Chandra 02 Some Issues of Cohesive zone

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Some issues in the application of cohesive zone models

for metal–ceramic interfaces

N. Chandra a,*, H. Li a, C. Shet a, H. Ghonem b

a Department of Mechanical Engineering, College of Engineering, Florida State University, Tallahassee, FL 32310, USAb Department of Mechanical Engineering, University of Rhode Island, Kingston, Rhode Island, USA

Received 29 November 2000

Abstract

Cohesive zone models (CZMs) are being increasingly used to simulate discrete fracture processes in a number of

homogeneous and inhomogeneous material systems. The models are typically expressed as a function of normal and

tangential tractions in terms of separation distances. The forms of the functions and parameters vary from model to

model. In this work, two different forms of CZMs (exponential and bilinear) are used to evaluate the response of in-

terfaces in titanium matrix composites reinforced by silicon carbide (SCS-6) fibers. The computational results are then

compared to thin slice push-out experimental data. It is observed that the bilinear CZM reproduces the macroscopic

mechanical response and the failure process while the exponential form fails to do so. From the numerical simulations,

the parameters that describe the bilinear CZM are determined. The sensitivity of the various cohesive zone parameters

in predicting the overall interfacial mechanical response (as observed in the thin-slice push out test) is carefully ex-amined. Many researchers have suggested that two independent parameters (the cohesive energy, and either of the

cohesive strength or the separation displacement) are sufficient to model cohesive zones implying that the form (shape)

of the traction–separation equations is unimportant. However, it is shown in this work that in addition to the two

independent parameters, the form of the traction–separation equations for CZMs plays a very critical role in deter-

mining the macroscopic mechanical response of the composite system. 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Cohesive zone models; Metal–ceramic interface; Push-out test; Thermal residual stress; Interface failure

1. Introduction

Metallic and intermetallic matrix composites (MMCs and IMCs) are being considered as potential

material systems of choice for advanced propulsion systems of future air and space craft because of their

high specific stiffness and strength. A critical issue in the successful application of these composites is the

mechanical characteristics of fiber–matrix interfaces. Interfaces are narrow regions separating well-defined

domains and are primarily responsible for a range of key properties including stiffness, strength, and

fracture behavior (Jayaraman et al., 1993). The role of interfaces is very vital to the stress transfer between

International Journal of Solids and Structures 39 (2002) 2827–2855

www.elsevier.com/locate/ijsolstr

* Corresponding author. Fax: +1-850-410-6420.

E-mail address: [email protected] (N. Chandra).

0020-7683/02/$ - see front matter

2002 Elsevier Science Ltd. All rights reserved.PII: S0 0 2 0 -7 6 8 3 (0 2 )0 0 1 4 9 -X


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the fiber and matrix and interfaces influence the mechanical performance and fracture behavior of com-

posites under various loading conditions (Kim and Mai, 1991a,b).

Interfaces have been modeled in a number of ways, e.g., a narrow region of continuum with graded

properties, an infinitely thin surface separated by springs, and cohesive zones with specific traction–sepa-

ration relations. One of the key research issues is to determine the best way to characterize interfaces within

the framework of continuum mechanics rather than using ad hoc methods just to facilitate numerical

implementations, e.g. springs in finite element methods. In the past few years, Chandra and co-workers

(Ananth and Chandra, 1995a,b; Chandra and Ananth, 1995; Mukherjee et al., 1997, 1998; Chandra and

Ghonem, 2001) simulated the interfacial mechanical behavior of composites in thin-slice push-out tests

incorporating the spring layer model and fracture mechanics approach. They analyzed both the initiation

and the propagation of interfacial failure to explain many of the experimental observations of fiber push-

out behavior at room and elevated temperatures. In the spring layer model (Ananth and Chandra, 1995b), a

stress based criterion for debonding and a frictional resistance based criterion for interfacial sliding have

been used to capture debonding and sliding. Debonding is postulated to occur under the combined action

of normal tensile stress (mode I) and shear stress (mode II) at the interface. In the fracture mechanics

approach strain energy release rate at the tip of the interface cracks is computed using the equivalentdomain integral method (Mukherjee et al., 1997). In this approach a critical strain energy release rate is

used as the criterion to determine the propagation of the crack.

Recently the cohesive zone approach is being increasingly used in describing fracture and failure be-

havior in a number of material systems. CZM has been used in the past to study crack tip plasticity and

creep under static and fatigue loading conditions, crazing in polymers, adhesively bonded joints, interface

cracks in bimaterials, and crack bridging due to fibers and ductile particles in composites. Interface mod-

eling using CZM has the distinct advantage compared to other global approaches (e.g. shear lag model), in

that it is based on a micro-mechanical approach. CZM was originally proposed by Barenblatt (1959, 1962)

as a possible alternative to the concept of fracture mechanics in perfectly brittle materials. Later, Dugdale

(1960) extended this concept to perfectly plastic materials by postulating the existence of a process zone at

the crack tip. Based on the Barenblatt and Dugdale models, Hillerborg and co-authors (van Mier, 1996)proposed the fictitious crack model (FCM) for analyzing crack growth in cementitious composites. FCM

has made rapid strides in the analysis of concrete structures in the field of civil engineering.

CZM has spawned a plethora of models in fracture of metals, ceramics, and polymers and their com-

posites (Needleman, 1990a,b; Rice and Wang, 1989; Wappling et al., 1998). It is not the purpose of this

paper to review all of those models and applications, but to outline some of the key works relevant to the

present study. Needleman was one of the first to use polynomial and exponential types of traction–sepa-

ration equations to simulate the particle debonding in metal matrices (Needleman, 1987, 1990a, 1990b,

1997). Xu and Needleman (1993, 1994, 1995) further used the above models to study the void nucleation at

the interface between particle and matrix, fast crack growth in brittle materials under dynamic loading, and

dynamic crack growth at the interface of bimaterials. Tvergaard (1990) used a quadratic traction–dis-

placement jump form to analyze interfaces. Tvergaard and Hutchinson (1992) used a trapezoidal shape of

the traction–separation model to calculate the crack growth resistance. Camacho and Ortiz (1996) em-

ployed a linear traction–separation equation with an additional fracture criterion to propagate multiple

cracks along arbitrary paths during impact damage in brittle materials. Geubelle and Baylor (1998) have

utilized a bilinear CZM to simulate the spontaneous initiation and propagation of transverse matrix cracks

and delamination fronts in thin composite plates subjected to low-velocity impact. In all the CZMs (except

Dugdale’s model and Camacho et al.’s model; see Table 1), the traction–separation relations for the in-

terfaces are such that with increasing interfacial separation, the traction across the interface reaches a

maximum, then decreases and eventually vanishes permitting a complete decohesion. The main difference

lies in the shape and the constants that describe that shape. In this context, the term ‘shape’ is used in a

loose sense to describe the normal (or tangential) traction vs. the normal (or tangential) separation response

2828 N. Chandra et al. / International Journal of Solids and Structures 39 (2002) 2827–2855


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Table 1

Various cohesive zone models and their parameters

Author

(year)

Proposed model Model parameters Problem

solved

Model c

Barenblatt

(1959, 1962) K ¼ R d þd 0

0

G 1ðt Þ dt ffit

p

¼ ffiffiffiffiffiffiffi

p ET 1m2

q (ductile)

¼ ffiffiffiffiffiffiffi

p ET 01m2q

(brittle)

T ¼ T 0 þT 1T 0 is work of separation

for brittle materials

T 1 is work of plastic deformation

Perfectly

brittle

materials

Dugdale

(1960) sa ¼ 2sin2 p

4T ry

For small value of T =ry

s1 ¼ 1:23 T

ry

2Yielding of

thin ideal

elastic–plastic

steel sheets

containing slits

Plastic z

ranges f

0.042 to

(in.)

