HOMOGENIZATION BASED CONTINUUM DAMAGE MODEL FOR COMPOSITES. Shinu Baby

Transcription

1 HOMOGENIZATION BASED CONTINUUM DAMAGE MODEL FOR COMPOSITES by Shinu Baby A thesis submitted to The Johns Hopkins University in conformity with the requirements for the degree of Master of Science. Baltimore, Maryland August, 2015 c Shinu Baby 2015 All rights reserved

2 Abstract Damage in composite material is inherently a multi-scale phenomena, coupling damage initiation and propagation at different length scales. Most damage models do not account for the effect of microstructure and different damage mechanisms at micro-scale. Models which account for micro-structure details are computationally expensive and hence cannot be used for large and realistic composite structures. The objective of this research is to develop a homogenization based, computationally efficient, continuum damage model (HCDM) which account for the micro-scale failure mechanisms. A robust micro-mechanical model with explicit representation of the damage mechanisms is developed. The fiber-matrix debonding at the interface is simulated using PPR cohesive zone model. Homogenized response from micromechanics model is used to develop the HCDM model. A damage evolution surface and damage evolution parameters are proposed. The HCDM model is expressed in principal damage coordinate system (PDCS), which enables the model to account for the non-proportional loading histories. This model introduces a fourth order damage tensor which characterizes the damage evolution in composite. This fourth order ii

3 ABSTRACT tensor is calibrated in terms of the principal values of the damage tensor and the dissipated damage work. Once these parameters are calibrated, composite structures having various micro-mechanical details can be analyzed without actually performing a micro-mechanical analysis. This makes this model computationally very efficient. The proposed model is validated by comparing the homogenized micro-mechanical results with the HCDM model results. The obtained HCDM model is implemented as a user subroutine UMAT in the commercially available FEM package ABAQUS. iii

4 Acknowledgments I express my sincere thanks and gratitude to Professor Somnath Ghosh for his guidance, encouragement and financial support through out my master s thesis. His supervision, patience and constructive criticism were really helpful during my research in Johns Hopkins University. I am confident that my experience as a researcher in the Computational Mechanics Research Laboratory (CMRL) will be an asset for me in all my future endeavors. I am also grateful to Dr. Dhirendra Kubair for the many productive discussions and helps throughout the course of my research. This work was sponsored by Army Research Lab through the CMEDE program and I gratefully acknowledge the sponsorship. I would like to thank my parents, brother and friends for their continued support and encouragement. I would also like to thank my fellow lab mates Coleman Alleman, Zhiye Li and Xiaofan Zhang for their continued support. iv

5 Dedication This thesis is dedicated to my parents, teachers and friends. v

6 Contents Abstract ii Acknowledgments iv List of Tables viii List of Figures ix 1 Introduction Thesis overview Anisotropic continuum damage mechanics Micromechanical RVE model Periodic Boundary Condition Cohesive zone model for interfacial debonding vi

7 CONTENTS PPR potential-based cohesive model for interfacial debonding Implementation of PPR Cohesive zone model in micromechanical model Micromechanical homogenization and stiffness evaluation Importance of Principal Damage Coordinate System HCDM Model formulation Damage evolution equations for the HCDM model Calibration and parametric representation of damage parameter P ijkl Implementation of the HCDM Model in ABAQUS Validation of the HCDM model Analysis of a composite structure with HCDM model Summary Future work Bibliography 50 Vita 55 vii

8 List of Tables viii

9 List of Figures 1.1 Micrographs of graphite-epoxy fiber-reinforced polymer matrix composite Composite failure by (a)matrix cracking, (b)fiber breakage and (c)fibermatrix debonding Traction separation law for PPR cohesive zone model D Cohesive Zone Element implementation in ABAQUS using the user subroutine UEL Homogenized micro-mechanics stress-strain response for uniaxial tensile loading Homogenized stiffness degradation for uniaxial tensile loading Orientation of Global coordinate system (x-y) and PDCS (x -y ) (a) when no rotation for loading e 11 0 and (b) when there is rotation for loading e 11 0 and e Angle of rotation vs effective strain for proportional and Non-proportional loading Variation of P ijkl d (a)unidirectional composite with rectangular fiber arrangement,(b) RVE mode for the composite Comparison of macroscopic stress-strain behavior obtained using HCDM and homogenized micromechanics (HMM) for e 11 loading Comparison of macroscopic stress-strain behavior obtained using HCDM and homogenized micromechanics (HMM) for e 22 loading Comparison of macroscopic stress-strain behavior obtained using HCDM and homogenized micromechanics (HMM) for biaxial loading ( e 11 and e 33 ) Comparison of macroscopic stress-strain behavior obtained using HCDM and homogenized micromechanics (HMM) for triaxial loading ( e 11, e 22 and e 33 ) Comparison of macroscopic stress-strain behavior obtained using HCDM and homogenized micromechanics (HMM) for proportional loading.. 41 ix

10 LIST OF FIGURES 2.14 Comparison of macroscopic stress-strain behavior obtained using HCDM and homogenized micromechanics (HMM) for non-proportional loading Composite plate subjected to a pressure wave (a)position dependent pressure loading, (b)amplitude function for the pressure application Quarter symmetry of the pressure plate with fiber orientation Pressure variation along the composite surface for a given time t 0 variation along the surface of of the composite plate Evolution of damage with time Dissipation of damage work with time Von-Mises stress at time (a) 0 (b) (c) (d) (e) (f) seconds Damage variable D 11 at time (a) 0 (b) (c) (d) (e) (f) seconds x

11 Chapter 1 Introduction Composite materials are made by combining two materials with distinct properties to create a unique and superior material. The history of composites start from ancient times where straw and mud are combined to make brick for construction applications. The straw gives strength and mud acts as a binder which holds the straw together. Composite science and technology has made tremendous amount of advancement in the last century. Researchers has developed different analysis and manufacturing techniques to create superior quality composite materials. Nowadays composite materials have a wide variety of applications and are replacing the conventional materials in automobile, aerospace and other high performance applications. Fiber reinforced composites are widely used in aerospace applications because of their high strength to weight ratio. The new Boeing 787 used carbon fiber composite material as the primary material in the construction of fuselage, airframe and wings. The 1

