a continuum theory of cohesive zone models: deformation and constitutive equations

A continuum theory of cohesive zone models: deformationand constitutive equations

Francesco Costanzo *

Engineering Science and Mechanics Department, The Pennsylvania State University, 227 Hammond Bldg.,

University Park, PA 16802, USA

Received 14 March 1997; accepted 15 December 1997

Abstract

A general three-dimensional continuum constitutive theory of cohesive zone models is presented. Thisdevelopment is part of a larger theory in which fracture is studied using elements of continuum theoriesof interfacial evolution. The main thrust of this work is the establishment of a systematic procedure forderiving cohesive interphase constitutive equations, where crack deformation is accounted for not onlyvia the opening displacement, but also via crack surface strain tensors. The material frame indi�erenceaxiom is used to show that in traditional cohesive zone models, where the opening displacement is theonly crack deformation descriptor, cohesive forces must be pointwise parallel to the openingdisplacement itself. This result indicates that the use of additional crack surface strain measures isrequired for the description of realistic crack-tip microstructures with possible anisotropic behavior.# 1998 Elsevier Science Ltd. All rights reserved.

1. Introduction

The essential feature distinguishing crack propagation problems from other types of initialvalue problems is the so-called fracture criterion. The latter is a relation which must beadjoined to the overall governing equations so that the motion of the crack can be properlycharacterized. A classical example is provided by the celebrated Gri�th [1] criterion. Thiscriterion states that in order for a crack to grow in a brittle material a quantity called energyrelease rate (ERR), traditionally denoted by G, must be equal to a critical value Gcr, the latterbeing a material property. The very nature of this statement requires that the ERR be afracture parameter, that is, a quantity capable of representing the crack readiness toward

International Journal of Engineering Science 36 (1998) 1763±1792

0020-7225/98/$19.00 # 1998 Elsevier Science Ltd. All rights reserved.PII: S0020-7225(98)00025-1

PERGAMON

* Tel.: 001 814 863 2030; fax: 001 814 863 7967; e-mail: [email protected]

motion and that a material property Gcr, homogeneous in nature with G, exist and be capableof representing the material resistance to crack motion. Furthermore, regardless of whetheror not the Gri�th criterion is a ``good criterion,`` its use clearly requires that G beunamibiguously de®ned, computable and non-trivially null.1 These considerations, and theanalysis of works such as those presented by Gurtin [2±4], Freund [5, 6], Gurtin and Yatomi [7],Nguyen [8], Chudnovsky [9] and Costanzo and Allen [10], lead to the conclusion that, from themathematical modeling viewpoint, the formulation of a fracture criterion and, therefore, of afracture theory as a whole, is not independent of what is meant by crack, and may not beindependent of: (a) what constitutive behavior has been selected for the bulk material of thesystem at hand, and/or (b) what the relevant regime of crack propagation is (e.g. quasi-staticor dynamic). For example, Gri�th's [1] de®nition of ERR requires one to construct the totalpotential energy of the system at hand and, subsequently, to di�erentiate such a function withrespect to crack front advancement, assuming that a derivative of this sort can be well de®ned.Thus, the calculation of Gri�th's ERR requires that all the assumptions necessary for apotential energy to exist and be (Gateaux) di�erentiable [8] be satis®ed, and that variationalcalculus be applicable. This indicates [8, 11] that the use of the Gri�th criterion, in its originalformulation, is essentially limited to the description of quasi-static sharp crack2 evolution inelastic materials. In fact, at least in a two-dimensional context, in order for the ERR to bebounded and non-trivially null, the pointwise strain energy function must be singular as 1/r,where r is the distance of a generic point from the crack front. As discussed by Nguyen [8],even when successful in extending the concept of total potential energy to the study ofdissipative materials [12, 13], the strength of the singularity due to the moving crack front for alarge number of dissipative systems is weaker than that found in elastic materials, thus makingthe ERR identically null under all circumstances, and consequently meaningless. Di�culties ofa di�erent type are encountered when extending the Gri�th criterion to elastodynamics. Infact, in this case a total potential energy as required in Ref. [1] cannot be constructed and, asshown by Freund [5, 6] and Gurtin and Yatomi [7], the extension strategy requires one torede®ne the ERR in a rather di�erent way, namely as the energy ¯ux into the moving crackfront. Note that this latter de®nition has been shown to be equivalent to the original one in thecase of quasi-static crack propagation [2, 3].The above discussion indicates that there is a need for a general fracture theory able to

remove the special assumptions and limitations that characterize present theories of crackpropagation. An additional concern regards the formulation of a strategy capable of``automatically'' generating the form of a fracture criterion so as to be consistent with allaspects of the problem at hand. A common strategy in dealing with this last issue is that ofchoosing a criterion that ``works'' in a speci®c context, say the Gri�th criterion in elastostatics,and extending it to the conditions of interest (see Ref. [8] for elasto-plastic materials, and

1 The requirement that G be ``non-trivially null'' indicates the necessity for a useful fracture parameter to bede®ned in such a way that its value is not always equal to zero, that is, it does not equal zero for any possiblebody's geometries, constitutive response functions and boundary conditions.2 By sharp crack is meant a simple cut acting as an internal surface across which tensile loads cannot be trans-

ferred. Furthermore, it is understood that the presence of sharp crack in a material causes most of the relevant ther-momechanical ®elds to be singular at the crack front.

F. Costanzo / International Journal of Engineering Science 36 (1998) 1763±17921764

Refs. [5±7] for elastodynamic crack propagation). However, as discussed by Costanzo and

Allen [14] and by Costanzo and Walton [15], the analysis of successful extension attempts

suggests that a fracture criterion is best understood as the constitutive response function of the

material to fracture. The latter can be seen as a type of phase transformation taking place on a

surface (the crack) evolving via the non-material3 motion of its boundary (the crack front).