Needleman

(1987)

/sep is work of separation

d are normalizing parametersrmax is cohesive strength

Particle-matrix

decohesion

d ¼ 109

108

m cenergy 1

rmax ¼ 11

ry ¼ 350

Rice and

Wang

(1989)

E 0 is initial Young’s modulus

h is normalizing parameter

rmax is maximum stress

a is constant E 0ha2 ¼ 2c

Solute

segregation

Needleman

(1990a)

/sep is work of separation

d are normalizing parameters

rmax is cohesive strength

Particle-matrix

decohesion

d ¼ 109108


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Table 1 (continued )

Author

(year)

Proposed model Model parameters Problem

solved

Model c

Needleman

(1990b)

/n, / t are work of normal

and shear separation

dn, d t are critical displacements


Decohesion of

interface under

hydrostatic

tension

dn ¼ dt ¼2 102 10

J IC=/n ¼0:57 – 2:

rmax=r0

Tvergaard

(1990)



Interfaces of

whisker rein-

forced metal

matrix com-

posites

dn ¼ dt ¼1 10

E ¼ 60 ðYoungry= E ¼ 0rmax=ry

Tvergaardand Hutch-

inson (1992)

C0 is work of separationdc is critical displacement

rmax is peak normal traction

=interface strength d1,d2 are factors governing shape

Crack growthin elasto-plas-

tic material,

peeling of ad-

hesive joints

Css=C0 ¼ðCss ¼ Pdcn=d

ct ¼

rmax=ry d2 ¼ 0:1ry= E ¼

Xu and

Needleman

(1993)

/n is work of normal separation

/t is work of shear separation



Particle-matrix

decohesion

dn ¼ dt ¼2 102 10


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Camacho

and Ortiz

(1996)

r0, s0 are normal and shear stress

at fracture initiation

drcr; dscr are critical normal and

shear opening displacement

G C ¼ fracture energy

Impact Alumin

r0 ¼ 40drcr ¼ 1

107 msteel:

r0 ¼ 15drcr ¼ 5

Geubelle

and Baylor

(1998)

/n, / t are work of normal and

tangential separation

Dn, D t are normal and tangential

displacement jump

dn, d t are normal and tangential

interface characteristic length

Delamination

by low-velocity

impact

rmax ¼ E E =10 crnormal

ment ju

Dcn ¼ 10


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as typically used in the literature. A more precise concept would be to examine the functional form of the

relation between the traction vector and displacement vector that exists between the separating surfaces

during the fracture process. However, for the sake of simplicity the term ‘shape’ will be used in this work.

The magnitude of the parameters in CZMs vary widely, ranging from MPa to GPa for traction, J to kJ

for energy, and nanometers to micrometers for separation distance. Table 1 describes the details of some

popular CZMs mentioned above with specific attention focused on the shape and the values of the model

parameters.

The basic question raised in this paper is whether the detailed form, i.e., the shape of traction–separation

equations, affects the macroscopic mechanical response of a material system. We attempt to answer this

question by comparing the predictions of two different models to the experimental results of thin slice push

out tests. In this research work, the metal–ceramic interfaces in Titanium Beta 21S/SCS composites have

been studied by using exponential and bilinear forms of CZMs. This paper is arranged as follows: in Section

2, formulation of the interface problem is presented; in Section 3, critical issues of various CZMs are

discussed; in Section 4, two CZMs used in the present work are outlined; following Section 4, the finite

element approach is given in Section 5; finally, the detailed results are discussed in Section 6, which includes

comparison between experimental observations and computational results and comparison between sim-ulation results from two different models. The effects of various parameters that describe CZMs are in-

vestigated systematically to examine the sensitivity of the interfacial parameters of CZM. Currently, many

researchers (e.g. Rahulkumar et al. (2000), Mohammed and Liechti (2000), Hutchinson and Evans (2000))

have assumed that two independent parameters (cohesive energy, and either of the cohesive strength or the

separation displacement) are sufficient to model interfaces using CZM. Series of parametric studies indicate

that this assumption leads to meaningless responses (see Figs. 10 and 12). The responses clearly show (by

the theory of contradiction) that those two parameters are not sufficient to represent the physics of the

problem of the interface separation process. In order to accurately simulate the interface and reproduce the

macroscopic mechanical behavior of composites, CZM should include, apart from those two parameters,

the shape of traction–separation law.

2. Interface problem

In general, continuum models of materials are stated as boundary value problems in which we seek

quantities such as displacements, velocities, stresses or temperatures at each point in a given domain. The

governing equations of the continuum are identified with the balance laws of classical physics such as

conservation of energy, mass and momentum. The geometries of deformation are prescribed in term of a

kinematic relationship between strain and displacements or deformation gradients. The specific choice of

the relationship depends on whether the body is undergoing finite deformation or not. To complete the

description of the boundary value problem a constitutive relationship between the kinematic quantity

(strain, deformation gradient) and the kinetic quantity (stress, stress rate) is postulated. This constitutive

equation is specific to the body under consideration within the range of temperature, rate of loading, type of

loading and environment and is usually obtained phenomenologically.

If the boundary value problem consists of domains where one set of equations cannot be applied, then

different sets are used to represent the individual domains. A case in point is the modeling of heterogeneous

composites in which different forms of constitutive relations are used. For such bodies, there is a distinct

separation between the various regions where the usual theories of continuous media are valid. Such a

separation leads to discontinuity in at least a few of the field quantities, e.g. strains. In the sense of con-

tinuum mechanics, this separation defines the interface that needs to be independently characterized. In

some cases, interfaces are not modeled explicitly and the two regions are rigidly bonded leading to con-

tinuous displacement fields with discontinuous strain and stress fields across the interface.



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Thus, the thermo-mechanical characterization of interfaces should be explicitly included for determining

the response of bodies represented by regions of different materials. This prescription is even more critical if

the interfaces were to physically separate (open up or slide) during the loading or unloading process.

2.1. Generic formulation

Consider the two solid bodies X1 and X2 separated by a common boundary S as shown in Fig. 1(a),

where S can be considered as the same surface S 1 2 X1 and S 2 2 X2, in the initial configuration, i.e. S 1 ¼S 2 ¼ S . Mathematically, we would like to define S as an infinitesimally thin 3-D domain with surfaces S 1and S 2 being the part of X1 and X2 respectively, before separation occurs. For all practical purposes the

surface S 1 or S 2 can be identified as a single surface as a part of either of the domains. A material particle

initially located (within either of the domains X1 and X2) at some position X , moves to a new location x x,

with a one to one correspondence between x x and X given by the equation of motion x x ¼ vð X ; t Þ or xi ¼ vð X j; t Þ.

In a generic sense, S defines the interface between any two domains. If X1 is a metal and X2 a ceramic,

then S represents a metal–ceramic interface; if X1 and X2 belong to the same material depicting grains of

different orientations then S is a grain boundary, and if X1 and X2 represent the same domain X1 [X2 ¼ X,then S is an internal surface not yet separated.

