HOMOGENIZATION BASED CONTINUUM DAMAGE MODEL …

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

c⃝ Shinu Baby 2015

All rights reserved

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 computation-

ally expensive and hence cannot be used for large and realistic composite structures.

The objective of this research is to develop a homogenization based, computation-

ally 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 micro-

mechanics 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 dam-

age tensor which characterizes the damage evolution in composite. This fourth order

ii

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

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

Dedication

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

v

Contents

Abstract ii

Acknowledgments iv

List of Tables viii

List of Figures ix

1 Introduction 1

1.1 Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 HOMOGENIZATION BASED CONTINUUM DAMAGE MODEL

9

2.1 Anisotropic continuum damage mechanics . . . . . . . . . . . . . . . 11

2.2 Micromechanical RVE model . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.1 Periodic Boundary Condition . . . . . . . . . . . . . . . . . . 14

2.2.2 Cohesive zone model for interfacial debonding . . . . . . . . . 15

vi

CONTENTS

2.2.3 PPR potential-based cohesive model for interfacial debonding 17

2.2.4 Implementation of PPR Cohesive zone model in micromechan-

ical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3 Micromechanical homogenization and stiffness evaluation . . . . . . . 22

2.4 Importance of Principal Damage Coordinate System . . . . . . . . . . 25

2.5 HCDM Model formulation . . . . . . . . . . . . . . . . . . . . . . . . 27

2.5.1 Damage evolution equations for the HCDM model . . . . . . 28

2.5.2 Calibration and parametric representation of damage parame-

ter P ′ijkl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.5.3 Implementation of the HCDM Model in ABAQUS . . . . . . . 34

2.6 Validation of the HCDM model . . . . . . . . . . . . . . . . . . . . . 34

2.7 Analysis of a composite structure with HCDM model . . . . . . . . . 36

2.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.8.1 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Bibliography 50

Vita 55

vii

List of Tables

viii

List of Figures

1.1 Micrographs of graphite-epoxy fiber-reinforced polymer matrix composite 21.2 Composite failure by (a)matrix cracking, (b)fiber breakage and (c)fiber-

matrix debonding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 Traction separation law for PPR cohesive zone model . . . . . . . . . 162.2 3D Cohesive Zone Element implementation in ABAQUS using the user

subroutine UEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3 Homogenized micro-mechanics stress-strain response for uniaxial ten-

sile loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.4 Homogenized stiffness degradation for uniaxial tensile loading . . . . 252.5 Orientation of Global coordinate system (x-y) and PDCS (x’-y’) (a)

when no rotation for loading e11 = 0 and (b) when there is rotation forloading e11 = 0 and e12 = 0 . . . . . . . . . . . . . . . . . . . . . . . . 27

2.6 Angle of rotation vs effective strain for proportional and Non-proportionalloading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.7 Variation of P ′ijkl with damage work Wd . . . . . . . . . . . . . . . . 33

2.8 (a)Unidirectional composite with rectangular fiber arrangement,(b) RVEmode for the composite . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.9 Comparison of macroscopic stress-strain behavior obtained using HCDMand homogenized micromechanics (HMM) for e11 loading . . . . . . . 39

2.10 Comparison of macroscopic stress-strain behavior obtained using HCDMand homogenized micromechanics (HMM) for e22 loading . . . . . . . 39

2.11 Comparison of macroscopic stress-strain behavior obtained using HCDMand homogenized micromechanics (HMM) for biaxial loading ( e11 ande33 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.12 Comparison of macroscopic stress-strain behavior obtained using HCDMand homogenized micromechanics (HMM) for triaxial loading ( e11, e22and e33 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.13 Comparison of macroscopic stress-strain behavior obtained using HCDMand homogenized micromechanics (HMM) for proportional loading . . 41

ix

LIST OF FIGURES

2.14 Comparison of macroscopic stress-strain behavior obtained using HCDMand homogenized micromechanics (HMM) for non-proportional loading 41