12 CHAPTER 1. INTRODUCTION ability of a composite to disperse stress waves and absorb the impact energy makes them ideal candidate for defense applications where composite material are used to protect army vehicles from explosion. Since composite materials are made of combining different materials, we can optimize the composite properties with careful selection and arrangement of individual components. For example, fiber reinforced composites are made of matrix (Epoxy, ceramic, polyester etc.) and reinforced with fiber (carbon, glass, boron, aramid etc.). Figure 1.1 show the micrograph of a fiber reinforced polymer matrix composite. The heterogeneities at the micro-scale affect the failure properties of a composites. There is enough experimental evidence to show that the damage in composite initiates at the micro-scale. Three dominant modes of failure mechanisms observed in composite materials are matrix cracking, fiber breakage and fiber-matrix debonding at the interface as shown in figure 1.2. These micro-scale failure mechanisms cause micro-crack and as a result crack bridging may occur, resulting in a dominant crack that causes the composite structure failure. Figure 1.1: Micrographs of graphite-epoxy fiber-reinforced polymer matrix composite 2

13 CHAPTER 1. INTRODUCTION Figure 1.2: Composite failure by (a)matrix cracking, (b)fiber breakage and (c)fibermatrix debonding Micro-scale failure mechanisms are sensitive to local morphology like volume fraction, size, shape, fiber orientation etc. To predict the composite failure accurately, these micro-scale details should be incorporated into the model. But modeling a composite structure with explicit representation of these micro-level details is computationally prohibitive. Most of the damage laws for composite such as Tsai-Hill, Tsai-Wu, Hashin etc., are developed from macroscopic observations, hence do not account for micro-scale details. Micro-mechanics based damage laws proposed by Voyiadjis et.al [1] do not account for complex damage paths such as non-proportional 3

14 CHAPTER 1. INTRODUCTION loadings. Two-scale model proposed by Fish et. al [2] accounts for micro-scale details but requires concurrent evaluations at the macro and micro scales, which make the model computationally very expensive. Structures with heterogeneous microstructures such as voids or inclusions are analyzed using effective properties obtained from homogenization at the micro-scale. Various homogenization methods such as Reuss and Voigt [3] based on the assumption of constant stress and strain, the concentric cylinder assemblage model, asymptotic homogenization theory, the Mori-Tanaka mean stress theory [4] etc. are proposed for obtaining the effective material properties. The basic assumption of all these models is the periodic representative volume element (RVE) in the microstructure and uniformity of the macroscopic field variable. Microstructure periodicity and uniformity of the macroscopic field variables may not be appropriate if the damage is evolving locally. This shortcoming is handled by hierarchical multi-scale damage model which differentiates between regions that require different resolutions. Such models require concurrent analysis of the macroscopic and microscopic damage model. Some of the popular hierarchical multi-scale models are proposed by Fish [2], Oden and Zohdi [5], Pagano and Rybicki, Oden and Vemaganti [6] etc. Most of these models are computationally very expensive and are limited to linear elastic analyses. Most of the phenomenological damage models for composite materials are developed from the macroscopic experimental observations. Hence these models do not account for the microstructure variability and micro-scale damage mechanisms and 4

15 CHAPTER 1. INTRODUCTION hence, applications of these model are limited in scope. Micro-mechanical models that solve boundary value problem of the representative volume element (RVE) predict the micro-scale damage mechanisms like fiber breakage, matrix cracking and fiber-matrix debonding accurately. Some examples of such model are those proposed by Budiansky and O Connell [7],Benveniste [4] and Nemat-Nasser [8]. RVE model with cohesive zone model to simulate fiber - matrix debonding is proposed by [9]. These analyses give complete insight into the micro-scale damage evolution. For practical engineering applications, it is impossible to perform detailed micro-mechanical analysis of the composite structure. The most convenient way is to develop a macroscopic constitutive law based on the homogenization of the micro-mechanical model. Continuum damage models (CDM) provide a constitutive frame work to accommodate damage induced stiffness degradation. CDM defines damage tensors of varying order (eg. scalar, vector, second order tensor) to account for the damage evolution and subsequent stiffness degradation [10]. Phenomenological CDM models identify the damage parameters from macroscopic experiments, hence do not account for micro-scale damage mechanisms. Micro-mechanics based CDM model [11] predicts the evolving damage parameters by conducting detailed micromechanical analysis of the representative volume element (RVE) with subsequent homogenization. Micromechanical based models make explicit connection between micro-scale damage mechanisms and macro-scale damage evolution. Most of the micromechanical based models do not account for damage evolution and the effect of loading histories. So 5

16 CHAPTER 1. INTRODUCTION these models incur significant error for non-proportional loading histories. Models proposed by Fish et al [2], Chaboche et al [12] and Massart et al [13] overcome this difficulty by performing concurrent macroscopic and microscopic analysis at each load step. This approach is computationally expensive since micromechanical analysis of the RVE is needed for each load step at every integration points in the elements of macroscopic structure. These shortcomings are overcome by the approach of Ghosh and coworkers [14][15] who developed a homogenization based anisotropic continuum damage model (HCDM) for composite undergoing interfacial debonding. A detailed micromechanical analysis of the representative volume element (RVE) is performed for a number of load cases. The homogenized results of this micromechanical analysis are used to calibrate the damage parameters for the HCDM model. The calibrated damage parameters account for the anisotropic damage evolution of the composite at the microscopic level. Once these damage parameters are calibrated for HCDM model, we can use this to analyze the composite damage at the macroscopic level. So HCDM model avoids micromechanical analysis of the RVE at every integration point for the macroscopic analysis. This makes HCDM model computationally efficient. HCDM model proposed by [16] does not account for effect of path dependency on the damage parameters. So this HCDM model cannot predict damage evolution accurately for non-proportional loading histories. Moreover this model requires the evolution and storage of the damage parameters at discrete points in strain space. 6

17 CHAPTER 1. INTRODUCTION Evolution and storage of these damage parameters for 3D HCDM model requires a large number of micromechanical RVE analysis followed by homogenization. This makes this model computationally expensive. Jain and Ghosh [17] have overcome these difficulties by modifying the HCDM formulation. The HCDM model uses the continuously evolving principal damage coordinate (PDCS) as the reference axis. This PDCS representation helps to account for damage induced anisotropy and predicts accurate results for non-proportional loading cases. The importance of PDCS is explained in detail in [17] and in section 2.4. This HCDM model introduces a functional form for the fourth order HCDM damage parameter in terms of the invariants of macroscopic strain components. Parametric representation of the damage parameters avoids the time consuming strain space interpolation in [14], hence enhances the computation efficiency significantly. In [17] the damage parameter P ijkl is defined in terms of the invariants of the strain tensor. This representation of the P ijkl makes the damage parameters load dependent which is not physically true. The damage parameters should be load independent and should evolve automatically as the damage evolves. Moreover this kind of P ijkl representation contradicts the fact that there can be elastic strain increment with no damage or zero damage tensor D ij. So representing P ijkl in terms of the strain invariants is not appropriate. The proposed model overcomes these difficulties by representing the damage parameters P ijkl in terms of eigen values of the damage tensor D ij and the dissipated 7

