석사학위논문 Master’s Thesis
손상경계면을 갖는 복합재료의 탄성 및 탄소성 다단계 손상 모델
Elastic and Elastoplastic Multi-Level Damage Models for Composites with Imperfect Interface
표
석 훈 (表 錫 勳 Pyo, Suk-Hoon) 건설 및 환경공학과
Department of Civil and Environmental Engineering 한 국 과 학 기 술 원 Korea Advanced Institute of Science and Technology 2008
손상경계면을 갖는 복합재료의 탄성 및 탄소성 다단계 손상 모델
Elastic and Elastoplastic Multi-Level Damage Models for Composites with Imperfect Interface
Elastic and Elastoplastic Multi-Level Damage Models for Composites with Imperfect Interface
Advisor: Professor Haeng-Ki Lee by Suk-Hoon Pyo Department of Civil and Environmental Engineering Korea Advanced Institute of Science and Technology
A thesis submitted to the faculty of the Korea Advanced Institute of Science and Technology in partial fulfillment of the requirements for the degree of Master of Science in Engineering in the Department of Civil and Environmental Engineering. Daejeon, Korea 2007. 12. 10. Approved by
_____________________ Professor Haeng-Ki Lee
손상경계면을 갖는 복합재료의 탄성 및 탄소성 다단계 손상 모델
표 석 훈
위 논문은 한국과학기술원 석사학위논문으로 학위논문 심사 위원회에서 심사 통과하였음.
2007 년 12 월 10 일
심사 위원장
이 행 기
(인)
심 사 위 원
김 진 근
(인)
심 사 위 원
곽 효 경
(인)
i
MCE
표 석 훈. Pyo, Suk-Hoon. Elastic and Elastoplastic Multi-Level Damage Models
20063601
for Composites with Imperfect Interface. 손상경계면을 갖는 복합재료의 탄성 및 탄소성 다단계 손상 모델. Department of Civil and Environmental Engineering. 2008. 134p. Advisor: Prof. Lee, Haeng-Ki. Text in English.
ABSTRACT Micromechanics-based multi-level damage models are presented to predict elastic and elastoplastic overall behavior and damage evolution in composite materials. First of all, Eshelby’s tensor for an ellipsoidal inclusion with slightly weakened interface is adopted to model particles with imperfect interface in particle reinforced composites and fibers with imperfect interface in fiber reinforced composites, respectively. Imperfect interfaces between inclusion and matrix in composites occur as deformations or loadings continue to increase. As an effort to realistically reflect the effect of loading history on the progression of imperfect interface, a multi-level damage model in accordance with the Weibull’s probabilistic function is developed. It is assumed that the progression of imperfect interface is governed by the average stress of inclusion. To the estimate overall elastoplastic behavior of ductile matrix composites, an effective yield criterion is derived based on the ensemble-volume averaging process and the first-order effects of eigenstrains due to the existence of spherical inclusions. As applications, the overall elastic and elastoplastic stress-strain curves of particle reinforced brittle matrix composites, particle reinforced ductile matrix composites, and unidirectional fiber reinforced brittle matrix composites under uniaxial, biaxial, and triaxial loading are investigated in detail. A series of parametric analysis are carried out to investigate the influence of model parameters. Furthermore, the proposed multi-level damage models are compared with available experimental data in the literature to verify the accuracy of the models. Finally, proposed micromechanics-based evolutionary multi-level damage model for unidirectional fiber reinforced composites is implemented into a finite element program ABAQUS to predict behavior of laminated composites under various loading conditions.
ii
TABLE OF CONTENTS ABSTRACT..................................................................................................................................................i TABLE OF CONTENTS ............................................................................................................................ ii LIST OF FIGURES .....................................................................................................................................v
CHAPTER 1 INTRODUCTION ..............................................................................................................1 1.1 Review and Scope .......................................................................................................................1 1.2 Overview of the present study .....................................................................................................3
CHAPTER 2 MICROMECHANICS-BASED ELASTIC DAMAGE MODELING OF PARTICULATE COMPOSITES WITH IMPERFECT INTERFACES ....................................................................................................................5 2.1 Introduction .................................................................................................................................5 2.2 A micromechanics-based constitutive model for particulate composites with imperfect interfaces ....................................................................................................................8 2.2.1 Recapitulation of the Eshelby's tensor for an ellipsoidal inclusion with slightly weakened interface ..............................................................................................8 2.2.2 Effective elastic moduli of three-phase particulate composites ........................................9 2.3 Damage modeling ..................................................................................................................... 12 2.4 Numerical simulation ................................................................................................................ 14 2.5 Comparisons with other numerical and analytical predictions and experiment ........................ 22 2.6 Appendix ................................................................................................................................... 27
CHAPTER 3 MULTI-LEVEL MODELING OF EFFECTIVE ELASTIC AND PROGRESSIVE IMPERFECT INTERFAVE IN PARTICULATE COMPOSITES ................................................................................................................. 28 3.1 Introduction ............................................................................................................................... 28 3.2 Multi-level damage modeling ................................................................................................... 31 3.2.1 Level 0 of two-phase composite state ............................................................................. 31 3.2.2 Level 1 of three-phase composite state ........................................................................... 31
iii
3.2.3 Level 2 of four-phase composite state ............................................................................. 34 3.2.4 Level 3 of five-phase composite state ............................................................................. 35 3.3 A micromechanics-based, multi-level elastic model for particulate composites with imperfect interfaces .......................................................................................................... 36 3.3.1 Level 0 of two-phase composite state ............................................................................. 36 3.3.2 Level 1 of three-phase composite state ........................................................................... 37 3.3.3 Level 2 of four-phase composite state ............................................................................. 37 3.3.4 Level 3 of five-phase composite state ............................................................................. 38 3.4 Numerical example ................................................................................................................... 39 3.5 Parametric analysis ................................................................................................................... 41 3.6 Experimental comparisons ........................................................................................................ 44 3.7 Appendix ................................................................................................................................... 48
CHAPTER 4 AN ELASTOPLASTIC MULTI-LEVEL DAMAGE MODEL FOR DUCTILE MATRIX COMPOSITES EVOLUTIONARY IMPERFECT INTERFACE ............................................................................................ 50 4.1 Introduction ............................................................................................................................... 50 4.2 Ensemble-volume-average homogenization process ................................................................ 52 4.2.1 Overview ......................................................................................................................... 52 4.2.2 A first-order formulation of the current stress norm of multi-phase ductile matrix composites ................................................................................................ 52 4.2.3 A first-order formulation of effective elastoplastic behavior of multiphase ductile matrix composites ...................................................................................... 54 4.3 Numerical simulation ................................................................................................................ 60 4.4 Parametric analysis ................................................................................................................... 66 4.5 Experimental comparison.......................................................................................................... 69
CHAPTER 5 A MICROMECHANICS-BASED EVOLUTIONARY DAMAGE MODEL FOR CONTINUOUS, FIBER-REINFORCED BRITTLE MATRIX COMPOSITES WITH MICROCRACKS AND IMPERFECT INTERFACES .......................................................................................... 71
iv
5.1 Introduction ............................................................................................................................... 71 5.2 Effective elastic moduli of continuous, fiber-reinforced brittle matrix composites with microcracks and imperfect interfaces ............................................................ 74 5.3 Damage modeling ..................................................................................................................... 80 5.4 Numerical simulations .............................................................................................................. 83 5.5 Experimental comparison.......................................................................................................... 90 5.6 Appendix ................................................................................................................................... 93
CHAPTER 6 NUMERICAL ANALYSIS OF LAMINATED COMPOSITE STRUCTURES BASED ON MICROMECHANICAL MULTI-LEVEL DAMAGE MODEL .......................................................................................................... 97 6.1 Introduction ............................................................................................................................... 97 6.2 Micromechanics-based constitutive damage model for off-axis unidirectional fiber reinforced composites ............................................................................... 99 6.3 Damage modeling ................................................................................................................... 101 6.4 Numerical simulations and experimental comparison ............................................................ 104 6.4.1 Verification of the computer code ................................................................................. 104 6.4.2 Experimental comparison of the response and damage behavior of a laminated composite under bending ............................................................................. 106 6.4.3 Experimental comparison of the response and damage behavior of a laminated composite under shear .................................................................................. 113 6.5 Appendix ................................................................................................................................. 118
CHAPTER 7 CONCLUSIONS ............................................................................................................. 119
SUMMARY (IN KOREAN) .................................................................................................................. 124
REFERENCES ....................................................................................................................................... 125
ACKNOWLEDGEMENTS CURRICULUM VITAE
v
LIST OF FIGURES 2.1
Schematics of a particulate composite subjected to uniaxial tension: (a) the initial state; (b) the damaged state .......................................................................................................................6
2.2
The present predicted stress-strain responses of particulate composites with imperfect interfaces under uniaxial tension with various S 0 values ............................................... 14 B
2.3
B
The predicted evolution of volume fractions of perfectly bonded particles and particles with imperfect interface corresponding to Figure 2.2 (S 0 = 1.09*σ y ) ............................................ 15 B
2.4
B
B
B
The predicted evolution of volume fractions of perfectly bonded particles and particles with imperfect interface corresponding to Figure 2.2 (S 0 = 2.18*σ y ) ............................................ 16 B
2.5
B
B
B
The predicted evolution of volume fractions of perfectly bonded particles and particles with imperfect interface corresponding to Figure 2.2 (S 0 = 3.27*σ y ) ............................................ 17 B
2.6
B
B
B
The present predicted stress-strain responses of particulate composites with imperfect interfaces under biaxial tension with various S 0 values ................................................................. 18 B
2.7
B
The predicted evolution of volume fraction of particles with imperfect interface corresponding to Figure 2.6 ........................................................................................................... 19
2.8
The present predicted stress-strain responses of particulate composites with imperfect interfaces under hydrostatic tension with various S 0 values .......................................................... 20 B
2.9
B
The predicted evolution of volume fraction of particles with imperfect interface corresponding to Figure 2.8 ........................................................................................................... 20
2.10
The present predicted elastic responses of particulate composites with imperfect interfaces under uniaxial tension with various α and β values ....................................................... 21
2.11
The comparison of predicted overall elastic uniaxial responses of 6061-T6 aluminum alloy matrix/silicon-carbide particle composites between the present perfect bonding and imperfect interface models and Ju and Lee’s (2001) partial debonding model .............................. 23
2.12
The comparison between the present predictions with various combinations of volume fractions of perfectly bonded particles and particles of imperfect interface and the Halpin-Tsai’s theoretical prediction for effective (normalized) Young’s modulus versus volume fraction of particles ........................................................................................................... 24
2.13
The comparison between the present prediction and experimental data (Zhou et al., 2004) for overall uniaxial tensile responses of Mg-Al matrix with Al 2 O 3 particulate B
B
B
B
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composites...................................................................................................................................... 25 2.14
The predicted evolution of volume fractions of perfectly bonded particles and particles with imperfect interface corresponding to Figure 2.13 .................................................................. 26
3.1
Schematics of a particulate composite subjected to uniaxial tension: (a) Level 0 of twophase composite state (initial state); (b) Level 1 of three-phase composite state; (c) Level 2 of four-phase composite state; (d) Level 3 of five-phase composite state ................................... 30
3.2
The present predicted stress-strain responses of particulate composites under uniaxial tension ............................................................................................................................................ 39
3.3
The predicted evolution of volume fractions of perfectly bonded particles various types of damaged particles corresponding to Figure 3.2 ......................................................................... 40
3.4
The present predicted stress-strain responses of particulate composites under uniaxial tension at the four-phase composite state with various S 0 values .................................................. 41 B
3.5
B
The predicted evolution of volume fractions of perfectly bonded particles, particles with mild imperfect interface and particles with severe imperfect interface corresponding to Figure 3.4 ....................................................................................................................................... 42
3.6
The present predicted stress-strain responses of particulate composites under uniaxial tension at four-phase composite state with various α 2 and β 2 values ............................................ 43 B
3.7
B
B
B
The comparison between the present prediction and experimental data (Sørensen, 1993) for overall uniaxial tensile responses of SiC fiber-reinforced CAS glass-ceramic matrix composites...................................................................................................................................... 44
3.8
The predicted evolution of volume fractions of perfectly bonded particles, particles with mild imperfect interface, severe imperfect interface and completely debonded particles corresponding to the present prediction in Figure 3.7 .................................................................... 45
3.9
The comparison between the present prediction and experimental data (Geiger and Welch, 1997) for overall uniaxial tensile responses of silicon carbide particle-reinforced 2009 aluminum matrix composites ................................................................................................ 46
3.10
The predicted evolution of volume fractions of perfectly bonded particles, particles with mild imperfect interface, severe imperfect interface and completely debonded particles corresponding to the present prediction in Figure 3.9 .................................................................... 47
4.1
The present predicted elastoplastic stress-strain responses of particulate composites under uniaxial tension .................................................................................................................... 61
vii
4.2
(a) The predicted evolution of volume fractions of perfectly bonded particles and particles with mild impefect interface of the composites at the three-phase composite state; (b) The predicted evolution of volume fractions of perfectly bonded particles and various types of damaged particles at the four-phase composite state; (c) The predicted evolution of volume fractions of perfectly bonded particles and various types of damaged particles at the five-phase composite state corresponding to Figure 4.1 ......................... 62
4.3
The present predicted elastoplastic stress-strain responses of particulate composites under biaxial tension ...................................................................................................................... 63
4.4
The predicted evolution of volume fractions of perfectly bonded particles and various types of damaged particles corresponding to Figure 4.3 ................................................................ 64
4.5
The present predicted elastoplastic stress-strain responses of particulate composites under triaxial tension (Γ = 0.6)....................................................................................................... 65
4.6
The predicted evolution of volume fractions of perfectly bonded particles and various types of damaged particles corresponding to Figure 4.5 ................................................................ 66
4.7
The present predicted elastoplastic stress-strain responses of particulate composites under uniaxial tension at the four-phase composite state with various S 0 values .......................... 67 B
4.8
B
The present predicted elastoplastic stress-strain responses of particulate composites under uniaxial tension at the four-phase composite state with various α 2 and β 2 values ............... 68 B
4.9
B
B
B
The comparison between the present prediction and experimental data (Papazian and Adler, 1990) for overall uniaxial tensile responses of SiC particulate-reinforced 5456 aluminum alloy matrix composites ................................................................................................ 70
4.10
The predicted evolution of volume fractions of perfectly bonded particles, particles with mild imperfect interface, severe imperfect interface and completely debonded particles corresponding to the present prediction in Figure 4.9 .................................................................... 71
5.1
The schematic of a continuous, fiber-reinforced composite subjected to uniaxial loading in the longitudinal direction ........................................................................................................... 73
5.2
The schematic of a penny-shape crack modeled as an oblate spheroidal void (a 1 = a 2 >> B
B
B
B
a 3 ) and aspect ratio a 3 /a 1 →0 ......................................................................................................... 75 B
5.3
B
B
B
B
B
The schematic of multi-level damage transition in a continuous, fiber-reinforced composite: (a) Level 0 of three-phase composite state (initial state); (b) Level 1 of threephase composite state; (c) Level 2 of five-phase composite state; (d) Level 3 of six-phase
viii
composite state ............................................................................................................................... 76 5.4
The present predicted stress-strain responses of unidirectional composites under uniaxial tension in the longitudinal direction with various S 0 values .......................................................... 84 B
5.5
B
The predicted evolution of volume fractions of perfectly bonded fibers, fibers with mild imperfect interface, fibers with severe imperfect interface and completely debonded fibers corresponding to Figure 5.4 ................................................................................................. 85
5.6
The present predicted stress-strain responses of unidirectional composites under uniaxial tension in the transverse direction with various S 0 values ............................................................. 86 B
5.7
B
The present predicted stress-strain responses of particulate composites under uniaxial tension in the transverse direction with various α 2 and β 2 values.................................................. 87 B
5.8
B
B
B
The present predicted stress-strain responses of unidirectional composites under uniaxial tension in the longitudinal direction with various c 1 values .......................................................... 88 B
B
5.9
The predicted evolution of microcrack density corresponding to Figure 5.8 ................................. 89
5.10
The present predicted stress-strain responses of unidirectional composites under uniaxial tension in the longitudinal direction with various c 2 values .......................................................... 90 B
B
5.11
The predicted evolution of microcrack density corresponding to Figure 5.10 ............................... 90
5.12
The comparison between the present predictions (with and without microcracks) and experimental data (Luo and Daniel, 2000) for overall uniaxial tensile responses in the longitudinal direction of silicon carbide/calcium aluminosilicate (SiC/CAS) composites ............ 91
5.13
The predicted evolution of volume fractions of perfectly bonded fibers, fibers with mild imperfect interface, severe imperfect interface and completely debonded fibers (fivephase composite state) corresponding to Figure 5.12 .................................................................... 92
5.14
The predicted evolution of microcrack density (six-phase composite state) corresponding to Figure 5.12 ................................................................................................................................. 93
6.1
Coordinates for a transversely isotropic material ........................................................................... 99
6.2
The comparison between the present prediction and experimental data (Weeks and Sun, 1998) for overall uniaxial tensile responses with three fiber orientations of AS4/PEEK composite ..................................................................................................................................... 105
6.3
The predicted evolution of volume fraction of perfectly bonded fibers, fibers with mild imperfect interface, fibers with severe imperfect interface, completely debonded fibers and microcrack corresponding to 15º fiber orientation case in Figure 6.2 ................................... 106
ix
6.4
The finite element discretization employed to simulate the bending response and damage evolution in laminated composite beam ....................................................................................... 108
6.5
The comparison of load-displacement curves between the present prediction and experimental data (Huang, 2007) under three-point bending ....................................................... 109
6.6
The predicted damage evolution versus displacement of laminated composite beam under three-point bending corresponding to Figure 6.5: (a) Segment A; (b) Segment B; (c) Segment C; (d) Segment D; (e) Segment E; (f) Segment F .................................................... 110
6.7
The sequence of the deformed shape and von-Mises effective stress of laminated composite beam during three-point bending ................................................................................ 113
6.8
FE model of Iosipescu shear test .................................................................................................. 114
6.9
The comparison of load-displacement curves between the present prediction and experimental data (Khashaba, 2004) under shear loading: (a) [75/-15] 2s laminated B
B
composite; (b) [45/-45] 2s laminated composite; (c) [30/-60] 2s laminated composite .................. 115 B
6.10
B
B
B
The sequence of the deformed shape and von-Mises effective stress of laminated composite under shear loading in case of [75/-15] 2s .................................................................... 116 B
6.11
B
The sequence of the deformed shape and von-Mises effective stress of laminated composite under shear loading in case of [45/-45] 2s .................................................................... 117 B
6.12
B
The sequence of the deformed shape and von-Mises effective stress of laminated composite under shear loading in case of [30/-60] 2s .................................................................... 118 B
B
1
CHAPTER 1 INTRODUCTION
1.1 Review and Scope This dissertation is primarily related to the mechanical behavior of composite materials. Composite structures offer an attractive alternative to more conventional forms of construction due to its high strengthto-weight ratio and resistance to corrosion (Park et al., 2006). Development of new composite materials and new applications of composite materials in various engineering fields are now accelerating. Composite materials mean that two or more materials, called constituents, are combined in a certain configuration on a macro- or micro-scopic scale to form a new material with specific properties that combine the best features of each constituent so as to maximize a given set of properties (stiffness, strength-to-weight ratio, tensile strength, etc.) and minimize others (weight and cost) (Schoutens, 1982). In terms of reinforcement profile, composite materials are classified as particulate composites (e.g., whisker) and fibrous composites (e.g., short-fiber reinforced composites, unidirectional composites, and woven fiber reinforced composites). Furthermore, in terms of matrix profile, composite materials are also classified as ductile matrix composites (e.g., metal matrix composites) and brittle matrix composites (e.g., ceramic matrix composites and polymer matrix composites). As composite materials undergo damage, the corresponding stress and strain responses do not remain constant even for the case of elastic deformation (Voyiadjis and Park 1996). Interfacial bonding condition is one of the most important factors that control the local elastic fields and the overall properties of composite materials (Duan et al., 2007). The imperfect interfacial bonding may be due to damage in a very thin interfacial layer known as interphase or interface (Nie and Basaran, 2005). Therefore, to evaluate accurately the effective property of a composite, the properties and structures of interfaces must be taken into consideration (Qu, 1993b). Most analytical and numerical models assume that the bond between the reinforcement and matrix is perfect and can be modeled using the continuity of tractions and displacements across a discrete interface. When the perfect interface conditions are present, the primary result is the well-known Eshelby (1957) solution of the ellipsoidal inclusion problem. Eshelby’s micromechanics model provides a relatively easy tool to solve three-dimensional problems in elasticity, and thus is suitable for such a composite materials. In the literature, two kinds of model are often used to simulate the mechanical properties of interface
Chapter 1. Introduction
2
regions in particle- or fiber-reinforced composites. The first kind of model can be referred to as interface models, for example, linear spring model (Hashin, 1991; Qu, 1993b; Zhong and Meguid, 1997; Duan et al., 2007), free sliding model (Ghahremani, 1980; Jasiuk et al., 1987), and interface stress model (Sharma et al., 2003; Duan et al., 2005), in which displacement and/or stress discontinuities are assumed to exist at an interface. The second is the interphase model which describes the interface region as a layer, called an interphase, between the particle and matrix. The interphase model (three-phase model) with two perfect bonding interfaces was first proposed by Walpole (1978). Later, many studies concerned with homogeneous and inhomogeneous interphases have been done in the damage analysis and the prediction of effective moduli of composite materials (Qiu and Weng, 1991; Wang and Jasiuk, 1998; Dai et al., 1999; Wu et al., 2004; Zhong et al., 2004). The objective of this study is to determine the overall elastic and elastoplastic behavior of particleand fiber-reinforced composite materials containing imperfect interface based on micromechanical formulation proposed by Ju and Chen (1994a, 1994b, 1994c). A multi-level damage model in accordance with the Weibull's probabilistic function is developed in this study to describe the aforementioned sequential, progressive imperfect interface in the composites. Furthermore, in order to model spherical particles and cylindrical fibers having imperfect interfaces in the composites, the Eshelby's tensor for an ellipsoidal inclusion with slightly weakened interface (Qu, 1993a, 1993b) is adopted. To verify proposed damage models, comparisons between numerical predictions and experimental data in the literature are conducted. Finally proposed damage models are implemented into ABAQUS non-linear FE code using a user-subroutine UMAT to solve various engineering boundary value problems and to numerically characterize the elastic behavior of laminated composites.
Chapter 1. Introduction
3
1.2 Overview of the present study An outline of the present work is as follows. In Chapter 2, a micromechanical framework is proposed to predict the effective elastic behavior and imperfect interface evolution of particulate composites. The Eshelby's tensor for an ellipsoidal inclusion with slightly weakened interface (Qu, 1993a, 1993b) is adopted to model spherical particles having imperfect interfaces in the composites and is incorporated into the micromechanical framework. Based on the Eshelby's micromechanics, the effective elastic moduli of three-phase particulate composites are derived. A damage model is subsequently considered in accordance with the Weibull's probabilistic function to characterize the varying probability of evolution of imperfect interface between the inclusion and the matrix. The proposed micromechanical elastic damage model is applied to the uniaxial, biaxial and triaxial tensile loadings to predict the various stress-strain responses. Comparisons between the present predictions with other numerical and analytical predictions and available experimental data in the literature are conducted to assess the potential of the present framework.
In Chapter 3, a multi-level elastic damage model is proposed based on a combination of a micromechanical formulation and a multi-level damage model to predict the effective elastic behavior and progressive imperfect interface in particulate composites. The progression of imperfect interface is assumed to be gradual and sequential. A multi-level damage model in accordance with the Weibull's probabilistic function is developed to describe a sequential, progressive imperfect interface in the composites. The Eshelby's tensor for an ellipsoidal inclusion with slightly weakened interface (Qu, 1993a, 1993b) is adopted to model particles having mild or severe imperfect interface and is incorporated into the micromechanical formulation by Ju and Chen (1994). A numerical example corresponding to uniaxial tension loading is solved to illustrate the potential of the proposed multi-level elastic damage model. A parametric analysis is also carried out to address the influence of model parameters on the progressive imperfect interface in the composites. Furthermore, the present predictions are compared with available experimental data in the literature to further illustrate the elastic damage behavior of the present framework and to verify the validity of the proposed multi-level elastic damage model.
