and disperses in a controlled way throughout the matrix to form a minor phase of toughening material. A new trend is the coating of polymeric composites with metals for multiple purposes ranging from electrical conductivity to thermal management to surface hardness. MesoScribe Technologies has a Direct Write Thermal Spray in which fine powders are injected into a small thermal plasma and accelerated through an aperture to make patterns on composites without a cure cycle or masking, and is adaptable to large and highly contoured surfaces.

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Integran has its Nanovar technology in which finely grained metal is applied to a composite surface to increase wear in leading edges and for Invar tool repair. The reduced grain size leads to increased hardness and strength by the inverse relationship to the square root of the grain size. Other methods are in development to apply monodisperse metallic coating to composites for weight reduction, improved surface finish, alternatives to environmentally unacceptable coatings, and greater design freedom. In the nano-scale composites, hybridization of nanoparticles offers the potential of a rich soup in which some additives e.g. CNTs, are added for mechanical and electrical properties, some, e.g. POSS (Polyhedral Oligomeric Silsesquioxane) or nanoclay can be added for flame resistance, and others, e.g. nano gold particles, can be added for color. As we become more fluent in the use of new materials and less prejudiced in the use of older materials hybridization will become a natural trend for those who want it all.

3. Alternatives to Carbon Fibers and Next Generation Carbon Fibers Alternatives to Carbon Fibers With the growth of applications of composites using the high specific strength and stiffness of carbon fibers, comes also the growing demand for those fibers in many sectors of industry: aerospace, energy, transportation, infrastructure, medical, sports equipment, etc. There is also the issue of galvanic corrosion in some applications where carbon forms one of the three elements of a circuit with metals like aluminum, and water. As a result there is a growing interest in non-carbon fibers, many of which have been looked at in the past, but now being considered with a hungrier eye, if some of their shortcomings can be overcome without sacrificing their benefits, especially in tensile strength. One example is liquid crystal polymers (LCP). Celanese developed a thermotropic polyester-polyarylate LCP in the mid 1970’s and commercialized the Vectra family of resins in 1985. Vectran® is an example of an LCP fiber. The molecules of the liquid crystal polymer are rigid and position themselves into randomly oriented domains. The polymer exhibits anisotropic behavior in the melt state, leading to the term thus the term “liquid crystal polymer.” The molten polymer is extruded through spinneret holes and the molecules align parallel to each other along the fiber axis. The highly oriented fiber structure results in high tensile properties. Vectran differs from other high-performance non-carbon fibers, aramid and ultra-high molecular weight polyethylene (UHMWPE), in that is thermotropic, melt-spun, and melts at a high temperature. Aramid fiber is lyotropic, solvent-spun and does not melt at high temperature. UHMWPE fiber is gelspun and melts at a relatively low temperature. In all these fibers the high modulus/high tensile strength is achieved through the oriented linear molecules called microfibrils. And all these fibers have an order of magnitude lower compression strength than tensile strength. In general, organic fibers, such as aramid fiber (e.g. Kevlar®), ultra high molecular weight polyethylene (UHMWPE) fiber (e.g. Spectra®), and Poly(p-phenylene-2,6benzobisoxazole) (PBO) fiber (e.g. Zylon®), have excellent mechanical and physical properties. Kevlar® provides excellent impact resistance and is one of the lightest structural

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fibers available on the market today, which has been widely used both in soft body armor applications, and as reinforcement for hard armor, helmets and electronic housing protection. UHWMPE fibers, such as Spectra® and Dyneema®, are a type of ultra lightweight, highstrength polyethylene fibers. High damage tolerance, non-conductivity and flexibility, a much higher specific strength and modulus and energy-to-break, low moisture sensitivity, and good UV resistance, all make this fiber a good aramid alternative. These fibers are typically used in ballistic and high impact composite applications. Zylon® consists of a rigid chain of molecules of poly(p-phenylene-2,6-benzobisoxazole, PBO. It has excellent tensile strength and modulus. Fabrics made from Zylon ® are found in both ballistic and composite applications. In addition to the attractive mechanical properties of organic fibers, albeit limited in their current form, it should be noted that they are highly valuable by the key industries of the U.S. and offer the significant advantage in the avoidance of galvanic corrosion with aluminum and certain other metals. With organic fiber composites, the corrosion threat is avoided and the cost and weight benefits would be enormous to the aerospace and other industries if other critical properties could be enhanced. In addition, there is the economic challenge now of the growing applications for carbon fibers and the resultant shortages of carbon composite materials, which may impact the economy of the US. Engineering conferences for the past two years have focused on that increasing shortage and what technical and scientific alternatives are available. How to make organic fibers a viable alternative to carbon fibers for structural applications is often discussed in these forums within the genre of nanocomposites. This category of research holds great promise to enable that technology sector and can accelerate the fulfillment of the promise of multi-functional materials. Typical characteristics of these organic fibers include non-polar chemical structures and crystalline chains. It is these structural characteristics that impart the advanced mechanical properties on to the fibers. For example, UHMWPE fiber obtains its high strength from the straightening of long polymer chains by taking advantage of the strong covalent bonds in the backbone of the monomer. The modulus of the fiber is proportional to the draw ratio which controls the degree of crystallinity. The main benefits of a UHMWPE continuous fiber include high specific strength and moduli, leading to a lower weight for a given design load. The chemical neutrality of the fiber surface leads to a high degree of corrosion resistance; there are no places to allow for a concentrated attack on the surface. In addition, the anisotropic nature of the fiber allows for low coefficients of thermal expansion, meaning dimensional stability of the finished composite product. However, the non-polar chemical structure and resulting lack of reactive groups on the organic fiber surface lead to low surface energy, and thus leads to difficulty in obtaining good wetting and adhesion at the fiber/matrix interface. This low surface energy requires that the matrix material be of an even lower energy to achieve sufficient wetting and adhesion, ultimately realizing strong bond at the fiber/matrix interface. This results in the limited applications of the organic fibers because many properties of the composites are determined by the transfer ability of the fiber/matrix interface. To tackle the problem, various surface treatments to improve the interfacial wetting and adhesion, are applied. There appears to be an absence of a good means to alter the fiber without sacrificing its desirable properties. It is concluded that novel cost-effective methods for improving the interfacial adhesion between the organic fibers and the polymer matrix are vital to the full realization of their potential as structural materials.

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For aramid and UHMWPE fibers, silane coupling treatments (effective for glass fibers) and oxidation treatments (effective for carbon fibers) are not effective in improving the interfacial strength [1-6]. For UHMWPE fibers, many treatment methods including nitrogen ion implantation, nitrogen plasma, fast atom beams, laser ablation, chain disentanglement, high power ion beam treatments, and cold plasma [7-13] have been used. Such approaches show some improvement in interfacial properties, but also can degrade the mechanical properties of the fibers by damaging the chain structure of the UHMWPE fiber and result in formation of amorphous hydrogenated carbon. Recently, the effectiveness of an atmospheric plasma was demonstrated for dramatic improvement in the adhesion of polyetheretherketone (PEEK) composites to epoxy [14]. This atmospheric plasma was shown to be an important potential strategy for improving the interfacial adhesion between organic fibers and polymer resins. A nanotechnology approach was recently developed by Dr. Zhong’s group in which conventional epoxy resins are converted into reactive nano-epoxy resins. Unlike the conventional epoxy resins that require carbon fibers to be surface oxidized/treated before being impregnated, the nano-epoxy resins contain reactive graphite nanofibers which can improve the wettability and adhesion properties between UHMWPE fibers and the resin matrix [15-19]. Next Generation Carbon Fibers – Continuous Nanoscale Carbon Fibers Traditional Carbon fibers have high strength, high modulus and attractively low density. The high strength-toweight ratio combined with superior stiffness has made carbon fibers the material of choice for high performance composite structures in the aerospace, defense and other industries. Polymer fibers, which leave a carbon residue and do not melt upon pyrolysis in an inert atmosphere, are generally considered candidates for carbon fiber production. It is known that the structural perfection of precursor fibers is the most crucial factors on the strength of carbon fibers. Imperfections (such as surface defects, bulk defects and others) in the precursor fibers are likely to be translated to the resulting carbon fibers, and the amounts and sizes of structural imperfections directly determine the final fiber strength. The fundamental approach/solution for improving the strength of carbon fibers is to reduce the amounts and sizes of numerous types of defects in the precursor and there has been a clear and continuing trend among commercial carbon fiber suppliers in achieving higher strengths through this approach. There is also a growing interest in having thinner plies of composite material without loss of strength or stiffness and thinner plies means thinner fibers. Historically the means of producing very small diameter (down to submicron range) has been electrospinning. Since the 1930s electrospinning has been used on nylon and other polymers to achieve small fibers to provide filtering media and other applications. In the process a strong electric field acts on a polymer solution resulting in a polymer stream which solidifies through the evaporation of the solvent. This can also be applied to a polyacrylonitrile (PAN) copolymer (precursor) to produce nanofibers with diameters in the nanoscale range with the potential of ultimately producing continuous nano-scaled carbon fibers with strengths and stiffness much higher than conventional micro scale carbon fibers. Additionally, since diameters of the electrospun PAN nanofibers can be further reduced by stretching and carbonization processes, the resulting nano-scaled carbon fibers can have diameters of less than 100 nm. When incorporated into a

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matrix composite this could yield a prepreg ply about two orders of magnitude less thickness with the same strength and stiffness as conventional prepreg offering benefits in weight critical structure.

4. Processing Technologies Out-of-Autoclave Processing: Although the autoclave has served the composite industry well providing structural integrity as well as thermal curing, there is a growing demand for a leaner means of curing parts. Being able to cure parts in a continuous stream like a pizza oven is the Lean Manufacturing teams dream. But even a common oven can provide leaner flow. The economic advantage of an oven process for structural composites is large. Autoclaves are an order of magnitude more expensive than ovens with the same temperature uniformity. The process flow and batch constraints of autoclaves can potentially be eliminated with ovens. And liquid nitrogen systems used to prevent fires would not be needed. Also there is the possibility of part growth that may limit the autoclave usefulness. Such an example is seen in the windmill blades in which designs are growing faster than an autoclave can be depreciated. Blades of necessity have been hot bag cured for many years and as they surpassed the 38-meter length the designs have been using carbon fiber in place of glass fiber composites. There is also the issue of large-scale complex parts such as racing sailboat hulls that would require an autoclave of the scale that not many groups can afford. These also have been “cooked in a tent” for many years. In the past there have been attempts to utilize UV curing, e-beam curing, X-ray curing, oven curing, tent curing, hot press curing, continuous pultrusion curing, many forms of resin infusion, etc., all with various degrees of success or the lack of significant applications. The key to success is the material. If they can be developed to have no voids in the uncured laminate and no volatile components in the resin system then the pressure element becomes moot. The concerns include matching the fiber volume associated with autoclave cures and processing variable that can impact design allowables. One of the noted successes in this trend is oven cured epoxy carbon fiber systems approved by the FAA for structural applications on civil aircraft (Agate Program allowables database is FAA approved) [20]. Some of these systems are improved with high (>30 in Hg). Generally if a void free uncured laminate can be achieved either through hot debulking, ultrasonic compaction or other means, vacuum bag pressure in an oven will be sufficient to achieve a structural part. There are many new methods that involve either single or double diaphragms to hold the part while being vacuum-cured. One of these is the double diaphragm resin infusion process RIDFT of Florida State University High Performance Materials Institute which should offer lower tooling costs and shorter cycle times. Another novel approach is that of Quickstep® developed jointly with the Australian research organization CSIRO, it is based on a liquid filled container in which a lightweight mold floats on one of the flexible faces of the pressure chambers [21]. The container is filled with a heat transfer fluid which is circulated through the chamber to rapidly heat and cool the mold. The process enables the cure cycle to be stopped and restarted at any time and parts of the laminate to be left uncured. Parts can be consolidated and formed and then final cured in-situ at a later time. It is being developed further with several universities and the National center for Composites.

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In-situ compaction and curing is another approach that will have great value for lean and rapid production of composites using unidirectional composite prepreg tape or tow forms. The uni-prepreg is wound onto a mandrel and heat applied from several IR sources to the material as it is placed. There is also work being done at NASA on E-Beam in-situ curing using low energy processing [22]. Resin Infusion Processes: Resin infusion although it is a generally an out-of-autoclave set of processes, is itself a highly significant trend in composites processing. It covers many processes such as RTM, VARTM, RFI and each of these major subsections has many variations (including some which go back into the autoclave), but the common factor among all of these is that the resins and the fibers are marketed separately and the customer fulfills the impregnation process within their own manufacturing planning. This can offer opportunity for customization and optimization for specific user needs and as well a significant cost savings in materials since the costly impregnation process is now done inhouse. Several of the major composite material suppliers forecast growing markets for the infusion resins and dry fiber preforms as compared to the more historical prepreg material forms. Resin infusion is a closed low pressure process for the manufacture of complex-shape, high-strength and lightweight composite parts for a wide range of aerospace, automotive, marine and satellite applications. In this process, a resin system is drawn into a dry fiber laminate in a mold where it can cure to form the finished part. RTM is an infusion process that employs an injection system to transfer a mixture of liquid resin and catalyst into a closed mold containing a preform, which is preset fiber mats. The resin is injected under a controlled pressure by a carefully designed pattern of inlet ports and vent holes. This guarantees that the fibers become fully wet and produce a low void content and high fiber-volume composite part. Fiber volumes approaching the 65% values, which is typical for prepreg lay-up techniques, have been achieved with RTM. Subsequent curing of the resin forms a net-shape part with good dimensional tolerances. The size and complexity of the part significantly influences the cycle times. VARTM is an adaptation of the RTM process and is generally used to manufacture parts for marine, ground transportation and infrastructure applications. The process uses an open mold cavity which is laid up with a preform and covered with a vacuum bag made of air impervious films such as nylon or silicone film. The air is expelled from the preform assembly using a vacuum pump. A liquid resin is allowed to infuse into the mold from an external reservoir after all air leaks are eliminated and the system is equilibrated. A high permeability resin distribution medium is placed on top of the preform to facilitate the resin flow over the lateral extent of the part. The system is kept under vacuum until the resin is completely gelled. The part may then be cured at room temperature or in an oven. Bag leaks and bridging are common problems in VARTM. Bag leaks take place at the sealant-bag-film interface or as a result of film failure due to improper handling. Bridging is the failure of the sealing bag to conform to the shape of the mold. This leads to the part failing to receive uniform pressure during the cure cycle. Matrix viscosity and process time are the two main differences between RTM and VARTM. For RTM, the resin must travel through the "X" and "Y" directions while for VARTM it travels on top and only needs to impregnate the "T" or "Z" direction. This requires

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shorter time and provides the advantage of the lower temperature and faster cure time combined with reduced thermal stress. Resin film infusion, RFI, is a composite manufacturing process which has advanced from earlier work on vacuum impregnation and RTM. In this process, a semi-cured resin film is liquefied and absorbed throughout the fiber. The mold filling is further assisted by vacuum to reduce the air voids remaining in the fabricated part. The resin and the fiber are generally placed together into the mold but are not initially combined. In some applications the fiber and the resin are placed in the mold in separate steps and are combined by applying pressure. Computer simulations are commonly used to determine processing details such as resin viscosity, preform permeability, resin/preform interactions, and the time to completely cure the composite part. The major difference between RFI and RTM is that former uses a hot melt resin film while the later utilizes a liquid resin. RFI does not require low minimum viscosity as in the RTM process. Orthophthalic, isophthalic polyesters and vinyl esters are primarily used in RTM processes. A variety of polymers are being developed specifically for RTM application, e.g. low-shrink and low-profile polyesters for improved surface appearance. New resins including epoxies, acrylic/polyester hybrids, urethanes, bismaleimides (BMI), and phenolic resins are also produced which require changes in the equipment and conditioning the resin prior to injection. These systems offer a whole new range of cost and performance options to the RTM process. Reinforcements used in RTM are normally glass fibers, continuous fiber mats and chopped strand preforms. Special mats that contain thermoplastic binders are heated and thermoformed into perfect preforms. Both woven and non-woven glass fibers and biaxial and triaxial mats have been produced for the RTM applications. Other high performance reinforcements such as carbon fiber and aramid can be incorporated in RTM laminates either alone or as part of a hybrid system. A typical RTM mold features oil connections, injection runner system, and self clamping/load devices. RTM surfaces offer high quality through a combination of appropriate resin reinforcements, molds and process conditions. A combination of mechanical clamping arrangements with special presses is necessary to secure the mold halves. This is required for applying pressure uniformly to the mold when pressurized resin is employed. RTM processing requires accurate and reliable injection of liquid resin. Resins must balance low viscosity at processing temperatures and long pot life without sacrificing the mechanical properties to alter the flow characteristics. The resin is injected until the mold is completely filled. Motionless or non-mechanical mixers are normally utilized to blend resin and catalyst. After curing for a required time the composite part is moved from the mold. A mixer flush system is also incorporated when required to purge non-disposable mixers. RTM is employed to reduce fill times and to fabricate large-scale composite structures with substantial laminate thicknesses. It fills the gap between hand lay-up and compression molding of sheet or bulk moldings in matched metal molds. In comparison with lay-up and spray-up processes RTM provides two finished surfaces on parts which can be similar or dissimilar, highly reproducible thickness, low monomer loss and higher output since it is less labor and material intensive. Compared with matched-metal-die compression molding, RTM enables the use of parts such as ribs and inserts, decreases lead times for molds, and has lower-cost molds and molding equipment.

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RTM provides a lot of processing flexibility. Shrink control systems can be employed to produce improved surfaces. The fibers and resins are utilized at their lowest cost plus resin content can be controlled to a significant degree and reinforcements can be easily incorporated. Cores and other insets can also be positioned prior to resin injection to yield complex parts in one fabrication step. RTM can also be used to prototype parts for market evaluation since the initial investment costs such as tooling and operating expenses are low. In this case, RTM's short lead time and lower cost tooling is a real advantage. It also offers production of near-net-shape parts which in turn leads to low material wastage. Closed RTM molds release fewer volatile materials than open molds. Added benefits of this closed-mold process are greatly reduced volatile organic compound (VOC) emissions. VARTM can be used at room temperature with no heated mold and compared to RTM more consistent and uniform coverage can be obtained. VARTM provides significant savings in the tooling cost as it requires only a one-piece mold. The use of the vacuum bag eliminates the need for making a precise matched metal mold as in the conventional RTM process and thus reduces the cost and design difficulties associated with large metal tools. VARTM also beats the 5-10% accuracy of hand lay-up. For a given resin, mold filling using RFI is relatively faster than most other liquidcomposite-molding processes since the RFI products are usually thin and the infusion distance is short. In addition, RFI can provide parts with high mechanical properties due to solid state of initial polymer material and elevated cure temperature. Along with the advantages, RTM also has inevitable disadvantages including inability to manufacture very large parts and permeability issues which lead to increased processing time. It is an intermediate volume production process and its tooling and equipment costs are higher than the hand lay-up and spray-up. Tooling design and fabrication to handle injection pressures, clamping and sealing are more complex and manufacture of complex parts require trial and error experimentations combined with flow simulation modeling. VARTM also is a relatively complex process to perform well. The flexible nature of the vacuum bag brings about the difficulty of controlling the final thickness of the preform, and hence the fiber volume fraction of the composite. Due to the complex nature of VARTM, the trial and error experimentation is not only inefficient but also expensive for the process design and optimization. RTM is being used for a number of products including aircraft components, recreational vehicle, truck and sports car bodies, automobile panels, medical equipment, dish antenna, storage tanks, electrical covers, windmill blades, plumbing parts, transportation seating, chemical pumps, marine components such as hatches and small boats, bicycle frames and doors. Resin Infusion processes enable us to manufacture a wide range of complex as well as simple fiber reinforced composite products. Improvements are being made on resins and tooling developments which further expand market opportunities. The combination of pre-positioning a variety of reinforcements and incorporating secondary reinforcements and other design details with good surface control on both sides makes RTM a primary process candidate for structural applications in the aerospace industry as well as other markets. Due to its versatility, growing use in industry, and being environmentally friendly, experts believe that the future of this useful composite process should continue to grow. In many applications its replacement of hand layup/ prepreg material forms/autoclave processing production has

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resulted in significant cost and rate benefits, further enhanced with the possibility of ganginfusion production scenarios and automated preform production and handling. Advanced Thermoplastic Composite Processing: Thermoplastics are having a resurgence of interest, largely due to a growing list of successful new processing methods. Reinforced Thermoplastic Laminates (RTL) is an economical means of producing a solid laminate if there is no constraint against constant thickness. The pre-consolidated laminate is heated above the melt temperature, usually by infrared lamps, and then automatically transferred into a pair of cool tools that rapidly close to form and cool the part. This achieved very rapid cycle time. Another process that has been growing in applications is the automated thermoplastic pultrusion process that can produce high volumes of long straight parts of various shapes [23]. These and other thermoforming processes are bringing new life to the applications of thermoplastics in significant structural applications that promise to realize the attractions of short cycle times and the possibility of recycling.

5. Other Trends Smart Composites: Smart materials have the ability to perform both sensing and actuation functions. The use of imbedded sensors such as piezoelectric, shape-memory alloys, magneto-strictive, or fiber optics with Bragg gratings (FOBG) to sense and mitigate the threats to the health of a structure, i.e. Structural Health Monitoring (SHM), holds great promise for the future of composite primary structure through the elimination of designed-in excess material for undetected damage events; being aware of damage when it happens and where it happens can eliminate much design conservatism. Other possibilities are the incorporation of self-healing or restorative abilities, active control of key functions such as vibration, etc. Smart composites face the challenges of effective dispersion and interfacial adhesion of the “smart” constituents. Smart composite materials can be obtained by mixing the polymer matrix with smart material used for health monitoring, active control and selfrestoration of structural and functional materials. Recent advances in optical glass fibers have produced a form which has the approximate same diameter as a carbon fiber so can be incorporated into a tape or fabric reinforcement without disruption of the load carrying capability. Bio-based composites: Increasing interest is developing in bio-based composite constituents. With shortages developing for the traditional petroleum based products, there are activities in US, China, Singapore and elsewhere to develop carbon fibers from renewable agricultural sources such as corn, soy, rice, wheat and other biomaterials that do not deplete the petroleum reserves. To date the efforts are still in their early stages of success, the quality is not that of the PAN or pitch based fibers but the costs are very attractive and the growing interest in greener processing will add impetus to these activities, particularly for those applications where lower performance is not critical and which are suffering from the current carbon fiber shortage. University of Delaware Affordable Composites from Renewable Resources (ACRES) program is one source of development in this area, having been awarded a USDA National Research Initiative to investigate the possibility of making circuit boards from soy resins and chicken feather based carbon fibers rather than the conventional epoxy, PAN-based

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composites. Michigan State University has a center for biocomposites with many projects underway on bio composites, green nanocomposites, biodegradeable thermoplastic polymers and soy based bioplastics among others [24]. Short fiber composites: Originally short fibers in composites were very short, basically just additives with aspect ratios only slightly greater than one. Subsequently, the long discontinuous fiber (LDF) composites appeared with fibers either chopped, stretch broken, or otherwise made discontinuous. These LDFs could approach the performance of continuous fiber composites with fiber lengths of 2” to 4” and some degree of alignment. For applications with complex geometry, these materials can offer relief from the limitations of continuous fiber prepregs difficulties in conforming to bends and reentrant shapes. In many cases these materials and processes can replace traditional metals in parts with complex shapes. In aerospace applications the advantages offered in replacing small metal parts and in some cases conventional composites materials and processes are significant and growing [25]. Chopped fibers can also be hybridized with continuous fibers to create an engineered form. Phoenix Composites selectively uses continuous fiber uni-directional and woven reinforcement locally introduced into a parent structure consisting of chopped random fiber reinforcement for a form that is comparable to continuous fiber composites in strength and stiffness but still retaining the geometric flexibility of a chopped fiber process. Chopped fiber composites when combined with thermoplastics can be a very cost-effective process. University of Alabama-Birmingham has produced effective bus seats using Long Fiber Reinforced Thermoplastics (LFT: PP + glass fiber) in a compression molding process.

Conclusions (Summary) Polymeric composites technology has been the vehicle of change in key industrial sectors for the past 30 years, growing from fiberglass reinforcement to more sophisticated polymeric fibers and the current champion, the carbon fiber/multi-phase matrix polymer composite materials. As applications have grown both in breadth and scale, new needs and visions have created strong and focused trends, in both materials and processing sciences and technologies, and emerging at an increasing rate. Market-driven pull and science-based push mechanisms have brought us to a richer landscape of increased dimensions and applications unimagined a few decades earlier. The composites community, unlike other industrial technolgies, is not complacent. Although autoclaves and prepreg have served us well, we want to get rid of them and process in a more optimized way with resin infusion and leaner manufacturing. While the current materials have enabled entire airplanes to be made of composite materials, we want those materials to now serve multiple functions, behave with intelligence, be greener, and are exploring the huge benefits of very small matters in the world of nanotechnology. Conventional materials such as thermoplastics take on a new life as vastly more efficient and focused processing methods are developed, and combining conventional materials in unconventional and novel ways is opening new possibilities. The compositeer has much to feel satisfied about but there is much to challenge them in the future.

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Acknowledgements Dr. Zhong acknowledges the support from NASA through grant NNM04AA62G and from NSF through NIRT Grant 0506531.

References [1] Delmonte, J.; Technology of Carbon and Graphitic Composites; Van Nostrand Reinholdt Co., 1980. [2] Subramanian, R. V.; Jukubowski J. J. Polym. Eng. Sci. 1978, 18, 590-600. [3] Broutman, L. J.; Agarwal, B. D. Polym. Eng. Sci. 1974, 14, 581-588. [4] William Jr, J. H.; Kousiounelos, P. N. Fibre. Sci. Tech. 1978, 11, 83-88. [5] Peiffer, D. G. J. Appl. Polym. Sci. 1979, 24, 1451-1455. [6] Arridge, R. G. C. Polym. Eng. Sci, 1975, 15, 757-760. [7] Chen, J. S.; Lau, S. P.; Sun, Z.; Tay, B.K.; Yu, G. Q.; Zhu, F. Y.; Zhu, D. Z.; Xu, H. J. Surf. Coat. Tech. 2001, 138, 33-38. [8] Kostov, K. G.; Ueda, M.; Tan, I. H.; Leite, N. F.; Beleto, A. F.; Gomes, G. F. Surf. Coat. Tech. 2004, 186, 287-290. [9] Ujvari, T.; Toth, A.; Bertoti, I.; Nagy, P. M.; Juhasz, A. Solid State Ionics, 2001, 141142, 225-229. [10] Torrisi, L.; Gammino, S; Mezzasalma, A. M.; Visco, A. M.; Badziak, J.; Parys, P.; Wolowski, J.; Woryna, E.; Krasa, J.; Laska, L.; Pfeifer, M.; Rohlena, K.; Boody; F. P. Appl. Surf. Sci. 2004, 227, 164-174. [11] Cohen, Y.; Rein, D. M.; Vaykhansky, L. E.; Porter, R. S. Composites Part A.1999, 30, 19-25. [12] Netravali, A. N. Fiber/resin interface modifiction techniques: A case study of ultra-high molecular weight polyethylene fibers, 50th Intl. SAMPE, Long Beach, CA, 2005. [13] Nguyen, H. X.; Riahi, G.; Wood, G.; Poursartip, A. in 33rd Intl. SAMPE Symp., Anaheim, CA, 1988. [14] Hicks, R. F.; Babayan, S. E.; Penelon, J.; Truong, Q.; Cheng, S. F.; Le, V. V.; Ghilarducci, J.; Hsieh, A.; Deitzel, J. M.; Gillespie, J. W. Atmospheric Plasma Treatment of Polyetheretherketone Composites for Improved Adhesion, SAMPE Fall Technical Conference Proceedings: Global Advances in Materials and Process Engineering, Dallas, TX, 2006; lCD-ROM, pp 9. [15] Neema, S.; Salehi-Khojin, A.; Zhamu, A.; Zhong, W. H.; Jana, S.; Gan,Y. X. J. Colloid Interf. Sci. 2006, 299, 332-341, [16] Jana, S.; Zhamu, A.; Zhong, W. H.; Gan,Y. X. J. Adhesion. 2006, 82, 1157-1175. [17] Salehi-Khojin, A.; Stone, J. J.; Zhong, W. H. J. Compos. Mater. 2007, 41, 1163-1176,. [18] Zhamu, A.; Wingert, M.; Jana, S.; Zhong, W. H.; Stone, J. J. Composites Part A. 2007, 38, 699-709. [19] Zhamu, A.; Zhong, W. H.; Stone, J. J. Compos. Sci. Tech. 2006, 66, 2736-2742. [20] Donnet, J. B.; Bansal, R. C. Carbon Fibers; 2nd Edition; Marcel Dekker: New York, NY, 1990. [21] Figueiredo, J. L. et al. Carbon Fibers, Filaments and Composites; Kluwer Academics Publishers: Netherlands, 1990.

128 [22] [23] [24] [25] [26] [27]

W.H. Zhong, R.G. Maguire, S.S. Sangari et al. http://www.tc.faa.gov/its/cmd/visitors/data/AAR-430/advanced.pdf http://www.quickstep.com.au http://www.sti.nasa.gov/tto/spinoff2001/ip7.html http://www.acm-fn.de/e_start.htm http://www.egr.msu.edu/cmsc/biomaterials/star/star http://www.hexcel.com/Products/Matrix+Products/Other+FRM/HexMC

In: Composite Materials Research Progress Editor: Lucas P. Durand, pp. 129-164

ISBN: 1-60021-994-2 © 2008 Nova Science Publishers, Inc.

Chapter 4

AN EXPERIMENTAL AND ANALYTICAL STUDY OF UNIDIRECTIONAL CARBON FIBER REINFORCED EPOXY MODIFIED BY SIC NANOPARTICLE Yuanxin Zhoua, Hassan Mahfuzb, Vijaya Rangaria and Shaik Jeelania a

b

Center for Advanced Materials at Tuskegee University, Tuskegee, AL, 36088 Department of Ocean Engineering, Florida Atlantic University, Boca Raton, FL 33431

Abstract In the present investigation, an innovative manufacturing process was developed to fabricate nanophased carbon prepregs used in the manufacturing of unidirectional composite laminates. In this technique, prepregs were manufactured using solution impregnation and filament winding methods and subsequently consolidated into laminates. Spherical silicon carbide nanoparticles (β-SiC) were first infused in a high temperature epoxy through an ultrasonic cavitation process. The loading of nanoparticles was 1.5% by weight of the resin. After infusion, the nano-phased resin was used to impregnate a continuous strand of dry carbon fiber tows in a filament winding set-up. In the next step, these nanophased prepregs were wrapped over a cylindrical foam mandrel especially built for this purpose using a filament winder. Once the desired thickness was achieved, the stacked prepregs were cut along the length of the cylindrical mandrel, removed from the mandrel, and laid out open to form a rectangular panel. The panel was then consolidated in a regular compression molding machine. In parallel, control panels were also fabricated following similar routes without any nanoparticle infusion. Extensive thermal and mechanical characterizations were performed to evaluate the performances of the neat and nano-phased systems. Thermo Gravimetric Analysis (TGA) results indicate that there is an increase in the degradation temperature (about 7 0C) of the nano-phased composites. Similar results from Differential Scanning Calorimetry (DSC) and Dynamic Mechanical Analysis (DMA) tests were obtained. An improvement of about 50C in glass transition temperature (Tg) of nano-phased systems were also seen. Mechanical tests on the laminates indicated improvement in flexural strength and stiffness by about 32% and 20% respectively whereas in tensile properties there was a nominal improvement between 7-10%. Finally, micro numerical constitutive model and damage constitutive equations were derived and an analytical approach combining the modified shear-lag model and Monte Carlo

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Yuanxin Zhou, Hassan Mahfuz, Vijaya Rangari et al. simulation technique to simulate the tensile failure process of unidirectional layered composites were also established to describe stress-strain relationships.

Introduction Carbon fiber reinforced polymer matrix composites due to their high specific strength and specific stiffness have become attractive structural materials not only in the weight sensitive aerospace industry, but also in marine, armor, automobile, railways, civil engineering structures, sport goods, etc. Generally, the in-plane tensile properties of the fiber/polymer composite are defined by the fiber properties, while the compression properties and properties along the thickness dimension are defined by the characteristics of the matrix resin. Epoxy resin is the most commonly used polymer matrix for advanced composite materials. Over the years, many attempts have been made to modify the properties of epoxy by the addition of either rubber particles [1-2] or fillers [3-4] so that the matrix-dominated composite properties are improved. The addition of rubber particles improves the fracture toughness of epoxy, but decreases its modulus and strength. The addition of fillers, on the other hand, improves the modulus and strength of epoxy, but decreases its fracture toughness. Usually, the typical filler content needed for significant enhancement of these properties can be as high as 10-20% by volume. At such high particle volume fractions, the processing of the material often becomes difficult, and since the inorganic filler has a higher density than the resin, the density of the filled resin is also increased. Nanoparticle filled resins are attracting considerable attention since they can produce property enhancement that are sometimes even higher than the conventional filled polymers at volume fractions in the range of 1 to 5%. It has been established that adding small amounts of nano-particles (<5 wt. %) to a matrix system can increase thermal and mechanical properties without compromising the weight or process-ability of the composite. These polymer-based nanocomposites derive their high properties at low filler volume fractions owing to the high aspect ratio and high surface area to volume ratio of the nano-sized particles. According to Reynaud et al. [5], an interface of 1 nm thick represents roughly 0.3% of the total volume of polymer in case of micro particle filled composites, whereas it can reach 30% of the total volume in the case of nanocomposites. In recent years, nano-scaled particles have been considered as filler material for fiber reinforced epoxy to produce high performance composites with enhanced properties. For example, Zhou et al. [6] used carbon nanofiber to modify carbon fiber reinforced epoxy. Sherman et al. [7] have modified unidirectional carbon/epoxy prepregs by using very thin alumina platelets. Mahfuz and his co-workers [8] have observed a 39% enhancement in flexural strength by infusing 1.5w% SiC nanoparticles in carbon/epoxy composite. Gojny et al. [9] have reported the influence of carbon nanotube on the mechanical and electrical properties of conventional fiber-reinforced composites. The primary objective of this research is to develop a novel manufacturing technique to fabricate unidirectional SiC nano-phased carbon prepregs. After successful manufacturing of these prepregs, we would evaluate the effects of nanoparticle infusion on the as-prepared prepregs. To do so, the easy way is to consolidate these manufactured nano-phased prepregs in nanocomposites laminates. Then we would compare the thermal-mechanical properties of the nano-phased systems with their neat counterparts, and finally, to develop an analytical

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Monte Carlo simulation on the tensile failure process of unidirectional carbon prepreg laminates.

Materials T700SC 12000-50C carbon fiber used in this research is manufactured by Toray Carbon Fibers America, Inc., USA. It is the highest strength, standard modulus fiber in the form of continuous filament tows with outstanding processing characteristics for filament winding, weaving and prepregging produced using the PAN (polyacrylonitrile) process. The epoxy system used was CR46T. It’s a high temperature cure prepreg resin system. To avoid degradation of its properties, the resin is kept under sub-zero temperatures in a sealed atmosphere. This resin system is well suited for prepreg applications, and its properties are shown in Table 1. Table 1. Resin properties Density

0.0457 lb/in3

Gell (min) @ 350 0F

6 – 10

GIC

0.733 lb/in2

Tg DRY 2hr @ 375 0F

437 0F

Tg WET 2hr @ 375 0F

295 0F

Tensile strength @ RT

10 ksi

Tensile modulus @ RT

643 ksi

Poisson ratio

0.36

Elongation

1.7 %

±

0.3

The nanosized fillers for this present investigation were chosen as nano-sized silicon carbide particles. These are highly complex material existing primarily in amorphous or crystalline states. The amorphous SiC is mainly used in coating industries. In functional and structural applications, crystalline SiC are extensively used due to their excellent thermomechanical properties such as high hardness and stiffness, good corrosion and oxidation resistance, high thermal conductivity and high chemical and thermal stability. [10-12]. Such SiC is available in two different phases, namely alpha (α) and beta (β) phases. The formation of these two structures depends on the molecular organization of the basic structural unit, a 2layer planner unit of Si and C in tetrahedral coordination. β−SiC is formed when the planes of Si and C are rearranged in a cubic symmetry with a lattice constant a = 0.4358 nm. On the other hand, heating of β−SiC to high temperature causes the transformation of the cubic symmetry to a mixture of hexagonal (6H) and rhombohedral (15R) polytypes known as αSiC. The corresponding lattice constant parameters of α-SiCs are: a = 0.3082 nm and c =

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1.5117 nm [13-14]. The α-SiC is chemically unstable and as a result their application is very limited. A schematic showing the crystal structure of α and β−SiC is given in Figure 1. The nano sized β-SiC particles were obtained from MER Corporation, USA. These particles are spherical in shape with average diameter of about 30 nm as shown in Figure 2. The bulk material contains more than 95% of SiC with small traces of Oxygen and Carbon.

(a)

(b) Figure 1. Crystal structure of (a) α-SiC, (b) β-SiC Particles [www.a-e/englisch/lexikon/ siliciumcarbid-bild2.htm]

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Figure 2. TEM micrograph of nano SiC sized particles.

Manufacturing Manufacturing of the Nano-phased Carbon Prepregs Online solution impregnation and filament winding were used as the method of manufacturing nano-phased unidirectional carbon prepregs. This method involves four principle steps: (1) uniform dispersion of nano particles in the resin system; (2) application of resin reaction mixture onto the reinforcing tows; (3) removal of solvent from the prepregs; and (4) filament winding. There are various techniques to disperse nanoparticles in the resin system. Acoustic cavitation is one of the efficient ways to disperse nano-particles into the virgin materials. In this case, the ultrasonic power supply (generator) converts 50/60 Hz voltage to a high frequency electrical energy. This voltage is applied to the piezoelectric crystals within the converter, where it is changed to small mechnical vibrations. The converter’s longitudinal vibrations are amplified by the probe (horn) and transmitted to the liquid as ultrasonic waves consisting of alternate compressions and rarefactions. These pressure fluctuations give rise to microscopic bubbles (cavities), which expand during the negative pressure excursions, and implode violently during the positive excursions. Some of these cavities oscillate at a frequency of the applied field (usually 20 kHz) while the gas content inside these cavities remains constant. As the bubble collapse, millions of shock waves, eddies and extremes in pressures and temperatures are generated at the implosion sites. Although this phenomenon known as cavitation, lasts but a few microseconds, and the amount of energy released by each individual bubble is minimal, the cumulative amount of energy generated is extremely high. During the operation, an active cavitation region is created close to the source of the ultrasound probe and that the ultrasonic processing produces high pitched noise in the form of harmonics which are above the human audible range, emanating from the container walls and the fluid surface. The development of cavitation processes in the ultrasonically processed melt creates favorable conditions for the intensification of various physio-chemical processes.

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Acoustic cavitation accelerates heat and mass transfer processes such as diffusion, wetting, dissolution, dispersion and emulsification [15-16]. Optimization of the solution prepregging process begins with the appropriate choice of solvent. A high degree of wetting can only be expected from solvents that possess favorable thermodynamics regarding wetting of the particular solid material (carbon filaments, in this case) [17-18]. The process of wetting entails the contact and spreading of the solvent over the surface of the solid, i.e., liquids that possess a low contact angle for a particular solid show considerable wetting behavior (as opposed to liquids that display high contact angles). This solvent should be chosen from a list of candidate solvents capable of dissolving the matrix polymer. The differences in wetting action, coupled with other relevant parameters such as boiling point and general practicality of the particular solvent choice usage, will lead to an appropriate choice of solvent. In particular, solvent characteristics should include a much lower boiling point than melt flow point of the resin and a lower density then that of the resin for ease of residual solvent removal [19]. An example of the preceding contact angle analysis can be found in a study by Patel and Lee [17]. In their study, fiberglass tows were subjected to contact angle analysis using the Wilhelmy plate method. A series of liquids was used (not polymer solutions), each having differing values of viscosity and surface tension. The equilibrium contact angles for all of these liquids were not observed to be a function of solvent viscosity (viscosity range = 0.33 mPa – 1499.0). Furthermore, the liquid surface tension was found to be positively correlated with the contact angle, i.e., increases in surface tension generally yielded larger contact angle measurements. It should be stressed that these results only indicate trends in contact angles; they may not imply favorable conditions for capillary flow (in addition to wetting), which is another important consideration in the prepreg process [15]. Once the appropriate solvent is identified for solution prepregging, prepregged tapes can be manufactured. The objective in solution prepregging is to prepare a uniform tape in which every fiber surface is uniformly wetted with the polymeric matrix material. Another objective in solution prepregging is maximizing the amount of matrix material pick-up. This is easily quantifiable as the amount of matrix material adhering to the fiber surface after a single immersion into the resin bath. The nature of the relationship between fiber dispersion and matrix pick up is expected to be competitive. This can be inferred from the extremes of the process. In a polymer solution with a concentration approaching zero, every filament can be expected to be wetted (resulting in a good fiber dispersion), assuming that the thermodynamics are favorable. But the matrix pick up in this case is nearly zero since there is no polymer in solution. At the other extreme, the polymer weight fraction in solution approaches one. In this case, the fiber wetting upon dipping will be very poor given the extremely high viscosity of the resin (kinetic limitation). But upon wetting, a large amount of polymer will remain on the fiber surface (high matrix pick up). Therefore, intuition states that there will exist an intermediate polymer solution concentration in which a balance is obtained between the fiber dispersion and matrix pick up. The concepts in the preceding paragraph can be more easily visualized by using a model that approximates the wetting process of a fiber tow by a polymer solution. By combining the Kelvin equation, which describes wetting of a solution in micro-capillaries and Darcy’s Law, which describes flow in porous media, the following equation is obtained:

t f = l 2 μVvoid /{2S b (2 / R)γ sizing cosθ }

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where: tf = tow wetting time l = tow thickness µ= solution viscosity Vvoid = tow void volume Sb = tow permeability (perpendicular to fiber direction) R = fiber-fiber separation γsizing= solution surface tension θ= contact angle A quick survey of the equation reveals the following three trends: • • •

As the solution viscosity increases, the time of tow wetting increases. As the surface tension of the solution increases, the time of tow wetting decreases. As the contact angle increases from 00(complete wetting) to 900(mostly nonwetting), the cosine term decreases and thus increases the time of tow wetting.