18 CHAPTER 1. INTRODUCTION damage work W d. Since P ijkl is a measure of the stiffness degradation, representation of the P ijkl in terms of damage tensor is physically more meaningful. In this model the P ijkl evolves with the damage in the composite making the HCDM model physically more accurate. 1.1 Thesis overview This thesis is organized as follows. Chapter 2 start with the concept of anisotropic continuum damage mechanics. Detailed derivation of anisotropic continuum damage mechanics (CDM) model with second order tensor is explained in this section. Section 2.2 explains the development of micromechanical representative volume element (RVE) model for the composite. Details about implementation of periodic boundary condition and fiber-matrix debonding using the PPR cohesive zone model is also explained in this section. Section 2.3 describes the homogenization method and section 2.4 explains the importance of principal damage coordinate system. Section 2.5 discusses the formulation and implementation of the HCDM model. Section 2.6 explains the numerical analysis and validation of the HCDM models for different loading histories. Practical application of the HCDM model is demonstrated for a composite plate subjected to a pressure wave in section 2.7 d. The thesis ends with the summary of the research findings and possible future work in section

19 Chapter 2 HOMOGENIZATION BASED CONTINUUM DAMAGE MODEL Experimental studies have proved that the failure of a composite is sensitive to microstructure variations like fiber spacing, fiber size, shape, volume fraction and dispersion etc. So the failure of composite materials is a multi-scale phenomena with damage initiation and evolution at different length scale. Most of the damage models do not account for this inherent connection between microscopic and macroscopic failure mechanics. Modeling a composite structure with explicit representation of the microstructure is computationally impossible. Usually composite structures are analyzed using the effective stiffness properties obtained from the homogenization 9

20 principles at the micro-scale which results in homogenization of the Representative Volume Element (RVE) of the composite structure. Homogenization model assumes periodic arrangement of the fiber in composite and also uniform macroscopic field variables. These assumptions allow the use of periodic boundary condition on RVE. Most of the homogenized models in literature are computationally expensive which make it impractical for using them for the analysis of realistic scale composite structure.the current research goal is to develop a computationally efficient damage model for composites accounting for microstructure failure mechanisms. This chapter develops a three dimensional homogenization based continuum damage model (HCDM) for the fiber reinforced composite with micromechanical damage. The proposed HCDM model forms a strong connection between microscopic and macroscopic damage mechanisms. The first step in developing the HCDM model is modeling micromechanical model or representative volume element (RVE) for the composite with the explicit representation of microscopic damage mechanisms like fiber breakage, matrix cracking and fiber-matrix debonding. The next step is to evaluate the homogenized responses of the RVE model. The homogenized micromechanical model (HMM) response is used to understand the damage evolution at the micro-scale. HMM response shows non uniform rate of stiffness degradation and continuously evolving principle damage axes. This kind of stiffness degradation changes the material symmetry of the composite from orthotropic to anisotropic[9]. The proposed HCDM model uses the continuously evolving Principal damage coordinate 10

21 system (PDCS) as the reference to account for anisotropic damage evolution and non-proportional loading histories. Once the complete information about the damage evolution at the micro-scale is explored by the micromechanical analysis, we define a damage constitutive equation at the macro-scale. Since this macroscopic law is developed from the micromechanical analysis, this model connect the micro-scale and macro-scale failure mechanisms. We use parametric homogenization to connect micro-scale and macro-scale. The proposed HCDM model is validated by comparing the HMM response and the HCDM response. HCDM model is incorporated into ABAQUS as a user subroutine UMAT, which allows the analysis of realistic scale composite structure. 2.1 Anisotropic continuum damage mechanics In continuum damage mechanics (CDM), the macroscopic damage model incorporates internal variables that evolve with the microscopic damage mechanisms. These internal variables can be a scalar, second order tensor or a fourth order tensor, depending on the application. Capabilities and limitation of each models are explained in detail in [9]. Continuum damage mechanics models (CDM) proposed by Kachanov (Kachanov,1987) introduce a fictitious stress Σ(= Σ ij e i e j ) acting on an effective area Ã, caused by 11

22 the reduction of the original resisting area A due to the presence of micro-cracks and stress concentration in the vicinity of cracks. The effective stress Σ is related to the Cauchy stress Σ(= Σ ij e i e j ) through a fourth order damage effect tensor M ijkl [9], expressed as: Σ ij = M ijkl (D)Σ kl (2.1) where D is the damage tensor. D can be zeroth, second or fourth order tensor depends on the CDM model employed. Different hypotheses are proposed by different authors to calculate the damage effective tensor M ijkl. Equivalent strain hypothesis proposed by Lemaitre and Chabhoche [12] assumes same strain state between the fictitous stress σ ij applied to the undamaged material and the actual stress σ ij applied to the damaged material. Hypothesis of strain equivalence leads to non-symmetric stiffness matrix [18]. Hypothesis of equivalent elastic energy proposed by Cordebois and Sidoroff [19]is used to evaluate M ijkl and to establish a relation between undamaged stiffness E ijkl and Eijkl 0. Equivalent elastic energy [20] hypothesis assumes that the elastic complimentary energy W C (Σ, D) in a damaged material with the actual stress Σ ij is equal to the the elastic complimentary energy W C ( Σ, 0) in a hypothetical undamaged material with a fictitious effective stress Σ ij W C (Σ, D) = 1 2 (E ijkl(d)) 1 Σ ij Σ kl = W C ( Σ, 0) = 1 2 (Eo ijkl) 1 Σij Σkl (2.2) From equations (2.1) and (2.2), the relation between the damaged stiffness E ijkl and 12

23 the undamaged stiffnesses E 0 ijkl is established in terms of damage effective tensor M ijkl ] as: E ijkl = (M ijpq ) 1 E o pqrs(m klrs ) T (2.3) With proper selection of the order of damage tensor D ij and the function of M ijkl, equation (2.3) can be used to evaluate the CDM model from the homogenized results of the micromechanics. Scalar damage model assumes uniform damage development in all directions. Homogenized response of the micromechanical analysis in section 2.3 shows non-uniform rate of stiffness degradation with continuously evolving damage. So scalar damage model is insufficient to represent this kind of stiffness degradation. Orthotropic damage using second order damage tensor D ij is proposed by Cordebois and Sidoroff [21]and used to describe the non uniform rate of stiffness degradation. Park and Voyajidis [22] has used second order damage tensor to describe the damage evolution in composite. Such models formulate the damage effective tensor M ijkl in terms of the symmetric tensor D ij as M ijkl = (δ ik D ik ) 1 δ jl (2.4) Substitution of equation (2.4) in (2.1) make the effective stress tensor unsymmetric. Voyajodis and Kattan [1] has discussed symmetrization technique. The symmetriza- 13