The main di�erence between fracture and (bulk) phase transitions is therefore one of

dimensionality, that is, fracture is, in essence a phase transformation of one dimension less

than (bulk) phase transitions. Hence, a theory of fracture can be formulated by de®ning a

crack as an evolving interphase and constructing a corresponding crack constitutive theory.

The construction of such a theory is the main goal of the present research e�ort. It should be

noticed that the notion of fracture as a type of phase transition has also been discussed in a

recent paper by Truskinovsky [17] and one by Gurtin and Podio-Guidugli [18]. For

convenience, by viewing crack evolution as a phenomenon which involves crack deformation

and crack growth, such a reformulation will be organized into two distinct phases: the ®rst

dealing with the crack response to deformation, and the second focusing on crack growth. The

present paper is devoted to the development of phase one, whereas phase two will be the object

of a forthcoming publication.

Traditionally, continuum theories such as the one studied herein have been extensively used

in the modeling of polymer crazing. Roughly speaking, the latter consists in the formation of

crack-like defects called crazes, found mainly in amorphous polymers, where the craze faces are

bridged by thin ®laments of polymer referred to as ®brils [19]. The latter are capable of bearing

load, thus providing ``cohesiveness'' to the craze. Other applications of cohesive models can be

found in the analysis of delamination in laminated composites and in the study of debonding

at the interphase of many composite material systems. New applications are currently under

consideration in the area of the continuum modeling of the so-called sculptured thin ®lms

(STF) [20±22]. Using various techniques of deposition, in a STF parallel columnar, structures

are grown upward o� of a substrate, thus forming a microstructure not entirely dissimilar in

appearance from that of crazes. For example, an element of similarity can be found in the fact

that the columns in a STF, like the ®brils in a craze, are separated by voids whose volume

fraction can be large. However, STFs are rather di�erent in that their microstructure can be

controlled, both by controlling the morphology of the ®lm's columns, which can be curved or

``corkscrew'' shaped, and by controlling the voids' volume fraction via techniques that include

in®ltration by ¯uids or polymeric resins. Furthermore, STF can be applied in very diverse

technology areas because of the fact that they o�er interesting mechanical, optical and

dielectric properties.

As mentioned earlier, the objective of the present research e�ort is to propose a general

approach to the formulation of a fracture theory within the general framework of continuum

thermodynamics and continuum thermodynamics theories of phase transition. In this paper,

the general framework proposed in Ref. [16] will be followed to generalize and perfect a theory

3 A non-material motion is a motion which is completely or partially unrelated to the motion of material points,the latter being (by de®nition) a deformation [16].

F. Costanzo / International Journal of Engineering Science 36 (1998) 1763±1792 1765

of cohesive zone models4 presented by Costanzo and Allen [14]. The theory in Ref. [14],although fairly general, is based on arguments that do not hold in a dynamic context and stillleaves unclear the nature of the forces associated with the process of cohesive zone nucleation.Furthermore, the results in Ref. [14] were con®ned to a two-dimensional case, and were notfully independent of the constitutive behavior of the bulk material surrounding the crack,whereas the present development removes such limitations. This study shows that it is possibleto formulate fracture theories in an essentially autonomous way, independently of propagationregime and, at least in the case of cohesive interfaces, independent of the assumptionsregarding the constitutive behavior of the bulk material.In addition to this introduction, the paper is organized in four other sections. Section two is

devoted to the introduction of some basic concepts and de®nitions. In particular, thede®nitions of crack and cohesive zone (or interphase) are discussed, along with regularityconditions of the various mechanical ®elds of interest and the concept of referential controlvolume. Section three is devoted to the formulation and analysis of fundamental balance laws.The force systems acting on the crack are de®ned and discussed. In particular, the notion oftwo-valued crack stress and deformation ®elds is introduced. The concept of cohesive force isintroduced and the di�erence between cohesive and internal forces is illustrated within adiscussion of the balance of angular momentum. The section is concluded by the formulationof a convenient statement of the entropy production inequality. Section four presents anexample intended to illustrate how a cohesive interphase constitutive theory can be practicallyconstructed. The paper is concluded by a discussion of the results obtained.

2. Background

This section deals with the description of a crack as a surface endowed with its ownthermodynamic potentials. Most of the material contained herein is taken from generalinterphase theories such as those discussed by Knowles and Abeyaratne [23, 24], Gurtin [25]and Cermelli and Gurtin [26]. The aforementioned material will be partially repeated here forconvenience and to more clearly identify where the present formulation departs from otherinterphase theories.

2.1. Crack geometry

With reference to Fig. 1 let Bk be a closed region of a three-dimensional Euclidian space E.@Bk and BÊk will denote the boundary and the interior of Bk, respectively. Adopting aLagrangian viewpoint, Bk is taken as the reference con®guration of the physical system ofinterest. Hereafter, the subscript k will denote quantities de®ned relative to the referencecon®guration. Points in Bk will be identi®ed by their position X relative to a ®xed (ambient)Cartesian reference frame. wwwk(X, t), where t$R identi®es time, will denote a deformation

4 The term ``cohesive zone model'' is used to identify a crack model where certain regions of the crack surface areassumed to be capable of bearing loads, even of tensile nature.


function de®ned over Bk. The variable t will be assumed to be contained in a convenientlyselected time interval T . Also, F=GRADwwwk (with det(F)>0) will denote the gradient of wwwkrelative to the position vector X. B(t)={xvx=wwwk(X, t), X $ Bk, t $T }.Bk is assumed to contain a growing crack Ck(t). Ck(t) is assumed to be a smooth, connected5

surface such that:

Ck�t� � Ck�t� � �Bk8tRt 2 T : �1�From a topological viewpoint, Ck(t) is understood to be a (closed) two-dimensional manifoldwith boundary, embedded in R3. Furthermore, Ck(t) is assumed to be oriented by a unitnormal vector ®eld n(X, t), or more simply n, with X$Ck(t). n_(X, t), or simply n_, withX$Ck(t), will denote the tangent space to Ck(t) at X, whereas N_ will identify the correspondingtangent bundle.The evolving line bk(t), de®ned by

bk�t� � fXjX 2 @Ck�t�g; �2�is the crack front (b(t) is the image of bk(t) under wwwk(X, t)). bk(t) is assumed to be a smoothline oriented by a unit normal vector ®eld nnnb(X, t) $ n_, with X $ bk(t), outward relative to thecrack surface.The crack will be assumed to have experienced full fracture on a portion Cfk(t) � Ck(t). The

remaining part of the crack, denoted CZk(t), is assumed to be cohesive, that is, capable oftransferring forces across itself, and it will be referred to as a cohesive zone, abbreviated c.z.,de®ned as follows:

CZk�t� � Ck�t�nCfk�t�; �3�

Fig. 1. Body containing a crack with a cohesive zone.