2.15 Composite plate subjected to a pressure wave . . . . . . . . . . . . . 422.16 (a)Position dependent pressure loading, (b)amplitude function for the

pressure application . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.17 Quarter symmetry of the pressure plate with fiber orientation . . . . 432.18 Pressure variation along the composite surface for a given time . . . . 432.19 t0 variation along the surface of of the composite plate . . . . . . . . 442.20 Evolution of damage with time . . . . . . . . . . . . . . . . . . . . . 442.21 Dissipation of damage work with time . . . . . . . . . . . . . . . . . . 452.22 Von-Mises stress at time (a) 0 (b) 0.0015 (c) 0.0025 (d) 0.003 (e) 0.0035

(f) 0.004 seconds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.23 Damage variable D11 at time (a) 0 (b) 0.0015 (c) 0.0025 (d) 0.003 (e)

0.0035 (f) 0.004 seconds . . . . . . . . . . . . . . . . . . . . . . . . . 47

x

Chapter 1

Introduction

Composite materials are made by combining two materials with distinct proper-

ties 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 conven-

tional 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 mate-

rial as the primary material in the construction of fuselage, airframe and wings. The

1

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 op-

timize 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 inter-

face 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 compos-ite

2


Figure 1.2: Composite failure by (a)matrix cracking, (b)fiber breakage and (c)fiber-matrix debonding

Micro-scale failure mechanisms are sensitive to local morphology like volume frac-

tion, 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 com-

putationally 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


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 an-

alyzed 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 uni-

formity 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 devel-

oped from the macroscopic experimental observations. Hence these models do not

account for the microstructure variability and micro-scale damage mechanisms and

4


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 Budi-

ansky 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 analy-

sis 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 accommo-

date 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. Mi-

cromechanical 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


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 ac-

count 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


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 Pijkl is defined in terms of the invariants of the strain

tensor. This representation of the Pijkl 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 Pijkl

representation contradicts the fact that there can be elastic strain increment with

no damage or zero damage tensor Dij. So representing Pijkl in terms of the strain

invariants is not appropriate.

The proposed model overcomes these difficulties by representing the damage pa-

rameters Pijkl in terms of eigen values of the damage tensor Dij and the dissipated

7


damage work Wd. Since Pijkl is a measure of the stiffness degradation, representation

of the Pijkl in terms of damage tensor is physically more meaningful. In this model the

Pijkl 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. Sec-

tion 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 ex-

plained 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 dis-

cusses the formulation and implementation of the HCDM model. Section 2.6 explains

the numerical analysis and validation of the HCDM models for different loading his-

tories. 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 2.8.

8

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

CHAPTER 2. HOMOGENIZATION BASED CONTINUUM DAMAGE MODEL

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 struc-

ture.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 dam-

age 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 microme-

chanical model (HMM) response is used to understand the damage evolution at the

micro-scale. HMM response shows non uniform rate of stiffness degradation and con-

tinuously 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


system (PDCS) as the reference to account for anisotropic damage evolution and

non-proportional loading histories. Once the complete information about the dam-

age 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 mechan-

ics

In continuum damage mechanics (CDM), the macroscopic damage model incorpo-

rates internal variables that evolve with the microscopic damage mechanisms. These

internal variables can be a scalar, second order tensor or a fourth order tensor, de-

pending 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(= Σijei ⊗ ej) acting on an effective area A, caused by

11


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(= Σijei ⊗ ej) through a fourth order damage effect tensor Mijkl [9],

expressed as:

Σij = Mijkl(D)Σkl (2.1)

where D is the damage tensor. D can be zeroth, second or fourth order tensor de-

pends on the CDM model employed. Different hypotheses are proposed by different

authors to calculate the damage effective tensor Mijkl. Equivalent strain hypothesis

proposed by Lemaitre and Chabhoche [12] assumes same strain state between the fic-

titous stress σij applied to the undamaged material and the actual stress σij 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 Mijkl and to establish a relation between undam-

aged stiffness Eijkl and E0ijkl. Equivalent elastic energy [20] hypothesis assumes that

the elastic complimentary energy WC(Σ, D) in a damaged material with the actual

stress Σij is equal to the the elastic complimentary energy WC(Σ, 0) in a hypothetical

undamaged material with a fictitious effective stress Σij

WC(Σ,D) =1

2(Eijkl(D))−1ΣijΣkl = WC(Σ,0) =

1

2(Eo

ijkl)−1ΣijΣkl (2.2)

From equations (2.1) and (2.2), the relation between the damaged stiffness Eijkl and

12


the undamaged stiffnesses E0ijkl is established in terms of damage effective tensor

Mijkl] as:

Eijkl = (Mijpq)−1Eo

pqrs(Mklrs)−T (2.3)

With proper selection of the order of damage tensor Dij and the function of Mijkl,

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. Ho-

mogenized 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 dam-

age using second order damage tensor Dij is proposed by Cordebois and Sidoroff

[21]and used to describe the non uniform rate of stiffness degradation. Park and Voy-

ajidis [22] has used second order damage tensor to describe the damage evolution in

composite. Such models formulate the damage effective tensor Mijkl in terms of the

symmetric tensor Dij as

Mijkl = (δik −Dik)−1δjl (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


tion technique suggested by [1] in global cordinate system is

Σij =Σik(δkj −Dkj)

−1 + (δil −Dil)−1Σlj

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 build-

ing block for this HCDM model. So modeling micromechanical RVE model with ac-

curate representation of the micro-scale details is essential. The main aspect of the

micromechanical RVE model is explained in subsequent subsections.

2.2.1 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 eij applied on the RVE boundary can be

decomposed into a periodic part ui and a macroscopic part as

ui = eijxj + ui (2.6)

14


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

(ui)p2 − (ui)p1 = eij∆xj (2.7)

where ∆xj is the relative coordinate of nodes on opposite periodic faces. The periodic

boundary condition is implemented in ABAQUS as a constraint equation. Macro-

scopic 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].

2.2.2 Cohesive zone model for interfacial debond-

ing

In this research a 3D micromechanical model is developed with the explicit repre-

sentation 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


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 inter-

facial 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(φn, φt), cohesive strength (σmax, τmax) and different soften-

ing behaviors. A complete derivation of the PPR model can be found in [27]. An

overview of the PPR CZM is given in section 2.2.3.

Figure 2.1: Traction separation law for PPR cohesive zone model

16


2.2.3 PPR potential-based cohesive model for in-

terfacial 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 (δn, δt )

Tn =

⎧⎪⎪⎪⎨⎪⎪⎪⎩Tn(∆n,∆t), if ∆n < δn and ∆t < δt

0, if ∆n ≥ δn or ∆t ≥ δt

(2.8)

2. Complete tangential failure occurs when normal or tangential separation (∆n,∆t)

reaches a critical length (δn, δt )

Tt =

⎧⎪⎪⎪⎨⎪⎪⎪⎩Tt(∆n,∆t), if ∆n < δn and ∆t < δt

0, if ∆n ≥ δn or ∆t ≥ δt

(2.9)

3. Area under the traction separation curve corresponds to the fracture energies.

17


Mode I (φn) and Mode II (φt) fracture energies are given by

φn =

δn∫0

Tn(∆n, 0)d∆n

φt =

δt∫0

Tt(∆t, 0)d∆t

(2.10)

4. Normal and tangential traction reaches maximum at a critical crack opening

displacement (δnc, δtc)

∂Tn

∂∆n

⏐⏐⏐⏐∆n=δnc

= 0

∂Tt

∂∆t

⏐⏐⏐⏐∆t=δtc

= 0

(2.11)

5. Maximum traction corresponds to cohesive strength (σmax, τmax)

Tn(δnc, 0) = σmax

Tt(δtc, 0) = τmax

(2.12)