24 tion technique suggested by [1] in global cordinate system is Σ ij = Σ ik (δ kj D kj ) 1 + (δ il D il ) 1 Σlj 2 (2.5) 2.2 Micromechanical RVE model Micromechanical analysis of the RVE is essential for the development of the HCDM model. Homogenized response of the micromechanical RVE model becomes the building block for this HCDM model. So modeling micromechanical RVE model with accurate representation of the micro-scale details is essential. The main aspect of the micromechanical RVE model is explained in subsequent subsections Periodic Boundary Condition In this research we assume periodic microstructure for the composite. Periodic boundary condition is enforced on opposite faces of the RVE as constrained equation. Any given macroscopic or average strain e ij applied on the RVE boundary can be decomposed into a periodic part ũ i and a macroscopic part as u i = e ij x j + ũ i (2.6) 14

25 Consider two nodes p1 and p2 on two opposite faces of the RVE. Using equation (2.6), we can relate the total displacement between these two nodes as (u i ) p2 (u i ) p1 = e ij x j (2.7) where x j is the relative coordinate of nodes on opposite periodic faces. The periodic boundary condition is implemented in ABAQUS as a constraint equation. Macroscopic strain are applied on the master nodes M1, M2 and M3 on the orthogonal faces and also by fixing the corner nodes as explained in [10] Cohesive zone model for interfacial debonding In this research a 3D micromechanical model is developed with the explicit representation of the fiber matrix debonding. Fiber matrix debonding is simulated using a potential based cohesive zone model known as Park-Paulino-Roesler (PPR) model [23]. Cohesive zone model (CZM) can be potential based or non-potential based. Non-potential based CZMs are easy to develop but they do not account for all the possible crack opening paths. Hence simulating crack growth using non-potential based CZMs are limited in scope. Needleman [24] proposed a polynomial potential to describe mode I fracture. Tvergaard [25] extended Needleman s potential by defining an effective displacement to account for mixed mode fracture. Needleman [25] also 15

26 proposed a CSM model for mixed mode fracture where exponential functions describe the normal and tangential traction-displacement relationship. Ortiz and coworkers [26] have developed irreversible cohesive laws for the unloading path after the interfacial softening. Park [27] has developed a unified potential-based CZM, known as PPR cohesive zone model, that considers high mode-mixity condition. PPR model can account for different fracture energies(φ n, φ t ), cohesive strength (σ max, τ max ) and different softening behaviors. A complete derivation of the PPR model can be found in [27]. An overview of the PPR CZM is given in section Figure 2.1: Traction separation law for PPR cohesive zone model 16

27 2.2.3 PPR potential-based cohesive model for interfacial debonding The traction-separation law for the PPR cohesive zone model is shown in figure 2.1. This traction-separation law is described by the following fracture boundary conditions. 1. Complete normal failure occurs when normal or tangential separation ( n, t ) reaches a critical length (δ n, δ t ) T n ( n, t ), if n < δ n and t < δ t T n = (2.8) 0, if n δ n or t δ t 2. Complete tangential failure occurs when normal or tangential separation ( n, t ) reaches a critical length (δ n, δ t ) T t ( n, t ), if n < δ n and t < δ t T t = (2.9) 0, if n δ n or t δ t 3. Area under the traction separation curve corresponds to the fracture energies. 17

28 Mode I (φ n ) and Mode II (φ t ) fracture energies are given by φ n = δ n T n ( n, 0)d n 0 δ t φ t = T t ( t, 0)d t (2.10) 0 4. Normal and tangential traction reaches maximum at a critical crack opening displacement (δ nc, δ tc ) T n n = 0 n=δnc T t t = 0 t=δtc (2.11) 5. Maximum traction corresponds to cohesive strength (σ max, τ max ) T n (δ nc, 0) = σ max T t (δ tc, 0) = τ max (2.12) For mixed-mode fracture, the normal and tangential traction separation law given by the PPR model [27] is T n ( n, t ) = Γ n δ n [ m ( 1 ) α ( n m δ n α + n δ n ( [Γ t 1 t δ t ) m 1 α ( 1 n ) β ( n β + t δ t ) α 1 ( m δ n α + ) ] m n δ n ) n + φ t φ n ] (2.13) 18

29 T t ( n, t ) = Γ n δ n [ n ( 1 ) β ( t n δ t β + t [Γ n ( 1 n δ n δ t ) n 1 ( β 1 ) β 1 ( t n β + ) ] n t ) α ( m α + n δ n δ t ) m ] t + φ n φ t t δ t (2.14) as The energy constants Γ n and Γ t are related to the mode I and II fracture energies Γ n = ( φ n ) φn φt /(φn φt) ( α m ) m, Γt = ( φ t ) φn φt /(φn φt) ( β n ) n, (2.15) The non-dimensional constants m and n in equation 2.13 are evaluated by satisfying the boundary condition 4. The initial slope indicators(λ n,λ t ) are defined as the ratio of the critical crack opening width to the final crack opening width as in equation Initial slope indicators control the elastic behavior, thus a small value of λ n,λ t decreases the artificial elastic deformation. m = α(α 1)λ2 n (1 αλ 2 n), n = β(β 1)λ2 t (1 βλ 2 t ), (2.16) λ n = δ nc δ n, λ t = δ nt δ t (2.17) Final crack opening length δ n,δ t are obtained by solving the boundary condition 3 and 5 in equation (2.10) 2.10 and

30 φ n ) m 1 ( α ) ( α δ n = αλ n (1 λ n ) α 1 σ max m + 1 m λ n + 1 δ t = φ ( ) ( ) n 1 (2.18) t β β βλ t (1 λ t ) β 1 τ max n + 1 n λ t + 1 For a positive crack opening displacement the traction increases to a maximum value coresponding to σ max, τ max as shown in figure 2.1. This region is called hardening region. After δ nc, δ tc traction reduces with displacement and reaches zero at δ n, δ t. This is called softening region. Unloading in the hardening zone have the same slope as the initial slope specified by λ n, λ t. Unloading in softening region traces a different path with reduced stiffness. Details about the PPR model formaulation can be found in [27] Implementation of PPR Cohesive zone model in micromechanical model Cohesive zone model is implemented in commercial finite element software ABAQUS using the user defined element (UEL) subroutine. Interface elements are made up of two eight-noded quadrilateral surfaces that are compatible with the standard twentynode brick elements as shown in figure 2.2. The corresponding cohesive interface elements have 16 nodes with a quadratic displacement interpolation, which leads to a total of 48 degrees of freedom per element. Integration in each element is con- 20