In any one of those cases, if S separates (fractures) to and as shown in Fig. 1(b), then the process

creates a new internal/external surface violating the fundamental laws of continuity. Obviously the newly

formed region cannot be uniquely mapped from the undeformed configuration. The equation of motion of

Fig. 1. Conceptual frame work of cohesive zone model for interface.

N. Chandra et al. / International Journal of Solids and Structures 39 (2002) 2827–2855 2833


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the body xi ¼ vð X j; t Þ cannot identify the new region. This is the fundamental problem in modeling fracture(creation of internal/external surface) in the framework of the mechanics of continuous media.

The surface S represented by the unit normal N ð N 1 2 S 1, and N 2 2 S 2 and N ¼ N 1 ¼ N 2) acting along theboundary separating the domain prior to deformation is as shown in Fig. 1(a). In the deformed configu-

ration as shown in Fig. 1(c) n̂n1 and n̂n2 represent the unit normal of the surfaces (separated or otherwise).

2.2. Domains X1 and X 2

Consider two neighboring particles located at X and X þ d X in the initial configuration. Let d X deformto d x x such that

d x x ¼ e F F d X ; ð1Þwhere e F F is the deformation gradient evaluated with respect to the material coordinates. When constitutiveequations of materials are formulated in the rate form (e.g. elastoplasticity, viscoplasticity) then it is

convenient to define velocity gradient e L L such thate L L ¼ ovvo x x

; ð2Þ

where the gradient of velocity vector vv is evaluated with respect to spatial coordinates. e L L is decomposed intoa symmetric part e D D and antisymmetric part eW W such thate L L ¼ e D D þ eW W ; ð3Þwheree D D ¼ 1

2ðe L L þ e L LT Þ and eW W ¼ 1

2ðe L L e L LT Þ; ð4Þ

where

e D D is the rate deformation tensor and

eW W is the spin tensor.

In order to formulate an appropriate constitutive relation to describe the thermo-mechanical behavior of

material, balance laws (conservation of mass, linear and angular momentum) and thermodynamic laws(first and second law) should be satisfied. Additionally, the equations should remain invariant under a

change of frame of reference. This condition can be satisfied by selecting quantities that are objective (frame

invariant) in nature.

If the material is governed by Cauchy elasticity, which states that the current stress depends only on the

current deformation, then the Cauchy’s stress can be written in terms of right stretch tensor eU U , such that~rr ¼ e R RG ð eU U Þe R RT ; ð5Þ

where G is the material response function and deformation gradient e F F ¼ e R R eU U and e R R is the orthogonalrotational tensor. However, if the material is hyperelastic (possesses a strain energy density function W ,

that depends on the current configuration), then

W ¼ W ðe E E Þ and eS S ¼ oW oe E E ; ð6Þ

where eS S is second Piola–Kirchoff stress tensor, and e E E is the Green–Lagrangian strain tensor. If the strainenergy density is W ¼ ð1=2Þ E ijC ijkl E kl, theneS S ¼ eC C e E E ; ð7Þwhere eC C is the fourth order elasticity tensor. In the finite deformation inelastic models, the constitutiveequations are formulated in the rate form based on hypoelasticity. In this case, it is assumed that the

material has no memory and that the Cauchy stress rate D~rr=Dt only depends on the current state ~rr andvelocity gradient

e L L. Thus



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D~rr

Dt ¼ g ð~rr; e L LÞ: ð8Þ

In order to fulfill the objectivity principle (principle of the material-frame indifference), the above equation

can be written as

r ¼ hð~rr; e D DÞ; ð9Þ

where r

is the objective Jaumann rate of Cauchy stress tensor, given by

r ¼ D~rr

Dt eW W ~rr ~rr eW W T : ð10Þ

It is noted that D~rr=Dt is the material derivative of the Cauchy stress. If r

is a linear function of the rate of

deformation, then

r ¼

eC C ð~rrÞ

e D D: ð11Þ

The above hypo-elastic formulation can be extended to inelastic material by adopting an additive de-

composition of the rate of deformation tensore D D ¼ e D DEl þ e D DIn; ð12Þwhere e D DEl and e D DIn are the elastic and inelastic parts of the rate of deformation tensor. Although the validityof the above equation has been questioned, for the present analysis the formulation is acceptable. Addi-

tionally, the equation has been implemented in most of the commercial codes including the one used in the

present work. The constitutive equation can then be written as

r ¼ eC C ðe D D e D DInÞ: ð13Þ

The elasticity tensor

eC C is assumed to be isotropic. Since the rate of deformation tensor

e D D is objective, the

above equation satisfies the principle of material frame indifference.

2.3. Interface S

If S continues to be a part of X1 and X2 (having points/particles common to both), then the motion of S

can be uniquely defined by the motion of either X2 or X1. Though the surface normal b N N would have rotatedand deformed to n̂n, in the sense of the kinematics b N N ¼ Ke F F 1n̂n, (K ¼ d s=dS ¼ stretch ratio, S and s are thelengths of a small segment in the original and deformed configurations, respectively). Thus, we have a one-

to-one relationship between the deformed and undeformed configurations.

However, if S is to be separated as shown in Fig. 1(b), then we have created a pair of new surfaces in the

traditional sense of the term. Consider the region bounded by and belonging to a new domain X.Assume that X is a 3-D domain made of extremely soft glue, which can be shrunk to an infinitesimally thinsurface but can be expanded to a 3-D domain. The constitutive relation of X is expressed quite differentlyfrom that of a typical 3-D solid (e.g. X1 or X2). The two surfaces that are initially part of X1 and X2 (S 1 and

S 2) have normal b N N 1 and b N N 2 in the undeformed configuration; b N N 1 and b N N 2 are equal and opposite. Duringdeformation the surfaces rotate to new normal n̂n1 and n̂n2. As the surfaces separate we have two surfaces

( 2 X1 \X) and ( 2 X2 \ X). The constitutive equation is written in terms of the normal andshear tractions and their corresponding separation displacements. For a narrow region (crack tip region)

the directions of n̂n1 and n̂n2 are approximated to be same. A typical constitutive relation of X is given by T – dd

relations (see Fig. 1(d)), such that

if j ddj


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Beyond a separation distance of jddj > jddsepj, the traction is identically zero within X,jddjP jddsepj; ~rrn̂n ¼ T ¼ 0: ð15Þ

It can also be construed that whenjddj >

jdd

sepjin the domain X, the stiffness C

ijkl 0. In order to implement

the vectorial inequalities given in Eqs. (14) and (15) typically two separate identities are postulated for the

normal and tangential components with limits set for each of them. Other methods of implementation are

equally possible.

In the numerical schemes, such as in finite element method, interfaces have been modeled as contact

surfaces or surfaces separated by spring elements. Such approaches usually have ad hoc criteria for sep-

aration and sliding and usually do not conserve energy. For example, when a nodal spring constant is

reduced to zero upon satisfying a failure criterion, stored energy is not appropriately dissipated.

In the cohesive zone approach, an independent specification of a local constitutive relation is established.