For mixed-mode fracture, the normal and tangential traction separation law given by

the PPR model [27] is

Tn (∆n, ∆t) =Γn

δn

[m

(1− ∆n

δn

)α(m

α+

∆n

δn

)m−1

− α

(1− ∆n

δn

)α−1(m

α+

∆n

δn

)m]

×

[Γt

(1− |∆t|

δt

)β (n

β+

|∆t|δt

)n

+ ⟨φt − φn⟩

](2.13)

18


Tt (∆n, ∆t) =Γn

δn

[n

(1− |∆t|

δt

)β (n

β+

|∆t

δt

)n−1

− β

(1− |∆t|

δt

)β−1(n

β+

|∆t|δt

)n]

×[Γn

(1− ∆n

δn

)α(m

α+

∆n

δn

)m

+ ⟨φn − φt⟩]

∆t

|∆t|(2.14)

The energy constants Γn and Γt are related to the mode I and II fracture energies

as

Γn = (−φn)⟨φn−φt⟩/(φn−φt)

( αm

)m, Γt = (−φt)

⟨φn−φt⟩/(φn−φt)

(β

n

)n

, (2.15)

The non-dimensional constants m and n in equation 2.13 are evaluated by sat-

isfying the boundary condition 4. The initial slope indicators(λn,λt) are defined as

the ratio of the critical crack opening width to the final crack opening width as in

equation 2.17. Initial slope indicators control the elastic behavior, thus a small value

of λn,λt decreases the artificial elastic deformation.

m =α(α− 1)λ2

n

(1− αλ2n)

, n =β(β − 1)λ2

t

(1− βλ2t )

, (2.16)

λn =δncδn

, λt =δntδt

(2.17)

Final crack opening length δn,δt are obtained by solving the boundary condition

3 and 5 in equation (2.10) 2.10 and 2.12

19


δn =φn

σmax

αλn(1− λn)α−1( αm

+ 1)( α

mλn + 1

)m−1

δt =φt

τmax

βλt(1− λt)β−1

(β

n+ 1

)(β

nλt + 1

)n−1(2.18)

For a positive crack opening displacement the traction increases to a maximum

value coresponding to σmax, τmax as shown in figure 2.1. This region is called hardening

region. After δnc, δtc traction reduces with displacement and reaches zero at δn, δt.

This is called softening region. Unloading in the hardening zone have the same slope

as the initial slope specified by λn, λt. Unloading in softening region traces a different

path with reduced stiffness. Details about the PPR model formaulation can be found

in [27].

2.2.4 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 twenty-

node 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


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 2.18. Details about the PPR finite element framework has been discussed in [27]

Figure 2.2: 3D Cohesive Zone Element implementation in ABAQUS using the usersubroutine UEL

21


2.3 Micromechanical homogenization and

stiffness evaluation

Homogenized micromechanical response (HMM) is obtained by volume averaging

the micromechanical response. The homogenized or macroscopic stress Σij and strain

eij are calculated as

Σij =1

Y

∫Y

σij(Y )dY (2.19)

eij =1

Y

∫Y

ϵij(Y )dY +1

2Y

∫∂Yint

([ui]nj + [uj]ni)dS (2.20)

where σij and ϵij are microscopic stresses and strains respectively. Y represents

the RVE domain and Yint corresponds to the interface domain. [ui] represents the

displacement jump across the interface along the direction ni. So the second term in

equation (2.20) represents the contribution in homogenized strain due to fiber-matrix

debonding.