31 ducted by Gaussian quadrature using nine integration points. The cohesive interface elements are compatible with the 20 noded quadratic brick elements that are used to model the fiber and matrix phases. Details about the UEL implementation can be found in [28]. A Micromechanical RVE model with cohesive elements is shown in figure 2.8. Initially the interface node on fiber and matrix share the the same coordinate. With the application of load, the interface surface separates from each other. The relative normal and tangential displacement of the cohesive elements are calculated. The normal and tangential traction on each cohesive elements are calculated from the normal and tangential displacement according to the traction displacement equation in Details about the PPR finite element framework has been discussed in [27] Figure 2.2: 3D Cohesive Zone Element implementation in ABAQUS using the user subroutine UEL 21

32 2.3 Micromechanical homogenization and stiffness evaluation Homogenized micromechanical response (HMM) is obtained by volume averaging the micromechanical response. The homogenized or macroscopic stress Σ ij and strain e ij are calculated as Σ ij = 1 Y Y σ ij (Y )dy (2.19) e ij = 1 Y Y ϵ ij (Y )dy + 1 2Y Y int ([u i ]n j + [u j ]n i )ds (2.20) where σ ij and ϵ ij are microscopic stresses and strains respectively. Y represents the RVE domain and Y int corresponds to the interface domain. [u i ] represents the displacement jump across the interface along the direction n i. So the second term in equation (2.20) represents the contribution in homogenized strain due to fiber-matrix debonding. Homogenized elastic stiffness tensor E ijkl is evaluate by solving six different boundary value problem of RVE with periodic boundary condition for unit strain components as explained in [11]. A three dimensional RVE with a circular fiber is subjected to uniaxial tension. The homogenized stress-strain response and the degradation of the homogenized stiffness components for a uniaxial tensile loading (e 11 0) are shown in figure 2.3 and 2.4 respectively. Homogenized stiffness components remains 22

33 constants until debonding initiates at a critical strain. Once debonding initiates, the homogenized stiffness decay rapidly during the initial stages of the debonding and saturates when complete debonding occurs. The saturated homogenized stiffness at the point of complete debonding corresponds to the stiffness of the RVE with void in it. The homogenized stiffness degradation response in figure 2.4 shows non-uniform rate of stiffness degradation with damage evolution. That is, for a uniaxial tensile loading e 11, stiffness component E 1111 decay more compared to the stiffness component E Scalar damage models are incapable of predicting this kind of stiffness degradation. Damage model with second order damage tensor D ij as in equation (2.4) can accurately predict non-uniform stiffness degradation. Detailed derivation about second order damage model can be found in [22]. Numerical studies by Prasana Ragavan and Ghosh[9] had proved that, in composites, the material symmetry is affected by damage evolution. As explained above, the RVE exhibit orthotropy in the undamaged configuration Eijkl 0. As damage evolves, the damaged stiffness E ijkl exhibits anisotropy for multi-axial loading cases. In the undamaged configuration, the material symmetry axes and global axes coincide with each other. For normal loading paths the initial material symmetry (orthotropy of the RVE) is preserved, ie damaged stiffness is orthotropic in global coordinate system. For multi-axial loading such as e xx = e xy and all other loading components equal to zero, the damaged stiffness E ijkl shows anisotropy. The damage induced anisotropy in E ijkl is due to the coupling 23

34 Figure 2.3: Homogenized micro-mechanics stress-strain response for uniaxial tensile loading between normal and shear strain component in the elastic energy expression in the global coordinate system. If the strains are represented in principal damage coordinate system (PDCS) the coupling term vanishes and the initial material symmetry is retained. This research assumes orthotropy for the damaged stiffness in principal damage coordinate system(pdcs). Determination of the orientation of PDCS requires calculation of the second order damage tensor D ij and subsequent evaluation of the eigen vectors at each load increment. For known values of Eijkl 0 and E ijkl, the damage tensor D ij is evaluated using equation (2.3). From equation (2.3), we have nine equations for each E ijkl and six unknowns for D ij. A nonlinear least square minimization solver is 24

35 Figure 2.4: Homogenized stiffness degradation for uniaxial tensile loading used to solve for D ij. The eigen values (D1, D2, D3) and eigen vectors (e D1, e D2, e D3 ) of D ij are evaluated. The transformation matrix [Q] D which transforms the global coordinate system to the principal coordinate system (PDCS) is formed using the eigen values of the damage tensor D ij as [Q] D = [e D1 e D2 e D3 ] T. 2.4 Importance of Principal Damage Coordinate System The importance of principle damage coordinate system (PDCS) is studied in detail by Jain and Ghosh in [11]. To understand the evolution of PDCS, micromechanical 25

36 analysis are performed on RVE for two different load histories. Two loading histories considered are 1. Proportional loading : equal to zero, e 12 e 11 = constant( 0) and all other strain components 2. Non-Proportional loading: (i) e 11 0 and all other strain components equal to zero for first half of the loading path (ii) e 12 e 11 = constant( 0) and all other strain components equal to zero for the second half of the loading path. The final state of strain e ij is same for both load cases. For each load cases the homogenized stiffness E ijkl is calculated and the orientation of the PDCS is determined. In figure 2.5 x-y represents the orientation of global coordinate system and x -y represents orientation of the PDCS. When there is rotation of the principal damage coordinate system, PDCS coincides with the global coordinate system as shown in figure 2.5(a). For proportional and non-proportional loading, PDCS rotate as shown in figure 2.5(b). The evolution of the PDCS for both loading is shown in figure 2.6. For proportional loading case the PDCS start rotating from the initiation of damage. However in non-proportional case PDCS coincides with global coordinate system ie during the first half, ie. for e 11 loading. PDCS start rotating once the shear loading is applied. The final orientation of the PDCS is different for both loading cases even though the final state of strain is same. In real life applications, non-proportional loadings are present and it is necessary to incorporate the continu- 26

37 ously evolving PDCS into the damage model. It is observed that the representation of damage parameters in PDCS gives accurate results compared to the case when parameters are not represented in PDCS. So the rotation of the damage parameters to PDCS is performed using the eigen vectors of the damage tensor D ij as shown in equation (2.22). Figure 2.5: Orientation of Global coordinate system (x-y) and PDCS (x -y ) (a) when no rotation for loading e 11 0 and (b) when there is rotation for loading e 11 0 and e HCDM Model formulation Homogenized response of the micromechanical model gives complete insight into the damage evolution at the micro-level. This section defines a constitutive damage law for the composite at the macro-level. Two important observations from the 27

38 Figure 2.6: Angle of rotation vs effective strain for proportional and Nonproportional loading micromechanical study being; the damage evolution is anisotropic in nature and that the continuously evolving PDCS should be incorporated into the HCDM model for the accurate representation of the loading histories. The proposed HCDM model incorporates both these observations Damage evolution equations for the HCDM model Damage evolution surface is defined in a space represented by the thermodynamic force conjugate of the damage Y ij. As explained in section 2.4, in order to account for non-proportional loading histories, damage evolution surface should be defined in principal damage ordinate system (PDCS). The proposed damage evolution surface 28