This relationship reflects the coupling of stresses (or tractions) at the interfaces to the displacements they

suffer. One class of interfacial constitutive model is based on the existence of a potential /ðDÞ that measuresthe energy cost to displace the adjacent planes across an interface by a relative displacement D (Needleman,

1992). The resulting tractions are given by

T ¼ o/ðDÞoðDÞ : ð16Þ

3. Critical issues in the application of cohesive zone models

Since CZMs provide a convenient way to model fracture (or for that matter any separation) process, it is

instructive to critically review the available models. It is our opinion that at this stage of their development,

CZMs should be truly called models rather than sanctifying them as constitutive equations or as cohesive

laws. We examine the most popular CZMs in terms of their forms, physical significance and applications.We examine not only the functional form of traction–separation distance relation but also the various

parameters that describe such relations, e.g. the rising part of the curve, the peak values (both the maximum

traction, rmax, and corresponding dmax), and the value of dsep when complete separation occurs. Usually the

T – dd relation is expressed in the form of T n vs. Dn=dc. The normalizing process ðDn=dcÞ is necessitated by therequirement that when traction is obtained as a function of cohesive strength, the multiplier has to be a

non-dimensional quantity. An important issue is whether ‘dc’ is just a scaling parameter or if it has any

other physical significance. It should be noted that dc varies anywhere from 1010 to 105 m and rmax ranges

from MPa to GPa. A final issue to be addressed is the physical significance of the area under the curve.

Does it represent just the surface energy or all the dissipative work associated with the separation process?

Barenblatt (1959, 1962) was the first to propose the concept of CZM while analyzing brittle fracture. He

postulated that the molecular force of cohesion exists near the edge of a crack. The intensity of the mo-

lecular force of cohesion ‘T n’ is found to vary as shown in row 1 of Table 1. Very interestingly, in this work

Barenblatt did not suggest any specific form of T – dd relationship. He postulated the existence of the cohesive

stress (thus reducing the singular stress). Crack propagation occurs when the integral effect of the cohesive

force acting ahead of the crack tip (along the crack plane) for the length equal to the cohesive zone size ‘ d ’

reaches a set value for the modulus of the cohesion, ‘ K ’. While investigating the yielding of an elastic–plastic

steel sheet, Dugdale (1960) identified the existence of cohesive force (termed as internal stresses) acting at

the tip of the crack for a length equal to plastic zone size.

Needleman (1987) was one of the earliest to apply the traction–separation cohesive relationship to model

void nucleation and debonding of inclusions in metallic materials. A number of potentials have been

suggested in order to obtain different forms of response between T and uu. For example, a polynomial



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function in 1987’s model (row 3, Table 1), an exponential type in 1990a’s model (row 5, Table 1), an ex-

ponential function for the normal traction and periodic function for the tangential traction in 1990b’s

model (row 6, Table 1) have all been proposed by Needleman and co-workers. In 1993, Needleman again

used exponential forms for both normal and shear tractions (row 9, Table 1). While the polynomial form

was chosen for analytical convenience, the exponential form was motivated by atomistic considerations

following the work of Rose et al. (1981). The periodic response for shear was postulated to model the

Pieriels shear stress associated with slips.

The area under the normal traction vs. separation distance is assumed to be the work of separation /sepand is in the range of 1 to 10 J/m2. In most of their (Needleman and co-investigators) work, the maximum

stress rmax is typically assumed to be about three times that of yield stress (ry) of the matrix material.

Typical values of rmax ¼ 1000–1400 MPa, and dmax ¼ 109 to 108 m have been used in their analysis. In thecase of the exponential model, separation dsep theoretically occurs as d ! 1 . However, for practical reasonit can be assumed that complete separation occurs when dsep is about 6–7 times dmax. For the polynomial

model dmax ¼ dsep=3.Rice and Wang (1989) analyzed embrittlement (or ductilization) effects of atomic scale separation of

solutes to grain boundaries. The normal traction vs. separation distance proposed in their work is shown inrow 4 of Table 1. The area under the curve was assumed to be

G ¼Z 1

0

rðdÞdd ¼ 2cint ¼ A; ð17Þ

where G is the strain energy release rate in an elastic-brittle material. Rice raises a very important question

regarding the area (A) under the curve. It can be interpreted as the adhesion energy or work of fracture, i.e.,

the energy consumed in opening up two new surfaces from a single one. As a first approximation, for

perfectly brittle materials the area under the curve can be assumed to be G . However, for ductile materials

we should recognize that if J -integral represents the strain energy release rate J ¼ oU =o A, then J IC

¼ W p

þ2cint

¼ W p

þ A;

ð18

Þwhere W p is the plastic (or viscoplastic) work ahead of the crack tip. While in brittle inclusions (carbides in

steel), W p is about 1.3–2.5 times that of A, in the failure of polycrystalline ductile materials W p is about 500–

1000 times that of A.

Tvergaard (1990) proposed a simplified interface model, where the traction shows a quadratic variation

with respect to the traction separation, having a steep ascending curve and gradual softening curve as

shown in row 7 of Table 1. Tvergaard and Hutchinson (1992) proposed a CZM as shown in row 8 of Table

1, for simulating crack growth processes in elastic–plastic solids. The T – d response is made up of linear

segments with separate slopes for rising and falling parts with a horizontal segment at the traction value of

rmax. For mode I cracks, in elastic–plastic solids, Tvergaard and Hutchinson (1992) have noted that the

shape parameters ðd1=dc; d2=dcÞ are relatively unimportant and only rmax and the area under the curve C0play a key role.

Camacho and Ortiz (1996) used CZM concepts in their finite element implementation of ballistic impact

problems. They departed from the earlier works and defined a separate multi-axial fracture criterion based

on local normal and tangential stresses. Once this criterion is satisfied a cohesive relationship, as shown in

row 10 of Table 1, is specified. The area under the curve is assumed to be the fracture energy, discounting

the energy consumed prior to the stress at the tip of the crack reaching rmax. The parameters used are

dsep ¼ 5:43 lm, rmax ¼ 1500 MPa, G ¼ 4074 J/m2 for steel (impactor), and dsep ¼ 17 lm, rmax ¼ 400 MPa,G ¼ 34 J/m2 for alumina. This work deviates once again from the others discussed in that it attempts tosolve a macroscopic boundary value problem using CZM only to model the separation and not the initi-

ation of cracks. Here G inherently includes the surface energy and all other inelastic energies associated with

the elastic–plastic processes.



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Geubelle and Baylor (1998) have simulated the delamination process in a composite plate subjected to

low velocity impact using CZMs. They proposed bilinear models both for normal and tangential separation

processes and we have followed this model in the present work. The area under the normal traction–

separation curve is considered as G IC

and that due to shear traction–separation is G IIC

. In applying this

model to graphite/epoxy delamination the following values have been adapted: for PMMA, rmax ¼ 324MPa, G IC ¼ 352:3 J/m2, dsep ¼ 2:17 lm; and for epoxy, rmax ¼ 91 MPa, G IC ¼ 88 J/m2, G IIC ¼ 315 J/m2(interface), G

plyIC ¼ 147 J/m2.

A number of other investigators have followed the above models in solving a variety of boundary value

problems. Rahulkumar et al. (2000) simulated large deformation peel tests of a fully relaxed polymer using

the model shown in row 8 of Table 1. They use /sep ¼ 5 – 10 J/m2 and dsep ¼ 25–50 lm and have a corre-sponding rmax ¼ 0:147 MPa. Espinosa et al. (2000) simulated dynamic impact of woven fiber compositesusing models shown in row 7, 8 and 11 of Table 1. They use a value of rmax ¼ 50 MPa, /sep ¼ 200–2000 J/m2, dsep ¼ 4–8 lm for the PMMA/epoxy interface. Foulk et al. (2000) modeled SCS-6/Timetal 21s metalmatrix composite interface using the model 8 in Table 1 with values of rmax ¼ 513–1500 MPa anddsep

¼ 0:1–0.2 lm. From the above discussion it is abundantly clear that there are many forms of cohesive

zone models and the parameters in each of them vary by orders of magnitude.