Prepreg residence time is also known to influence both the fiber dispersion and efficiency. In a study by Lacroix et al. [20], ultra-high modulus polyethylene fiber bundles were prepregged with a xylene/ low-density polyethylene solution. For a prepregging time range of 8 min. – 19.5 hours, it was noted that increasing prepreg time increased the layer thickness of deposited polymer around the fiber surfaces. Similar results were obtained in a study by Moon et al. [19] in which solvent prepregged fiber bundles were prepared from glass fibers and a high-density polyethylene/ toluene solution. After the fiber tapes are prepregged with the nano-phased resin, the solvent has to be driven off. In this case, since the tapes are not to be wound around a storage spool following prepregging, solvent elimination should be complete. This represents a crucial step in the overall composite manufacturing process, as residual solvent can result in voids during the melt consolidation process. How the solvent interaction with the fiber/matrix/nanoparticle interface is an important consideration, given the influence of the quality of the interface in determining the final mechanical properties of the composite. The presence of solvent is generally known to reduce the quality of the matrix/fiber interface. The reasons for this phenomenon are unclear, but can be explained by the following hypothesis [21]: • • •

Solvent extraction can cause separation of the fiber/matrix interface Solvent concentration at the interface will interfere with fiber/matrix contact; and Phase separation of low molecular weight species at the interface may form a weak interface between the fiber and matrix.

Solvent removal, in part, is regarded to proceed by solvent concentration at the interface, followed by solvent traversing the fiber surface and escaping from the ends of the composite. Obviously this will result in poor interfacial quality if this is to occur during melt consolidation or autoclave processing, as the case maybe. A study conducted by Wu et al. [22] illustrates how residual solvent negatively affects composite mechanical property quality. Solution prepregged carbon fiber reinforced polyethersulphone composites were prepared and compared with strictly hotmelt processed composites of the same nominal fiber

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content. The transverse flexural strength of the solution prepregged material was only half that of the melt-processed material. Upon analysis of the solution prepregged material using differential scanning calorimetry (DSC), it was found that residual solvent remained in the sample, despite hotmelt consolidation of the prepreg. Residual solvent can most likely be attributable to difficulty in solvent diffusion during the consolidation process. The reasons for poor interfacial quality are thought to be attributable in the reasons outlined in the preceding paragraph. Commercially available high temperature prepreg resin CR46T was first dissolved in acetone (Dimethyl ketone, class 1B, Fisher Scientific Co. LLC, USA), at a ratio of 65:35 by mechanical stirring at 1500 RPM for about 4 hours as shown in Figure 3. Spherical shaped silicon carbide nanoparticles were carefully measured to have a 1.5% loading by weight of the resin and mechanically mixed with the liquid resin. The mixture was then irradiated with high intensity Sonic Vibra Cell ultrasonic liquid processor (Ti-horn, 20 kHz, 100W/cm2) at 50% amplitude for 30 minutes. This ensured uniform mixing of nanoparticles over the entire volume of the resin. To avoid temperature rise during sonication, cooling was employed by submerging the mixing beaker in a water bath maintained at 50 0F as shown in Figure 4. The nano-phased resin reaction mixture was then transferred into a heating bath maintained at a constant temperature of 80 0F throughout the fabrication as shown in Figure 5. A continuous strand of carbon fiber from a spool attached in the spindle bracket assembly was allowed to pass through the resin bath at a rate of about 1 meter per minute. In this case, the resin reaction mixture individually wet each filament within the fiber tow. Once the fiber was coated with nano-phased resin the excess solvent was removed from the prepreg by passing the wet strand through a high temperature heater maintained at 160 0F. The nano-phased prepreg tow was then routed and fed through a fiber delivery system and was precisely hoopwound on a rotating foam mandrel on the filament winding machine. Figure 6 represents the schematic of solution impregnation and filament winding setup. During the fiber placement, the winding angle was kept at 89.8750 to avoid excessive gaps or overlaps between adjacent courses.

Figure 3. CR46T resin mixed with acetone.

An Experimental and Analytical Study of Unidirectional Carbon Fiber…

Figure 4. Ultrasonic cavitation.

Figure 5. Nano-phased resin.

Figure 6. Schematic of solution impregnation and filament winding.

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Manufacturing of Nano-phased Carbon Prepreg Unidirectional Laminates The process of unidirectional laminate fabrication started with online prepregging. By this it means that when eight layers of the manufactured prepregs were hoop wound on the rotating foam mandrel during prepreg manufacturing, the remaining tow was cut and the mandrel was removed from the filament winding machine. During prepreg stacking, care was taken to place tows without allowing the previous layer to dry. The prepregs on the cylinder was then longitudinally cut open into a rectangular sheet as shown in Figure 7. These rectangular sheets were arranged in a compression molding setup by putting symmetric layers of plastic film, bleeder cloth and teflon on the top and bottom. The whole setup was then placed in Tetrahedron MTP press compression molder as shown in Figure 8. Mold temperature was ramped to get 350 0F while the mold pressure was kept as 40 Psi and consolidated for about 4 hours to obtain a 2mm thick SiC-carbon-epoxy nanophased unidirectional laminate (as shown in Figure 9). A typical consolidation cycle is shown in Figure 10.

Figure 7. Schematic of unidirectional laminate preparation.

Figure 8. Compression molder.

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Figure 9. Nano-phased unidirectional laminate.

Figure 10. Consolidation cycle for laminates.

Experimental Results and Discussion Differential Scanning Calorimetry The Differential Scanning calorimetric (DSC) studies have been carried out to understand the effect of nanoparticles on glass transition temperature of cured carbon-epoxy prepregs based on consolidated samples. Figure 11 represents the DSC curve of as-fabricated neat system. The curve exhibits an endothermic baseline shift at about 220 0C and highly exothermic peak at about 390 0C. The baseline shift at 220 0C is assigned to the glass transition temperature and exothermic peak at 390 0C is assigned to the decomposition temperature of the epoxy resin. These results match well with the supplier materials data sheet. Figure 12 represents the DSC curve of as-fabricated 1.5 wt.% SiC nano-phased system. The curve showed only one exothermic peak at about 397 0C which is assigned to the decomposition temperature of the cured epoxy resin. The baseline shift corresponding to glass transition temperature was almost disappeared. This clearly indicated that the epoxy was highly cross-linked due to catalytic effect caused by SiC nanoparticles. The shift was not observed, we believe, because of the equipment sensitivity to the higher cross-linked polymers. To further validate the

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Yuanxin Zhou, Hassan Mahfuz, Vijaya Rangari et al. 0.1000 0.0812 0.0625 0.0437 0.0250 0.0062

Heat Flow (W/g)

-0.0125 -0.0313 -0.0500 -0.0688 -0.0815 -0.1063 -0.1250 -0.1438 -0.1625 -0.1812 -0.2000 50

100

150

200

250

300

350

400

Temperature (°C)

Figure 11. DSC graph of neat prepreg system.

0.0500

0.0188

Heat Flow (W/g)

-0.0816

-0.1563

-0.2250

-0.2938

-0.3625

-0.4312

-0.5000 50

100

150

200

250

300

350

Temperature (°C)

Figure 12. DSC graph of 1.5 wt.% SiC nano-phased prepreg system.

400

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Figure 13. DMA graphs of neat and 1.5 wt.% SiC nano-phased prepreg systems.

results, dynamic mechanical analysis (DMA) tests were also carried out under single cantilever test environment. The results indicated the existence of glass transition (Tg) for the nano-phased system which is about 5 0C higher than the neat counterpart as shown in Figure 13. The increase in Tg may be attributed to a loss in the mobility of chain segments of epoxy resin resulting from the high nanoparticle/matrix interaction. Impeded chain mobility is possible if the nanoparticles are well dispersed in the matrix. The particle surface-to-surface distances (‘matrix bridges’) should then be relatively small and chain segment movement may be restricted. Good adhesion of nanoparticles with the surrounding polymer matrix additionally may have benefited the dynamic modulus by hindering molecular motion to some extend. The hard particles incorporated into the polymer may also have acted as additional virtual ‘‘network nodes’’. In either situation it can be deduced that Tg increased as a result of more number of cross-linked polymer chains and restricted mobility of the chain segments in the presence of SiC nanoparticles.

Thermo Gravimetric Analysis Thermo gravimetric analysis (TGA) has been carried out to find the degradation temperature or to estimate the thermal stability of neat and 1.5 wt.% nano-phased prepreg systems. Figure 14 shows that in as-fabricated neat system, the resin decomposed at about 390 0C which is represented by the peak of the derivative curve. The TGA curve shown in Figure 15 indicated that the decomposition temperature of the nano-phased system was about 397 0C, which is almost 7 0C higher than the neat counterpart. From the results it is clear that in nano-phased systems, epoxy was amply cross-linked and had minimum particle-to-particle interaction,

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which resulted in increase in thermal stability of the system. These results are consistent with the DSC results as well. 0.5

110

o

390 C 100

0.3 90 0.2

Weight (%)

o

Deriv. of Weight (%/ C)

0.4

80 0.1

0.0

70 200

300

400

500

600

Temperature (o C) Figure 14. TGA graph of neat prepreg system. 0.8

110

o

0.6

100

0.4

90

0.2

80

0.0

70 200

300

400

500

600

Temperature (o C) Figure 15. TGA graph of 1.5 wt.% SiC nano-phased prepreg system.

Weight (%)

o

Deriv. of Weight (%/ C)

397 C

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Flexure Response of Layered Composites A typical flexure stress-strain plot of neat and nano-phased laminates is shown in Figure 16. The curves show considerable non-linear deformation and the irregularities in the curves were attributed to random fiber breakage with pinging noise during the test. The specimens failed rapidly after reaching the point of maximum stress. In general, the composites exhibited brittle-type failure. The curves also revealed that by infusing 1.5 wt.% SiC nanoparticles, strength and modulus significantly improved. Nano-phased system showed approximately 32% increase in flexural strength and 20% in modulus when compared to the neat ones as shown in Table 2. This result was as expected because of the strong bonding between filler particles and matrix in nano-phased specimen which resulted in the static adhesion strength as well as the interfacial stiffness to transfer stresses and elastic deformation efficiently from the matrix to the fillers via the interface. In other words, large contact areas which translated into high interfacial stiffness and homogeneous dispersion of nanoparticles assisted in an efficient stress transfer between polymer and nanoparticles which lead the particles to carry a part of the external load and resulted in improved flexural strength and stiffness. In addition, the nanoparticles may have acted as stoppers to crack growth by pinning the cracks. It is also observed for the nanocomposites in the present study that the strain-to-break tends rather to slightly higher values in comparison with the neat systems. This increase suggests that the nanoparticles are able to introduce additional mechanisms of failure and energy consumption without blocking matrix deformation. Standard deviation for the set of neat and nano-phased system experimental data were shown to have lower values (Table 3). Lower standard deviation indicated the stability and consistency in the results as well.

Figure 16. Engineering Stress-Strain curves of flexure test.

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Material Neat Sample 1 Neat Sample 2 Neat Sample 3 Neat Sample 4 Neat Sample 5 1.5wt% Sample1 1.5wt% Sample2 1.5wt% Sample3 1.5wt% Sample4 1.5wt% Sample5

Flexural Strength (MPa) 765.32 806.46 789.38 782.62 793.45 1043.35 995.28 1012.59 1088.92 1071.55

Average Strength (MPa)

Gain/Loss Strength (%)

787.45

--

1042.34

+32.37

Flexural Modulus (GPa) 82.14 80.43 82.05 80.29 76.91 93.93 97.02 98.25 90.65 99.36

Average Modulus (GPa)

Gain/Loss Modulus (%)

80.36

--

95.84

+19.26

Table 3. Standard deviation and Coefficient of variation of flexure test data

Neat system Strength Neat system Modulus +1.5wt% Nano-phased system Strength +1.5wt% Nano-phased system Modulus

Standard Deviation (±) 13.52 1.89 34.99 3.17

Co-eff. of Variation (%) 1.72 2.36 3.36 3.31

Tensile Response of Layered Composites Typical curves for the tensile behavior of both neat and 1.5 wt.% nano-phased specimen are shown in Figure 17. The in-plane tensile behavior of both the composites shows linear behavior up to approximately 1.2% strain where initial fiber failure occurred. The behavior continued to be linear again till the final specimen failure. Both the elastic modulus and the strength of nano-phased composites were between 7-10% higher than their neat counterparts. The reason for such small improvement could be visualized in the sense that, in tension the fiber took maximum load and the nanoparticle infusion in the matrix did not contribute much in improving the tensile properties. The improvement of modulus in this study was mainly because of the improvement of the matrix modulus by filler dispersion. Therefore it can be deduced that higher tensile properties in the nanocomposite is due to higher nano-phased matrix properties. Average mechanical properties and their deviation are shown in Table 4 and Table 5, respectively.

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Figure 17. Engineering Stress-Strain curves of tensile test.

Table 4. Tensile test data for carbon prepreg laminates Material Neat Sample 1 Neat Sample 2 Neat Sample 3 Neat Sample 4 Neat Sample 5 1.5wt% Sample1 1.5wt% Sample2 1.5wt% Sample3 1.5wt% Sample4 1.5wt% Sample5

Tensile Strength (GPa) 1.32 1.41 1.39 1.42 1.34 1.48 1.39 1.51 1.49 1.55

Average Strength (GPa)

Gain/Loss Strength (%)

1.38

--

1.48

+7.25

Tensile Modulus (GPa) 86.44 84.35 85.03 88.57 86.61 96.39 90.68 93.13 97.90 94.37

Average Modulus (GPa)

Gain/Loss Modulus (%)

86.20

--

94.49

+9.62

Table 5. Standard deviation and Coefficient of variation of tensile test data

Neat system Strength Neat system Modulus +1.5wt% Nano-phased system Strength +1.5wt% Nano-phased system Modulus

Standard Deviation (±) 0.04 1.49 0.05 2.51

Co-eff. of Variation (%) 2.86 1.69 3.56 2.66

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SEM Analysis SEM analysis was carried out on JSV 5800 JOEL Scanning Electron Microscope. Specimen from failed samples of flexure tests were selected, prepared and attached to the sample holder with a silver paint and coated with gold to avoid charge build-up by the electron. Figures 18 and 19 show the SEM micrographs obtained. It is observed from Figures 18a-18b that most of the damage was located in the loading zone, including large intra and inter-layer delamination cracks as well as fiber/bundle failures. The damage and the fracture processes were mainly due to local shear components. They also show interfiber micro-cracks and delamination cracks. These micrographs also reveal that the carbon fibers were highly oriented with uniform resin distribution. Figure 19b shows the SEM micrograph in which SiC nanoparticles (white dots) are distributed uniformly without agglomeration. Also revealing the size of the filament to be 8-10 microns in diameter and SiC nanoparticle to be of about 30-40 nm range.

(a)

(b)

Figure 18. SEM of failed flexure samples in thickness direction.

(a)

(b)

Figure 19. SEM of failed flexure samples.

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Numerical Simulation of Tensile Failure Process of Layered Composites The tensile failure of fiber reinforced composite material involves a complicated damage accumulation process resulting from random fiber breakage, stress transfer form broken to intact fiber, and interface debonding between the fiber and matrix. It is difficult to analyze such a complicated probabilistic failure phenomenon precisely by means of analytical methods. The Monte Carlo simulation technique coupled with a stress analysis method is one of the most effective tools for understanding the tensile failure process [23-27]. In past Monte Carlo simulations, a micro-composite unit with a coarse mesh and a few fibers of short length was always used as the numerical model. In practice, structural composites usually contain large quantities of fibers, when such micro-composite unit is applied to simulate the failure process of practical composites, it may result in a lack of statistical effects and magnification of boundary effects, causing errors in calculations of the stress concentration. Yuan et al. [28] presented a two-dimensional large-fine numerical micro-composite model with fine mesh, sufficient fibers and adequate length instead of the aforementioned model and developed a new Monte Carlo simulation method to study the tensile failure process of unidirectional composites. Based on the new model and method, the average statistical evolution of the composites deformation and failure, caused by the accumulation of the random breakages of large quantities of fibers, matrices and interfaces, is successfully simulated. By taking account of the inertial effect, strain-rate effect of components and the softening effect caused by the thermo-mechanical coupling in the simulation model, the tensile stress–strain curves of unidirectional fiber reinforced resin matrix composites CFRP and GFRP at different high strain-rates were successfully predicted, which agree well with the experimental results [29-31]. All above Monte Carlo simulations were coupled with the classical shear-lag model. It is assumed that the fibers bear the whole axial load and the matrix only carries the shear stress. Ochiai et al. [32, 33] proposed a modified shear-lag model, which takes the axial load born by the matrix into account, to study the stress concentration in the elastic and elastic-plastic matrix caused by single fiber breakage. In the present study, Monte Carlo numerical constitutive model according to Ochiai's modified shear-lag model with fine mesh, sufficient fibers and adequate length was established to study the failure process of unidirectional layered composites, to predict the mechanical behavior of these composites with the prepreg epoxy matrix and to study the relationship between the interface and composite strengths.

Model of Composites Figure 20 shows the large-fine numerical model of unidirectional composite that consists of n fibers and n+1 matrices. Each fiber or matrix, at the length of L, is composed of m elements of length Δx=L/m in the longitudinal direction. The cross-section of the fiber and the matrix are simplified as rectangle, considering that the simplified fiber has the same sheared area as that of the actual one, we have

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H = πD / 2

df = D / 2

dm =

(1 − V ) f

Vf

df

(1)

where H, df and dm are the thickness of composite, the width of fiber and the width of matrix, respectively (as shown in Figure 20). D and Vf are the diameter and the volume fraction of fiber. The displacement component at node (i,j) is expressed as u i , j .

Figure 20. Model of Unidirectional carbon fiber reinforced matrix resin.

The composite is pulled at one end and fixed at the other end. In figure 20, the left boundary is fixed, and the right moves with a constant speed V, namely

⎧⎪u ik, 0 = 0 ⎨ k ⎪⎩u i ,m = VkΔt

(1 ≤ i ≤ 2n + 1) (1 ≤ i ≤ 2n +

(2)

The initial condition is

u i0, j = 0 (1 ≤ i ≤ 2n + 1) and (0 ≤ j ≤ m )

(3)

Constitutive Assumptions It is assumed that the fibers are homogeneous and linear elastic, the fiber strength is described statically by single Weibull distribution [34]:

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149

⎡ L ⎛ σ ⎞β ⎤ P(σ ) = exp ⎢− ⎜⎜ ⎟⎟ ⎥ ⎢⎣ L0 ⎝ σ 0 ⎠ ⎥⎦

(4)

where, P is the survive probability of fiber at stress σ , and the

σ 0 and β are Weibull scale

parameter and Weibull shape parameter, L and L0 length and reference length of fiber. In addition, it is assumed that epoxy matrix is homogeneous and linear elastic.

⎧σ m = E m ε ⎨ ⎩τ m = Gm γ

(5)

where, E m and G m are tensile modulus and shear modulus.

Shear Stress on the Interface The shear stress at i−1/i,j interface (shown in Figure 21) τ i−1/i,j can be expressed as a function of the fiber displacement ui,j and the interface displacement ui−1/i,j:

τ i −1 / i , j = G f (u i , j − u i −1 / i , j ) /(df / 2)

(6a)

where G f is the shear modulus of the fiber. τ i−1/i,j can be also expressed as :

τ i −1 / i , j = Gm (u i −1 / i , j − u i −1, j ) / (dm / 2)

(6b)

If the interface does not break, combine (6A) and (6B) to eliminate ui−1/i,j, then we get

τ i −1 / i , j =

2Gm G f Gm df + G f dm

(u

i, j

− u i −1, j )

(7)

when the interface breaks, we have

τ i −1 / i , j = τ c where, τ c is the friction between the fiber and the matrix when the matrix cracks or the interface debonds.

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Figure 21. Interface shear stress between fiber element and matrix element.

Governing Equation For the fiber element

dfE f

d 2ui, j dx 2

+ (τ i / i +1, j − τ i −1 / i , j ) = 0

(8a)

+ (τ i / i +1, j − τ i / i −1, j ) = 0

(8b)

For the matrix

dmE m

d 2ui, j dx 2

dmE m

dmE m

d 2 u1, j dx 2

d 2 u 2 n +1, j dx 2

+ (τ 1 / 2, j − 0 ) = 0

(8c)

+ (0 − τ 2 n / 2 n +1, j ) = 0

(8d)

Equation 8A-D can be expressed by the governing equation as follow

Ai dfE f

d 2ui, j dx

2

+

2Gm G f dfGm + dmG f

(u

i +1, j

(9)

where,

⎛1 + μ ⎞ ⎛1 + μ ⎞ ⎟+⎜ ⎟( −1) i Ai = ⎜ ⎝ 2 ⎠ ⎝ 2 ⎠

− 2u i , j + u i −1, j ) = 0

μ=

(

E m dm E m 1 − V f = E f df E fV f

)

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Finite Difference Method We define the non-dimensional displacement coordinate as

U i, j = u i, j / ξ

,

X = x /ξ ,

where

τ i / i +1, j = τ i / i +1, j ξ / E f df

(

dfE f G f dm + G m df

ξ=

(10)

)

2G f G m

(11)

The Equation 9 can be rewriten as

Ai

d 2U i , j dX 2

+ (U i +1, j − 2U i , j + U i −1, j ) = 0

(12)

The second-order derivative at point (i,j) is: ⎧ 1 k k ⎪ 2 ( U i,j−1 − 2U i,j + U ( ) Δ x ⎪ d 2 U ik,j ⎪ 4 k k 2 = ⎨ 2 ( U i,j − U i,j−1) dX ⎪ 3( Δx) 4 ⎪ k k ⎪⎩ 3( Δx) 2 ( U i,j+1 − U i,j)

)

k i,j+ 1

(13)

no element break element (i,j)~(i,j+1) break element (i,j-1)~(i,j) break Substituting Equation (13) into (12), one can obtain:

Ai (UL + UR ) + C 2 (ΔX ) (C 4 UD + C 5 UU ) 2

U

k i, j

=

C1 Ai + C 2 C 3 (ΔX )

⎧ U ik, j −1 UL = ⎨ ⎩ 0 element (i,j-1)~(i,j) unbroken element (i,j-1)~(i,j) broken

⎧⎪ UR = ⎨ ⎪⎩

U

0

k i , j +1

2

(14)

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Yuanxin Zhou, Hassan Mahfuz, Vijaya Rangari et al. element (i,j)~(i,j+1) unbroken element (i,j)~(i,j+1) broken

⎧τ *i /ki +1, j UU = ⎨ ⎩ 0 interface (i,j)~(i+1,j) unbroken interface (i,j)~(i+1,j) broken

⎧τ *i −k1 / i , j UD = ⎨ ⎩ 0 interface (i-1,j)~(i,j) unbroken interface (i-1,j)~(i,j) broken

C

1

⎧ 1 =⎨ ⎩ 2 element broken no element broken

C

2

⎧ 0.75 =⎨ ⎩ 1 element broken no element broken

⎧0 ⎪ C3 = ⎨1 ⎪2 ⎩

both side matrix broken single side matrix brokes no matrix broken

C

4

⎧1 i ≠ 1 =⎨ ⎩0 i = 1

⎧1 i ≠ n = C 5 ⎨⎩0 i = n Using the successive over-relaxation, we have

[U ]

k q i, j

where,

[ ]

= λ U ik, j

q

[ ]

+ (1 − λ ) U ik, j

q −1

(15)

λ is the relaxation factor, which controls the convergence speed of solution, and q is k

k

the times of iteration. U i , j is the right side of Equation (14). After U i , j is obtained, the stress of the segment of fiber and matrix can be calculated from following expression:

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153

For fiber element:

σ i, j = E f

U i +1, j − U i , j

(16a)

X

For matrix element

σ i, j = Em

U i +1, j − U i , j

(16b)

X

The strain and stress of composites were calculated from average stress of all elements.

σc =

n m 1 ⎡n m ⎤ × ⎢∑∑ σ 2i +1, j (1 − V f ) + ∑∑ σ 2i , jV f ⎥ (2 N + 1)M ⎣ i −0 1 1 1 ⎦

(17)

VKΔt L

(18)

εc =

Strength of Fiber Element and the Failure Criterion Strength assignment to the fiber elements In simulating, the strength of the fiber elements should be predetermined. According to the Weibull statistical constitutive model the strength of the fiber follow Equation (4). If we assume L in Equation (4) equal to mesh length Δx, here σΔx can be obtained from the scale parameter σ0 at experimental length Lo. n×m random array ηi,j, equally distributed in the range of (0,1), are produced by the computer, and we let

ηi, j

⎡ Δx ⎛ S i , j ⎜⎜ = P (Δx, S i , j ) = exp ⎢− ⎢⎣ L0 ⎝ σ 0

⎞ ⎟⎟ ⎠

β

⎤ ⎥ ⎥⎦

(19)

From the Equation (19), we can get the strength of fiber element 1

S i, j

⎡ L ⎤β = ⎢− 0 ln (η i , j )⎥ σ 0 ⎣ Δx ⎦

(20)

The failure criterion The failure criterion of fiber is

σ i, j ≥ S i, j

fiber element broken

(21a)

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σ i, j ≤ S i, j

fiber element unbroken

(21b)

τ i, j ≥ τ m

interface broken

(22a)

τ i, j ≤ τ m

interface unbroken

(22b)

The failure criterion of interface is

where, τm is the ultimate shear stress of matrix. The failure criterion of matrix is

ε i, j ≥ ε m

matrix element broken

(23a)

ε i, j ≤ ε m

matrix element unbroken

(23b)

where, εm is the failure strain of the matrix

Simulation Procedure Based on the above numerical constitutive model, a computational program is compiled to simulate the microscopic dynamic failure process of unidirectional composites. The simulation procedure is illustrated as following: (A) Randomly assign a statistical strength Si,j (i=2,4,...2n, J=1,2,…m) to the fiber element, definitely assign the tensile strength to the matrix and the shear strength to the interface. (B) Solve Equation (14) by iteration formula (15) using Ui,jk−1, Ui,jk−2 and the boundary conditon at time t=kΔt, obtain the displacement field Ui,jk (i=1,2,...2n+1,j=1,2,...m). (C) Determine whether the element or the interface breakage (or the matrix element unloading) has happened, if new breakage occurs, take the breakage into account and repeat steps (B) and (C) until no new breakage occurs, else calculate the apparent stress σc and strain εc using Equation (17) and (18). (D) Increase a time step and repeat step (B) and (C) till the composites failure happens. Here the "composites failure" is defined as the state when the stress σc drops from σmax to about 50% of σmax.

Experimental Results of Fiber, Matrix and Composite The statistical parameters of fiber were obtained from tension tests of T700 fiber bundles. The tensile stress-strain curve of fiber bundle in Figure 22 shows considerable amount of nonlinearity. The specimen failed gradually after reaching the maximum stress due to the tensile strength distribution of fibers. Three parameters were determined from each stress-strain

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155

curve: elastic modulus (E), tensile strength ( σ b ), and failure strain ( ε b ). Elastic modulus or Young's modulus is the initial slope of the stress-strain curve. Tensile strength is the maximum stress at the peak load and the strain corresponding to the tensile strength is the failure strain. The average values of these three properties are:

σ b = 1.93GPa

E = 210GPa

ε b = 1.07%

Based on fiber bundles model and statistical theory of fiber strength [34], the Weibull parameters for tensile strength of carbon fibers also can be obatined:

σ 0 = 2.70GPa

β = 9.03

L0 = 100mm

2.00

Stress (GPa)

1.60

1.20

0.80

T700 Carbon Fiber 0.40

Experimental Results Simulated Results 0.00 0.000

0.005

0.010

Strain Figure 22. Stress strain curve of carbon fiber.

0.015

0.020

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Stress (MPa)

80

40

Neat Epoxy +1.% SiC Nano Particle Filled Epoxy 0 0.00

0.02

0.04

0.06

Strain (mm/mm) Figure 23. Stress-strain curves of the epoxy.

Figure 23 shows the stress-strain curves of epoxy and nanophased epoxy. It can be observed that both the modulus and strength have been improved by filling nano particle into matrix system. All parameters of fiber and matrix were listed in Table 6. Table 6. Parameters of T700 fiber and matrix Material E (GPa) G (GPa) D ( μm )

T700 carbon fiber 210 87.5 5

Neat epoxy 2.45 1.02 --------

Nano-phased epoxy 3.32 1.38 --------

Vf (or Vm)

β

49% 9.03

51% --------

51% --------

σ 0 (GPa)

2.70

--------

--------

σ b ( MPa)

--------

89

110

τ max (GPa)

--------

45

55

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Simulation Result and Discussion Figure 24 shows the simulation results of neat and 1.5wt.% SiC nano-phased layered composites along with the experimental tensile results. It can be observed that the numerical simulation results agree well with the experimental results. This indicates that the numerical model, the simulation method and the program are reliable. Simulated results and experimental results also show that the strength of nanocomposite was about 7-10% higher than that of the neat one. As already discussed in the previous chapter, it may have been contributed from the improvement of matrix and interface properties.

Figure 24. Simulated Stress-strain curves of tensile test.

The failure strain of matrix is higher than that of fiber in unidirectional composites, the fiber element with the lowest strength is first broken. Then, with increasing applied load, the breakage of fiber elements occurs randomly. On the other hand, high shear stress are generated in the matrix elements due to the fiber breakages, so that the matrix cracking and interfacial debonding occur, leading to the final failure of composites. Figure 25 shows the simulated failure process of composite from the above results. Figure 25a and 25b indicate the initial fracture occurring at fiber at low stress level. Stress concentration in the matrix

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elements results in matrix cracking and interfacial debonding. One can observe the damage zone of nano-phased composite is smaller than that of neat system. Figure 25c and 25d show the failure appearances at about 60% Peak load of the composites. Figure 25e and 25f show the finial failure appearances of composites. In these figures, the fiber element breakage of two composite are almost same, but matrix element breakage of nano-phased composite is smaller than that of neat system.

(a)

(b)

(c)

(d)

Figure 25. Continued on next page.

An Experimental and Analytical Study of Unidirectional Carbon Fiber…

(e)

(f)

Figure 25. Simulated tensile failure process of neat and nano-phased composites.

Figure 26. Stress strain curves of CFRP with different interface strengths.

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Figure 27. Relationship between composite’s tensile strength and interface strength.

Interface strength is a very important parameter affecting the strength of composite. Based on the current numerical model, the effect of interface strength on the tensile failure process of unidirectional composites has been studied. Figure 26 shows the tensile stress– strain curves of composite with different interface strength (100, 50, 20 and 10MPa). The relationship between the strength of composites and interface strength are shown in Figure 27. From Figure 26 and 27, it can be observed that the tensile strength of the composites increases with interface strength, and when the value of interface strength is over 50 MPa, the tensile strength of composites tends to be a constant and not affected by interface strength. Figure 28 shows the simulated micro damage patterns of the composites with weak interface and strong interface. Comparing these patterns with the macro stress–strain curves, it is evident that: when the interface strength gets weaker, the adjacent interfaces get easier to break after the fibers break, the load transfer along the traverse direction gets more ineffective, the reinforced function of fiber gets more insufficient, the damage area gets larger, the damage pattern gets more disordered, the negative effect gets greater, thus the composite’s strength gets lower; when the interface strength gets stronger, the fiber breakage propagates throughout the composite along the straight line and the composites strength gets higher.

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161

(b)

Figure 28. Simulated micro damage patterns. (a) Weak interface, (b) Strong interface.

Conclusion Using the solution impregnation and filament winding techniques, an innovative manufacturing method was introduced to manufacture nano-phased carbon/epoxy prepreg tapes/tows from a nanoparticle modified epoxy resin system. It was seen from TGA that the as-prepared nanocomposites were thermally more stable than their neat counterparts. An improvement of about 7 0C was noticed. The improvement in thermal stability was believed to have been caused by increased cross-linking and better interactions between the epoxy and SiC nanoparticles. As seen with DSC and DMA, an improvement in glass transition temperature (Tg) of the nano-phased system was about 5 0C. The improvement is due to the restricted mobility of the chain segments in the presence of SiC nanoparticles, as a result of a greater number of crosslinked polymer chains. Response of nano SiC infused composites under flexure loading showed significant improvements in strength as well as stiffness over the neat ones. On an average the increment in strength and stiffness was 32% and 20% respectively over the neat systems. It is believed that the higher strength of the SiC systems is attributed to better interfacial bonding and the resistance offered by the nanoparticles to crack propagation. In respect of tensile strength and stiffness, the nanocomposite systems offered improvements between 7 and 10%. It was seen that all the test coupons (neat and nanophased) failed in the gage length and failure modes were as described in the ASTM standard. This nominal improvement in the tensile properties was believed to have occurred because in tension, stresses in fibers were more dominant than in the nano-phased matrix and the property improvement is merely due to the improved properties of the resin. Monte Carlo simulation technique has been established to simulate unidirectional layered neat and nano-phased composites. The simulation results are in agreement with the experimental results. Tensile failure process was simulated and the damage zone and micro-

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damage patterns were studied. The damage zone for the nano-phased system was found to be less than that of the neat system. It was also shown that the interface strength played an important role in the composite’s tensile failure.

Acknowledgements The authors would like to gratefully acknowledge the support of USACERL through grant no.:W9132T-07-p-0011 and National Science Foundation.

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[31] Xia Y., Wang Y. 2001. An improved dynamic Monte Carlo microscopic numerical constitutive model incorporating thermal-mechanical coupling for unidirectional GRP under tensile impact. Composites Science and Technology 61: 997–1003. [32] Ochiai S., Tokinori K., Osamura K., Nakatani E., Yamatsuta Y. 1991. Stress concentration at a notch-tip in unidirectional metal matrix composites. Metall Trans 22A: 2085–2095. [33] Ochiai S., Osamura K. 1986. Stress distribution in a segmented coating film on metal fiber under tensile loading. Journal of Materials Science 21: 2744-2752. [34] Zhou Y., Jiang D., Xia Y. 2001. Tensile mechanical behavior of T300 and M40J fiber bundles at different strain rates. Journal of Materials Science 36: 919-922. [35] Mahesh S., Hanan J. C., Üstündag E., Beyerlein I. J. 2004. Shear-lag model for a single fiber metal matrix composite with an elasto-plastic matrix and a slipping interface. International Journal of Solids and Structures 41: 4197-4218. [36] Huang W., Nie X., Xia Y. 2003. An experimental study on the in situ strength of SiC fibre in unidirectional SiC/Al composites. Composites Part A: Applied Science and Manufacturing 34: 1161-1166. [37] Zhou Y., Wang Y., Xia Y., Mallick P. K. 2003. An experimental study on the tensile behavior of a unidirectional carbon fiber reinforced aluminum composite at different strain rates. Materials Science and Engineering A, 362: 112-117.

In: Composite Materials Research Progress Editor: Lucas P. Durand, pp. 165-207

ISBN: 1-60021-994-2 © 2008 Nova Science Publishers, Inc.

Chapter 5

DAMAGE EVALUATION AND RESIDUAL STRENGTH PREDICTION OF CFRP LAMINATES BY MEANS OF ACOUSTIC EMISSION TECHNIQUES Giangiacomo Minak1 and Andrea Zucchelli2 Department of Mechanical Engineering, Alma Mater Studiorum - Università di Bologna, Viale Risorgimento 2, 40135 Bologna, Italia

Abstract A new approach that integrates acoustic emission (AE) and the mechanical behaviour of composite materials is presented. Usually AE information is used to evaluate qualitatively the damage progression in order to assess the structural integrity of a wide variety of mechanical elements such as pressure vessels. From the other side, the mechanical information, e.g. the stress-strain curve, is used to obtain a quantitative description of the material behaviour. In order to perform a deeper analysis, a function that combines AE and mechanical information is introduced. In particular, this function depends on the strain energy and on the AE events energy, and it was used to study the behaviour of CFRP composite laminates in different applications: (i) to describe the damage progression in tensile and transversal load testing; (ii) to predict residual tensile strength of transversally loaded laminates (condition that simulates a low velocity impact).

Introduction Long fibre reinforced composite laminates are a complex structure at the meso-scale. The fibres embedded in the matrix constitute the lamina and the overlapping of different laminas makes the composite laminate. A consequence of this architecture is the complex behaviour during loading and servicing of components realized by such material, and the multiplicity of different failure mechanisms that determine the damage progression.

1 2

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

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In particular the prevision of the performances (e.g. stiffness, damping) and the strength limits (e.g. tensile, compressive and fatigue) of this kind of material is an important task for the material scientists and engineers. Numerical models for the composite laminate progressive failure are currently developed by the researchers for different applications [1-3] . These models require an experimental validation by means of tests in which damage progression is monitored in a suitable way. Nowadays, different techniques are proposed such as electrical resistance [4], fibre Bragg grating sensors [5], photo-elasticity [6] and acoustic emission [7-9]. On the other hand, residual strength evaluation after fatigue or impact loading is important for the determination of composite components reliability. In fact, laminate composite materials have a wide application in light-weight structural members. In particular fibre-reinforced plastics are increasingly used in airborne structures and the long range passenger airplanes of the future may include many important parts of the fuselage and components made with Carbon Fibre Reinforced Plastic (CFRP). This class of materials is characterized by outstanding strength-to-weight and stiffness-to-weight ratios. Nevertheless their resistance to accidental damage is an important issue for the designer. In particular, CFRPs are very susceptible to internal damage caused by transverse loads such as indentation and impact, while the probability of such loadings occurring during the manufacture, service or maintenance of composite structures is very high [10]. This lack of resistance to low velocity and low energy impact damage [11-13] is one of the main obstacles to a more widespread application of these composite materials, especially in the case of a thermo-set matrix like epoxy. A threshold, conventionally located at 20 m/s, divides the impact problems into two fields, high and low velocity, due to the different types of induced damage [14]. In the low-velocity impact field, a quasi-static loading can simulate the actual behaviour, since the vibrational effects are negligible [15, 16]. In fact, many researchers [17-23] use load-displacement histories to compare structural responses from impact and quasi-static tests and they find that both the dynamic and static responses have corresponding load drops due to failures in the laminates. Low velocity and low energy impact damage usually consists of matrix cracking [24, 25] and delamination [23, 26, 27], while debonding and fibre breaking occur for higher impact energy values [28, 29]. As said, besides the behaviour of the material during an impact, an issue of great interest is the evaluation of the post-impact resistance characteristics of CFRP. In fact, damage due to impact often can be present in the component before it is put into service and loaded. To detect the damage level present in the laminate and the damaged zone area, several techniques are used, such as simple visual inspection , C-scan and X-ray. AE event counts are also utilized to predict the residual tensile strength (RTS) after impact [30]. Due to the importance of delamination, which decreases locally the buckling load, much effort has been spent in researching the compression after impact (CAI) performances of composites [31, 32]. Nevertheless tensile [30, 33, 35] and fatigue properties [36, 37] are also important to predict the component failure.

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In this chapter after a brief introduction about AE, we define a function of the acoustic energy and of the strain energy that allowed us to characterize damage in two different load configurations: tensile testing, on different types of CFRP laminate specimens; transversal loading of quasi-isotropic CFRP laminate plates. Moreover in the second case, the residual tensile strength was related to the AE recorded during the transversal loading phase.