In Chapter 4, an elastoplastic multi-level damage model considering evolutionary imperfect interface is developed in this work to predict the effective elastoplastic behavior and multi-level damage evolution in particle reinforced ductile matrix composites (PRDMCs). The elastoplastic multi-level damage model is micromechanically derived on the basis of the ensemble-volume averaging procedure and the first-order effects of eigenstrains. The Eshelby's tensor for an ellipsoidal inclusion with slightly weakened interface
Chapter 1. Introduction
4
(Qu, 1993a, 1993b) is adopted to model particles having mild or severe imperfect interface, and a multilevel damage model (Lee and Pyo, in press) in accordance with the Weibull's probabilistic function is employed to describe the sequential, progressive imperfect interface in the composites. Numerical examples corresponding to uniaxial, biaxial and triaxial tension loadings are solved to illustrate the potential of the proposed micromechanical framework. A series of parametric analysis are carried out to investigate the influence of model parameters on the progression of imperfect interface in the composites. Furthermore, the present prediction is compared with available experimental data in the literature to verify the proposed elastoplastic multi-level damage model.
In Chapter 5, a 3D micromechanics-based evolutionary damage model is proposed to predict the effective elastic behavior of continuous, fiber-reinforced brittle matrix composites with microcracks and imperfect interfaces. The Eshelby's tensor for a circular cylindrical inclusion with slightly weakened interface (Qu, 1993a, 1993b) is adopted to model continuous fibers with imperfect interfaces. The nucleation of microcracks is simulated by employing the continuum damage model proposed by Karihaloo and Fu (1989). A multi-level damage modeling process in accordance with the Weibull's probabilistic function is incorporated into the micromechanical framework to describe the sequential evolution of imperfect interfaces in the composites. Numerical examples corresponding to uniaxial loadings in the longitudinal and transverse directions are solved to illustrate the potential of the proposed damage model. Furthermore, the present prediction is compared with available experimental data in the literature to verify the accuracy of the proposed damage model.
In Chapter 6, a 3D micromechanics-based evolutionary damage model (Lee and Pyo, submitted for publication) is implemented into a finite element program to numerically predict the overall behavior and damage evolution of laminated composite structures. A multi-level damage modeling process in accordance with the Weibull's probabilistic function is incorporated into the micromechanical framework to describe the sequential evolution of imperfect interfaces in the composites. To verify the implemented computational model, computational studies are undertaken by comparing the predicted stress-strain curve and experimental data available in literature under uniaxial tension with various fiber orientations. The implemented computational model provides results that show good agreement with available benchmark problems having different material compositions, thickness and boundary conditions.
In Chapter 7, concluding remarks are summarized.
5
CHAPTER 2 MICROMECHANICS-BASED ELASITC DAMAGE MODELING OF PARTICULATE COMPOSITES WITH IMPERFECT INTERFACES
2.1 Introduction Interfacial bonding condition is one of the most important factors that control the local elastic fields and the overall properties of composites (Duan et al., 2007). The imperfect interfacial bonding may be due to damage in a very thin interfacial layer known as interphase or interface (Nie and Basaran, 2005). The term imperfect interface is used to characterize a situation in which the displacements are discontinuous at the interface between the inclusion and the matrix (Crouch and Mogilevskaya, 2006). The linear springlayer of vanishing thickness, which is the simplest model of an imperfect interface, assumes that interfacial traction becomes continuous, but the displacements at the two sides of the interphase-layer become discontinuous (Crouch and Mogilevskaya, 2006). Refer to Benveniste (1985), Aboudi (1987), Achenbach and Zhu (1989, 1990), Hashin (1990, 1991), Zhu and Achenbach (1991), Qu (1993a, 1993b), Sudak et al. (1999), and Mogilevskaya and Croush (2002) for a further literature review on imperfect interface. Micromechanical damage models for composites considering interfacial debonding were proposed by many researchers (e.g., Zhao and Weng, 1997; Ju and Lee, 2000, 2001; Lee, 2001; Liang et al., 2006; Liu et al., 2006; Lee and Kim, 2007). They replaced the isotropic debonded inclusions by the perfectly debonded inclusions with yet unknown transversely isotropic properties to describe the loss of the loadtransfer capacity of debonded interface between inclusion and matrix. Schjudt-Thomsen and Pyrz (2000) studied creep modeling of short fiber reinforced composites with weakened interfaces and complex fiber orientation. Their approach was based upon the Mori-Tanaka mean field theory and the modified Eshelby's tensor for an inclusion with weakened interface proposed by Qu (1993a, 1993b). A similar work was previously conducted by Jun and Haian (1997). Zhong et al. (2004) studied three -dimensional micromechanical modeling of particulate composites with imperfect interface. In their derivations, the imperfect interface was characterized by a spring-type model assuming that the traction continuity remains intact, while the displacement experienced a jump proportional to the interfacial traction. Recently, Ju and Ko (in press) proposed a new micromechanical elastoplastic progressive damage model where the partial
Chapter 2. Elastic damage modeling of composites with imperfect interfaces
6
debonding at the fiber interface was represented by the growing debonding angles of arc microcracks. A micromechanical framework is proposed in the present study to predict the effective elastic behavior and imperfect interface evolution of particulate composites. The schematic of a particulate composite with imperfect interfaces is illustrated in Figure 2.1. In order to model spherical particles having imperfect interfaces in the composite, the Eshelby's tensor for an ellipsoidal inclusion with slightly weakened interface (Qu, 1993a, 1993b) is adopted. Based on the Eshelby's micromechanics, the effective elastic moduli of three-phase particulate composites are derived. A damage model is subsequently considered in accordance with the Weibull's probabilistic function to describe the varying probability of evolution of imperfect interface between the inclusion and the matrix. The novelty in the present derivation is in the simplicity and flexibility of the model to represent the imperfect interface evolution with limited number of model parameters. It is also straightforward to extend the proposed model to be used as a multi-level damage model, where some initially perfectly bonded particles may be transformed to particles with mild imperfect interface, some particles with mild imperfect interface may then be transformed to particles with severe imperfect interface, and all particles may be transformed to completely debonded particles asymptotically. In our derivations, the particles are assumed to be elastic spheres that are randomly dispersed in an elastic matrix. All particles are assumed to be non-interacting and initially embedded firmly in the matrix with perfect interfaces. It is also assumed that the evolution of imperfect interface is controlled by the
Matrix (phase 0)
Particle with imperfect interface (phase 2)
Perfectly bonded particle (phase 1)
(a)
(b)
Figure 2.1 Schematics of a particulate composite subjected to uniaxial tension: (a) the initial state; (b) the damaged state
Chapter 2. Elastic damage modeling of composites with imperfect interfaces
7
average internal stresses of the particles as well as the Weibull parameters (Weibull, 1951). The proposed micromechanical elastic damage model is then applied to the uniaxial, biaxial and triaxial tensile loadings to predict the various stress-strain responses. Comparisons between the present predictions with other numerical and analytical predictions and available experimental data are conducted to further assess the potential of the present framework.
Chapter 2. Elastic damage modeling of composites with imperfect interfaces
8
2.2 A micromechanics-based constitutive model for particulate composites with imperfect interfaces
2.2.1 Recapitulation of the Eshelby's tensor for an ellipsoidal inclusion with slightly weakened interface Qu (1993a, 1993b) derived the Eshelby's tensor for an ellipsoidal inclusion with slightly weakened interface in an elastic matrix of infinite extent. The imperfect interface between the inclusion and the matrix was modeled by a spring layer of vanishing thickness. The Eshelby's tensor for an ellipsoidal inclusion with slightly weakened interface (Qu, 1993a, 1993b) is adopted to model spherical particles having imperfect interfaces in particulate composites and is incorporated into the micromechanical framework. The summary of the Eshelby's tensor derived by Qu (1993a, 1993b) is repeated here for completeness of the proposed elastic damage model. With the help of the Green's function in the unbounded domain, the strain field inside an elliptical inclusion in a homogeneous linearly elastic solid can be represented by (Qu, 1993b)
ij x Sijkl kl* Tijst x I stkl Sstkl kl*
(2.1)
M Sijkl x kl*
where S is the (original) Eshelby inclusion tensor, ε * is the uniform eigenstrain, I is the fourth-rank identity P
P
tensor and the fourth-rank tensor T is given by
Tijpq x Lklmn Lpqst
S ksGijmn ξ xnt nl dS ξ
(2.2)
in which L is the fourth-rank elasticity tensor, G(x) is related to the Green's function given in Eq. (2.6a) and the Appendix of Qu (1993b), and n denotes the unit outward normal vector. In addition, the second-order tensor ij denoting the compliance of the interface spring layer is given by (Qu, 1993a)
ij ij ni n j
(2.3)
Chapter 2. Elastic damage modeling of composites with imperfect interfaces
9
where and represent the compliance in the tangential and normal directions of the interface and
ij signifies the Kronecker delta. Since Sijkl Lklmn Gijkl ξ x dV ξ , one arrive at (Qu, 1993b)
M Sijkl
1 Sijkl x dV x Sijkl Iijpq Sijpq H pqrs Lrsmn I mnkl Smnkl
(2.4)
where
H pqrs
1 ik n j nl jk ni nl il n j nk jl ni nk dS 4 S
(2.5)
In the case of spherical inclusions, H ijkl can be simplified as (Qu, 1993b)
Hijkl Pijkl Qijkl
(2.6)
in which
Pijkl
1 Iijkl a
Qijkl
1 2 Iijkl ij kl 5a
(2.7)
(2.8)
Here, a denotes the radius of inclusion. Further details of the derivation of the Eshelby's tensor for an ellipsoidal inclusion with slightly weakened interfaces can be found in Qu (1993a, 1993b).
2.2.2 Effective elastic moduli of three-phase particulate composites Let us consider an initially perfectly bonded two-phase composite consisting of an elastic matrix (phase 0) with bulk modulus 0 and shear modulus 0 , and randomly dispersed elastic spherical particles (phase 1) with bulk modulus 1 and shear modulus 1 . As loadings or deformations are applied, some
Chapter 2. Elastic damage modeling of composites with imperfect interfaces
10
particles may be imperfect in their interfaces (phase 2). The effective stiffness tensor C* for the three-phase composite can be derived based on the governing equations for linear elastic composites containing arbitrarily non-aligned and/or dissimilar inclusions (Ju and Chen, 1994a) as
2 1 1 1 C* C0 I r A r Sr I r S r A r S r r 1
(2.9)
where C0 is the elasticity tensor of the phase 0 (matrix), “ ” is the tensor multiplication, r denotes the volume fraction of the r-phase inclusion, and S r signifies the Eshelby's tensor for the r-phase. The fourthrank tensor A r is defined as 1
Ar Cr C0 C0
(2.10)
where Cr is the elasticity tensor of the r-phase. The components of Eshelby's tensor S1 for a (perfectly bonded) spherical particle embedded in an isotropic linear elastic and infinite matrix can be derived as
S1 ijkl
1 (5 0 1) ij kl (4 5 0 ) ( ik jl il jk ) 15 1 0
(2.11)
where 0 denotes the Poisson's ratio of the matrix. By carrying out the lengthy algebra, the Eshelby's tensors S 2 for a spherical particle with imperfect interface embedded in an isotropic linear elastic and infinite matrix given in Eq. (2.4) can be rephrased as
S2 ijkl
1 1200a 1 0
2
80 480 400 2 a 500 1 2 2 0 0 0 0
98 140 50 268 1240 1300 320 720 400 a 3 2 7 5 2
0
2
0
0
0
0
0
ij kl
2
2
0
0
0
0
ik
jl
il
jk
(2.12)
Chapter 2. Elastic damage modeling of composites with imperfect interfaces
11
where 0 denotes the Lame constant of the matrix. Substituting Eqs. (2.11) and (2.12) into Eq. (2.9) yields the effective stiffness tensor C* for the three-phase particulate composite as
C* *ij kl * ik jl il jk
(2.13)
Accordingly, the effective Lame constant * and the shear modulus * for the three-phase composite can be expressed as
1 2
* 30 20 1 3 20 2 4 1 2
* 2 0 2 4
(2.14)
(2.15)
where the components 1 , ... , 4 are listed in the Appendix. Alternatively, the effective Young's modulus E * and Poisson's ratio * of the three-phase particulate composite are easily obtained through the following relations
E *
*
* 3 * 2 * *
*
2 * *
*
(2.16)
(2.17)
Chapter 2. Elastic damage modeling of composites with imperfect interfaces
12
2.3 Damage modeling Imperfect interfaces between inclusion and matrix in particulate composites may occur as deformations or loadings continue to increase, and may affect the load-carrying capacity and the overall stress-strain behavior of the composites. For convenience, following Tohgo and Weng (1994) and Zhao and Weng (1995, 1996, 1997), the probability of imperfect interface is modeled as a two-parameter Weibull process. Assuming that the Weibull (1951) statistics governs and the average internal stresses of particles are the controlling factor of the Weibull function, one can express the current volume fraction of particles with imperfect interface 2 at a given level of 11 1 (see also Ju and Lee, 2001)
M 2 Pd 11 1 1 exp 11 1 S0
(2.18)
where is the original particle volume fraction, P d denotes the cumulative probability distribution B
B
function of imperfect interface for the uniaxial tensile loading (in the 1-direction), 11 1 is the internal stress of particles (phase 1) in the 1-direction, the subscript 1 denotes the particle phase, and S 0 and M are B
B
the Weibull parameters. For the biaxial and hydrostatic tensile loadings, 11 1 in Eq. (2.18) are replaced by
22 33 / 2 and 11 22 33 / 3 , respectively. 1 1 1 1 1 Within the context of the first-order (noninteracting) approximation, the stresses inside particles should be uniform. The internal stresses of particles required for the initiation of the imperfect interface were explicitly derived by Ju and Lee (2000) as
2 1 1 1 C1 I S1 A1 S1 I r S r A r S r r 1
1
:ε U : ε
(2.19)
By carrying out the lengthy algebra, the components of the positive definite fourth-rank tensor U for the present model are explicitly given by
Uijkl U1ij kl U 2 ik jl il jk
(2.20)
Chapter 2. Elastic damage modeling of composites with imperfect interfaces
13
with
U1
1 22 24 1 3 1 2 1 1 1 22 2 11 3 1 22 11 22 24 1 31 22 33 24
(2.21)
U2
1 1 2 2 11 2 2 2 4
(2.22)
where 1 is the current volume fraction of perfectly bonded particles, 1 is the Lame constant of particles, and the components 1 , ... , 4 are listed in the Appendix. In the case of tensile loading, the averaged internal stresses of particles can be obtained as follows (see also Ju and Lee, 2000):
11 1 U1 2U2 11 U1 22 U133
(2.23)
22 1 U111 U1 2U2 22 U133
(2.24)
33 1 U111 U1 22 U1 2U2 33
(2.25)
where 11 , 22 and 33 are the total strains in the 1, 2 and 3 directions, respectively.
Chapter 2. Elastic damage modeling of composites with imperfect interfaces
14
2.4 Numerical simulation To illustrate the proposed micromechanical elastic damage model for particulate composites, let us first consider the uniaxial tension loading. In this case, the components of the macroscopic stress can be expressed as 11 0 and ij 0 for all other stress components. According to the linear elasticity, the macroscopic incremental elastic strain for this case reads (see also Ju and Lee, 2001)
1 0 0 * 0 0
0 0 *11 E *
(2.26)
For convenience, we adopt the material properties and volume fraction of particles for the 6061-T6 aluminum alloy matrix/silicon-carbide particle composites according to Ju and Lee (2001) and Zhao and
800
700
600
Stress [σ11] (MPa)
500
400
300
200
perfect bonding perfectly bonded imperfect σy*3.27 interface (S 0 = 3.27*σ y ) σy*2.18 interface (S 0 = 2.18*σ y ) imperfect σy*1.09 interface (S 0 = 1.09*σ y ) imperfect
100
B
B
B
B
B
B
B
B
B
B
B
B
B
0 0.000
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
Strain [ε11]
Figure 2.2 The present predicted stress-strain responses of particulate composites with imperfect interfaces under uniaxial tension with various S 0 values B
B
Chapter 2. Elastic damage modeling of composites with imperfect interfaces
15
Weng (1995): E 0 = 68.3 GPa, 0 = 0.33, E 1 = 490 GPa, 1 = 0.17, y = 250 MPa; = 0.2. The compliance B
B
B
B
parameters and given in Eq. (2.3), which are related to the components of Eshelby's tensor S 2 for spherical inclusions with imperfect interfaces, are assumed to be = 2.0 and = 3.0. Moreover, to address the influence of the Weibull parameter S 0 on the evolution of imperfect interface and to evaluate the B
B
proposed elastic damage model sensitivity to the Weibull parameter S 0 , three different sets of Weibull B
B
parameters are used: S 0 =1.09× y and M = 5; S 0 = 2.18× y and M = 5; S 0 = 3.27× y and M = 5. B
B
B
B
B
B
We plot the present predicted stress-strain responses of particulate composites with imperfect interfaces (imperfect interface model) under uniaxial tension with various S 0 values in Figure 2.2. We also B
B
plot the present predicted response of the perfectly bonded particulate composites (perfect bonding model) in the figure for comparison. Figures 2.3 – 2.5 exhibit the evolutions of perfectly bonded particles and
0.2
Particle Volume Fraction
0.15
0.1
0.05
perfect bonding imperfect interface 0 0.000
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
Strain [ε11]
Figure 2.3 The predicted evolution of volume fractions of perfectly bonded particles and particles with imperfect interface corresponding to Figure 2.2 (S 0 = 1.09*σ y ) B
B
B
B
Chapter 2. Elastic damage modeling of composites with imperfect interfaces
16
particles with imperfect interface as functions of the uniaxial strains corresponding to Figure 2.2. From Figures 2.2 – 2.5, it is clear that lower S 0 leads to faster evolution of imperfect interface, indicating that B
B
most particles are imperfect in their interfaces in early stage with lower S 0 and the composites show the B
B
nonlinear stress-strain behavior. It is also noted that the responses with the imperfect interface are lower than the response without the imperfect interface. To further examine the effect of varying S 0 values on the stress-strain responses of the composites B
B
under different loading cases, we consider biaxial and triaxial loading cases on next. In biaxial tensile loading case, the components of the macroscopic stress can be expressed as 22 33 0 and ij 0 for all other components. From the linear elasticity, the macroscopic incremental elastic strain for a biaxial tensile loading is given by (see also Ju and Lee, 2001)
0.2
Particle Volume Fraction
0.15
perfect bonding
0.1
imperfect interface
0.05
0 0.000
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
Strain [ε11]
Figure 2.4 The predicted evolution of volume fractions of perfectly bonded particles and particles with imperfect interface corresponding to Figure 2.2 (S 0 = 2.18*σ y ) B
B
B
B
Chapter 2. Elastic damage modeling of composites with imperfect interfaces
17
2 * 0 0 * 0 1 0 *22 E 0 0 1 *
(2.27)
In triaxial tensile loading case, the components of the macroscopic stress can be expressed as 22 33 11 0 and ij 0 for all other components, where is the proportional loading ratio. In particular, = 1 indicates a pure hydrostatic loading. From the linear elasticity, the macroscopic incremental elastic strain for a triaxial tensile loading can be expressed as (see also Ju and Lee, 2001)
0.2
Particle Volume Fraction
0.15
0.1
perfect bonding imperfect interface 0.05
0 0.000
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
Strain [ε11]
Figure 2.5 The predicted evolution of volume fractions of perfectly bonded particles and particles with imperfect interface corresponding to Figure 2.2 (S 0 = 3.27*σ y ) B
B
B
B
Chapter 2. Elastic damage modeling of composites with imperfect interfaces
1 2 * 0 0
0 1
18
*11 0 E 1 * 0
*
0
(2.28)
Figures 2.6 and 2.8 show the present predicted stress-strain responses of particulate composites with imperfect interfaces under biaxial and hydrostatic tension loadings with various S 0 values, respectively. B
B
Figures 2.7 and 2.9 exhibit the evolution of particles with imperfect interface as functions of the biaxial and hydrostatic strains corresponding to Figures 2.6 and 2.8, respectively. Similar to the uniaxial tensile loading case, lower S 0 leads to faster evolution of imperfect interface and lower stress-strain response. It is also B
B
observed that hydrostatic tension results in higher stress-strain behavior, which corresponds to findings from Ju and Lee's (2001) observations.
700
600
Stress [σ22(=σ33)] (MPa)
500
400
300
200
perfect bonding perfectly bonded imperfect interface (S 0 = 3.27*σ y ) 3.27 imperfect interface (S 0 = 2.18*σ y ) 2.18 imperfect interface (S 0 = 1.09*σ y ) 1.09
100
B
B
B
B
B
B
B
B
B
B
B
B
B
0 0.000
0.001
0.002
0.003
0.004
0.005
Strain [ε22(=ε33)]
Figure 2.6 The present predicted stress-strain responses of particulate composites with imperfect interfaces under biaxial tension with various S 0 values B
B
Chapter 2. Elastic damage modeling of composites with imperfect interfaces
19
Lastly, to examine the effect of varying and values on the stress-strain response of the proposed micromechanical framework, we conduct a parametric analysis of and . Four different sets of the compliance parameters are used: = 2.0 and = 3.0; 2.0 104 and 3.0 104 ; 2.0 105 and 3.0 105 ; 2.0 106 and 3.0 106 . Figure 2. 10 exhibit the present predicted stress-strain responses of particulate composites with imperfect interfaces under uniaxial tension with various values of and . It is seen from the figure that as the values of and become higher, the effect of imperfect interface on the stress-strain behavior is more pronounced. Further, it is shown from the parametric analysis that the converged values of the parameters for the lower bound are = 2.0 and = 3.0. It is also worthy mentioning that if = = 0, the material shows a perfect bonding material characteristic.