5 The connectedness of the crack surface is not an essential assumption: the theory presented herein holds whether

or not the crack and/or the cohesive zone are connected manifolds. Such an assumption is meant to simplify the the-ory presentation because it allows one to deal with a crack with just one front and a cohesive zone with just twofronts.


where the overline denotes set closure. CZk(t) will also be assumed to be a connected set withboundary (c.f. footnote 5):

@CZk�t� � ak�t� [ bk�t�; �4�where ak(t) is assumed to be a smooth line oriented by a unit vector ®eld nnna(X, t) $ n_, outwardrelative to Cfk(t). Clearly, if Cfk(t) = {;} then CZk(t)= Ck(t) and ak(t) is not de®ned.

2.2. Crack ®elds, jump conditions and regularity assumptions

To study the motion of Bk and of the crack contained therein, the use of various scalar,vector and tensor ®elds is required. Some of these ®elds will be de®ned as bulk quantities, thatis, as quantities de®ned in the interior of Bk, whereas others will be de®ned only on Ck(t). Inorder to avoid confusion, the former will be referred to as bulk ®elds, whereas the latter will bereferred to as crack ®elds. In turn, crack ®elds can be further classi®ed into two categories:``intrinsic'' ®elds, that is, quantities which are not related in any obvious way to bulk ®elds;and ``limit'' ®elds, that is, quantities de®ned as the limits of bulk ®elds while approaching thecrack. Clearly, as soon as the notion of limit is introduced, the issue must be confrontedregarding the regularity of bulk ®elds in the vicinity of the crack. Intrinsic crack ®elds will beassumed to be bounded and su�ciently smooth everywhere in Ck(t). The limit of a generic bulk®eld f(X, t) de®ned in Bk\Ck(t) as the crack is approached is de®ned as:

F2�X; t� � limx40�

f�X2xn�X; t�; t� 8X 2 Ck�t�; �5�

where x$R. The quantities F+ and Fÿ are (non-intrinsic) crack ®elds. Furthermore, it isexpected that F+$Fÿ, that is, bulk ®elds will, in general, experience a jump discontinuityacross Ck(t). Given a crack ®eld de®ned as in Eq. (5), it is, therefore, useful to de®ne the othertwo associated crack ®elds:

�F�X; t�� F��X; t� ÿ Fÿ�X; t� and hF�X; t�i � 12 �F��X; t� � Fÿ�X; t��; �6�

where [F] and hFi will be referred to as the jump and the average of f across Ck(t). In turn,the ®elds F+ and Fÿ can be uniquely represented in terms of the jump and average ®elds, thatis,

F� � hFi � 12 �F� and Fÿ � hFi ÿ 1

2 �F�: �7�Contrary to theories regarding sharp cracks, it will be assumed that jump discontinuities arethe only type of discontinuities bulk ®elds will su�er while the crack is approached, regardlessof how it is approached [18]. In other words, one of the main goals that a cohesive zone theoryis intended to achieve is that of removing the singular thermomechanical behavior thatcharacterizes more traditional fracture theories.A new hypothesis is now introduced concerning the existence of two-valued intrinsic crack

®elds. The de®nition in Eq. (5) shows that the notion of two-valued crack ®elds is rathernatural if introduced via a limit operation which yields two di�erent values on the two sides ofthe crack. In contrast, one does not, in general, anticipate that an intrinsic crack ®eld, say for


example the crack surface energy c, may take two di�erent values on the two sides of thecrack, e.g. c+ and cÿ. However, it is well known that under deformation the surface Ck(t) willbe mapped into two surfaces, C+(t) and Cÿ(t), respectively, usually referred to as the crackfaces. The latter will have di�erent deformed con®gurations and it is likely that their surfaceenergy at points www+k (X, t) and wwwÿk(X, t), originating from a single point X in the referencecon®guration, will be di�erent. For this reason, some of the intrinsic crack ®elds de®ned later,necessary for describing the crack behavior under deformation, will be allowed to be two-valued, that is, to take on di�erent values on the two sides of the crack.

2.3. The projection and inclusion maps

In theories describing the behavior of a moving surface, one may need to regard vector ®eldsde®ned over Ck(t) taking values in N_ as vector ®elds taking values in R3 and, similarly, onemay need to decompose vector ®elds taking values in R3 as the sum of ®elds normal andtangent to Ck(t). In performing these operations two tensor ®elds I and P, called the inclusionand projection maps, respectively, turn out to be very useful. These issues have been analyzedin depth in several articles and texts on manifold theory. Hence, the discussion reported hereinis limited to mere de®nitions.The inclusion map, denoted by I, is a linear transformation in Lin(n_, R3) and is de®ned as

follows:

I : n?4R3; Ia � a; �8�where Eq. (8) simply states that the map I takes a vector a $ n_ into itself as a member of R3.The projection map P$Lin(R3, n_) is de®ned as follows:

IPa � �1ÿ n n�a 8a 2 R3; �9�where 1 is the identity map in Lin(R3, R3). From Eqs. (8) and (9) follows that:

P � IT and P � I � 1s; �10�where the superscript T indicates the transpose6 and 1s denotes the identity map in Lin(n_, n_).