4. The cohesive zone models

In this work, two CZMs, exponential (Xu and Needleman, 1993) and bilinear (Geubelle and Baylor,

1998) forms are used to simulate the interfacial mechanical response by considering push-out test process as

the example.

For the exponential CZM, the interfacial potential is given by

/ðDn;DtÞ ¼ /n þ /n expðDn=dnÞf½1 r þ Dn=dn½ð1 qÞ=ðr 1Þ

½q

þ ½ðr qÞ=ð

r

1ÞD

n=d

nexp

ðD2

n=d

2

t Þg; ð19

Þwith q ¼ /t=/n, r ¼ Dn=dn, where /n and /t are the work of normal and shear separation, respectively; Dnand Dt are the normal and tangential displacement jumps, respectively; dn and dt are the normal and

tangential interface characteristic-lengths; and Dn is the value of Dn after complete shear separation takesplace under the condition of normal tension being zero, T n ¼ 0.

Using Eq. (19) in Eq. (16) the interfacial tractions are obtained as

T n ¼ ð/n=dnÞ expðDn=dnÞfðDn=dnÞ expðD2t =d2t Þþ ½ ð1 qÞ=ðr 1Þ½1 expðD2t =d2t Þ½r Dn=dng;ð20Þ

T t ¼ ð/n=dnÞð2dn=dtÞDt=dtfq þ ½ðr qÞ=ðr 1ÞDn=dng expðDn=dnÞ expðD2

t =d

2

t Þ: ð21ÞThe works of normal and shear separations are related to rmax and smax, respectively,

/t ¼ ffiffiffiffiffiffiffi

e=2p

smaxdt; /n ¼ rmaxedn; ð22Þwhere e ¼ expð1Þ.

With Dt ¼ 0, the normal traction–separation relation obtained from Eq. (20) is shown in Fig. 2(a), andthe variation of T t with Dt given by Eq. (21) for Dn ¼ 0 is shown in Fig. 2(b).

The interfacial constitutive relations for the bilinear CZM are given below:

For dn > 0,



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T n ¼rmaxdmax

dn; ðd6 dmaxÞrmaxd

1d1dmax

dn;

ðd > dmax

Þ;

ð23Þ

T t ¼rmaxdmax

DcnDctdt; ðd6 dmaxÞ

rmaxd

1d1dmax

DcnDctdt; ðd > dmaxÞ:

( ð24Þ

For dn ¼ 0,

T t ¼rmaxdmax

DcnDctdt; ðd6 dmaxÞ

rmaxd

1d1dmax

DcnDctdt; ðd > dmaxÞ;

( ð25Þ

where rmax and smax are interface normal and tangential strength, respectively; dmax is interface charac-

teristic length parameter; Dcn and Dct are the critical normal and tangential separations at which complete

separation is assumed; and dn, dt and d denote the non-dimensional normal, tangential and total dis-placement jumps respectively, defined by

dt ¼ DtDct

; dn ¼ DnDcn

; d ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid

2t þ d2n

q : ð26Þ

For pure opening ðDt ¼ 0Þ and pure shear separation ðDn ¼ 0Þ, the variation of normal and tangentialtractions with respect to Dn and Dt are shown in Fig. 3(a) and (b), respectively. The normal ðCnÞ andtangential ðCtÞ works of separation per unit area of interface are given by

Fig. 2. (a) Variation of normal traction, T n, with Dn for D t ¼ 0. (b) Variation of shear traction, T t, with Dt for Dn ¼ 0.

Fig. 3. (a) Normal traction T n as a function of the normal separation dn for dt ¼ 0, (b) shear traction T t vs. shear separation d t fordn ¼ 0.



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Cn ¼ rmaxDcn=2; Ct ¼ smaxDct=2: ð27Þ

5. Finite element approach

To evaluate the interfacial properties of composites, the single fiber push-out test has emerged as the de-

facto experimental tool. The objective of any test for characterizing the interface is to extract qualitative as

well as quantitative information about the properties. But many factors make it clear that extracting

quantitative data from the single push-out test is not trivial.

The thin-slice push-out test is modeled as an axisymmetric problem (Kallas et al., 1992), since the

loading (both thermal and mechanical) produces axially symmetric displacement fields. The discretized

mesh has 4800 axisymmetric four-node elements to model the fiber and matrix. Duplicate nodes are created

at the interface on the fiber and matrix sides. 240 axisymmetric cohesive elements with each having four

nodes and zero thickness in the direction normal to the interface are used to model the interface behavior.

The general-purpose commercial code ABAQUS (1998) is employed to carry out the analysis due to itscapability of handling non-linear problems and also, because of its flexibility in allowing user-defined

subroutines to be linked to the main program. The cohesive element model is input as a user-defined el-

ement subroutine UEL into ABAQUS. The mesh used in the simulation is shown in Fig. 4.

To simulate the single fiber push-out test, the elastic constitutive behavior is assumed for the fiber, and

the matrix is assumed to be a rate independent elastic–plastic material. The temperature dependency of the

elastic and inelastic properties (Nimmer et al., 1991; Kroupa and Neu, 1994) of the constituent phases is

included in a piece-wise linear manner. The material properties used in the analysis are given in Fig. 5. The

analysis is done in the following three steps:

1. In step 1, the cooling process after the composite consolidation at high temperature is modeled. A ref-

erence temperature T ref is assumed above which the composite is stress free, which is taken as 815 C.

Thermal residual stress (TRS) is induced due to the coefficient of thermal expansion mismatch whenthe sample is cooled down from T ref to ambient temperature. The boundary conditions for this step

are shown in Fig. 6(a). The nodes on the top surface are tied to have equal displacement to create a gen-

eralized plane strain condition.

2. When a thin slice is cut out of the bulk composite plate, the residual stresses tend to relax. This is sim-

ulated by removing the existing tying constraints and boundary conditions from the top and bottom, and

Fig. 4. The model used for push-out test simulation and details of interface element.



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allowing the stresses in the specimen to relax and reach equilibrium. The symmetry boundary conditions

used for this step are shown in Fig. 6(b). This process results in shear stresses at the interface due to the

differential axial residual strains between the fiber and matrix, as shown in the Fig. 7.

3. Axial displacement is applied to the punch shown in Fig. 6(c) until the fiber slides out of the matrix after

the entire length of the fiber debonds.

6. Results and discussion

6.1. Distribution of thermal residual stresses

The role of TRS is often ignored in analytical and experimental considerations of interfacial effects in

metallic based composite materials. This oversight is unfortunate because the resultant interpretation of

properties and behaviors is usually misleading. TRS are inherent characteristics of composite materials

arising from the differing coefficients of thermal expansion of the component materials. Since composites

Fig. 5. Variation of material properties with temperature.



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are invariably used at a different temperature than that at which they are fabricated, the differing thermal

expansions or contractions of the fiber and matrix set up the thermally induced residual stresses around the

interface on cool down from consolidation temperature. Typical MMCs and IMCs are fabricated at

temperatures which are quite high relative to ambient (above 815 C for Ti-MMCs (Yang et al., 1991; Sohi

et al., 1991)), and thus hold the possibility of producing very high stress levels that influence the fracture

and failure process of the material system.