Acoustic Emission Technique Nearly every scientist who makes mechanical experimental testing is familiar with the acoustic emission produced by the material during the loading phase, which sometimes can be heard simply by naked ears. In fact, during a material test or in general when a component is subject to external loads, a rapid stress redistribution can occur due to permanent and irreversible phenomena, caused by damage mechanisms, such as a matrix crack onset or growth, delamination and fibre fracture. During this redistribution, part of the strain energy stored in the material is released in the form of heat and of elastic waves that propagate in the material until they reach the free surface. These transient elastic waves are commonly detected as acoustic waves. Some acoustic emission can be also produced by mechanisms different from damage (such as sliding and friction of two surfaces in contact) and this must be taken into account. The elastic waves propagating at the component surfaces are detected by means of piezoelectric devices that convert the mechanical signal into an electrical one. Even if the AE physical principle is very simple and immediate, the use of this technique is not so straightforward because the acoustic wave propagation in solids, especially in the anisotropic ones as CFRP, is quite complicated. Multiple waves that propagate with different velocities, reflection, refraction, dispersion, and attenuation, may affect the measured signal. Nevertheless some advantages with respect to other non destructive testing techniques can be found in the possibility to monitor a large volume of material by means of few sensors able to locate the damage by triangulation and to make it continuous during real life service. In reality, the acoustic emission is produced within the material itself once loaded at a level that produces some form of damage. In this sense, it is not strictly a non destructive testing method since it is based on passive monitoring of acoustic energy released by the material or structure itself while under load. AE is also used to monitor damage onset and progression in laboratory tests [7-9, 38, 39]. The most difficult AE analysis task is the identification of the damage mechanism, particularly when multiple damage mechanisms are present as in the case of CFRP. In fact, changes in AE due to propagation in the material and to the measurement system may mask characteristics that are related to the damage mechanism. If the AE source is known, as in the case of uniaxial specimens, the quality of AE data can be improved by noise discrimination and rejection.

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Numerous methods have been attempted to identify damage mechanisms from AE data [40,41] and in general many carefully controlled laboratory experiments are necessary to develop relationships between measured AE signals and a specific damage mechanism. The results from AE monitoring have been used in attempts to estimate the residual strength or life of a structure [34]. Most strength assessments from AE are based on empirical correlations developed from failure tests on a large number of nominally identical structures [42]. Even if recently the research on AE, especially as regards composite laminates focused on modal analysis [43-45] in this chapter we consider classical feature-based (also known as parametric) AE analysis [38], in which for each acoustic emission a set of meaningful parameters (shown in figure 1) are detected such as: -

progressive event number counts per event maximum amplitude within the event event duration event energy

Figure 1. Acoustic emission parameters.

This approach has been used in composite laminates with different AE interpretations. Many authors (e.g. Siron and tsuda [40] ) report that fibre breakage produces large amplitude signals while matrix cracking results in much smaller amplitudes and delamination is thought to produce medium amplitude signals. Other studies conclude that matrix cracking causes large amplitude signals while fibre breakage produced low amplitudes [47]. In reality the amplitude depends on a number of factors including the local stress conditions and the energy released. In fact, for example, in [43] a very small increment of matrix crack growth produces a much smaller amplitude signal than a large matrix crack. Moreover, long duration events are attributed to delamination [44].

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More details on AE methods can be found for example in [46]. Our contribution in this field is the introduction of a novel function able to combine the acoustic energy released during an event and the strain energy stored in the material in that moment. This function will be called “sentry” function because it signals important material damage events during the tests, and its integral is related to the total damage and to the residual strength as it will be shown in the next paragraph and in the examples.

The Sentry Function In order to perform a deeper analysis of the laminate behaviour, a function that combines both the mechanical and acoustic energy information [35, 47] is introduced. This function is expressed in terms of the logarithm of the ratio between the strain energy (Es) and the acoustic energy (Ea),

⎛ E (x ) ⎞ f (x ) = Ln⎜⎜ s ⎟⎟ ⎝ E a (x ) ⎠

(1)

where x is the test driving variable (usually displacement or strain). The function f(x) takes into account the continuous balancing between the stored strain energy and the released acoustic energy due to damage. The function f(x) is generally discontinuous and can be described by the combinations of four types of function, shown in figure 2: (I) an increasing function PI(x), (II) a sudden drop function PII(x), (III) a constant function PIII(x) and (IV) a decreasing function PIV(x). These functions are defined over an “acoustic emission domain” (ΩAE) that correspond to the displacement range over which the AE events were recorded. For all AE quantities ΩAE represents the definition domain and outside the function of AE cumulative events, cumulative counts, events energy and all other quantities related to the AE information are null. Dividing the AE domain ΩAE in sub-domain as reported in figure 2 it possible to write the following relation:

Ω AE = Ω AE ,I U Ω AE ,II U Ω AE ,III U Ω AE ,IV

(2)

In that condition the function f can be written as follow:

⎧ PI (x ) ⇔ x ∈ Ω AE ,I ⎪ P (x ) ⇔ x ∈ Ω AE ,II ⎪⎪ II f = ⎨ PIII (x ) ⇔ x ∈ Ω AE ,III ⎪ P (x ) ⇔ x ∈ Ω AE ,IV ⎪ IV ⎪⎩ 0 x ∉ Ω AE

(3)

Giangiacomo Minak and Andrea Zucchelli

⎛E ⎞ f = Ln ⎜⎜ s ⎟⎟ ⎝ Ea ⎠

170

PI

P II

P III

P IV

…. …. ….

ΩAE,I

ΩAE,III

ΩAE,IV

ΩAE,II ΩAE Displacement

Figure 2. The basic functions PI , PII , PIII and PIV, used to describe the function f.

If there is more than one sub-domain ΩAE,k, k∈{I,II,III,IV}, it is possible to write the following relation:

⎞ ⎞ ⎛ n IV ⎞ ⎛ n III ⎞ ⎛ n II ⎛ nI Ω AE = ⎜⎜ U ΩiAE ,I ⎟⎟ U ⎜⎜ U ΩiAE ,II ⎟⎟ U ⎜⎜ U Ω iAE ,III ⎟⎟ U ⎜⎜ U ΩiAE ,VI ⎟⎟ ⎠ ⎝ i =1 ⎠ ⎝ i =1 ⎠ ⎠ ⎝ i =1 ⎝ i =1

(4)

where nI, nII, nIII, and nIV are the number sub-domain corresponding to the trend type I,II, III and IV respectively. Then functions PI,PII, PIII and PIV can be written as follow:

⎧ PI ,1 (x ) / x ∈ Ω1AE ,I ⎪ PI (x ) = ⎨ ... ⎪ P (x ) / x ∈ Ω n I AE ,I ⎩ I ,n I

⎧ PII ,1 (x ) / x ∈ Ω1AE ,II ⎪ PII (x ) = ⎨ ... ⎪P (x ) / x ∈ Ω n II AE ,II ⎩ II ,n II

⎧ PIII ,1 (x ) / x ∈ Ω1AE ,III ⎪ PIII (x ) = ⎨ ... n III ⎪P ⎩ III ,n III (x ) / x ∈ Ω AE ,III

⎧ PIV ,1 (x ) / x ∈ Ω1AE ,IV ⎪ PIV (x ) = ⎨ ... n IV ⎪P ⎩ IV ,n IV (x ) / x ∈ Ω AE ,IV

(5)

From the physical point of view the part of f(x) characterized by an increasing trend, type I, represents the strain energy storing phases. The slope of PI,i (x) functions decreases during

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the test because the energy stored in the material tends to its limit and the AE cumulate energy (Ea) increases due to the progression of material damage. When a significant internal material failure happens there is an instantaneous release of the stored energy that produces an AE event with a high energy content. This fact is highlighted by the sudden drops of the function f(x) that can be described by functions of type II: PII(x). In figure 3 are reported some examples of combinations of parts of the function f(x), and in particular in figure 3a it possible to note three trends of type I (PI,1, PI,3 and PI,2) respectively connected by two functions of type II (PII,1 and PII,2).

S1-

P II,1

P I,1

A

S2+

S1+

f

P II,2

S2-

P I,3

P I,2

Displacement

P II,3

P II,4

B

P I,4

f P III, 1

P IV,1

Displacement

P II,5

C

P I,5 f P IV,2 Displacement

Figure 3. Examples of compositions of the basic function describing common parts of the function f.

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During the first phase, PI,1, the material is storing the strain energy, but when an internal limit is reached a failure happens and the function f presents a first sudden drop (PII,1). After this first drop the material starts again to store strain energy and it is interesting to notice that the slope S1- of PI,1, before the event, is equal to the initial lope of PI,2 (S1+). The failure that caused the first fall in this case does not affect significantly the material integrity (i.e. PII,1 is not related to an important material modification). So the value of f(x) is reduced due to the internal energy release, but the material strain energy capability is not compromised. Different is the case represented by the second fall PII,2 in which the slope before (S2-) and after (S2+) are different. This means that after PII,2 the material strain energy storing capability is changed and because of an important material modification. In particular it is interesting to notice that S2->S2+. After this event it is also reasonable to hypothesize that the material damage increases. It was also previously observed [47], that typically after one or two drops of f(x) characterized by the slope variation of PI(x) (S->S+) the next drop is followed by a function of type III or IV. In figure 3B and figure 3C are reported two possible situations that happen when the material damage has reached an important level. In the case reported in figure 3B at the end of the storing energy phase PI,4 there is an instantaneous strain energy release that causes the falling phase PII,3. The following constant behaviour of f(x), described by PIII,1, is due to a progressive strain energy storing phase that is superimposed to an equivalent material damage progression. The next fall PII,4 is followed by a decreasing function PIV,1. The decreasing function type PIV, is related to the fact that the AE activity is greater then the material strain energy storing capability: the damage has reached a maximum and the material has no resources to bear the load. Then the phase PIV,1 in figure 3B indicates that the material is totally damaged. In figure 3B between PIII,1 and PIV,1 there is a drop function PII,4 that is due to an important failure inside the material, but this situation is not the general one. In fact, sometimes it happens that after a constant trend of f(x) there are no more drops and f(x) gradually decreases. This behaviour is due to a gradual damage progression inside the material and no important failure events happen. At the opposite, the case represented in figure 3C is related to a critical event inside the material: the strain energy storing phase PI,5 is followed by a sudden drop PII,5 and then it follows a decreasing phase PIV,2. This situation generally happens at the end of a test or, if it happens at the beginning it reveals the presence of some defects inside the material. The described analysis shows that the function f(x) can be usefully implemented to describe the material damage progression because it takes into account both mechanical and acoustic information. Summarizing we have that the increasing part of f(x) reveals the material strain energy storing capability, the falls reveal the instantaneous release of the stored energy due to failures, the constant and decreasing trends of f(x) prelude and describe an important failure of the material structure. In the following sections two examples regarding the application of different strategies about the use of the function f(x) to describe the composite material damage are developed. In the first example the function f(x) is used to determine the damage progression of five different types of laminates under in plane tensile loading. In particular considering the function f(x) it was possible to identify the most important material failure highlighted by the sub function PII,i. Then considering the stressstrain information limited in the sub-domain ΩAE,I, ΩAE,III and ΩAE,IV, it was determined the progressive stiffness reduction of the material. In the first example the local structure of the function f(x) is used to interpret the stressstrain data in order to determine the material damage model. While in the second example

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the function f(x) is used to estimate the overall material damage when composite laminate are subjected to an out-of-plane load. Because of its synthesis capability the function f(x) can be used to summarize the whole material damage history and in the case of the transversal load tests the integral of the function f(x), Int(f), over the domain ΩAE was calculated: Int ( f ) =

⎛E Ln ⎜ s ⎝ Ea Ω AE

∫

⎞ ⎟ dΩ ⎠

(6)

The values of Int(f) are influenced by the complete trend of f(x) and by the extension of the AE domain.

Applications Case study 1: materials and method The composite laminates used for the tensile tests were different in terms of lay-up (unidirectional laminates (UD), angle-ply laminates (AP) and quasi-isotropic laminates (QI), fibre volume percentage and laminates thickness. The pre-pregs were made by T-300 graphite fibres and epoxy matrix. Specimens were cured in autoclave then cut by a diamond saw. The characteristics of these three types of laminates are reported in Table 1. For each type ten specimens were tested. Table 1. Types of laminates used for the tensile tests

Laminate type

ID

Lay-up

Fibre Volume (%)

Thickness (mm)

Unidirectional

UD

[0°]8

60

1.4

Angle ply

AP

[±45°]4S

30

2.8

QI1

[0°,90°, ±45°]4S

60

1.4

QI2

[0°, ±45°, 90°]4S

60

1.4

QI3

[0°,90°, ±45°]4S

30

2.8

Quasi isotropic

Specimen dimensions were 250 mm in length and 25 mm in width for all types of laminates as recommended by ASTM 3039M for AP and QI lay-ups and the gage length was 140 mm, as shown in figure 4A. Uniaxial tensile tests were done under displacement control using an INSTRON 8032 with a 100 kN load-cell, and speed of 0.05 mm/sec. In order to reduce the acquisition of spurious acoustic external signals [47] two noise gates were assembled in a series configuration with the specimens as shown in figure 4B.

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During the test, the AE were monitored by a Physical Acoustic Corporation (PAC) PCIDSP4 device with two transducers PAC R15 setting up the amplitude threshold at 40 dB. 140 mm 110 mm 25 mm

A

AE transducers Grips AE transducers

Specimen Grips 250 mm

B

Figure 4. (A) specimen for tensile test and test set-up with AE transducers position, (B) the specimen fixing grips and external AE noise insulation devices.

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The AE transducers were placed in a linear configuration located at a distance of 110 mm, as shown in figure 4A. The Elastic modulus was measured by means of HBM LY416/350 strain-gauges. Case study 1: results and discussion The experimental results analysis and discussion is here organized in two phases: the first one in which the only classical mechanical information (stress and strain) are considered, and the second one in which the AE information is analyzed and related to the material mechanical response. The stress-strain curves in Figure 5 show the effect of the fibre orientation and volume percentage and also the small, but not negligible, influence of the plies sequence in the laminates. In particular the AP diagram was qualitatively different from the other laminate ones because of the different failure mechanism that was fibre-dominated for UD and QI and matrix-dominated for AP. In fact, the mechanical response of AP laminates with respect to in plane tensile load was strongly nonlinear and the ultimate stress value was about 60 MPa. The two aspects that determined the behaviour of AP laminate were the fibre direction (±45°) and the low volume percentage of fibres. Visual inspection revealed also a marked necking due to the high percentage of matrix forming the composite and the sliding of fibres in it. 2500

Stress (MPa)

2000

UD QI2 QI1 QI3 AP

1500

1000

500

0 0.000

0.010

0.020

0.030

0.040

0.050

0.060

0.070

0.080

0.090

Strain Figure 5. Examples of stress-strain diagram for the different types of laminates.

From the stress-strain diagram it was possible to determine some of the most significant information regarding the mechanical response of laminates. In particular the elastic modulus (EX) and the ultimate strength were estimated considering all tested specimens. In table 2 are summarized the theoretical Elastic modulus, estimated by means of the lamination theory, and

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the corresponding experimental values, while in table 3 the ultimate strain and stress values are reported. Table 2. Theoretical calculation and experimental measure of Elastic modulus of each laminate type Ex (MPa) Theoretical

Experimental

ID

M.V.

M.V.

S.D.

UD

197000

190000

16000

AP

23000

22000

3000

QI1

75000

74000

6000

QI2

75000

76000

3000

QI3

77000

78000

3000

Table 3. Ultimate strain and stress values of tested laminates Ultimate Strain

Ultimate Stress (MPa)

ID

M.V.

S.D.

M.V.

S.D.

UD

0.019

0.001

2157

104

AP

0.084

0.008

62

7

QI1

0.012

0.002

571

35

QI2

0.010

0.001

600

44

QI3

0.012

0.001

322

19

The main acoustic parameters that have been considered are: counts per AE event and AE event energy (Ea). Both parameters have been related by means of double entry diagram with the stress-strain behaviour. In the following figures 6-10 examples of these diagrams, one for each type of laminate, are reported.

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Figure 6. UD laminate type, cumulative counts, AE energy, function f and stress versus displacement.

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Figure 7. AP laminate type cumulative counts, AE energy, function f and stress versus displacement.

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Figure 8. QI1 laminate type, cumulative counts, AE energy, function f and stress versus displacement.

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Figure 9. QI2 laminate type, cumulative counts, AE energy, function f and stress versus displacement.

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Figure 10. QI3 laminate type, cumulative counts, AE energy, function f and stress versus displacement.

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From figures 6 to 10 it is possible to observe that the five types of laminates have different behaviours from the AE point of view. A preliminary observation can be done considering the AE domain that can be used to identify the Free Failure Domain (FFD), i.e. the strain domain over which no failures are detected. Considering the ratio between ΩAE and the strain at rupture it can be seen that the percentages of the FFD over the all strain domain are the following: 11% in the case of UD laminates, 0.5% in the case of AP laminates, 1% in the case of QI1 laminates, 30% in the case of QI2 laminates and 1.4% in the case of QI3 laminates. The estimated percentage values of FFD indicate the different attitude to the damage onset of each type of laminate, and in particular it is interesting to note that the QI2 laminate type is the one that has the greater capability to be strained without significant damage. On the contrary the AP laminate types are the most sensitive to the applied strain and reveal an early damage onset, probably due to the high matrix percentage content and to fibre orientation (±45°). Considering the diagram of AE event cumulative counts, figures 6A to 10A, it can be observed that only in the case of AP laminates the slope of the diagram is quite constant during the all test. For the other laminate types, UD and QI, the cumulative counts reveal an initial trend with low slope values that progressively increase during the test. Such behaviour can be interpreted considering the different damage attitude of the laminates and their structure. In the case of AP laminates the mechanical behaviour was dominated by the matrix deformation and cracking. The effect of fibres in AP laminates did not influence the material behaviour and, on the contrary, as observed during experiments at the early stage of tests, fibres promoted matrix breakage and spalling. Such interpretation of AP laminates behaviour is also supported by the AE energy diagram, figure 7B, where it is noticeable the presence of AE events with an energy content (the maximum AE event energy is about 4.0⋅10-4 J) that is typical of composite laminate matrix failures [47]. Different behaviour was observed in the case of UD, QI1 and QI2 laminates. Considering, for example, diagrams of figures 6A, 8A and 9A, for the UD, QI1 and QI2 laminates respectively, it is possible to note the presence of a strain domain where the cumulative count rate is quite low. For these laminates, during the initial test stage no significant failures can be detected and considering also the energy diagrams, figures 6B, 8B and 9B it is possible to assume that the sources of AE event are mainly due to matrix cracks onset. Comparing in particular the cumulative counts and energy diagrams of QI1 and QI2 laminates it is interesting to note that in the case of QI2 the maximum number of cumulative counts (∼ 3⋅105 counts) is lower than the one of QI1 laminate (∼ 2⋅106 counts), but, at the same time, the maximum AE event energy of QI2 (∼ 1.2⋅10-3 J) is comparable to the one of QI1 laminate (∼ 2.2⋅10-3 J). This behaviour can be understood considering the different delamination strength of the two laminates. In fact, as reported in [47, 48], delamination is a possible failure mechanism for laminates of type QI1, and, on the contrary, it is not a typical failure for laminates of type QI2. So in the case of QI1 the maximum number of counts is greater than in the case of QI2 thanks to the contribution of events caused by inter-laminar fractures and delamination. Nevertheless the maximum AE energies for the QI1 and QI2 laminates are comparable because the final crisis of both materials is characterized by a fibre breaking process that determines the release of an AE event with an high energy content. The behaviour of QI3 laminate is quite different if compared to QI1 and QI2. In particular the cumulative counts trend, figure 10A, shows a consistent release of AE events at the early test stage, but the total number of cumulative

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counts over all test (∼ 1.4⋅104 counts) is lower than in the case of QI1 and QI2. Considering the AE energy diagram of QI3, figure 10B, it is possible to observe that events with a high energy content happens also in the early-middle stage of the test, differently than in the case of QI1 and QI2 where events with a high energy content appear only at the final test stage. All these facts are related to the QI3 ply composition: the matrix volumes of each ply influences the dominant material failure mode, the matrix cracking, as in the case of AP. All the previous considerations can be effectively summarized and completed considering the diagrams of the sentry functions f. In the case of UD laminate the construction of the function f reveals the initial material damage, figure 6C, that is highlighted by the sub-function PI,1 and PII,1. This initial damage can be related to some material internal adjustment and to the onset of some internal cracks but, as revealed by the following PI,2, such damages do not compromise the material capability of storing strain energy. The following PIV,1 sub-function reveals that the internal cracks propagate and that the material storing energy capability is progressively reduced until a sequence of drops due to splitting (PII,2, PII,3 and PII4) mixed with some strain energy storing phases (PI,3, PI,4, PI,5) that prelude to the first and most critical drop PII,5. This drop is mainly related to the free edge delamination and after this a new but small strain energy storing phase starts: PI,6. During this phase all the laminate plies work as springs in parallel and go on storing strain energy. After this phase next drops, PII,6 and PII,7, mixed with a slowly increasing part of f, PI,7, and two constant trends for f, PIII,1 and PIII,2, prelude the final crisis of the laminate. Considering the diagram in figure 7C of the AP laminate it can be observed that the structure of the sentry function is simpler if compared to the UD case. In fact only three subdomains of strain energy storing phases, PI,1, PI,2 and PII,6, and three drops, PI,1, PI,2 and PI,3, characterize the structure of the sentry function for the AP laminate reported in figure 10C. The smooth trend of the sub-function of type I reveal the modest capability of the material to store the strain energy and this fact is due to the high matrix volume percentage in each ply and to the fibre orientation. The initial trend of the sentry function of QI1 is characterized by a first material crisis, PIV,1, followed by a drop, PII,1. Such behaviour is due to an initial material adjustment and to some inter-laminar cracks onset that will contribute to delamination process. The next phase is characterized by two strain energy storing phases, PI,1 and PI,2, respectively followed by a sudden drop, PI,2, and a decreasing sub-function, PIV,2. In particular the sudden drop and the decreasing sub-function indicate the onset of the material crisis due to delamination and transversal cracks. After the sub-function PIV,2 the sentry function is characterized by a complex combination of sudden drops and constant sub-functions. This behaviour indicates that the material integrity is compromised and that each ply is progressively damaged until the final crisis. In this way it is interesting to notice that after the PIV,2 there are four sudden drops indicating the important crisis of each basic ply type (0°, +45°, -45°, 90°) that constitutes the original laminate QI1. In the case of QI2 the sentry function has a different trend with respect to the case of QI1. In fact at the test beginning damage is not appreciable and a sub-function of type I, PI,1, indicates a strain energy storing phase. After the first drop, PII,1, a consistent material damage indicates the reduced strain energy storing capability. Such material damage is mainly due to a global laminate loss of strength: the absence of delamination contributes to the cracks distribution and growth inside all the deformed material volume and this creates the condition

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for a general crisis of the laminate. In fact the PII,1 is followed by a small series of strain energy storing phases, PI,2 and PI,3, and a combination of sub-function of type II, III and IV highlighting a great material damage. The QI3 is characterized by a sentry function that mixes the behaviour of the studied UD and AP laminates. At the test beginning is visible a combination of sub-functions of type I and of type II with a predominant strain energy storing phase. This phase is characterized by the sequence of the following sub-functions: PI,1, PII,1, PI,2, PII,2 and PI,3. A delamination failure is revealed by the slope change of f corresponding to PII,3. After this failure a reduced energy storing capability is revealed by the PI,4 and the following drop PII,4 is due to the failure of the weakest ply inside the laminate. This first ply crisis is followed by a subfunction of type I, PI,5, that has a smooth trend due to the previous material damage. The damage corresponding to the sub-function PII,6 is due to the crisis of one of the stronger plies. After this laminate crisis the energy storing phase represented by the PI,6 is due to the stresses redistribution between the undamaged plies that are now working as springs in parallel. The analysis here described has been used to perform a quantitative estimation of the laminate damage and in particular the sentry function of each laminate has been used to perform a discretization of the stress-strain curve. As an example of this analysis we report the case of UD laminate type. 2500

40 35

2000

f

Stress-Strain

f

25

1500

20 1000

15

Stress (MPa)

30

10

Drops

500

5 0 0.000

0.003

0.005

0.008

0.010

0.013

0.015

0.018

0 0.020

Strain Figure 11. Stress and f diagram versus strain, the most key drops of f are highlighted by means of gaps diagram.

The analysis of the sentry function in figure 11 allows the identification of 10 drops on the basis of which the stress-strain curve was divided in order to calculate the Elastic modulus.

Damage Evaluation and Residual Strength Prediction of CFRP Laminates …

185

2500

Stress (MPa)

2000

1500

Experimental 1000

Model

500

0 0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

Strain Figure 12. Stress-strain diagram and its discretization according to the key-drops. 2500

2.0E+05

2000

1.6E+05

Stress (MPa)

1.4E+05 1500

1.2E+05 1.0E+05

1000

8.0E+04 6.0E+04 Stress - Strain Model Young Modulus

500

4.0E+04

Young modulus (MPa)

1.8E+05

2.0E+04 0 0

0.0025

0.005

0.0075

0.01

0.0125

0.015

0.0175

0.0E+00 0.02

Strain Figure 13. Discretized stress-strain curve and Elastic modulus according to the key-drops analysis.

By means of a linear regression the Elastic modulus of the stress-strain curve corresponding to each strain interval was estimated. In figure 12 the discretized stress-strain curve is reported and overlapped to the original diagram. In figure 13 the discretized stressstrain curve and Elastic modulus values are reported.

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Table 4. Lower, upper and intermediate strain values used for the stress-strain curve discretization, Elastic modulus (Young) and Damage (D)

Mean Values Low strain

Up strain

Ref. Strain

Young [Mpa]

D

UD

0.0000 0.0031 0.0104 0.0120 0.0154

0.0030 0.0097 0.0109 0.0154 0.0190

0.0015 0.0064 0.0106 0.0137 0.0172

191200 137500 105653 88522 31211

0.029 0.302 0.464 0.551 0.842

AP

0.0000 0.0012 0.0085 0.0252 0.0308

0.0006 0.0037 0.0137 0.0307 0.0839

0.0003 0.0025 0.0111 0.0280 0.0573

22865 6340 1666 329 182

0.006 0.724 0.928 0.986 0.992

QI1

0.0000 0.0015 0.0031 0.0060 0.0081 0.0089

0.0015 0.0031 0.0040 0.0072 0.0086 0.0115

0.0008 0.0023 0.0035 0.0066 0.0083 0.0102

74300 71600 62000 45175 32976 22151

0.047 0.082 0.205 0.421 0.577 0.716

QI2

0.0000 0.0034 0.0044 0.0080 0.0087

0.0029 0.0041 0.0075 0.0087 0.0099

0.0015 0.0038 0.0060 0.0084 0.0093

72100 67050 59050 41022 28195

0.051 0.118 0.223 0.460 0.629

QI3

0.0000 0.0003 0.0008 0.0016 0.0041 0.0050 0.0114

0.0003 0.0007 0.0016 0.0035 0.0048 0.0114 0.0120

0.0001 0.0005 0.0012 0.0025 0.0045 0.0082 0.0117

77608 53487 45048 27500 24973 20900 12997

0.005 0.314 0.422 0.647 0.680 0.732 0.833

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187

It has to be noticed that building up the diagram in figure 13 the Elastic modulus values have been associated to the mean values of the strain range that have been used to discretize the stress-strain curve. Using the calculated values of the Elastic modulus it was also estimated the values of the damage by means of its conventional definition: D = 1- E(s)/E0, where E0 is the Young modulus of the undamaged material. In Table 4 are summarized the strain intervals, low-strain and up-strain, that have been used to discretize the stress-strain curves of all specimens for each type of laminates, the corresponding strain mean value to which the estimated Elastic modulus and Damage values are associated. 2500

1 0.9

2000

0.8

1500

0.6 0.5

1000

0.4

Damage

Stress (MPa)

0.7

0.3 Stress - Strain Model Damage

500

0.2 0.1

0 0

0.0025

0.005

0.0075

0.01

0.0125

0.015

0.0175

0 0.02

Strain Figure 14. Discretized stress-strain curve and Damage according to the key-drops analysis.

The information summarized in Table 4 can be usefully implemented in FEA software to model the considered composite laminate progressive failure behaviour. In figure 15 are graphically represented the Elastic modulus and the Damage values plotted considering as strain values the mean values reported in Table 4

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A 250000

E (MPa)

200000

150000 UD 100000

AP QI1 QI2

50000

QI3 0 0.000

0.010

0.020

0.030

0.040

0.050

0.060

0.070

0.080

0.090

Strain

B 1.0 0.9 0.8 0.7

D

0.6 0.5 UD

0.4

AP

0.3

QI1

0.2

D(s ) = 1 −

0.1 0.0 0.000

0.010

0.020

0.030

0.040

0.050

E(s ) E0

0.060

QI2 QI3

0.070

0.080

0.090

Strain Figure 15. Trends of (A) Elastic modulus, E, and (B) Damage, D, versus strain for each type of laminate; for the damage calculation, of each laminate, E0 is equal to the mean value of the Young modulus as reported in Table 2 that correspond to the Young modulus of the undamaged laminate.

Details about the damage of each type of laminate are reported in figures 16 to 20.

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189

1.0 0.9 0.8 0.7

D

0.6 0.5 0.4 0.3 0.2

UD

0.1 0.0 0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

0.018

0.020

Strain Figure 16. Damage for UD laminate types.

1.0 0.9 0.8 0.7

D

0.6 0.5 0.4 0.3 0.2 AP

0.1 0.0 0.000

0.010

0.020

0.030

0.040

0.050

Strain Figure 17. Damage for AP laminate types.

0.060

0.070

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Giangiacomo Minak and Andrea Zucchelli 1.0 0.9 0.8 0.7

D

0.6 0.5 0.4 0.3 0.2 QI1

0.1 0.0 0.000

0.002

0.004

0.006

0.008

0.010

0.012

Strain Figure 18. Damage for QI1 laminate types.

1.0 0.9 0.8 0.7

D

0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.000

QI2 0.002

0.004

0.006

0.008

Strain Figure 19. Damage for QI2 laminate types.

0.010

0.012

Damage Evaluation and Residual Strength Prediction of CFRP Laminates …

191

1.0 0.9 0.8 0.7

D

0.6 0.5 0.4 0.3 0.2 QI3

0.1 0.0 0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

Strain Figure 20. Damage for QI3 laminate types.

Case study 2: materials and method Eighteen graphite/epoxy composite square laminate plates 250x250 mm2 were studied. Their thickness was 1.6 mm. They have been made in autoclave from pre-pregs by stacking eight unidirectional plies with quasi-isotropic orientations [0,90,45,-45]s. The specimens were placed in a circular clamping fixture with an internal diameter of 200 mm and they were loaded at the centre: an hemispherical hardened steel ball with a radius of 7 mm was indented in the top centre point of the laminate by means of a servo-hydraulic Instron 8033 testing machine controlled by an MTS Teststar II system and equipped by a 25kN load cell. The specimens were loaded monotonically out-of-plane in control of displacement and the head speed was 0.05 mm/s. The tests were stopped at three different damage levels, one (Low) corresponding to the load value of 2 kN, the second (Medium) corresponding to the first load drop in the loaddisplacement curve and the third (High) to the complete perforation of the plate. During the test, the AE has been monitored by a Physical Acoustic Corporation (PAC) PCI-DSP4 device with four transducers PAC R15 setting up the amplitude threshold at 40 dB. In figure 21a it is possible to see the fixture system equipped with AE piezoelectric sensors and in figure 21b the complete experimental setup. After each quasi-static test, the damaged plate was sliced by a diamond saw to obtain tensile specimens with the same geometry suggested by ASTM D 5766 for open hole testing of CFRP, a width of 40 mm and a length of 250 mm. The indented zone was in the centre of these tensile specimens and the whole damaged zone was included in the specimen width. The damaged zones size was previously identified by means of the localization tool of the AE system as it is shown in figure 22 for the High damage level.

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Figure 21. (A) fixture system, (B)experimental device.

AE sensors

Detected AE sources

External fibers direction

Cutting directions

40 mm 250 mm Figure 22. Damaged zone area detected by AE emission and tensile specimens cutting directions for the High damage level.

Some plates were also analyzed by C-Scan and MicroCT and in these cases the value of damaged area evaluated by AE was confirmed. Nine plates had the external ply fibres oriented in the direction of the specimen axis and other nine had the external ply fibres orthogonal to the specimen axis, so that two different

Damage Evaluation and Residual Strength Prediction of CFRP Laminates …

193

stacking sequences were produced, respectively [0,90,45,-45]s and [90,0,45,-45]s , as shown in figure 22. Analogous tensile tests were run, for each stacking sequence, on nine undamaged specimens cut from undamaged zones of the same plates in order to get reference values. Tensile tests were performed by means of the same servo-hydraulic machine in displacement control with a head speed of 0.02 mm/s and a gauge length of 150 mm. Also in the case of tensile tests the AE have been monitored by a Physical Acoustic Corporation (PAC) PCI-DSP4 device with two transducers PAC R15, setting the amplitude threshold at 40 dB. Case study 2: Results and Discussion Transversal load test In figure 23 are shown macro photos of the loaded side (a) and of the back side (b) of damaged plates for the three different damage levels.

Figure 23. Damaged zones (inside the dotted circles) for the three damage levels (Low, Medium, High) on the loaded surface (a) and on the back surface (b).

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In the picture of the low damage level lamina, both in the loaded side (L-a) and in the back side (L-b) the indentation is barely visible by the naked eye. The medium level damage laminas present a slightly larger mark on the loaded side (M-a) and some fibre breakage with matrix leakage on the back side (M-b). Finally the highly damaged lamina pictures (H-a) and (H-b) show fibre failure on both sides.

Figure 24. Fibres failure on the loaded surface for the Low load level: fractures on fibres are evidenced by the arrows.

Figure 25. Tensile fibre failure on the back surface for the Medium load level: fracture on one fibre is evidenced by the arrows.

Damage Evaluation and Residual Strength Prediction of CFRP Laminates …

195

Investigating more deeply the loaded side of the low damage level laminate it was possible to find a number of broken fibres due to local shear [29], as it is shown in the SEM image of figure 24. A different failure mode for the fibre is shown in the SEM image of figure 25. 3

Load (kN) 2.5

3

1.5

2

2.5

1

1.5

2

P

2

2.5

3

M 0.5

1

1.5

B

1

0

0.5

0

2

4

6

8

10

12

14

0

0.5

0

2

4

6

8

10

12

14

0 0

2

4

6

8

10

12

Displacement (mm)

14

3.0

30.0

2.5

25.0

2.0

20.0

1.5

15.0 Load

1.0

10.0

Energy 0.5

5.0

0.0 0.0

2.0

4.0

6.0

8.0

10.0

12.0

Indentation: Total Strain Energy (J)

Indentation: Maimum Load (kN)

Figure 26. Transversal load-displacement curves for the three damage levels.

0.0 14.0

Indentation: Maximum displacement (mm) Figure 27. Maximum displacement, maximum load and total strain energy of the transversal loading tests.

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In this case, referred to the back side of a medium damage level lamina, matrix-fibre debonding is evident and the tensile fibre fracture surfaces are orthogonal to their axis and widely separated. Load-displacement curves for the three damage levels are reported in figure 26 while the maximum load, maximum displacements and total strain energy recorded in each test can be found in figure 27. Acoustic emission analysis of the transversal load test A parametric AE analysis was performed considering the acoustic energy, the cumulative events and the cumulative counts per event. As an example, in figure 28 these three AE parameters are plotted together with the respective load-displacement diagrams. For the AE diagrams reported in figure 28, the ΩAE is defined in the displacement range [0.7; 11.1] mm. It is interesting to notice the presence of the FFD that represents a phase of the test during which no damages are induced in the laminate (i.e. [0; 0.7]mm). The diagrams of cumulative events and cumulative counts versus the displacement are characterized by a general monotonic increasing trend, a quite similar shape and, for each diagram, at the same displacement value there are significant slope variations. In particular from figure 28A and figure 28B, inside the ΩAE, it is possible to notice a first part of the cumulative events and counts diagram characterized by low values and a smooth trend and dominated by AE events with a low number of counts and a low AE event rate. In figure 28 this first test phase has been identified in an AE sub-domain marked as Z1 limited, for the considered example, in the displacement range of [0.7; 6.3] mm. Considering also the energy diagram, figure 28C, it is possible to observe that only one event in Z1, at the displacement value of 3.6 mm, has an appreciable acoustic energy (over 3.0 10-6 J) and from the statistical analysis it was observed that only 5 events have an energy over 1.0 10-6 J, that can be considered typical for fibres breakage in bending. The sub-domain Z1 is physically dominated by material adjustment (especially fibre alignment) and by matrix deformation and matrix crack onset that are typically related to AE events with a small number of counts and low energy content [47]. The presence of few AE events with a high value of energy (over 1.0 10-6 J) can be physically related to the breakage of some fibres as was previously noticed by the SEM image (figure 23) even if probably the energy content of these events in most cases should be low since these fibres are not loaded in tension. So during this first test phase (Z1) no significant damage is induced to the material, as it was noticed during the visual inspection of laminate surfaces, a small hemispherical mark in the matrix is appreciable (figure 23 L-a & L-b), and as pointed out by means of the SEM image, figure 24, a certain amount of fibres are broken. After this first phase there is a considerable increase of the AE activity represented by increased values of the slope in both cumulative event and counts diagrams. The test phase characterized by a high AE activity presents two other slope variations that can be used to define three other sub-domains: Z2 over the displacement range [6.3; 7.2] mm, Z3 over the displacement range [7.2; 8.1] mm and the Z4 over the displacement range [8.1; 11.1] mm. It is important to point out, as initially noticed, that all the slope variations in the cumulative event and counts diagram correspond to the same indenter displacement values. This is mainly related to the fact that, generally, the damage growth inside the material causes an increase of the total AE activity (events with an increased rating and an increased number of counts per

Damage Evaluation and Residual Strength Prediction of CFRP Laminates … 3.0

1.0E+04 Event CUM Load

9.0E+03 8.0E+03

A 2.5

7.0E+03

2.0

6.0E+03 1.5

5.0E+03 4.0E+03

Load (kN)

Event CUM

197

1.0

3.0E+03 Z1

2.0E+03

Z2 Z3

Z4

0.5

1.0E+03 0.0E+00

0.0 0

2

4

4.5E+05

8

10

12

14 3.0

Displacement (mm) Count CUM

4.0E+05

B

2.5

Load

3.5E+05 3.0E+05

2.0

2.5E+05 1.5 2.0E+05 1.5E+05

Load (kN)

Count CUM

6

1.0

1.0E+05 0.5 5.0E+04 0.0E+00 9.0E-04

0.0 0

2

4

6

8

10

12

Eac Displacement (mm) Load

8.0E-04

14

3.0

C 2.5

7.0E-04 2.0

5.0E-04 1.5 4.0E-04 3.0E-04

Load (kN)

Eac (J)

6.0E-04

1.0 Ω AE

2.0E-04

0.5 1.0E-04 0.0E+00

0.0 0

2

4

6

8

10

12

14

Displacement (mm)

Figure 28. Main AE parameter and load diagram versus displacement; (A) cumulate of AE events, (B) cumulate of AE counts per event, (C) AE energy of each event.

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Giangiacomo Minak and Andrea Zucchelli

event [47]). The slope variations in all diagrams are associated to events with a high energy content. In fact, as shown in figure 28C, the transitions between Z1 and Z2 and then between Z2 and Z3 are determined by two events with an energy content higher then 1.0 10-6 J probably caused by fibres breakage. It is also interesting to notice that in the sub-domain Z2 there is a considerable number of events with a high energy content (22 events with an energy higher then 1.0 10-6 J) probably related to the breakage of quite large number of fibres. This fact was also confirmed by the visual and SEM analysis (figure 23 M-a & M-b, figure 25). Similar considerations can be developed considering the sub-domains Z3 and Z4 with a number of 27 and 100 events with energy content higher then 1.0 10-6 J respectively, that probably reveal the progressive fibres breaking process. Besides the analysis of the AE information it is interesting to relate the AE diagrams to the mechanical response of the laminate. In particular it is possible to notice that both the first zone Z1 and the second zone Z2 end at the two important load drops. In the sub-domain Z1 the load-displacement diagram has a monotonic increasing trend with an increasing slope and this is a direct consequence of the fact that the system stiffness is increased by the transition from the bending to the membrane behaviour [35]. So this sub-domain characterizes the test phase during which no important damage is induced and the main part of the mechanical energy is stored in the material as strain energy, in fact only a small part of the mechanical energy is dissipated by fibres adjustment or alignments and matrix crack onset. After the first load drop, the sub-domain Z2 begins, where the load displacement diagram is again increasing monotonically, but contrary to what is observed in Z1 the slope is decreasing. This is related to the material damage corresponding to the first load drop. In fact, as noticed by visual inspection and SEM analysis, after the first load drop the fibres breakage and the brittle matrix leakage reduce the local resistance of the laminate. So in the subdomain Z2 the strain energy storing capability of the laminate is reduced if compared with the laminate behaviour in Z1. In the sub-domain Z3 the load-displacement diagram is characterized by a monotonic increasing trend with a consistent decreasing of the slope. After the second load drop delamination and fibre breakages compromise the local out-of-plane resistance of the laminate and the energy storing capability is significantly reduced. The third zone ends when the load reaches a relative maximum value and then it decreases. The sub-domain Z4 is characterized by a slowly decreasing trend of the load with the total penetration of the indenter in the laminate and the AE event are mainly caused by the delamination, matrix cracking and leakage, fibre breaking and bending-pull-out. At the end of the loaddisplacement diagram there is a new increasing trend due to the contact of the support of the spherical indenter with the laminate surface. The physical evidence of the failure modes that happens during the loading history in Z4 can be reconstructed by visual and SEM inspection as shown in figure 23 (H-a& H-b) and figure 24. An example of implementation of the function f(x) for this case study is shown in figure 29 where the strain energy (Es), the cumulative AE event energy (Ea), the load and the f(x) diagrams relative to the same test reported in figure 28 are shown.