2.4. Referential control volumes

In continuum mechanics it is customary to state the fundamental conservation laws withrespect to an arbitrary subregion of the physical system at hand, usually referred to as acontrol volume. More precisely, a control volume is taken as a subregion of B, that is, theimage of Bk under deformation. However, in the present discussion, one must be able to easilyseparate that part of the motion due to bulk deformation and that due to crack growth, thelatter being a non-material motion. This problem is typical of the study of moving interfacesand, as eloquently discussed by Gurtin [16, 18, 25], is better managed if the conservation laws

6 From a more formal viewpoint, I and P are not endomorphisms. Thus, the notation IT should be understood asdenoting the adjoint of I.


are stated relative to control volumes carved out of Bk, that is, relative to the referencecon®guration. These control volumes will be referred to as referential control volumes (RCVs).Clearly, some special care must be taken when selecting RCVs, and, in particular, whenselecting RCVs whose boundary moves together with an evolving (non-material) surface orline. In other words, it is possible and useful to de®ne two basic types of RCVs. The ®rst is atime independent type (TIRCV in abbreviated form), that is, a control volume whose boundarydoes not move relative to the reference con®guration. The velocity ®eld characterizing aTIRVC boundary under deformation is only that induced by the deformation. In other words,a deformation function maps a TIRVC into the control volumes traditionally used incontinuum mechanics for the studies of connected regular bodies. The second type of RCVs isa time dependent one (TDRCV in abbreviated form). In this case, the boundary of the RCVwill be characterized by a velocity ®eld selected so as to accomplish some speci®c task. Forexample, in the case of a growing crack, a TDRCV can be easily constructed as the envelopeof all spheres of radius d centered along bk(t). Such a RCV will move with the crack front bk(t)so as to always contain the moving crack front and therefore will be time dependent. In thisexample the TDRCV boundary velocity ®eld can be directly inferred from the crack frontvelocity ®eld. However, it should be noticed that the velocity ®eld characterizing the boundaryof a TDRCV is, in reality, somewhat arbitrary as long as it is compatible with the non-material motion under study. This arbitrariness is an important concept. In fact, sinceTDRCVs are used to state conservation laws one must make sure that such laws will beindependent of the particular choice of TDRCV. This leads to the introduction of newinvariance principles, such as the principle of invariance under surface reparametrizationdiscussed by Gurtin in Ref. [16]. The material presented in this paper will only make use ofTIRCVs.

3. Cohesive zone balance laws

This section deals with the derivation of the local statements of various conservation lawsfor the cohesive zone, that is, that part of the crack capable of o�ering resistance to opening.Balance laws for the rest of the crack surface can be obtained as a special case of those for thecohesive zone, simply by setting the cohesive forces to zero. For simplicity, the theorypresented herein is purely mechanical.

3.1. Force systems

Fig. 2 shows a time independent referential control volume (TIRCV) Rk and its image underdeformation. @Rk will denote the boundary of Rk and it will be assumed oriented by an outwardunit normal vector ®eld m. Rk contains a portion of the crack which will be denoted by CRk

withboundary @CRk

. The latter is oriented by a unit normal vector ®eld nnn(X, t) $ n_ outward withrespect to CRk

. The image under deformation of Rk will be denoted by R=R+[Rÿ. Forconvenience, let R+

k and Rÿk denote the inverse images of R+ and Rÿ, respectively.As discussed earlier, the crack will cause a discontinuity in various mechanical ®elds. In

particular, the discontinuity in the deformation function will cause a RCV to be separated into


two pieces. Hence, while still using RCVs to state and analyze conservation laws, it is useful towrite these laws in two ways. First, the balance laws will be stated relative to a RCVconsidered as one piece. Second, the aforementioned laws will be restated, relative to a RCVseen as the union of the inverse images of the RCV's two pieces as they appear in thedeformed con®guration. Clearly, the two statements must be satis®ed simultaneously. Asshown later, this dual formulation combined with the consistency requirement just stated,serves the purpose of better justifying the need for two-valued intrinsic crack ®elds and, in thecase of the balance of linear momentum, yields non-trivial results.With reference to Fig. 2, it is assumed that the TIRCV Rk is acted upon by three force

systems:

1. A system of surface forces acting over @Rk; these forces (per unit area) will be assumed tocoincide with the usual traction ®eld of continuum mechanics, denoted by s, and thereforeexpressed via the Cauchy theorem, that is, s = Tkm, where Tk is the ®rst Piola±Kirchho�stress tensor.

2. A system of body forces per unit mass, denoted by b, acting in Rk.3. A system of line forces distributed along @CRk

and denoted by t(X, t) (X $Ck(t)). Theexistence of such a line force density has been previously postulated in theories of interphaseevolution [16], but this is the ®rst time that it is used in a fracture theory. In analogy withCauchy's postulate, the intrinsic crack ®eld ttt will be assumed to be a function of the type

ttt � ttt�nnn�X; t�;X; t�; �11�that is, a function of the unit normal orienting the boundary @CRk

. The assumption inEq. (12) can be shown to imply [27, 28] the existence of an intrinsic crack tensor ®eldSSSk(X, t), with X $Ck(t), such that

Fig. 2. A time independent control volume containing a portion of the crack surface in its reference and deformedcon®gurations.


ttt�X; t� � SSSk�X; t�nnn�X; t�; X 2 @CRk : �12�The ®eld SSSk can be thought of as a crack stress ®eld and will be considered a member ofLin(n_, R3), thus implying that the ®eld ttt is not necessarily tangent to the crack surface,unless proven otherwise.