The distribution of TRS induced during the cooling process (step 1) is shown in Fig. 7. It can be seen that

the fiber and matrix are under uniform axial compression and tension respectively to satisfy the force

equilibrium. Also, the fiber has similar distribution for radial and tangential stresses and the matrix is

subjected to compressive radial and tensile tangential stresses.

Fig. 7. Distribution of TRS along cross-section after cooling at ambient for sample thickness of 0.63 mm.

Fig. 6. Boundary conditions used in the finite element analysis.



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The evolution of shear stress distribution along the interface from step 2 to 3 (see Fig. 6) of the push-out

test is shown in Fig. 8.

6.2. Comparison between experimental results and numerical predictions

The non-aged Timetal-21S/SCS-6 composite system was fabricated and push-out tested as a part of this

work. The reinforcing fibers are made of SCS-6 materials with 142 lm diameter, with the volume fraction of

fiber being 35%. The following three cases selected from a set of experimental data of Osborne et al. (2000),

are used in the current study:

test temperature: 25 C; sample thickness: 0.63 mm,

test temperature: 500 C; sample thickness: 1.07 mm,

test temperature: 650 C; sample thickness: 1.10 mm.

Force–displacement data obtained from the experiments are shown in Fig. 9, and are the average of at

least two data sets in each case.

Finite element simulation was carried out on the model shown in Fig. 4 as per the three steps discussed

earlier. The bilinear CZM (see Eqs. (23)–(27)) for the interface is characterized by four parameters: dmax,

smax, Dct and D

cn. Since shear failure is by far the dominant failure mechanism in current study, it is assumed

that Dcn is equal to Dct . Therefore, only three parameters need to be determined in the simulation. To de-

termine smax and Dct , the work of shear separation ð/tÞ is first estimated based on the area under the force–

displacement curves obtained from the experiment (Fig. 9). The values of smax for various temperatures are

chosen so that the experimental response is closely matched. And then Dct is determined by using Eq. (27).

The values used in the simulation are listed in Table 2.

Fig. 8. Shear stress distribution along the interface. (a) After slicing, (b) beginning of loading, (c) debonding occur and (d) complete

debonding. Note: the displacement is applied at the left side of fiber (see arrow).



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The experimental and computational results for various temperatures (i.e. 25, 500 and 650 C) are shown

in Fig. 9. As noted above, the samples for 500 and 650 C are approximately of the same thickness and can

be compared directly. However, due to maximum force limitations, the sample for room temperature is

thinner. First, it can be seen from both experiments and simulations that the peak load decreases with the

increase of temperature due to the relaxation of compressive residual stresses which act on the fiber as a

clamping force at elevated temperatures (Davidson, 1992). The simulation response for various tempera-

tures matches the experimental results very well. Also, it can be seen that it is linear for room temperature

until the maximum load is reached. This is followed by catastrophic and complete debonding. For elevated

temperatures, each curve has an initial linear portion, and as the load continues to increase, a reduction in

slope can be observed (especially for the plot at 650 C). It was reported that the change in slope from

linearity for elevated temperatures corresponds to the initiation of the interfacial debond crack (Kerans and

Parthasarathy, 1991). However, from the current simulation results, it is found that the debonding does not

take place until the peak load is reached. It was reported (Davidson, 1992) that the matrix material behaves

elastically at lower temperature. At higher temperature (>400 C), the ability of the matrix phase to flowunder the applied stress is enhanced. It is shown by simulation that the plastic deformation in the matrix

indeed takes place along the entire interface during the loading process at elevated temperatures. Even at

ambient temperature, the plastic deformation occurs near the bottom along the interface. However, it is just

not enough to cause the slope change in macroscopic force–displacement curve. Hence, it should be rea-

sonable to conclude that the change of slope in the curves during loading at elevated temperatures is due

not to the initiation of interfacial debonding, but to plastic deformation of the matrix material at the in-

Fig. 9. Experimental measurement and numerical predictions, at different temperatures. Note: displacement for each temperature is

relative to the starting position.

Table 2

The values of CZM parameters used for simulation

Temperature dmax smax (MPa) Dct (m) D

cn (m) /t (J m

2)

25 C 1.0 200 3.5E5 3.5E5 3000500 C 1.0 100 1.3E5 1.3E5 650650 C 1.0 90 1.42E5 1.42E5 640



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terface. It is clear that the choice of smax is very critical to reproduce the macroscopic mechanical response

of the material system.

6.3. Comparison between two cohesive zone models

At the outset it appears that the CZM for the interface and conventional field equations with appro-

priate constitutive equations for the bulk (matrix and fiber) will reproduce the test results. Consequently,

the agreement leads to a set of cohesive zone parameters (see Table 2) that can then be used for other

geometric and loading configurations. Before such a positive assertion can be made, it is worthwhile to

analyze the uniqueness of the shape of the CZM and the sensitivity of the chosen parameters on matching

the experimental data.

For the purpose of examining whether the form of the CZM is important if the energy (area under force–

displacement curve) is maintained at the same level, two forms (exponential and bilinear) are chosen. An

energy value of /n ¼ /t ¼ 3000 J/m2 was used for the two forms. The simulation results for bilinear andexponential forms are shown in Fig. 10(a) and (b), respectively. The value of smax was selected as 200 MPa(as in Table 2 for ambient temperature case), and then altered to 300 and 400 MPa, which corresponds to

curves 1–3 (Fig. 10(a) and (b)), respectively. It should be noted that a bilinear CZM with smax ¼ 200 MPamatches the experiment very well.

Fig. 10(a) and (b) bring home some very important conclusions regarding a number of issues. Fig. 10(a)

illustrates that even though the cohesive energy is the same, increasing the interfacial shear strength changes

the F – DU response of the push-out test. This result is in contrast to the common belief that the separation

process is governed mainly by the energy of separation. Again, focusing on smax ¼ 300 and 400 MPa in Fig.10(a), there is a non-linear response when the force is above 50 N. On closer examination, it is observed that

matrix yields under those conditions, leading to the slope change in the macroscopic response. Arrows in

the Fig. 10(a) indicate the loading at which the initial plastic deformation occurs. The initial debonding for

all three cases, however, takes place at the peak load. Hence, the average shear stress required for failure

savg ¼ P peak=2pd L is different in the three cases. The present discussions again clearly reveal that selection of smax is critical in determining the macroscopic mechanical response.

Fig. 10. Load–displacement plots simulated using exponential and bilinear CZMs. (a) Bilinear model (Geubelle et al., 1998), (b) ex-

ponential model (Xu and Needleman, 1993). Note: Pd –– plastic deformation; Db –– debonding.



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Fig. 10(b) shows the overall response of the push-out test for the exponential model keeping the cohesive

energy and interfacial strength identical to those of Fig. 10(a). It is immediately clear that the predicted

response is in no way similar to the experimental observations. Arrows in the Fig. 14(b) indicate the loading

at which the initial debonding occurs. The important question is why there is such a drastically different

response. In fact, no amount of parametric variations on the exponential CZM can make it match with

experimental results making it unsuitable for modeling the interface in this problem. Focusing on Fig.

10(b), for smax ¼ 200 MPa, the peak load (about 53 N) does not correspond to either the initial debondingor the matrix yield stress. As the push-out load is increased (truly the push-out displacement, see Fig. 8(a)),

the interface responds to the external force according to its constitutive equation. When the peak stress in

each element is reached (Fig. 10(b)), the slope is reversed and the interface becomes more and more

compliant. Thus the overall stiffness of all the interfaces in tandem is continually altered with different

cohesive elements changing their stiffness as the push out displacement increases. Since the separation in

this model takes place with a significantly large displacement, the first debonding does not occur until very

late in the process.