Damage Evaluation and Residual Strength Prediction of CFRP Laminates … 3.0

1.8E+01 1.6E+01

Es Load

1.4E+01

2.5

A

1.0E+01 1.5 8.0E+00 6.0E+00

Load (kN)

2.0

1.2E+01 Es (J)

199

1.0

Z1

4.0E+00 Z2 Z3

2.0E+00

0.5

Z4

0.0E+00 2.5E-03

0.0 0

2

4

6

8

10

12

14

3.0

Displacement (mm) Eac CUM Load

2.5

B

2.0 1.5E-03 1.5 1.0E-03

Load (kN)

Ea Cumulate (J)

2.0E-03

1.0 5.0E-04

0.5

Ω AE 0.0E+00

0.0 0

2

4

6

8

10

12

14

Displacement (mm)

2.5E+01

3.0

C 2.5

2.0E+01 PII,2

2.0

P II,3

1.5E+01

1.5

f

PII,4 1.0E+01

PI,1

P I,2

P I,3

1.0

P III,1 5.0E+00

Load (kN)

P II,1

P III,2

P IV,1

f Load

0.5

0.0E+00

0.0 0

2

4

6

8

10

12

14

Displacement (mm)

Figure 29. (A) f general behaviour, (B) example of f diagram for an experimental indentation test.

In figure 29A the sub-domain Z1,Z2, Z3 and Z4, as previously analysed and in figure 29B the AE domain, ΩAE, are reported. The strain energy diagram, figure 29A, is continuous,

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monotonic with an increasing trend and it is characterized by four main parts: the first part has an exponential trend, the other parts have nearly linear trends with different slopes. The slope variations coincide to the sub-domain Z2, Z3 and Z4 limits. The cumulative AE event energy is a discrete function that monotonically increases. The sub-domain previously cited delimits some of the Ea diagram variations: the first part of Ea is characterized by low values and a smooth trend (inside Z1), the second domain, Z2, is characterized by an Ea increasing trend and slope and at the end of Z2 there is an evident gap in the Ea value. This gap, as it is clear in figure 28C, is due to an AE event with a high energy content and from the mechanical point of view is related to the second load drop: the material damage has reached a high level. In the sub-domain Z3 and Z4 the Ea diagram is characterized by an increasing trend but no particular behaviour can be noticed. On the contrary, it is interesting to notice that in the Z2 domain the Ea trend is increasing with a general slope greater than the Ea slope in Z3 and Z4. In particular comparing the diagrams in figure 28 and the Ea diagram in figure 29B it is evident that the AE activity in Z2 is characterized by an increased AE event rate than in Z1 and in Z2 the events have a mean number of counts and a mean energy content higher than the ones in Z1. This analysis of the AE information indicates that the test phase coinciding to the sub-domain Z2 is the prelude to the main material crisis. Considering the f(x) diagram it is possible to notice the presence of three increasing diagram parts (type I: PI,1; PI,2; PI,3), four falls (diagram of type II: PII,1; PII,2; PII,3; PII,4), two constant parts (PIII,1; PIII,2) and one decreasing part (PIV,1). The three increasing diagram parts of f(x) are limited in the sub-domain Z1 indicating that at this first test stage the material has a moderate attitude in storing the mechanical energy. Analysing the f(x) diagram in the subdomain Z1 it is possible to note that the first fall PII,1 that connect PI,1 and PI,2 is not related to an important material failure: there is no slope variation of f(x) before and after the fall PII,1 (S1-=S1+). On the contrary, considering the second fall (PII,2) it is possible to note that the final slope of PI,2 and the starting slope of PI,3 are different (S2->s2+). This fact is due to the first important material damage and, as noticed during 28C analysis, it is related to the fibres breakage. It is worth noting that that the simple analysis of the load-displacement diagram does not single out this first damage even though it is important because it definitely influences the material strain energy storing capability and indicates the displacement value at which the damage significantly begins. The other three sub-domains are characterized by a general decreasing and constant trends of f(x). In particular in Z2, after the third fall (PII,3), the f(x) is characterized by an initial constant trend directly followed by a decreasing trend (no falls connect PIII,1 and PIV,1). This behaviour can be explained by the fact that the cumulated damage during the test phase in sub-domain Z1 and the first fall is great enough to compromise the material strain energy storing capability. The fourth fall, PII,4, that follows the decreasing trend of f(x) (PIV,1) is due to the final material local degradation. It is interesting to notice that in sub-domains Z3 and Z4 the f(x) is characterized by a constant trend while the load-displacement diagram reveals the presence of a local load maximum value. So even if the load diagram indicates a residual stiffness of the laminate the f(x) clearly indicates that the material damage is definitive. The constant behaviour of f(x) indicates that despite the local load maximum the material energy release is continuous and great enough to compensate the material strain energy storing attitude: the mechanical energy propagates the material damage.

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3.3. Residual Tensile Strength From the experimental results it was verified that the RTS of the laminates with [0,90,+45,45]s lay-up was greater than the [90,0,+45,-45]s one for each damage level. In [30] this result is put into relation with the greater tensile fibre damage of the ply adjacent to the most external one on the back side. In figure 30 the tensile tests results are summarized in terms of RTS and the correspondent values of the displacement. RTS tests showed a reduction respect to undamaged specimens [35] even for the barely visible indentations corresponding with Low damage levels. This is explained by the shear fibre breakage shown in figure 24. This fibre failure mode, different from the tensile one, is characterized by low strain energy level and, consequently, low acoustic energy emission. Nevertheless the study of the function f can be useful to identify this kind of failure and an example of this analysis was previously cited and it is reported in figure 29 B. In particular from the diagram in figure 29B the important drop of f at a displacement of 3.5 takes into account a low value of strain energy stored in the laminate (Es = 1.3 J) and an AE event with low energy content (Ea = 3.810-6 J). In order to take into account the material damage it is necessary to evaluate all events that cause loss of structural integrity. Since the function f amplifies the most important material damage events and it is able consider at the same time the strain energy storing capability and the released internal energy, its integral was utilized as a damage indicator. In figure 31 the RTS data are plotted versus the respective values of Int(f) for each laminate type. In particular it is evident the negative relation between the RTS and the values of the f integrate, confirming, as presented by other authors using different damage indicators [49-52], that the variable Int(f) is a reliable instrument to evaluate the material damage during the indentation process. To represent mathematically the relations between RTS and the damage indicator many different approaches are utilized: discontinuous relations are composed by linear[49] or non linear [50] equations and they present a threshold at the damage indicator, so values of damage indicator lower to a specified threshold value do not change the RTS that is so equal to the virgin material tensile strength; on the contrary continuous relations[51, 52] have a plateau which value is equal to the virgin material tensile strength when the damage indicator in zero and they have a curvature inversion. In the present work in order to relate the Int(f) and the RTS a continuous relation was considered having the following form: RTS = Ae

− B ( Int ( f ) )

C

(2)

Where the constant A is related to the ultimate load of the virgin material, and the constant B and C can be obtained by means of a linear regression based on the experimental data. Implementing the model in (2) to the experimental data it was estimated the following values for the coefficient of the continuous model: -

A = 39 kN laminate configuration [0,90,+45,-45]s: B = 8.8 10-5; C = 2.0 (mm-1); laminate configuration [90,0,+45,-45]s: B = 8.3 10-7; C = 2.8 (mm-1);

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In figure 32 the mathematical continuous models implementing the previous coefficient are represented by means of the continuous line showing a good fit. 3.0

14.0

2.5

10.0 2.0 8.0 1.5 6.0 1.0 4.0 MAX Displacement 2.0

0.5

Max Load

A 0.0 0.0

Maximum Load (kN)

Maximum Displacement (mm)

12.0

25.0

50.0

75.0

100.0

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0.0 150.0

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1.2E+06

1.6E+04

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1.4E+04

Max Count

1.2E+04

8.0E+05

1.0E+04 6.0E+05 8.0E+03 6.0E+03

4.0E+05

4.0E+03 2.0E+05

Maximum Count Cum

Maximum Event Cum

1.0E+06

2.0E+03

A 0.0E+00 0.0

25.0

50.0

75.0

100.0

125.0

0.0E+00 150.0

Int(f) (mm)

Figure 30. Scatter diagrams of Int(f) and the main mechanical variables (A) and AE parameters (B).

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Tensile Test: Load at rupture (kN)

40 35 30 25 20 15 10

[0,90,+-45]

5

[90,0,+-45]

0 0

1

2

3

4

5

6

Tensile Test: Displacement at rupture (mm) Figure 31. Ultimate load and ultimate displacement from the residual strength tensile tests.

Tensile Residual Strength (kN)

45.0 40.0 35.0 30.0 25.0 20.0 15.0 10.0 [0,90,+-45]

5.0

[90,0,+-45]

0.0 0.0

20.0

40.0

60.0

80.0

100.0

120.0

140.0

160.0

Int(f ) (mm) Figure 32. Tensile residual strength versus the integrate of the function f and the representation of the respective correlation model for the two damaged laminate configurations [0°,90°.+45°,-45°]s and [90°,0°.+45°,-45°]s

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Conclusion In this chapter, a new approach to the evaluation of damage progression and of the residual strength of CFRP was presented. This approach is based on standard parametric AE and in particular on the acoustic energy. A function of the acoustic energy and of the strain energy, called Sentry function, was introduced and its application was illustrated in the case of: 1) damage progression in tensile testing of different types of CFRP laminates; 2) damage progression and residual strength evaluation in the case of CFRP plates loaded at the centre. In the first case, the Sentry function allowed us to single out important material failures and to calculate the corresponding damage values, while in the second case, after the damage identification phase, the residual tensile strength was related to the integral of the Sentry function over the acoustic domain defined in the transversal load test.

References [1] Slight DW, Progressive Failure Analysis Methodology for Laminated Composite Structures, NASA/TP-1999-209107, 1999. [2] Basu S, Wass AM, Ambur AR, Prediction of progressive failure in multidirectional composite laminated panels, International Journal of Solids and Structures, 44 (2007) 2648-2676 [3] Lapczyk I, Hurtado JA, Progressive damage modeling in fibre-reinforced materials, Composites Part A, 38 (2007) 2333-2341 [4] Abry JC, Bochard S, Chateauminois A, Salvia M, Giraud G, In situ detection of damage in CFRP laminates by electrical resistance measurements, Composites Science and Technology, 59 (1999) 925-935 [5] Tsuda H, Lee JR, Strain and damage monitoring of CFRP in impact loading using a fibre Bragg grating sensor system, Composites Science and Technology, 67 (2007) 13531361 [6] Deuschle HM, Wittel FK, Gerard H, Busse G, Kroplin BH, Investigation of progressive failure in composites by combined simulated and experimental photoelasticity, Computational Material Science, 38 (2006) 1-8 [7] Benmedakhene S, Kenane M, Benzeggagh ML, Initiation and growth of delamination in glass/epoxy composites subjected to static and dynamic loading by acoustic emission monitoring, Composites Science and technology, 59 (1999) 201-208 [8] Bourchak M, Farrow IR, Bond IP, Rowland CW, Menan F, Acoustic Emission energy as a fatigue damage parameter for CFRP composites, International Journal of Fatigue, 29 (2007) 458-470 [9] Loutas TH, Kostopulos V, Ramirez-Jimenez C, Pharaoh M, Damage evolution in center-holed glass/polyester composites under quasi static loading using time-frequency

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[11] [12] [13] [14] [15]

[16] [17] [18] [19]

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analysis of acoustic emission monitored waveforms, Composites Science and Technology, 66 (2006) 1366-1375 Hosur MV, Murty CRL, Ramamurthy TS, Shet A, Estimation of impact-induced damage of CFRP laminates through ultrasonic imaging, NDT&E International, 31(5), (1998) 359-374 Abrate S, Impact on laminated composite materials. Applied Mechanics Review, 44(4), (1991) 155–90. Abrate S, Impact on laminated composites: recent advances. Applied Mechanics Reviews , 47(11), (1994) 517–44. Abrate S, Impact on composite structures. Cambridge: Cambridge University Press, (1998). Cantwell WJ, Morton J, The impact resistance of composite materials a review, Composites, 22(5), (1991) 347-362 Caprino G, Langella A, Lopresto V, Prediction of first failure energy of circular carbon fibre reinforced plastic plates loaded at the centre, Composites Part A, 34 (2003) 349-357 Caprino G, Langella A, Lopresto V, Elastic behaviour of circular composite plates transversely loaded at the centre, Composites Part A , 33 (2002) 1191–1197 Found MS, Holden GJ, Swamy RN. Static indentation and impact behaviour of GRP pultruded sections, Composite Structures, 39 (1997) 223-228 Lee SM, Zahuta P, Instrumented impact and static indentation of composites, Journal of Composite Materials, 25(2), (1991) 204–22 Kwon YS, Sankar BV, Indentation-flexure and low-velocity impact damage in graphite epoxy laminates, Journal of Composites Technology and Research, 15(2), (1993) 101–110 Wardle BL, Lagace PA, On the use of quasi-static testing to assess impact damage resistance of composite shell structures, Mechanics of Composite Materials and Structures, 5(1), (1998) 103–121 Sjöblom PO, Hartness TM, Cordell TM, On low-velocity impact testing of composite materials, Journal of Composite Materials, 22(1),(1988) 30–52. Lagace PA, Williamson JE, Tsang PHW, Wolf E, Thomas S, A preliminary proposition for a test method to measure impact damage resistance, Journal of Reinforced Plastics and Composites, 12(5), (1993) 584–601 Symons DD, Characterisation of indentation damage in 0/90 lay-up T300/914 CFRP, Composites Science and Technology 60, (2000) 391-401 Alderson KL, Evans KE. Failure mechanisms during the transverse loading of filamentwound pipes under static and low velocity impact conditions, Composites, 23(3), (1992) 167–73 Hirai Y, Hamada H, Kim JK. Impact response of woven glass-fabric composites. I. Effect of fibre surface treatment, Composites Science and Technology, 58(1), (1998) 91105 Matemilola SA, Stronge WJ. Low speed impact damage in filament wound CFRP composite pressure vessels, Journal of Pressure Vessel Technology;119(4), (1997) 435–43 Schoeppner GA, Abrate S, Delamination threshold loads for low velocity impact on composite laminates, Composites Part A, 31(9), (2000) 903-915

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[28] Found MS, Lamb JR, Damage assessment of impacted thin CFRP panels, Composite: Part A, 35(9), (2004) 1039–1047 [29] Luo RK, Green ER, Morrison CJ, Impact damage analysis of composite plates, International Journal of Impact Engineering, 22 , (1999) 435-447 [30] Caprino G, Teti R. Residual strength evaluation of impacted GRP laminates with acoustic emission monitoring, Composites Science and Technology, 53, (1995) 13-19 [31] Cartié DDR, Irving PE. Effect of Resin and Fibre Properties of Impact and Compression after Impact Performances of CFRP, Composites Part A, 33 (4), (2002) 483-493 [32] Zhang X, Davies GAO, Hitchings D, Impact damage with compressive preload and post-impact compression of carbon composite plates, International Journal of Impact Engineering, 22 , (1999) 485-509 [33] Caprino G, Residual strength prediction of impacted CFRP laminates, Journal of Composite Materials,18, (1984) 508-518. [34] Caprino G, Teti R, de Iorio I, Predicting residual strength of pre-fatigued glass fibrereinforced plastic laminates through acoustic emission monitoring, Composites Part B, 36 (2005) 365-371 [35] Cesari F, Dal Re V, Minak G, Zucchelli A, Damage and residual strength of laminated graphite-epoxy composite circular plates loaded at the centre, Composites Part A, 38 (2007), 1163-1173 [36] Symons DD, Davis G, Fatigue testing of impact-damage T300/914 carbon-fibrereinforced plastic, Composite Science and Technology, 60(3), 2000, 379-389 [37] Minak G, Morelli P, Zucchelli A, Fatigue residual strength of laminated graphite-epoxy circular plates damaged by transversal load, Proceedings of the 12th European Conference on Composite Materials, Biarritz, (2006) [38] Dunegan HL, Harris DO, Tatro CA, Fracture analysis by use of acoustic emission Engineering Fracture Mechanics, 1 (1), (1968) 105-110 [39] Shippen NC, Adams DF, Acoustic Emission monitoring of damage progression in graphite/epoxy laminates, Journal of Reinforced Plastics and Composites, 4(1985) 242-261 [40] Siron O, Tsuda H, Acoustic Emission in Carbon Fibre-Reinforced Plastics Materials, Annales De Chimie et Science des Matériaux, 25, 7, (2000) 533-537, [41] Mizutani Y, Nagashima K, Takemoto M, Ono K, Fracture mechanism characterization of cross-ply carbon-fibre composites using acoustic emission analysis, NDT&E International, 33 (2000) 101-110 [42] Walker JL, Hill E, Workman GL, Russell SS, A Neural Network-Acoustic Emission Analysis of Impact Damaged Graphite-Epoxy Pressure Vessels, Proceedings of the ASNT 1995 Spring Conference, Las Vegas (1995) 106–108. [43] Prosser WH, Waveform Analysis of AE from Composites. Proceedings of the Sixth International Symposium on Acoustic Emission from Composite Materials (AECM-6), San Antonio (1998) 61–70. [44] Prosser WH, Jackson KE, Kellas S, Smith BT, McKeon J, Friedman A, Advanced, Waveform Based Acoustic Emission Detection of Matrix Cracking in Composites. Materials Evaluation 53(9),(1995) 1052–1058 [45] Woo SC, Choi NS, Analysis of fracture process in single-edge-notched laminated composites based on the high amplitude acoustic emission events, Composites Science and Technology, 67 (2007) 1451-1458

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[46] Shull PJ, Non destructive Evaluation: Theory, Techniques, and Applications, Marcel Dekker Inc., 2002. [47] Zucchelli A, Dal Re V, Experimental analysis of composite laminate progressive failure by AE monitoring, Proceedings ICEM12 - 12th International Conference on Experimental Mechanics, Bari (2004) [48] MIL-HDBK-17-3E; Department of Defence Handbook, Polymer Matrix Composites, volume 3. Materials Usage, Design, and Analysis. [49] Shim VPW, Yang LM, Characterization of the residual mechanical properties of woven fabric reinforced composites after low-velocity impact, International Journal of Mechanical Sciences, 47, 2005, 647-665 [50] Caprino G, Lopresto V, The significance of indentation in the inspection of carbon fibre-reinforced plastic panels damaged by low velocity impact, Composites Science and Technology, 60, (2000) 1003-1012 [51] Sanchez-Saez S, Barbero E, Zaera R, Navarro E., Compression after impact of thin composite laminates, Composites Science and Technology, 65, (2005) 1911-1919 [52] Qi B, Herszberg I, An engineering approach for predicting residual strength of carbon/epoxy laminates after impact and hygro-thermal cycling, Composite Structures, 47, (1999) 483-490

In: Composite Materials Research Progress Editor: Lucas P. Durand, pp. 209-236

ISBN: 1-60021-994-2 © 2008 Nova Science Publishers, Inc.

Chapter 6

RESEARCH DIRECTIONS IN THE FATIGUE TESTING OF POLYMER COMPOSITES W. Van Paepegem*, I. De Baere, E. Lamkanfi, G. Luyckx and J. Degrieck Dept. of Mechanical Construction and Production, Sint-Pietersnieuwstraat 41, 9000 Gent, Belgium

Abstract For a long time, fatigue testing of composites was only focused on providing the S-N fatigue life data. No efforts were made to gather additional data from the same test by using more advanced instrumentation methods. The development of methods such as digital image correlation (strain mapping) and optical fibre sensing allows for much better instrumentation, combined with traditional equipment such as extensometers, thermocouples and resistance measurement. In addition, validation with finite element simulations of the realistic boundary conditions and loading conditions in the experimental set-up must maximize the generated data from one single fatigue test. This research paper presents a survey of the authors’ recent research activities on fatigue in polymer composites. For almost ten years now, combined fatigue testing and modelling has been done on glass and carbon polymer composites with different lay-ups and textile architectures. This paper wants to prove that a synergetic approach between instrumented testing, detailed damage inspection and advanced numerical modelling can provide an answer to the major challenges that are still present in the research on fatigue of composites.

1. Introduction The research on fatigue in composites in general has been largely inspired by the research on fatigue in metals. Despite the advantages that this knowledge transfer has provided, it has also brought about that there is still a widespread belief that the fatigue behaviour of metals and composites is indeed very similar. As a consequence the aim of most fatigue tests on

*

E-mail address: [email protected]

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composites is still to establish the S-N curve for that particular composite. The efforts to combine such fatigue tests with a variety of online and offline monitoring techniques and detailed numerical simulations of the experimental boundary conditions and observed material degradation, are much more limited. This paper wants to give a general overview of the different types of fatigue tests, the available online and offline monitoring techniques and the indispensable need of finite element calculations to understand the outcomes of these tests. As such, it should become clear that one single experimental fatigue test, if properly instrumented and simulated, can provide a lot more information about the fatigue behaviour of the tested composite material. The next paragraphs will discuss: • • •

the different fatigue test set-ups and related online monitoring techniques, the inspection of fatigue damage, the finite element simulation of experimental boundary conditions.

2. Fatigue Test Set-ups and Online Monitoring Techniques In this paragraph, a general overview of the most relevant fatigue test set-ups is given: (i) tension-tension fatigue, (ii) bending fatigue, and (iii) shear dominated fatigue. The related online monitoring techniques are discussed and some examples of measurements are briefly presented. An elaborate discussion of all types of fatigue testing, including tension-compression fatigue, biaxial fatigue and torsional fatigue, can be found elsewhere [1].

2.1. Tension-Tension Fatigue The uni-axial tension-tension fatigue test is the most widely used fatigue test. The coupon geometry is a parallel-sided specimen, instrumented with tabs. The choice of the tabbing material differs among the testing laboratories. Some prefer steel or aluminium tabs, but most of them use glass/epoxy tabs, where the glass reinforcement has a [+45°/-45°]ns stacking sequence. In most cases, the tabs are straight-sided non-tapered tabs. A fatigue test is usually conducted with a servo-hydraulic testing machine, equipped with grips that clamp the specimen. The alignment of the specimen is very important. No bending loads must be induced in the specimen due to misalignment. In tension-tension fatigue tests, the stress ratio R (= σmin/σmax) is often chosen to be 0.1. The test frequency is always chosen as high as possible to limit the duration of the test and minimize the cost, but the fatigue response of some composites strongly depends on the frequency (especially in case of fibre-reinforced thermoplastics). In the international standards, the number of cycles to failure is considered as the main outcome of the tension-tension fatigue test. Yet it is worth the effort to use online instrumentation methods. The most simple and effective online measurement is the axial stiffness evolution. The axial stiffness can be directly calculated from the axial stress (loadcell) and the axial strain (extensometer). The axial strain must never be calculated from the axial displacement and the

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gauge length, because the inevitable slip in the clamps can lead to serious errors in the strain calculation. Depending on the fibre and matrix type and the stacking sequence, the stiffness degradation can range from a few percent to several tens of percent [2-7]. If the transverse strain is measured additionally, the Poisson’s ratio νxy can be followed as well. It has been recently showed by Van Paepegem et al. [8] that the evolution of the Poisson’s ratio is a very sensitive parameter for fatigue damage. Figure 1 shows the evolution of the Poisson’s ratio for a [0°/90°]2s unidirectional glass fabric/epoxy composite in tensiontension fatigue. The νxy – εxx curves in strain-controlled fatigue between 0.0006 and 0.006 show a highly nonlinear behaviour and are upper-bounded by the static degradation of the Poisson’s ratio.

νxy versus εxx for [0°/90°]2s fatigue test W_090_8 static [0°/90°]2s test IF4 static [0°/90°]2s test IF6 [0°/90°]2s fatigue test W_090_8: cycle 600 + 5 [0°/90°]2s fatigue test W_090_8: cycle 3600 + 5 [0°/90°]2s fatigue test W_090_8: cycle 37200 + 5

0.20 0.15 0.10

νxy [-]

0.05 -0.00 0.000

0.005

0.010

0.015

0.020

-0.05 -0.10 -0.15 -0.20

εxx [-]

Figure 1. Evolution of the Poisson’s ratio νxy in function of the longitudinal strain εxx for a glass/epoxy [0°/90°]2s specimen at three chosen intervals in the fatigue test [8].

Another online technique is the use of embedded optical fibre sensors with a Bragg grating. The Bragg grating is a periodical variation of the optical refractive index that is written in the core of the glass fibre and is typically a few millimetres in length (Figure 2). When broadband light is transmitted into the optical fibre, the Bragg grating acts as a wavelength selective mirror. For each grating only one wavelength, the Bragg wavelength, λB is reflected with a Full Width at Half Maximum of typically 100 pm, while all other wavelengths are transmitted. The Bragg wavelength is directly proportional with the period of the Bragg grating. If the optical fibre sensor is embedded in a composite laminate, the strain in the loaded laminate will cause the period of the Bragg grating to change, and hence the value of the reflected Bragg wavelength.

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Figure 2. Measurement principle of the optical fibre sensor.

The advantages are numerous: • • • •

the measurement is absolute and does not drift in time, fibre optic sensors are rugged passive components resulting in a high lifetime (>20 years) and are insensitive to electromagnetic interference, the fibre Bragg grating forms an intrinsic part of the optical fibre and has very small dimensions which makes it very suitable for embedding in composite plates, many fibre Bragg gratings can be multiplexed employing only one optical line so more sensing points can be read out at the same time.

Doyle et al. [9] experimented on the use of fibre optic sensors for tracking the cure reaction of a fibre reinforced epoxy, with success. They also successfully demonstrated the feasibility of these sensors for monitoring the stiffness reduction due to fatigue damage, for thermosetting matrix. De Baere et al. [10,11] have shown that the optical fibre sensors also survive the production process for carbon fabric thermoplastics (both autoclave and compression moulding) and that the correspondence between the axial strain measurements from the extensometer and the optical fibre sensor were identical in tension-tension fatigue tests (see Figure 3). That means that the adhesion of the embedded optical fibre sensor to the surrounding thermoplastic material is very good. The accumulation of permanent strain is another important phenomenon to monitor. Especially in composites with large residual stresses built up during manufacturing, the relief of thermal stresses due to fatigue cracking can result in accumulation of permanent strain. There again, optical fibre sensors are very sensitive sensors to measure these permanent strains. Figure 4 shows the stress-strain curves of intermediate static tensile tests during a tension-tension fatigue test on a carbon thermoplastic. The accumulation of permanent strain can be clearly seen.

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Figure 3. Comparison of the longitudinal strain εxx measurement from the optical fibre and the extensometer in a tension-tension fatigue test of a carbon fibre-reinforced thermoplastic [11].

Figure 4. Intermediate static tests in a tension-tension fatigue experiment of a carbon fibre-reinforced thermoplastic [11].

Resistance measurement is a well-established damage detection technique for unidirectional carbon composites [12]. For a long time, there has been disagreement between researchers whether the resistance should increase or decrease when local fibre fractures occur [13-15]. In a recent series of articles, it has been clearly demonstrated that the resistance must increase with increasing damage to the fibre yarns, but a lot of researchers observe a decrease of resistance, due to bad contact of the electrodes.

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Recently, De Baere et al. [16,17] showed that resistance measurement also works very well for monitoring damage in carbon fabric reinforced thermoplastics under tension-tension fatigue loading. Current injection has been done with an innovative technique. Behind the tabs, in the strain-free area of the specimens, the current is injected by means of two rivets at both ends of the specimen, as shown in Figure 5.

Figure 5. Use of rivets for electrical resistance measurement in carbon fibre composites [17].

Figure 6 shows the evolution of relative resistance change ρ and axial fatigue stress σxx during fatigue cycles 4025 till 4030.

Figure 6. Correspondence between applied sine wave of stress σxx and measured resistance in a tensiontension fatigue test of a carbon fabric/PPS composite [17].

2.2. Bending Fatigue Uni-axial fatigue experiments in tension/compression are most often used in fatigue research [18-20] and accepted as a standard fatigue test, while bending fatigue experiments are scarcely used to study the fatigue behaviour of composites [21-23]. Bending fatigue tests differ in several aspects:

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•

•

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the bending moment is (piecewise) linear along the length of the specimen (3-point bending, 4-point bending, cantilever beam bending). Hence stresses, strains and damage distribution vary along the gauge length of the specimen. On the contrary, with tension/compression fatigue experiments, the stresses, strains and damage are assumed to be equal in each cross-section of the specimen, due to the continuous stress redistribution, the neutral fibre (as defined in the classic beam theory) is moving in the cross-section because of changing damage distributions. Once a small area inside the composite material has moved for example from the compressive side to the tensile side, the damage behaviour of that area is altered considerably, the finite element implementation of related damage models gives rise to several complications, because each material point is loaded with a different stress, strain and possibly stress ratio, so that damage growth can be different for each material point. In tension/compression fatigue tests, the stress- or strain-amplitude is constant during fatigue life and differential equations describing decrease of stiffness or strength, can often be simply integrated over the considered number of loading cycles, smaller forces and larger displacements in bending allow a more slender design of the fatigue testing facility.

Basically, three types of bending fatigue tests can be distinguished: (i) three-point bending [24,25], (ii) four-point bending [26], and (iii) cantilever bending [22,27-30]. The success of these tests for fatigue of polymer composites is quite limited, because the interpretation of the results is more difficult and in case of stiffness degradation, stress redistribution across the specimen height comes into play. Moreover, as long as the bending stiffness of the laminate is high enough (e.g. sandwich composites), the deflections are small and linear beam theory still applies, but once that the bending stiffness of the composite decreases (e.g. thin laminates), the deflections are large and geometric nonlinearities and friction at the roller supports affect the fatigue results. The authors designed a test set-up for cantilever bending fatigue tests as depicted in Figure 7. The power of the motor is transmitted by a V-belt to a second shaft. The second shaft bears a mechanism with crank and connecting rod, which imposes an alternating displacement on the hinge (point C in Figure 7) that connects the connecting rod with the lower clamp of the composite specimen. At the upper end the specimen is clamped (point A in Figure 7). Hence the sample is loaded as a composite cantilever beam. A full Wheatstone bridge on the connecting rod is used to measure the force acting on the composite specimen. Due to the (bending) stiffness degradation of the specimen during fatigue life, the measured force will gradually decrease as the amplitude umax of the prescribed displacement remains constant. In order to record the out-of-plane displacement profile, it was necessary to develop a mechanism to hold the specimen fixed in this state, because recording the profile while the test keeps running at a frequency of 2.2 Hz, gives rise to some practical problems. A rotary digital encoder was attached to the second shaft. Its angular position (relative to a certain reference angle) is directly related with the loading path of the

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composite specimen. The frequency inverter reads the signal from the rotary encoder and can stop and hold the motor at a predetermined angular position of the encoder.

Figure 7. Test set-up for cantilever bending fatigue [28].

A digital photograph of the out-of-plane displacement profile is taken from the side view. To enhance the contrast, the edge of the composite specimen has been painted white. An example of such a digital photograph is given in Figure 8. When the number of pixels for a known distance is counted, the out-of-plane displacement profile can be calculated. Thereto an edge-detection algorithm is used which detects the edges of the composite specimen on the digital photograph. Figure 8(right) shows an example of the edge detection algorithm. Of course, the calculated out-of-plane displacement profile applies to the deformation of the specimen surface, not to the out-of-plane displacement of the midplane of the laminate.

Figure 8. Use of image processing algorithms to track the out-of-plane displacement profile in cantilever bending fatigue.

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2.3. Shear Dominated Fatigue Fatigue testing in pure shear is very difficult. Lessard et al. [31] modified the static three-rail shear test (ASTM D 4255/D 4255M – 01) to do fatigue testing on carbon/epoxy plates. A much more common method are the tension-tension fatigue tests on a [+45°/-45°]ns laminate, This test is based on the ASTM D3518/D3518M-94(2001) Standard Test Method for ‘In-Plane Shear Response of Polymer Matrix Composite Materials by Tensile Test of a ±45° Laminate’. This standard explains how the shear stress-strain curve can be derived from a static tensile test on a ±45° laminate, by measuring the longitudinal and transverse strain. The test is also called a bias tension test, because the bias (or cross-grain) direction is the 45° direction between warp and weft direction in case of fabric reinforced composites. In both pure shear and shear dominated fatigue, the test frequency is a very important parameter. The shear stresses can lead to significant autogeneous heating and once the temperature exceeds the glass transition temperature, the deformations can be very large. Figure 9 shows the localized yielding of a [(+45°,-45°)]2s carbon fabric/PPS composite in tension-tension fatigue at 2 Hz. Temperature rises up to 90 °C were measured with a thermocouple at the top surface.

Figure 9. Localized yielding of a [(+45°,-45°)]4s carbon fabric/PPS composite in tension-tension fatigue at 2 Hz. Shear stress-strain curve for cyclic [+45°/-45°]2s test IH2 60

Shear stress τ12 [MPa]

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IH2 cyclic test IH4 static test

10

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0.01

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Shear strain γ12 [-]

Figure 10. Shear stress-strain curve for the cyclic tensile test on a [+45°/-45°]2s glass/epoxy specimen and the envelope of the corresponding static test [32].

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A hardly studied phenomenon is the accumulation of permanent strain during shear dominated fatigue loading. For composite materials with a thermoplastic matrix, creep effects seem to be dominant, while in case of thermosetting materials, permanent strain is simply neglected in most reported literature. Moreover, for both types of material, the phenomenon is not understood quite well. Van Paepegem et al. [32,33] studied the accumulation of permanent shear strain in [+45°/-45°]2s glass/epoxy laminates under cyclic loading. They showed that the shear modulus significantly degrades, but that the accumulation of permanent shear strain is even more important. Figure 10 shows the accumulation of permanent shear strain in cyclic loading of unidirectional glass fabric/epoxy composites.

3. Visualization of Fatigue Damage 3.1. Micrographs The most easy inspection technique is visual inspection. Depending on the difference in optical refractive index of the matrix and fibre materials, the transparency of the composite laminate can be very high. Gagel et al. [34] reported an extraordinary high transparency of Eglass multi-axial non-crimp fabric epoxy laminates. Matrix cracks, voids and inclusions could be detected easily by transmitted light. Optical or light microscopy provides a direct path from observations made with the naked eye, to what is visible at magnifications up to about 1000 × [35]. Fracture surfaces are embedded in resin and polished before observation. Figure 11 shows a microscopic image of the damage in a plain weave glass/epoxy composite loaded in bending fatigue [36].

1 mm

Figure 11. Micrograph of the fatigue damage at the clamped end of a composite specimen loaded in cantilever bending fatigue [36].

3.2. Ultrasonic Inspection A very common inspection technique for fatigue damage in (textile) composites is ultrasonics. Ultrasonics can be performed in various modes of operation, but the most common for fatigue damage detection is the through-transmission (C-scan) technique.

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Through-transmission ultrasonics basically consists of a transducer for emitting ultrasonic pulses that is placed at or near one surface and a receiver sensor that is located at the opposite surface. The technique applies to relatively low frequency sound beams, typically 0.5 MHz to 15 MHz, having a small aperture. The transducer and receiver are coupled to the surfaces or they are immersed in water together with the composite. The ultrasound waves are attenuated by defects in the composite and the acoustic attenuation is monitored using the receiver [37]. Figure 12 shows the C-scan of a thermoplastic composite specimen tested in three-point bending fatigue. The central area is clearly damaged.

Figure 12. C-scan of the central damaged zone in a composite specimen loaded in three-point bending fatigue.

With classical ultrasonic C-scans, the surface of the object under investigation is scanned point by point in order to detect and to localise possible defects or possible anomalies. In a Cscan the transducer is normally kept perpendicular and at a constant distance to the surface of the object. A less known but promising technique is the ultrasonic polar scan. With the use of polar scans we do not aim at the detection and localisation of defects or anomalies, but rather at the characterisation of the material. Therefore in a polar scan a single representative point of the object is scanned, under all possible angles θ and ϕ of incidence of the ultrasonic beam, as is shown in Figure 13. Due to the dimensions of a real ultrasonic beam, a small zone, rather than a single point of the object is scanned. The distance between transducer and scanned point is again kept constant, and an acoustic coupling medium, such as water, is used. As is also the case with classical C-scans, scanning is performed using pulsed signals. Obliquely incident ultrasonic waves have already been used more or less frequently for purposes of material characterisation. In each case wave velocities or arrival times of ultrasonic pulses were measured [38-40]. In a polar scan however, the amplitude of the transmitted beam is measured. Amplitude measurements are much easier to perform, and can be done with the most simple ultrasonic apparatus, an advantage for the possible application of the technique in industrial circumstances. In the early eighties Van Dreumel and Speijer [41] have shown that ultrasonic polar scans in principle can visualise in a non-destructive way fibre orientations of the layers in laminates stacked from unidirectional layers. Unfortunately, after these experiments, polar scans have been hardly studied or used any more, the reasons for this being mainly the complexity of the "formation" of a polar scan, and the lack of means at that time for the numerical simulation of a polar scan.

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Figure 13. Schematic drawing of the polar scan set-up (left) and example of an experimentally measured polar scan of a unidirectional carbon/epoxy composite [42].

Yet, Maes [43] showed that the recorded polar scans of a glass fabric/epoxy composite before and after fatigue damage clearly differ, as shown in Figure 14. Due to the degradation of the elastic properties, the propagation speed of ultrasound in the respective directions has changed.

Figure 14. Polar scan of a glass fabric/epoxy composite before fatigue loading (left) and after fatigue loading (right) [43].

3.3. X-ray Micro-tomography High-resolution 3D X-ray micro-tomography or micro-CT is a relatively new technique which allows scientists to investigate the internal structure of their samples without actually opening or cutting them [44]. Without any form of sample preparation, 3D computer models of the sample and its internal features can be produced with this technique. In order to perform tomography, digital radiographs of the sample are made from different orientations by rotating the sample along the scan axis from 0 to 360 degrees. After collecting all the

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projection data, the reconstruction process is producing 2D horizontal cross-sections of the scanned sample. These 2D images can then be rendered into 3D models, which enable to virtually look into the object. Figure 15 shows the micro-tomography images of a fatigue damaged 5-harness satin weave carbon/PPS (left) and the embedded optical fibre sensors in a carbon thermoplastic composite (right).

Figure 15. Micro-tomography images of a fatigue damaged carbon/PPS composite (left) and three composite samples with embedded optical fibre sensor (right) [11].

4. Finite Element Simulation of Experimental Boundary Conditions In this paragraph it is shown that finite element simulations should support the experimental work in order to be able to discriminate between intrinsic material behaviour and induced effects by (insufficiently understood) experimental boundary conditions. Four examples are given where the strong correlation between experimental measurements and finite element simulations is proven.

4.1. Clamping Conditions in Tension-Tension Fatigue As stated before, the composite coupons for tension-tension fatigue testing are parallel-sided specimens, instrumented with aluminium or composite tabs. One of the main concerns in tension-tension fatigue testing of composites is tab failure, i.e. the specimen fails just next to or inside the tabbing area. Such failures are due to the inevitable stress concentration near the clamped edges. In Figure 16, two types of standard tensile machine fixtures are shown, with the dimensions of the grips. In order to optimize the shape and length of the tabs, it is important to simulate the stress state near the clamped region. Therefore a simulation of part of the clamping mechanism has been done in ABAQUS™/Standard v6.6-2.

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Figure 16. InstronTM mechanical grips (left) and InstronTM servohydraulic grips (right).

Figure 17 illustrates the simulated parts in the finite element model. Because of symmetry, only half of the clamps is modelled, which reduces calculation time. The corresponding symmetry boundary conditions have been imposed on the specimen. To further reduce computation time, a rigid body constraint is placed on part of the housing cylinder of the wedge grips, only the area where the cylinder makes contact with the grip is left deformable. Furthermore, a part that models the wedge is added, also with a rigid body constraint to reduce calculation time. The reference point of this part is given a certain downward displacement. This part represents the hydraulic plunger in the hydraulic clamps.

Figure 17. The simulated clamps in the finite element model [45].