0.2
Particle Volume Fraction
0.15
0.1
0.05
S3.27 0 = 3.27*σ y S2.18 0 = 2.18*σ y S1.09 0 = 1.09*σ y
0 0.000
0.001
0.002
0.003
B
B
B
B
B
B
B
B
B
B
B
0.004
0.005
Strain [ε22(=ε33)]
Figure 2.7 The predicted evolution of volume fraction of particles with imperfect interface corresponding to Figure 2.6
Chapter 2. Elastic damage modeling of composites with imperfect interfaces
20
1400
1200
Stress [σ11(=σ22=σ33)] (MPa)
1000
800
600
400
perfect bonding bonded 3.27 imperfect interface (S 0 = 3.27*σ y ) 2.18 imperfect interface (S 0 = 2.18*σ y ) 1.09 imperfect interface (S 0 = 1.09*σ y )
200
0 0.000
0.001
0.002
B
B
B
B
B
B
B
B
B
B
B
B
0.003
0.004
B
0.005
Strain [ε11(=ε22=ε33)]
Figure 2.8 The present predicted stress-strain responses of particulate composites with imperfect interfaces under hydrostatic tension with various S 0 values B
B
0.2
Particle Volume Fraction
0.15
0.1
0.05
S3.27 0 = 3.27*σ y S2.18 0 = 2.18*σ y S1.09 0 = 1.09*σ y B
B
B
B
B
B
B
B
B
B
B
0 0.000
0.001
0.002
0.003
0.004
0.005
Strain [ε11(=ε22=ε33)]
Figure 2.9 The predicted evolution of volume fraction of particles with imperfect interface corresponding to Figure 2.8
Chapter 2. Elastic damage modeling of composites with imperfect interfaces
21
500
450
400
Stress [σ11] (MPa)
350
300
250
200
150
perfect bonding perfectly bonded α=2.0, β=3.0 standard α=2.E-4, case 1 β=3.E-4 α=2.E-5, case 2 β=3.E-5 α=2.E-6, case 3 β=3.E-6
100
B
50
B
B
0 0.000
0.001
0.002
0.003
0.004
0.005
Strain [ε11]
Figure 2.10 The present predicted elastic responses of particulate composites with imperfect interfaces under uniaxial tension with various α and β values
Chapter 2. Elastic damage modeling of composites with imperfect interfaces
22
2.5 Comparisons with other numerical and analytical predictions and experiment To verify and further assess the proposed micromechanical framework, we compare the present predictions with other numerical (Ju and Lee, 2001) and analytical (Kerner, 1956; Halpin and Kardos, 1976) predictions and available experimental data (Zhou et al., 2004). We first compare the present prediction with the numerical prediction given in Ju and Lee (2001) for 6061-T6 aluminum alloy matrix/silicon-carbide particle composites under uniaxial tension in Figure 2.11. Similar to Section 2.4, the elastic properties of the composites used in this simulation are (see Arsenault, 1984; Nieh and Chellman, 1984; Zhao and Weng, 1995; Ju and Lee, 2001) E 0 = 68.3 GPa, 0 = 0.33, E 1 = B
B
B
B
490 GPa, 1 = 0.17, y = 250 MPa; = 0.2. The compliance parameters and are assumed to be =2.0 and = 3.0 (lower bound) to clearly show the effect of imperfect interface on the stress-strain behavior of imperfect particulate composites. The Weibull parameters S 0 and M are assumed to be S 0 = 1.09× y , M = B
B
B
B
5. Clearly, the present imperfect interface model shows slightly high stress values during the evolution of imperfect interface in comparison with the partial debonding model (Ju and Lee, 2001). It should be noted that we do not display our predictions beyond the strain value 11 = 0.004, which corresponds to the yielding stress of the Ju and Lee's (2001) predicted response, in the figure since the present model does not consider plastic formulations. Our analytical predictions are compared with Halpin-Tsai's analytical solutions (Kerner, 1956; Halpin and Kardos, 1976) for particulate composites to validate the proposed micromechanical framework. The Halpin-Tsai's analytical equations predict the elastic moduli of composites with different reinforcement geometries changing from a sphere (aspect ratio of one in the principal material direction) to a long fiber (aspect ratio approaching infinity) (Halpin and Kardos, 1976). The analytical equation for the Young's modulus of particulate composites is given by (Kerner, 1956; Halpin and Kardos, 1976; Lee and Simunovic, 2001)
E* 2 1 *
f f m (7 5 ) (8 10 ) 15(1 ) m m f m f m m (7 5 m ) (8 10 m ) f 15(1 f )
(2.29)
Chapter 2. Elastic damage modeling of composites with imperfect interfaces
23
where subscripts m and f denote the matrix and particle phases, respectively, and the Poisson ratio * is given by a rule of mixtures
* m m f f
(2.30)
The following constituent elastic properties for carbon fiber polymer matrix composites are considered for comparison: E 0 = 3 GPa, 0 = 0.35, E 1 = 380 GPa and 1 = 0.25 (Lee and Simunovic, 2001). B
B
B
B
Figure 2.12 shows the effective (normalized) Young's modulus versus volume fraction of particles predicted from the present perfect bonding and imperfect interface models. In particular, various combinations of volume fractions of perfectly bonded particles (phase 1) and particles with imperfect
400
partial debonding (Ju and Lee, 2001) imperfect interface (present prediction) perfect bonding (present prediction)
Stress [σ11] (MPa)
300
200
100
0 0.000
0.001
0.002
0.003
0.004
Strain [ε11]
Figure 2.11 The comparison of predicted overall elastic uniaxial responses of 6061-T6 aluminum alloy matrix/silicon-carbide particle composites between the present perfect bonding and imperfect interface models and Ju and Lee’s (2001) partial debonding model
Chapter 2. Elastic damage modeling of composites with imperfect interfaces
24
interface (phase 2) are considered in the imperfect interface model as: 1 / 2 = 9/1, 7/3, 5/5, 0/10 (lower bound). The compliance parameters are assumed to be = 2.0, = 3.0. It is also assumed that the present imperfect interface model is stationary (no damage evolution). The theoretical prediction based on the Halpin-Tsai's analytical solution is also plotted in the figure. It is seen from Figure 2.12 that the present prediction from the perfect bonding model (upper bound) is almost identical with the Halpin-Tsai's analytical solution. The present predictions from the imperfect interface model are well within the bounds. To further assess the potential of the present framework, we compare the present prediction with experimental data reported by Zhou et al. (2004) for the uniaxial stress-strain behavior of Mg-Al matrix with Al 2 O 3 particulate composites. Here, we adopt the elastic properties according to Zhou et al. (2004) as B
B
B
B
140
120
Halpin-Tsai’s analytical solution Kerner perfect bonding bonding imperfect interface ( 1: 2 =9:1) 9:1 imperfect interface ( 1: 2 =7:3) 7:3 imperfect interface ( 1: 2 =5:5) 5:5 imperfect interface ( 1: 2 =0:10) 0:10
100
B
B
B
B
E*/E0
80
60
40
20
0 0
0.2
0.4
0.6
0.8
1
Volume fraction of particle
Figure 2.12 The comparison between the present predictions with various combinations of volume fractions of perfectly bonded particles and particles of imperfect interface and the Halpin-Tsai’s theoretical prediction for effective (normalized) Young’s modulus versus volume fraction of particles
Chapter 2. Elastic damage modeling of composites with imperfect interfaces
25
follow: E 0 = 73 GPa, 0 = 0.33, E 1 = 400 GPa, 1 = 0.24, y = 250 MPa, = 0.48. It is emphasized that we B
B
B
B
need to estimate the compliance parameters and given in Eq. (2.12) as well as the Weibull parameters S 0 and M in Eq. (2.18) due Zhou et al. (2004) did not report those parameters. In accordance with the B
B
experimental data reported by Zhou et al. (2004), the above compliance and Weibull parameters are estimated to be: = 2.0, = 3.0; S 0 = 625 MPa, M = 5. B
B
Based on the above material properties and parameters, we depict our uniaxial stress-strain predictions on the composites with and without evolution of imperfect interface against experimental data provided by Zhou et al. (2004) in Figure 2.13. Overall, the present prediction from the imperfect interface model and the experimental data match well. The predicted evolution of imperfect interface corresponding to Figure 2.12 is shown in Figure 2.14. The good agreement between the present prediction and the experiment encourages for possible use of the proposed micromechanical framework to modeling the imperfect interface in (quasi-brittle) particulate composites. 400
Stress [σ11] (MPa)
300
200
100
experiment imperfect interface perfect bonding 0 0.0000
0.0005
0.0010
0.0015
0.0020
0.0025
Strain [ε11]
Figure 2.13 The comparison between the present prediction and experimental data (Zhou et al., 2004) for overall uniaxial tensile responses of Mg-Al matrix with Al 2 O 3 particulate B
composites
B
B
B
Chapter 2. Elastic damage modeling of composites with imperfect interfaces
26
0.5
Fiber Volume Fraction
0.4
0.3
0.2
perfect bonding 0.1
imperfect interface
0 0.0000
0.0005
0.0010
0.0015
0.0020
0.0025
Strain [ε11]
Figure 2.14 The predicted evolution of volume fractions of perfectly bonded particles and particles with imperfect interface corresponding to Figure 2.13
Chapter 2. Elastic damage modeling of composites with imperfect interfaces
27
2.6 Appendix. Parameters Λ 1 , . . , Λ 4 in Eqs. (2.14), (2.15), (2.21), and (2.22) B
B
B
B
1 22 1 1 31 2 1 1 15 1 0 1 2 1 31 2 1 1 2 2 1 31 2 2 2
(2.31)
15 1 0 1
(2.32)
4 1 1 2 2
2 24 1 3 32 2 2 2 3 600a 1 0 2 2 32 2 2 1 24 1 33 24
(2.33)
300a 1 0 2 2
4
(2.34)
2 1 24
in which
1
1 51 1 0 5 0 1 , 2 1 3 2 1 1
2
3
2 2 2
2 31 2 2 1 , 3 2 2 2
4
1 15 1 0
(2.36)
2 2
1 0 3 1 0 2 1 0 2
(2.35)
2 2
1 0 0 1
2 1200a 1 0 1
1 4 5 0 2 1
5 0 1
(2.37)
2 0 0 2 1 2 0 3 2 0 2 2 0
(2.38)
7 5 0 0 8 10 0 1 2 1 0
2 600a 1 0
2
0
2 0
(2.39)
2
(2.40)
with
1 80 480 0 400 02 a 500 1 2 0 0 98 140 0 50 02 0
2
268 1240 0 1300 02 0
2 320 720 0 400 02 a 0 3 2 7 5 0
2
(2.41)
(2.42)
28
CHAPTER 3 MULTI-LEVEL MODELING OF EFFECTIVE ELASTIC AND PROGRESSIVE IMPERFECT INTERFAVE IN PARTICULATE COMPOSITES
3.1 Introduction Debonding phenomenon existed in between particles and matrix is one of well-known damage mechanisms in particulate composites and its effect on the mechanical behavior of the composites has to be well addressed for an accurate analysis of the composites. The progressive interfacial debonding between particles and matrix may occur under increasing deformation or stresses from the viewpoint of failure analysis. One-step Weibull distribution has been used by many researchers to describe the interfacial debonding between particles (fibers) and matrix (e.g., Tohgo and Weng, 1994; Zhao and Weng, 1995, 1996, 1997; Ju and Lee, 2000, 2001; Lee and Simunovic, 2000, 2001, 2002; Liu et al., 2005; Liang et al., 2006; Lee and Pyo, 2007). However, a sequential probabilistic debonding analysis is necessary to realistically reflect the effect of loading history on the interfacial debonding. Gurvich and Pipes (1995, 1996), Tabiei and Sun (1999, 2000) and Hwang et al. (2003) introduced sequential multi-step failure models to model the sequential probabilistic failure between fibers and matrix in laminated composites. In their sequential multi-step failure models, a structure was modeled as a parallel arrangement of elements based on the fiber bundle theory (Hwang et al., 2003). When an element fails, the load is redistributed among the remaining elements and final failure occurs when all of the elements fail (Hwang et al., 2003). Another approach to model the sequential probabilistic failure is proposed by Liu et al. (2004, 2006) and Ju and Ko (in press). In their formulations, a progressive interfacial debonding model was developed based on Weibull distribution function (Liu et al., 2004, Ju and Ko, in press) and one damage mode was transformed to another as applied external loads increased. That is, another damage mode is initiated when the previous local principal radial normal stress reaches the critical interfacial debonding strength (Liu et al., 2004, Ju and Ko, in press). Recently, a multi-level model for damage evolution in microstructurally debonded composites was developed by Ghosh et al. (2007) for multi-scale analysis of damaged composite structures due to microstructural damage induced by interfacial debonding.
Chapter 3. Multi-level modeling of particulate composites
29
In this study, a multi-level elastic damage model based on a combination of a micromechanical formulation and a multi-level damage model is proposed to predict the effective elastic behavior and progressive imperfect interface in particulate composites. Since the progression of imperfect interface is assumed to be gradual and sequential, as loads or deformations continue to increase, some initially perfectly bonded particles are transformed to particles with mild imperfect interface. Some particles with mild imperfect interface are then transformed to particles with severe imperfect interface, and all particles are transformed to completely debonded particles asymptotically within the proposed framework. A multilevel damage model in accordance with the Weibull's probabilistic function is developed to describe the aforementioned sequential, progressive imperfect interface in the composites. The Eshelby's tensor for an ellipsoidal inclusion with slightly weakened interface (Qu, 1993a, 1993b) is adapted to model particles having mild or severe imperfect interface and is incorporated into the micromechanical formulation by Ju and Chen (1994a). The schematic of a particulate composite under uniaxial tension, illustrating a sequential progression of imperfect interface in the composite, is shown in Figure 3.1. In particular, the following four-level elastic damage model is considered for a complete description of the sequential progression of imperfect interface in the composite: (1) level-0 of two-phase composite state consisting of matrix and perfectly bonded particles; (2) level-1 of three-phase composite state consisting of matrix, perfectly bonded particles and particles having mild imperfect interface; (3) level-2 of four-phase composite state consisting of matrix, perfectly bonded particles, particles having mild imperfect interface and particles having severe imperfect interface; (4) level-3 of five-phase composite state consisting of matrix, perfectly bonded particles, particles having mild imperfect interface, particles having severe imperfect interface and completely debonded particles. We assume that the level of severity of imperfect interface can be classified simplistically as either mild (moderately) imperfect interface or severe imperfect interface. It is obvious that the severity of imperfect interface may vary as loads or deformations continue to increase. However, our premise is that its cumulative effect on the effective stress-strain behavior can be quantitatively described by using this simple approximation. The two imperfect states, mild imperfect interface and severe imperfect interface, are also assumed to occur sequentially. Further, particles are assumed to be randomly dispersed, non-interacting elastic spheres that are initially embedded firmly in an elastic matrix with perfect interfaces. It is also assumed that the progression of imperfect interface is governed by the average internal stresses of the particles as well as the Weibull parameters (Weibull, 1951). A numerical example corresponding to uniaxial tension loadings is solved to illustrate the potential of the proposed multi- level elastic damage model. A parametric analysis is also carried out to address the influence of model parameters on the progressive imperfect interface in the
Chapter 3. Multi-level modeling of particulate composites
30
composites. Comparisons between the present prediction with available experimental data are conducted to further illustrate the elastic damage behavior of the present framework and to verify the validity of the proposed multi-level elastic damage model.
a
Perfectly bonded
b
particle (phase 1)
Matrix (phase 0)
Particle with mild imperfect interface (phase 2)
c
d
Particle with severe
Void (phase 4)
imperfect interface (phase 3)
Figure 3.1 Schematics of a particulate composite subjected to uniaxial tension: (a) Level 0 of twophase composite state (initial state); (b) Level 1 of three-phase composite state; (c) Level 2 of four-phase composite state; (d) Level 3 of five-phase composite state
Chapter 3. Multi-level modeling of particulate composites
31
3.2 Multi-level damage modeling Imperfect interfaces between particles and matrix in particulate composites may occur as deformations or loadings continue to increase, and may affect the load-carrying capacity and overall stressstrain behavior of the composites. As an effort to realistically reflect the effect of loading history on the progression of imperfect interface, a multi-level damage model in accordance with the Weibull's probabilistic function is developed. Specifically, a four-level damage model is considered in the present study, in the order of sequence of progressive imperfect interface, for a complete description of the sequential, progressive imperfect interface in the composite as illustrated in Figure 3.1 and is discussed in detail in this section.
3.2.1 Level 0 of two-phase composite state As illustrated in Figure 3.1 (a), in the initial state, an initially perfectly bonded two-phase composite state consists of an elastic matrix (phase 0), and randomly dispersed elastic spherical particles (phase 1) with the volume fraction .
3.2.2 Level 1 of three-phase composite state Following Tohgo and Weng (1994) and Zhao and Weng (1995, 1996, 1997), the probability of mild imperfect interface is modeled as a two-parameter Weibull process. Assuming that the Weibull (1951) statistics governs and the average internal stresses of perfect bonded particles (phase 1) are the controlling factor of the Weibull function, the current volume fractions of particles having mild imperfect interface 2 and perfectly bonded particles 1 in the three-phase composite at a given level of 11 1 are (see also Ju and Lee, 2001)
M 11 1 S0
2 Pd 11 1 1 exp
(3.1)
1 2
(3.2)
Chapter 3. Multi-level modeling of particulate composites
32
where is the original particle volume fraction, P d signifies the cumulative probability distribution B
B
function of mild imperfect interface for the uniaxial tensile loading (in the 1-direction), internal stress of particles (phase 1) in the 1-direction, the subscript
11 1 is
the
1 denotes the particle phase, and S 0 B
B
and M are the Weibull parameters. For the biaxial and hydrostatic tensile loadings, 11 1 in Eq. (3.1) are replaced by 22 1 33 1 / 2 and 11 1 22 1 33 1 / 3 , respectively. The internal stresses of particles required for the initiation of the imperfect interface were explicitly derived by Ju and Lee (2000) as
n 1 1 1 C1 I S1 A1 S1 I r S r A r S r r 1
1
:ε U : ε
(3.3)
where C 1 is the elasticity tensor of the perfectly bonded particles, “ ” is the tensor multiplication, I is the B
B
fourth-rank identity tensor, and r denotes the volume fraction of the r-phase inclusion. The fourth-rank tensor A r is defined as B
B
1
Ar Cr C0 C0
(3.4)
The components of Eshelby's tensor S 1 for a (perfectly bonded) spherical particle and S 4 for a B
B
B
B
(completely debonded) spherical particle embedded in an isotropic linear elastic and infinite matrix can be derived as
S1,4 ijkl 15 11 (5 0 1) ij kl (4 5 0 ) (ik jl il jk )
(3.5)
0
where 0 denotes the Poisson's ratio of the matrix and ij signifies the Kronecker delta. Following Chapter 3, the Eshelby's tensors S m+1 for a spherical particle with imperfect interface B
B
embedded in an isotropic linear elastic and infinite matrix can be expressed as
Chapter 3. Multi-level modeling of particulate composites
Sm1 ijkl
1 1200a 1 0
2
33
80 480 400 2 a 500 1 2 2 0 0 0 0 m
98 140 50 268 1240 1300 320 720 400 a 3 2 7 5 2
0
2
0
0 m
0
0
ij kl
(3.6)
2
2
0
0 m
0
0
m
m
0
ik
jl
il
jk
where m = 1, 2 in which 1 indicates the mild imperfect interface stage, while 2 denotes the severe imperfect interface stage. In addition, a denotes the radius of particles, 0 and 0 denote the Lame constants of the matrix, and and represent the compliance parameters in the tangential and normal directions of the interface. Solutions to the case of small with = 0 yield approximations to the free-sliding interfaces, whereas the free-sliding can be achieved by setting with = 0 (Qu, 1993b). Following Qu's (Qu, 1993b) definitions on and , the severity of imperfect interface is assumed to be controlled by the values of and in the present study. It should also be noted that those parameters are not position dependent, indicating that no interfacial partial debonding is considered in this framework. By carrying out the lengthy algebra, the components of the positive definite fourth-rank tensor U for the multi-phase composite state are explicitly given by
Uijkl U1ij kl U 2 ik jl il jk
(3.7)
with
U1
1 22 24 1 3 1 2 1 1 1 22 2 11 3 1 22 11 22 24 1 31 22 33 24
(3.8)
U2
1 1 2 2 11 2 2 2 4
(3.9)
where 1 and 1 are the Lame constants of particles and the components 1 , ... , 4 are listed in the Appendix.
Chapter 3. Multi-level modeling of particulate composites
34
3.2.3 Level 2 of four-phase composite state Similar to the Level 1 of three-phase composite state, assuming that the Weibull (1951) statistics governs and some particles with mild imperfect interface are transformed to particles with severe imperfect interface as deformations or loads increase, the current volume fractions of particles having severe imperfect interface 3 , particles having mild imperfect interface 2 and perfectly bonded particles 1 in the four-phase composite state at a given level of 11 1 can be derived through the following two-step Weibull approach
M 2 1 exp 11 1 S0
(3.10)
M 3 2 1 exp 11 1 S 0
(3.11)
2 2 3
(3.12)
1 2
(3.13)
The components of the positive definite fourth-rank tensor U for the four-phase composite state in Eq. (3.7), denoted by U 1 and U 2 , can be expressed as B
B
B
B
U1
1 22 24 26 1 3 1 2 1 1 1 22 2 11 3 5 1 22 11 22 24 26 1 31 22 33 24 35 26
U2
1 1 22 11 2 2 2 4 26
where the components 1 , ... , 6 are listed in the Appendix.
(3.14)
(3.15)
Chapter 3. Multi-level modeling of particulate composites
35
3.2.4 Level 3 of five-phase composite state Similar to the Level 2 of four-phase composite state, assuming that the Weibull (1951) statistics governs and some particles with severe imperfect interface are transformed to completely debonded particles as deformations or loads increase, the current volume fractions of completely debonded particles 4 particles having severe imperfect interface 3 , particles having mild imperfect interface 2 and perfectly bonded particles 1 in the five-phase composite at a given level of 11 1 can be derived through the following three-step Weibull approach
M 2 1 exp 11 1 S0
(3.16)
M 3 2 1 exp 11 1 S 0
(3.17)
M 11 1 S0
4 3 1 exp
(3.18)
3 3 4
(3.19)
2 2 3
(3.20)
1 2
(3.21)
The components of the positive definite fourth-rank tensor U for the five-phase composite state in Eq. (3.7), denoted by U 1 and U 2 , can be expressed as B
U1
B
B
B
1 22 24 26 28 1 3 1 2 1 1 1 22 2 11 3 5 7 1 22 11 22 24 26 28 1 31 2 2 33 2 4 35 26 37 28
U2
1 1 22 11 22 24 26 28
where the components 1 , ... , 8 are listed in the Appendix.
(3.22)
(3.23)
Chapter 3. Multi-level modeling of particulate composites
36
3.3 A micromechanics-based, multi-level elastic model for particulate composites with imperfect interfaces We start with considering an initially perfectly bonded two-phase composite consisting of an elastic matrix (phase 0) with bulk modulus 0 and shear modulus 0 , and randomly dispersed elastic spherical particles (phase 1) with bulk modulus 1 and shear modulus 1 . As loads or deformations continue to increase, some initially perfectly bonded particles are transformed to particles with mild imperfect interface, some particles with mild imperfect interface are then transformed to particles with severe imperfect interface, and all particles are transformed to completely debonded particles asymptotically within the proposed framework. The effective stiffness tensor C* for the multi-phase particulate composite can be derived based on the governing equations for linear elastic composites containing arbitrarily non-aligned and/or dissimilar inclusions (Ju and Chen, 1994a) as
n 1 1 1 C* C0 I r A r Sr I r S r A r S r r 1
(3.24)
where Cr is the elasticity tensor of the r-phase. Substituting Eqs. (3.5) and (3.6) into Eq. (3.24) yields the effective stiffness tensor C* for the multiphase particulate composite as
C* *ij kl * ik jl il jk
(3.25)
3.3.1 Level 0 of two-phase composite state The effective Lame constants * and * in Eq. (3.25) for the two-phase composite state can be expressed as
1 2
* 30 20 1 20 2
(3.26)
Chapter 3. Multi-level modeling of particulate composites
1 2
37
* 2 0 2
(3.27)
where the components 1 and 2 are listed in the Appendix.
3.3.2 Level 1 of three-phase composite state The effective Lame constants * and * in Eq. (3.25) for the three-phase composite state can be expressed as
1 2
* 30 20 1 3 20 2 4 1 2
(3.28)
* 2 0 2 4
(3.29)
where the components 1 ,.., 4 are listed in the Appendix.
3.3.3 Level 2 of four-phase composite state The effective Lame constants * and * in Eq. (3.25) for the four-phase composite state can be expressed as
1 2
* 30 20 1 3 5 20 2 4 6 1 2
* 2 0 2 4 6
where the components 1 ,.., 6 are listed in the Appendix.
(3.30)
(3.31)
Chapter 3. Multi-level modeling of particulate composites
38
3.3.4 Level 3 of five-phase composite state The effective Lame constants * and * in Eq. (3.25) for the five-phase composite state can be expressed as
1 2
* 30 20 1 3 5 7 20 2 4 6 8 1 2
* 2 0 2 4 6 8
where the components 1 ,.., 8 are listed in the Appendix.