Homogenized elastic stiffness tensor Eijkl is evaluate by solving six different bound-

ary value problem of RVE with periodic boundary condition for unit strain compo-

nents 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 (e11 = 0) are

shown in figure 2.3 and 2.4 respectively. Homogenized stiffness components remains

22


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 e11, stiffness component E1111 decay more compared to the stiffness compo-

nent E2222. Scalar damage models are incapable of predicting this kind of stiffness

degradation. Damage model with second order damage tensor Dij 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 af-

fected by damage evolution. As explained above, the RVE exhibit orthotropy in the

undamaged configuration E0ijkl. As damage evolves, the damaged stiffness Eijkl ex-

hibits 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

exx = exy and all other loading components equal to zero, the damaged stiffness Eijkl

shows anisotropy. The damage induced anisotropy in Eijkl is due to the coupling

23


Figure 2.3: Homogenized micro-mechanics stress-strain response for uniaxial tensileloading

between normal and shear strain component in the elastic energy expression in the

global coordinate system. If the strains are represented in principal damage coordi-

nate 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 calcu-

lation of the second order damage tensor Dij and subsequent evaluation of the eigen

vectors at each load increment. For known values of E0ijkl and Eijkl, the damage tensor

Dij is evaluated using equation (2.3). From equation (2.3), we have nine equations for

each Eijkl and six unknowns for Dij. A nonlinear least square minimization solver is

24


Figure 2.4: Homogenized stiffness degradation for uniaxial tensile loading

used to solve for Dij. The eigen values (D1, D2, D3) and eigen vectors (eD1, eD2, eD3)

of Dij 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 Dij as [Q]D = [eD1 eD2 eD3]T .

2.4 Importance of Principal Damage Co-

ordinate 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


analysis are performed on RVE for two different load histories. Two loading histories

considered are

1. Proportional loading : e12e11

= constant(= 0) and all other strain components

equal to zero,

2. Non-Proportional loading: (i) e11 = 0 and all other strain components equal

to zero for first half of the loading path (ii) e12e11

= constant(= 0) and all other

strain components equal to zero for the second half of the loading path.

The final state of strain eij is same for both load cases. For each load cases the

homogenized stiffness Eijkl is calculated and the orientation of the PDCS is deter-

mined. 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 coordi-

nate system ie during the first half, ie. for e11 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


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 Dij 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 e11 = 0 and (b) when there is rotation for loadinge11 = 0 and e12 = 0

2.5 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


Figure 2.6: Angle of rotation vs effective strain for proportional and Non-proportional 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.

2.5.1 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


in PDCS is

F′=

1

2Y

′

ijP′

ijklY′

kl − 1 (2.21)

where the prime subscript represents quantities expressed in PDCS using the trans-

formation laws

P′

ijkl = QipQiqQkrQlsPpqrs and Y′

ij = QikQjlYkl (2.22)

and Qij is the transformation matrix obtained from the eigen vectors of the damage

tensor Dij as explained in section 2.3. Thermodynamic force conjugate of the damage

Yij is expressed as

Y ′ij = −1

2e′pq

∂E ′pqrs

∂Dij

e′rs (2.23)

The fourth order tensor P ′ijkl introduced in equation (2.21) accounts for the damage

initiation and it characterizes the damage evolution in the HCDMmodel. The damage

evolution equation is obtained by differentiating damage surface equation (2.21) with

the thermodynamic force conjugate of the damage Y ′ij.

Dij = λ∂F ′

∂Y ′ij

=λ

2

(P ′ijklY

′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


multaneously at all times,

F ′ ≤ 0 (2.25a)

λ ≥ 0 (2.25b)

λF ′ = 0 (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 evolu-

tion is characterized by λ > 0 which requires F = 0 from equation (2.25c) to satisfy

the yield criteria.

2.5.2 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


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

)n+1

(Y

′

kl

)n+1

− 1 = 0 (2.26)

(Dij)n+1 − (Dij)n =λ

2

((P ′ijkl

)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 evalu-

ated, the components of P ′ijkl is evaluated from (2.27) using non-linear least square

minimization. The damage tensor Dij for different loading cases are calculated us-

ing equation (2.3). The damage increment (Dij)n+1 − (Dij)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 Dij. All the components of P ′ijkl is calculated

from the equation (2.27) using non-linear least square minimization. P ′ijkl values ob-

tained form the homogenized micromechanical analysis shows exponential decay as

damage progresses. The variation of P ′ijkl with Wd is shown in figure 2.7.