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Two time steps were implemented: in the first, the wedge was given a downward motion of 0.75 mm, simulating the pre-stressing of the grips; in the second, the bottom of the specimen was pulled down over 1 mm, simulating a tensile test. Contact conditions were imposed between the surfaces of the specimen and the grip, the grip and the cylinder and the grip and the wedge. Since the grip first follows the movement of the wedge and then the movement of the specimen, the slave surfaces of all contact conditions mentioned, were placed on the grips. Between specimen and grip, the tangential behaviour ‘rough’ was implemented, which means that no slip occurs once nodes make contact. For the other contact conditions, the ‘lagrange’ condition was used, which means that the tangential force is μ times the normal force, μ being the friction coefficient. The same friction coefficient was used for both conditions. The grip was meshed with a C3D8R element, a linear brick element with reduced integration, whereas all other parts were meshed with C3D20R, a quadratic brick element with reduced integration. The C3D8R of the grip is required instead of the C3D20R, since the slave surfaces require midface nodes and the C3D20R do not have one. For the grip, the wedge and the cylinder, steel was implemented with a Young’s modulus of 210000 MPa and a Poisson’s ratio of 0.3. The specimen was modelled in a composite material with the following elastic properties (Table 1). Table 1. The implemented engineering constants in the finite-element model for the specimen. E11 [MPa] E22 [MPa] E33 [MPa]

56000 57000 9000

ν12 [-] ν13 [-] ν23 [-]

0.033 0.3 0.3

G12 [MPa] G13 [MPa] G23 [MPa]

4175 4175 4175

In [46], the authors have derived an analytical formula for the clamping setup illustrated in Figure 18 that describes the interaction between the load F on the specimen, the force RA of the plunger (see Figure 18) represented by part A and the contact force P on the specimen.

Figure 18. Symbolic representation of the gripping principle of a clamp [45].

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The following equation was derived, with μij the coefficient of friction between parts i and j (i,j=A,B,C):

P=

(1 − μ AC μ BC ) cos α − ( μ BC − μ AC ) sin α F cos α − μ BC sin α + RA 2 sin α + μ BC cos α sin α + μ BC cos α

(1)

For both grips displayed in Figure 16, the angle α is set equal to 10 degrees. In [45] it was shown that the grips in Figure 17 can be replaced by their equivalent contact pressure, calculated from Equation (1), because the simulated contact pressure with the finite element model of Figure 17 perfectly corresponds with the analytically calculated contact pressure. Next a detailed finite element model of the end tab region has been developed. Figure 19 shows the model of this setup, both mesh and boundary conditions are illustrated. The specimen is meshed using C3D20R elements using a global element size of 2 mm. Where stress concentrations were expected, the element size was reduced to 0.5 mm. The thickness of the specimen was 2.4 mm, which is also the thickness of the tabs, as has already been mentioned. The material properties for the composite specimen are given in Table 1. For the boundary conditions, the displacement along the 1 and 2 axis was inhibited for planes B1 (on top) and B2 (at the bottom), simulating the ‘rough’ boundary condition from the previous paragraph. Since contraction of the specimen is possible in the 3-direction due to the Poisson effect, the movement along the 3-axis was allowed for both planes. In order to prevent movement of the entire sample along the 3 axis, the central line of plane C (at the back) was fixed.

Figure 19. Illustration of the model for the end tab region [45].

Two time steps were modelled. In the first, the contact pressure p, calculated from Equation (1), was imposed. In the second, a tensile stress of 600 MPa was applied on surface A. The exact value of the stress does not matter, since the stress concentration factors are compared.

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In that way, a detailed analysis of the stress concentration factors in the end tab region was studied for several clamping conditions and end tab geometries. Detailed results can be found in [45].

4.2. Friction in Single-Sided 3-Point Bending Fatigue In uni-axial fatigue tests on fibre-reinforced composites, the stress-strain hysteresis loop can be used as a measure for stiffness degradation and energy dissipation. In case of three-point bending fatigue tests, the hysteresis loop of the bending force versus midspan displacement can yield similar information. In this numerical study, it is shown that the shape of the hysteresis loop can be affected significantly by friction at the supports, especially for large deflections. As such, the area of the closed hysteresis loop is no longer a measure for energy dissipation and damage growth. Figure 20 shows typical hysteresis loops of the bending force versus midspan deflection at several times during fatigue life for the [90°/0°]2s carbon/PPS laminate. The amplitude of the midspan deflection was 14.5 mm and the testing frequency was 2.0 Hz. The hysteresis loops are gone through in clockwise direction (loading – unloading). The problem treated here, is the typical shape of the hysteresis curve. One would expect that the dissipated energy during such a hysteresis cycle is used for initiation and propagation of microscopic fatigue damage, but in the case of this material, ultrasonic C-scans could not detect any significant fatigue damage in the specimens (apart from the last stage in fatigue life). Therefore it was assumed that the effect could be induced by friction at the supports, given the very large midspan displacements for a short span length. Typical hysteresis curves in bending for [90°/0°]2s carbon/PPS laminate 700

Cycle 1 Cycle 80 000 Cycle 160 000 Cycle 191 000

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Figure 20. Typical hysteresis curves in bending for [90°/0°]2s carbon/PPS laminate.

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Finite element simulations have been done to prove the hypothesis of friction affecting the shape of the hysteresis loop. The simulations have been done with the commercial implicit finite element code SAMCEFTM. The finite element mesh is shown in Figure 21. Eight layers of composite have been modelled with isoparametric volumic elements, one element per layer through the thickness. The end supports and the load striking edge have been modelled as rigid body cilinders with radius 5 mm. The contact conditions between supports and composite elements can have a different friction coefficient. The material is assumed to behave in a linear elastic manner, but the geometric nonlinearity is taken into account.

Figure 21. SAMCEFTM finite element model of the three-point bending test.

Figure 22. Simulated displacement contours for a three-point bending test on a [90°/0°]2s carbon/PPS laminate.

Figure 22 shows the simulated deflection of the [90°/0°]2s specimen for a prescribed midspan displacement of 14.5 mm (in agreement with the imposed displacement in the three-point bending fatigue tests).

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Simulated hysteresis curves for different friction conditions 700

μ = 0.0 μ = 0.1 μ = 0.2 μ = 0.3

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Figure 23. Simulated hysteresis curves for a [90°/0°]2s carbon/PPS laminate with different friction conditions at the supports: (i) μ = 0.0, (ii) μ = 0.1, (iii) μ = 0.2 and (iv) μ = 0.3. Simulated force-displacement history for three-point bending test (μ = 0.3) 700

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Figure 24. Detailed simulation of the force-displacement curve of the [90°/0°]2s carbon/PPS laminate for μ = 0.3.

In Figure 23, the simulated hysteresis curves are plotted for different friction conditions. The complete loading-unloading path has been simulated, where the imposed midspan

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displacement increases from 0.0 to 14.5 mm and decreases back to 0.0 mm. The curve of bending force versus midspan deflection is shown for four different friction conditions at the two end supports: (i) μ = 0.0, (ii) μ = 0.1, (iii) μ = 0.2 and (iv) μ = 0.3. It can be clearly seen that for μ = 0.0, there is no hysteresis. However, the curve is slightly nonlinear due to the geometric nonlinearity (large deflection). For μ = 0.3, the typical shape of the hysteresis curve is found back, although no material damage was taken into account. As a consequence, the shape variation is only due to the friction coefficient. The simulation for μ = 0.3 has been done again with a very small time step at the transition from loading to unloading. The effect is even more pronounced, as can be seen in Figure 24. It is worth to mention that the value of the maximum bending force is in very good agreement with the experimentally measured one during the three-point bending fatigue tests (see Figure 20). Finally, the simulated stress-strain history in Figure 25 for one of the integration points of the finite element at the tensile side in midspan proves that there is no material hysteresis. It has been shown that the friction between the composite specimen and the supports was the predominant cause of this phenomenon. Static bending tests with different support conditions were performed and three-dimensional finite element analyses were done with different friction coefficients. These tests confirmed the hypothesis. Simulated stress-strain history for three-point bending test (μ = 0.3) 700

Stress σ11 [MPa]

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Strain ε11 [%]

Figure 25. Simulated stress-strain history of [90°/0°]2s carbon/PPS laminate for μ = 0.3.

Therefore, it can be concluded that the information from hysteresis loops in bending fatigue must be considered very carefully.

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4.3. Fully-Reversed 3-Point Bending Fatigue This study investigates whether a three-point bending setup with fully reversed loading can be used for the validation of (a combination of) damage models for thin composite laminates in static or fatigue loading conditions. When fully reversed bending is used, each side of the specimen is successively loaded in tension as well as in compression. As a result, the material in the beam sees alternating tension and compression, which makes this setup ideal for the validation of tensioncompression fatigue models. If fully reversed bending is considered, some changes must be made to the original threepoint bending setup in Figure 26.

Figure 26. Single-sided three-point bending test [47].

Figure 27. The central roll and the rotating outer support (left) and the mounted fully reversed threepoint bending setup [47].

(i) for the central indenter as well as the outer supports, two contact cylinders are required, one for the upward and one for the downward motion. Since the centre of the specimen remains horizontal (see Figure 26, right), no additional modifications are needed for the central indenter; ii) because the specimen rotates at its ends (see Figure 26, right), the outer supports need to allow for this rotation. Otherwise, this would induce unwanted reaction forces in the specimen, corrupting the fatigue data. The indenter and used supports for the

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developed fully-reversed bending fatigue set-up are shown in Figure 27 and the total assembly in Figure 28.

Figure 28. Total assembly of the fully reversed three-point bending setup [47].

The correct modelling of the boundary conditions applied in this fully-reversed threepoint bending set-up is not straightforward. Below the detailed finite element model is discussed. Since the central indenting rolls do not require any rotation, they are modelled by two separate rolls (Figure 29, left). To reduce calculation time, rigid body conditions are applied on all areas that do not make contact with the specimen. The element type is the same linear brick element with reduced integration, C3D8R, as before, the element size is 0.5 mm. The easiest way to model the rotating support is by modelling it as a single part, which is depicted in Figure 29, right. The rolls are slightly longer than the width of the specimen, so that the specimen does not make contact with the connecting part between the two rolls. Extra partitions are created resulting in a better mesh. The distance between the two rolls is equal to the thickness of the specimen, the rolls have a diameter of 10, as was the case in the experimental setup.

Figure 29. The model of the central indenter as two separate rolls (left) and the model of the rotating support as one part (right) [47].

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Again there is a rigid-body constraint on all the partitions that do not make contact with the specimen, in order to save calculation time (Figure 29, left and right). The part is meshed with C3D8R elements; the element size is also 0.5 mm. The latter is done to assure that the calculation does not diverge as a result of contact problems. Figure 30 shows the final model of the fully-reversed three-point bending set-up. For the boundary conditions of the rotating support, only the rotation of the support around its ‘natural axis’ is allowed, all other movement is constrained. For the contact conditions, the slave surface is put on the support and the master surface is on the specimen. The latter helps the rotating of the support, since normally, the slave surface follows the movement of the master surface. The specimen is a beam with dimensions 2.4 mm x 15 mm x 80 mm and it is meshed with quadratic brick elements with reduced integration, C3D20R. The global size of the elements is 3 mm. However, in the zones of contact, the size is 1 mm to ensure that no convergence problems occur due to the contact conditions. The material model is the same linear elastic model as in paragraph 4.1 (see Table 1).

Figure 30. Illustration of the mesh and the boundary conditions for the three-point bending setup with rotating supports [47].

With this model, the bending stresses in the experimental fatigue tests could be simulated very well. More details can be found in [47].

4.4. Cantilever Bending Fatigue This paragraph deals with the correct modelling of the set-up for cantilever bending fatigue that was already shown in Figure 7. It is tempting to model the clamped side of the specimen by a number of fully constraint nodes in the finite element mesh. However, this model appears to add unwanted stiffness to the specimen and the predicted force is considerably higher than the measured one. The calculations were done for a plain weave glass/epoxy specimen. The prescribed displacement umax (= 34.4 mm) was chosen rather large in order to assess the effect of geometrical non-linearities. The corresponding maximum bending force for the specimen,

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measured by a strain gauge bridge, was 117.5 Newton at the first loading cycle. Table 2 illustrates the influence of several modelling assumptions. 2-D and 3-D meshes have been used with “complete fixation” (fixing all nodes in the clamped cross-section) and “clamping surfaces” (modelling of the clamping plates with prestressing force). Due to symmetry conditions, the 3-D simulations were performed for the half width of the specimen and they were indicated in the table as “3-D symmetry models”. All simulations are quasi-static analyses, except the fourth simulation, which takes into account the inertia forces during fatigue loading. From the third and fourth simulation it is confirmed that a quasi-static analysis is sufficient. Indeed, since the fatigue experiments are performed at a frequency of 2.23 Hz and the mass of the reciprocating parts is very small due to the limited forces in bending, the inertia forces are negligible. Table 2. Comparison of the different finite element models for the bending fatigue setup.

2D plane strain, complete fixation

No. of elements 445

Bending force [N] 155.1

2D plane strain, clamping surfaces

517

141.2

0’46’’

3D symmetry model, complete fixation

1461

139.2

32’45’’

3D symmetry model, complete fixation, inertia forces

1461

138.9

4 h 35’57’’

3D symmetry model, clamping surfaces, geometrical non-linearities 3D symmetry model, clamping surfaces

1765

146.7

21’53’’

1765

120.8

40’44’’

FE model type

no

Y

Z

X

Figure 31. Finite element model of the bending fatigue setup [36].

CPU time 0’17’’

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Figure 31 shows the finite element mesh for a 3-D analysis with full modelling of the clamped surfaces and the prescribed displacement. The diagonal lines in the left part of the mesh are used by the SAMCEF preprocessor to indicate the presence of clamping conditions. Due to the symmetry conditions with respect to the (x,y)-plane, only one half of the specimen width has to be modelled. The lines in the bottom right part make up the rigid body part where the prescribed bending displacement is applied.

5. Conclusion This paper has presented a collection of research efforts in the field of (i) fatigue test set-ups and related online monitoring techniques, (ii) inspection of fatigue damage and (iii) the finite element simulation of experimental boundary conditions. It has been shown that an integrated approach of these three research fields can benefit the knowledge and insight into the fatigue testing of fibre-reinforced composites.

Acknowledgements The author W. Van Paepegem gratefully acknowledges his finance through a grant of the Fund for Scientific Research – Flanders (F.W.O.), and the advice and technical support of the Ten Cate company. The author I. De Baere is highly indebted to the university research fund BOF (Bijzonder Onderzoeksfonds UGent) for his research grant.

References [1] Harris, B. (ed.) (2003). Fatigue in composites. Science and technology of the fatigue response of fibre-reinforced plastics. Cambridge, Woodhead Publishing Ltd., 742 pp. [2] Hashin, Z. (1985). Cumulative damage theory for composite materials: residual life and residual strength methods. Composites Science and Technology, 23, 1-19. [3] Whitworth, H.A. (2000). Evaluation of the residual strength degradation in composite laminates under fatigue loading. Composite Structures, 48(4), 261-264. [4] Highsmith, A.L. and Reifsnider, K.L. (1982). Stiffness-reduction mechanisms in composite laminates. In : Reifsnider, K.L. (ed.). Damage in composite materials. ASTM STP 775. American Society for Testing and Materials, pp. 103-117. [5] Yang, J.N., Jones, D.L., Yang, S.H. and Meskini, A. (1990). A stiffness degradation model for graphite/epoxy laminates. Journal of Composite Materials, 24, 753-769. [6] Yang, J.N., Lee, L.J. and Sheu, D.Y. (1992). Modulus reduction and fatigue damage of matrix dominated composite laminates. Composite Structures, 21, 91-100. [7] Kedward, K.T. and Beaumont, P.W.R. (1992). The treatment of fatigue and damage accumulation in composite design. International Journal of Fatigue, 14(5), 283-294. [8] Van Paepegem, W., De Baere, I., Lamkanfi, E. and Degrieck, J. (2007). Poisson’s ratio as a sensitive indicator of (fatigue) damage in fibre-reinforced plastics. Fatigue and Fracture of Engineering Materials & Structures, 30, 269–276.

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[9] Doyle, C., Martin, A., Liu, T., Wu, M., Hayes, S., Crosby, P.A., Powell, G.R., Brooks, D. and Fernando, G.F. (1998). In-situ process and condition monitoring of advanced fibre-reinforced composite materials using optical fibre sensors. Smart Materials & Structures, 7(2), 145-158 APR 1998. [10] De Baere, I., Voet, E., Van Paepegem, W., Vlekken, J., Cnudde, V., Masschaele, B. and Degrieck, J. (2007). Strain monitoring in thermoplastic composites with optical fibre sensors: embedding process, visualization with micro-tomography and fatigue results. Journal of Thermoplastic Composite Materials, 20 (September 2007), 453-472. [11] De Baere, I., Voet, E., Luyckx, G., Van Paepegem, W., Vlekken, J., Cnudde, V., Masschaele, B. and Degrieck, J. (2007). On the feasibility of optical fibre sensors for strain monitoring in thermoplastic composites under fatigue loading conditions. Proceedings of the Third international workshop on Optical Measurement Techniques for Structures and Systems (OPTIMESS 2007), May 28-30, 2007, Leuven, Belgium. [12] Abry, J.C., Bochard, S., Chateauminois, A., Salvia, M. and Giraud, G. (1999). In situ detection of damage in CFRP laminates by electrical resistance measurements. Composites Science and Technology, 59, 925-935. [13] Angelidis, N., Wei, C.Y. and Irving, P.E. (2004). The electrical resistance response of continuous carbon fibre composite laminates to mechanical strain. Composites Part A, 35, 1135–47. [14] Chung, D.D.L. and Wang, S. (2006). Discussion on paper ‘The electrical resistance response of continuous carbon fibre composite laminates to mechanical strain’ by Angelidis N, Wei CY, Irving PE, Composites: Part A 2004;35:1135–47. Composites Part A, 37, 1490–1494. [15] Angelidis, N., Wei, C.Y. and Irving, P.E. (2006). Short communication - Response to discussion of paper: The electrical resistance response of continuous carbon fibre composite laminates to mechanical strain. Composites Part A, 37, 1495–1499. [16] De Baere, I., Van Paepegem, W. and Degrieck, J. (2007). The use of rivets for electrical resistance measurement on carbon fibre-reinforced thermoplastics. Smart Materials and Structures, 16(5), 1821-1828. [17] De Baere, I., Van Paepegem, W. and Degrieck, J. (2007). Electrical resistance measurement on carbon fibre-reinforced thermoplastics with rivets as electrodes. Proceedings of the Fourth International Conference Emerging Techonologies in NonDestructive Testing (ETNDT 4), Stuttgart, Germany, 2 – 4 april 2007. [18] Fujii, T., Amijima, S. and Okubo, K. (1993). Microscopic fatigue processes in a plainweave glass-fibre composite. Composites Science and Technology, 49, 327-333. [19] Schulte, K., Reese, E. and Chou, T.-W. (1987). Fatigue behaviour and damage development in woven fabric and hybrid fabric composites. In : Matthews, F.L., Buskell, N.C.R., Hodgkinson, J.M. and Morton, J. (eds.). Sixth International Conference on Composite Materials (ICCM-VI) & Second European Conference on Composite Materials (ECCM-II) : Volume 4. Proceedings, 20-24 July 1987, London, UK, Elsevier, pp. 4.89-4.99. [20] Hansen, U. (1997). Damage development in woven fabric composites during tensiontension fatigue. In : Andersen, S.I., Brøndsted, P., Lilholt, H., Lystrup, Aa., Rheinländer, J.T., Sørensen, B.F. and Toftegaard, H. (eds.). Polymeric Composites - Expanding the Limits. Proceedings of the 18th Risø International Symposium on Materials Science, 1-5 September 1997, Roskilde, Denmark, Risø International Laboratory, pp. 345-351.

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[21] Ferry, L., Gabory, D., Sicot, N., Berard, J.Y., Perreux, D. and Varchon, D. (1997). Experimental study of glass-epoxy composite bars loaded in combined bending and torsion loads. Fatigue and characterisation of the damage growth. In : Degallaix, S., Bathias, C. and Fougères, R. (eds.). International Conference on fatigue of composites. Proceedings, 3-5 June 1997, Paris, France, La Société Française de Métallurgie et de Matériaux, pp. 266-273. [22] Herrington, P.D. and Doucet, A.B. (1992). Progression of bending fatigue damage around a discontinuity in glass/epoxy composites. Journal of Composite Materials, 26(14), 2045-2059. [23] Chen, A.S. and Matthews, F.L. (1993). Biaxial flexural fatigue of composite plates. In : Miravete, A. (ed.). ICCM/9 Composites : properties and applications. Volume VI. Proceedings of the Ninth International Conference on Composite Materials, 12-16 July 1993, Madrid, Spain, Woodhead Publishing Limited, pp. 899-906. [24] Sidoroff, F. and Subagio, B. (1987). Fatigue damage modelling of composite materials from bending tests. In : Matthews, F.L., Buskell, N.C.R., Hodgkinson, J.M. and Morton, J. (eds.). Sixth International Conference on Composite Materials (ICCM-VI) & Second European Conference on Composite Materials (ECCM-II) : Volume 4. Proceedings, 2024 July 1987, London, UK, Elsevier, pp. 4.32-4.39. [25] El Mahi, A., Berthelot, J.-M. and Bezazi, A. (2002). The fatigue behaviour and damage development in cross-ply laminates in flexural tests. Proceedings of the Tenth European Conference on Composite Materials (ECCM-10), Brugge, Belgium, 3-7 June 2002. [26] Caprino, G. and D'Amore, A. (1998). Flexural fatigue behaviour of random continuousfibre-reinforced thermoplastic composites. Composites Science and Technology, 58, 957-965. [27] Van Paepegem, W. and Degrieck, J. (2002). A New Coupled Approach of Residual Stiffness and Strength for Fatigue of Fibre-reinforced Composites. International Journal of Fatigue, 24(7), 747-762. [28] Van Paepegem, W. and Degrieck, J. (2001). Fatigue Degradation Modelling of Plain Woven Glass/epoxy Composites. Composites Part A, 32(10), 1433-1441. [29] Van Paepegem, W. and Degrieck, J. (2002). Modelling damage and permanent strain in fibre-reinforced composites under in-plane fatigue loading. Composites Science and Technology, 63(5), 677-694. [30] Van Paepegem, W. and Degrieck, J. (2005). Simulating Damage and Permanent Strain in Composites under In-plane Fatigue Loading. Computers & Structures, 83(23-24), 1930-1942. [31] Lessard, L.B., Eilers, O.P. and Shokrieh, M.M. (1995). Testing of in-plane shear properties under fatigue loading. Journal of Reinforced Plastics and Composites, 14, 965-987. [32] Van Paepegem, W., De Baere, I. and Degrieck, J. (2006). Modelling the nonlinear shear stress-strain response of glass fibre-reinforced composites. Part I: Experimental results. Composites Science and Technology, 66(10), 1455-1464. [33] Van Paepegem, W., De Baere, I. and Degrieck, J. (2006). Modelling the nonlinear shear stress-strain response of glass fibre-reinforced composites. Part II: Model development and finite element simulations. Composites Science and Technology , 66(10), 14651478.

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[34] Gagel, A., Fiedler, B. and Schulte, K. (2006). On modelling the mechanical degradation of fatigue loaded glass-fibre non-crimp fabric reinforced epoxy laminates. Composites Science and Technology, 66(5), 657-664. [35] Hull, D. (1999). Fractography: observing, measuring and interpreting fracture surface topography. Cambridge, Cambridge University Press, 366 pp. [36] Van Paepegem, W. (2002). Development and finite element implementation of a damage model for fatigue of fibre-reinforced polymers. Ph.D. thesis. Gent, Belgium, Ghent University Architectural and Engineering Press (ISBN 90-76714-13-4), 403 p. [37] Mouritz, A.P. (2003). Non-destructive evaluation of damage accumulation. In: Harris, B. (ed.). Fatigue in Composites. Cambridge, Woodhead Publishing and CRC Press, 2003, pp. 242-266. [38] Audoin, B. and Baste, S. (1994). Ultrasonic Evaluation of Stiffness Tensor Changes and Associated Anisotropic Damage in a Ceramic Matrix Composite. Journal of Applied Mechanics 61:309-316. [39] Kriz, R.D. & Stinchcomb, W.W. (1979). Elastic Moduli of Transversely Isotropic Graphite Fibers and Their Composites. Experimental Mechanics 19:41-49. [40] Rokhlin, S.I. and Wang, W. (1992). Double throughtransmission bulk wave method for ultrasonic phase velocity measurements and determination of elastic constants of composite materials. J. Acoust. Soc. Am. 91:3303-3312. [41] van Dreumel, W.H. and Speijer, J.L. (1981). Nondestructive Composite Laminate Characterisation by Means of Ultrasonic Polar-Scan. Materials Evaluation 39:922-925. [42] Degrieck J. (1995). Some Possibilities for Non-Destructive Characterisation of Composite Plates by Means of Ultrasonic Polar Scans. Proceedings First Joint BelgianHellenic Conference on Non Destructive Testing, Patras, Greece, 22-23 May 1995. [43] Maes, K. (1998). Non-destructive evaluation of degradation in a fibre-reinforced plastic. Master thesis (in Dutch). Ghent University, Ghent, 105 pp. [44] Cnudde, V., Masschaele, B., Dierick, M., Vlassenbroeck, J., Van Hoorebeke, L. and Jacobs, P. (2006). Recent progress in X-ray CT as a geosciences tool. Applied Geochemistry, 21(5), 826-832. [45] De Baere, I., Van Paepegem, W. and Degrieck, J. (2007). On the design of end tabs for quasi-static and fatigue testing of fibre-reinforced composites. Accepted for Polymer Composites. [46] De Baere, I., Van Paepegem, W. and Degrieck, J. (2007). Design of mechanical clamps with extra long wedge grips for static and fatigue testing of composite materials in tension and compression. Accepted for Experimental Techniques. [47] De Baere, I., Van Paepegem, W. and Degrieck, J. (2007). On the feasibility of a threepoint bending setup for the validation of (fatigue) damage models for thin composite laminates. Accepted for Polymer Composites.

In: Composite Materials Research Progress Editor: Lucas P. Durand, pp. 237-256

ISBN: 1-60021-994-2 © 2008 Nova Science Publishers, Inc.

Chapter 7

DAMAGE VARIABLES IN IMPACT TESTING OF COMPOSITE LAMINATES Maria Pia Cavatorta and Davide Salvatore Paolino Mechanical Engineering Department – Politecnico di Torino, Corso Duca degli Abruzzi, 24 – 10129 Torino (Italy)

Abstract The Chapter presents an overview of the damage variables proposed in the literature over the years, including a new variable recently introduced by the Authors to specifically address the problem of thick laminates subject to repeated impacts. Numerous impact data are used as a basis to address and comment potentials and limitations of the different variables. Impact data refer to single impact events as well as repeated impact tests performed on laminates with different fiber and matrix combinations and various lay-ups. Laminates of different thickness are considered, ranging from tenths to tens of millimeters. The analysis shows that some of the variables can indeed be used for assessing the damage tolerance of the laminate. In single impact tests, results point out the existence of an energy threshold at about 40-50% of the penetration energy, below which the damage threat is quite negligible. Other variables are not directly related to the amount of damage induced in the laminate but rather give an indication of the laminate efficiency of energy absorption.

Introduction Composite laminates are expected to absorb low velocity impacts either during assembling or use. Even when the impact damage is barely visible, the incurred micro-damage may have a significant effect on the laminate strength and durability. The impact energy can be absorbed at any point of the laminate, well away from the point of impact, and by means of various laminate level failure mechanisms including front face indentation (indicative of local matrix crushing and local fiber breakage), interlaminar delamination, back face splitting and fiber peeling. In the literature [1-11], it is acknowledged that matrix cracking is the first type of damage introduced during impact; however, the presence of matrix cracks per se does not significantly change the overall laminate stiffness. Rather, the matrix crack tips may act as

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initiation point for delaminations and fiber breaks which may dramatically reduce the local and or global laminate stiffness thus affecting the load-time response. The literature also acknowledges that more damaging energy absorption mechanisms (such as delamination, fiber pull-out, fiber/matrix debonding and fiber fracture) follows matrix cracking and that they significantly reduce the strength and stiffness of the laminate. Considering the importance of damage assessment, there have been several attempts in the literature to look for measurable test quantities that could be correlated to the damage process [6, 12-19]. Under low-velocity impact loading conditions, the time of contact between the impactor and the target is relative long. Even though vibratory load responses from the composite sample, the impactor and the specimen supporting fixture are common features of impact loading history, the load history can still yield important information concerning damage initiation and growth [20]. Several authors have used the force-time history to compare the structural response from impact tests: in particular, values of the First Damage Force (FDF) [14-21], the Delamination Threshold Load (DTL) [6] or Hertzian Failure Load (Ph) [15] as well as of the Peak Force (Fpeak) [2-3, 17, 22-24] have often been used to rank laminate performance. Identification of the FDF poses no troubles as it corresponds to the first load drop which can be detected in the load-time history. However, comparison of laminate performance on such basis can be risky since the level of laminate damage associated with the first load drop may be quite different for a given laminate tested under different impact energies, or for different laminates tested at a given impact energy. In this respect, definition of the DTL and of the Ph appear more suitable for ranking laminate performance as they are intended to identify a more specific damage condition, that is the initiation of significant damage. In the case of the DTL, significant damage is defined as predominately delamination, while for the Ph energy absorption mechanisms other than matrix cracking are considered. The DTL and Ph do not necessarily correspond to the first load drop; rather, they are associated to the load drop at which a significant change in the slope of the forcedisplacement curve may be detected and which signals a change in laminate stiffness. Experimental determination of these load thresholds, which are shown to vary with the laminate thickness to the 3/2 power, may prove helpful for damage tolerant design: no significant damage threat is associated to impact events for which Fpeak is below the laminate DTL or Ph. On the contrary, for impact events for which Fpeak is above the laminate DTL or Ph, a damage threat exists, even if no information can be obtained on the final amount of cumulative damage that will occur. Difficulties and possible ambiguities in determining the DTL or the Ph have often led researchers to use Fpeak instead, considering it as the turning point between rather limited and more significant forms of damage. In [17,25], Liu suggested that for any composite laminate there exists a maximum value of Fpeak. When the impact energy is such that Fpeak is below this maximum value, the laminate suffers indentation and local matrix cracking, whereas when loaded by the maximum Fpeak significant delamination starts to take place. The idea of Fpeak as the signaling point of significant damage initiation is the basis of the dimensionless parameter introduced in 1975 by Beaumont et al. [16] called the Ductility Index (DuI). The DuI, which is proposed as a useful tool for ranking the impact performance of different materials under similar testing conditions, is defined as the ratio between the propagation energy Epropagation and the initiation energy Einitiation and it is given by the expression:

Damage Variables in Impact Testing of Composite Laminates

DuI =

E propagation Einitiation

239

(1)

where Einitiation and Epropagation correspond to the energies absorbed before and after Fpeak, respectively. The ductility index is small for brittle materials, where most of the energy is absorbed before Fpeak, and high for ductile materials, where most of the energy is absorbed after Fpeak is exceeded. The energy absorption mechanisms before Fpeak are crazing and microcracking of the matrix; whereas, after Fpeak, crack growth is observed via fiber pull-out, fiber/matrix debonding and fiber fracture [26-29]. Other energy variables have been introduced since the DuI to rank impact performance. In [12-13] Belingardi and Vadori introduced the Damage Degree (DD) defined as the ratio between the absorbed energy (Ea) and the impact energy (Ei). Ei is the kinetic energy of the impactor right before contact takes place and it is indeed the energy introduced into the specimen. Ea can be calculated from the force-displacement curve as the area surrounded by the curve in case of closed force-displacement curves (impact event with rebound) or the area bounded by the force-displacement curve up to a constant level of force and the horizontal axis in case of open force-displacement curves (impact event with no rebound). Based on the energy viewpoint, penetration should take place the first time Ea reaches Ei. Therefore the DD is below one for impact events with rebound while it reaches the value of one in case the impactor is stopped with no rebound or specimen penetration is achieved. In [13-14], it was shown that the relationship between the DD and the impact energy increases monotonically until saturation and a fairly good data interpolation was achieved by a linear regression curve [14]. A saturation energy level (Esa) was defined as the impact energy at which the DD regression curve reaches the value of one. This energy threshold is of practical and theoretical interest since it defines the maximum energy level the laminate can dissipate with no penetration and by means of internal damage mechanisms only [12]. In synthesis, the DD is defined as: In [17,18], Liu proposed a second-order polynomial regression curve to describe the absorbed energy vs. impact energy curve up to penetration (named by Liu the energy profile):

Ea = aEi2 + bEi + c

(2)

Depending on the laminate under study, the linear term and the constant c can be smaller than the quadratic term so that equation (2) can be simplified as:

Ea ≅ aEi2

(3)

From the energy profile, Liu was able to define a Penetration Threshold (Pn) (in a series of “continuous” impacts at increasing impact energies, it represents the first condition of no impactor rebound and therefore of equality between impact and absorbed energy), and a Perforation Threshold (Pr) (first condition of laminate complete perforation). Between the penetration and perforation thresholds, there exists a range, named by Liu “the range of the penetration process”, in which the impact energy and the absorbed energy are equal to each

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other but which represent different stages of the penetration process with the impactor moving deeper and deeper into the specimen as the impact energy increases. Penetration and perforation thresholds increase with thickness, so does the range of the penetration process. In other words, while for thin laminates the difference between the penetration and the perforation thresholds can be negligible, for thick laminates the same can become quite significant. For cross-ply glass-epoxy composite laminates, Liu [17] found:

Pn = 0.8t 0.0247 ≈ 0.8 Pr

(4)

where t is the laminate thickness. Equation (4) indicates that for the investigated glass/epoxy laminates, the penetration threshold is about 80% of the perforation threshold. In case of 3mm thin laminates, the range of penetration process (Pr–Pn) is less than 2 J. For 6-mm laminates, a difference of 15J is found, while for 12-mm thick laminates, (Pr –Pn) exceeds 100J and by far can not be neglected. In addition to identification of the laminate Pn, Pr and range of penetration process, the energy profile was used by Liu to define a coefficient η, named the Efficiency of Energy Absorption. The coefficient is defined as the ratio between the area bounded by the polynomial regression line of equation (2) up to Pn and the horizontal axis and the area of the rectangular triangle having for hypotenuse the bisector from zero to Pn. The bisector of the energy profile represents the equal energy between impact and absorption; therefore, the triangular zone corresponds to the highest energy-area the material can possibly have. As all materials have an energy absorption capability less than 100%, the regression curve is always below the bisector. However, the closer the regression curve to the bisector, the higher the energy absorption capability of the laminate. An interesting analysis of the energy profile was provided in [19]. By normalizing the impact energy and the absorbed energy by the laminate Pn, Mian and Quaresimin were able to obtain a single master curve which proved to work very well when thin laminates were investigated. A direct consequence of the existence of a master curve is that, when normalized by the laminate Pn, the efficiency of energy absorption is basically constant for all laminates, i.e. η varies linearly with the laminate Pn. The range of penetration process yet remained to be investigated. To this aim, a new variable, named the Damage Index (DI), was recently introduced by the Authors [30-32]. The DI definition aroused considering that in the range of the penetration process, the impactor moves deeper and deeper into the specimen as the impact energy increases. On the contrary, pure energy variables as the DD by definition saturates to one over the entire penetration process.

DI = DD

sMAX sQS

(5)

The value sMAX in equation (5) refers to the displacement value recorded at the instant when the force approximately reaches a constant value, in case of impact tests that cause

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241

specimen perforation; while it corresponds to the maximum displacement recorded during the test below the perforation threshold Pr. In order to make the DI a non-dimensional quantity, the displacement sMAX was normalised by the corresponding displacement sQS measured in quasi-static perforation tests. The choice of normalising by sQS was taken so to define an absolute reference test and leave apart possible strain-rate effects on the sMAX values. For all the laminates investigated by the Authors, sMAX of perforation tests was constant regardless of the impact velocity and equal to the sQS value.

Experimental Experimental impact tests were performed according to ASTM 3029 standard [33] using an instrumented free-fall drop dart testing machine. The impactor has a total mass of 20 kg; its head is hemispherical with a radius of 10 mm. Stainless steel was chosen for its high hardness and resistance to corrosion. The maximum falling height of the testing machine is 2 m, which corresponds to a maximum impact energy of 392 J. The drop-weight apparatus was equipped with a motorized lifting track. By means of a piezoelectric load cell, force-time curves were acquired. The acceleration history was calculated dividing the force term by the impactor mass. The displacement was obtained by double integration of the acceleration and thus force-displacement curves were plotted. By integration of the force-displacement curves, deformation energy-displacement curves were then obtained. Initial conditions were given with the time axis having its origin at the time of impact. At time t=0, the dart coordinate is zero and its initial velocity can be obtained by the well known relationship:

v0 = 2 gΔh

(6)

where Δh is defined as the height loss of the centre of mass of the dart with respect to the reference surface [12]. The impact velocity was also measured by an optoelectronic device. Agreement between measured and theoretical values was very good. The collected data were stored after each impact and the impactor was returned to its original starting height. Using this technique, the chosen impact velocity was consistently obtained in successive impacts. Because, the target holder was rigidly attached to the frame of the testing device, the tup struck the specimen each time at the same location. Square specimen panels, with 100 mm edge, were clamped through rigid plates having a central hole 76.2 mm in diameter, and fixed to a rigid base to prevent slippage of the specimen. The clamping system was designed to provide an uniform pressure all over the clamping area. Prior to impact tests, a series of quasi-static perforation tests were performed to get information on the laminate strength characteristics. Specimens were tested using a servohydraulic machine with maximum loading capacity of 100 kN. The hydraulic actuator was electronically controlled in order to perform constant velocity tests. Signals of the force applied by the actuator and of the actuator displacement were acquired in time with an appropriate sampling rate. Table 1 reports main characteristics of the laminates used in the study.

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Maria Pia Cavatorta and Davide Salvatore Paolino Table 1. Main characteristics of the laminates analyzed in the chapter Ref.

Fiber /Matrix

Lay-up

[30]

Glass/Vinylester +Polyester

[32]

Glass/Epoxy

[21]

Glass/Epoxy

[14]

Carbon/Epoxy

[random/02/90/random /90/02/random] [0/90]15 [0/90]30 [03 /903]5 [03 /903]10 [random/-45/+45/02]2 [0/90]4 [0/90]8 [0/90]16 [0/60/-60]4 [0/60/-60]8 [0/60/-60]16

Thickness [mm] 12.31

Acronym used GVP90_12.31

4.00 8.00 4.00 8.00 4.50 0.35 0.75 1.55 0.40 0.85 1.75

GE90s_4.00 GE90s_8.00 GE90m_4.00 GE90m_8.00 GE45_4.50 CE90_0.35 CE90_0.75 CE90_1.55 CE60_0.40 CE60_0.85 CE60_1.75

Results for Single Impact Tests Figures 1-3, 5-6 reports the results obtained for the analyzed laminates. For sake of comparison among the different laminates, in all graphs the impact energy Ei is divided by the penetration threshold Pn [34]. 1.0 0.9 0.8

GVP90_12.31 GE90s_8.00 GE90m_8.00 GE45_4.50 GE90s_4.00 GE90m_4.00 CE60_1.75 CE90_1.55 CE60_0.85 CE90_0.75 CE60_0.40 CE90_0.35 1st linear trend 2nd linear trend polynomial trend

Fpeak/Fpen

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Ei/Pn Figure 1. Normalized peak force plotted against non-dimensional impact energy Ei/Pn. Linear and polynomial trends.

Figure 1 reports data for Fpeak vs. the non-dimensional impact energy Ei/Pn. Values of Fpeak are normalized by the peak force Fpen registered for an impact energy equal to Pn.

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243

Considering the two-order difference in laminate thickness, data scattering is fairly limited. Data for the thinner laminates are the most dispersed and show the lowest values. In [35], the impact force was shown to depend on the flexibility of the laminate: values of Fpeak decrease with increasing laminate flexibility. A general trend for Fpeak can be envisaged. In [36], Found et al. proposed a relationship between Fpeak and the square root of the impact velocity, i.e. between Fpeak and the impact energy to the ¼ power. The proposed relationship (dotted curve in Figure 1) well interpolates the experimental data. Interpolation by two straight lines also appears rather good, allowing to point out that, for impact energies above 40%-50% Pn, the rate of increase of Fpeak with increasing impact energies slows down, with the value of Fpeak approaching an asymptote. The asymptotic trend of Figure 1 well agrees with the idea of a maximum value for Fpeak [17,25]. In this respect, data of Figure 1 seems to suggest that no real damage threat is associated to impact events for which the impact energy is below 40%-50% of the laminate Pn. A concept of impact threshold energy has been put forward by many researchers [35, 37-39]. This threshold has been defined as a measure of the ability of a composite laminate to resist initial strength degradation [35]. 1.0 0.9 0.8 0.7

DD

0.6 0.5

GVP90_12.31 GE90m_8.00 GE90s_4.00 CE60_1.75 CE60_0.85 CE60_0.40

0.4 0.3 0.2 0.1 0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

GE90s_8.00 GE45_4.50 GE90m_4.00 CE90_1.55 CE90_0.75 CE90_0.35 0.8

0.9

1.0

Ei/Pn Figure 2. DD values plotted against non-dimensional impact energy Ei/Pn.