(3.32)
(3.33)
Chapter 3. Multi-level modeling of particulate composites
39
3.4 Numerical example To illustrate the elastic damage behavior of the present framework, we consider the uniaxial tension loading case ( 11 0 and ij 0 for all other stress components). For brevity, we adopt the material properties for the 6061-T6 aluminum alloy matrix/silicon-carbide particle composites according to Zhao and Weng (1995) and Ju and Lee (2001): E 0 = 68.3 GPa, 0 = 0.33, E 1 = 490 GPa, 1 = 0.17, y = 250 B
B
B
B
MPa, = 0.2. As a result of the parametric analysis on compliance parameters and reported in Chapter 2, the compliance parameters of the mild imperfect interface, denoted by 1 and 1 , and the severe imperfect interface, denoted by 2 and 2 , in Eq. (3.6) are chosen to be 1 2.0 105 , 1 3.0 105 ; 2 = 2.0, 2 = 3.0. Moreover, the Weibull parameters in Eq. (3.1) are assumed to be S 0 = 1.09× y and M = 5. B
B
500
450
2-phase composite state 3-phase composite state
400
4-phase composite state 5-phase composite state
Stress [σ11] (MPa)
350
300
250
200
150
100
50
0 0.000
0.001
0.002
0.003
0.004
0.005
Strain [ε11]
Figure 3.2 The present predicted stress-strain responses of particulate composites under uniaxial tension
Chapter 3. Multi-level modeling of particulate composites
40
To investigate the influence of the level of damage on the stress-strain response of the composites, four different phase composite states illustrated in Figure 3.1 are considered in this simulation. Figure 3.2 shows the present predicted stress-strain responses of particulate composites under uniaxial tension. The predicted evolution of volume fractions of perfectly bonded particles and various types of damaged particles (e.g., particles with mild imperfect interface, particles with severe imperfect interface, completely debonded particles) at the three-phase, four-phase and five-phase composite states are exhibited in Figure 3.3. At the early stage of loading, the stress-strain curves are shown to be linear since the volume fractions of the damaged particles are low (Figure 3.2). As strains or stresses continue to increase, the volume fractions of the damaged particles increase gradually and the composites show nonlinear stress-strain responses as shown in Figure 3.2. Clearly, the high-level composite states give rise to substantial weaker stress-strain response in comparison with the low-level composite states. It is demonstrated through this simulation that the current multi-level elastic damage model naturally captures the gradual transition from previous damage level to next damage level and various types of damaged particles exist simultaneously within the proposed framework. 0.2
Particle Volume Fraction
0.15
perfectly bonded (3-phase)
0.1
mild imperfect (3-phase) perfectly bonded (4-phase) mild imperfect (4-phase) severe imperfect (4-phase) perfectly bonded (5-phase) mild imperfect (5-phase) severe imperfect (5-phase)
0.05
completely debonded (5-phase)
0 0.000
0.001
0.002
0.003
0.004
0.005
Strain [ε11]
Figure 3.3 The predicted evolution of volume fractions of perfectly bonded particles various types of damaged particles corresponding to Figure 3.2
Chapter 3. Multi-level modeling of particulate composites
41
3.5 Parametric analysis The Weibull parameter S 0 in Eq. (3.1) is closely related to the strength at the particle-matrix interface B
B
in particulate composites. A parametric analysis is conducted to examine the influence of the Weibull parameter S 0 on the progression of imperfect interface in the composites. As primary parameters governing B
B
the progressive imperfect interface, three representative sets of Weibull parameters are used: S 0 = 1.09× y , B
B
M = 5; S 0 = 2.18× y , M = 5; S 0 = 3.27× y , M = 5. For comparison, a uniaxial tensile simulation is B
B
B
B
conducted using the same material properties and model parameters as used in Chapter 3.4. The present predicted stress-strain responses of particulate composites under uniaxial tension at the four-phase composite state with various S 0 values are shown in Figure 3.4. Figure 3.5 exhibits the B
B
predicted evolution of volume fractions of perfectly bonded particles, particles with mild imperfect
500
Stress [σ11] (MPa)
400
300
200
S3.27 0 = 3.27*σ y B
100
B
B
S2.18 0 = 2.18*σ y B
B
B
S1.09 0 = 1.09*σ y B
B
B
B
B
0 0.000
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
Strain [ε11]
Figure 3.4 The present predicted stress-strain responses of particulate composites under uniaxial tension at the four-phase composite state with various S 0 values B
B
Chapter 3. Multi-level modeling of particulate composites
42
interface, and particles with severe imperfect interface, respectively, as a function of the uniaxial strain corresponding to Figure 3.4. It is clearly shown from Figures 3.4 and 3.5 that if the strength at the particlematrix interface is low (lower S 0 ), most particles are mild imperfect or severe imperfect in their interfaces B
B
in early stage and the composites this parametric analysis that the influence of the Weibull parameter S 0 on B
B
the progression of show a nonlinear stress-strain behavior even in early stage of loading. It can be concluded from imperfect interface and stress-strain behavior of the composites is quite significant; thus, an accurate characterization of S 0 would be essential to realistically predict the progressive imperfect B
B
interface in particulate composites. Another parametric analysis is conducted to evaluate the proposed multi-level elastic damage model sensitivity to the compliance parameters and and to address the influence of varying and values on the stress-strain response of the model. As parameters controlling the severity of imperfect interface, four
0.3 perfectly bonded (S 0 = 1.09*σ y ) B
m ild im perfect severe im perfect perfectly bonded m ild im perfect
0.25
severe im perfect perfectly bonded m ild im perfect severe im perfect
(S 0 (S 0 (S 0 (S 0 (S 0 (S 0 (S 0 (S 0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
= 1.09*σ y ) = 1.09*σ y ) = 2.18*σ y ) = 2.18*σ y ) = 2.18*σ y ) = 3.27*σ y ) = 3.27*σ y ) = 3.27*σ y ) B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
Particle Volume Fraction
0.2
0.15
0.1
0.05
0 0.000
0.002
0.004
0.006
0.008
0.010
Strain [ε 11]
Figure 3.5 The predicted evolution of volume fractions of perfectly bonded particles, particles with mild imperfect interface and particles with severe imperfect interface corresponding to Figure 3.4
Chapter 3. Multi-level modeling of particulate composites
43
sets of the compliance parameters for the severe imperfect interface are used: 2 = 2.0, 2 = 3.0; 2 2.0 104 , 2 3.0 104 ; 2 6.0 105 , 2 7.0 105 ; 2 2.0 105 , 2 3.0 105 . For simplicity, fixed values of the compliance parameters for the mild imperfect interface are used as follows: 1 2.0 105 , 1 3.0 105 . The present predicted stress-strain responses of particulate composites with severe imperfect interfaces under uniaxial tension with various values of 2 and 2 are shown in Figure 3.6. As the values of the compliance parameters 2 and 2 become higher, the effect of the severe imperfect interface on the stress-strain behavior of the composites is more pronounced.
350
300
Stress [σ11] (MPa)
250
200
150
100
α2.0,3.0 2 =2.0, β 2 =3.0 α2E-4, 3E-4β 2 =3.E-4 2 =2.E-4, α6E-5,7E-5 =6.E-5, β 2 =7.E-5 2 α2E-5,3E-5 =2.E-5, β 2 =3.E-5 2
50
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
0 0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0.004
Strain [ε11]
Figure 3.6 The present predicted stress-strain responses of particulate composites under uniaxial tension at four-phase composite state with various α 2 and β 2 values B
B
B
B
Chapter 3. Multi-level modeling of particulate composites
44
3.6 Experimental comparisons The proposed multi-level elastic damage model is further exercised to predict the behavior of brittle ceramic composites. The uniaxial stress-strain behavior of unidirectional SiC fiber-reinforced CAS glassceramic matrix composites tested by Sørensen (1993) is numerically predicted using the multi-level elastic damage model. The unidirectional fibers in the composites are modeled as spherical particles in this simulation. The elastic properties of the composites used in this simulation are as follows (Sørensen, 1993; Sørensen and Talreja, 1993; Sørensen et al., 1993; Vanswijgenhoven and Van Der Biest, 1997): E 0 = 98 B
B
GPa, 0 = 0.3, E 1 = 200 GPa, 1 = 0.15, = 0.35. According to the experimentally obtained stress-strain B
B
curve in the longitudinal direction (Sørensen, 1993), the compliance parameters 1 , 1 , 2 and 2 given in
500
450
400
Stress [σ11] (MPa)
350
300
250
200
150
experiment 100
present prediction (5-phase composite)
50
0 0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
Strain [ε11]
Figure 3.7 The comparison between the present prediction and experimental data (Sørensen, 1993) for overall uniaxial tensile responses of SiC fiber-reinforced CAS glass-ceramic matrix composites
Chapter 3. Multi-level modeling of particulate composites
45
Eq. (3.6) as well as the Weibull parameters S 0 and M in Eq. (3.1) are estimated to be: 1 2.0 105 , B
B
1 3.0 105 , 2 = 2.0, 2 = 3.0; S 0 = 450 MPa, M = 2. B
B
The present predicted uniaxial stress-strain curve of the composites at the five-phase composite state is compared with the experimentally obtained stress-strain curve in the longitudinal (fiber) direction (Sørensen, 1993) in Figure 3.7. It is noted that the present prediction and the experimental data match well within the small strain region (the corresponding threshold strain th = 0.005), but the predicted stress-strain curve starts to deviate from the experimentally obtained one after the threshold strain. The experiment in Sørensen (1993) was designed to show the brittle behavior of ceramic composites reinforced with unidirectional fibers. It was shown from Figure 3.7 that the experimentally obtained stress of the composites was increased substantially after the threshold strain due the unidirectional fibers may play the
0.4
perfectly bonded mild imperfect severe imperfect completely debonded
0.35
Particle Volume Fraction
0.3
0.25
0.2
0.15
0.1
0.05
0 0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
Strain [ε11]
Figure 3.8 The predicted evolution of volume fractions of perfectly bonded particles, particles with mild imperfect interface, severe imperfect interface and completely debonded particles corresponding to the present prediction in Figure 3.7
Chapter 3. Multi-level modeling of particulate composites
46
primary role in the increase of stress. On the other hand, inclusions are limited to be spherical particles in our framework and therefore our predictions cannot be compared with his experimental data after the threshold strain. The predicted evolution of volume fractions of particles corresponding to Figures 3.7 is shown in Figures 3.8. To assess and demonstrate the accuracy of the proposed multi-level elastic damage model, the present predictions are compared with experimental data reported by Geiger and Welch (1997) for the uniaxial stress-strain behavior of silicon carbide particulate-reinforced 2009 aluminum matrix composites. The matrix is assumed to behave elastically. We adopt the same material properties for the composites as those in Geiger and Welch (1997) as follows: E 0 = 72.4 GPa, 0 = 0.33, E 1 =450 GPa, 1 = 0.19, = 0.294. Since B
B
B
B
model parameters of the proposed model were not reported by Geiger and Welch (1997), one needs to
600
500
Stress [σ11] (MPa)
400
300
200
experiment 100
present prediction (5-phase composite state)
0 0
0.002
0.004
0.006
0.008
0.01
0.012
Strain [ε11]
Figure 3.9 The comparison between the present prediction and experimental data (Geiger and Welch, 1997) for overall uniaxial tensile responses of silicon carbide particle-reinforced 2009 aluminum matrix composites
Chapter 3. Multi-level modeling of particulate composites
47
estimate the compliance parameters 1 , 1 , 2 and 2 given in Eq. (3.6) as well as the Weibull parameters S 0 and M in Eq. (3.1). B
B
Those compliance and Weibull parameters are fitted according to the experimentally obtained stressstrain curve (Geiger and Welch, 1997). The fitted model parameters are: 1 2.0 105 , 1 3.0 105 ,
2 = 2.0, 2 = 3.0; S 0 = 825 MPa, M = 3. The predicted uniaxial stress-strain curve of the composites at the B
B
five-phase composite state based on the above material properties and parameters is shown in Figure 3.9. The experimentally obtained stress-strain curve is also plotted in the figure for comparison. As a whole, the present prediction and the experimental data match well. The predicted evolution of volume fractions of particles corresponding to Figure 3.9 is shown in Figure 3.10.
0.3
perfectly bonded mild imperfect severe imperfect completely debonded
0.25
Particle Volume Fraction
0.2
0.15
0.1
0.05
0 0
0.002
0.004
0.006
0.008
0.01
0.012
Strain [ε11]
Figure 3.10 The predicted evolution of volume fractions of perfectly bonded particles, particles with mild imperfect interface, severe imperfect interface and completely debonded particles corresponding to the present prediction in Figure 3.9
Chapter 3. Multi-level modeling of particulate composites
48
3.7 Appendix. Parameters Λ 1 , . . , Λ 8 in Eqs. (3.8) - (3.23), and (3.26) - (3.33) B
B
B
B
1 22 1 1 31 2 1 1 15 1 0 1 2 1 31 2 1 1 2 2 1 31 2 2 2
15 1 0 1
(3.34)
(3.35)
4 1 1 2 2
2 24 1 3 32 2 2 2 3 600a 1 0 2 2 32 2 2 1 24 1 33 24 300a 1 0 2
(3.36)
2
4
(3.37)
2 1 24
3 26 1 5 33 2 3 2 5 600a 1 0 3 3 33 2 3 1 26 1 35 26 300a 1 0 3
(3.38)
2
6
(3.39)
3 1 26
4 28 1 7 34 2 4 7 15 1 0 4 2 3 2 1 2 1 3 2 4 4 4 8 7 8 8
15 1 0 4
(3.40)
(3.41)
4 4 1 28
in which
1
1 51 1 0 5 0 1 , 2 1 31 2 1
2
3
2 2 2
2 31 2 2 1 , 3 2 2 2
4
5
3 2 3
3 33 2 4 3 , 3 2 3 3
6
7
4 54 1 0 5 0 1 , 2 4 3 4 2 4
8
1 4 5 0 2 1 2 2 2 2
3 4 2 3
4 4 5 0 2 4
(3.42)
(3.43)
(3.44)
(3.45)
Chapter 3. Multi-level modeling of particulate composites
1 15 1 0
1 0 0 1
1 0 3 1 0 2 1 0
2 1200a 1 0
3 1200a 1 0
2
2
49
5 0 1
(3.46)
2 0 0 2 1 2 0 3 2 0 2 2 0
(3.47)
3 0 0 3 3 3 0 3 3 0 2 3 0
(3.48)
4 5 0 1 1
(3.49)
7 5 0 0 8 10 0 1 2 1 0
(3.50)
0 2 2 0
(3.51)
2 600a 1 0
3 600a 1 0 4
2
0
2
3 0
4
(3.52)
5 0 7 2
(3.53)
with
1 80 480 0 400 02 a 500 1 2 0 0 1 98 140 0 50 02 01
2
268 1240 0 1300 02 0 1
80 480 400 a 500 1 2 268 1240 1300
2 320 720 0 400 02 a 0 31 21 7 5 0
3
2
0
0
0
2
2 0 2 98 140 0 50 02 0 2
2
0
0
(3.54)
(3.55)
(3.56)
0 2
4 320 720 0 400 02 a 0 3 2 22 7 5 0
2
(3.57)
50
CHAPTER 4 AN ELASTOPLASIC MULTI-LEVEL DAMAGE MODEL FOR DUCTILE MATRIX COMPOSITES EVOLUTIONARY IMPERFECT INTERFACE
4.1 Introduction Ductile matrix composites (e.g., metal matrix composites, MMCs) reinforced with various shapes of particles, fibers or whiskers have been used in aerospace, electronics and a variety of other engineering applications. In particular, MMCs combine the properties of high strength and high modulus ceramics with those of high ductile metals or alloys to produce enhanced mechanical properties of the composite over metal or alloys (Ravi et al., 2007). Particle or fiber reinforced ductile matrix composites are in general subjected to a number of damage modes on the microscale (Drabek and Bohm, 2004). Debonding phenomenon existed in between particles and the matrix is one of the major damage modes in particle reinforced ductile matrix composites (PRDMCs) and its effect on the mechanical behavior of the composites has to be well addressed for an accurate analysis of the composites. Ju and Chen (1994c) developed a micromechanical formulation to predict the effective elastoplastic behavior of two-phase PRDMCs under arbitrary loading histories by considering the first-order stress perturbations of elastic particles on the ductile matrix. Ju and Tseng (1996, 1997) further improved Ju and Chen's (1994c) work by incorporating second-order stress perturbations due to pairwise particle interactions. Qu (1993a, 1993b) derived the Eshelby's tensor for an ellipsoidal inclusion with slightly weakened interface in an elastic matrix of infinite extent where the imperfect interface between the inclusion and the matrix was modeled by a spring layer of vanishing thickness. Lee and Pyo (2007) proposed a micromechanics-based elastic damage model to predict the effective elastic behavior and imperfect interface evolution in particle composites. The Eshelby's tensor for an ellipsoidal inclusion with slightly weakened interface (Qu, 1993a, 1993b) was adopted in their formulations to model the imperfect interface. Most recently, Lee and Pyo (in press) proposed a multi-level elastic damage model based on a combination of a micromechanical formulation and a multi-level damage model to predict the effective elastic behavior
Chapter 4. Elastoplastic multi-level modeling for ductile matrix composites
51
and progressive imperfect interface in particulate (brittle) composites. Their multi-level damage model in accordance with the Weibull's probabilistic function described the sequential, progressive imperfect interface in the composites. The primary objective of the present work is the extension of the framework of Lee and Pyo (in press) to accommodate the elastoplastic behavior and multi-level damage evolution in PRDMCs. Following Lee and Pyo (in press), the four-level elastic damage model, which was illustrated in Figure 3.1, is adopted for a complete description of the sequential progression of imperfect interface in the composite: (1) level-0 of two-phase composite state consisting of a ductile matrix and perfectly bonded particles; (2) level-1 of three-phase composite state consisting of a ductile matrix, perfectly bonded particles and particles with mild imperfect interface; (3) level-2 of four-phase composite state consisting of a ductile matrix, perfectly bonded particles, particles with mild imperfect interface and particles with severe imperfect interface; (4) level-3 of five-phase composite state consisting of a ductile matrix, perfectly bonded particles, particles with mild imperfect interface, particles with severe imperfect interface and completely debonded particles. Since two different damage modes of imperfect interface (mild imperfect interface and severe imperfect interface) are assumed to occur sequentially, imperfect interface would be developed and transformed from one mode to another as deformations or loadings continue to increase. Accordingly, there may be two different modes of imperfect interface (mild imperfect interface and severe imperfect interface) occurring simultaneously at the Level 2 of four-phase composite state. It will be followed by three different modes of imperfect interface (mild imperfect interface, severe imperfect interface and complete debonding) occurring simultaneously at the Level 3 of five-phase composite state at the next step. The two-parameter Weibull statistics is adopted to estimate the volume fraction of particles with different modes of imperfect interface. Particles are assumed to be non-interacting, randomly dispersed, elastic spheres that are initially embedded firmly in the ductile matrix with perfect interfaces. It is also assumed that the progression of imperfect interface is governed by the average internal stresses of perfect bonded particles (phase 1) as well as the Weibull parameters (Weibull, 1951). Numerical examples corresponding to uniaxial, biaxial and triaxial tension loadings are solved to illustrate the potential of the proposed micromechanical framework. A series of parametric analysis are carried out to investigate the influence of model parameters on the progression of imperfect interface in the composites. Furthermore, the present prediction is compared with available experimental data in the literature to further illustrate the elastoplastic damage behavior of the present framework and to verify the proposed elastoplastic multi-level damage model.
Chapter 4. Elastoplastic multi-level modeling for ductile matrix composites
52
4.2 Ensemble-volume-average homogenization process
4.2.1 Overview The homogenized elastoplastic damage responses of ductile matrix composites with imperfect interfaces are formulated in this section. Specifically, five-phase composites consisting of a ductile matrix (phase 0), perfectly bonded particles (phase 1), particles with mild imperfect interface (phase 2), particles with severe imperfect interface (phase 3) and completely debonded particles (voids) (phase 4) are considered. The von-Mises yield criterion with isotropic hardening law is adopted here for the ductile matrix; thus, the stress σ and the equivalent plastic strain e p at any matrix point satisfy the following yield function (see, e.g., Ju and Lee, 2000; Ju and Ko, in press):
F σ, e p H σ K 2 e p 0
(4.1)
where K e p denotes the isotropic hardening function of the ductile matrix and H σ σ : I d : σ signifies the square of the deviatoric stress norm in which I d denotes the deviatoric part of the fourth-rank B
B
identity tensor. Following the work of Ju and Chen (1994c), Ju and Lee (2000, 2001), Sun et al. (2003), Ju et al. (2006) and Ju and Ko (in press), a micromechanical framework, where an ensemble-averaged yield criterion is constructed for the entire composite and only the first order effects are considered in the formulation of effective plastic response, is employed.
4.2.2 A first-order formulation of the current stress norm of multi-phase ductile matrix composites Following Ju and Chen (1994c), Ju and Tseng (1996) and Ju and Lee (2000), the square of the “current stress norm” at the local point x, which determines the plastic strain in ductile matrix composites for a given phase configuration , can be written as
Chapter 4. Elastoplastic multi-level modeling for ductile matrix composites
53
x| : I d : x| , if x in the matrix; H |x 0, otherwise.
(4.2)
The ensemble average of H |x over all possible realizations, denoted by H
m
x ,
can be
obtained by integrating H over all possible particle and void configurations (cf. Ju and Lee, 2000)
H
m
H x| H P d H x| H P d H x| H P d H x| H P d
x H 0
0
0
1
1
1
2
2
0
0
3
3
2
3
4
4
(4.3)
4
where P q is the probability density function for finding the q-phase (q = 1, 2, 3, 4) configuration q in the composites and H 0 σ0 : I d : σ0 is the square of the far-field stress norm in the matrix. Moreover, the total stress at any point x in the matrix can be decomposed into two parts: the far-field stress σ 0 and the perturbed stress σ ' due to the presence of the particles and voids as:
σ x σ0 σ ' x
(4.4)
with σ0 C0 : ε0 where C0 denotes the fourth-rank elasticity tensor of the matrix and ε 0 is the elastic strain field induced by the far-field loading. Following Ju and Chen (1994c) and Ju and Lee (2000), the perturbed stress for any matrix point x due to a typical isolated q-phase inhomogeneity centered at xq(1) reads
: ε*0 σ ' x|x(1) C0 G x x(1) q q q
(4.5)
* where ε*0 q is the solution of the eigenstrain ε q for the single inclusion problem of the q-phase, and
G rˆq
1 q3H1 q5 H 2 30 1 0
The components of H 1 and H 2 are given by P
P
P
P
(4.6)
Chapter 4. Elastoplastic multi-level modeling for ductile matrix composites
54
(4.7)
(4.8)
H1ijkl rˆq 5Fˆijkl 15,3 0 ,3,3 6 0 , 1 2 0 ,1 2 0 2 Hijkl rˆq 3Fˆijkl 35, 5, 5, 5,1,1
ˆq , q a / rˆq , a is the radius of a spherical particle or void, and 0 is the where rˆq x x(1) q , rˆq r Poisson's ratio of the matrix material. The components of the fourth-rank tensor Fˆ - which depends on six scalar quantities B 1 , B 2 , B 3 , B 4 , B 5 , B 6 - were previously defined by (Ju and Lee, 2000): B
B
B
B
B
B
B
B
B
B
B
B
Fˆijkl Bm B1nˆi nˆ j nˆk nˆl B2 ik nˆ j nˆl il nˆ j nˆk jk nˆi nˆl jl nˆi nˆk B3 ij nˆk nˆl
B4 kl nˆi nˆ j B5 ij kl B6 ik jl il jk
(4.9)
with the unit outer normal vector nˆ rˆq / rˆq and index m = 1 to 6. In addition, the noninteracting eigenstrain ε*0 q in Eq. (4.5) is given by (see, e.g., Ju and Chen, 1994a)
ε*0 q Aq Sq
1
: 0,
q 1, 2,3, 4
(4.10)
where S q is the Eshelby's tensor for the q-phase inclusion and the fourth-rank tensor A q is defined as B
B
Aq Cq C0
B
1
B
C0 in which C q signifies the elasticity tensor of the q-phase. B
B
4.2.3 A first-order formulation of effective elastoplastic behavior of multi-phase ductile matrix composites Since a matrix point receives the perturbations from particles and voids, the ensemble-average stress norm for any matrix point x can be evaluated by collecting and summing up all the current stress norm perturbations produced by any typical particle or void centered at x(1) q in the particle or void domains and averaging over all possible locations of x(1) q (cf. Ju and Lee, 2000). Accordingly, H be rephrased as
m
x in Eq. (4.3) can
Chapter 4. Elastoplastic multi-level modeling for ductile matrix composites
H
m
55
(1) (1) 0 x H 0 xx(1) a H x|x1(1) H 0 P x1(1) dx1(1) x x(1) a H x|x(1) 2 H P x 2 dx 2 1
(1) x x3 a
H x|x H P x dx (1) 3
0
(1) 3
2
(1) 3
(1) x x4 a
H x|x H P x dx (1) 4
0
(1) 4
(1) 4
...
(4.11)
where P x(1) signifies the probability density functions for finding a particle or a void centered at x(1) q . It q
is assumed that the probability density functions take the form P x(1) q
Nq
, where N q is the total B
V
B
numbers of particles or voids dispersed in a representative volume V. Further, Eq. (4.11) can be recast into a more simplified form:
H
m
x H 0
N3 V
N1 V
0 0 2 rˆ1 drˆ1 A(rˆ1) H rˆ1 H dA V rˆ 2 drˆ2 A(rˆ 2 ) H rˆ2 H dA
N
rˆ 3 A(rˆ 3 ) drˆ3
H rˆ3 H 0 dA
N4 V
rˆ 4 A(rˆ 4 ) drˆ4
H rˆ4 H 0 dA ...