39 in PDCS is F = 1 2 Y ijp ijkly kl 1 (2.21) where the prime subscript represents quantities expressed in PDCS using the transformation laws P ijkl = Q ip Q iq Q kr Q ls P pqrs and Y ij = Q ik Q jl Y kl (2.22) and Q ij is the transformation matrix obtained from the eigen vectors of the damage tensor D ij as explained in section 2.3. Thermodynamic force conjugate of the damage Y ij is expressed as Y ij = 1 E 2 e pqrs pq e rs (2.23) D ij The fourth order tensor P ijkl introduced in equation (2.21) accounts for the damage initiation and it characterizes the damage evolution in the HCDM model. The damage evolution equation is obtained by differentiating damage surface equation (2.21) with the thermodynamic force conjugate of the damage Y ij. Ḋ ij = λ F Y ij = λ ( ) P 2 ijkl Y kl + Y klp klij (2.24) The yield criteria and the loading-unloading condition can be expressed as Kuhn- Tucker condition in equation (2.25). These constraint equitations should satisfy si- 29

40 multaneously at all times, F 0 λ 0 λf = 0 (2.25a) (2.25b) (2.25c) Inequality (2.25a) corresponds to the elastic domain and confines the stress within the elastic domain, i.e to satisfy the consistency condition. For any loading state, all conditions in equation (2.25) should be satisfied simultaneously. For F < 0, equation (2.25c) requires λ = 0, which represents elastic loading. Loading with damage evolution is characterized by λ > 0 which requires F = 0 from equation (2.25c) to satisfy the yield criteria Calibration and parametric representation of damage parameter P ijkl The fourth order tensor P ijkl in damage surface equation (2.21) is the parameter introduced to connect the micro-scale and macro-scale failure mechanisms. This damage parameter P ijkl evolves with the damage at the macroscopic level. So it is essential to get the accurate representation of P ijkl which accounts for all damage mechanisms and loading histories. P ijkl is calculated by solving equation (2.21), (2.24) 30

41 and the Kuhn-Tucker condition in equation (2.25) simultaneously. In the incremental formulation for the evolving damage, backward Euler method is used to evaluate the P ijkl. For a load increment from step n to n + 1, the damage surface equation and damage evolution equation are F = 1 2 ( Y ij ) n+1 ( ) ( ) P ijkl Y kl 1 = 0 (2.26) n+1 n+1 (D ij ) n+1 (D ij ) n = λ ( (P ijkl 2 )n+1 (Y ( ) kl) n+1 + (Y kl) n+1 P klij )n+1 (2.27) Substitution of equation (2.27) into (2.26) gives the parameter λ. Once λ is evaluated, the components of P ijkl is evaluated from (2.27) using non-linear least square minimization. The damage tensor D ij for different loading cases are calculated using equation (2.3). The damage increment (D ij ) n+1 (D ij ) n in equation (2.27) is evaluated for all load cases and all load increments. The corresponding Y ij is also calculated for the given damage state D ij. All the components of P ijkl is calculated from the equation (2.27) using non-linear least square minimization. P ijkl values obtained form the homogenized micromechanical analysis shows exponential decay as damage progresses. The variation of P ijkl with W d is shown in figure 2.7. The damage parameter P ijkl is represented in terms of dissipated damage work W d and the eigen values of the damage tensor D ij. The dissipation of the strain energy density due to stiffness degradation is calculated by the equation (2.28). To uniquely 31

42 determine a particular damage state, all four damage parameters (W d, D 1, D 2, D 3 ) are necessary. With only W d value, we cannot determine the damage tensor D ij uniquely. Moreover different D ij tensors can give same dissipated damage work W d. This is the reason for defining P ijkl as a function of W d and eigen values of the damage tensor D ij. So P ijkl value at a given material point can give complete information about the damage state or the stiffness degradation. W d = 1 2 e ij E ijkle kl (2.28) The functional representation of P ijkl is shown in equation (2.29). Since the P ijkl values calculated from the micromechanical analysis shows exponential decay, we assume exponential function for P ijkl. P ijkl (W d, D 1, D 2, D 3 ) = P ijkl (W d ) P ijkl (D 1, D 2, D 3 ) ( ( ) ) ( 2 ( ) ) 2 P ijkl Wd a 1 Wd a 4 (W d ) = a 0 exp + a 3 exp a 2 a 5 (2.29a) (2.29b) P ijkl (D 1, D 2, D 3 ) = exp ( (α D 1 + β D 2 + γ D 3 )) (2.29c) Micromechanical analysis are performed for different loading cases and the corresponding ( ) P ijkl ref micromechanical analysis. Once ( P ijkl values are obtained, where ref represents the data obtained from ) ref values are obtained from the micromechan- 32

43 Figure 2.7: Variation of P ijkl with damage work W d ical analysis, the constants in equation (2.29) (a 0, a 1,...a 5, α, β, γ) are calibrated using non-linear least square minimization as minimize N ref i=1 [ (P ] 2 ijkl )ref P ijkl (W d, D 1, D 2, D 3 ) i (2.30) The accuracy of the HCDM model depends on the accuracy of the parametric representation of the P ijkl. All the constants in equation (2.29) is calibrated and the error observed is less than 4%. These coefficients can be used for the analysis of the composite structure at the macroscopic level. 33

44 2.5.3 Implementation of the HCDM Model in ABAQUS The HCDM model is implemented in ABAQUS using the user material subroutine UMAT. This allows to simulate a large, realistic scale composite structure with microscopic failure details. In finite element analysis frame work, state variable updates are taking place at Gauss points for a given strain. For n th converged configuration, state variables (ϵ ij ) n, (σ ij ) n, (D ij ) n, (W d ) n, (E ijkl ) n are known. The problem is to update the state variables in the converged n th configuration to their corresponding values in the updated (n+1) th configuration, i.e to (ϵ ij ) n+1, (σ ij ) n+1, (D ij ) n+1, (W d ) n+1, (E ijkl ) n+1. We assume that the incremental strain for the geometric update, n n+1 is known. For this incremental solution process, we use return mapping algorithm which satisfy the incremental consistency requirement. 2.6 Validation of the HCDM model The 3D homogenization based continuum damage model was validated by comparing the homogenized micromechanics (HMM) results of the RVE with the HCDM model response. The validation of the HCDM model was performed for unidirectional composite with rectangular fiber arrangement as shown in figure 2.8(a). The finite element RVE model for this composite microstructure is shown in figure 2.8(b). Micromechanical analysis of the RVE was carried out and the HCDM damage parameters were calibrated from the homogenized response of the micromechanical analysis. 34