It is interesting to note that in both Fig. 10(a) and (b), the shape of the overall force–displacement re-

sponse ( F – DU , along the longitudinal fiber axis) is similar to that of input traction–separation curve of

CZM (T – d, normal to the interface in the radial direction, see Figs. 2 and 3). This enigmatic coincidence can

be explained on the basis of the geometry of the push-out test and the loading process. A simplified model is

presented in the next section to explain why the two responses show a close resemblance.

6.4. A simplified model for push-out test based on cohesive zone model

In this section, we develop a simplified model of the push-out test problem. The circular ceramic fiber is

connected to the metallic matrix by means of interfaces. While the ceramic domain X1 is modeled as purely

elastic, the metallic matrix X2 is modeled as temperature-dependent elastic-linearly strain hardening plastic

material governed by von Mises yield criterion. X is modeled by cohesive zone elements discretized as N

four-node axisymmetric elements governed by traction–separation laws, through T – d relations in the finite

element analysis. When the push-out load F is applied, the end of the fiber moves by DU . We would like to

establish a relationship between the global F – DU and the cohesive zone T – d relations.

As shown in Fig. 11, we have simplified cohesive zone finite elements as nonlinear springs. We have pairs

of nodes connected by springs at f1; 10g, f2; 20g; . . . ; fn; n0g co-located in the fiber and the matrix before theapplication of load. Let T i be the traction at a spring i after some global load F is applied. Without loss of

generality, let f i be the shear force given by f i ¼ pdT ili, where d is the fiber diameter, T i is the average sheartraction in the region of spring ði ði 1ÞÞ=2 to ði þ ði þ 1ÞÞ=2 and li the axial length of the region. Theshear displacement of spring i is governed by T – d relation.

From the equilibrium of force, we have

F ¼ X N

i¼1 f i: ð28Þ

6.4.1. Case (1): bilinear model d < dmaxAssume that the matrix has not yielded; for ith spring, T i ¼ k idi and f i ¼ ðpdiliÞk id , so that

F ¼ pdk iX N i¼1

dili: ð29Þ

Since the effective length of the spring remains constant, the total length can be expressed as a linear

combination of each of them. Also

P N

1 di ¼ DU , leading to



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F ¼ K 0 DU : ð30ÞThus the global F – DU curve will be linear with a different slope, K 0.

6.4.2. Case (2): exponential model d < dmax

As shown in Fig. 11, since k ¼ k ðdÞ, we have the following equation, F ¼ pd

X N i¼1

k ðdiÞdili;

F ¼ K ðDÞDU :ð31Þ

It is thus seen that we have a monotonically increasing curve, with the slope changing with DU , i.e. K ðDÞ.

6.4.3. Case (3): dP dmax (for some springs)

Since only a few springs will have dP dmax, the traction of others will still be below the peak value. Thus

in both cases (bilinear and exponential) the slope of F – DU will change when any of the spring exceeds

d > dmax.

6.4.4. Case (4): matrix yielding

Whenever the matrix yields, the stress–strain response of the matrix becomes non-linear. Consequently

the slope of the F – DU curve also changes.

6.4.5. Case (5): spring failure, dP dmaxSince the number of loaded springs will be less than n, the slope of F – DU will change.

Thus, the simplified analysis shows that a change in the global response of F – DU may be caused by (1)

change in slope (change in sign) of traction–separation (T – d) law of CZM, (2) matrix yielding, or (3) co-

hesive zone failure.

Fig. 11. (a) The schematic free body diagram of all cohesive zone elements. (b) The bilinear and exponential traction–separation

relations of CZM.



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6.5. Parametric study

In Section 6.2, we evaluated a set of parameters (Table 2) within the framework of the bilinear CZM that

matched the experimental results. However, numerical simulation clearly shows that there are many other

parameters that significantly alter the progression of debonding and hence the load–displacement behavior.

In this section, the effects of various parameters, i.e., work of shear separation ð/tÞ, interfacial shearstrength ðsmaxÞ, test temperature and interface characteristic length ðdmaxÞ are investigated systematically byusing bilinear CZM in terms of load–displacement response of the push-out test. The main purpose is not

only to simulate a specific material system, but also to provide some insights on the effects of the chosen

parameters on the deformation and failure behavior of the materials in general.

6.5.1. Effect of interface characteristic length, dmaxIn this section, the effect of interface characteristic length, dmax, on the force–displacement curve is

studied and the results are shown in Fig. 12. The values of dmax are selected to be 0.1, 0.4, 0.7 and 1.0; the

corresponding F –

DU curves are I, II, III and IV, respectively. It is interesting to find that the shape of theforce–displacement curve is dependent on the value of interface characteristic length, which determines the

shape of shear traction vs. shear separation. The arrows in the figure indicate the loading at which deb-

onding occurs at the interface. Comparing the numerical results with the experimental data, it can be

concluded that for the current material system, the suitable value for dmax is 1.0. Therefore, it is reasonable

to conclude that the shape of the traction–separation curve for the CZM is an important factor in deter-

mining the macroscopic mechanical behavior of the interface in addition to the other interfacial parameters

such as the cohesive energy. Thus, it is obvious that setting dmax ¼ 1:0 fixes the shape of the CZM; in otherwords, any other values of dmax corresponding to different shapes will yield unacceptable results for the

present problem of metal–ceramic interfaces. This is an important point to be considered while selecting

specific shapes of CZM for a given problem.

Fig. 12. Effect of interface characteristic length dmax on load–displacement plots (I, II, III and IV represent corresponding force–

displacement curves to different values of characteristic lengths 0.1, 0.4, 0.7 and 1.0, respectively).



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Having established that the shape is critical, we proceed to conduct parametric study of other parameters

(/t, smax or dsep) that describe the bilinear model.

6.5.2. Effect of work of shear separation, /t The effect of the work of shear separation, /t, on the force–displacement curve is discussed in the fol-

lowing section and the results are shown in Fig. 13. The work of shear separation is varied from 100 to 3000

J/m2 and the curves 1–4 correspond to the work of shear separation of 100, 500, 1000 and 3000 J/m2,

respectively. The interfacial shear strength smax (200 MPa) is kept constant. The total area under the force–

displacement curve represents the overall work required to separate the fiber from the matrix (complete

debonding). It is found from the simulation that the debonding begins to occur at the right end (Fig. 8) just

when the peak load is reached. And the subsequent debonding process along the interface from right to the

left takes place at a very fast pace (corresponding to the unloading portion of curves in Fig. 13). The slope

of the F – DU curve is steeper for the case with lower /t and this slope reduces with an increase in /t. As

happens in the push-out test the entire fiber is debonded soon after the first cohesive element (at the left

end) fails. The actual critical separation distances D

c

t at which the cohesive element debonds completely areDct ¼ 1 106, 5 106, 1 105, 3 105 m, corresponding to /t ¼ 100, 500, 1000, 3000 J/m2 respec-tively. Thus, sufficiently large tangential displacement occurs at the left element before the entire fiber is

being pushed out.

6.5.3. Effect of shear strength, smaxIn this section, simulations were carried out by keeping /t constant and varying smax (and corre-

spondingly critical tangential displacement Dct through the Eq. (27) for the bilinear CZM). The value of dmaxis set equal to 1 as explained earlier.