(4.12)
where A rˆq is a spherical surface of radius rˆq (q = 1,2,3,4). Using the two identities Eqs. (28) and (29) in Ju and Chen (1994c) and the perturbed stress given in Eq. (4.5), the ensemble-averaged current stress norm at any matrix point can be obtained as:
H
m
x σ0 : T(q) : σ0
(4.13)
The components of the positive definite fourth-rank tensor T (q) for the “q”-phase composite state read P
(q) Tijkl T1( q)ij kl T2( q) ik jl il jk
P
(4.14)
where T1(5) and T2(5) for the five-phase composite state are given by
2 2 1 802 a 2 1 0 2 802 a 2 1 0 3 4 2 (4.15) 3T1(5) 2T2(5) 50 1 2 0 2 2 2 3 2 2 3 2 3 2 3 2 1 1 2 2 4 4 3 3
Chapter 4. Elastoplastic multi-level modeling for ductile matrix composites
T2(5)
2 2 402 a 2 1 0 2 402 a 2 1 0 3 4 1 1 23 50 0 35 02 2 4 1 2 22 32 4 42
where the volume fraction for the q-phase is defined as q
56
(4.16)
4 a3 N q , and the parameters 1 , ... , 3 V
4 and 1 , ..., 4 are listed in Appendix of Chapter 3. Similarly, T1(2) and T2(2) , T1(3) and T2(3) , T1(4) and T2(4) for the two-phase, three-phase, and four-phase composite states, respectively, read:
3T1(2) 2T2(2) 50 1 2 0
T2(2)
1
2
31 21 2
1 1 23 50 0 35 02 2 4 12
2 1 802 a 2 1 0 2 2 3T1(3) 2T2(3) 50 1 2 0 2 3 2 2 3 2 1 1 2 2
T2(3)
2 402 a 2 1 0 2 1 2 1 23 50 0 35 0 4 12 2 22
3T1(4)
2T2(4)
T2(4)
50 1 2 0
2
(4.18)
(4.19)
(4.20)
2 2 1 802 a 2 1 0 2 802 a 2 1 0 3 2 2 3 2 2 3 2 3 2 3 1 1 2 2 3
(4.21)
2 2 402 a 2 1 0 2 402 a 2 1 0 3 1 1 23 50 0 35 02 2 4 1 2 22 32
(4.17)
(4.22)
Following Ju and Chen (1994c), the relation between the far-field stress σ 0 and the macroscopic stress σ takes the form
σ 0 P( q ) : σ
where the components of P( q ) for the “q”-phase composite state are
(4.23)
Chapter 4. Elastoplastic multi-level modeling for ductile matrix composites
(q) Pijkl P1( q)ij kl P2( q) ik jl il jk
57
(4.24)
with
3P1( q ) 2 P2( q ) P2( q )
1
(4.25)
1 1( q )
1
2 1 2( q )
(4.26)
and the coefficients 1(5) and 2(5) for the five-phase composite state read:
101 1 2 0
1(5)
31 2 1
2(5)
2 31 2 2 1200a 1 0
2
32 2 2
10 4 1 2 0
3 2 3
3
4
1200a 1 0
33 2 3
2
(4.27)
34 2 4
1 7 5 0 2 600a 1 0 2 2 1
2
2
600a 1 3
0
3
2
4
7 5 4
0
2 4
(4.28)
where the parameters 1 , .., 4 are listed in Appendix of Chapter 3. Similarly, the coefficients 1(2) and 2(2) ,
1(3) and 2(3) , 1(4) and 2(4) for the two-phase, three-phase, four-phase composite states, respectively, read:
1(2)
2(2)
1(3)
2(3)
101 1 2 0
(4.29)
31 2 1
1 7 5 0
(4.30)
2 1 101 1 2 0 31 2 1
2 31 2 2 1200a 1 0 32 2 2
1 7 5 0 2 600a 1 0 2 2 1
2
2
2
(4.31)
(4.32)
Chapter 4. Elastoplastic multi-level modeling for ductile matrix composites
1(4)
2(4)
101 1 2 0
31 2 1
2 31 2 2 1200a 1 0 32 2 2
1 7 5 0 2 600a 1 0 2
2 1
2
2
2
3 2 3
3
58
4
1200a 1 0
2
33 2 3
600a 1 3
0
2
4
3
(4.33)
(4.34)
By combining Eqs. (4.13) and (4.23), the ensemble-averaged current stress norm in a matrix point can be rephrased as:
H
m
x σ : T(q) : σ
(4.35)
where the positive definite fourth-rank tensor T( q ) for the “q”-phase composite state is defined as
T
T( q ) P( q ) T( q ) P( q)
(4.36)
and can be shown to be
(q) Tijkl T1( q)ij kl T2( q) ik jl il jk
(4.37)
where
3T1( q ) 2T2( q )
T2( q )
T2( q )
1 2( q )
3T1( q ) 2T2( q )
1 (q) 1
2
2
(4.38)
(4.39)
Following Ju and Lee (2000), the effective yield function for the multi-phase ductile matrix composites can be proposed as
Chapter 4. Elastoplastic multi-level modeling for ductile matrix composites
F 1 1 : T : K 2 e p 2
59
(4.40)
where 1 is the current (perfectly bonded) particle volume fraction. The effective equivalent plastic strain rate for the composites is defined as (Ju and Lee, 2000)
ep
2 p 1 :T : 3
p
2 1 1
2
2 σ:T:σ 3
(4.41)
in which denotes the plastic consistency parameter. The together with the yield function F must obey the Kuhn-Tucker loading/unloading conditions. In our derivations, the simple power-law type isotropic hardening function is employed (see also Ju and Lee, 2000):
K2 e p
2 p y h e 3
q
(4.42)
where y denotes the initial yield stress, and h and q are the linear and exponential isotropic hardening parameters, respectively, for the multi-phase composites. Furthermore, a four-level damage model in the order of sequence of progressive imperfect interface as illustrated in Figure 3.1 is employed here to model the sequential, progressive imperfect interface in ductile matrix composites (see detail descriptions in Chapter 3.).
Chapter 4. Elastoplastic multi-level modeling for ductile matrix composites
60
4.3 Numerical simulations To show the potential of the proposed micromechanical framework, we first predict the behavior of silicon-carbide particle reinforced 6061-T6 aluminum alloy matrix composites under uniaxial tension loading ( 11 0 and ij 0 for all other stress components). The elastic properties of the composites are adopted according to Zhao and Weng (1995), Ju and Lee (2001) and Lee and Pyo (2007) as: E 0 = 68.3 B
B
GPa, 0 = 0.33, E 1 = 490 GPa, 1 = 0.17, = 0.2. Following the parametric analysis on the compliance B
B
parameters and reported in preceding Chapters, the compliance parameters of the mild imperfect interface, denoted by 1 and 1 , and the severe imperfect interface, denoted by 2 and 2 , in Eq. (3.6) are chosen to be 1 2.0 105 , 1 3.0 105 ; 2 = 2.0, 2 = 3.0. Moreover, the plastic parameters and
600
500
Stress [σ11] (MPa)
400
300
200
2-phase composite state 3-phase composite state
100
4-phase composite state 5-phase composite state 0 0.000
0.002
0.004
0.006
0.008
0.010
Strain [ε11]
Figure 4.1 The present predicted elastoplastic stress-strain responses of particulate composites under uniaxial tension
Chapter 4. Elastoplastic multi-level modeling for ductile matrix composites
0.2
0.2
a
b
0.15
0.15
Particle Volume Fraction
Particle Volume Fraction
61
perfectly bonded
0.1
mild imperfect
0.1
perfectly bonded mild imperfect severe imperfect
0.05
0.05
0 0.000
0 0.000
0.002
0.004
0.006
0.008
0.010
0.002
0.004
0.006
0.008
0.010
Strain [ε 11 ]
Strain [ε11]
0.2
c
Particle Volume Fraction
0.15
0.1
perfectly bond mild imperfect severe imperfect 0.05
perfectly debonded
0 0.000
0.002
0.004
0.006
0.008
0.010
Strain [ε11]
Figure 4.2 (a) The predicted evolution of volume fractions of perfectly bonded particles and particles with mild impefect interface of the composites at the three-phase composite state; (b) The predicted evolution of volume fractions of perfectly bonded particles and various types of damaged particles at the four-phase composite state; (c) The predicted evolution of volume fractions of perfectly bonded particles and various types of damaged particles at the five-phase composite state corresponding to Figure 4.1
Chapter 4. Elastoplastic multi-level modeling for ductile matrix composites
62
Weibull parameters are assumed to be y = 400 MPa, h = 220 MPa, q = 0.45; S 0 = 380 MPa, M = 5. B
B
Four different phase composite states as illustrated in Figure 3.1 are considered in this simulation to examine the effect of the level of damage on the elastoplastic stress-strain response of the composites. The present predicted elastoplastic stress-strain responses of the particulate composites under uniaxial tension considering the four different phase composite states are plotted in Figure 4.1. Figure 4.2(a) exhibits the predicted evolution of volume fractions of perfectly bonded particles and particles with mild imperfect interface of the composites at the three-phase composite state versus uniaxial strains, which is corresponding to Figure 4.1. The predicted evolution of volume fractions of perfectly bonded particles and various types of damaged particles at the four-phase and five-phase composite states versus uniaxial strains corresponding to Figure 4.1 are exhibited in Figures 4.2(b) and 4.2(c), respectively. In Figures 4.1 and 4.2, the elastoplastic stress-strain responses corresponding to various phase
600
500
Stress [σ22(=σ33)] (MPa)
400
300
200
2-phase composite state 3-phase composite state
100
4-phase composite state 5-phase composite state 0 0
0.002
0.004
0.006
0.008
0.01
Strain [ε22(=ε33)]
Figure 4.3 The present predicted elastoplastic stress-strain responses of particulate composites under biaxial tension
Chapter 4. Elastoplastic multi-level modeling for ductile matrix composites
63
composite states are shown to be bounded by the responses of the two-phase and five-phase composite states. It is observed that the responses with the higher phase composite states are lower than those with the lower phase composite states. It is also noted from the figures that a smooth and gradual transition from the perfectly bonded interface to the perfectly debonded interface occurs within the present framework. In the present numerical simulations corresponding to Figure 4.2, the lower value of the Weibull parameters, which relates to the interfacial strength, (S 0 = 380 MPa and M = 5.0) is used to clearly show B
B
the effect of imperfect interface on the stress-strain behavior of imperfect particulate composites. Experimental characterization of the parameters is, however, required for more accurate prediction of imperfect interface evolution. Details of the relationship between the parameter and interfacial strength can be found in Lee (2001). The proposed multi-level elastic damage model is further exercised to predict the behavior of the composites under biaxial tension ( 22 33 0 and ij 0 for all other components) and triaxial tension
0.2
Particle Volume Fraction
0.15
0.1
perfectly bonded (3-phase) mild imperfect (3-phase) perfectly bonded (4-phase) mild imperfect (4-phase) severe imperfect (4-phase) perfectly bonded (5-phase) mild imperfect (5-phase) severe imperfect (5-phase) completely debonded (5-phase)
0.05
0 0.000
0.002
0.004
0.006
0.008
0.010
Strain [ε22(=ε33)]
Figure 4.4 The predicted evolution of volume fractions of perfectly bonded particles and various types of damaged particles corresponding to Figure 4.3
Chapter 4. Elastoplastic multi-level modeling for ductile matrix composites
64
( 22 33 11 0 and ij 0 for all other components) loading cases. The proportional loading ratio Γ is chosen to be 0.6 for the axisymmetric triaxial tension in this simulation. The predicted stress-strain responses of the composites considering the four different phase composite states under biaxial tension and triaxial tension loading cases are exhibited in Figures 4.3 and 4.5, respectively. The predicted evolution of volume fractions of perfectly bonded particles and various types of damaged particles under biaxial tension loading corresponding to Figure 4.3 are shown in Figure 4.4. Figure 4.6 exhibits the predicted evolution of volume fractions of perfectly bonded particles and various types of damaged particles under triaxial tension loading corresponding to Figure 4.5. It is observed from Figures 4.1, 4.3 and 4.5 that triaxial tension results in a higher elastoplastic stress-strain behavior, which corresponds to findings from Ju and Lee's (2000) and Lee and Pyo's (2007) observations.
1400
2-phase composite state 1200
3-phase composite state 4-phase composite state
Stress [σ11(=σ22/Γ=σ33/Γ)] (MPa)
1000
5-phase composite state
800
600
400
200
0 0.000
0.002
0.004
0.006
0.008
0.010
Strain [ε11]
Figure 4.5 The present predicted elastoplastic stress-strain responses of particulate composites under triaxial tension (Γ = 0.6)
Chapter 4. Elastoplastic multi-level modeling for ductile matrix composites
65
0.2
Paricle Volume Fraction
0.15
perfectly bonded (3-phase) mild imperfect (3-phase)
0.1
perfectly bonded (4-phase) mild imperfect (4-phase) severe imperfect (4-phase) perfectly bonded (5-phase) mild imperfect (5-phase) 0.05
severe imperfect (5-phase) completely debonded (5-phase)
0 0.000
0.002
0.004
0.006
0.008
0.010
Strain [ε11]
Figure 4.6 The predicted evolution of volume fractions of perfectly bonded particles and various types of damaged particles corresponding to Figure 4.5
Chapter 4. Elastoplastic multi-level modeling for ductile matrix composites
66
4.4 Parametric analysis A series of parametric analysis are conducted to examine the influence of the Weibull parameters S 0 B
B
and M and the compliance parameters and on the elastoplastic behavior and progression of imperfect interface in PRDMCs. First, following Lee's (2001) investigation on the Weibull parameters, three sets of Weibull parameters that are closely related to the strength at the particle-matrix interface in the PRDMCs are used: S 0 = 1.09× y , M = 5; S 0 = 2.18× y , M = 5; S 0 = 3.27× y , M = 5. For simplicity, a uniaxial B
B
B
B
B
B
tensile simulation is conducted using the same elastic properties as used in Section 4.3. The compliance parameters and the plastic parameters are assume to be 1 2.0 105 , 1 3.0 105 , 2 2.0 ,
2 3.0 ; y = 250 MPa, h = 220 MPa, q = 0.45.
400
350
300
Stress [σ11] (MPa)
250
200
150
S3.27 0 = 3.27*σ y
100
B
B
B
S2.18 0 = 2.18*σ y B
B
B
B
S1.09 0 = 1.09*σ y B
50
B
B
2-phase composite compositesstate state
0 0.000
0.002
0.004
0.006
0.008
0.010
Strain [ε11]
Figure 4.7 The present predicted elastoplastic stress-strain responses of particulate composites under uniaxial tension at the four-phase composite state with various S 0 values B
B
Chapter 4. Elastoplastic multi-level modeling for ductile matrix composites
67
Figure 4.7 shows the predicted elastoplastic stress-strain responses of the PRDMCs under uniaxial tension at the four-phase composite state with various S 0 values. It is clear from the figure that higher B
B
interfacial strength parameter S 0 leads to higher elastoplastic stress-strain response. It is also found from B
B
this parametric analysis that the influence of the Weibull parameter S 0 on the elastoplastic response is quite B
B
influential. Thus, the Weibull parameter S 0 needs to be experimentally characterized via the measurement B
B
of the interfacial strength of PRDMCs. The methodology for determining model parameters including the Weibull parameters can be found in Chapter 3. A parametric analysis conducted to determine lower and upper limits of the Weibull parameters can also be found in Lee (2001). To evaluate the proposed elastoplastic multi-level damage model sensitivity to the compliance parameters and , a parametric analysis of and is also carried out. Following Chapter 3, four sets of the compliance parameters for the severe imperfect interface are used: 2 2.0 , 2 3.0 ; 2 2.0 104 ,
300
250
Stress [σ11] (MPa)
200
150
100
α2.0,3.0 2 =2.0, β 2 =3.0 α2E-4, 2 =2.E-4, 3E-4β 2 =3.E-4 α6E-5,7E-5 2 =6.E-5, β 2 =7.E-5 α2E-5,3E-5 2 =2.E-5, β 2 =3.E-5
50
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
0 0
0.002
0.004
0.006
0.008
0.01
Strain [ε11]
Figure 4.8 The present predicted elastoplastic stress-strain responses of particulate composites under uniaxial tension at the four-phase composite state with various α 2 and β 2 values B
B
B
B
Chapter 4. Elastoplastic multi-level modeling for ductile matrix composites
68
2 3.0 104 ; 2 6.0 105 , 2 7.0 105 ; 2 2.0 105 , 2 3.0 105 . The compliance parameters for the mild imperfect interface are fixed as follows: 1 2.0 105 , 1 3.0 105 . Moreover, the Weibull parameters and plastic parameters are assumed to be S 0 = 1.09× y , M = 5; y = 250 MPa, h B
B
= 220 MPa and q = 0.45. The predicted stress-strain responses of PRDMCs with severe imperfect interfaces under uniaxial tension with various values of 2 and 2 are shown in Figure 4.8. It is seen from the figure that the effect of the severe imperfect interface on the elastoplastic stress-strain behavior of the composites become more noticeable as 2 and 2 become higher.
Chapter 4. Elastoplastic multi-level modeling for ductile matrix composites
69
4.5 Experimental comparison To assess the predictive capability of the proposed elastoplastic multi-level damage model, the present prediction is compared with the experimental data on SiC particulate-reinforced 5456 aluminum alloy matrix composites reported by Papazian and Adler (1990). We adopt the material properties of the composites as follows (Papazian and Adler, 1990; Bonfoh and Lipinski, 2007): E 0 =73 GPa, 0 = 0.33, E 1 B
B
B
B
=485 GPa, 1 = 0.2, = 0.2, y = 230 MPa. Since the model parameters of the proposed model were not reported by Papazian and Adler (1990), the model parameters are estimated by fitting the experimentally obtained stress-strain curve (Papazian and Adler, 1990) with the prediction. The fitted model parameters are: 1 2.0 105 , 1 3.0 105 , 2 2.0 , 2 3.0 ; h = 1,350 MPa and q = 0.24; S 0 = 530 MPa, M B
B
500
450
400
Stress [σ11] (MPa)
350
300
250
200
150
experiment
100
present prediction (5-phase composite) 50
0 0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
Strain [ε11]
Figure 4.9 The comparison between the present prediction and experimental data (Papazian and Adler, 1990) for overall uniaxial tensile responses of SiC particulate-reinforced 5456 aluminum alloy matrix composites
Chapter 4. Elastoplastic multi-level modeling for ductile matrix composites
70
= 5. Figure 4.9 shows the comparison between the present prediction and the experimental data on the overall elastoplastic stress-strain response of the composites at the five-phase composite state on the based of the above material properties and parameters. Overall, the present prediction and the experimental data match well. The predicted evolution of volume fractions of particles corresponding to Figure 4.9 is shown in Figure 4.10.
0.2
Particle Volume Fraction
0.15
0.1
perfectly bonded mild imperfect severe imperfect completely debonded 0.05
0 0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
Strain [ε11]
Figure 4.10 The predicted evolution of volume fractions of perfectly bonded particles, particles with mild imperfect interface, severe imperfect interface and completely debonded particles corresponding to the present prediction in Figure 4.9
71
CHAPTER 5 A MICROMECHANICS-BASED EVOLUTIONARY DAMAGE MODEL FOR CONTINUOUS, FIBER-REINFORCED BRITTLE MATRIX COMPOSITES WITH MICROCRACKS AND IMPERFECT INTERFACES
5.1 Introduction The mechanical behavior of continuous, fiber-reinforced brittle matrix composites (e.g., ceramic matrix composites, CMCs) is shown to be affected by several factors such as fiber bridging, crack branching, microcracking, delamination, fiber-matrix debonding, fiber slip, and fiber fracture, etc. (Miller et al., 2001). The fiber/matrix interfacial debonding usually dominates the damage behavior at a lower level of loading, and will be followed by the subsequent nucleation of microcracks as the load or displacement continues to increase (Lee et al. 2004). Many studies have been conducted to understand the influence of the fiber/matrix interface and matrix cracking on the mechanical behavior of the composites (e.g., Ahn et al., 1998; Curtin et al., 1998; Paar et al., 1998; Lamon, 2001; González and LLorca, 2007). One-step or multi-step damage models based on the Weibull probabilistic function have been proposed to predict the progressive interfacial debonding and its effect on the mechanical behavior of a wide range of composites (Zhao and Weng, 1995, 1996, 1997; Liang et al., 2006; Lee and Kim, 2007; Ju and Ko, in press, etc.). Another approach has also been proposed to model the interfacial behavior of particle or fiber-reinforced composites with imperfect interface (Qu, 1993a, 1993b; Jun and Haian, 1997; Lee and Pyo, 2007, in press). In their derivations, the imperfect interface was characterized by a spring-type model assuming that the traction continuity remains intact, while the displacement experienced a jump proportional to the interfacial traction (Qu, 1993a, 1993b; Lee and Pyo, 2007, in press). Crack nucleation is another important damage mechanism affecting the mechanical behavior of brittle matrix composites. The (stochastic) crack nucleation process is related to the distribution of the weak sites in the composites and the microcracks initiate uniformly across the material volume (Herrmann and Roux, 1990; Isichenko, 1992; Chakrabarti and Benguigui, 1997; Krajcinovic and Vujosevic, 1998; Lee and
Chapter 5. Evolutionary damage model for fiber-reinforced composites
72
Simunovic, 2006), allowing us to model the nucleation process using a simple scalar field that implies uncorrelated crack position and orientation (Lee and Simunovic, 2006). Lee and Simunovic (2006) developed a constitutive model based on a combination of a fracture-mechanics based model and micromechanical formulations for the prediction of the crack evolution and effective mechanical behavior of damage-tolerant brittle composites. Liang et al. (2006) employed a continuum damage model proposed by Karihaloo and Fu (1989) to simulate the nucleation of microcracks in laminated composites. In the present study, a three-dimensional (3D) micromechanics-based evolutionary damage model is proposed to predict the effective elastic behavior of continuous, fiber-reinforced brittle matrix composites with microcracks and imperfect interfaces. The Eshelby's tensor for a circular cylindrical inclusion with slightly weakened interface (Qu, 1993a, 1993b) is adopted to model continuous fibers with imperfect interfaces and is incorporated into the micromechanical framework. The progression of imperfect interfaces is assumed to be governed by the average internal stresses of the fibers as well as the Weibull parameters (Weibull, 1951), and nucleation of microcracks is simulated by employing the continuum damage model proposed by Karihaloo and Fu (1989). The multi-level elastic damage model proposed by Lee and Pyo (in press) is considered for a complete description of the sequential progression of imperfect interface in the composites. In the present derivation, continuous fibers are modeled to be elastic cylindrical inclusions that are randomly dispersed yet unidirectionally aligned in an elastic matrix. Microcracks are assumed to be located perpendicular to the fibers and the size of microcracks is uniform (see Figure 5.1). Numerical examples
1
3 2 Figure 5.1 The schematic of a continuous, fiber-reinforced composite subjected to uniaxial loading in the longitudinal direction
Chapter 5. Evolutionary damage model for fiber-reinforced composites
73
corresponding to uniaxial loadings in the longitudinal and transverse directions are solved to illustrate the potential of the proposed damage model. A series of parametric analysis are also carried out to address the influence of model parameters on the elastic damage behavior of the present framework. Comparisons between the present prediction with available experimental data are conducted to verify the accuracy of the proposed damage model.