Figure 2 reports data for the DD, which appear fairly more dispersed, apart from the data points of the three thicker laminates (GVP90_12.31, GE90s_8.00; GE90m_8.00) that are basically overlapping. As a general rule, it can be said that the DD increases for increasing impact energies and shows notably higher values for thicker laminates. In this respect, it is important to note that high values of the DD do not imply severe damage within the laminates. Indeed, the DD is a measure of the percentage of impact energy absorbed by the laminate whereas no distinction is made on the absorption mechanisms as it is the case for the DuI. High values of the absorption energy Ea can indeed be desirable, for example in crash

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events [40]. Strictly speaking, the DD is not a damage variable but rather a point by point measurement of the laminate efficiency of energy absorption, whereas the coefficient η proposed by Liu averages the efficiency of energy absorption over a wide range of impact energies (from very low energies to Pn). 1

GVP90_12.31 GE90s_8.00 GE90m_8.00 GE45_4.50 GE90s_4.00 GE90m_4.00 CE60_1.75 CE90_1.55 CE60_0.85 CE90_0.75 CE60_0.40 CE90_0.35

0.9 0.8

Ea/Pn

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Ei/Pn Figure 3. Energy absorption Master Curve.

Figure 3 plots the Master Curve proposed in [19]. Data for the three thicker laminates are again superimposed and very close to the bisector, meaning that the absorbed energy is about equal the impact energy (DD almost one). For thin laminates, the difference is more enhanced pointing out a lower efficiency of energy absorption [3]. As said, no indications can be evinced from Figures 2 and 3 in terms of laminate damage tolerance. In this respect, definition of the DuI, which differentiates between the nature of energy absorption mechanisms, appears rather appealing. In its original definition, the DuI is meant to rank different laminates tested under similar impact conditions, on the basis of a more fragile or more ductile behavior under impact loading. It is worthwhile noticing that in the literature the DuI is basically used to rank different laminates at perforation or fracture (Charpy tests), for which determination of Einitiation and Epropagation poses no trouble. The forcedisplacement curves are open curves and Einitiation is calculated without ambiguity as the area bounded by the force-displacement curve up to Fpeak and the horizontal axis, while Epropagation is calculated as the area bounded by the force-displacement curve from Fpeak to a constant level of force and the horizontal axis (Figure 4a). In the attempt of proving the DuI against other damage variables, computation of the DuI was also applied to impact tests with rebound. When the dart rebounds, the force-displacement curves are closed curves and computation of Einitiation and Epropagation becomes troublesome as different areas could be considered. No references of DuI computation in impact tests with rebound were found in the literature. Considering that in impact events with rebound the energy absorbed by the laminate is equal to the area bounded by the force-displacement curve, in computing the DuI

Damage Variables in Impact Testing of Composite Laminates

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it was decided to calculate Einitiation as the area bounded by the force-displacement curve up to Fpeak and Epropagation as the area bounded by the force-displacement curve after Fpeak so that the sum of the two energies is equal to the overall energy absorbed by the laminate (Figure 4b). In this way, only the portion of impact energy in fact dissipated by the laminate was taken into account and differentiated by the nature of energy absorption mechanisms. 12

Fpeak

Force [kN]

10 8

Initiation Energy

6

Propagation Energy

4 2 0 0

5

10

15

20

25

Displacement [mm] Figure 4a. An example of force-displacement curve with perforation: filled areas correspond to Einitiation (Initiation Energy) and Epropagation (Propagation Energy). 30

Fpeak

Force [kN]

25 20 15

Initiation Energy

10

Propagation

5

Energy

0 0

2

4

6

8

10

Displacement [mm] Figure 4b. An example of force-displacement curve with rebound: filled areas correspond to Einitiation (Initiation Energy) and Epropagation (Propagation Energy).

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Maria Pia Cavatorta and Davide Salvatore Paolino 2.5

GVP90_12.31 GE90s_8.00 GE90m_8.00 GE45_4.50 GE90s_4.00 GE90m_4.00 CE60_1.75 CE90_1.55 CE60_0.85 CE90_0.75 CE60_0.40 CE90_0.35

2.0

DuI

1.5 1.0 0.5 0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Ei/Pn Figure 5. DuI values plotted against non-dimensional impact energy Ei/Pn.

Figure 5 reports data for the DuI plotted against the non-dimensional impact energy Ei/Pn. By looking at the DuI values at penetration (Ei=Pn), thicker laminates appear to exhibit a more ductile behavior. However, considering the elevated heterogeneity of the laminates under study (in terms of type of fiber and matrix, orientation and percentage of fibers as well as laminate thickness), it is the Authors’ opinion that caution must be taken in ranking laminate performance. It should also be reminded that for the thicker laminates, penetration and perforation thresholds do not coincide but are quite distant from each other. Significance of Figure 5 is to show that, by extending computation of the DuI to impact energies below Pn, the DuI can be used as a damage variable. In particular, data on Figure 5 show that for impact energies up to 40% Pn, the amount of Epropagation is almost null meaning that the main energy absorbing mechanism is matrix cracking. Above 40% Pn, and especially in the case of thick laminates, the contribution of Epropagation becomes more and more important. This implies that Fpeak occurs at a value of displacement significantly lower than the maximum displacement reached by the laminate before dart rebound. As the impact energy increases, contribution of delamination and of fiber breakage to the energy absorption mechanisms becomes more and more important. Figure 6 reports data in terms of the DI. By taking into account the value of the maximum displacement, the DI is more of a damage variable than the DD was. Indeed, Figure 6 shows that at very low impact energies the DI is almost null to then increase monotonically for increasing impact energies. Up to impact energies of about 40-50% Pn, signaled by graphs of Fpeak and DuI as energy thresholds, the difference in the value of the DI for different laminates is quite limited. Above this threshold, the difference in DI data for different laminates increases significantly. Data on Figure 6 also show that DI values do not saturate to one at Pn, thus allowing to monitor the range of the penetration process. The effectiveness of the variable in distinguishing between the penetration and the perforation thresholds can be evinced from Figures 7 and 8 which report DD and DI data for two different laminates.

Damage Variables in Impact Testing of Composite Laminates

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Interestingly, up to Pn the DI increases linearly with the impact energy to then grow quite abruptly over the range of the penetration process. A linear relationship also exists between the DD data and the impact energy; however, the DD data can not be used beyond penetration as (by definition) the DD stays at the value of one over the entire range of the penetration process. 1.0

GVP90_12.31 GE90m_8.00 GE90s_4.00 CE60_1.75 CE60_0.85 CE60_0.40

0.9 0.8 0.7

DI

0.6

GE90s_8.00 GE45_4.50 GE90m_4.00 CE90_1.55 CE90_0.75 CE90_0.35

0.5 0.4 0.3 0.2 0.1 0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Ei/Pn Figure 6. DI values plotted against non-dimensional impact energy Ei/Pn 1.0 0.9

y = 0.68x + 0.28

0.8

R2 = 0.96

DD, DI

0.7 0.6 0.5 0.4 0.3

y = 0.52x - 0.14

0.2

R2 = 0.99

0.1

DD

DI

0.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Ei/Pn Figure 7. Comparison between DD and DI values for impact tests on glass/epoxy 6.25 mm thick laminates. Impact data taken from reference [17].

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Maria Pia Cavatorta and Davide Salvatore Paolino 1.0

y = 0.80x + 0.21

0.9

R2 = 0.90

0.8

DD, DI

0.7 0.6 0.5

y = 0.58x - 0.01

0.4

R2 = 0.98

0.3 0.2 0.1

DD

DI

0.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Ei/Pn Figure 8. Comparison between DD and DI values for impact tests on GE45_4.50 laminates.

Results for Repeated Impact Tests Apart from monitoring the range of the penetration process, the DI has proven to provide important pieces of information in case of repeated impact tests, a loading conditions of particular relevance in marine applications [4,30-32,38-39,41-42]. Figures 9-14 report data obtained on two different laminates (GVP90_12_31, CE90m_4.00) tested under repeated impacts. The two depicted impact energies were selected to represent tests of no laminate perforation within 40 impacts and tests of laminate perforation. Figures 9-10 reports data for Fpeak. As it can be observed, for impact energies that cause no perforation within test duration, values of Fpeak slightly increase in the first few impacts to then reach an asymptote. On the contrary, for energies that cause perforation, values of Fpeak decrease impact after impact as a consequence of damage accumulation. For a given laminate, initial values of Fpeak reported in Figure 10 are obviously higher than those of Figure 9 due to the higher impact energy used in the test (Figure 1), while values just before perforation are lower than the asymptotic values of Figure 9 due to the significant damage induced in the laminate. With respect to the initial values of Fpeak, it is worthwhile noticing that for the 25 J tests and the 98 J test performed on the GVP90_12.31 laminate, the maximum in Fpeak is not reached at the first impact. This effect has already been observed in the literature. In a series of repeated impact tests run on carbon/epoxy composite laminate, Wyrick and Adams [22] commented the initial increase in Fpeak as the result of the compaction process of the thin layer of unreinforced resin at the impacted surface. At low impact energy levels, damage to the fibers near the surface is minimal and the compaction process provides a harder surface for the next impact. In this respect, it is worthwhile noticing that this initial increase in Fpeak was observed when the impact energy was below 40% Pn, once more confirming the existence of an energy threshold level.

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16 14

Fpeak [kN]

12 10 8 6 4

GVP90_12.31

2

CE90m_4.00

0 1

4

7

10

13

16

19

22

25

28

31

34

37

40

Impact Number Figure 9. Values of peak force in repeated impact tests performed on GVP90_12.31 and CE90m_4.00 laminates. Impact energy: 25 J. 30

Fpeak [kN]

25 20 15 10 GVP90_12.31

5

CE90m_4.00 0 1

3

5

7

9

11

13

15

17

19

Impact Number Figure 10. Values of peak force in repeated impact tests performed on GVP90_12.31 and CE90m_4.00 laminates. Impact energy: 98 J.

Figures 11-12 report data in terms of the DuI. As in Figure 5, for impact tests with rebound, contribution of Einitiation and Epropagation is calculated according to the definition illustrated in Figure 4b. Figure 11 shows that for impact energies that cause no perforation, the DuI is very low and constant throughout the test, signaling no significant damage

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Maria Pia Cavatorta and Davide Salvatore Paolino

accumulation. For impact energies that cause perforation, the DuI maintains a very low value up to a few impacts before perforation when it rapidly increases (Figure 12). 0.10 GVP90_12.31

0.08

CE90m_4.00

DuI

0.06 0.04 0.02 0.00 1

4

7

10

13

16

19

22

25

28

31

34

37

40

Impact Number Figure 11. DuI data for repeated impact tests performed on GVP90_12.31 and CE90m_4.00 laminates. Impact energy: 24.5 J. 2.0 GVP90_12.31

DuI

1.5

CE90m_4.00

1.0

0.5

0.0 1

3

5

7

9

11

13

15

17

19

Impact Number Figure 12. DuI data for repeated impact tests performed on GVP90_12.31 and CE90m_4.00 laminates. Impact energy: 98 J.

Data in terms of DD and DI are reported in Figures 13 and 14. To avoid confusion, data are organized for single impact energies and single laminates. DD and DI data are plotted together to favor a comparison between the two variables. For energies that cause no

Damage Variables in Impact Testing of Composite Laminates

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perforation (Figure 13), the DD data show an initial decrease thus suggesting a reduction in the percentage of impact energy that the laminate is able to absorb. For the thicker laminate (GVP90_12.31), this initial reduction is quite significant, going from a percentage of energy absorption of about 85% in the first impact to a quite stable value of about 70% in subsequent tests. Visual observation of the laminate after each impact pointed out that in the first few impacts the impactor indents the laminate and that the size of this indentation does not significantly change in subsequent tests. Existence of an initial localized damage that does not appear to significantly grow in subsequent tests is also what is conveyed by the DuI as well as the DI data, which keep to a constant low level throughout the test. 1.0 0.9

DD

DI

0.8

DD, DI

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 1

4

7

10

13

16

19

22

25

28

31

34

37

40

Impact Number

Figure 13a. Comparison between DD and DI data in repeated impact tests performed on CE90m_4.00 laminates. Impact energy: 24.5 J. 1.0

DD

0.9

DI

0.8

DD, DI

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 1

4

7

10

13

16

19

22

25

28

31

34

37

40

Impact Number

Figure 13b. Comparison between DD and DI data in repeated impact tests performed on GVP90_12.31 laminates. Impact energy: 24.5 J.

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Maria Pia Cavatorta and Davide Salvatore Paolino 1.0 0.9 0.8

DD, DI

0.7 0.6

y = 1.1E-01x + 3.2E-01

0.5

2

R = 1.00

0.4 0.3 0.2 0.1

DD

DI

0.0 1

2

3

4

5

Impact Number

Figure 14a. Comparison between DD and DI data in repeated impact tests performed on CE90m_4.00 laminates. Impact energy: 98 J. 1.0 0.9 0.8

DD, DI

0.7 0.6 0.5

y = 9.3E-03x + 1.9E-01

0.4

R2 = 0.96

0.3 0.2 0.1

DD

DI

0.0 1

3

5

7

9

11

13

15

17

19

Impact Number

Figure 14b. Comparison between DD and DI data in repeated impact tests performed on GVP90_12.31 laminates. Impact energy: 98 J.

Figure 14a reports data for the CE90m_4.00 laminate tested at an impact energy of 98J. Perforation is achieved at the 5th impact. As it can be observed, the DD constantly increases impact after impact to reach a value of about one at the 4th impact, where laminate penetration is achieved. DI values increase at a constant rate up to the 3rd impact to then grow very rapidly and reach the value of one at laminate perforation. This trend is more evident in Figure 14b, for which perforation is achieved at the 19th impact. Up to the 17th impact, the DI slowly increases at a constant rate impact after impact. In the last two impacts before

Damage Variables in Impact Testing of Composite Laminates

253

perforation, the increase is on the contrary very rapid. Differently from the DuI which keeps to a constant low level up to a few impacts before perforation (Figure 12), the DI allows to monitor the initial phase of steady damage accumulation helping foreseen perforation. The initial slow decrease of DD values from the first to the second impact is followed by a constant phase up to the 17th impact, after which the DD increases rapidly and reaches a value of one at perforation. Likewise the DuI, DD data are not very sensitive for predicting laminate perforation as, apart from the last 2-3 impacts, the constant phase of Figure 14b does not differ from the asymptotic trends of Figures 13a and 13b, where no perforation is achieved. Also, DD values at the first impact are about the same, regardless of the level of impact energy.

Conclusion Impact test data obtained on different laminates are used to compare damage variables which have been proposed in the literature over the years. To this aim, definition of the two energy contributions used to compute the DuI has been extended to analyze impact tests with rebound. In single impact tests performed at different impact energies, data for Fpeak and DuI point out the existence of an impact energy threshold at about 40-50% Pn, below which the energy absorption mechanism is mainly matrix cracking. Graphs of Fpeak vs. impact energy show a bi-linear trend with a change in slope around the energy threshold; while values of the DuI are almost null below the energy threshold to then increase quite abruptly up to penetration. Interestingly, the energy threshold is about the same for all the laminates analyzed in the study, whose thickness varies from tenths to tens of millimeters. DI data increase monotonically for increasing impact energies and show very limited scattering up to the energy threshold. DD values and data in the Master Curve give no indications on the laminate damage tolerance; rather, they provide a measure of the absorption capability of the laminate. Results show that thicker laminates are characterized by a higher efficiency of energy absorption. Main advantage of the DI variable is the possibility to distinguish between the penetration and perforation energy thresholds. The distinction is essential when dealing with thick laminates, for which the impact energy that causes laminate perforation can by far exceed the penetration energy. In the range of the penetration process, the DI effectively monitors the impactor moving deeper and deeper into the laminate. Also in case of repeated impact tests, the DI provides important pieces of information. For impact energies that cause no laminate perforation within test duration, the DI stays at a constant low value throughout the test, owing to a negligible damage accumulation besides initial laminate indentation. For impact energies that cause perforation, the DI shows an initial phase of linear growth with the number of impacts, owing to a steady accumulation of damage. A few impacts before perforation, the DI starts raising quite abruptly, helping foreseeing laminate failure. Results for Fpeak show that it maintains a constant value when perforation is not achieved while it decreases rapidly otherwise. However, graphs of Fpeak versus impact number do not signal any change in the rate of damage accumulation. DuI values are almost null throughout the test when no perforation occurs. Low and constant values also characterize tests at higher

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energies up to a few impacts before perforation, when data show a rapid increase. Therefore, by looking at the DuI values in the first impacts, no prediction can be made as to whether or not laminate perforation will occur within test duration. In case of repeated impacts, DD data can monitor the efficiency of energy absorption impact after impact. Results show that DD values at the first impact are about the same, regardless of the impact energy. Moreover, regardless of the final output of the test (no perforation/perforation), DD values show an initial slight decrease followed by a rather constant phase. When perforation is to be achieved, DD values start to increase a few impacts before final failure.

Acknowledgments The Authors wish to acknowledge valuable discussions with professor Giovanni Belingardi.

References [1] Roy, R; Sarkar BK; Bose NR. Impact fatigue of glass fibre-vinylester resin composites. Composites Part A, 2001, 32, 871-876. [2] Zhou, G. Static behaviour and damage of thick composite laminates. Composite Structures, 1996, 36, 13-22. [3] Zhou, G; Davies, GAO. Impact response of thick reinforced polyester laminates. International Journal of Impact Engineering, 1995, 3, 357-374. [4] Sutherland, LS; Guedes Soares, C. Effects of laminate thickness and reinforcement type on the impact behaviour of E-glass/polyester laminates. Composites Science and Technology, 1999, 59, 2243-2260. [5] Azouaoui, K; Rechak, S; Azari, Z; Benmedakhene, S; Laksimi, A; Pluvinage, G. Modelling of damage and failure of glass/epoxy composite plates subject to impact fatigue. International Journal of Fatigue, 2001, 23, 877-885. [6] Schoeppner, GA; Abrate, S. Delamination threshold loads for low velocity impact on composite laminates. Composites Part A, 2000, 31, 903-915. [8] Guden, M; Yildirim, U; Hall, IW. Effect of strain rate on the compression behaviour of a woven glass fiber/SC-15 composite. Polymer Testing, 2004, 23, 719-725. [9] Baucom, JN; Zikry, MA. Low-velocity impact damage progression in woven E-glass composite systems. Composites Part A, 2005, 36, 658-664. [10] Ambu, R; Aymerich, F; Ginesu, F; Priolo, P. Assessment of NDT interferometric techniques for impact damage detection in composite laminates. Composites Science and Technology, 2006, 66, 199-205. [11] Okoli, OI. The effect of strain rate and failure modes on the failure energy of fibre reinforced composites. Composite Structures, 2001, 54, 199-303. [12] Belingardi, G; Grasso, F; Vadori, R. Energy absorption and damage degree in impact testing of composite materials. Proceedings XI ICEM (Int. Conf. Experimental Mechanics), Oxford (UK), 1998, 279-285. [13] Belingardi, G; Vadori, R. Low velocity impact tests of laminate glass-fiber-epoxy matrix composite materials plates. International Journal of Impact Engineering, 2002, 27, 213-229.

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[14] Belingardi, G; Vadori, R. Influence of the laminate thickness in low velocity impact behaviour of composite material plate. Composite Structures, 2003, 61, 27-38. [15] Shyr, TW; Pan YH. Impact resistance and damage characteristics of composite laminates. Composite Structures, 2003, 62, 193-203. [16] Beaumont, PWR; Reiwald, PG; Zweben, C. Methods for improving the impact resistance of composite materials. in Foreign object impact behaviour of composites. ASTM Spec Tech Publ, 1974, 568, 134–58. [17] Liu, D. Characterization of Impact Properties and Damage Process of Glass/Epoxy Composite Laminates. Journal of Composite Materials, 2004, 38, 1425-1442. [18] Liu, D; Raju, BB; Dang, X. Size effects on impact response of composite laminates. International Journal of Impact Engineering, 1988, 21, 837-854. [19] Mian, S; Quaresimin, M. A model for the energy absorption capability of composite laminates. Proceedings 8th ASME Conference on ESDA (Engineering Systems Design and Analysis), Turin (Italy), July 4-7, 2006, Paper Number: 95788. [20] Lee, SM; Zahuta, P. Instrumented impact and static indentation of composites. Journal of Composite Materials, 1991, 25, 204-222. [21] Belingardi, G; Cavatorta, MP; Duella, R. Material characterization of a composite-foam sandwich for the front structure of a high speed train. Composite Structures, 2003, 61, 13-25. [22] Wyrick, DA; Adams, DF. Residual Strength of a Carbon/Epoxy Composite Material Subjected to Repeated Impact. Journal of Composite Materials, 1988, 22, 749-765. [23] Caprino, G; Lopresto, V; Scarponi, C; Briotti, G. Influence of material thickness on the response of carbon-fabric/epoxy panel to low velocity impact. Composites Science and Technology, 1999, 59, 2279-2286. [24] Zhou, G; Davies, GAO. Characterization of thick glass woven roving/polyester laminates: 1. Tension, compression and shear. Composites, 1995, 26, 579-586. [25] Liu, D; Raju, BB; Dang, X. Impact perforation resistance of laminated and assembled composites plates. International Journal of Impact Engineering, 2000, 24, 733-746. [26] Harmia, T; Friedrich, K. Mechanical and thermomechanical properties of discontinuous long glass fiber reinforced PA66/PP blends. Plastics, Rubber and Composites Processing and Applications, 1995, 23, 63-69. [27] Kishore; Kulkarni, SM; Sharathchandra, S. Sunil, D. On the use of an instrumented setup to characterize the impact behaviour of an epoxy system containing varying fly ash content. Polymer Testing, 2002, 21, 763-771. [28] Jang, J; Han, S. Mechanical properties of glass-fibre mat/PMMA functionally gradient composite. Composites Part A, 1999, 30, 1045-1053. [29] Pegoretti, A; Zanolli, A; Migliaresi, C. Flexural and interlaminar mechanical properties of unidirectional liquid cristalline single-polymer composites. Composites Science and Technology, 2006, 66, 1953-1962. [30] Belingardi, G; Cavatorta, MP; Paolino, DS. Comparative Response in Repeated Impact Tests of Hand Lay-up and Vacuum Infusion Glass Reinforced Composites. International Journal of Impact Engineering, 2007. doi:10.1016/j.ijimpeng.2007.02.005 [31] Belingardi, G, Cavatorta, MP; Paolino, DS. A new damage index to monitor the range of the penetration process in thick laminates. Composites Science and Technology. In press.

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[32] Belingardi, G, Cavatorta, MP; Paolino, DS. On the rate of growth and extent of the steady damage accumulation phase in repeated impact tests. Composites Science and Technology. Submitted. [33] ASTM D3029. Standard Test Method for Impact Resistance of Rigid Plastic Sheeting or Parts by means of a Tup (Falling Weight). American Society for Testing Materials, 1982. [34] Lopresto, V; Melito, V; Leone, C; Caprino, G. Effect of stitches on the impact behaviour of graphite/epoxy composites. Composites Science and Technology, 2006, 66, 206-214. [35] Ying, Y. Analysis of the impact threshold energy for carbon fiber and fabric reinforced composites. Journal of Reinforced Plastics and Composites, 1998, 17, 1056-1075. [36] Found, MS; Howard, IC. Single and multiple impact behaviour of a CFRP laminate. Composite Structures, 1995, 32, 159-163. [37] Luo, RK; Green, ER; Morrison, CJ. An approach to evaluate the impact damage initiation and propagation in composite plates. Composites Part B, 2001, 32, 513-520. [38] Supratik Datta; Vamsee Krishna, A; Rao RMVGK. Low Velocity Impact Damage Tolerances Studies on Glass-Epoxy Laminates – Effects of Material, Process and Test Parameters. Journal of Reinforced Plastics and Composites, 2004, 23, 327-345. [39] Jang, BP; Huang, CT; Hsieh, CY; Kowbel W; Jang, BZ. Repeated Impact Failure of Continuous Fiber Reinforced Thermoplastic and Thermoset Composites. Journal of Composite Materials, 1991, 25, 1171-1203. [40] Sutherland, LS; Guedes Soares, C. Effect of laminate thickness and of matrix resin on the impact of low fibre-volume, woven roving E-glass composites. Composites Science and Technology, 2004, 64, 1691-1700. [41] Kawaguchi, T; Nishimura, H; Ito, K; Sorimachi, H; Kuriyama, T; Narisawa, I. Impact fatigue properties of glass fiber- reinforced thermoplastics. Composite Science and Technology, 2004, 64, 1057-1067. [42] Bijoysri Kahn; Rao, RMVGK; Venkataraman, N. Low Velocity Fatigue Studies on Glass Epoxy Composites Laminates with Varied Material and Test Parameters – Effect of Incident energy and Fibre Volume Fraction. Journal of Reinforced Plastics and Composites, 1995, 14, 1150-1159.

In: Composite Materials Research Progress Editor: Lucas P. Durand, pp. 257-273

ISBN 1-60021-994-2 c 2008 Nova Science Publishers, Inc.

Chapter 8

E LECTROMECHANICAL F IELD C ONCENTRATIONS AND P OLARIZATION S WITCHING BY E LECTRODES IN P IEZOELECTRIC C OMPOSITES Yasuhide Shindo and Fumio Narita Department of Materials Processing, Graduate School of Engineering, Tohoku University Abstract The electromechanical field concentrations due to electrodes in piezoelectric composites are investigated through numerical and experimental characterization. This work consists of two parts. In the first part, a nonlinear finite element analysis is carried out to discuss the electromechanical fields in rectangular piezoelectric composite actuators with partial electrodes, by introducing models for polarization switching in local areas of the field concentrations. Two criteria based on the work done by electromechanical loads and the internal energy density are used. Strain measurements are also presented for a four layered piezoelectric actuator, and a comparison of the predictions with experimental data is conducted. In the second part, the electromechanical fields in the neighborhood of circular electrodes in piezoelectric disk composites are reported. Nonlinear disk device behavior induced by localized polarization switching is discussed.

1.

Introduction

Sensor and actuator applications take advantage of the piezoelectric coupling converting electrical energy into mechanical energy and vice versa. Piezoelectric ceramics and composites play a significant role as active electronic components in many areas of science and technology, such as smart and MEMS devices. In some actuator applications, high values of stress and electric field arise in the neighborhood of an electrode tip in piezoelectric ceramics [1] and composites [2], and the field concentrations can result in electromechanical degradation [3, 4]. One of the limitations for practical use of piezoelectric ceramics and composites is also their nonlinear behavior, which occurs due to polarization switching and/or domain wall motion at high electromechanical field levels near the electrode tip. In order to optimize the performance of the piezoelectric devices, it is important to

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Yasuhide Shindo and Fumio Narita

understand the electromechanical field concentrations due to electrodes in piezoelectric ceramics and composites. Recently, Yoshida et al. [5] discussed the electromechanical field concentrations due to circular electrodes in piezoelectric ceramics through theoretical and experimental characterizations. Their model quantitatively predicted the nonlinear electromechanical fields induced by polarization switching near the circular electrode tip. Also, numerical predictions of strain concentration were in relatively good agreement with measured values. The main aim of this work is to evaluate the electromechanical fields in the neighborhood of surface and internal electrodes in piezoelectric composites. First, we study the effect of applied voltage on the electromechanical field concentrations near the electrodes in rectangular piezoelectric composite actuators. A nonlinear finite element analysis is performed to calculate the strain, stress, electric field and electric displacement by introducing models for polarization switching in local areas of the field concentrations. Two criteria based on the work done by electromechanical loads and the internal energy density are used and compared. Strain measurements are also presented to validate the predictions using a four layered piezoelectric actuator. A comparison of strain concentration is made between measurements and calculations, and a nonlinear behavior induced by localized polarization switching is discussed. The device performance and polarization switching zone near the electrodes are further predicted for some electrode configurations in the rectangular piezoelectric composites. Next, we discuss the electromechanical field concentrations due to circular electrodes in piezoelectric disk composites. The effects of applied voltage and localized polarization switching on the disk device performance are examined.

2.

Basic Equations

Consider a piezoelectric material with no body force and free charge. The governing equations in the Cartesian coordinates xi (i = 1, 2, 3) are given by σji,j = 0

(1)

Di,i = 0

(2)

where σij is the stress tensor, Di is the electric displacement vector, a comma denotes partial differentiation with respect to the coordinate xi , and the Einstein summation convention over repeated indices is used. The relation between the strain tensor εij and the displacement vector ui is given by 1 (uj,i + ui,j ) (3) εij = 2 and the electric field intensity vector is Ei = −φ,i

(4)

where φ is the electric potential. In a ferroelectric, polarization switching leads to a change in the remanent strain εrij and remanent polarization Pir . The total strain and electric displacement are = εlij + εrij

(5)

Di = Dil + Pir

(6)

εij

Eletromechanical Field Concentrations and Polarization Switching...

259

where the superscript l denotes the linear contribution to the strain and electric displacement, and the linear piezoelectric relationships are given by εlij

= sijkl σkl + dkij Ek

(7)

Dil = dikl σkl + ik Ek

(8)

In Eqs. (7) and (8), sijkl , dkij and ik are the elastic compliance tensor, direct piezoelectric tensor and dielectric permittivity tensor, which satisfy the following symmetry relations: sijkl = sjikl = sijlk = sklij , dkij = dkji , ik = ki

(9)

σij and Dil are related to εlij and Ei by σij Dil

= cijkl εlkl − ekij Ek =

eikl εlkl

(10)

+ ik Ek

(11)

where cijkl and eikl are the elastic and piezoelectric tensors, and cijkl = cjikl = cijlk = cklij , ekij = ekji

(12)

For piezoceramics which exhibit symmetry of a hexagonal crystal of class 6 mm with respect to principal x1 , x2, and x3 axes, the constitutive relations can be written in the following form:   σ1          σ2      σ  



    3 =    σ 4           σ  5     

σ6

c11 c12 c13 0 0 0 0 0 c12 c11 c13 0 0 0 c13 c13 c33 0 0 0 0 0 c44 0 0 0 0 0 c44 0 0 0 0 0 0 c66

                  

εl1 εl2 εl3 εl4 εl5 εl6

        



    −           

0 0 e31 0 0 e31 0 0 e33 0 e15 0 0 e15 0 0 0 0



   E   1   E2      E3  

(13)

  l   D1  



0 0 0 0 e15  D2l = 0 0 0 e15 0  l    D3 0 e31 e31 e33 0

       0    0    0      

εl1 εl2 εl3 εl4 εl5 εl6

        







 11 0 0   E1    +  0 11 0  E2      0 0 33  E3      

(14) where σ1 = σ11, σ2 = σ22 , σ3 = σ33 σ4 = σ23 = σ32, σ5 = σ31 = σ13 , σ6 = σ12 = σ21

)

εl1 = εl11, εl2 = εl22 , εl3 = εl33 εl4 = 2εl23 = 2εl32, εl5 = 2εl31 = 2εl13, εl6 = 2εl12 = 2εl21

(15) )

(16)

260

Yasuhide Shindo and Fumio Narita 

c11 = c1111 = c2222, c12 = c1122, c13 = c1133 = c2233, c33 = c3333  1  c44 = c2323 = c3131, c66 = c1212 = (c11 − c12) 2 e15 = e131 = e223, e31 = e311 = e322, e33 = e333

(17) (18)

The direction of the spontaneous polarization P s of each grain can change by 90 ◦ or 180◦ for ferroelectric switching induced by a sufficiently large electric field. In order to develop a non-linear model incorporating the polarization switching mechanisms with the electromechanical fields calculations, two criteria are used. The first criterion for polarization switching is based on work down, and the second is internal energy density switching criterion. The first criterion [6] states that a polarization switches when the electrical and mechanical work exceeds a critical value σij ∆εij + Ei∆Pi ≥ 2P s Ec

(19)

where ∆εij and ∆Pi are the changes in the spontaneous strain and polarization during switching, respectively, and Ec is a coercive electric field. The changes in ∆εij = εrij and ∆Pi = Pir for 180◦ switching can be expressed as ∆ε11 = 0, ∆ε22 = 0, ∆ε33 = 0, ∆ε12 = 0, ∆ε23 = 0, ∆ε31 = 0 ∆P1 = 0, ∆P2 = 0, ∆P3 = −2P s

(20) (21)

For 90◦ switching in the x3 x1 plane, there results ∆ε11 = γ s, ∆ε22 = 0, ∆ε33 = −γ s, ∆ε12 = 0, ∆ε23 = 0, ∆ε31 = 0 (22) ∆P1 = ±P s , ∆P2 = 0, ∆P3 = −P s

(23)

For 90◦ switching in the x2 x3 plane, ∆ε11 = 0, ∆ε22 = γ s, ∆ε33 = −γ s, ∆ε12 = 0, ∆ε23 = 0, ∆ε31 = 0 (24) ∆P1 = 0, ∆P2 = ±P s , ∆P3 = −P s

(25)

The polarization switching criterion based on internal energy density (second criterion) [7] is defined as U = Uc

(26)

where U is the internal energy density and Uc is a critical value of internal energy density corresponding to the switching mode. The internal energy density associated with 180◦ switching can be written as U=

1 D3E3 2

(27)

In the case of 90◦ switching in the x3 x1 plane, the internal energy density is U=

1 (σ11ε11 + σ33ε33 + 2σ31ε31 + D1 E1) 2

(28)

Eletromechanical Field Concentrations and Polarization Switching...

261

For 90◦ switching in the x2 x3 plane, U=

1 (σ22ε22 + σ33ε33 + 2σ32ε32 + D2 E2) 2

(29)

The critical value of internal energy density is assumed in the following form: 1 (Ec)2 Uc = T 2 33

(30)

where T 33 is the dielectric permittivity at constant stress. The constitutive equations (10) and (11) during polarization switching are σij Dil

= cijkl εlkl − e0kij Ek =

e0ikl εlkl

(31)

+ ik Ek

(32)

The new piezoelectric constant e0ikl is related to the elastic and direct piezoelectric constants by e0111 e0122 e0133 e0123 e0131 e0112 e0211 e0222 e0233 e0223 e0231 e0212 e0311 e0322 e0333 e0323 e0331 e0312

= d0111c11 + d0122c12 + d0133c13 = d0111c12 + d0122c11 + d0133c13 = d0111c13 + d0122c13 + d0133c33 = 2d0123c44 = 2d0131c44 = 2d0112c66 = d0211c11 + d0222c12 + d0233c13 = d0211c12 + d0222c11 + d0233c13 = d0211c13 + d0222c13 + d0233c33 = 2d0223c44 = 2d0231c44 = 2d0212c66 = d0311c11 + d0322c12 + d0333c13 = d0311c12 + d0322c11 + d0333c13 = d0311c13 + d0322c13 + d0333c33 = 2d0323c44 = 2d0331c44 = 2d0312c66

(33)

The components of the piezoelectricity tensor d0ikl are 0

dikl





1 = d33ni nk nl + d31(ni δil − ni nk nl ) + d15(δik nl − 2ni nk nl + δil nk ) 2

(34)

where ni is the unit vector in the poling direction, δij is the Kroneker delta, and d33 = d333, d31 = d311, d15 = 2d131 are the direct piezoelectric constants.

262

3. 3.1.

Yasuhide Shindo and Fumio Narita

Rectangular Piezoelectric Composite Actuators Computational Model

We performed 3D finite element calculations to present the electromechanical fields distributions around the electrode tip. The geometry used was a four layered piezoelectric composite actuator, as shown in Fig. 1. A rectangular Cartesian coordinate system ( x, y, z) is used with the z-axis coinciding with the poling direction. Electrodes with length a and width W are embedded in the piezoelectric actuator of length L and width W . An external electrode is attached on both sides of the actuator to address each electrode. The thickness of the layer h is chosen, and a L − a tab region exists on both sides of the layer. The total thickness is 4h. Because of the geometric and loading symmetry, only a half of the specimen needs to be analyzed. The electric potential on two electrode surfaces (−L/2 ≤ x ≤ −L/2 + a, |y| ≤ W/2, z =0, 2h) equals the applied voltage, φ = V0 . The electrode surface (L/2 − a ≤ x ≤ L/2, |y| ≤ W/2, z = h) is connected to the ground, so that φ =0. The normal displacement and shear stress on the surface (|x| ≤ L/2, y = −W/2, |z| ≤ 2h) are zero. The surface (|x| ≤ L/2, y = W/2, |z| ≤ 2h) is stress free.

W

y

Electrode

x

O

z

Poling x

a a

h

h

O

a L Figure 1. A rectangular piezoelectric composite actuator. Each element consists of many grains, and each grain is modeled as a uniformly polarized cell that contains a single domain. The model neglects the domain wall effects and interaction among different domains. In reality, this is not true, but the assumption does not affect the general conclusions drawn. The polarization of each grain initially aligns as closely as possible with the z- direction. The polarization switching is defined for each

Eletromechanical Field Concentrations and Polarization Switching...

263

element in a material. The electric potential φ is applied, and the electromechanical fields of each element are computed from the finite element analysis (FEA). The switching criterion of Eq. (19) or (26) is checked for every element to see if switching will occur. After all possible polarization switches have occurred, the piezoelectric tensor of each element is rotated to the new polarization direction. The electroelastic fields are re-calculated, and the process is repeated until the solution converges. The macroscopic response of the material is determined by the finite element model, which is an aggregate of elements. The spontaneous polarization P s and strain γ s are assigned representative values of 0.3 C/m 2 and 0.004, respectively. Our previous experiments [8] verified the accuracy of the above scheme, and showed that the results obtained are of general applicability.

3.2.

Experiments

The actuator discussed in this section was fabricated using a soft lead zirconate titanate (PZT) C-91 [9]. The material properties are listed in Table 1, and the corcive electric field is approximately Ec = 0.35 MV/m. The dimensions of the specimen are L = 30 mm, W = 10 mm, and 4h = 20 mm. The electrode length is a = 20 mm. The specimen was placed on the rigid floor. The high-voltage amplifier was limited to 1.25 kV so that a 0.25 MV/m field corresponded to a layer thickness of 5.0 mm. Strain gauges were placed around the electrode tip region. The sensors have an active length of 0.2 mm.

Table 1. Material properties of C-91.

C-91

3.3.

Elastic stiffnesses (×1010N/m2) c11 c12 c13 c33 12.0 7.7 7.7 11.4

c44 2.4

Piezoelectric coefficients (C/m2) e31 e33 e15 −17.3 21.2 20.2

Dielectric permittivities (×10−10C/Vm) 11 33 226 235

Results and Discussion

We first present analytical and experimental results for L = 30 mm, W = 10 mm, and 4h = 20 mm. The electrode length is a = 20 mm. Fig. 2 shows the finite element analysis results for the strain εzz versus electric field E0 = V0/h at the face of the actuator (at y = 5 mm plane) for x = 5 mm and z = 0.8 mm. For the polarization switching effect, the predictions based on work (Eq. (19)) and energy density (Eq. (26)) are shown. Also plotted are the experimental data in the range approximately ± 0.18 MV/m. Calculation results show that a monotonically increasing negative electric field causes polarization reversal. Polarization switching in a local region leads to a significant increase of compressive strain within the actuator when compared to the linear case. After the electric field reaches about −0.20 (0.24) MV/m, local polarization switching, based on work (energy density), can cause an unexpected decrease in compressive strain near the electrode tip during switching.

264

Yasuhide Shindo and Fumio Narita

Strain,Hzz (u10-6)

200 100

L = 30 m m W = 10 m m 4h = 20 m m a = 20 m m

0

x = 5.0 m m y = 5.0 m m z = 0.8 m m

Test

-100

FEA W ork Energy density

-200 -0.3

-0.2 -0.1 0 0.1 Electric field,E0 (M V /m )

0.2

Figure 2. Strain versus electric field for laminated actuator.

E0 = -0.34 M V /m Poling

W ork

Energy density

o

90 sw itching 180o sw itching

Figure 3. Polarization switching zone induced by electric field for laminated actuator. Fig. 3 shows the 180 ◦ and 90◦ switching zones near the electrode tip of the actuator under E0 = −0.34 MV/m. Predictions by different criteria are presented. 90 ◦ switching zone based on energy density are larger than that based on work. Fig. 4 displays the distribution of the normal component of stress σzz as a function of x at y = 0 mm and z = 0, 1 and 9 mm for laminated actuator with L = 30 mm, W = 10 mm, 4h = 20 mm and a = 20 mm under E0 = 0.2 MV/m from the finite element analysis. The solid line represents the

Eletromechanical Field Concentrations and Polarization Switching...

265

D isplacem ent,uz (Pm )

normal stress at the interface, the dashed line represents the normal stress near the internal electrode tip, and the alternate long and short dashed line denotes the value near the surface electrode tip. The normal stress at the interface is singular at the electrode tip. The stress ahead of the electrode tip is tensile, while the stress behind the electrode tip is compressive. The values of the normal stress near the internal electrode tip are higher than those near the

2

L = 30 m m W = 10 m m 4h = 20 m m

FEA W ork

1

x= 0 m m y= 0 m m z = 10 m m

0 -1

a = 20 m m 24 28

-2 -1000 0 V oltage,V0 (V )

1000

Figure 4. Normal stress versus x for laminated actuator under E0 = 0.2 MV/m. surface electrode tip. Fig. 5 shows the comparison of the distribution of the shear stress σzx against x near the internal electrode tip with that near the surface electrode tip for the same laminated actuator. The shear stress peaks at about x = 5.5 mm, and the peak value near the internal electrode tip is higher than that near the surface electrode tip. z Surface electrode

h

b

h

O

x

bT b

c

r

y

Internalelectrode

Figure 5. Shear stress versus x for laminated actuator under E0 = 0.2 MV/m.