The force ®eld ttt is a new and ``unusual'' addition to traditional c.z. theories. Its introduction isintended to correct a shortcoming of current cohesive zone models that has been identi®ed byrecent studies of dynamic crack propagation in lattices. Studies such as those by Marder andGross [29], Zhou et al. [30] and Gao [31], have demonstrated that the onset of instabilities incrack growth are strongly dependent on the stretching of bonds parallel to the fracture plane,that is, on the A±A and B±B bonds depicted in Fig. 3. Hence, although su�cient for theanalysis of equilibrium and quasi-static growth of cracks, traditional cohesive zone modelsseem intrinsically de®cient for the study of dynamic crack propagation since the crack tipdeformation description is limited to the crack opening displacement, that is, the stretching ofthe A±B bonds. This de®ciency has also been discussed in a recent study of polymercrazing [32] where, with reference to Fig. 4, a microscopic model of a polymer craze isproposed which, in addition to the vertical polymer ®brils, includes a secondary network of

Fig. 3. Crack in a lattice.

Fig. 4. Crack in a polymer craze.


polymer chains, called ``cross-tie'' ®brils, whose action has a component parallel to the crackplane. A craze model with cross-tie ®brils was ®rst proposed by Brown [33] in order to explainthe fact that there is a signi®cant load transfer between a ®bril and its neighbors causing astress concentration inside the craze itself, responsible for the craze failure.In Section 4.1 it will be shown that traditional c.z. models which do not include any

description of mechanical actions parallel to the crack plane have another intrinsic de®ciency:they seem limited to the description of isotropic cohesive zones. In fact, using an invarianceprinciple under a change of observer, one can show that elastic as well as dissipative c.z.models based on the crack opening displacement as the only kinematic state variable and thecohesive force (de®ned in Section 3.2) as the only kinetic parameter, require the cohesive forceto be always parallel to the opening displacement. This clearly limits the capability ofdescribing microstructures such as that depicted in Fig. 4.The crack ®eld ttt is being proposed in the present theory as a possible modi®cation of

current cohesive zone models to account for the mechanical actions parallel to the crack plane,present within a craze-like region such as that depicted in Fig. 4. In this sense, ttt should beinterpreted as a surface deformation force. Clearly, in modifying traditional cohesive zonemodels to account for deformation parallel to the crack plane, a di�erent strategy can befollowed. For example, the c.z. kinematic and/or constitutive equations can be made dependenton convenient c.z. deformation measures parallel to the crack plane such as F+ and Fÿ, wherethe latter are the two values of the surface deformation gradient on the positive and negativesides of the crack surface, respectively. Then, the generalized (global) thermodynamic forcesconjugate to the new kinematic descriptors F+ and Fÿ can be identi®ed using a standardvariational argument, similar to that used to de®ne the energy release rate of a sharp crack [14].In this paper, the author has chosen a strategy which postulates the existence of the ®eld tttbecause sometimes, as in dynamic crack propagation, variational arguments are not applicable.It should be noticed that, as shown later in the paper, the generalized velocities thermodynamicconjugate to ttt are, roughly speaking, the time derivatives of F+ and Fÿ.

3.2. Conservation of linear momentum

Now that the force systems acting on the selected TIRCV have been introduced, theprinciple of conservation of linear momentum can be stated. As mentioned earlier, eachconservation law will be stated twice: ®rst relative to Rk and second relative to each of the twoparts R+

k and Rÿk. Hence, the statement of conservation of linear momentum for the TIRCVRk reads:

d

dt

�Rk

rk _x dV ��@Rk

Tkm dA��Rk

rkb dV��@CRk

SSSkv dL; �13�

where rk is the mass density in the reference con®guration. Using the surface divergencetheorem [27, 28] and letting the TIRCV shrink to the crack surface, for example, by taking thelimit as E 4 0, E being depicted in Fig. 2 and de®ned as the height of the pillbox Rk, oneobtains:


�CRk�T�k ÿ Tÿk �n dA�

�CRk

DivSSSSk dA � 0; �14�

where DivS is the surface divergence operator. By classical localization arguments, one ®nallyobtains:

�Tk�n�DivSSSSk � 0: �15�Now, with reference to Fig. 2, consider the referential control volumes R+

k and Rÿk. Theboundaries of these two domains consist of the union of the crack portion CRk

and part of theboundary of Rk. The parts of @Rk above and below the crack surface will be denoted by S+

and Sÿ, respectively. Thus, the boundaries of R+k and Rÿk are characterized by

@R�k � S� [ CRk and @Rÿk � Sÿ [ CRk : �16�Furthermore, the line @CRk

will be denoted by @C�Rkand @CÿRk

when considered part of R+k and

Rÿk, respectively.The force systems acting on R+

k and Rÿk certainly include those de®ned earlier at points (1)and (2). However, the force density ttt de®ned at point (3), cannot be used in stating the balanceof linear momentum for R+

k and Rÿk because there is no clear way to distinguish how ttt isdistributed over @C�Rk

and @CÿRk, respectively. For this reason, two additional force density ®elds,

denoted by ttt2, are now introduced such that:

ttt2�X; t� � SSS2k �X; t�nnn�X; t�;X 2 @C2Rk

�t�: �17�Furthermore, since CRk

now appears as part of the external boundary of both R+k and Rÿk, one

must address the issue regarding whether or not there exists an additional force ®eld acting onR+k and Rÿk through CRk

. It will be assumed that such a force system does indeed exist. Thelatter will be regarded as a system of crack internal forces per unit area, denoted by sssk.Clearly, relative to R+

k and Rÿk, sssk must be considered as a system of external surface forces.These forces will be referred to as cohesive, and they represent the essential ``ingredient'' of anyfracture theory of cohesive failure zones. Cohesive forces can be understood as the resistanceo�ered by a crack to an opening action. From this viewpoint, full fracture can be de®ned asthat crack state where no resistance is o�ered by the crack faces when opening is experienced,that is, sssk=0. The action of the ®eld sssk over CRk

when the latter is considered part of R+k will

be assumed to be equal and opposite to that over CRkwhen the latter is viewed as part of Rÿk.