Chapter 5. Evolutionary damage model for fiber-reinforced composites
74
5.2 Effective elastic moduli of continuous, fiber-reinforced brittle matrix composites with microcracks and imperfect interfaces Let us consider an initially perfectly bonded three-phase composite consisting of an elastic matrix (phase 0) with bulk modulus 0 and shear modulus 0 , and randomly located unidirectionally aligned elastic circular fibers (phase 1) with bulk modulus 1 and shear modulus 1 , and penny-shaped microcracks (phase 5) of radius c (see Figure 5.1). The penny-shaped microcrack is modeled as an oblate spheroidal void (a 1 = a 2 >> a 3 ) and aspect ratio a 3 /a 1 →0 as shown in Figure 5.2. The fibers are assumed B
B
B
B
B
B
B
B
B
B
to be non-interacting and initially embedded firmly in the matrix with perfect interface. As loads or deformations continue to increase, some initially perfectly bonded fibers are transformed to fibers with mild imperfect interface (phase 2), some fibers with mild imperfect interface are then transformed to fibers with severe imperfect interface (phase 3), and all fibers are transformed to completely debonded fibers that are regarded as cylindrical voids (phase 4), and microcracks are nucleated asymptotically within the proposed framework. The schematic of this transition is shown in Figure 5.3.The effective stiffness tensor
C* for multi-phase, linear elastic composites containing arbitrarily non-aligned and/or dissimilar inclusions can be derived as (Ju and Chen, 1994a)
a2 B
a3 B
B
a1 B
B
B
Figure 5.2 The schematic of a penny-shape crack modeled as an oblate spheroidal void (a 1 = a 2 >> B
a 3 ) and aspect ratio a 3 /a 1 →0 B
B
B
B
B
B
B
B
B
Chapter 5. Evolutionary damage model for fiber-reinforced composites
5 1 1 1 C* C0 I r A r Sr I r S r A r S r r 1
75
(5.1)
where Cr is the elasticity tensor of the r-phase, “ ” is the tensor multiplication, I is the fourth-rank identity tensor, and r denotes the volume fraction of the r-phase inclusion, and S r signifies the Eshelby's tensor for the r-phase. The fourth-rank tensor A r is defined as
a
b
Matrix (phase 0)
Fiber with mild imperfect interface (phase 2)
Perfectly bonded fiber (phase 1) Penny-shape microcrack (phase 5)
d
Void (phase 4)
c
1
2
Fiber with severe imperfect interface (phase 3)
Figure 5.3 The schematic of multi-level damage transition in a continuous, fiber-reinforced composite: (a) Level 0 of three-phase composite state (initial state); (b) Level 1 of three-phase composite state; (c) Level 2 of five-phase composite state; (d) Level 3 of six-phase composite state
Chapter 5. Evolutionary damage model for fiber-reinforced composites
76
1
Ar Cr C0 C0
(5.2)
The fourth-rank Eshelby's tensor S1 for perfectly bonded cylindrical fibers and S 4 for completely debonded cylindrical voids embedded in an isotropic linear elastic and infinite matrix can be derived as (Ju et al., 2006)
S1,4 ijkl 4 1 1 SIK(1) ij kl SIJ(2) ik jl il jk
(5.3)
0
where 0 denotes the Poisson's ratio of the matrix and ij signifies the Kronecker delta. The second-rank (1) tensor S IJ(2) is symmetric, while the second-rank tensor S IK is not symmetric. Details of the definition of (1) the second-rank tensors S IK and S IJ(2) can be found in Ju and Sun (1999). The components of the second(1) rank tensors S IK and S IJ(2) are given by (Ju et al., 2006)
(1) (1) (1) (1) S11 S22 S12 S21
1 4 0 1 , 2
(1) (1) S13 S23 2 0 ,
(2) (2) (2) (2) (1) (1) (1) S22 S12 S21 S31 S32 S33 0 , S11 (2) (2) (2) (2) S13 S23 S31 S32 1 0 ,
1 3 4 0 , 2
(5.4)
(2) S33 0
The Eshelby's tensor for a circular cylindrical inclusion with slightly weakened interface (Qu, 1993a, 1993b) is adopted to model continuous fibers with imperfect interface. The Eshelby's tensor for an ellipsoidal inclusion with slightly weakened interface can be written as (Qu, 1993a, 1993b)
M Sijkl
1 O O O O Sijkl (x) dV (x) Sijkl Iijpq Sijpq H pqrs Lrsmn I mnkl Smnkl
(5.5)
where Ω denotes an ellipsoidal subdomain in a homogeneous and linearly elastic solid, L signifies the O fourth-rank elasticity tensor, Sijkl is the original Eshelby's tensor and H is given by
Chapter 5. Evolutionary damage model for fiber-reinforced composites
H pqrs
1 nik n j nl n jk ni nl nil n j nk n jl ni nk dS 4 S
77
(5.6)
In case of circular cylindrical inclusions (a 1 = a 2 = a, a 3 →∞), H ijkl can be simplified as (Qu, 1993b) B
B
B
B
B
B
B
B
Hijkl Pijkl Qijkl
(5.7)
in which
P1111 P2222 4P2323 4P1313 2P1212
3 , 8a
Q1111 Q2222 3Q1122 3Q2211 3Q1212
9 , 32a
(5.8)
others = 0
By carrying out the lengthy algebra, the Eshelby's tensors S l+1 for a circular cylindrical inclusion with B
B
imperfect interface embedded in an isotropic linear elastic and infinite matrix in Eq. (5.5) can be recast as
Sl 1 ijkl
1 256a 1 0
2
S
M (2l 1) ij IK
kl S IJM (2l ) ik jl il jk
(5.9)
where l = 1, 2 in which 1 indicates the mild imperfect interface stage, while 2 denotes the severe imperfect M (2l 1) interface stage. In addition, the components of the second-rank tensors S IK and S IJM (2l ) are given by
M (2l 1) M (2l 1) M (2l 1) M (2l 1) S11 S22 S12 S21 32a 1 5 0 4 02
30 l 24 1 2 0 0 l 3 7 32 0 32 02 2
M (2l 1) S13
M (2l 1) S23
M (2l ) S22
0 l
128a 0 1 0 48 1 2 0 0 1 2 0 0 0 0 ,
M (2l 1) M (2l 1) M (2l 1) S31 S32 S33 0, M (2l ) S11
,
M (2l ) S12
M (2l ) S21
32a
3 7 0 4 02
3
0
(5.10)
l l ,
M (2l ) M (2l ) M (2l ) M (2l ) S13 S23 S31 S32 4 1 0 16a 30l , 2
M (2l ) S33 0
Chapter 5. Evolutionary damage model for fiber-reinforced composites
78
Following Sun and Ju (2004), the fourth-rank Eshelby's tensor S 5 for an oblate spheroidal void (a 1 = B
B
B
B
a 2 >> a 3 ) is employed to model penny-shaped microcracks (the aspect ratio a 3 /a 1 → 0) as B
B
B
B
B
B
B
B
S5 ijkl SIK(3) ij kl SIJ(4) ik jl il jk
(5.11)
(3) The components of the second-rank tensors S IK and S IJ(4) are given by
4 2 1 (3) (3) (3) (3) S11 S22 S12 S21 2 0 4 2 1
2 g , 2 2 1
2 2 1 2 2 (3) (3) , S13 S23 2 0 2 g 2 1 1 2 2 1 2 2 (3) (3) S31 S32 4 0 2 , g 4 0 2 1 1 2 4 (3) , S33 4 0 2 g 4 0 2 1 3 1
(5.12)
4 2 7 (4) (4) (4) (4) S11 S22 S12 S21 2 0 4 2 1
2 g , 2 2 1
2 2 2 (4) (4) (4) (4) S13 S23 S31 S32 0 2 , g 2 0 2 1 1 4 2 2 12 2 8 (4) S33 4 0 2 g 4 0 1 3 2 1
where γ denotes the aspect ratio of the spheroidal void and g(γ) is given by (Sun and Ju, 2004) 1/ 2 1 2 cos 1 , 1 3/ 2 1 2 g ( ) 1/ 2 cosh 1 2 1 , 1 3/ 2 2 1
(5.13)
Chapter 5. Evolutionary damage model for fiber-reinforced composites
79
Substituting Eqs. (5.3), (5.9) and (5.11) into Eq. (5.1) yields the effective stiffness tensor C* for the six-phase unidirectional composite as
(1) C* CIK ij kl CIJ(2) ik jl il jk
(5.14)
where 3
(1) (2) (1) (1) , CIK 20 KK 20 IK 0 nK n 1
(2) CIJ(2) 2 0 IJ
(5.15)
(1) in which 0 denotes the Lame constant of the matrix, and the components IK and IJ(2) are listed in
Appendix. Accordingly, the stress-strain relation for the (unidirectional) continuous, fiber-reinforced brittle matrix composites can be written as
(2) C (1) 2C11 11 11 (1) C21 22 (1) 33 C31 0 23 0 13 12 0
(1) C12
(1) C13
0
0
(1) (2) C22 2C22
(1) C23
0
0
0
0
(1) C32
(1) C33
(2) 2C33
0
0
(2) C23
0
0
0
0
(2) C13
0
0
0
0
0 11 0 22 0 33 0 2 23 0 213 (2) 212 C12
(5.16)
Chapter 5. Evolutionary damage model for fiber-reinforced composites
80
5.3 Damage modeling The multi-level elastic damage model proposed by Lee and Pyo (in press) is considered here for a complete description of the sequential progression of imperfect interface in the composites. The probability of imperfect interface is modeled as a two-parameter Weibull process and the average internal stresses of perfect bonded fibers (phase 1) are chosen to be the controlling factor of the Weibull function as
M 2 1 exp 11 1 S0
(5.17)
M 3 2 1 exp 11 1 S0
(5.18)
M 4 3 1 exp 11 1 S 0
(5.19)
3 3 4 ,
(5.20)
2 2 3 ,
1 2
where is the original particle volume fraction,
11 1 is the internal stress of fibers (phase 1) in the 1-
direction, the subscript 1 denotes the particle phase, and S 0 and M are the Weibull parameters. In case of B
the uniaxial loading in the longitudinal direction,
B
11 1 is replaced by 33 1 .
The internal stresses of fibers required for the initiation of the imperfect interface were explicitly derived by Ju and Lee (2000) as
n 1 1 1 C1 I S1 A1 S1 I r S r A r S r r 1
1
:ε U : ε
(5.21)
By carrying out the lengthy algebra, the components of fourth-rank tensor U for the present model are explicitly given by
(1) Uijkl U IK ij kl U IJ(2) ik jl il jk
(5.22)
Chapter 5. Evolutionary damage model for fiber-reinforced composites
81
(1) where the second-rank tensors U IK and U IJ(2) can be expressed as
(1) U IK
(1) EIK
1 2 (2) KK
2 EII(2)WIK 1 2 (2) II
3
(1) EIn WnK
(2) n 1 1 2 nn
U IJ(2)
,
EIJ(2)
(5.23)
1 2 (2) IJ
in which
(1) EIK
2 (2) 1 1 KK 1
(1) 21 IK 1 3 (1) nK , 1 1 n 1
2 (2) EIJ(2) 1 1 IJ 1
(1) (2) (1) (1) 31 WI 1 1 11 211 21 (1) (1) (2) (1) 12 1 22 2 22 32 WI 2 W (1) (1) (2) 13 (1) 1 33 2 33 I 3 23
(1) (3) (5) (7) (1) IK IK IK IK IK ,
1
,
(1) I1 (1) I 2 , (1) I 3
(2) (4) (6) (8) (2) IJ IJ IJ IJ IJ
(5.24)
(1) where the subscript I = 1, 2, 3 and the components IK , ..., IJ(8) in Eq. (5.24) are listed in Appendix.
In case of tensile loading, the averaged internal stresses of fibers can be obtained as
(1) (2) (1) (1) 11 U11 2U11 U12 U13 1 (1) (1) (2) (1) U 21 U 22 2U12 U 23 22 1 (1) (1) (1) (2) U 31 U 32 U 33 2U 33 33 1
1
11 22 33
(5.25)
In accordance with the continuum damage model for the nucleation of microvoids or microcracks in brittle materials (Karihaloo and Fu, 1989), the density of nucleated microcracks in the composite is modeled using the following isotropic scalar function (Liang et al., 2006)
v 0 , 5 th v 0 c1 1 a
a th c
2 a th ,
(5.26)
Chapter 5. Evolutionary damage model for fiber-reinforced composites
82
where v 0 is the initial density of microcracks. th ijth ijth is the threshold effective strain below which no nucleation occurs, c 1 and c 2 are material (nucleation) parameters depending on the specific shape and B
B
B
B
1/ 2
2 2 2 2 2 2 distribution of microcracks, and a ij ij 11 22 33 2 23 13 12
accumulated effective strain (Karihaloo and Fu, 1989).
is the current
Chapter 5. Evolutionary damage model for fiber-reinforced composites
83
5.4 Numerical simulations To illustrate the elastic damage behavior of the present framework and to address the effect of the Weibull parameter S 0 on the evolution of imperfect interface in uncracked, continuous, fiber-reinforced brittle B
B
matrix composites (five-phase composite state), we first consider a uniaxial tension in the longitudinal direction case ( 33 0 and ij 0 for all other stress components). We adopt the material properties for the Nicalon fiber-reinforced Barium Magnesium Aluminosilicate (BMAS) glass ceramic composite in accordance with Dannemann and Mandell (1994) in this simulation: E m = 106 GPa, m = 0.222, E f = 200 B
B
B
B
GPa, f = 0.299 and = 0.4, where the subscripts m and f represent the matrix and fibers, respectively. The compliance parameters 1 , 1 , 2 and 2 given in Eq. (5.10) are assumed to be 1 2.0 107 ,
300
250
Stress [σ33] (MPa)
200
150
100
So S 0 == 100 100MPa MPa B
50
B
B
So S 0 == 200 200MPa MPa B
B
S 0 == 300 300MPa MPa So B
B
0 -0.003
-0.002
[ε 11 ] B
B
-0.001
0 Strain
0.001
0.002
0.003
0.004
0.005
[ε 33 ] B
B
Figure 5.4 The present predicted stress-strain responses of unidirectional composites under uniaxial tension in the longitudinal direction with various S 0 values B
B
Chapter 5. Evolutionary damage model for fiber-reinforced composites
84
1 3.0 107 , 2 2.0 , 2 3.0 . Moreover, three different sets of Weibull parameters are used to evaluate the proposed elastic damage model sensitivity to the Weibull parameter S 0 : S 0 = 100 MPa and M B
B
B
B
= 2; S 0 = 200 MPa and M = 2; S 0 = 300 MPa and M = 2. B
B
B
B
The present predicted stress-strain responses of the composites with imperfect interfaces under uniaxial tension in the longitudinal direction with various S 0 values are shown in Figure 5.4. Figure 5.5 B
B
exhibits the evolutions of perfectly bonded fibers, fibers with mild imperfect interface, severe imperfect interface and completely debonded fibers as functions of the uniaxial strains corresponding to Figure 5.4. From Figure 5.5, it is clear that lower S 0 leads to faster evolution of imperfect interface, indicating that B
B
most fibers are damaged in their interfaces in early stage with lower S 0 and the composites exhibit a B
B
nonlinear stress-strain behavior. We also plot the present predicted stress-strain responses of unidirectional
0.6
0.5
perfect bonding (So=100)
mild imperfect (So=100)
severe imperfect (So=100)
complete debonded (So=100)
perfect bonding (So=200)
mild imperfect (So=200)
severe imperfect (So=200)
complete debonded (So=200)
perfect bonding (So=300)
mild imperfect (So=300)
severe imperfect (So=300)
complete debonded (So=300)
Fiber Volume Fraction
0.4
0.3
0.2
0.1
0 0
0.001
0.002
0.003
0.004
Strain [ε 33]
Figure 5.5 The predicted evolution of volume fractions of perfectly bonded fibers, fibers with mild imperfect interface, fibers with severe imperfect interface and completely debonded fibers corresponding to Figure 5.4
Chapter 5. Evolutionary damage model for fiber-reinforced composites
85
composites with imperfect interfaces under uniaxial tension in the transverse direction( 11 0 and
ij 0 for all other stress components) with various S 0 values in Figure 5.6. As shown in Figures 5.4 and B
B
5.6, the stress-strain responses due to uniaxial tension in the transverse direction are lower than those due to uniaxial tension in the longitudinal direction. It is concluded that the progressive evolution of imperfect interface affects significantly the overall elastic damage behavior of the composites. To further examine the proposed evolutionary damage model sensitivity to the compliance parameters
and , we conduct a parametric analysis of and . As parameters controlling the severity of imperfect interface, three sets of the compliance parameters for the severe imperfect interface condition are used: 2 = 2.0, 2 = 3.0; 2 2.0 104 , 2 3.0 104 ; 2 2.0 107 , 2 3.0 107 . For simplicity, fixed values of the compliance parameters for the mild imperfect interface condition and
400
Stress [σ11] (MPa)
300
200
= 100 100 MPa SSo MPa 0 =
100
B
B
B
= 200 200 MPa SSo MPa 0 = B
B
= 300 300 MPa SSo MPa 0 = B
B
0 -0.003
-0.001
[ε 33 ] B
B
0.001 Strain
0.003
0.005
0.007
0.009
0.011
[ε 11 ] B
B
Figure 5.6 The present predicted stress-strain responses of unidirectional composites under uniaxial tension in the transverse direction with various S 0 values B
B
Chapter 5. Evolutionary damage model for fiber-reinforced composites
86
Weibull parameters are used: 1 2.0 107 , 1 3.0 107 ; S 0 = 200 MPa, M = 2. The present predicted B
B
stress-strain responses of the composites under uniaxial tension in the transverse direction with various values of 2 and 2 are shown in Figure 5.7. As the values of the compliance parameters 2 and
2 become higher, the effect of the severity of imperfect interface on the stress-strain behavior of the composites is more pronounced. Another parametric study is also conducted using the same material properties as used in the previous simulations to investigate the influence of nucleation parameters c 1 and c 2 in Eq. (5.26) on the elastic B
B
B
B
damage behavior of cracked, continuous, fiber-reinforced brittle matrix composites (six-phase composite state). Six different sets of c 1 and c 2 are used in the simulation: c 1 = 0.2 and c 2 = 1.5; c 1 = 0.25 and c 2 = B
B
B
B
B
B
B
B
B
B
B
B
1.5; c 1 = 0.3 and c 2 = 1.5; c 1 = 0.3 and c 2 = 1.1; c 1 = 0.3 and c 2 = 1.6; c 1 = 0.3 and c 2 = 2.3. For brevity, B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
fixed values of the compliance parameters and Weibull parameters are used: 1 2.0 107 ,
300
Stress [σ11] (MPa)
200
100
α 2 =2.E-7, β 2 =3.E-7 So = 100 MPa B
B
B
B
α 2 =2.E-4, β 2 =3.E-4 So = 200 MPa B
B
B
B
α 2 =2.0, So = 300βMPa 2 =3.0 B
B
B
B
B
B
B
0 0
0.002
0.004
0.006
0.008
Strain [ε11]
Figure 5.7 The present predicted stress-strain responses of particulate composites under uniaxial tension in the transverse direction with various α 2 and β 2 values B
B
B
B
Chapter 5. Evolutionary damage model for fiber-reinforced composites
87
1 3.0 107 , 2 = 2.0, 2 = 3.0; S 0 = 180 MPa, M = 1.6. th in Eq. (5.26) is assumed to be th = B
B
0.001. We plot the present predicted stress-strain responses of the composites with microcracks and imperfect interfaces under uniaxial tension in the longitudinal direction with various c 1 and c 2 values in B
B
B
B
Figures 5.8 and 5.10, respectively. Figures 5.9 and 5.11 exhibit evolutions of microcrack density corresponding to Figures 5.10 and 5.12, respectively. As the values of the material constants c 1 becomes B
B
higher and c 2 becomes lower, the effect of nucleation of microcracks on the stress-strain behavior of the B
B
composites is shown to be more noticeable.
300
250
Stress [σ33] (MPa)
200
150
c 1 =0.2 c1 = 0.2 B
100
B
B
c 1 =0.25 c1 = 0.25 B
B
c 1 =0.3 c1 = 0.3 B
B
B
50
0 -0.004
-0.002
[ε 11 ] B
B
0 Strain
0.002
0.004
0.006
0.008
[ε 33 ] B
B
Figure 5.8 The present predicted stress-strain responses of unidirectional composites under uniaxial tension in the longitudinal direction with various c 1 values B
B
Chapter 5. Evolutionary damage model for fiber-reinforced composites
88
0.300
0.250
Microcrack density
0.200
0.150
0.100
ccrack1 1 =0.2 B
B
B
ccrack2 1 =0.25 B
0.050
B
ccrack3 1 =0.3 B
B
B
0.000 0
0.002
0.004 Strain [ε
3
0.006 3
0.008
]
Figure 5.9 The predicted evolution of microcrack density corresponding to Figure 5.8
Chapter 5. Evolutionary damage model for fiber-reinforced composites
89
300
250
Stress [σ33] (MPa)
200
150
100
cc22 =1.1 = 1.1 B
B
B
= 1.6 cc22 =1.6 B
B
= 2.3 cc22 =2.3
50
B
B
B
0 -0.004
-0.002
[ε 11 ] B
B
0
0.002
0.004
0.006
0.008
[ε 33 ]
Strain
B
B
Figure 5.10 The present predicted stress-strain responses of unidirectional composites under uniaxial tension in the longitudinal direction with various c 2 values B
B
0.300
0.250
Microcrack density
0.200
0.150
ccrack1 2 =1.1 B
0.100
B
B
ccrack2 2 =1.6 B
B
ccrack3 2 =2.3 B
B
B
0.050
0.000 0
0.002
0.004 Strain [ε
3
0.006 3
0.008
]
Figure 5.11 The predicted evolution of microcrack density corresponding to Figure 5.10
Chapter 5. Evolutionary damage model for fiber-reinforced composites
90
5.5 Experimental comparison The proposed evolutionary damage model is further exercised to predict the behavior of continuous, fiber-reinforce brittle matrix composites, and the prediction is compared with experimental data reported by Luo and Daniel (2000) to verify the accuracy of the proposed model. The stress-strain behavior of unidirectional silicon carbide/calcium aluminosilicate (SiC/CAS) composites tested by Luo and Daniel (2000) is numerically predicted using the proposed evolutionary damage model. The elastic properties of the composites used in this simulation are (Luo and Daniel, 2000): E m = 97.9 GPa, m = 0.25, E f = 188 B
B
B
B
GPa, f = 0.3, = 0.36. According to the experimentally obtained stress-strain curve in the longitudinal
500
Stress [σ33] (MPa)
400
300
200
experiment 100
present prediction (no-microcracks) present prediction (with microcracks)
0 0
0.002
0.004 Strain [ε33
0.006
0.008
]
Figure 5.12 The comparison between the present predictions (with and without microcracks) and experimental data (Luo and Daniel, 2000) for overall uniaxial tensile responses in the longitudinal direction of silicon carbide/calcium aluminosilicate (SiC/CAS) composites
Chapter 5. Evolutionary damage model for fiber-reinforced composites
91
direction (Luo and Daniel, 2000), the compliance parameters 1 , 1 , 2 and 2 given in Eq. (5.10) as well as the Weibull parameters S 0 and M in Eqs. (5.17)-(5.19) for the uncracked, five-phase composite state are B
B
estimated to be: 1 2.0 107 and 1 3.0 107 , 2 = 2.0, 2 =3.0; S 0 = 280 MPa, M = 1.3. For the B
B
cracked, six-phase composite state, additional parameters related to nucleation of microcracks c 1 , c 2 , B
B
B
B
v 0 and th in Eq. (5.26) are estimated to be: c 1 = 0.25, c 2 = 1.6, v 0 = 0.0 and th =0.002. B
B
B
B
The present predicted (longitudinal) uniaxial stress-strain responses of the two different composite states are compared with the experimentally obtained stress-strain curve by Luo and Daniel (2000) in Figure 5.12. The predicted evolutions of volume fractions of fibers (five-state composite state) and microcrack density (six-phase composite state) corresponding to Figures 5.12 are shown in Figures 5.13 and 5.14, respectively. It is observed that the response with the model considering nucleation of
0.4
0.35
Fiber Volume Fraction
0.3
0.25
0.2
perfect bonding mild imperfect interface
0.15
severe imperfect interface completely debonded
0.1
0.05
0 0
0.002
0.004 Strain [ε33
0.006
0.008
]
Figure 5.13 The predicted evolution of volume fractions of perfectly bonded fibers, fibers with mild imperfect interface, severe imperfect interface and completely debonded fibers (fivephase composite state) corresponding to Figure 5.12
Chapter 5. Evolutionary damage model for fiber-reinforced composites
92
microcracks is lower than that without nucleation of microcracks. Clearly, in comparison with the model without nucleation of microcracks, the present prediction with nucleation of microcracks matches well with the experimentally obtained stress-strain curve. 0.250
Microcrack density
0.200
0.150
0.100
0.050
0.000 0
0.002
0.004
0.006
0.008
Strain [ε33]
Figure 5.14 The predicted evolution of microcrack density (six-phase composite state) corresponding to Figure 5.12
Chapter 5. Evolutionary damage model for fiber-reinforced composites
93
(1) (10) (2) (1) 5.6 Appendix. Parameters IK and IJ in Eq. (5.15), and IK , … , IJ in
Eq. (5.24) These parameters in Eqs. (5.15) and (5.24) take the form:
IJ(2)
(1) (3) (5) (7) (9) IK (1) IK IK IK IK IK ,
1 (4) (6) (8) (10) (2) IJ IJ IJ IJ IJ 2
(5.27)
with 3 (2r 1) (2 r ) r 1) (2 r ) (2 r 1) (2 r 1) (2 r 1) (2 2 2 In nK (r = 1, 2, 3, 4, 5) r IK IK KK II IK n 1
(5.28)
r) (2 r ) (2 r ) (2 IJ 2 rIJ IJ
(5.29)
(r = 1, 2, 3, 4, 5)
and (2 r 1) IK
(IKr ) 1 2 II(2r )
,
IJ(2r )
1
(5.30)
2 4 IJ(2r )
(2 r 1) (2 r ) (2 r 1) (2 r 1) (Ir1) 1 11 211 21 31 (r ) (2 r 1) (2 r 1) (2 r ) (2 r 1) 12 1 22 2 22 32 I 2 (r ) (2 r 1) (2 r 1) (2 r 1) (2 r ) I3 13 23 1 33 233
1
I(21 r 1) (2r 1) I 2 (2r 1) I 3
(5.31)
(1) where I = 1, 2, 3. In addition, the parameters IK , .... , IJ(10) read (1) IK
(3) IK
(5) IK
(7) IK
3 1 (1) (2) (2) (1) (1) (1) 2S IK KK 2S II IK S In nK , 4 1 0 n 1
2
256a 1 0
2
3 256a 1 0
2
IJ(2)
1S IJ(2) IJ(2) 2 1 0
3 M (1) (4) M (2) (3) M (1) (3) 2S IK KK 2S II IK S In nK , n 1
IJ(4)
3 M (3) (6) M (4) (5) M (3) (5) 2S IK KK 2S II IK S In nK , n 1
IJ(6)
3 4 (1) (8) (2) (7) (1) (7) 2S IK KK 2S II IK S In nK , 4 1 0 n 1
3
n 1
(9) (3) (10) (9) (3) (9) IK 5 2S IK KK 2S II(4)IK S In nK ,
(8) IJ
(5.32)
2 S IJM (2) IJ(4) 128a 1 0
3 S IJM (4) IJ(6) 128a 1 0
(2) (8) 4 S IJ IJ 2 1 0
(10) (4) (10) IJ 2 5 SIJ IJ
2
2
(5.33)
(5.34)
(5.35)
(5.36)
Chapter 5. Evolutionary damage model for fiber-reinforced composites
94
with
(1) IK
(3) IK
(5) IK
(7) IK
(9) IK
(1) YIK
1 2 1' S II(2) 4 1 0
,
IJ(2)
(2) YIK
1 M (2) 2 2' S II 2 256a 1 0 (3) YIK
1 M (4) 2 3' S 2 II 256a 1 0 (4) YIK , 1 (2) 1 S 2 1 0 II
(5) YIK
1 2S II(4)
,
IJ(10)
1 ' 1 4 1 S IJ(2) 4 1 0
,
IJ(4)
,
IJ(6)
IJ(8)
1 1 M (2) 4 2' S IJ 2 256a 1 0 1 1 M (4) 4 3' S 2 IJ 256a 1 0
1 1 2 S (2) 1 0 IJ
1 2 4S IJ(4)
(5.37)
(5.38)
(5.39)
(5.40)
(5.41)
in which
YI(1r ) ( r ) YI 2 Ar B r (r ) YI 3
(5.42)
with
' 1 1 1 (1) (2) (1) (1) ' S11 2S11 1' S21 1' S31 1 21 4 1 4 1 4 1 0 0 0 1 1 1 (1) (1) (2) (1) ' S22 2S22 A1 1' S12 1' 21' S32 1 4 1 0 4 1 0 4 1 0 1 1 1 (1) (1) (1) (2) ' ' ' ' S33 2S33 1 S13 1 S23 1 21 4 1 0 4 1 0 4 1 0
1
(5.43)
Chapter 5. Evolutionary damage model for fiber-reinforced composites
95
1 1 1 M (2) S M (1) 2S11 2' 22' 2' S M (1) 2' S M (1) 2 11 2 21 2 31 256a 1 0 256a 1 0 256a 1 0 1 1 1 M (2) M (1) ' S M (1) 2S22 A2 2' S M (1) 2' 22' S32 2 2 12 2 22 2 256a 1 0 256a 1 0 256a 1 0 1 1 1 M (1) M (1) M (1) M (2) ' ' ' ' 2 S 2 S 2 22 S 2S33 2 13 2 23 2 33 256a 1 0 256a 1 0 256a 1 0
1
(5.44)
1 1 1 M (4) S M (3) 2S11 3' 23' 3' S M (3) 3' S M (3) 2 11 2 21 2 31 256a 1 0 256a 1 0 256a 1 0 1 1 1 M (4) S M (3) 2S22 A3 3' S M (3) 3' 23' 3' S M (3) 2 12 2 22 2 32 256 a 1 256 a 1 256 a 1 0 0 0 1 1 1 M (3) M (3) M (3) M (4) ' ' ' ' 3 S 3 S 3 23 S 2S33 2 13 2 23 2 33 256a 1 0 256a 1 0 256a 1 0
1
(5.45)
1 1 1 (1) (2) (1) (1) S11 2S11 S21 S31 1 4 1 0 4 1 0 4 1 0 1 1 1 (1) (1) (2) (1) S22 A4 S12 1 2S22 S32 4 1 0 4 1 0 4 1 0 1 1 1 (1) (1) (1) (2) S33 S13 S23 1 2S33 4 1 0 4 1 0 4 1 0
(3) (4) 1 S11 2S11 (3) A5 S12 (3) S13
(3) S21 (3) (4) 1 S22 2S22
' 1 S I(1) 1 1 4 1 0 1 B1 1' S I(1)2 , 4 1 0 1 1' S I(1) 4 1 0 3
(3) S23
(3) S32 (3) (4) 1 S33 2S33 (3) S31
1
(5.46)
1
1 M (1) 2' S 2 I1 256a 1 0 1 M (1) B2 2' S 2 I2 256a 1 0 1 M (1) ' S 2 256a 1 2 I 3 0
(5.47)
(5.48)
Chapter 5. Evolutionary damage model for fiber-reinforced composites
1 1 3' S M (3) S I(1) 2 I1 1 256a 1 0 4 1 0 S I(3) 1 (3) 1 1 ' M (3) (1) , , B3 3 S B S B S 4 5 I 2 I 2 I 2 2 256a 1 0 (3) 4 1 0 S I 3 1 (1) 1 M (3) ' S S I3 3 256a 1 2 I 3 4 1 0 0
96
(5.49)
where
r'
r 0 0 r
r 0 3 r 0 2 r 0
,
r'
0
2 r 0
(5.50)
97
CHAPTER 6 NUMERICAL ANALYSIS OF LAMINATED COMPOSITE STRUCTURES BASED ON MICROMECHANICAL MULTI-LEVEL DAMAGE MODEL
6.1. Introduction Due to their great potential for weight savings, fiber reinforced laminated composites are becoming increasingly important for structural applications concerned with light weight design, e.g. in the fields of aerospace, transportation and other industrial applications. Despite these desirable physical properties, composites are fragile and susceptible to damage from a number of sources, both during initial processing and in service (Lapczyk and Hurtado, 2007). Even though the damage does not cause catastrophic failure, it significantly reduced structural stiffness (Hassan and Batra, 2008). In order to exploit the advantages of laminated composites, reliable prediction of the mechanical behavior is essential. Many micromechanical models have been developed in order to obtain the homogenized elastic properties of composite materials (Qu, 1993a, 1993b, Tohgo and Weng, 1994; Zhao and Weng, 1995, 1996, 1997; Ju and Lee, 2000, 2001; Liu et al., 2004, 2006; Lee and Pyo, 2007, in press) or to study the effect of microcracks on the elastic response of brittle solids (Karihaloo and Fu, 1989; Lee and Liang, 2004; Lee and Simunovic, 2006; Lee and Pyo, submitted for publication). Damage propagation cannot easily be modelled by more conventional finite element models due to the size and complexity of these microstructures (Grufman and Ellyin, in press). Many finite element analyses about damage and failure in laminated composites under various loading cases have been carried out (Davalos et al., 1996; Luciano and Zinno, 2000; Maa and Cheng, 2002; Kilic and Haj-Ali, 2003; Rodriguez and Ochoa, 2004; Liang et al., 2006; Guo, 2007; Lee and Kim, 2007; Maimi et al., 2007; Teng, 2007). Davalos et al. (1996) conducted the analysis of pultruded FRP beams in bending using finite element (FE) analysis. Liang et al. (2006) and Lee and Kim (2007) implemented micromechanics based damage model into FE code to simulate laminated plates containing a cutout under uniaxial compression. Guo (2007) conducted numerical and experimental studies to investigate the effect of reinforcements around cutouts on the stress concentration and buckling behavior of a carbon/epoxy composite panel under in-plane shear load.