266

Yasuhide Shindo and Fumio Narita

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( )

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' &

Figure 6. Displacement versus voltage for laminated actuator.

Next, the effect of electrode length on the performance of the actuator with L = 30 mm, W = 10 mm and 4h = 20 mm is discussed. Fig. 6 shows the predictions of displacement uz at x = 0 mm, y = 0 mm and z = 10 mm as a function of applied voltage V0, based on work, for a = 20, 24 and 28 mm. Non-linearity in the displacement versus voltage curves depends on the electrode length. The displacement increases with an increase of a from 20 mm to 24 mm. Little difference is observed between the results for a = 24 mm and 28 mm.

4. 4.1.

Piezoelectric Disk Composites Problem Statement and Solution Procedure

Consider a two-layered piezoelectric disk composite with radius c and thickness h as shown in Fig. 7. The origin of the coordinates (r,θ,z) is located at the center of the interface considered as z = 0, 0 ≤ r ≤ c. The z axis is assumed to coincide with the six fold axis of symmetry in the class of a 6mm crystal class, or with the poling axis in the case of poled piezoelectric ceramics. Three parallel circular electrodes of radius b lie in the planes z = 0, ±h. The bonding at z = 0 is assumed to be perfect so that the stresses and displacements are continuous along the interface of the composite. Let the voltage applied to the internal electrode surface be denoted by V0 . The surface electrodes are grounded. The axisymmetric models were generated using the commercial FE method software package. The electrode layers were not incorporated into the model.

Eletromechanical Field Concentrations and Polarization Switching...

267

" # $%" #

!

Figure 7. A piezoelectric disk composite actuator.

4.2.

Numerical Results and Discussion

We consider C-91+ /C-91− and C-91+ /C-91+ with c = 10 mm and 2h = 2 mm, corresponding to the tension and bending actuator models, respectively. The superscripts − and + denote, respectively, the situations for negative and positive poling directions. Plotted in Fig. 8 are the numerical values of radial strain εrr near the circular internal electrode tip (r = 9 mm and z = 0.2 mm) as a function of electric field E0 = V0 /h for

Strain,Hrr (u10-6)

60 C-91+/C-91+

40

c = 10 m m 2h = 2 m m b= 8mm

20 0

FEA W ork Energy density

r= 9 m m z= 0.2 m m

-0.2 0 0.2 Electric field,E0 (M V /m ) Figure 8. Strain versus electric field for disk composite tension actuator.

268

Yasuhide Shindo and Fumio Narita E0 = -0.22 M V /m

+0.22 M V /m

Poling

-0.30 M V /m

+0.30 M V /m

W ork o

0.5 m m

Figure 9. MV/m.

90 sw itching 180o sw itching

Normal stress versus r for disk composite tension actuator under E0 = 0.2

C-91+ /C-91− disk tension actuator with b = 8 mm. The predictions based on work (Eq. (19)) and energy density (Eq. (26)) are shown. As the electric field is reduced from zero, the compressive strain increases. Local polarization switching can cause a decrease in compressive strain near the circular electrode tip. Little difference is observed between two criteria. As the positive electric field increases, polarization switching did not occur. Fig. 9 shows the distribution of the normal stress σzz as a function of r at z = 0 and 0.2 mm for C-91+ /C-91− disk tension actuator with b = 8 mm under E0 = 0.2 MV/m. Near

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Figure 10.

#

Shear stress versus r for disk composite tension actuator under E0 = 0.2

Eletromechanical Field Concentrations and Polarization Switching...

269

MV/m.

" # $%" #

!

Figure 11. Displacement versus voltage for disk composite tension actuator.

the circular electrode tip, the normal stress at the interface is singular, and the stress ahead of the circular electrode tip is tensile, while the stress behind the electrode tip is compressive. The normal stress, apart from the interface, near the electrode tip has smaller value than the interface stress. Fig. 10 gives the distribution of the shear stress σzr as a function of r at z = 0.01 and 0.2 mm for the same disk tension actuator. The magnitudes of the shear stress increase toward the circular electrode tip as is expected. Fig. 11 shows the predictions of displacement uz at r = 0 mm and z = 1 mm as a function of applied voltage V0, based on work, for b = 8 and 10 mm. There is a small influence of the electrode radius on the displacement versus voltage curves. Fig. 12 shows the computed strain εrr of C-91+ /C-91+ disk bending actuator corresponding to Fig. 8. The negative electric field increases the compressive strain, similar to C-91+ /C-91− disk tension actuator. After the electric field reaches about −0.25 MV/m, polarization switching leads to a decrease in the compressive strain. As the electric field E0 continues to be reduced, the strain becomes tensile. On the other hand, as the positive electric field is increased, the strain near the electrode tip increases gradually due to the piezoelectric effect and then sharply increases as switching occurs due to electromechanical field concentrations. Little difference is observed between two criteria. Fig. 13 shows the predicted switching zones, based on work, near the circular electrode tip. As the electric fields increase, the area of the switched region grows. Fig. 14 shows the normal stress distribution σzz as a function of r at z = 0 and 0.2 mm for C-91+ /C-91+ disk bending actuator with b = 8 mm. The interface normal stress of the disk bending actuator is singular at the circular electrode tip, similar to the disk tension actuator. The stress ahead of the circular electrode tip is tensile, while the stress behind the circular electrode tip changes from tensile to compressive in the neighborhood of the electrode tip.

270

Yasuhide Shindo and Fumio Narita

"#

$

"#

$

% % %&

' ( !

%# %

Figure 12. Strain versus electric field for disk composite bending actuator.

Figure 13. Polarization switching zone induced by electric field for disk composite bending actuator.

Eletromechanical Field Concentrations and Polarization Switching...

271

!

Figure 14. MV/m.

Normal stress versus r for disk composite bending actuator under E0 = 0.2

Figure 15. MV/m.

Shear stress versus r for disk composite bending actuator under E0 = 0.2

272

Yasuhide Shindo and Fumio Narita

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#

Figure 16. Tip deflection versus voltage for disk composite bending actuator. Fig. 15 shows the similar results for the shear stress distribution σzr . A singularity in the interface shear stress also develops at the circular electrode tip. Note that for the C-91 + /C91+ disk bending actuator, since the problem is unsymmetrical to the r-axis, the shear stress does not become zero along the whole r-axis. Fig.16 gives a plot of the tip deflection uz at r = 10 mm and z = 0 mm with applied voltage V0, based on work, for C-91+ /C-91+ disk bending actuator with b = 8 and 10 mm. The curve rises steeply at first when the voltage is increased from zero. The tip deflection then gradually levels off when the voltage reaches about 220 V, because of switching in the lower layer (see Fig. 13). A similar phenomenon can be observed for negative voltage. The bending actuator for b = 10 mm exhibits higher deflection.

5.

Conclusions

The electromechanical field distributions in the neighborhood of the electrodes in piezoelectric composites were investigated. Two criteria for polarization switching in piezoelectric materials were incorporated into a finite element procedure. The results indicated that high values of electromechanical fields cause the localized polarization switching near the electrode tip, and the strain vs electric field curves show the non-linear behavior. Also, the size of the switching zone in the piezoelectric composites increased with increasing electric fields. As a remark, we note that this study may be useful in designing advanced piezoelectric composite actuators.

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Acknowledgements This work was partially supported by the Grant-in-Aid for Scientific Research (B) and Young Scientists (B) from the Ministry of Education, Culture, Sports, Science and Technology, Japan.

References [1] Shindo, Y., Narita, F. & Sosa, H. (1998). Electroelastic analysis of piezoelectric ceramics with surface electrodes. Int. J. Eng. Sci., 36, 1001-1009. [2] Narita, F., Yoshida, M. & Shindo, Y. (2004). Electroelastic effect induced by electrode embedded at the interface of two piezoelectric half-planes. Mech. Mater., 36, 9991006. [3] Dos Santos e Lucato, S. L., Lupascu, D. C., Kamlah, M., R¨odel, J. & Lynch, C. S. (2001). Constraint-induced crack initiation at electrode edges in piezoelectric ceramics. Acta Mater., 49, 2751-2759. [4] Qiu, W., Kang, Y.-L., Qin, Q.-H., Sun, Q.-C. & Xu, F.-Y. (2007). Study for multilayer piezoelectric composite structure as displacement actuator by Moir´e interferometry and infrared thermography experiments. Mater. Sci. Eng. A, 15, 452-453. [5] Yoshida, M., Narita, F., Shindo, Y., Karaiwa, M. & Horiguchi, K. (2003). Electroelastic field concentration by circular electrodes in piezoelectric ceramics. Smart Mater. Struct., 12, 972-978. [6] Hwang, S. C., Lynch, C. S. & McMeeking, R. M. (1995). Ferroelectric/ferroelastic interactions and a polarization switching model. Acta Metall. Mater., 43, 2073-2084. [7] Kalyanam, S. & Sun, C. T. (2005). Modeling of electrical boundary condition and domain switching in piezoelectric materials. Mech. Mater., 37, 769-784. [8] Narita, F., Shindo, Y. & Hayashi, K. (2005). Bending and polarization switching of piezoelectric laminated actuators under electromechanical loading. Comput. Struct., 83, 1164-1170. [9] Shindo, Y., Yoshida, M., Narita, F. & Horiguchi, K. (2004). Electroelastic field concentrations ahead of electrodes in multilayer piezoelectric actuators: experiment and finite element simulation. J. Mech. Phys. Solids, 52, 1109-1124.

In: Composite Materials Research Progress Editor: Lucas P. Durand, pp. 275-296

ISBN: 1-60021-994-2 © 2008 Nova Science Publishers, Inc.

Chapter 9

RECENT ADVANCES IN DISCONTINUOUSLY REINFORCED ALUMINUM BASED METAL MATRIX NANOCOMPOSITES S.C. Tjong* Department of Physics and Materials Science, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong

Abstract Aluminum-based alloys reinforced with ceramic microparticles are attractive materials for many structural applications. However, large ceramic microparticles often act as stress concentrators in the composites during mechanical loading, giving rise to failure of materials via particle cracking. In recent years, increasing demand for high performance materials has led to the development of aluminum-based nanocomposites having functions and properties that are not achievable with monolithic materials and microcomposites. The incorporation of very low volume contents of ceramic reinforcements on a nanometer scale into aluminumbased alloys yields remarkable mechanical properties such as high tensile stiffness and strength as well as excellent creep resistance. However, agglomeration of nanoparticles occurs readily during the composite fabrication, leading to inferior mechanical performance of nanocomposites with higher filler content. Cryomilling and severe plastic deformation processes have emerged as the two important processes to form ultrafine grained composites with homogeneous dispersion of reinforcing particles. In the present review article, recent development in the processing, structure and mechanical properties of the aluminum-based nanocomposites are addressed and discussed.

Introduction Discontinuously reinforced aluminum (DRA) based metal matrix composites are of increasing interest because of their high specific stiffness and strength, high isotropic and excellent wear resistance as well as cost effective manufacturing. DRA composites have been *

E-mail address: [email protected]

276

S.C. Tjong

developed in the past two decades for various automobile, aerospace, electronic packaging and other structural applications. Many factors affect the mechanical properties of DRA composites including matrix alloy composition, reinforcement material, reinforcement size, shape, volume fraction and distribution, nature of the matrix-reinforcement interface, etc. The reinforcement materials generally should possess significantly higher specific and specific strength, as well as high melting temperature compared to the matrix alloy. Ceramic reinforcement has the advantage of a relatively low density and high elastic modulus. Typical ceramic particles commonly used to reinforce aluminum and its alloys including SiC, B4C, Si3N4, AlN, Al2O3, TiC, TiB2, etc Particle reinforced composites are conventionally prepared either via powder metallurgy (PM) or liquid metallurgy, in which the reinforcing particles with sizes of several microns are directly incorporated into solid or liquid aluminum, respectively. The composites thus prepared can be viewed as ex-situ MMCs. However, ceramic microparticles fracture readily during mechanical loading, leading to low toughness of the composites [1-3]. Figs. 1(a)-1(b) show typical fracture morphology of ceramic microparticles in Al-based composites during tensile loading. Furthermore, reinforcement material such as SiC is not thermodynamically stable and thus can react with aluminum matrix during the composite fabrication and service at elevated temperatures. Efforts have been made to overcome the occurrence of such difficulties by developing novel in-situ processing. In the process, the reinforcing particles are directly formed in a metallic matrix by chemical reactions between constituent elements during the composite fabrication [4, 5]. Accordingly, very fine in-situ particles with diameters down to submicrometer scale ( > 100 nm) can be synthesized and dispersed more uniformly within aluminum matrix [6]. The formation of clean, ultrafine and thermally stable ceramic reinforcements rendering the in-situ composites exhibit excellent mechanical properties.

Figure 1. SEM fractographs showing fracture and decohesion of alumina particles of (a) 6061 Al/20vol.%Al2O3 and (b) 7005 Al /10vol.% Al2O3 composites tensile tested at room temperature [3].

The successful synthesis of large-scale ceramic, metallic and intermetallic nanoparticles in recent years has motivated materials scientists to develop novel metal-matrix nanocomposites with excellent mechanical properties for advanced structural engineering

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applications. Nanoparticles can be synthesized by several processes such as gas phase condensation, laser ablation, aerosol route, mechanochemical processing is well established [5, 7-9]. They reveal unique physical and mechanical properties that are different from those of bulk solids and microparticles. Due to their high specific surface area, nanoparticles exhibit a high reactivity and strong tendency towards agglomeration. It is necessary to disperse exsitu nanoparticles more uniformly in aluminum matrix in order to obtain desired mechanical properties. In the case of liquid metallurgy processing, high-intensity ultrasonic waves can be employed to disperse the SiC nanoparticles more uniformly in molten aluminum alloy [10, 11]. In powder metallurgy route, mechanical alloying, particularly cryomilling has been used to refine and disperse the ceramic phase in the Al matrix [12 -16]. In most cases, ceramic particles with original sizes of several micrometers can be reduced to nanometer level after cryomilling [17]. Recently, there has been a growing interest in the application of severe plastic deformation (SPD) such as high pressure torsion (HPT) and equal channel angular pressing (ECAP) for producing materials with ultrafine grain structure in submicrometer levels [18 29]. ECAP is more attractive for industrial applications because it can be employed to produce large fully-dense samples or products. It consists of pressing the sample through a die into an L-shaped channel without changing its cross-section. The sample deforms by simple shear, thereby inducing a high density of dislocations that are subsequently arranged to the meta-stable sub-grains of high-angle boundaries. By repeating the pressing process, the strain is accumulated during each increment cycle. The ultra-fine grained composites processed by ECAP exhibit high yield strength and good ductility [27].

Agglomeration of Particles Generally, ceramic particles of micrometer sizes are prone to cluster during the composite fabrication. Particle clustering is more prevalent in cast than in PM microcomposites [30, 31]. This leads to the mechanical properties of microcomposites are far below the theoretical values. For the PM microcomposites, the particle size ratio of the matrix and reinforcement is the main factor controlling the degree of microstructural homogeneity [32-35]. Furthermore, secondary processing technique such as ECAP and HPT are reported to be very effective to improve the dispersion of reinforcing ceramic particles in the PM DRA composites [20, 24, 27]. Figs. 2(a) -2(c) show the effect of ECAP extrusion cycles on the particle distribution in PM 6061 Al/20% Al2O3 composite. The composite in the as-fabricated condition shows extensive particle clustering as expected. The clusters are aligned along the extrusion direction (Fig. 2(a)). These clusters begin to dissolve and disperse into individual particles after four ECAP passes at 370 ºC. The particle distribution appears homogeneous after pressing for seven passes. In addition to declustering, ECAP treatment also yields grain refinement of the aluminum alloy matrix. It is well recognized that nanoparticles tend to agglomerate into large clusters during composite processing even under low loading levels of reinforcement. In this respect, appropriate processing procedures are needed to improve the dispersion of nanoparticles in aluminum matrix. Recently, Yang et al. used high-intensity ultrasonic waves to assist the dispersion of SiC nanoparticles (average size ≤ 30 nm) in molten aluminum alloy A356 [10, 11]. Fig. 3 shows a typical experimental setup for the ultrasonic assisted melting. The

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ultrasonic waves generate nonlinear effects in molten metal such as transient cavitation and acoustic streaming. Acoustic cavitation involves the formation, growth, pulsating and collapsing of tiny bubbles, thereby yielding transient local hot spots and implosive impacts to break up the clustered particles. The strong impact and local high temperatures enhance the wettability between molten metal and nanoparticles. Consequently, cast Al-based nanocomposite with better dispersion of ceramic nanoparticles can be prepared (Fig. 4).

(a)

(b) Figure 2. Continued on next page.

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(c) Figure 2. Microstructure of the PM 6061 Al /20% Al2O3 composite: (a) as-extruded condition. The reinforcing alumina size is ~ 1-5 μm, (b) after four ECAP passes and (c) after seven ECAP passes [24].

Figure 3. Experimental setup of ultrasonic assisted melting [10].

In PM nanocomposites, clustering of nanoparticles often occurs during processing and the degree of agglomeration increases with increasing filler content [36] (Fig. 5). Through a solid-state cryomilling route, better dispersion of nanoparticles in aluminum matrix can be achieved. In the process, collisions between the grinding media lead to repeated fracture and welding of the raw powders in a high-energy ball mill. Low temperature (liquid nitrogen) environment suppresses the recovery and recrystallization of matrix grains during milling, thereby yielding finer grain structures. The nature of the process allows the incorporation of large volume fractions of reinforcement into aluminum matrix with a homogeneous distribution [12,13, 15, 37, 38]. Consolidation of cryomilled powders via hot pressing, cold isostatic pressing (CIP), hot isostatic pressing (HIP), extrusion, spark plasma sintering, etc. is necessary to produce bulk composites with full density, useful shapes and sizes for practical

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applications. Goujon et al. prepared the Al 5000/AlN (4-30 vol.%) nanocomposites though cryomilling of 5000 Al powder (380 nm) and AlN particle (150 nm) followed by hot pressing [12, 13]. Cryomilling for 6 h is required to obtain a good homogeneity of powder mixtures. The crystallite sizes of Al and AlN in the powders are reduced to about 49 and 30 nm, respectively. Hot pressing leads to homogeneous dispersion of the AlN phase in the Al alloy matrix and to an increase of the crystallite size of Al to submicrometer regime but not of AlN (Fig. 6). The microstructure of this composite consists of UFG aluminum grains free of AlN particles and regions dispersed with AlN nanoparticles.

Figure 4. SEM micrograph showing the microstructure of as-cast A356/2%SiC nanocomposite [10].

Figure 5. TEM micrograph showing agglomeration of particulates at the grain boundary of Al/5 vol.% Al2O3 nanocomposite. The mean size of alumina nanoparticles is ~ 50 nm [36].

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Figure 6. Microstructure of Al 5000/20.6vol.% AlN nanocomposite prepared by cryomilling and hot pressing [13].

Structure-Property Relationship Aluminum-based nanocomposites can be classified into two categories according to the size dimensions of reinforcing particle and aluminum matrix employed, i.e. micrograined matrix composites reinforced with nanoparticles and UFG matrix composites reinforced with submicron- or nanoparticles. In the former case, ceramic nanoparticles are introduced directly into aluminum matrix having grain sizes in micrometer level via PM or ingot casting. The latter relates the use of cryomilling to refine the reinforcing particles and aluminum matrix down to submicrometer of nanoscale regime. Alternatively, the matrix grains of the composites can also be refined to submicrometer level using the SPD process. It is well recognized that the deformation behavior of nanocrystalline metals is quite different from their micro-grained counterparts. According to the Hall-Petch relation, a substantial increase in yield strength can be achieved by reducing the grain size of metals to the submicrometer or nanometer regime. Nanocrystalline metals generally have very low tensile ductility, and exhibit creep and superplasticity at lower temperatures compared to their micro-grained counterparts [9]. This is attributed to large volume (more than 50%) of atoms are located at the grain boundaries or interfacial boundaries of nanometals. Consequently, grain boundary activity is a dominant factor for controlling the mechanical properties. It is of practical interest to understand the effect of particle additions on the mechanical properties of aluminum and its alloys having submicrometer or nanometer grain sizes.

Micro-grained Matrices Tjong et al. investigated the microstructure and mechanical properties of pure aluminum reinforced with low loading levels of Si3N4 (15 nm) or Si-N-C (25 nm) nanoparticles. Such

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nanoparticles were prepared by means of the laser induced gas-phase reactions [39-41]. They reported that the mechanical strength of nanoparticle strengthened composites is far superior to that of microparticle reinforce composite with a similar volume content of particulate. In other words, the tensile strength of Al/1vol% Si3N4 (15 nm) and Al/1vol.% Si-N-C (25 nm)nanocomposites is comparable to that of Al/15vol% SiC (3.5μm) composite, but the yield stress of such nanocomposites is significantly higher than that of the microcomposite. The tensile ductility of nanocomposites is also higher than that of microcomposite (Table 1). However, increasing the Si-N-C nanoparticle content to 5 vol.% leads to deterioration of mechanical properties as a result of the particle agglomeration. The strengthening mechanism of nanocomposites is derived from the Orowan stress. It is well known that the Orowan strengthening results from interaction between dislocation and the dispersed particles during mechanical loading. Recently, Kang and Chan [36] also reported that the tensile strength of the Al/1vol.% Al2O3 nanocomposite is similar to that of the Al/10 vol.%SiCp (13 μm) composite, and the yield strength of the former is higher than that of the latter (Fig. 7). This figure reveals that the yield and tensile strengths of Al reinforced with Al2O3 nanoparticles increase with increasing filler content up to 4 vol.% Al2O3 at the expense of tensile ductility. Above 4 vol.%, the strengthening effects level off owing to the agglomeration of alumina nanoparticles as shown in Fig. 5. The main strengthening effect in such nanocomposites also arises from the Orowan stress. It is worth-noting that both the tensile strength and tensile ductility of cast Al-based composites are improved considerably as a result of better dispersion of nanoparticles in the alloy matrix via laser assisted melting [10,11].

Figure 7. Tensile properties of Al/Al2O3 nanocomposites prepared by conventional powder metallurgy method. The tensile properties of Al/10 vol.% SiC (10 μm) microcomposites are also show for the purposes of comparison [36].

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Table 1. Tensile properties of Al-based micro- and nanocomposites [41]. Specimen Pure Al Al/15 vol.% SiC (3.5 μm) Al/1 vol.% Si3N4 (15 nm) Al/1 vol.% Si-N-C (25 nm) Al/5 vol.% Si-N-C (25 nm)

Tensile Strength, MPa 70 176 180 178 153

Yield Strength, MPa 30 94 144 134 114

Elongation at Break, % --14.5 17.4 19.7 6.2

It is widely known that DRA microcomposites exhibit higher creep resistance than their unreinforced matrix materials because the particulates acting as barriers to dislocation movement. There is no plastic flow occurs within ceramic reinforcing particles. Accordingly, plastic deformation of DRA composites is controlled exclusively by flow in the metallic matrices. The high temperature creep behavior of coarse-grained Al and its alloys reinforced with microparticles is characterized by high values of n and Q. The creep activation energy of microcomposites is often much larger than that for aluminum lattice self-diffusion (142 kJ/mol) [42-45]. Such anomalous behavior can be rationalized by introducing a threshold stress (σo) opposing creep flow. In this respect, the observed creep deformation is not driven by the applied stress σ but rather by an effective stress σc (σc = σ -σo). The threshold stress may originate from several sources such as Orowan bowing between particles, attractive attraction between dislocations and particles as well as back-stress associated with local dislocation climb [5, 42]. The rate controlling equation can be written as follows:

ε& = A(

σ −σo G

) n exp( −

Q ) RT

[1]

where ε& is the creep rate, A is a constant, G is shear modulus, R is Universal gas constant and T is absolute temperature. The creep behavior of Al-based microcomposites is related to modified creep behavior of aluminum solid solution alloys, and the equations developed for solid solution alloys can be used to described the creep behavior of composites provided that the applied stress is replaced by an effective stress. Thus, the threshold stress for creep in Albased microcomposites is associated with interactions between dislocations and fine dispersion of particles. These particles may be fine oxides in PM MMCs or precipitates in the matrix alloy of cast composites [42-44]. Introducing a threshold stress and its temperature dependence into the creep rate analysis yields a true stress exponent, n, of 3, 5 or 8, and true creep activation energy. For composites with a true exponent close to 3, dislocation viscous glide is rate controlling with an activation energy for creep is associated with interdiffusion of the solute atoms. On the other hand, dislocation climb process predominates for n = 5 in which the activation energy for creep is associated with aluminum lattice self-diffusion. For the composites with a true exponent close to 8, creep deformation is controlled by the lattice diffusion and its rate is proportional to the third power of substructure grain size λ [45, 46]. Mathematically, the phenomenological creep rate equation for n = 8 can be written as:

ε& = S (DL/b2) (λ/b)3 [(σ- σo)/E]8

[2]

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where DL is lattice self-diffusion coefficient, λ sub-grain size, E Young’s modulus and S a numerical constant. A substructure is formed due to an increase in the dislocation density as a consequence of the thermal mismatch between the matrix and the reinforcement. The size of substructure is controlled by the interparticle spacing. Generally, subgrains can be generated more easily in pure Al than in the Al solid solution alloys during creep deformation [45, 46].

Figure 8. Creep rate vs applied stress for the Al/1vol.% Si-N-C nanocomposite at 573 -673 K [41].

Figure 9. Arrhenius plot of steady creep rate against 103/T for Al/1vol.%Si-N-C (25 nm) nanocomposite [41].

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Figure 10. Comparison of creep behavior between Al/1vol.%Si-N-C (25 nm) nanocomposite (open symbol) and Al/15vol% SiC (3.5μm) composite (solid symbol) at 573 and 623 K [41].

The creep behavior of the nanoparticle reinforced composites is mainly depended on the matrix materials selected, i.e pure aluminum and aluminum solid solution alloy. The high temperature creep strength of micrograined aluminum is also greatly improved by the addition of low volume content of ceramic nanoparticles. Tjong et al. demonstrated that the creep resistance of the Al/1vol.%Si3N4 (15nm) and Al/1vol.%Si-N-C (25 nm) nanocomposites is about two orders of magnitude higher than that of the [40, 41]. Fig. 8 shows the variation of steady creep rate vs applied stress for the Al/1vol.%Si-N-C (25 nm) nanocomposite. The Arrhenius plot of creep rate against 103/T for this nanocomposite is shown in Fig. 9. The nanocomposite exhibits an apparent stress exponent (n) varying from 15.7 to 23.0 and an apparent creep activation energy (Q) of 248 kJ/mol. The apparent activation energy of the Al/1vol.%Si-N-C nanocomposite is much higher than that for lattice diffusion of aluminum (142 kJ/mol). Similar high apparent values of n and Q values are also observed for the Al/1vol.% Si3N4 nanocomposite. For the purposes of comparison, the creep rates of the Al/15vol.% SiCp (3.5 µm) microcomposite and the Al/1vol.%Si-N-C (25 nm) nanocomposite at 573 and 623 K are presented in Fig. 10. It is evident that the creep rate of the Al/1vol.%Si-N-C nanocomposite is about two orders of magnitude lower comparing to the Al/15vol.% SiCp (3.5 µm) microcomposite. To rationalize the high apparent values of n and Q of the Al/1vol.%Si-N-C nanocomposite, a slip creep mechanism of constant substructure as given in Eq (2) is applied to the Al/1vol.%Si-N-C nanocomposite (Fig. 11). It appears that the datum points at three temperatures can be fitted linearly. It is considered that the nanoparticles of very small volume content (1 vol.%) pin the subgrain boundaries effectively. Consequently, the microstructure of nanocomposite remains unchanged during creep deformation. The threshold stress can be determined from Fig. 12 by extrapolating the linear regression line to zero strain rates. The values of threshold stress are determined to be 36.3, 25.7 and 17.3 MPa at 573, 623 and 673 K, respectively. It is obvious that the threshold

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stress is temperature dependent. By plotting the lattice diffusion compensated creep rate ( ε& /DL) against the modulus-compensated effective stress (σ -σo/E), the datum points for all temperatures merge into a straight line with a slope of 8 (Fig. 12). This implies that the creep rate of the Al/1vol.%Si-N-C nanocomposite is subgrain forming dislocation creep controlled by lattice-diffusion.

Figure 11. Variation of nanocomposite [41].

ε& 1 / 8 with applied stress on double linear scales for Al/1vol.%Si-N-C (25 nm)

Figure 12. Variation of diffusivity normalized steady creep rate, ε& /DL, with modulus-compensated effective stress, σ -σo/E, on double logarithmic coordinates for Al/1vol.%Si-N-C (25 nm) nanocomposite [41].

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Figure 13. TEM micrograph showing interaction between dislocations and alumina nanoparticles during creep of PM 2024 Al/Al2O3 nanocomposite at 10 MPa, 678 K [47].

Figure 14. Creep rate vs effective stress on a logarithmic scale for PM 2014Al/Al2O3 nanocomposites prepared at (a) 0.3 and (b) 1.0 % oxygen levels [47].

Recently, Mohamed and coworkers studied the creep mechanism of the PM aluminum solid solution alloy (2014 Al) reinforced with alumina nanoparticles [47]. The alumina nanoparticles of 30 and 35 nm are intentionally formed in 2014 Al during sintering via the introduction of water moisture with oxygen levels controlled at 0.3 and 1.0 wt%, respectively. Analysis of the creep data of this alloy reveals the presence of temperature- dependent

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threshold stress resulting from the interaction between moving dislocations and alumina oxide nanoparticles (Fig. 13). Such dislocation-particle interaction would impede lattice dislocation movement, thereby reducing creep rate of the composite. By incorporating the threshold stress into analysis, plots of creep rate versus effective stress yield straight lines with different slopes, i.e. n = 3 for low stress region and 5 for high stress regime (Fig. 14). Hence, the creep behavior of nanocomposite is consistent with the behavior of Al-Cu solid solution alloy (2014 Al) that exhibits a transition from viscous glide (n = 3) to the high-stress region (n = 5) where dislocations break away from the solute atom atmosphere. They indicated that the true creep characteristics of PM 2024 Al/Al2O3 nanocomposite are consistent with those reported for aluminum solid-solution alloys [43]. Therefore, the creep deformation of nanocomposite with a matrix containing solutes is controlled by a viscous glide slip mechanism.

Ultrafine Grained Matrices As mentioned above, conventional PM blending method yields Al nanocomposites with inhomogeneous distribution of reinforcing particles within the metal matrix. Cryomilling can provide a homogeneous dispersion of reinforcing particles in submicrometer or nanocrystalline matrix. The subsequent hot consolidation of cryomilled nanopowders into final bulk products causes the composites to have an UFG structure as a result of grain growth of the matrix. Schoenung and coworkers investigated the microstructure and tensile behavior of bulk nanostructures 5083 Al/5 vol.% SiC (25 nm) composite prepared by cryomilling followed by hot isostatic pressing and hot rolling [48]. They reported that the hot rolled composite consists of regions dispersed with SiC nanoparticles (100 -200 nm) and regions free of SiC nanoparticles (~ 700 nm). Fig. 15 shows the TEM micrograph of the SiCdispersed region in which SiC nanoparticles are distributed homogeneously within the ultrafine grains of the 5053 al matrix. The tensile properties of such composite from room temperature to 573 K are shown in Fig. 16. The composite exhibits very high tensile strength at room temperature but extremely low ductility. The strength decreases but the ductility increases with increasing test temperatures. In a nanocomposite with an UFG matrix, the dislocation movement in the matrix is restricted by the high density of grain boundaries. Consequently, the composite exhibits high tensile strength but very low tensile ductility. It is well known that nanocrystalline (NC) materials exhibit very low tensile ductility and toughness due to the lack of strain hardening [9]. The presence of coarser grains within the nanocrystalline matrix can enhance the ductility of nanostructured materials at the expense of mechanical strength [49-51]. Different toughening approaches have been proposed to enhance the ductility of NC materials either via thermomechanical treatment or cryomilling. Recently, Lavernia and coworkers reported that the UFG Al-Mg alloys with a bimodal microstructure exhibit a combination of high strength and good ductility [52- 55]. Such alloys were synthesized by consolidation of a mixture of cryomilled Al-Mg and unmilled powders. Consequently, strain hardening is regained in CG regions while maintaining high strength in NC regions. The CG grains can provide more dislocation activity than the NC grains. Ductilephase toughening in bi-modal structured Al-Mg alloys is attributed to the occurrence of crack bridging as well as delamination between UFG and CG regions during plastic deformation [51].

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Figure 15. Bright field TEM micrograph of the hot rolled 5053 Al/5vol.% SiC composite showing the dispersion of SiC nanoparticles in ultrafine grains [48].

Figure 16. Tensile stress-strain curves for the hot rolled 5053 Al/5 vol.% SiC composite at various temperatures [48].

Based on this approach, Schoenung and coworkers prepared bimodal 5083Al/10 wt%B4C composite by blending cryomilled composite powders with an equal amount of CG 5083 Al followed by CIP and extrusion [15]. Figs. 17(a) -17(b) show the microstructure of bimodal composite consisting of UFG (NC) Al and CG Al. The B4C are uniformly distributed in the

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NC Al, and the NC Al and the CG Al are alternately distributed. (Fig.17(a)). This bimodal composite exhibits very high compressive yield strength of 1065 MPa comparing to 504 MPa of bimodal 5083 alloy. However, the composite still exhibits low compressive ductility (0.8%). Annealing the composite at 723 K improves its ductility to 2.5%. Fig. 18 shows the temperature dependence of the yield strength for the tri-modal composite. The yield strength decreases rapidly with test temperatures up to 473 K, followed by a relatively slow decrease at higher temperatures. At 473 K, the compressive yield stress of the tri-modal composite is 282 MPa, being higher than that of the heat-treated 5083 alloy at room temperature [15]. In another study, Scheonung and coworkers fabricated bimodal 5083Al/6.5vol.% SiC (25 nm) composite by blending cryomilled composite powders with an equal amount of CG 5083 Al followed by HIP and hot rolling [56]. Such nanocomposite exhibits improved tensile ductility of 2.6% when compared with the nanocomposite consolidated from 100% of the cryomilled composite powders having a tensile ductility of 0.5%. This is because the ductile coarsegrains can undergo larger extent of plastic deformation, while ultrafine grains exhibit limited deformation.

Figure 17. Bright field TEM images for the bimodal 5083Al/10 wt% B4C composite in the (a) extrusion direction and (b) transverse direction, with the inset being the selected area diffraction patterns taken at the interface between the NC Al and B4C [15].

The creep behavior of the UFG composites is now considered. Presently, little is known regarding the high temperature creep behavior and deformation mechanism of such composites. What is the effect of reinforcing particles on the UFG composites having large grain boundary areas? Would these particles act as effective obstacles to the dislocation movement and hinder grain boundary sliding and diffusional flow during high temperature creep? If they do, the creep rates of UFG composites would reduce dramatically. More research work is needed in this area in near future to elucidate these problems. Nevertheless, proper understanding of the creep behavior of near-nanostructured Al-based alloy shed light on the creep deformation of UFG composites. Very recently, Chauhan et al. investigated the creep behavior of an UFG Al 5083 alloy at 573 – 648 K [57]. The alloy was prepared by consolidating cryomilled powders via HIP and extrusion. Analysis of the creep date reveals the presence of a temperature dependence threshold stress. Incorporation of this threshold

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stress into a modified creep equation yields a true stress of ~ 2 and a true activation energy close to that for boundary diffusion for Al, indicating that the rate controlling process is related to grain boundary sliding. In other words, the grain boundary activity becomes dominant in an UFG 5083 Al alloy during creep deformation at high temperatures. Processing of DRA composites through SPD is an effective route to refine the grain size of composites to submicrometer level and to disperse the reinforcing particles homogeneously within the UFG matrix. Langdon and coworkers studied microstructural development in an Al-6061 composite reinforced with 10 vol.% Al2O3 particulates by means of the HPT and ECAP techniques [23]. The average size of particulates is ~ 10 μm. For HPT, the samples were strained at room temperature to a total strain of ~ 7 under a pressure of 3.5 GPa. For ECAP, samples were pressed for eight passes at 673 K, and two additional passes at 473 K, giving a total strain of ~ 10. Substantial grain refinement of aluminum alloy matrix can be achieved using both techniques, i.e., a mean grain size of ~0.2 μm is attained after HPT and ~0.6 μm after ECAP. The microstructures of strained composites consist of an array of very small grains with poorly defined boundaries. There is no refinement for the alumina microparticles after HPT or ECAP treatment. The strength of the ECAP 6061 Al/Al2O3 composite is increased by almost two fold by ECAP and close to a factor of ~3 by torsion straining due to the grain refining of aluminum alloy matrix [23]. In general, ECAP treatment does not cause fracture of reinforcing particles during plastic straining, especially for finer particulates [28]. However, limited particle cracking is found for the 6061 Al composite reinforced with large alumina particulate (7.4 μm) subjected to ECAP at room temperature [58].

Figure 18. Temperature dependence of the compressive yield strength for the tri-modal 5083Al/10 wt% B4C composite. The inset shows true stress-strain curves tested at elevated temperatures [15].

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(a)

(b) Figure 19. TEM micrograhs of (a) unreinforced pure Al after eight passes and (b) Al/5 vol.% Gr composite after four passes at room temperature [29].

ECAP treatment of Al-based composites at room temperature is particularly attractive from the economic viewpoint. Proper selection of reinforcing particulates that experience no cracking is of technological interest. Very recently, Saravanan et al. used ECAP to refine the matrix grains of the Al/5 vol.% Gr (65 μm) composite at room temperature [29]. The soft and self lubricating nature of graphite can prevent the fracture of particulates during ECAP treatment. Figs. 19(a) shows a typical TEM micrograph of pure aluminum after eight ECAP passes at room temperature. The microstructure is characterized by well defined subgrains with a size of ~ 620 nm. In contrast, a significant grain refinement, down to the submicrometer level of ~ 300 nm can be achieved by pressing the Al/5 vol.% Gr composite at

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room temperature for only four passes (Fig. 19(b)). Moreover, the grain boundaries of submicron grains of the composite are diffused comparing to a well defined structure of Al grains. The selected area electron diffraction (SAD) patterns of pure Al and the composite reveal numerous spot features indicating the presence of an array of many ultarfine grains having random distribution of orientations (insets of Figs. 1(9a)-(b)). The tensile strength of the composite increases from 97 to 249 MPa after four ECAP passes.

Figure 20. TEM micrograph of Al/5 vol.% Al2O3 nanocomposite fabricated by HPT consolidation of raw material powders under 1.5 GPa [21].

Figure 21. Tensile stress-strain curves of Al samples (1,2) and Al/5 vol.% Al2O3 samples (3, 4, 5) fabricated by HPT consolidation under the pressure of 1.5 GPa and tested at 300 ºC at strain rates of 104 -1 s (1,3) and 10-3 s-1 (2, 4) and at 400 ºC at a strain rate of 10-4 s-1 (5) [21].

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Apart from forming ultrafine grains in composites, ECAP treatment can also consolidate ultrafine raw powders to produce fully dense (> 98%) bulk composite materials. Alexandrov et al. used the HPT technique to consolidate the Al powder (50 μm) and Al2O3 nanoparticle (50 nm) to form the Al/5 vol.% Al2O3 nanocomposite under a pressure of 1.5 GPa at room temperature. The powder mixture of nanocomposite was ball-milled for 30 min to ensure a uniform distribution of ceramic particles [21]. Fig. 20 shows the TEM micrograph of the HPT consolidated Al/5 vol.% Al2O3 nanocomposite. The nanocomposite exhibits an UFG structure having an average gain size of 120 nm. Room temperature tensile tests showed that the Al/5 vol.% Al2O3 nanocomposite have limited ductility of 1 to 2%. At 300 ºC, the nanocomposite tested at a strain rate of 10-3 s-1 had a plastic flow stress of ~ 66 MPa and a tensile ductility of ~ 20 % (Fig. 21). In contrast, pure Al had a flow stress of ~ 60 MPa and a tensile ductility of ~ 40 % tested at the same strain rate. However, the Al/5 vol.% Al2O3 nanocomposite showed a high strain-rate sensitivity of flow stress at 400 K; the strain-rate sensitivity (m) was 0.35. Strain rate sensitivity defined as the slope of logarithmic plot of the flow stress vs. strain rate. It is an inverse of stress exponent (n) and an important parameter in superplasticity. The Al/5 vol.% Al2O3 nanocomposite exhibited a low flow stress of 20 MPa but a high tensile ductility of ~ 200 %. The enhanced tensile ductility observed in the HPT consolidated nanocomposite with a total elongation of ~ 200 % indicating the occurrence of superplastic-like flow behavior. According to the literature, high strain rate super-plasticity can be achieved in ECAP processed aluminum alloys with UFG structures [59]. High strain rate superplasticity in the sub-micron metals is often characterized by very high flow stresses or pronounced strengthening. Grain boundary sliding is considered to be the dominant deformation mode for superplasticity in the sub-micron and nanocrystalline metals [60]. Future challenges for materials scientists are to elucidate the underlying creep and superplastic deformation mechanisms of aluminum based nanocomposites having UFG and nanocrystalline matrices.