In other words, the existence of two-valued internal crack ®elds will not be considered in thepresent formulation. It will be shown that this assumption permits to establish a simplerelationship between the ®elds ttt2 and ttt. Physically, saying that the overall action of the ®eldsssk on R+

k is equal and opposite to that over Rÿk implies that sssk is some sort of internal force®eld. However, contrary to internal force ®elds strictly speaking, it will be argued in Section3.3 that sssk o�ers a non-null contribution to the overall moment of the external forces.In order to state the balance of linear momentum an additional step is necessary: the

establishment of a sign convention for the force sssk. The latter will be assumed ``positive'' whenacting on the positive side of CRk

, that is, when CRkis considered part of Rÿk. Hence, the

statement of conservation of linear momentum, relative to R+k and Rÿk, takes on the form


d

dt

�R�k

rk _x dV ��R�k

rkb dV��S�

Tkm dAÿ�CRk

sssk dA��@C�Rk

SSS�k nnn dL; �18�

d

dt

�Rÿk

rk _x dV ��Rÿk

rkb dV��Sÿ

Tkm dA��CRk

sssk dA��@CÿRk

SSSÿk nnn dL: �19�

Using the same arguments that led to Eq. (15), the above equations yield

T�k nÿ sssk �DivS�SSS�k � � 0; �20�

ÿTÿk n� sssk �DivS�SSSÿk � � 0: �21�Summing Eqs. (20) and (21) and comparing the result with Eq. (15), one concludes that

SSSk � SSS�k � SSSÿk ) ttt � ttt� � tttÿ: �22�This result is essentially a consequence of the assumption excluding the existence of two-valuedintrinsic internal crack ®elds, and makes it appropriate to refer to ttt and to SSSk as two-valuedintrinsic crack ®elds.Subtracting Eq. (21) from Eq. (20) and multiplying both sides of the resulting equation

by 1/2, one obtains

12DivS�DSSSk� � hTkin � sssk; �23�

where

DSSSk � SSS�k ÿ SSSÿk : �24�It should be noticed that the equilibrium equation just obtained is not equivalent to that inEq. (15). In other words, in the study of cracks stating conservation laws relative to areferential control volume (Rk) and to its parts (R+

k and Rÿk) is indeed useful and yields piecesof information which cannot be obtained by considering conservation laws written in just oneway.If SSS+

k =SSSÿk=0, that is, if a theory is constructed which does not include the crack surfacedeformational stress ®elds, Eqs. (15) and (23) become

�Tk�n � 0 and T�k n � Tÿk n � sssk: �25�Using a somewhat di�erent approach, Eq. (25) were ®rst derived and discussed by Gurtin inRef. [4].

3.3. Conservation of angular momentum

With reference to Fig. 2, the equation expressing the balance of angular momentum for theTIRCV Rk takes on the form


d

dt

�Rk

r� rk _x dV ��Rk

r� rkb dV��@Rk

r� Tkm dA��@CRk�r� � SSS�k nnn� rÿ � SSSÿk nnn� dL

ÿ�CRk�r� � sssk ÿ rÿ � sssk� dA; �26�

where

r�X; t� � wwwk�X; t� ÿ o �27�is the position of a point X at time t in the deformed con®guration relative to an arbitrary®xed point o in E. The fact that the last term in the above equation should be included in theoverall moment of the external forces is not obvious. A discussion to justify this choice ispresented in Remark 1 following the derivation of the restrictions imposed by the balance ofangular momentum.The function r(X, t) is discontinuous across the crack surface and, for future reference, it is

convenient to denote such a discontinuity in a special way:

ddd�X; t� � �wwwk�X; t�� r�X; t�� 8X 2 Ck�t�: �28�The function ddd is usually referred to as the crack opening displacement. Using Eq. (28),Eq. (26) takes on the form:

d

dt

�Rk

r� rk _x dV ��Rk

r� rkb dV��@Rk

r� Tkm dA��@CRk�r� � SSS�k nnn� rÿ � SSSÿk nnn� dL

ÿ�CRk

ddd� sssk dA: �29�

When R+k and Rÿk are used instead of Rk, the corresponding statement of the principle of

conservation of angular momentum reads:

d

dt

�R2

k

r� rk _x dV ��R2

k

r� rkb dV��S2

r� Tkm dA

��@CRk

r2 � SSS2k nnn dL3

�CRk

r2 � sssk dA: �30�

Contrary to what was seen in the discussion of the balance of linear momentum, the statementof the balance of angular momentum relative to Rk as a whole, i.e. Eq. (29), can be obtainedby simply adding the ``+'' and ``ÿ'' statements in Eq. (30). Hence, in this case, no particularbene®t results from the technique used in the previous section and, for convenience, thediscussion of the balance of angular momentum will be limited to Eq. (30).The main result yielded by Eq. (30) is expressed by the following proposition:

Proposition 1 (conservation of angular momentum). The balance of angular momentum is satis®edif and only if 8X $ Ck(t)

SSS�k �F��T � F��SSS�k �T and SSSÿk �Fÿ�T � Fÿ�SSSÿk �T; �31�


where F2$Lin(n_, R3) and are de®ned as follows:

F2 � F2I: �32�Proof. As a ®rst step, recall that the balance of momenta can be stated in an equivalentvariational form using the principle of virtual work (PVW).

Lemma 1 (balance of momentaÐprinciple of virtual work). Necessary and su�cient conditions forthe satisfaction of the balance of momenta are that�

Rk

rk�bÿ �x� � w dV��@Rk

Tkm � w dAÿ�CRk

ooo � ddd� sssk dA

��@CRk�SSS�k nnn � w� � SSSÿk nnn � wÿ� dL � 0 �33�

and �R2

k

rk�bÿ �x�w dV��S2

Tkm � w dA��@C2

Rk

SSS2nnn � w2 dL3�CRk

sssk � w2 dA � 0; �34�

for every in®nitesimal rigid displacement ®eld w, that is, 8wo, ooo $ R3 and

w � w0 � ooo� r: �35�The proof of lemma 1 is omitted since it follows the same arguments of classical proofsconcerning the equivalence of the PVW and balance of momenta such as that presented inRef. [34]. Lemma 1 is a convenient technical step that allows one to reformulate the problemat hand using the variational statements in Eq. (34). The latter, after shrinking the controlvolume Rk down to the crack surface and applying the surface divergence theorem, become�

CRk�2T2

k n3sssk� � w2 dA��CRk

DivS��SSS2k �Tw2� dA � 0; �36�

8wo, ooo$R3. Using the identity [27, 28]:

DivS�STu� � u �DivS�S� � S �GRADS�u� 8S 2 Lin�n?;R3�; �37�Eq. (36) can be rewritten as:�

CRk�2T2

k n3sssk �DivS�SSS2k �� w2 dA�

�CRk

SSS2k �GRADS�w2� dA � 0: �38�

Using Eqs. (20) and (21), Eq. (38) can be simpli®ed to�CRk

S2k �GRADS�Wr2� dA � 0 8W 2 Skw�R3;R3�; �39�

where the fact has been used that given a vector ooo $ R3 there exists a unique skew-symmetrictensor W (that is W $ Skw(R3, R3)) such that ooo�r =Wr 8r $ R3. Now, recalling that Rk andtherefore CRk

are arbitrary, Eq. (39) becomes:


SSS�k �WF� �W � SSS�k �F��T � 0 and SSSÿk �WFÿ �W � SSSÿk �Fÿ�T � 0; �40�8W $ Skw(R3, R3), where it can be shown [16] that F2=GRADS(r

2). Clearly, Eq. (40) issatis®ed if and only if

SSS�k �F��T � F��SSS�k �T and SSSÿkkk �Fÿ�T � Fÿ�Sÿk �T; �41�that is, if and only if the transformations SSS+

k (F+)T and SSSÿk(Fÿ)T are symmetric. q

Eq. (41) is a statement analogous to that expressing the symmetry of the Cauchy stresstensor, when the latter is expressed in terms of the ®rst Piola±Kirchho� stress tensor.Remark 1. Fig. 5 shows an assemblage of linear springs intended to be a (very) simpli®edanalogical model of a ``periodic cell'' of the craze microstructure depicted in Fig. 4. As athought experiment, let the rigid slabs at the top and bottom of the spring assemblage in Fig. 5,originally at a vertical distance d0, be kept separated at a distance d>d0 along the y-directionand in such a way that no displacement along the x-direction is permitted. Under theseconditions, if the springs' sti�ness constants k1, k2 and k3 are as depicted in Fig. 5, that is, ifthe assemblage is symmetric with respect to the y-axis, the resultants of the (springs') ``cohesiveforces'' over the two slabs will be equal and opposite as well as aligned with the y-axis. Hence,the total moment of the spring forces will be null. However, if the symmetry of the spring'ssti�ness is altered, the resultant of the springs' forces over the two slabs, while remaining equaland opposite, will have horizontal components which will result in a net moment. In otherwords, in the absence of a particular material symmetry, in order to vertically separate the twoslabs without causing a horizontal shear motion, one must apply a vertical separating force aswell as a moment, to balance the moment due to the non-symmetric response of the springassemblage. This simple argument is meant to physically justify the inclusion of the term ddd�sssk

in Eq. (29). In fact, if the term ddd�sssk were not to be accounted for, the implication would bethat ddd�sssk=0 and this strongly limits the symmetry properties of the cohesive zone. Anotherway of justifying the presence of the said term is to say that while sssk can be considered as aninternal force because of its cohesive action, sssk acts at a surface of displacement discontinuitythat is a surface that must be considered as part of the external boundary R in the deformedcon®guration, i.e. sssk also acts as an externally applied force ®eld.As mentioned earlier, the condition ddd�sssk=0 is essentially equivalent to a special

constitutive assumption in that it restricts the symmetry properties of the cohesive zone. Infact, ddd�sssk=0 implies that:

Fig. 5. Simpli®ed linear spring analogical model of a craze microstructure.


sssc � sdddjjdddjj ; �42�

where s is a scalar function. In other words, the condition ddd�sssk=0 restricts the cohesive forceto be always parallel to the opening displacement thus forcing the c.z. to be isotropic. Theinteresting element of this discussion is that, using the principle of material frame indi�erence,in Section 4.1 it will be shown that a c.z. theory, where the opening displacement ddd is chosenas the only kinematic descriptor for the c.z. motion, leads to the condition ddd�sssk=0. Thismeans that the response of a cohesive zone where forces or deformation tangent to the crackplane are not accounted for cannot be anisotropic. Hence, they may be inadequate to properlydescribe microstructures such as that depicted in Fig. 4. These considerations are of extremeimportance in view of the growing popularity of cohesive zone models in the numericalanalysis of fracture [15, 35±38], and, as mentioned earlier, they will be given a rigoroustreatment in Section 4.

3.4. Entropy production inequality

The present theoretical development is being carried out under the assumption that thesystem at hand is characterized by a uniform temperature ®eld which is kept constant in time.This assumption makes it essentially impossible to properly introduce a precise notion ofentropy or internal energy. Consequently, one cannot rely on the classical statements of eitherthe conservation of energy principle or the Claussus±Duhem inequality [39]. However, thisdoes not prevent one from postulating a convenient statement of the second law ofthermodynamics (or entropy production inequality), nor from analyzing constitutive behaviorswith dissipation. One way to do this is to postulate the existence of a free energy functionrepresenting the amount of strain energy reversibly stored in the system, and to use as entropyproduction inequality that to which the full statement of the second law would reduce underthe current assumptions. Thus, letting h(X, t) (de®ned for X $ BÊk) and c(X, t) (de®ned forX $ Ck(t)) be the Helmholtz free energies per unit mass of the bulk material and per unitsurface of the crack, respectively, the inequality that will be used as second law ofthermodynamics reads [16, 18]:

d

dt

�Rk

h� 1

2rk _x � _x

� �dV�

�CRk

c dA

( )R�@Rk

Tkm � _x dA��Rk

rb � _x dA

��@CRk�SSS�k nnn � _x� � SSSÿk nnn � _xÿ� dL: �43�

Eq. (43) states that, for any arbitrarily selected time independent referential control volume,the total rate of change of kinetic and free energy cannot exceed the amount of mechanicalpower supplied by the external forces. Whenever the latter exceeds the left hand side ofEq. (43), the excess amount of power will be considered dissipated.Shrinking the TIRCV down to the crack surface and applying the surface divergence,