45 The macroscopic finite element model implementing the constitutive relations of the HCDM model has a single 8-noded quadrilateral element. Material properties of matrix are E m = 4.6 GPa, ν m = 0.4, and that of the fiber are E f = 210 GPa, ν f = 0.3. The Cohesive zone parameters in the interface are δ c = 5e-5m, δ e = 2e-3m and σ m = 0.02 GPa. HCDM model was validated for different load combination such as uniaxial, biaxial and triaxial loadings. The comparison between HCDM model response and the HMM model response are shown in figure from 2.9 to Excellent match between the HCDM model and HMM model response proves the satisfactory performance of the HCDM model. The HCDM model predicts the damage evolution with an accuracy of 3%. The main source of error in the HCDM model is the error in the parametric representation of the damage parameter P ijkl. HCDM model was also validated for proportional and non-proportional loadings and the agreement between HCDM model and HMM model is shown in figure 2.13 and The advantage of the HCDM model is the incorporation of the continuously evolving PDCS. This allows to account for damage induced anisotropy and load histories. The computational efficiency of the HCDM model is evident from the CPU time it takes to solve the problem. The HCDM model took 117 seconds where the RVE model took seconds, i.e HCDM model is 140 times faster than the RVE model. This high computational efficiency makes the HCDM model suitable for the analysis 35

46 of large realistic composite structure. 2.7 Analysis of a composite structure with HCDM model Once the HCDM model is calibrated and validated for a given microstructure we can use this for the simulation of a composite structure. HCDM model is implemented in the commercial finite element software ABAQUS as a user subroutine UMAT. Details about the implementation of the HCDM model is explained in section This analysis involves the simulation of a composite plate subjected to a pressure wave as shown in figure In this example a pressure source is kept at distance H from the composite plate. The pressure source can be an explosive with a radius R e. In figure 2.15 R p and h p are the radius and thickness of the composite plate.the time taken by the pressure wave to reach the plate (t 0 ) varies with respect tot the radial distance r. The time t 0 can be calculated from the pressure wave speed (v) and height H as H, if r R v e t 0 = (2.31) H, if R e r R p (r R e) 2 + H v where R p is the radius of the composite plate. For this analysis, we assume an exponentially decaying, position dependent pressure loading which is shown in figure 36

47 2.16(a). The pressure load at a radial distance r is calculated as P 0, if r R e P max (r) = (2.32) P 0 e k(r Re), if R e r R p The amplitude function for the pressure load application is shown in figure 2.16(b). Here t 0 is the time taken by the pressure wave to reach a particular point in the composite plate. Pressure at that point rises to a maximum value P max at a time t 1 and it is kept constant till t 2 and then reduces to zero at time t 3. So by controlling the time parameters (t 1, t 2, t 3 ) we can simulate different pressure loading. The 3D finite element model used for this analysis is shown in figure The composite plate has radius of 10cm and 0.5cm thickness. We assume the quarter symmetry of the model as shown in figure Fixed boundary condition is applied on the outer surface of the plate and symmetric boundary condition is applied on the to symmetric sides of the composite plate. Pressure wave is applied along the z axis axis and the fiber direction is assumed to be along the z-axis as shown in figure We also assume that the pressure source is kept at a distance of 5cm and the pressure wave speed is 1000m/sec. The variation of the pressure along the composite plate surface at a given time is shown in figure The initial time for the pressure wave to reach the composite plate surface varies as shown in the figure The analysis is carried out in ABAQUS using the user subroutine UMAT. The stress wave propagation in the composite plate at different time intervals are shown 37

48 in figure The corresponding damage evolution in the composite plate is shown in figure The evolution of damage tensor D ij with time at the corner point of the composite plates is shown in figure The dissipation of the damage work is also shown in figure From figures 2.20 and 2.21 it is evident that, the damage in the composite plate initiates after it reaches a critical state and increases rapidly and then saturates. Same kind of behavior is observed in the homogenized response of the micromechanical analysis in section 2.3 and figure 2.5. So the macroscopic response is in accordance with the homogenized micromechanical analysis. Figure 2.8: (a)unidirectional composite with rectangular fiber arrangement,(b) RVE mode for the composite 38

49 Figure 2.9: Comparison of macroscopic stress-strain behavior obtained using HCDM and homogenized micromechanics (HMM) for e 11 loading Figure 2.10: Comparison of macroscopic stress-strain behavior obtained using HCDM and homogenized micromechanics (HMM) for e 22 loading 39

50 Figure 2.11: Comparison of macroscopic stress-strain behavior obtained using HCDM and homogenized micromechanics (HMM) for biaxial loading ( e 11 and e 33 ) Figure 2.12: Comparison of macroscopic stress-strain behavior obtained using HCDM and homogenized micromechanics (HMM) for triaxial loading ( e 11, e 22 and e 33 ) 40

51 Figure 2.13: Comparison of macroscopic stress-strain behavior obtained using HCDM and homogenized micromechanics (HMM) for proportional loading Figure 2.14: Comparison of macroscopic stress-strain behavior obtained using HCDM and homogenized micromechanics (HMM) for non-proportional loading 41

52 Figure 2.15: Composite plate subjected to a pressure wave Figure 2.16: (a)position dependent pressure loading, (b)amplitude function for the pressure application 42

53 Figure 2.17: Quarter symmetry of the pressure plate with fiber orientation Figure 2.18: Pressure variation along the composite surface for a given time 43

54 Figure 2.19: t 0 variation along the surface of of the composite plate Figure 2.20: Evolution of damage with time 44

55 Figure 2.21: Dissipation of damage work with time 45

56 Figure 2.22: Von-Mises stress at time (a) 0 (b) (c) (d) (e) (f) seconds 46

57 Figure 2.23: Damage variable D 11 at time (a) 0 (b) (c) (d) (e) (f) seconds 47

58 2.8 Summary A robust and accurate parametric, homogenization based continuum damage model (HCDM) is developed for a unidirectional composite. In this HCDM model, damage parameters are characterized as internal variables which evolves with the damage. The validation study proves the satisfactory performance of the HCDM model.the incorporation of the PDCS allows this model to simulate non-proportional loading histories and damage induced anisotropy very effectively. Since the damage parameters for this HCDM model is calibrated from the detailed micromechanical analysis of RVE, this model forms a strong connection between micro-scale and macro-scale failure mechanisms. The functional representation of the damage parameter makes this model computationally very efficient and suitable for the analysis of realistic scale composite structures. This model can be used to compare dissipation of damage for different RVE for specific applications. Hence the proposed HCDM model can be used as a effective tool for the design of the composite microstructure Future work The main failure mechanism included in this research is the fiber matrix debonding. Inclusion of other micro-scale failure mechanisms such as fiber breakage and matrix cracking will give more insight into the failure mechanism of the composite. This research developed a HCDM model for a unidirectional composite with rect- 48