Chapter 6. Damage modeling of laminate composite structures
98
However, there still remain various uncertainties about the failure mechanism (Suemasu et al., 2006). Development of computational methodologies for the accurate prediction of damage accumulation and growth in continuous fiber composite laminates is presently an active area of research (Basu et al., 2007). In the present study, continuous fibers are modeled to be elastic cylindrical inclusions that are randomly dispersed yet unidirectionally aligned in an elastic matrix. Microcracks are assumed to be located perpendicular to the fibers and the size of microcracks is uniform (see also Lee and Pyo, submitted for publication). It is also assumed that the progression of imperfect interface is governed by the average internal stresses of the fibers as well as the Weibull parameters (Lee and Pyo, in press). A micromechanics based constitutive model incorporating damage models was implemented into ABAQUS non-linear FE code (ref. 116) using a user-subroutine UMAT to solve boundary value problems. Numerical examples corresponding to various loading conditions and geometries are solved to illustrate the potential of the proposed implemented damage model. Comparisons between the present prediction with available experimental data in the literature are conducted to verify the accuracy of the proposed computational model.
X2 B
X1 B
X3 B
Figure 6.1 Coordinates for a transversely isotropic material
Chapter 6. Damage modeling of laminate composite structures
6.2
Micromechanics-based
constitutive
99
damage
model
for
off-axis
unidirectional fiber reinforced composites The 3-direction is chosen to be fiber direction and the plane 1-2 corresponds to the transversely isotropic plane in the unidirectional fiber reinforced composites in this study (see Figure 5.1). Now we consider rotations through an angle about the X 2 -axis to derive the stiffness of off-axis unidirectional B
B
fiber-reinforced composites (see Figure 6.1). Following the stiffness transformation law by Herakovich (1998), we have the following stress and strain transformations (see also Liang et al., 2006):
C : T1 :
(6.1)
T2 :
(6.2)
where the subscript ρ signifies stress and strain matrices in the principal coordinate system, the subscript ς denotes stress and strain matrices in the global coordinate system, and the transformation matrices T 1 and B
B
T 2 are given as B
B
cos 2 0 sin 2 T1 0 cos sin 0 cos 2 0 sin 2 T2 0 2 cos sin 0
0
sin 2
0
2 cos sin
1
0
0
0
0
cos 2
0
2 cos sin
0
0
cos
0
0
cos 2 sin 2
sin
0
0 cos sin 0
0
0 0 0 sin 0 cos
0
sin 2
0
cos sin
1
0
0
0
0
cos
0
cos sin
0
0
cos
0
0
cos 2 sin 2
sin
0
2
0 2 cos sin 0
0
(6.3)
0 0 0 sin 0 cos
(6.4)
By combining Eqs. (6.1) and (6.2), the stiffness matrix of off-axis unidirectional fiber-reinforced composites C , which is the transformed stiffness matrix through an arbitrary angle about X 2 -axis, is B
B
Chapter 6. Damage modeling of laminate composite structures
100
derived as
T11 C T2 : C :
(6.5)
where the stiffness matrix of the unidirectional fiber-reinforced composites with the fibers oriented off-axis takes the form:
C11 C12 C21 C22 C C32 C 31 0 0 C C52 51 0 0
C13
0
C15
C23
0
C25
C33
0
C35
0
C44
0
C53
0
C55
0
C64
0
0 0 0 C46 0 C66
The components of C are given in Appendix.
(6.6)
Chapter 6. Damage modeling of laminate composite structures
101
6.3 Damage modeling The multi-level elastic damage model proposed by Lee and Pyo (in press) is considered here for a complete description of the sequential progression of imperfect interface in the composites. The probability of imperfect interface is modeled as a two-parameter Weibull process and the average internal stresses of perfect bonded fibers (phase 1) are chosen to be the controlling factor of the Weibull function as
f 2 1 exp S0
m
M
(6.7)
f 3 2 1 exp S0
(6.8)
f 4 3 1 exp S0
(6.9)
m m
3 3 4 ,
2 2 3 ,
M
M
1 2
where is the original particle volume fraction,
(6.10)
f m is the internal stress of fibers, and S
0
B
B
and M are
the Weibull parameters. The internal stresses of fibers required for the initiation of the imperfect interface were explicitly derived by Ju and Lee (2000) as 1
1 1 1 C1 I S1 A1 S1 I S1 A1 S1 : ε U : ε
(6.11)
By carrying out the lengthy algebra, the components of fourth-rank tensor U for the present model are explicitly given by
(1) Uijkl U IK ij kl U IJ(2) ik jl il jk
(6.12)
Chapter 6. Damage modeling of laminate composite structures
102
(1) where the second-rank tensors U IK and U IJ(2) can be expressed as
3
(1) (1) f (2) (1) f (1) U IK 2 EIK KK 2 EII(2)IKf (1) EIn nK , n 1
U IJ(2) 2EIJ(2)IJf (2)
(6.13)
in which
(1) EIK
2 f (2) 1 1 KK
IKf (1)
f (1) IK
1 2 IIf (2)
21 IKf (1) 1
IJf (2)
,
3
nKf (1) , n 1
2 f (2) EIJ(2) 1 1 IJ
1
(6.14)
(6.15)
2 4 IJf (2)
If1(1) 1 11f (1) 211f (2) 21f (1) 31f (1) f (1) 12f (1) 1 22f (1) 2 22f (2) 32f (1) I 2 f (1) 13f (1) 23f (1) 1 33f (1) 233f (2) I 3
IKf (1)
3 (1) (2) (2) (1) (1) (1) 2S IK KK 2S II IK S In nK , 4 1 0 n 1
IJf ( 2 )
1
If1(1) f (1) I 2 f (1) I 3
(6.16)
S IJ( 2) IJ ( 2 ) 2 1 0
(6.17)
(1) (1) where the subscript I = 1, 2, 3 and the components S IK , S IJ(2) , IK and IJ(2) in Eq. (6.17) are listed in
Chapter 5. In case of tensile loading, the averaged internal stresses of fibers can be obtained as
σ f m f 11 f 22 f 33 2 f 23 f 13 f 12 2
with
2
2
2
2
2
1/ 2
(6.18)
Chapter 6. Damage modeling of laminate composite structures
f 11 (1) (2) U11 2U11 f 22 U (1) 21 f 33 U31(1) f 23 0 f 13 0 0 f 12
(1) U12
103
(1) U13
0
0
(1) U 23 (1) (2) U 33 2U 33
0
0
0
0
0
0
(2) U 23
0
0
0
0
(2) U13
0
0
0
0
(2) 2U12 (1) U 32
(1) U 22
0 11 0 22 0 33 0 2 23 0 213 (2) 212 U12
(6.19)
In accordance with the continuum damage model for the nucleation of microvoids or microcracks in brittle materials (Karihaloo and Fu, 1989), the density of nucleated microcracks in the composite is modeled using the following isotropic scalar function (Liang et al., 2006)
v 0 , 5 th v 0 c1 1 a
a th c
2 a th ,
(6.20)
where v 0 is the initial density of microcracks. th ijth ijth is the threshold effective strain below which no nucleation occurs, c 1 and c 2 are material (nucleation) parameters depending on the specific shape and B
B
B
B
1/ 2
2 2 2 2 2 2 22 33 2 23 13 12 distribution of microcracks, and a ij ij 11
accumulated effective strain (Karihaloo and Fu, 1989).
is the current
Chapter 6. Damage modeling of laminate composite structures
104
6.4 Numerical simulations and experimental comparison In this section some comparisons with both numerical and experimental data available in literature are carried out to show the effectiveness of the proposed constitutive and damage model presented previous sections. The proposed constitutive and damage model for laminate composites are implemented into ABAQUS non-linear FE code using a user-subroutine UMAT.
6.4.1 Verification of the computer code The implemented computational model is validated by comparing the predicted stress-strain curve and experimental data under uniaxial tension with various fiber orientations. The stress-strain curve of
400
350
Stress [σ33] (MPa)
300
prediction-15º
experiment-15º
prediction-30º
experiment-30º
prediction-45º
experiment-45º
250
200
150
100
50
0 0
0.005
0.01
0.015
Strain [ε33]
Figure 6.2 The comparison between the present prediction and experimental data (Weeks and Sun, 1998) for overall uniaxial tensile responses with three fiber orientations of AS4/PEEK composite
Chapter 6. Damage modeling of laminate composite structures
105
AS4/PEEK composite test by Weeks and Sun (1998) is numerically predicted using implemented computational model. The elastic properties of the composites used in this simulation are (Weeks and Sun, 1998; Hassan and Batra, 2008): E m = 6.14 GPa, m = 0.356, E f = 214 GPa, f = 0.263 and = 0.6, B
B
B
B
where the subscript m and f represent the matrix and fibers, respectively. According to the experimentally obtained stress-strain curve (Weeks and Sun, 1998), the model parameters (see detail descriptions of Chapter 5) are estimated as: 1 2.0 107 and 1 3.0 107 , 2 = 2.0, 2 = 3.0; S 0 = 290 MPa, M = 0.8, B
B
c 1 = 0.25, c 2 = 1.1, v 0 = 0.0 and th = 0.005. B
B
B
B
The three-dimensional, eight-node, linear brick solid element (C3D8) in ABAQUS is used for modeling the laminated composites. Following Hassan and Batra (2008), four-ply composites of overall dimensions 216 mm × 15.9 mm × 2.54 mm, divided into a uniform 84 × 6 × 4 FE mesh, were loaded by pulling the opposite end faces. Figure 6.2 exhibits the comparison between present predicted uniaxial stress-strain responses and the experimentally obtained stress-strain curve (Weeks and Sun, 1998) with 0.7
0.6
Volume Fraction
0.5
perfect bonding 0.4
mild imperfect interface severe imperfect interface completely debonded
0.3
microcrack
0.2
0.1
0 0
0.005
0.01
0.015
Strain [ε33]
Figure 6.3 The predicted evolution of volume fraction of perfectly bonded fibers, fibers with mild imperfect interface, fibers with severe imperfect interface, completely debonded fibers and microcrack corresponding to 15º fiber orientation case in Figure 6.2
Chapter 6. Damage modeling of laminate composite structures
106
various fiber orientations (15°, 30° and 45°). Overall, good predictions are shown by the FE models compared with the experimental results. This confirms the ability of the proposed implemented computational model to predict the overall behavior of laminated composites. The predicted evolutions of volume fractions of fibers and microcrack corresponding to 15° fiber orientation case of Figure 6.2 are shown in Figure 6.3. Smooth and gradual transitions from the perfectly bonded fibers to various types of damaged fibers and evolution of microcracks occur within the present framework; as a result, nonlinear stress-strain response can be obtained during the damage evolution.
6.4.2 Experimental comparison of the response and damage behavior of a laminated composite under bending To further examine the proposed implemented computational model, we conduct comparison between proposed model and 12-ply carbon fiber reinforced epoxy matrix laminated beam, [0/±45/0/90/0] s , B
B
subjected to three-point bending test by Huang (2007). The beam had a span of 84 mm, a width of 15.1 mm and a thickness of 2.76 mm (Huang, 2007). Figure 6.4 illustrates the finite element discretization employed to simulate the bending response and damage evolution in laminate composite. The material properties of laminated composite are (Huang, 2007): E m = 3.685 GPa, m = 0.35, E f = 210 GPa, f = 0.2 and = B
B
B
B
0.44. According to the experimentally obtained load-displacement curve (Huang, 2007), the model parameters are estimated as: 1 2.0 107 and 1 3.0 107 , 2 = 2.0, 2 = 3.0; S 0 = 450 MPa, M = B
B
3.0, c 1 = 0.35, c 2 = 1.15, v 0 = 0.0 and th = 0.005. B
B
B
B
The predicted load-displacement curve is compared with experimental data (Huang, 2007) under three-point bending in Figure 6.5. It is worth noting that the prediction of the proposed implemented computational model for laminated composites agrees well with the experimental data. The predicted evolution of volume fractions of perfectly bonded fibers, various types of damaged fibers and microcracks versus displacement corresponding to various segments depicted in Figure 6.5 are exhibited in Figure 6.6. The damage evolutions of segment A are shown to be more noticeable compared with those of the other segment. Figure 6.7 shows the sequence of the deformed shape and von-Mises effective stress of laminated composite beam during three-point bending.
Chapter 6. Damage modeling of laminate composite structures
107
Segment D Segment C Segment B Segment A
Segment A
Segment E Segment F
Figure 6.4 The finite element discretization employed to simulate the bending response and damage evolution in laminated composite beam
Chapter 6. Damage modeling of laminate composite structures
108
0.4
0.35
0.3
Load (KN)
0.25
0.2
0.15 prediction experiment #1
0.1
experiment #2 experiment #3 experiment #4
0.05
0 0
1
2
3
4
5
displacement (mm)
Figure 6.5 The comparison of load-displacement curves between the present prediction and experimental data (Huang, 2007) under three-point bending
Chapter 6. Damage modeling of laminate composite structures
a
109
0.5
Segment A
0.45 0.4
perfect bonding mild imperfect interface severe imperfect interface completely debonded microcrack
volume fraction
0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0
1
2
3
4
5
displacement (mm)
b
0.5
Segment B
0.45 0.4
perfect bonding mild imperfect interface severe imperfect interface completely debonded microcrack
volume fraction
0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0
1
2
3
4
5
displacement (mm)
Figure 6.6 The predicted damage evolution versus displacement of laminated composite beam under three-point bending corresponding to Figure 6.5: (a) Segment A; (b) Segment B; (c) Segment C; (d) Segment D; (e) Segment E; (f) Segment F
Chapter 6. Damage modeling of laminate composite structures
110
0.5
c
Segment C
0.45 0.4
volume fraction
0.35 0.3 perfect bonding
0.25
mild imperfect interface 0.2 severe imperfect interface 0.15
completely debonded
0.1
microcrack
0.05 0 0
1
2
3
4
5
4
5
displacement (mm)
0.5
d
Segment D
0.45 0.4
volume fraction
0.35 0.3
perfect bonding mild imperfect interface
0.25
severe imperfect interface 0.2
completely debonded
0.15
microcrack
0.1 0.05 0 0
1
2
3
displacement (mm)
Figure 6.6 The predicted damage evolution versus displacement of laminated composite beam under three-point bending corresponding to Figure 5 (continued)
Chapter 6. Damage modeling of laminate composite structures
e
0.5
Segment E
0.45
perfect bonding mild imperfect interface
0.4
severe imperfect interface
0.35 volume fraction
111
completely debonded
0.3
microcrack
0.25 0.2 0.15 0.1 0.05 0 0
1
2
3
4
5
displacement (mm)
f
0.5
Segment F
0.45 0.4
volume fraction
0.35 0.3
perfect bonding mild imperfect interface
0.25
severe imperfect interface
0.2
completely debonded
0.15
microcrack
0.1 0.05 0 0
1
2
3
4
5
displacement (mm)
Figure 6.6 The predicted damage evolution versus displacement of laminated composite beam under three-point bending corresponding to Figure 5 (continued)
Chapter 6. Damage modeling of laminate composite structures
112
10/50
20/50
30/50
40/50
50/50 Figure 6.7 The sequence of the deformed shape and von-Mises effective stress of laminated composite beam during three-point bending
Chapter 6. Damage modeling of laminate composite structures
113
6.4.3 Experimental comparison of the response and damage behavior of a laminated composite under shear Another numerical study is also conducted to predict shear behavior of laminated composites. In this simulation, the Iosipescu (V-notch) shear test was selected, as one of the most efficient method, to determine the shear properties of laminated composites (Broughton et al., 1990; Bhatnagar et al., 1995; Khashaba, 2004). Figure 8 illustrates FE model of Iosipescu shear test in present simulations. The material properties of glass fiber reinforced epoxy matrix laminated composite are (Khashaba, 2004): E m = 3.2 GPa, B
B
m = 0.36, E f = 72.4 GPa, f = 0.22 and = 0.4267. The comparisons are conducted with the cross-ply B
B
laminates with different stacking sequences ([75/-15] 2s , [45/-45] 2s and [30/-60] 2s ). According to the B
B
B
B
B
B
experimentally obtained load-displacement curve (Khashaba, 2004), the model parameters are estimated as: 1 2.0 107 and 1 3.0 107 , 2 = 2.0, 2 = 3.0; S 0 = 1400 MPa, M = 3.0, c 1 = 0.35, c 2 = 1.1, B
B
B
B
B
B
v 0 = 0.0 and th = 0.0001. Figures 6.9 shows the comparisons of load-displacement curves between the present prediction and experimental data (Khashaba, 2004) under shear loading for cross-ply laminated composite. It is evident that computed results are close to those experimentally obtained data. Figures 6.10 – 6.12 illustrate the sequence of the deformed shape and von-Mises effective stress of cross-ply laminated composite under shear loading.