Fig. 14 shows the macroscopic response of force–displacement curves with different interfacial shear

strengths, smax. It is seen that when smax6 200 MPa, there is a linear relationship between force and dis-

placement. For cases when smax ¼ 300 or 400 MPa, the slope of the ascending part of the curve is found to

Fig. 13. Effect of work of shear separation on force–displacement curves.



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change when the load reaches about 55 N. It is observed from the results that yielding in the matrix occurs

at this load and is responsible for the change in slope. It is worthwhile to note that at critical points where

debonding initiates, the matrix yields when the stress due to applied loading combined with the TRS

reaches the yield stress of 1043 MPa for a given external load (55 N in this case). It can also be noted that

the force–displacement curve response for smax ¼ 200 MPa is one closer to the experimental result. It isgenerally assumed that given the cohesive energy /t and the cohesive strength smax, the behavior of the

CZM can be uniquely described. But it is clear from the present work, apart from /t and smax, the shapeðdmax ¼ 1Þ is required to uniquely and completely describe the CZM. On the other hand when dmax 6¼ 1, orother CZMs are used, the response will be quite different from that of experimental results (see Figs. 10(b)

and 12).

6.5.4. Effect of temperature

For sample with thickness of 0.63 mm, the simulation is performed to study the effect of temperature on

the force–displacement curve and the results are shown in Fig. 15(a) and (b) for cohesive energies of 3000

and 1000 J/m2, respectively. As expected, the peak value of load decreases as the temperature increases. It

can also be seen that magnitude of peak force reduces with the decrease in the cohesive energy (Fig. 15(b)

compared to Fig. 15(a)). For the simulation at room temperature, the load keeps increasing linearly with

displacement until the load reaches the peak value, and then debonding initiates from the left end. On theother hand, at higher temperatures the load increases linearly initially; then the slope changes with the onset

of plastic deformation in the matrix. Also, it should be noted that at higher temperatures, the temperature-

dependent yield stress value is considerably lower than that at room temperature which results in plastic

deformation in the matrix at a much lower external load (


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that there is a significant amount of residual shear stress even before the application of any load. So it is

interesting to examine the absorption of cohesive energy by various elements along the length, especially at

both ends. The growth of interfacial shear traction and work of shear separation in the cohesive elements

during mechanical loading (step 3), are shown in Figs. 16 and 17 for two cases (case 1: dmax ¼ 0:1; case 2:dmax ¼ 1:0). The evolution of shear traction T t and interfacial cohesive energy /t in Figs. 16 and 17 isplotted along the interface at various stages of loading as indicated by 1–5 in Fig. 12.

Figs. 16(a) and 17(a) show that the distribution of shear traction at early stages of mechanical loading is

anti-symmetric with respect to the sample center; the right side (top) is in positive shear and the left side(bottom) is in negative shear, indicated by the curve 1. With slight increase in loading, the shear traction of

Fig. 15. The effect of temperature on force–displacement curve (a) / t ¼ 3:0 103 and (b) / t ¼ 1:0 103.

Fig. 16. (a) The evolution of shear traction and (b) work of shear separation for case 1 ðdmax ¼ 0:1, /t ¼ 3000, smax ¼ 200Þ (1–5 indicatevarious loading positions shown on curve I in Fig. 12).



26/29

all elements along the interface turns to negative shear traction and also reaches the maximum value smax as

indicated by 2 and 3 in Figs. 16(a) and 17(a). These stages indicated by 1–3 lie along the ascending part of

the T – d curve (Fig. 3). After reaching the peak value smax the shear traction begins to reduce to zero and the

interface debonds as, indicated by curves 4 and 5 in Fig. 16(a) and curve 4 in Fig. 17(a). The stages in-

dicated by 4 and 5 lie along the descending part of T – d curve. It is interesting to note that mechanical

loading is applied at the right end (right side in the plot) and the shear traction first reaches zero in a few

elements on the bottom (left side on the plot), indicating that the fracture has initiated from the left end,which is consistent with the experimental results (Ghosn et al., 1994).

Figs. 16(b) and 17(b) show the evolution of the work of shear separation (cohesive energy) in the co-

hesive elements along the interface. In both figures the cohesive energy along the interface is zero at the

beginning of loading and gradually increases with the increase in the mechanical loading, as indicated by

curves 1–5 in Fig. 16(b) and curves 1–4 in Fig. 17(b). As observed earlier, with the increase of mechanical

loading, the work of shear separation reaches its critical value /t first for a few elements near the bottom

end (left side in the plot) rather than that at right side (top) indicating that the debonding always begins to

occur at the bottom for the current material system. The cause of bottom fracture initiation during thin-

slice push-out tests of MMCs has been found to be predominantly due to TRSs (Ananth et al., 1995;

Chandra et al., 1995).

7. Summary and conclusions

This work can be summarized as follows:

• The concept of continuum mechanics is extended to include a zone of discontinuity modeled by cohesive

zones. A comprehensive analysis of some of the popular CZMs has been presented with respect to the

form of the equations and the magnitude of the parameters. It is clearly demonstrated that energy values

ranging from J to kJ, traction values ranging from MPa to GPa and separation distances ranging from

nm to lm are employed to describe interfaces. We illustrate some of the outstanding issues, e.g., physical

Fig. 17. (a) The evolution of shear traction and (b) work of shear separation for case 1ðdmax ¼ 1:0, /t ¼ 3000, smax ¼ 200). (1–4 indicatevarious loading positions shown on curve IV in Fig. 12.)



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interpretation of the area under the curve, form of the equations (shape of the traction curves), cohesive

strength and scales in CZMs.

• Two CZMs (exponential and bilinear) have been employed to model the interface failure in metal matrix

composites. The bilinear CZM is suitable to simulate the push-out test process and the calculated results

match the experiment data very well. The macroscopic mechanical behavior ( F – DU curve shape) of

interfaces in the material system has a striking resemblance to that of CZM (shape of interfacial trac-

tion–separation law) provided no plastic deformation in the matrix takes place. This implies that the

macroscopic mechanical response of the interface of the material system is very sensitive to the shape

of the interfacial constitutive relation of the CZM in addition to the other interfacial parameters.

• Debonding of cohesive elements takes place when the accumulated cohesive energy reaches the critical

value, as prescribed in the model. It has been observed during the push-out tests, when the load/displace-

ment is applied at the top end, a few cohesive elements near the bottom end first attain the critical en-

ergy, leading to initiation of fracture at the bottom end.

• CZMs can be used to describe a broad range of fracture processes in a wide variety of material systems.

However, in order to have truly predictive capabilities, this phenomenological failure model must rely on

carefully designed and conducted experiments to select a proper cohesive zone constitutive relation. Sucha relation should include the shape and appropriate material parameters such as interfacial strength,

work of separation and critical displacement discontinuities, and response during loading and unloading.

• It is our view that the CZM represents the physics of the interface separation process and hence the

shape of the CZM should in some sense depend on the inelastic processes occurring at the micromechan-

ical level. Hence when using CZMs to model separation in a given material system, an appropriate shape

(form), depending on the type of material system and the inelastic micromechanical processes, should be

used. Otherwise, the CZM based modeling and simulation will not yield meaningful results.

Acknowledgements

The authors wish to acknowledge the Air Force Office of Scientific Research (F49620-99-10275) for

providing partial financial assistance in support of this research.

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Chandra 02 Some Issues of Cohesive zone

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