The damage parameter P ′ijkl is represented in terms of dissipated damage work Wd

and the eigen values of the damage tensor Dij. The dissipation of the strain energy

density due to stiffness degradation is calculated by the equation (2.28). To uniquely

31


determine a particular damage state, all four damage parameters (Wd, D1, D2, D3) are

necessary. With only Wd value, we cannot determine the damage tensor Dij uniquely.

Moreover different Dij tensors can give same dissipated damage work Wd. This is the

reason for defining P ′ijkl as a function of Wd and eigen values of the damage tensor

Dij. So P ′ijkl value at a given material point can give complete information about the

damage state or the stiffness degradation.

Wd = −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 (Wd, D1, D2, D3) = P ′

ijkl (Wd) ∗ P ′ijkl (D1, D2, D3) (2.29a)

P ′ijkl (Wd) = a0exp

(−(Wd − a1

a2

)2)

+ a3exp

(−(Wd − a4

a5

)2)

(2.29b)

P ′ijkl (D1, D2, D3) = exp (− (α ∗D1 + β ∗D2 + γ ∗D3)) (2.29c)

Micromechanical analysis are performed for different loading cases and the corre-

sponding(P ′ijkl

)ref

values are obtained, where ref represents the data obtained from

micromechanical analysis. Once(P ′ijkl

)ref

values are obtained from the micromechan-

32


Figure 2.7: Variation of P ′ijkl with damage work Wd

ical analysis, the constants in equation (2.29) (a0, a1, ...a5, α, β, γ) are calibrated using

non-linear least square minimization as

minimize

Nref∑i=1

[(P ′ijkl

)ref

− P ′ijkl (Wd, D1, D2, D3)

]2i

(2.30)

The accuracy of the HCDM model depends on the accuracy of the parametric rep-

resentation 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


2.5.3 Implementation of the HCDMModel 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 micro-

scopic failure details. In finite element analysis frame work, state variable updates are

taking place at Gauss points for a given strain. For nth converged configuration, state

variables (ϵij)n , (σij)n , (Dij)n , (Wd)n , (Eijkl)n are known. The problem is to update

the state variables in the converged nth configuration to their corresponding values in

the updated (n+1)th configuration, i.e to (ϵij)n+1 , (σij)n+1 , (Dij)n+1 , (Wd)n+1 , (Eijkl)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 com-

paring the homogenized micromechanics (HMM) results of the RVE with the HCDM

model response. The validation of the HCDM model was performed for unidirec-

tional 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 param-

eters were calibrated from the homogenized response of the micromechanical analysis.

34


The macroscopic finite element model implementing the constitutive relations of the

HCDM model has a single 8-noded quadrilateral element. Material properties of ma-

trix are Em = 4.6 GPa, νm = 0.4, and that of the fiber are Ef = 210 GPa, νf = 0.3.

The Cohesive zone parameters in the interface are δc = 5e-5m, δe = 2e-3m and σm =

0.02 GPa.

HCDMmodel 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 2.14. 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 Pijkl.

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 2.14. The advantage of the HCDM model is the incorporation of the continu-

ously 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 16704 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


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. HCDMmodel 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 2.5.3

This analysis involves the simulation of a composite plate subjected to a pressure

wave as shown in figure 2.15. 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

Re. In figure 2.15 Rp and hp are the radius and thickness of the composite plate.The

time taken by the pressure wave to reach the plate (t0) varies with respect tot the

radial distance r. The time t0 can be calculated from the pressure wave speed (v)

and height H as

t0 =

⎧⎪⎪⎪⎨⎪⎪⎪⎩Hv, if r ≤ Re

H√(r−Re)2+

Hv

, if Re ≤ r ≤ Rp

(2.31)

where Rp 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


2.16(a). The pressure load at a radial distance r is calculated as

Pmax(r) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩P0, if r ≤ Re

P0e−k(r−Re), if Re ≤ r ≤ Rp

(2.32)

The amplitude function for the pressure load application is shown in figure 2.16(b).