Figure 6.8 FE model of Iosipescu shear test
Chapter 6. Damage modeling of laminate composite structures
114
9
a
8
[75/-15] 2s
7
B
B
Load (KN)
6 5 4 3
prediction
2
experiment
1 0 0
1
2
3
4
5
displacement (mm)
10
b
9
[45/-45] 2s
8
B
B
Load (KN)
7 6 5 4 prediction 3 experiment
2 1 0 0
1
2
3
4
displacement (mm)
10 9
c
[30/-60] 2s
8
B
B
Load (KN)
7 6 5 4 prediction
3
experiment
2 1 0 0
1
2
3
4
displacement (mm)
Figure 6.9 The comparison of load-displacement curves between the present prediction and experimental data (Khashaba, 2004) under shear loading: (a) [75/-15] 2s laminated B
B
composite; (b) [45/-45] 2s laminated composite; (c) [30/-60] 2s laminated composite B
B
B
B
Chapter 6. Damage modeling of laminate composite structures
115
[75/-15] 2s B
B
10/40
20/40
30/40
40/40
Figure 6.10 The sequence of the deformed shape and von-Mises effective stress of laminated composite under shear loading in case of [75/-15] 2s B
B
Chapter 6. Damage modeling of laminate composite structures
116
[45/-45] 2s B
B
10/40
20/40
40/40
30/40
Figure 6.11 The sequence of the deformed shape and von-Mises effective stress of laminated composite under shear loading in case of [45/-45] 2s B
B
Chapter 6. Damage modeling of laminate composite structures
117
[30/-60] 2s B
B
10/40
20/40
40/40
30/40
Figure 6.12 The sequence of the deformed shape and von-Mises effective stress of laminated composite under shear loading in case of [30/-60] 2s B
B
Chapter 6. Damage modeling of laminate composite structures
118
6.5 Appendix. Components of C in Eq. (6.7) These components are
(1) (2) (1) (1) (2) (1) (2) C11 c4 C11 2C11 c 2 s 2 C13 C31 4 C13 s 4 C33 2C33
(6.21)
(1) (1) C12 c2C12 s 2C32
(6.22)
(1) (1) (2) (1) (2) (2) (1) C13 c4 C13 c 2 s 2 C11 2C11 C33 2C33 4 C13 s 4 C31
(6.23)
(1) (2) (1) (2) (1) (1) (2) (2) C15 cs c 2 C11 2C11 C13 2C13 s 2 C31 C33 2C33 2 C13
(6.24)
(1) (1) C21 c2C21 s 2C23
(6.25)
(1) C22 C22
(2) 2C22
(6.26)
s
(1) C21
(6.27)
(1) C21 )
(6.28)
C23 c
2
(1) C23
(1) C25 cs(C23
2
(1) (1) (2) (1) (2) (2) (1) C31 c4 C31 c 2 s 2 C11 2C11 C33 2C33 4 C13 s 4 C13
(6.29)
(1) (1) C32 c2C32 s 2C12
(6.30)
(1) (2) (1) (1) (2) (1) (2) C33 c4 C33 2C33 c 2 s 2 C13 C31 4 C13 s 4 C11 2C11
(6.31)
(1) (1) (2) (2) (1) (2) (1) (2) C35 cs c2 C31 C33 2C33 2C13 s 2 C11 2C11 C13 2 C13
(2) (2) C44 c2C23 s 2C12
(2) (2) C46 cs C23 C12
(6.33)
(6.34)
(1) (2) (1) (2) (1) (1) (2) (2) C51 cs c 2 C11 2C11 C31 2C13 s 2 C13 C33 2C33 2 C13
(1) (1) C52 cs C32 C12
(6.35) (6.36)
(2) (1) (2) (1) (1) (1) (2) (2) C55 c 4 s 4 C13 c 2 s 2 C11 2C11 C13 C31 C33 2C33 2C13
(1) (1) (2) (2) (1) (2) (1) (2) C53 cs c2 C13 C33 2C33 2C13 s 2 C11 2C11 C31 2 C13
(6.32)
(6.37) (6.38)
(6.39)
(2) (2) C66 c2C12 s 2C23
(6.40)
(2) (2) C64 cs C23 C12
in which c = cos and s = sin
119
CHAPTER 7 CONCLUSIONS
Elastic and elastoplastic multi-level damage models for composites with imperfect interface have been presented.
In Chapter 2, a micromechanical framework to predict the effective elastic behavior and imperfect interface evolution of particulate composites is presented. The Eshelby's tensor for an ellipsoidal inclusion with slightly weakened interface (Qu, 1993a, 1993b) is employed to model spherical particles having imperfect interfaces in the composites and is incorporated into the micromechanical framework. The effective elastic moduli of three-phase particulate composites are derived based on the Eshelby's micromechanics. A damage model is subsequently considered in accordance with the Weibull's probabilistic function to characterize the varying probability of evolution of imperfect interface between the inclusion and the matrix. The proposed micromechanical elastic damage model is applied to the uniaxial, biaxial and triaxial tensile loadings to predict the various stress-strain responses. Further, the present predictions are compared with other numerical and analytical predictions and available experimental data to verify and assess the proposed micromechanical framework. The successful incorporation of the weakened interface model by a spring layer of vanishing thickness (Qu, 1993a, 1993b) into governing micromechanical ensemble-volume averaged field equations (Ju and Chen, 1994a) yields computational frameworks that enable us to predict the effective elastic behavior and imperfect interface evolution of particulate composites, which can be portrayed as the first noninteracting, micromechanics-based approximation for imperfect interface modeling. It is observed from Figures 2.2 – 2.10 that the effect of imperfect interfaces on the effective stress-strain responses is more pronounced as the damage evolution becomes rapid. It is found from the parametric study on the compliance and Weibull parameters that higher and and lower S 0 , which is related the B
B
interfacial strength of composites, lead to faster evolution of imperfect interface. By contrast, if the and are low and the S 0 is high, then the effects of imperfect interface are not pronounced when B
B
compared with the perfect bonding model. The proposed micromechanical framework is assessed by comparing the present predictions with
Chapter 7. Conclusions
120
other numerical (Ju and Lee, 2001) and analytical (Kerner, 1956; Halpin and Kardos, 1976) predictions as well as experimental data (Zhou et al., 2004). The predicted stress-strain behavior of particulate composites featuring the current evolution in imperfect interface is observed to be in good qualitative agreement with the experimental data as shown in Figure 13. However, further comparisons between numerical predictions based on the proposed elastic damage model and other numerical simulations and experiments will be needed for the calibration of the model parameters ( , , S 0 , M). B
B
In Chapter 3, a multi-level elastic damage model based on a combination of a micromechanical formulation and a multi-level damage model has been presented to predict the effective elastic behavior and progressive imperfect interface in particulate composites. A four-level damage model based on the Weibull's probabilistic function is developed to describe the progressive imperfect interface in the composites. The Eshelby's tensor for a slightly weakened ellipsoidal inclusion (Qu, 1993a, 1993b) to model particles having mild or severe imperfect interface is incorporated into a micromechanics-based multi-level elastic model. The proposed micromechanical elastic damage model is applied to the uniaxial tension loading to predict the corresponding stress-strain responses. A parametric analysis is also carried out to address the influence of the compliance and Weibull parameters on the progressive imperfect interface in the composites. Finally, the present predictions are compared with experimental data (Sørensen, 1993; Geiger and Welch, 1997) to further illustrate the elastic damage behavior of the present framework and to verify the validity of the proposed multi-level elastic damage model. The stress-strain curves of the composites are linear at the early stage of loading since the volume fractions of damaged particles are low. As strains or stresses continue to increase, the volume fractions of damaged particles increase gradually, resulting in a nonlinear stress-strain response. The proposed multilevel elastic damage model naturally captures the gradual transition from previous damage level to next damage level and various types of damaged particles exist simultaneously within the proposed framework. If the strength at the particle-matrix interface is low (lower S 0 ), most particles are mild imperfect, severe B
B
imperfect or completely debonded in their interfaces even in early stage of loading. The influence of the Weibull parameter S 0 on the progression of imperfect interface and stress-strain behavior of the composites B
B
is quite significant. As the values of the compliance parameters 2 and 2 become higher, the effect of the severe imperfect interface on the stress-strain behavior of the composites is more pronounced. The predicted stress-strain behavior of particulate composites featuring the multi-level damage progression of imperfect interface is observed to be in good qualitative agreement with experimental data (Geiger and Welch, 1997).
Chapter 7. Conclusions
121
In Chapter 4, an elastoplastic multi-level damage model considering evolutionary imperfect interface to predict the effective elastoplastic behavior and multi-level damage evolution in particle reinforced ductile matrix composites (PRDMCs) has been presented. The Eshelby's tensor for an ellipsoidal inclusion with slightly weakened interface (Qu, 1993a, 1993b) is adopted to model particles having mild or severe imperfect interface, and a multi-level damage model (Lee and Pyo, in press) in accordance with the Weibull's probabilistic function is employed to describe a sequential, progressive imperfect interface in the composites. The proposed micromechanical elastoplastic damage model is applied to the uniaxial, biaxial and triaxial tension loadings to predict the corresponding elastoplastic stress-strain responses. A series of parametric analysis are carried out to investigate the influence of model parameters on the elastoplastic behavior and progression of imperfect interface in the composites. Furthermore, the present prediction is compared with available experimental data in the literature to verify the proposed elastoplastic multi-level damage model. The elastoplastic stress-strain responses corresponding to various phase composite states are bounded by the responses of the two-phase and five-phase composite states. The higher phase composite states are lower than those with the lower phase composite states. A smooth and gradual transition from the perfectly bonded interface to the perfectly debonded interface occurs within the present framework, resulting in a nonlinear stress-strain response within the elastic range. The influence of the Weibull parameters S 0 and M B
B
and the compliance parameters and on the elastoplastic response and progression of imperfect interface in the composites is shown to be significant. Most particles are mild imperfect, severe imperfect or completely debonded in their interfaces even in early stage of loading with the lower S 0 . As the values B
B
of the compliance parameters 2 and 2 become higher, the effect of the severe imperfect interface on the elastoplastic stress-strain behavior of the composites is more pronounced. The predicted elastoplastic stress-strain behavior of PRDMCs is shown to have a very good correlation with the experimental data (Papazian and Adler, 1990). It should be noted that particles having different levels of imperfect interface (different levels of homogeneously distributed interfacial damage), giving rise to isotropic behavior, under general loading conditions are a certainly modeling assumption made in this framework. However, as deformations proceed under a uniaxial or biaxial loading, the composites progressively become transversely isotropic after evolutionary partial interfacial debonding as stated in Ju and Lee (2001). This issue is beyond the scope of the present work; nevertheless, it should be subject of future research for more accurate prediction of particulate composites behavior. It is obvious that the probabilities of perfectly bonded particles transforming to particles with mild imperfect interface or particles with mild imperfect interface transforming to particles with severe
Chapter 7. Conclusions
122
imperfect interface are dependent upon the two Weibull parameters S 0 and M and the internal stress of B
B
those particles. For simplicity, the internal stress of perfectly bonded particles has been considered as the controlling factor of the Weibull statistical function for the initiation of various stages of imperfect interface of the multi-phase composite state in the present study. The above assumption would, however, tend to lead to a slight overestimation of the volume fraction of particles that have reached Levels 2 and 3.
In Chapter 5, a 3D micromechanics-based evolutionary damage model has been proposed to predict the effective elastic behavior of continuous, fiber-reinforced brittle matrix composites with microcracks and imperfect interfaces. The Eshelby's tensor for a circular cylindrical inclusion with slightly weakened interface (Qu, 1993a, 1993b) is adopted to model continuous fibers with imperfect interfaces. A multi-level damage modeling process in accordance with the Weibull's probabilistic function is incorporated into the micromechanical framework to describe the sequential evolution of imperfect interfaces in the composites. The nucleation of microcracks is simulated by employing the continuum damage model proposed by Karihaloo and Fu (1989). Numerical examples are solved to illustrate the potential of the proposed damage model and comparisons between the present prediction with available experimental data are conducted to verify the accuracy of the proposed damage model. The progressive evolution of imperfect interface has significant influence on the overall elastic damage behavior of continuous, fiber-reinforced brittle matrix composites. The stress-strain responses due to uniaxial tension in the transverse direction are lower than those due to uniaxial tension in the longitudinal direction. It is found from the parametric study on the Weibull and compliance parameters that lower S 0 and higher 2 and 2 , which is related the interfacial strength of composites, lead to faster B
B
evolution of imperfect interface. Furthermore, as the values of the nucleation parameters c 1 becomes B
B
higher and c 2 becomes lower, the effect of the penny-shape microcrack nucleation on the stress-strain B
B
behavior of the composites is more noticeable. In comparison with the model without nucleation of microcracks, the present prediction with nucleation of microcracks is observed to be in good qualitative agreement with the experimentally obtained stress-strain curve (Luo and Daniel, 2000). It is observed from the numerical simulations and parametric analysis that the influence of the nucleation parameters c 1 and c 2 , Weibull parameter S 0 , and compliance parameters 2 and 2 is quite B
B
B
B
B
B
remarkable. Therefore, experimental verifications on those parameters will be needed for more accurate prediction of the elastic damage behavior of continuous, fiber-reinforced brittle matrix composites.
In Chapter 6, The micromechanics based constitutive model (Lee and Pyo, submitted for publication) for laminated composites, the multi-level damage model (Lee and Pyo, in press) for sequential progression
Chapter 7. Conclusions
123
of imperfect interface, and the continuum damage model for the nucleation of microcracks (Karihallo and Fu, 1989) are implemented into ABAQUS non-linear FE code using a user-subroutine UMAT to solve boundary value problems and to numerically characterize the elastic behavior of laminated composites. To verify the implemented computational model, computational studies are undertaken by comparing the predicted stress-strain curve and experimental data available in literature under uniaxial tension with various fiber orientations. Furthermore, three-point bending test and Iosipescu (V-notch) shear test are also numerically conducted and compared with experiment data to predict overall behavior and damage evolution. Numerical validations for widely different loadings and geometries have shown that the model can obtain satisfactory solutions that agree well with results available in the literature. In particular, multi-level elastic damage model naturally captures the gradual transition from previous damage level to next damage level and various types of damaged fibers exist simultaneously within the proposed framework. However, the damage model needs to be extended to accommodate other damage mechanism (e.g., fiber breakage, delamination) to realistically characterize the damage mechanisms in laminated composites.
As an extension of the present study, a series of characterization tests on a particulate composite and fiber reinforced composites need to be conducted in the future for an accurate calibration of the model parameters (S 0 , M, 1 , 1 , 2 and 2 ). Data points will be sampled from the tests (e.g., stress-strain curves, B
B
load-deflection curves, etc.) and will be used to obtain an optimal set of the parameters by minimizing the sum of the squares of differences between the predicted and measured loads at all data points. Since the lower and upper limits of the parameters can be determined from a series of parametric analysis, this would be a nonlinear least-square constrained minimization problem. Alternatively, these parameters can be determined simultaneously with a trial-and-error procedure.
124
SUMMARY (IN KOREAN) 요 약 문
손상경계면을 갖는 복합재료의 탄성 및 탄소성 다단계 손상 모델
복합재료의 탄성 및 탄소성 거동과 손상 진전을 예측하기 위해 미세역학을 기반으로 한 다단계 손상 모델을 제안하였다. 먼저, 손상경계면을 갖는 입자강화 복합재료와 섬유강화 복 합재료의 손상을 모사하기 위해 미소한 손상경계면을 갖는 타원체형 강화입자 Eshelby 텐서 를 적용하였다. 변형이나 외부하중이 증가함에 따라 강화입자와 기지재료 사이의 손상경계면 이 발생한다. 이러한 손상경계면의 손상의 변화 정도를 실제적으로 모사하기 위하여 Weibull 확률함수를 기본으로 한 다단계 손상모델을 개발하였다. 다단계 손상 확률함수에서 손상의 진전은 강화입자의 내부응력에 의해 지배된다고 가정하였다. 연성의 기지재료를 갖는 복합재 료의 탄소성 거동을 예측하기 위하여 Ensemble-volume average와 first-order effects of eigenstrain 의 개념을 적용하여 유효 항복 기준을 유도하였다. 제안된 다단계 손상모델들을 적용하였다. 먼저, 취성의 기지재료를 갖는 입자강화 복합재 료, 연성의 기지재료를 갖는 입자강화 복합재료, 그리고 취성의 기지재료를 갖는 섬유강화 복 합재료를 1축, 2축, 3축 응력 하에서의 응력-변형률 관계를 조사하였다. 모델상수들이 거동에 미치는 영향을 조사하기 위해 변수검증을 실시하였다. 또한, 문헌조사를 통하여 얻은 기존 실 험결과와의 비교를 통하여 제안된 모델들의 검증을 실시하였다. 최종적으로 비선형 유한요소 프로그램인 ABAQUS에 적용하여 다양한 경계조건을 갖는 적층된 복합재료의 거동을 예측하 였다.
125
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ACKNOWLEDGEMENTS I rarely plan my research it plans me.
Max Perutz
I gratefully acknowledge the supervision and guidance of my advisor, Prof. Haeng-Ki Lee, and committee members, Prof. Jin-Keun Kim and Prof. Hyo-Gyoung Kwak. I am thankful to Prof. Taehyo Park of Hanyang University, who is the first person to instruct me on how to do academic study. I want to express my love and gratitude to my family and my love, Seong-Kum, for their support and care during the development of this work. I also wish to express my gratitude to my colleagues, Mr. Su-Yong Song, BongRae Kim, Sung-Kug Ha, Sang-Hyeon Cheong, Muhammad Afzal, Rudy Tawie, and Hyeong-Ki Kim at the KAIST for their support throughout this study.
감사의 글 어느덧 지난 2년을 돌아보는 시간이 되었습니다. 항상 훗날 뒤를 돌아보았을 때 후회하 는 일이 없어야겠다는 생각을 하지만 처음생각과 같은 결과를 얻기란 쉽지 않다는 것을 새삼 깨닫게 됩니다. 가장 먼저 2년 동안 부족한 제자에게 학문적 소양과 사회적인 가르침을 주신 이행기 교 수님께 깊은 감사의 마음을 전합니다. 많이 부족한 석사학위논문에 많은 가르침을 주신 김진 근 교수님과 곽효경 교수님께도 감사 드립니다. 또한 석사과정 동안 많은 지도를 아끼지 않 으신 카이스트 건설 및 환경공학과 모든 교수님들께도 감사의 말씀을 드립니다. 저에게 공부의 큰 뜻을 일깨워 주시고, 이끌어 주신 한양대 토목공학과 박대효 교수님께 머리 숙여 감사의 뜻을 전합니다. ‘훨훨 날아라’라고 말씀 해주시는 아버지, 항상 사랑으로 돌봐주시고 어머니, 저를 믿 고 후원해주는 형제들, 우리 가족 항상 감사하고 사랑합니다. 항상 바쁘다는 핑계로 자주 못 가도 투정하지 않고 나 하나만 믿고 따라준 성금양 사랑합니다. 지난 대학원 생활 동안 기쁜 일, 힘든 일, 슬픈 일, 즐거운 일을 항상 함께 해준 SAM Lab. 식구들에게 진심으로 고마움을 전합니다. 먼저 졸업하신 수용형(실험동에서 웃고 떠들 면서 실험했던 기억이 생생합니다. 얼렁 2세 만드셔요), 이 긴 학위논문이 시작할 수 있도록
도와준 랩장 봉래형, 연구실 궂은일 도맡아서 해준 연구실 살림꾼 성국이, 나이 어린 동생을 동기로 둔 죄로 고생 많았던 상현형(형 마음 다 압니다), 어디로 튈지 모르고 사고 잘치는 형기, 차세대 연구실 대표주자 은아, 많은걸 전수해 주지 못해 미안한 은화, 모든 식구들 여 러분 덕분에 이렇게 마무리를 웃으면서 할 수 있는 것 같습니다. I’m grateful for all the help given me by Brother Afzal & Rudy. I’ll pray for you & your family. 감사의 글에 제일 먼저 이름을 올리고 싶었지만 일에는 순서가 있는 것 같습니다. 광식 형! 형 덕분에 대전생활이 외롭지 않았습니다. 제게 큰 꿈을 그릴 수 있도록 도와주고, 형의 에너지를 직접 느낄 수 있어서 너무 좋았습니다. 고마운 나의 든든한 응원군 정식이, 열심히 군복무중인 종혁이, 열심히 공부중인 대환이, 병조, 세홍이, 항상 동네에서 받겨주던 종훈이, 종삼이, 승준이, 성민이, 원태, 영수, 영일이, 나의 소중한 친구들에게 감사의 말을 전하고 싶습니다. 석사 2년을 함께 해준 한밭회 식구들 준래형님, 범섭형, 강욱형, 영훈형, 준호형, 형주형, 영훈이, 승훈이, 효정이, 잊지 못할 추억이 많은 동기들 상민형, 주원형, 낙타문주, 성우, 인엽 이, 엠마, 윤정이, 지영이, 신애, 지현이, 저희 연구실 식구인 것 같은 호영형, 옥상에서 담배 같이 피던 창길이 여러분 덕분에 카이스트 생활이 외롭지 않았던 것 같습니다. 한양 토목쟁이들 근영이 아빠 인환형, 승환형, 수현형, 형섭형, 희찬이, 문규, 지원이, 현 진이, 승현형, 우헌형, 배수용이, 최석이, 현준, 재홍, 정훈, 류상, 무철, 병민, 수진, 준우 다들 토목계에서 한자리씩 하실 거라 믿습니다. 파이팅 입니다. 왕십리 술집에 받겨주는 마음씨 좋은 동아리 동기들 의태, 현태, 용민, 성철, 태한, 현정, 연희, 관희, 정혁, 현진, 희창, 호형 다들 바쁠 때라 자주 연락도 못하고 미안하구나 위에 이름이 빠졌다고 서운해 하거나 미워하지 말아주셔요. 저의 불찰이니 너그러운 마 음으로 이해해 주시고, 제가 이 자리에 있게 해주신 모든 분들 감사합니다. 사랑합니다.
Friendship is the inexpressible comfort of feeling safe with a person, having neither to weigh thoughts nor measure words. George Eliot
CURRICULUM VITAE Name
: Pyo, Suk-Hoon
Date of Birth
: February 18, 1982
Place of Birth
: Gochang, Korea
EDUCATION M.S.
Department of Civil and Environmental Engineering Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea March 2006-February 2008
B.S.
Department of Civil Engineering Hanyang University, Seoul, Korea March 2000-February 2006
EXPERIENCE Graduate Research Assistant
March 2006 to December 2007
Department of Civil and Environmental Engineering, KAIST, Korea
LICENSE Engineer
Civil Engineering, Human Resources Development Service of Korea (License No. 05201070585T)
Engineer
Information Processing, Human Resources Development Service of Korea (License No. 052020219630)
LIST OF PUBLICATIONS Thesis 1.
Pyo, Suk-Hoon, Elastic and Elastoplastic Multi-Level Damage Models for Composites with Imperfect Interface, Master’s Thesis, Department of Civil and Environmental Engineering, Korea Advanced Institute of Science and Technology, Daejeon, Korea, February 2008.
Journal Papers 1.
Lee, H. K. and Pyo, S. H., “Micromechanics-based elastic damage modeling of particulate composites with weakened interfaces,” International Journal of Solids and Structures, Vol. 44, Issue 25-26, pp. 8390-8406, 2007.
2.
Lee, H. K. and Pyo, S. H., “Multi-level modeling of effective elastic behavior and progressive weakened interface in particulate composites,” Composites Science and Technology, in press.
3.
Lee, H. K. and Pyo, S. H., “An elastoplastic multi-level damage model for ductile matrix composites considering evolutionary weakened interface,” International Journal of Solids and Structures, in press.
4.
Lee, H. K. and Pyo, S. H., “A micromechanics-based evolutionary damage model for continuous, fiber-reinforced brittle matrix composites with microcracks and imperfect interfaces,” Composites Science and Technology, submitted for possible publication
5.
Lee, H. K., Pyo, S. H., Kim, B. R., “On joint strength, peel stress and failure modes in adhesively bonded double-strap and supported single-lap GFRP joints,” Composite Structures, submitted for possible publication
6.
Lee, H. K., Pyo, S. H., Kim, H. K., “A Micromechanics-based Elastic Constitutive Model for ParticleReinforced Composites Containing Weakened Interfaces and Microcracks,” Journal of the Computational Structural Engineering Institute of Korea, in press (in Korean).
Conference Papers 1.
Lee, H. K., Pyo, S. H., Kim, B. R., Cho, J. R., “Experimental study on mechanical behavior of adhesively bonded GFRP joints,” Conference on Korean Society of Composite Mechanics, Kumi, Korea, pp. 269-272, November 23, 2006 (in Korean).
2.
Lee, H. K. and Pyo, S. H., “A Micromechanics-based Elastic Model for Particle-Reinforced Composites Containing Slightly Weakened Interfaces,” Proceedings of COSEIK Annual Conference 2007, Seoul, Korea, pp. 441-444, April 12-13, 2007 (in Korean).
3.
Lee, H. K., Pyo, S. H., Kim, B. R., “Investigation on joint strength of adhesively bonded, supported single-lap GFRP joints,” Conference on Korean Society of Composite Mechanics, Seoul, Korea, pp. 45-48, April 28, 2007 (in Korean).
4.
Lee, H. K., Pyo, S. H., Kim, B. R., “Investigation on peel stresses of adhesively bonded double-lap and supported single-lap GFRP joints,” Third International Conference on Structural Engineering, Mechanics and Computation, Cape Town, South African, September 10-12, 2007.
5.
Lee, H. K., and Pyo, S. H., “A Numerical Approach to Multi-Level, Elastic Modeling of Weakened Particle-Reinforced Brittle Composites,” Proceedings of 20th KKCNN Symposium on Civil Engineering, Jeju, Korea, pp. 283-286, October 4-5, 2007.
6.
Lee, H. K., and Pyo, S. H., “Micromechanics-based Elastoplastic Multi-Level Damage Modeling of Ductile Polymeric Composites with Weakened Interface,” Proceedings of the 6th Korea-Japan Joint Symposium on Composite Materials, Pohang, Korea, pp. 115-116, November 1, 2007.
7.
Lee, H. K., and Pyo, S. H., “Study on Micromechanics-based Elastic and Damage Model for fiberreinforced composites,” 14th Guided Weapon Conference 2007, Daejeon, Korea, pp. 644-647 (in CDROM), November 1, 2007 (in Korean).