59 angular arrangement. This model can be extended for other microstructure such as cross-ply, hexagonal arrangement, elliptical fiber composites and braided composites. This allows the comparison of the damage dissipation at the macro-scale for different composite microstructure and helps in selection of the microstructure for a given application. This model can be extended to incorporate other microstructure characteristic such as fiber orientation, fiber arrangements, volume fraction etc. The HCDM model can also be extended for fatigue and dynamic loading cases. 49

60 Bibliography [1] G. Z. Voyiadjis, P. I. Kattan, and Z. N. Taqieddin, Continuum approach to damage mechanics of composite materials with fabric tensors, Int. J. Damage Mech., vol. 16, no. 3, pp , [2] J. Fish, Q. Yu, and K. Shek, Computational damage mechanics for composite materials based on mathematical homogenization, Int. J. Numer. Meth. Engrg, vol. 45, pp , [3] R. Christensen and K. Lo, Solutions for effective shear properties in three phase sphere and cylinder models, Journal of the Mechanics and Physics of Solids, vol. 27, no. 4, pp , [4] Y. Benveniste, On the mori-tanaka s method in cracked bodies, Mechanics Research Communications, vol. 13, no. 4, pp , [5] J. Oden and T. I. Zohdi, Analysis and adaptive modeling of highly heterogeneous elastic structures, Computer Methods in Applied Mechanics and Engineering, vol. 148, no. 34, pp ,

61 BIBLIOGRAPHY [6] J. Oden and K. S. Vemaganti, Estimation of local modeling error and goaloriented adaptive modeling of heterogeneous materials: I. error estimates and adaptive algorithms, Journal of Computational Physics, vol. 164, no. 1, pp , [7] B. Budiansky, On the elastic moduli of some heterogeneous materials, Journal of the Mechanics and Physics of Solids, vol. 13, no. 4, pp , [8] M. Hori and S. Nemat-Nasser, Interacting micro-cracks near the tip in the process zone of a macro-crack, Journal of the Mechanics and Physics of Solids, vol. 35, no. 5, pp , [9] P. Raghavan and S. Ghosh, A continuum damage mechanics model for unidirectional composites undergoing interfacial debonding, Mech. Mater, vol. 37, no. 9, pp , [10] J. R. Jain and S. Ghosh, Damage evolution in composites with a homogenization based continuum damage mechanics model, Int. Jour. Damage. Mech., vol. 18, no. 6, pp , [11], Homogenization based 3d continuum damage mechanics model for composites undergoing microstructural debonding, ASME Jour. Appl. Mech, vol. 75, no. 3, pp , [12] J. L. Chaboche, S. Kruch, and T. Pottier, Micromechanics versus macromechan- 51

62 BIBLIOGRAPHY ics: a combined approach for metal matrix composite constitutive modelling, Eur. J. Mech. A/Solids, vol. 17, pp , [13] S. Maiti and P. H. Geubelle, A cohesive model for fatigue failure of polymers, Eng. Fract. Mech., vol. 72, pp , [14] S. Ghosh, Y. Ling, M. B., and R. Kim, Interfacial debonding analysis in multiple fiber reinforced composites, Mech. Mater, vol. 32, pp , [15] S. Ghosh and J. R. Jain, A homogenization-based continuum damage mechanics model for cyclic damage in 3d composites, The Aero. J. of the Royal Aero. Society, [16] S. Ghosh, J. Bai, and P. Raghavan, Concurrent multi-level model for damage evolution in microstructurally debonding composites, Mech. Mater, vol. 39(3), pp , [17] S. Ghosh and J. R. Jain, A homogenization based continuum damage mechanics model for cyclic damage in 3d composites, The Aeronaut. Jour., vol. 113, no. 1144, pp , [18] G. Z. Voyiadjis and P.I.Kattan, A plasticity-damage theory for large deformation of solids. part i: Theoretical formulation, Int. J. Engrg. Sci, vol. 30(9), pp ,

63 BIBLIOGRAPHY [19] G. T. Camacho and M. Ortiz, Computational modelling of impact damage in brittle materials, Int. Jour. Solids Struct, vol. 33, pp , [20] G. Z. Voyiadjis and P. I. Kattan, On the symmetrization of the effective stress tensor in continuum damage mechanics, Jour. Mech. Beh. Mater., vol. 7, no. 2, pp , [21] J. Cordebois and F. Sidoroff, Endommagement anisotrope en elasticite et plasticite, J. de Mecanique Theorique et Appliquee, vol. Numero Special, pp , [22] G. Z. Voyiadjis and P. I. Kattan, Advances in damage mechanics: Metals and metal matrix composites with an introduction to fabric tensors. Elsevier, [23] T. Park and G. Voyiadjis, Damage analysis and elasto-plastic behavior of metal matrix composites using the finite element method, Eng. Fract. Mech., vol. 56, no. 5, pp , [24] A. Needleman, An analysis of decohesion along an imperfect interface, Int. J. Fracture, vol. 42, pp , [25], Micromechanical modelling of interfacial decohesion, Ultramicroscopy, vol. 40, pp , [26] M. Ortiz and A. Pandolfi, Finite-deformation irreversible cohesive element for 53

64 BIBLIOGRAPHY three-dimensional crack-propogation analysis, Int. Jour. Numer. Meth. Engng, vol. 44, pp , [27] K. Park and G. H. Paulino, Computational implementation of the {PPR} potential-based cohesive model in abaqus: Educational perspective, Engineering Fracture Mechanics, vol. 93, pp , [28] S. Swaminathan, N. J. Pagano, and S. Ghosh, Analysis of interfacial debonding in three-dimensional composite microstructures, Jour. Engng. Mater. Tech., vol. 128, pp ,

65 Vita Shinu Baby received Bachelor of Technology degree in Mechanical Engineering from College of Engineering Trivandrum in After the Bachelors degree he worked in IBM from 2007 to 2009 as a system software engineer. In 2009 he joined Indian Institute of Technology (IIT) Kanpur and obtained his Master of Technology degree in His major was aerospace engineering and specialization was aerospace structure analysis. During this time he also worked as graduate research assistant in structure analysis laboratory in IIT Kanpur aerospace engineering department. From 2011 to 2012 he worked as an engineer in Eaton corporation. In April 2012 he joined Johns Hopkins University as a exchange scholar and enrolled into the graduate study in August In Johns Hopkins University he worked in Computational Mechanics and Research Laboratory (CMRL) as a graduate assistant. 55