Here t0 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 Pmax at a time t1

and it is kept constant till t2 and then reduces to zero at time t3. So by controlling

the time parameters (t1, t2, t3) we can simulate different pressure loading.

The 3D finite element model used for this analysis is shown in figure 2.17. The

composite plate has radius of 10cm and 0.5cm thickness. We assume the quarter

symmetry of the model as shown in figure 2.17. 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 2.17.

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 2.17. The initial time for the pressure wave

to reach the composite plate surface varies as shown in the figure 2.19.

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


in figure 2.22. The corresponding damage evolution in the composite plate is shown

in figure 2.23. The evolution of damage tensor Dij with time at the corner point of

the composite plates is shown in figure 2.20. The dissipation of the damage work is

also shown in figure 2.21. 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


Figure 2.9: Comparison of macroscopic stress-strain behavior obtained using HCDMand homogenized micromechanics (HMM) for e11 loading

Figure 2.10: Comparison of macroscopic stress-strain behavior obtained usingHCDM and homogenized micromechanics (HMM) for e22 loading

39


Figure 2.11: Comparison of macroscopic stress-strain behavior obtained usingHCDM and homogenized micromechanics (HMM) for biaxial loading ( e11 and e33 )

Figure 2.12: Comparison of macroscopic stress-strain behavior obtained usingHCDM and homogenized micromechanics (HMM) for triaxial loading ( e11, e22 ande33 )

40


Figure 2.13: Comparison of macroscopic stress-strain behavior obtained usingHCDM and homogenized micromechanics (HMM) for proportional loading

Figure 2.14: Comparison of macroscopic stress-strain behavior obtained usingHCDM and homogenized micromechanics (HMM) for non-proportional loading

41


Figure 2.15: Composite plate subjected to a pressure wave

Figure 2.16: (a)Position dependent pressure loading, (b)amplitude function for thepressure application

42


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


Figure 2.19: t0 variation along the surface of of the composite plate

Figure 2.20: Evolution of damage with time

44


Figure 2.21: Dissipation of damage work with time

45


Figure 2.22: Von-Mises stress at time (a) 0 (b) 0.0015 (c) 0.0025 (d) 0.003 (e) 0.0035(f) 0.004 seconds

46


Figure 2.23: Damage variable D11 at time (a) 0 (b) 0.0015 (c) 0.0025 (d) 0.003 (e)0.0035 (f) 0.004 seconds

47


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 dam-

age parameters for this HCDM model is calibrated from the detailed micromechan-

ical analysis of RVE, this model forms a strong connection between micro-scale and

macro-scale failure mechanisms. The functional representation of the damage param-

eter 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.

2.8.1 Future work

The main failure mechanism included in this research is the fiber matrix debond-

ing. 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


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 differ-

ent composite microstructure and helps in selection of the microstructure for a given

application. This model can be extended to incorporate other microstructure charac-

teristic such as fiber orientation, fiber arrangements, volume fraction etc. The HCDM

model can also be extended for fatigue and dynamic loading cases.

49

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. 301–330, 2007.

[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. 1657–1679, 1999.

[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. 315 – 330, 1979.

[4] Y. Benveniste, “On the mori-tanaka’s method in cracked bodies,” Mechanics

Research Communications, vol. 13, no. 4, pp. 193 – 201, 1986.

[5] J. Oden and T. I. Zohdi, “Analysis and adaptive modeling of highly heteroge-

neous elastic structures,” Computer Methods in Applied Mechanics and Engi-

neering, vol. 148, no. 34, pp. 367 – 391, 1997.

50

BIBLIOGRAPHY

[6] J. Oden and K. S. Vemaganti, “Estimation of local modeling error and goal-

oriented adaptive modeling of heterogeneous materials: I. error estimates and

adaptive algorithms,” Journal of Computational Physics, vol. 164, no. 1, pp. 22

– 47, 2000.