Conclusions The development of aluminum nanocomposites is still in embryonic stage and there are many challenges in this field in the years ahead. Considerable progress has been made in the fabrication, microstructural and mechanical characterization of novel aluminum-based metal matrix nanocomposites in recent years. The nanocomposites can be simply prepared by incorporating very low volume contents of ceramic nanoparticles into aluminum matrix via PM or ingot casting. The nanocomposites thus prepared exhibit excellent mechanical properties including high yield strength and superior creep resistance. However, agglomeration of nanoparticles occurs readily during the composite fabrication, leading to poorer mechanical performance of composites with higher filler content. This problem can be eliminated in cast nanocomposites by using high-intensity ultrasonic waves to disperse the nanoparticles in molten aluminum. In the case of PM nanocomposites, cryomilling and severe plastic deformation processes have emerged as the two major processes to produce aluminum based composites having ultrafine grained matrix structures and homogeneous dispersion of reinforcing particles within the matrices.

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INDEX A Aβ, 283 accounting, 44 accuracy, 69, 124, 263 acetone, 136 acidity, 116 acoustic waves, 167 activation energy, 283, 285, 291 actuation, 125 actuators, xi, 106, 115, 257, 258, 272, 273 adaptation, 122 additives, 110, 118, 126 adhesion, ix, 11, 110, 112, 119, 120, 125, 141, 143, 212 adhesion properties, 120 adhesion strength, 143 adhesives, 162 adjustment, 183, 196, 198 aerospace, vii, viii, 2, 109, 110, 111, 112, 118, 119, 120, 122, 124, 126, 130, 276 age, ix, 109 Alabama, 126 algorithm, 52, 54, 65, 73, 75, 81, 103, 105, 216 alkalinity, 116 alloys, xi, 14, 47, 125, 163, 275, 276, 281, 283, 284, 288, 294 alternative(s), 6, 13, 36, 58, 118, 119 aluminium, 4, 162, 210, 221 aluminium alloys, 4 aluminum, xi, 14, 116, 117, 118, 119, 164, 275, 276, 277, 279, 280, 281, 283, 285, 287, 288, 289, 291, 292, 293, 294, 295 ambiguity, 244 amplitude, 136, 168, 174, 191, 193, 206, 215, 219, 225 anisotropy, 22, 42, 46, 117 annealing, 290

AP, 173, 175, 176, 178, 182, 183, 184, 186, 188, 189 Arborite, vii arithmetic, 5, 6, 41 ash, 255 aspect ratio, 44, 113, 115, 126, 130 asphalt, vii assessment, 206, 238 assignment, 154 assumptions, 20, 23, 232 asymptotic, 243, 248, 253 atmosphere, 120, 131, 288 atoms, 281, 283 attention, ix, 110, 130 automation, 113, 117 averaging, 6 avoidance, 119 awareness, 110

B barriers, 283 beams, 120, 219 behavior, xi, 4, 13, 49, 73, 75, 78, 105, 110, 118, 134, 144, 164, 244, 246, 257, 258, 272, 281, 283, 285, 288, 290, 294 Belgium, 51, 101, 102, 104, 105, 106, 209, 234, 235, 236 bending, 49, 59, 64, 72, 106, 196, 198, 210, 214, 215, 216, 218, 219, 225, 226, 227, 228, 229, 230, 231, 232, 233, 235, 236, 267, 269, 270, 271, 272 benefits, ix, 109, 116, 118, 119, 121, 124, 125, 126 bias, 217 biomaterials, 125, 128 blends, 255 BMI, 123 bonding, 143, 266 bounds, 12, 69 Bragg grating, 125, 204, 211, 212

298

Index

braids, 115 broadband, 211 bubbles, 133, 278 bulk materials, 115

C candidates, 38, 120 capillary, 134 carbon, vii, viii, ix, x, 2, 4, 6, 10, 11, 12, 18, 19, 22, 23, 27, 31, 36, 38, 42, 44, 46, 47, 48, 49, 50, 58, 81, 92, 109, 110, 112, 113, 115, 117, 118, 119, 120, 121, 123, 125, 126, 127, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 141, 143, 144, 145, 147, 149, 151, 153, 155, 156, 157, 159, 161, 162, 163, 164, 166, 205, 206, 207, 209, 212, 213, 214, 217, 220, 221, 225, 226, 227, 228, 234, 242, 248, 255, 256 carbon nanotubes, 44, 46, 49, 112 carbonization, 120 case study, 104, 127, 198 cast(ing), 277, 278, 280, 281, 282, 283, 294 catalyst, 122, 123 catalytic effect, 139 cell, 173, 191, 241, 262 cellulose, vii ceramic(s), vii, xi, 2, 6, 163, 257, 258, 266, 273, 275, 276, 277, 278, 281, 283, 285, 294 chain mobility, 141 chemical composition, 116 chemical properties, vii, 116 chemical reactions, 276 chemical reactivity, 115 chemical structures, 119 chicken, 125 China, 125 civil engineering, 130 classes, 115 clustering, 277, 279 clusters, 277 coatings, 112, 115, 118 collagen, vii collisions, 279 commercial, 72, 106, 117, 120, 163, 226, 266 communication, 234 community, 126 compatibility, 26 complexity, 105, 122, 219 compliance, 72, 84, 85, 93, 259 complications, 215 components, vii, 7, 18, 19, 22, 23, 28, 32, 33, 36, 38, 42, 113, 115, 121, 124, 147, 148, 165, 166, 212, 257, 261

composites, vii, viii, ix, x, xi, 1, 2, 3, 4, 6, 10, 13, 14, 15, 21, 22, 28, 31, 38, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 61, 69, 70, 101, 102, 103, 106, 109, 110, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 125, 126, 129, 130, 135, 143, 144, 148, 149, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 166, 204, 205, 206, 207, 209, 210, 212, 213, 214, 215, 217, 218, 221, 225, 233, 234, 235, 236, 254, 255, 256, 257, 258, 272, 275, 276, 277, 279, 281, 282, 283, 285, 288, 290, 291, 292, 294 composition(s), 43, 171, 183, 276 compounds, 114 computation, 28, 67, 94, 222, 244, 246 computer simulation, 123 computing, 104, 244 concentration, viii, 1, 2, 25, 26, 28, 115, 134, 135, 148, 149, 158, 164, 221, 224, 225, 258, 273 conception, 101 concrete, vii concurrent engineering, 104 condensation, 277 conditioning, 123 conductivity, 112, 113, 119, 132 confidence, 34 configuration, 3, 34, 65, 69, 73, 78, 81, 88, 91, 173, 175, 201 confusion, 250 Congress, 101, 106, 107 consolidation, 135, 136, 138, 288, 293 constant rate, 252 constraints, 52, 53, 54, 55, 56, 67, 69, 72, 73, 80, 81, 89, 90, 98, 99, 102, 107, 121 construction, 51, 111, 183 continuity, 71 control, x, 62, 116, 117, 124, 125, 129, 173, 191, 193 conventional composite, 126 convergence, 74, 77, 78, 83, 85, 94, 95, 97, 100, 102, 154, 231 convex, viii, 51, 52, 54, 55, 61, 64, 65, 66, 72, 74, 75, 79, 101, 103 cooling, 136 corn, 125 correlation(s), x, 168, 203, 209, 221 corrosion, 2, 110, 118, 119, 132, 241 cosine, 135 cost saving, 122 costs, ix, 110, 114, 121, 124, 125 Coulomb, 42 coupling, viii, 1, 44, 45, 59, 63, 64, 120, 148, 164, 219, 257 covalent bond, 119 coverage, 124 CPU, 56, 73, 98, 232

Index crack, 52, 99, 100, 116, 143, 162, 167, 168, 196, 198, 237, 239, 273, 288 creep, xi, 115, 218, 275, 281, 283, 284, 285, 286, 287, 288, 290, 291, 294 critical value, 260, 261 cross-linked polymers, 139 crystal polymers, 118 crystal structure, 132 crystalline, 47, 119, 132, 163 crystals, 133 curing, 113, 117, 121, 122, 123 cybernetics, 104 cycles, 54, 73, 84, 210, 214, 215, 277 cycling, 207

D damping, 112, 113, 114, 115, 166 database, 121 DD, 190, 205, 206, 239, 240, 243, 244, 246, 247, 248, 250, 251, 252, 253, 254 decomposition, 139, 141 decomposition temperature, 139, 141 defects, 120, 172, 219 defense, 120 definition, 3, 32, 59, 61, 63, 66, 77, 80, 82, 98, 113, 169, 187, 238, 240, 244, 247, 249, 253 deformability, 162 deformation, 105, 143, 148, 164, 182, 196, 216, 241, 277, 281, 283, 284, 285, 288, 290, 291, 294 degradation, x, 129, 131, 141, 200, 211, 215, 220, 225, 233, 236, 243, 257 degree of crystallinity, 119 Delaware, 125 delivery, 136 demand, xi, 118, 121, 275 Denmark, 102, 103, 104, 234 density, 11, 26, 44, 62, 76, 77, 79, 82, 96, 116, 117, 120, 130, 134, 135, 260, 264, 267, 276, 277, 279, 284, 288 derivatives, 75, 76, 77, 78, 103 detection, 204, 213, 216, 218, 219, 234, 254 deterministic, 52, 54, 56, 72 deviation, 10, 18, 41, 143, 144, 147 diamond, 173, 191 diaphragm, 121 dielectric, 259, 261, 263 dielectric permittivity, 259, 261 differential equations, 215 differential scanning calorimetry (DSC), x, 129, 136, 139, 140, 142, 161 differentiation, 258 diffraction, 45, 46, 48, 290

299

diffusion, 2, 25, 28, 44, 46, 134, 136, 283, 284, 285, 286, 291 diffusion process, 28, 44 diffusivity, 286 discontinuity, 235 discrete variable, 101 discretization, 184, 185, 186 discrimination, 167 discs, 102, 106 dislocation, 282, 283, 284, 286, 288, 290 dispersion, xi, 69, 115, 125, 133, 134, 135, 143, 144, 167, 275, 277, 278, 279, 280, 282, 283, 288, 289, 294 displacement, 26, 27, 59, 64, 87, 88, 149, 150, 152, 155, 166, 169, 173, 177, 178, 179, 180, 181, 191, 193, 195, 196, 197, 198, 200, 201, 203, 210, 215, 216, 222, 224, 225, 226, 227, 228, 231, 233, 238, 239, 240, 241, 244, 245, 246, 258, 259, 262, 266, 269, 273 distribution, 67, 68, 115, 122, 147, 156, 164, 183, 215, 264, 265, 268, 269, 272, 276, 277, 279, 288, 293, 294 doors, 124 double logarithmic coordinates, 286 dream, 121 ductility, 239, 277, 281, 282, 288, 290, 294 durability, 112, 113, 237 duration, 168, 210, 248, 253, 254 dynamic mechanical analysis, 141

E ears, 167 EI, 5, 7, 15, 22 Einstein, Albert, 258 elastic deformation, 143 elasticity, 12, 23, 48, 96, 166 elastomers, 44, 47 electric field, 120, 257, 258, 260, 263, 264, 267, 268, 269, 270, 272 electric potential, 258, 262, 263 electrical conductivity, 113, 116, 117 electrical properties, 118, 163 electrical resistance, 166, 204, 214, 234 electrodes, xi, 213, 234, 257, 258, 266, 272, 273 electromagnetic, 115, 212 electron, 147 electrospinning, 120 elongation, 294 embryonic, 294 emission, x, 165, 166, 167, 168, 169, 192, 196, 204, 205, 206 emulsification, 134

300

Index

endothermic, 139 endurance, 32 energy, x, xi, 61, 62, 72, 76, 77, 79, 82, 83, 84, 89, 91, 92, 99, 105, 110, 115, 118, 119, 122, 133, 143, 165, 166, 167, 168, 169, 170, 171, 172, 176, 177, 178, 179, 180, 181, 182, 183, 184, 195, 196, 197, 198, 199, 200, 201, 204, 205, 225, 237, 238, 239, 240, 241, 243, 244, 245, 246, 248, 249, 250, 251, 252, 253, 254, 255, 256, 257, 258, 260, 261, 263, 264, 268, 279 energy consumption, 143 energy density, xi, 61, 62, 76, 82, 83, 89, 91, 92, 257, 258, 260, 261, 263, 264, 268 energy emission, 201 environment, 110, 141, 279 environmental conditions, 2, 44 environmental control, 113 epoxy, vii, ix, 2, 6, 10, 11, 12, 13, 18, 19, 20, 22, 23, 27, 31, 34, 35, 36, 39, 41, 42, 44, 46, 48, 58, 81, 84, 92, 112, 116, 117, 120, 121, 125, 129, 130, 131, 138, 139, 141, 149, 150, 157, 161, 162, 163, 166, 173, 191, 204, 205, 206, 207, 210, 211, 212, 217, 218, 220, 231, 233, 235, 236, 240, 247, 248, 254, 255, 256 epoxy resins, 31, 34, 48, 120 equal channel angular pressing, 277 equality, 107, 239 equilibrium, 26, 27, 134 equipment, x, 110, 118, 123, 124, 139, 209 esters, 123 estimating, 5, 6, 8, 10, 12, 13, 42, 43, 47 European, viii, 49, 51, 52, 101, 102, 111, 162, 163, 206, 234, 235 evaporation, 120 evidence, 198 evolution, 44, 82, 88, 148, 204, 210, 211, 214 exclusion, 5 exothermic, 139 exposure, 117 extraction, 135 extrusion, 277, 279, 289, 290

195, 198, 200, 201, 204, 205, 207, 210, 221, 237, 253, 254, 275 family, 72, 101, 113, 118 fatigue, x, 2, 29, 110, 117, 166, 204, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 225, 226, 228, 229, 230, 231, 232, 233, 234, 235, 236, 254, 256 fiber bundles, 135, 156 fiber optics, 125 fibers, vii, viii, ix, 2, 3, 10, 11, 12, 13, 14, 18, 19, 20, 23, 27, 28, 29, 31, 35, 36, 42, 44, 49, 51, 52, 53, 54, 56, 57, 58, 61, 62, 65, 66, 67, 69, 70, 72, 73, 75, 81, 84, 85, 86, 88, 96, 97, 100, 101, 109, 110, 112, 113, 114, 115, 116, 117, 118, 119, 120, 122, 123, 124, 125, 126, 127, 135, 147, 148, 149, 150, 156, 161, 162, 192, 246, 248 filament, ix, 48, 129, 131, 133, 134, 136, 137, 138, 147, 161, 205 filled polymers, 130 fillers, 112, 114, 115, 116, 130, 132, 143 film(s), 50, 115, 122, 123, 138, 164 finance, 233 finite element method, 67 fires, 121 First World, 106, 107 fishing, vii fixation, 232 flame, 113, 114, 115, 118 flexibility, 110, 117, 119, 124, 126, 243 flexural strength, x, 129, 130, 136, 143 fluctuations, 133 fluid, 25, 27, 121, 133 focusing, 8 Formica, vii fracture processes, 147 fractures, 182, 194, 213 France, 1, 235 freedom, 118 friction, 151, 167, 215, 223, 224, 225, 226, 227, 228 fuel cell, 115 fulfillment, 119 functionalization, ix, 109, 112

F G FAA, 121 fabric, 125, 205, 207, 211, 212, 214, 217, 218, 220, 234, 236, 255, 256 fabrication, xi, 124, 136, 138, 275, 276, 277, 294 failure, viii, x, xi, 2, 31, 32, 33, 34, 35, 36, 38, 39, 40, 41, 43, 46, 47, 48, 53, 122, 130, 131, 143, 144, 148, 149, 155, 156, 158, 159, 160, 162, 164, 165, 166, 168, 171, 172, 175, 182, 183, 184, 187, 194,

GAO, 206, 254, 255 gas phase, 277 gauge, 193, 211, 215, 232 generation, 44, 112, 113 genetic algorithms, 52, 71, 104, 106 genre, 119 Germany, 50, 102, 106, 107, 234

Index glass, x, 47, 48, 84, 111, 116, 117, 120, 121, 123, 125, 126, 129, 135, 139, 141, 161, 204, 205, 206, 209, 210, 211, 217, 218, 220, 231, 234, 235, 236, 240, 247, 254, 255, 256 glass transition, x, 129, 139, 141, 161, 217 glass transition temperature, x, 129, 139, 161, 217 GNP, 115 gold, 118, 147 grain boundaries, 281, 288, 293 grain refinement, 291, 292 grains, 262, 277, 279, 280, 281, 288, 289, 290, 291, 292, 293, 294 graph, 140, 142 graphite, 112, 115, 117, 120, 173, 191, 205, 206, 233, 256, 292 gravimetric analysis, 141 Greece, 236 groups, 121 growth, 110, 116, 117, 118, 121, 143, 167, 168, 183, 196, 204, 215, 225, 235, 238, 239, 253, 256, 278, 288

H hardness, 113, 117, 118, 132, 241 head, 191, 193, 241 healing, 125 health, 110, 125 heat(ing), 111, 121, 122, 132, 134, 136, 167, 217, 290 heat transfer, 121 height, 215, 241 hemicellulose, vii heterogeneity, 246 high fat, 116 homogeneity, 277, 280 Hong Kong, 275 hot pressing, 279, 280 hot spots, 278 house(ing), 105, 119, 222 humidity, 116 hybrid, vii, 115, 116, 117, 123, 163, 234 hybridization, ix, 109, 116, 118 hydroxyapatite, vii hypothesis, 35, 135, 226, 228 hysteresis, 225, 226, 227, 228 hysteresis loop, 225, 226, 228

I identification, viii, 1, 2, 3, 14, 15, 17, 30, 31, 39, 40, 42, 43, 44, 167, 184, 204, 240

301

identity, 7 images, 221, 290 imaging, 205 immersion, 134 impact energy, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 251, 252, 253, 254 implementation, 98, 198, 215, 236 impregnation, ix, 122, 123, 129, 133, 136, 137, 161, 163 in situ, 117, 164 incidence, 219 inclusion, 8, 9, 11, 43, 45, 49, 112 independent variable, 63 indication, xi, 237 indicators, 201 indices, 258 industrial application, 52, 67, 73, 277 industrial sectors, 126 industry, ix, 110, 116, 117, 118, 121, 124, 130 inelastic, 44 inertia, 232 infinite, 9, 11, 28, 33, 40 infrastructure, 110, 118, 122 initiation, 225, 238, 256, 273 insight, 233 inspection, 218 Instron, 191 insulation, 174 integration, 223, 228, 230, 231, 241 integrity, x, 121, 165, 172, 183, 201 intelligence, 126 intensity, 136, 258, 277, 294 interaction(s), 4, 9, 32, 33, 34, 48, 50, 115, 123, 135, 141, 161, 223, 262, 273, 282, 283, 287, 288 interface, ix, 99, 110, 112, 119, 122, 127, 130, 135, 143, 148, 149, 150, 151, 153, 155, 156, 157, 160, 161, 162, 164, 265, 266, 269, 272, 273, 276, 290 interfacial adhesion, ix, 109, 110, 112, 119, 120 interfacial bonding, 162 interfacial properties, 120 interference, 115, 212 intermetallic nanoparticles, 276 international standards, 210 interpretation, 14, 182, 215 interval, 185 intuition, 134 invariants, 58, 63 inversion, 14, 15, 16, 31, 201 investment, 124 ion implantation, 120 IR, 122, 204 iron, 14, 48 irradiation, 163

302

Index

isostatic pressing, 279, 288 Italy, 237, 255 iteration, 52, 55, 77, 83, 84, 94, 95, 98, 154, 155

J Japan, 273

K kinetic energy, 239 knowledge transfer, 209

L labor, 123 laminar, 182, 183 laminated composites, 42, 103, 106, 205 lamination, 56, 63, 64, 65, 66, 67, 73, 101, 103, 104, 116, 175 laser, 120, 277, 282 laser ablation, 120, 277 laws, 24, 45, 47 lead, 15, 22, 74, 77, 119, 123, 124, 134, 143, 211, 217, 263, 279 leakage, 194, 198 leaks, 122 lifetime, 212 lignin, vii limitation, 134 liquid nitrogen, 121, 279 liquids, 134 literature, viii, x, 1, 2, 3, 4, 10, 14, 18, 21, 22, 31, 32, 33, 35, 38, 42, 43, 44, 45, 51, 52, 71, 72, 73, 100, 218, 237, 238, 244, 248, 253, 294 localization, 10, 11, 13, 17, 20, 30, 42, 45, 191 location, 60, 99, 241 London, 46, 163, 234, 235

M machine learning, 103 magnetic properties, 116 management, 113, 115, 117 manipulation, 80 manufacturer, 113 manufacturing, ix, 2, 3, 43, 61, 69, 102, 104, 105, 109, 110, 112, 113, 122, 123, 126, 129, 130, 133, 135, 138, 161, 212, 275 mapping, x, 209 market(s), 110, 116, 119, 122, 124

masking, 117 mass loss, 44 mass transfer process, 134 material degradation, 210 materials science, 6, 14 mathematical programming, 54, 65, 72, 102 mathematics, 5, 46 matrix, vii, viii, ix, x, 1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 28, 29, 30, 31, 35, 36, 37, 38, 39, 40, 41, 42, 43, 46, 48, 56, 57, 58, 63, 96, 109, 110, 112, 115, 116, 117, 119, 120, 121, 126, 130, 134, 135, 141, 143, 144, 148, 149, 150, 151, 153, 154, 155, 156, 157, 158, 162, 163, 164, 165, 166, 167, 168, 173, 175, 182, 183, 194, 196, 198, 211, 212, 218, 233, 237, 238, 239, 246, 253, 254, 256, 275, 276, 277, 279, 280, 281, 282, 283, 284, 285, 288, 291, 292, 294 measurement, x, 44, 46, 167, 209, 210, 212, 213, 214, 234, 244 mechanical behavior, 46, 149, 162, 164 mechanical degradation, 236 mechanical energy, 198, 200, 257 mechanical properties, ix, xi, 44, 61, 69, 109, 111, 112, 113, 117, 119, 124, 130, 135, 144, 207, 255, 275, 276, 277, 281, 282 mechanical stress, 27 media, 120, 279 melt(s), 117, 118, 120, 123, 125, 133, 134, 135, 136, 163 melting, 276, 277, 279, 282 melting temperature, 276 membranes, 59, 66, 73, 86, 105 MEMS, 257 metal oxide, 115 metallurgy, 276, 277, 282 metals, vii, 2, 47, 69, 114, 116, 117, 118, 119, 126, 209, 281, 294 Micarta, vii micrometer, 277, 281 microscopy, 218 microstructure(s), 2, 4, 22, 28, 41, 42, 43, 280, 281, 285, 288, 289, 291, 292 military, vii Ministry of Education, 272 mixing, 78, 115, 125, 136 MMA, 76, 77, 78, 79, 83, 88, 89, 90, 100, 101, 106 MMCs, 276, 283 mobility, 141, 161 modeling, 4, 43, 52, 105, 116, 124, 204 models, vii, viii, xi, 1, 2, 3, 4, 5, 13, 14, 18, 20, 30, 41, 42, 43, 44, 45, 47, 50, 103, 105, 166, 202, 215, 220, 221, 222, 229, 232, 236, 257, 258, 266, 267

Index modulus, 5, 12, 26, 42, 44, 46, 49, 96, 116, 118, 119, 120, 130, 131, 135, 141, 143, 144, 150, 156, 157, 175, 176, 184, 185, 186, 187, 188, 218, 223, 276, 283, 284, 286 moisture, vii, 1, 2, 4, 5, 8, 9, 10, 11, 13, 15, 17, 19, 20, 21, 23, 25, 26, 27, 42, 43, 44, 45, 48, 119, 287 moisture content, 5, 9, 11, 20, 21, 27, 43 moisture sorption, 44 mold, 121, 122, 123, 124, 138 moldings, 123 molecular weight, 118, 127, 135 molecules, 118, 119 monomer, 119, 123 Monte Carlo, x, 129, 131, 148, 149, 162, 163, 164 Moon, 135, 163 morphology, 2, 4, 7, 10, 11, 22, 42, 43, 46, 111, 276 Moscow, 163 motion, 141, 223, 229, 257 moulding, 212 movement, 141, 223, 224, 231, 283, 288, 290 MTS, 191 multiple factors, 43 multiplicity, 31, 165 multiwalled carbon nanotubes, 46

N nanocomposites, xi, 112, 113, 114, 115, 116, 119, 126, 130, 143, 161, 162, 163, 275, 276, 279, 280, 281, 282, 283, 285, 287, 288, 294 nanocrystalline metals, 281, 294 nanofibers, 112, 114, 115, 120 nanofillers, 115 nanometer, xi, 113, 275, 277, 281 nanometer scale, xi, 113, 275 nanoparticles, ix, xi, 113, 115, 116, 118, 129, 130, 133, 136, 139, 141, 143, 147, 161, 162, 275, 277, 278, 279, 280, 281, 282, 285, 287, 288, 289, 294 nanostructured materials, 288 nanostructures, 116, 288 nanotechnology, ix, 109, 110, 111, 120, 126 nanotube(s), 48, 50, 114, 130 nanowires, 114 NASA, 122, 127, 204 National Science Foundation, 162 negative relation, 201 negativity, 80 Netherlands, 49, 103, 105, 106, 116, 127 network, 46, 141 neutrons, 45 New Orleans, 103 New York, 47, 101, 127, 163, 295 Newton, 72, 232

303

next generation, ix, 109 nickel, 163 nitrogen, 120 nodes, 223, 231, 232 noise, 133, 143, 167, 173, 174 non-linear, 143, 231, 232, 260, 272

O observations, 34, 218 oil, 123 one dimension, 80 optimization, viii, 51, 52, 53, 54, 55, 61, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 78, 79, 80, 81, 82, 83, 84, 85, 88, 89, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 110, 117, 122, 124 optimization method, 52, 61, 69, 89, 100 optoelectronic, 241 organic fibers, 112, 118, 119, 120 organization, 121, 132 orientation, 42, 57, 58, 61, 62, 65, 66, 67, 70, 72, 73, 75, 78, 87, 88, 96, 97, 99, 103, 105, 111, 175, 182, 183, 246 oxidation, 44, 120, 132 oxide(s), 14, 283, 288 oxygen, 132, 287

P packaging, 276 PAN, 112, 120, 125, 131 parameter, 64, 66, 80, 89, 90, 91, 103, 150, 154, 160, 197, 204, 211, 217, 238, 294 Paris, 46, 101, 103, 235 particles, xi, 9, 110, 114, 115, 118, 130, 132, 133, 141, 143, 162, 275, 276, 277, 278, 280, 281, 282, 283, 288, 290, 291, 294 passive, 167, 212 PEEK, 120 pendulum, 49 percolation, 116 perforation, 191, 239, 240, 241, 244, 245, 246, 248, 249, 250, 251, 252, 253, 254, 255 performance, vii, xi, 2, 88, 110, 111, 113, 115, 117, 118, 120, 123, 125, 126, 130, 238, 239, 246, 257, 258, 266, 275, 294 permeability, 113, 122, 123, 124, 135 phenolic resins, 123 physical and mechanical properties, 277 physical properties, ix, 109 piezoelectric, xi, 125, 133, 167, 191, 241, 257, 258, 259, 261, 262, 263, 266, 267, 269, 272, 273

304

Index

piezoelectricity, 261 pitch, 125 planning, 122 plasma, 117, 120, 279 plastic deformation, xi, 275, 283, 288, 290, 294 plastic strain, 291 plasticity, 294 plastics, 166, 233 platelets, 115, 130 plywood, vii PM, 276, 277, 279, 281, 283, 287, 288, 294 PMMA, 255 Poisson ratio, 131 polarization, xi, 257, 258, 260, 261, 262, 263, 268, 269, 272, 273 polyarylate, 118 polycarbonate, 163 polycrystalline, 4, 14, 15, 48, 163 polyester(s), 47, 118, 123, 204, 254, 255 polyetheretherketone, 120 polyethylene, vii, 50, 118, 119, 127, 135, 163 polymer(s), vii, ix, x, 1, 2, 4, 6, 15, 41, 44, 48, 110, 111, 112, 114, 115, 116, 118, 119, 120, 123, 124, 125, 126, 130, 134, 135, 141, 143, 161, 209, 215, 236, 255 polymer chains, 119, 141, 161 polymer composites, x, 209, 215, 255 polymer materials, ix, 110, 115 polymer matrix, 6, 41, 44, 115, 119, 125, 130, 141 polymer solutions, 134 polymer systems, 112 polymer-based composites, 44 polymeric composites, 110, 117 polymeric materials, 112 polymeric matrices, ix, 110 polystyrene, 163 poor, 78, 134, 135, 136 porous media, 134 ports, 122 Portugal, 102 power, 110, 120, 133, 215, 238, 243, 283 PPS, 214, 217, 221, 225, 226, 227, 228 prediction, 13, 30, 31, 32, 35, 44, 47, 206, 254 pressure, x, 64, 92, 104, 121, 122, 123, 133, 138, 165, 205, 224, 241, 277, 291, 293, 294 probability, 150, 166 probe, 133 production, ix, 110, 120, 122, 124, 125, 212 program(ming), viii, 51, 52, 56, 101, 105, 106, 125, 155, 157 proliferation, 112 propagation, 69, 162, 167, 220, 225, 238, 256 proposition, 205

prototype, 124 PTFE, 162 pulses, 219 pultrusion, 121, 125 pumps, 124 PVC, 46 pyrolysis, 120

Q qualifications, 114 quantitative estimation, 184

R radial distance, 28 radiation, 115 radius, 28, 53, 191, 226, 241, 266, 269 range, 6, 13, 31, 34, 42, 44, 49, 120, 122, 123, 124, 130, 133, 134, 135, 147, 154, 166, 169, 187, 196, 211, 239, 240, 244, 246, 247, 248, 253, 255, 263 reactive groups, 119 reactivity, 277 realism, 4, 44 reality, 167, 168, 262 reasoning, 17, 22 recalling, 52 recognition, ix, 110 reconstruction, 221 recovery, 279 recrystallization, 279 recycling, 125 redistribution, 167, 184, 215 reduction, 172, 201, 212, 233, 251 reference frame, 8, 32 refining, 291 reflection, 167 refractive index, 211, 218 regression, 185, 201, 239, 240, 285 regression line, 240, 285 reinforcement, ix, 3, 15, 17, 104, 109, 110, 112, 119, 125, 126, 210, 254, 276, 277, 279, 284 reinforcing fibers, 18, 41, 42 rejection, 167 relationship(s), x, 116, 118, 130, 134, 149, 160, 168, 239, 241, 243, 247, 259 relaxation, 81, 154 relevance, 248 reliability, viii, 2, 32, 43, 44, 166 renewable energy, 111, 112, 115 repair, 117, 118, 163 reserves, 125

Index resin reaction, 133, 136 resins, 39, 41, 43, 46, 118, 120, 122, 123, 124, 125, 130 resistance, x, xi, 2, 104, 111, 113, 116, 117, 118, 119, 132, 162, 166, 198, 205, 209, 213, 214, 234, 241, 255, 275, 283, 285, 294 resolution, 220 resources, 54, 172 revolutionary, 111 rheological properties, 50 Rhode Island, 104 rice, 125 robust design, 102 rods, vii rolling, 288, 290 room temperature, 122, 124, 276, 288, 290, 291, 292, 293 Royal Society, 46 RTS, 166, 201 rubber, 130, 162

S SA, 81, 102, 105, 163, 205, 233 SAD, 293 safety, 94 sample(ing), 8, 31, 34, 136, 147, 215, 220, 221, 224, 238, 241, 277 satellite, 122 satisfaction, 54 saturation, 239 savings, 70, 124 sawdust, vii scattering, 243, 253 science, ix, 110, 126, 257 scull, vii search, 66, 101, 103 selected area electron diffraction, 293 selecting, 100 SEM micrographs, 147 sensing, x, 125, 209, 212 sensitivity, 17, 56, 61, 66, 81, 94, 103, 105, 119, 139, 294 sensors, 115, 125, 166, 167, 191, 192, 211, 212, 221, 234, 263 separation, 135 series, 134, 173, 184, 213, 239, 241, 248 shape, 22, 42, 49, 53, 54, 69, 74, 101, 103, 104, 114, 115, 122, 124, 125, 132, 150, 196, 221, 225, 226, 228, 276 shape-memory, 125 shear, x, 29, 31, 32, 33, 35, 46, 48, 49, 59, 94, 103, 104, 129, 147, 148, 149, 150, 151, 155, 158, 164,

305

195, 201, 210, 217, 218, 235, 255, 262, 265, 269, 272, 277, 283 shear strength, 155 shock waves, 133 shortage, 119, 125 Si3N4, 276, 281, 282, 283, 285 SIC, 129 sign, 32 signaling, 238, 249 signals, 168, 169, 173, 219, 238 silane, 120 silica, 115 silicon, ix, 14, 129, 132, 136, 163 Silicon carbide, 163 silver, 147 simulation, x, 10, 39, 45, 124, 130, 131, 148, 155, 157, 162, 163, 164, 210, 219, 221, 227, 228, 232, 233, 273 sine wave, 214 Singapore, 125 sintering, 279, 287 sites, 133 smart materials, 113 society, ix, 109 software, 45, 72, 106, 187, 266 solid state, 124 solvent(s), 111, 118, 120, 133, 134, 135, 136 Spain, 235 species, 135 specific surface, 115, 277 speed, 85, 94, 97, 111, 116, 150, 154, 173, 191, 193, 205, 220, 255 spindle, 136 sports, 110, 111, 118, 124 stability, 71, 113, 119, 143 stages, 125, 240 statistical analysis, 196 statistics, 5, 6 steel, 96, 191, 210, 223, 241 storage, vii, 124, 135 strain, x, 9, 10, 11, 13, 14, 21, 23, 31, 32, 34, 35, 36, 39, 43, 44, 45, 47, 56, 61, 62, 67, 72, 82, 89, 91, 92, 105, 130, 143, 144, 148, 154, 155, 156, 157, 158, 160, 161, 164, 165, 167, 169, 170, 172, 175, 176, 182, 183, 184, 185, 186, 187, 188, 195, 196, 198, 199, 200, 201, 204, 209, 210, 211, 212, 213, 214, 215, 217, 218, 225, 228, 232, 234, 235, 241, 254, 258, 259, 260, 263, 267, 268, 269, 272, 277, 285, 288, 291, 293, 294 strategies, 172 stratification, 68 strength, viii, xi, 2, 3, 10, 18, 19, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 46, 51, 53, 66,

306

Index

67, 69, 70, 71, 72, 73, 84, 94, 101, 102, 103, 105, 106, 111, 113, 116, 117, 118, 119, 120, 121, 122, 126, 130, 131, 143, 144, 150, 154, 155, 156, 157, 158, 160, 161, 162, 163, 164, 166, 168, 169, 175, 182, 183, 203, 204, 206, 207, 215, 233, 237, 238, 241, 243, 275, 276, 277, 281, 282, 285, 288, 290, 291, 294 stress, viii, ix, x, xi, 2, 21, 24, 25, 27, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 47, 48, 49, 56, 59, 71, 96, 102, 110, 115, 117, 123, 130, 143, 148, 149, 150, 151, 154, 155, 156, 157, 158, 161, 165, 167, 168, 172, 175, 176, 177, 178, 179, 180, 181, 184, 185, 186, 187, 210, 212, 214, 215, 217, 221, 224, 225, 228, 235, 257, 258, 261, 262, 264, 265, 268, 269, 271, 272, 275, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 293, 294 stress-strain curves, 157, 175, 187, 212, 289, 291, 293 stretching, 120 structural characteristics, 119 suffering, 125 Sun, 127, 273 superplasticity, 281, 294 suppliers, 120, 122 supply, 113, 133 supply chain, 113 surface area, 130 surface energy, 119 surface tension, 134, 135 surface treatment, 205 Sweden, 103 switching, xi, 257, 258, 260, 261, 262, 263, 264, 268, 269, 270, 272, 273 symbols, 76, 78 symmetry, 7, 132, 222, 232, 233, 259, 262, 266 synthesis, 106, 173, 239, 276 systems, vii, x, 110, 113, 115, 116, 117, 121, 123, 124, 129, 130, 141, 143, 162, 254

T tanks, vii, 124 Taylor series, 55, 74, 75 technology, viii, ix, 109, 110, 112, 117, 118, 119, 126, 204, 233, 257 teflon, 138 TEM, 133, 280, 287, 288, 289, 290, 292, 293, 294 temperature, ix, x, 2, 14, 21, 43, 44, 115, 116, 117, 118, 121, 123, 124, 125, 129, 131, 132, 136, 138, 141, 217, 279, 283, 285, 286, 287, 288, 290, 294 temperature dependence, 290

tensile strength, x, 28, 44, 46, 118, 119, 155, 156, 160, 161, 162, 164, 165, 166, 167, 201, 204, 282, 288, 293 tensile stress, 28, 33, 148, 156, 160, 224 tension, 32, 33, 35, 36, 41, 48, 134, 144, 156, 162, 196, 210, 212, 213, 214, 215, 217, 221, 229, 234, 236, 267, 268, 269 test data, 31, 144, 145, 147, 253 Texas, 163 textiles, vii TGA, x, 129, 141, 142, 161 theory, 9, 25, 34, 43, 46, 56, 59, 79, 156, 175, 215, 233 thermal expansion, vii, 1, 4, 10, 13, 14, 16, 17, 42, 43, 45, 113, 117, 119 thermal properties, viii, 1, 13, 42 thermal stability, 113, 132, 141, 142, 161 thermodynamics, 44, 134 thermo-mechanical, 11, 12, 14, 43, 49, 148, 164 thermomechanical treatment, 288 thermoplastic(s), vii, ix, 110, 113, 115, 117, 123, 125, 126, 163, 210, 212, 213, 214, 218, 219, 221, 234, 235, 256 Third World, 101 threat(s), xi, 119, 125, 237, 238, 243 threshold(s), xi, 35, 116, 166, 174, 191, 193, 201, 205, 237, 238, 239, 240, 241, 242, 243, 246, 248, 253, 254, 256, 283, 285, 288, 290 titanium, 14, 46, 112, 117 Tokyo, 105 toluene, 135 topology, 52, 54, 67, 68, 71, 72, 74, 78, 81, 96, 97, 101, 102, 106 tracking, 212 transducer, 219 transformation, 11, 21, 45, 132, 163 transition(s), vii, viii, 1, 2, 3, 4, 6, 8, 9, 10, 12, 13, 14, 17, 18, 19, 20, 21, 35, 36, 41, 42, 43, 44, 45, 47, 198, 228, 288 transmission, 110, 218, 219 transparency, 218 transport, 44 transportation, 118, 122, 124 treatment methods, ix, 110, 115, 120 trend, ix, 110, 113, 117, 118, 120, 121, 122, 170, 172, 173, 182, 183, 184, 196, 198, 200, 242, 243, 252, 253 trial and error, 124 triangulation, 167

U UK, 234, 235, 254

Index ultrasonic waves, 133, 219, 277, 278, 294 ultrasound, 133, 163, 219, 220 uniaxial tension, 31 uniform, 21, 45, 64, 92, 122, 124, 133, 134, 136, 147, 241, 294 universities, 121 updating, 77 USDA, 125 UV, 119, 121

V vacuum, 121, 122, 123, 124 validation, x, 166, 209, 229, 236 values, viii, 2, 4, 8, 13, 18, 19, 20, 41, 52, 54, 58, 61, 64, 65, 66, 70, 76, 77, 79, 81, 84, 88, 89, 90, 95, 110, 116, 122, 134, 143, 156, 166, 173, 176, 182, 185, 186, 187, 193, 196, 200, 201, 204, 238, 241, 243, 246, 247, 248, 252, 253, 254, 257, 258, 263, 265, 267, 272, 277, 283, 285 vapor, 46 variable(s), viii, x, xi, 26, 51, 52, 53, 54, 55, 61, 62, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84, 88, 89, 90, 94, 95, 96, 97, 98, 99, 100, 101, 103, 121, 169, 201, 202, 237, 239, 240, 244, 246, 250, 253 variation, 44, 57, 61, 82, 83, 84, 89, 90, 91, 99, 100, 144, 147, 172, 200, 211, 228, 285 vector, 25, 52, 59, 258, 261 vehicles, 115 velocity, x, 165, 166, 205, 207, 236, 237, 238, 241, 243, 254, 255 versatility, 124 vessels, x, 165, 205 vibration, 53, 64, 66, 71, 125

307

Vietnam, 105 vinylester, 254 Viscoelastic, 49 viscosity, 115, 122, 123, 134, 135 visualization, 234

W Washington, 105 wave propagation, 167 wavelengths, 211 wear, 113, 118, 162, 275 Weibull distribution, 150 weight ratio, 2, 51, 166 weight reduction, 118 welding, 279 wettability, 120, 278 wetting, 112, 119, 134, 135 wheat, 125 wind, 110 wood, vii workers, 130 writing, 6, 22, 45

X X-ray, 47, 121, 166, 220, 236 X-ray diffraction (XRD), 47 xylene, 135

Y yield, 32, 42, 45, 121, 124, 225, 238, 277, 281, 282, 288, 290, 291, 294