[7] B. Budiansky, “On the elastic moduli of some heterogeneous materials,” Journal

of the Mechanics and Physics of Solids, vol. 13, no. 4, pp. 223 – 227, 1965.

[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. 601 – 629, 1987.

[9] P. Raghavan and S. Ghosh, “A continuum damage mechanics model for unidi-

rectional composites undergoing interfacial debonding,” Mech. Mater, vol. 37,

no. 9, pp. 955–979, 2005.

[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. 533–568, 2008.

[11] ——, “Homogenization based 3d continuum damage mechanics model for

composites undergoing microstructural debonding,” ASME Jour. Appl. Mech,

vol. 75, no. 3, pp. 031 011–1–031 011–15, 2008.

[12] J. L. Chaboche, S. Kruch, and T. Pottier, “Micromechanics versus macromechan-

51

BIBLIOGRAPHY

ics: a combined approach for metal matrix composite constitutive modelling,”

Eur. J. Mech. A/Solids, vol. 17, pp. 885–908, 1998.

[13] S. Maiti and P. H. Geubelle, “A cohesive model for fatigue failure of polymers,”

Eng. Fract. Mech., vol. 72, pp. 691–708, 2005.

[14] S. Ghosh, Y. Ling, M. B., and R. Kim, “Interfacial debonding analysis in multiple

fiber reinforced composites,” Mech. Mater, vol. 32, pp. 561–591, 2000.

[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, 2009.

[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. 241–266, 2007.

[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. 371–383, 2009.

[18] G. Z. Voyiadjis and P.I.Kattan, “A plasticity-damage theory for large deforma-

tion of solids. part i: Theoretical formulation,” Int. J. Engrg. Sci, vol. 30(9), pp.

1089–1108, 1992.

52

BIBLIOGRAPHY

[19] G. T. Camacho and M. Ortiz, “Computational modelling of impact damage in

brittle materials,” Int. Jour. Solids Struct, vol. 33, pp. 2899–2938, 1996.

[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. 139–165, 1996.

[21] J. Cordebois and F. Sidoroff, “Endommagement anisotrope en elasticite et plas-

ticite,” J. de Mecanique Theorique et Appliquee, vol. Numero Special, pp. 45–60,

1982.

[22] G. Z. Voyiadjis and P. I. Kattan, Advances in damage mechanics: Metals and

metal matrix composites with an introduction to fabric tensors. Elsevier, 2006.

[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. 623–646, 1997.

[24] A. Needleman, “An analysis of decohesion along an imperfect interface,” Int. J.

Fracture, vol. 42, pp. 21–40, 1990.

[25] ——, “Micromechanical modelling of interfacial decohesion,” Ultramicroscopy,

vol. 40, pp. 203–214, 1992.

[26] M. Ortiz and A. Pandolfi, “Finite-deformation irreversible cohesive element for

53

BIBLIOGRAPHY

three-dimensional crack-propogation analysis,” Int. Jour. Numer. Meth. Engng,

vol. 44, pp. 1267–1282, 1999.

[27] K. Park and G. H. Paulino, “Computational implementation of the {PPR}

potential-based cohesive model in abaqus: Educational perspective,” Engineer-

ing Fracture Mechanics, vol. 93, pp. 239 – 262, 2012.

[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. 96–106, 2006.

54

Vita

Shinu Baby received Bachelor of Technology degree

in Mechanical Engineering from College of Engineer-

ing Trivandrum in 2007. After the Bachelors degree

he worked in IBM from 2007 to 2009 as a system soft-

ware engineer. In 2009 he joined Indian Institute of

Technology (IIT) Kanpur and obtained his Master of

Technology degree in 2011. 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 en-

gineer in Eaton corporation. In April 2012 he joined Johns Hopkins University as

a exchange scholar and enrolled into the graduate study in August 2012. In Johns

Hopkins University he worked in Computational Mechanics and Research Laboratory

(CMRL) as a graduate assistant.

55

Documents

HOMOGENIZATION BASED CONTINUUM DAMAGE MODEL …