Eq. (43) becomes:


d

dt

�CRk

c dA

( )R�CRk�T�k n � _x� ÿ Tÿk n � _xÿ� dA

��CRk�DivS�SSS�k � � _x� �DivS�SSSÿk � � _xÿ� dA

��CRk�SSS�k �GRADS� _x�� SSSÿk �GRADS� _xÿ�� dA: �44�

Finally, by standard localization arguments and using Eqs. (20), (21) and (28), Eq. (44) can beshown to take on the following local form:

_cÿ sssk � _dddÿ 12DSSSk � _�F� ÿ SSSk � _hFiR0; �45�

where_�� stands for

d

dt��:

The above inequality is instrumental in developing a thermodynamically consistent as well aspurely mechanical cohesive zone constitutive theory, with or without dissipation.

4. C.Z. constitutive theory: an example

Section 3 was devoted to the derivation of convenient statements of the conservation lawsand the entropy production inequality for the crack surface and for the cohesive zone inparticular. Clearly, the assumption underlying this analysis is that the crack is an entityendowed with its own thermodynamic functions and, therefore, whose behavior can becharacterized via a convenient set of constitutive equations. This section is devoted toillustrating, via a particular example, a general procedure to determine what restrictions areplaced by thermodynamics' second law on the response functions of a proposed cohesive zoneconstitutive theory. Before proceeding any further, it should be noticed that the laws derived sofar are completely independent of the crack mode of propagation. This fact is implicit inhaving derived the aforementioned laws using time independent referential control volumes, i.e.RCVs that do not limit the crack propagation in any way, except for taking place so that aconvenient time interval can be found during which the selected RCV is not swept by any ofthe crack fronts. Furthermore, no mention whatsoever was necessary concerning the behaviorof the bulk material. These remarks imply that the constitutive equations derived herein willalso be independent of both the crack propagation mode and of the bulk material constitution.Along with the discussion of the constraints placed by the second law, this section has also

the purpose of deriving the additional constraints placed on the response functions by thesatisfaction of the principle of material frame indi�erence. Although only a speci®c example isillustrated, the procedure shown herein is general and in no way constrained by the selectedc.z. type. The results presented include the extension of a two-dimensional quasi-static theoryproposed by Costanzo and Allen [14] to a full three-dimensional and dynamic case. Also, it


will be shown that the enforcement of the principle of material frame indi�erence leads to non-trivial results, particularly with regard to the response function that relates the cohesive forcesssk to the opening displacement ddd. These results also have a certain importance, because someof the models discussed in Ref. [14] have been recently utilized in the solution of dynamiccrack propagation problems [15], that is, under conditions not accounted for Ref. [14]. Thiswork therefore provides a formal proof of the validity of the theory in Ref. [14] under anypropagation mode and extends it to a full three-dimensional context.

4.1. Hyperelastic cohesive zone

As already indicated, the crack Ck(t) is assumed to be an entity endowed with its ownresponse functions. In particular, the behavior of the crack surface at points within thecohesive zone, that is for X $ CZk(t), is assumed to be fully characterized in terms of thefollowing crack ®elds c(X, t) sssk, SSSk and DSSSk, where c(X. t) denotes the crack Helmholtz freeenergy function (for a purely mechanical theory such as the one presented herein, this functionreduces to the stored strain energy function). Clearly, in order for the c.z. characterization tobe complete one must specify the variables on which the response functions depend, that is, theso-called (thermodynamic) independent state variables. This choice determines the nature ofthe constitutive model selected.By analogy with the constitutive theory of a hyperelastic solid, the following de®nition is

therefore introduced:

De®nition 1 (hyperelastic cohesive zone). A hyperelastic cohesive zone is de®ned to be a c.z.whose constitutive response is only a function of the current c.z. deformation state, that is, theset of variables ddd, [F] and hFi:

c�X; t� � cc�ddd�X; t�; �F�X; t��; hF�X; t�i�; �46�sssk�X; t� � sssc�ddd�X; t�; �F�X; t��; hF�X; t�i�; �47�SSSk�X; t� � SSSc�ddd�X; t�; �F�X; t��; hF�X; t�i�; �48�

DSSSk�X; t� � DSSSc�ddd�X; t�; �F�X; t��; hF�X; t�i�; �49�where the subscript ``c'' has been used to distinguish the response functions from the physicalquantities characterized by said functions. Furthermore, the latter are assumed to satisfy therequirements imposed by the balance of angular momentum and to be su�ciently smooth topermit the derivations that will follow.Clearly, the constitutive relations in the above equations contain the implicit assumption that

the c.z. is spatially homogeneous, although this hypothesis is not essential to the developmentof the proposed theory.Given the unusual nature of the above constitutive equations, before the analysis of the

restriction imposed by the second law and by the principle of material frame indi�erence, it isnecessary to show that the choice of thermodynamic state variables is ``proper,`` that is, non-redundant. In fact, at this stage it is not obvious whether or not the information carried by,say, the pair (ddd, F+) is any less than that carried by the selected triple of state variables (ddd, [F],hFi). To properly address the question concerning whether or not the information contained in


the triple (ddd, [F], hFi) can be expressed in some more compact form which avoids redundancy,

one needs to address a more fundamental question: is it possible to construct a real or

hypothetical experiment where the variables (ddd, [F], hFi) are independently controlled? The

following lemma proves that one can indeed construct an entire class of thought experiments

where each of the selected state variables can be varied arbitrarily, thus proving that the choice

of independent state variables in the de®nition 1 is correct.

Proposition 2 (admissible c.z. opening processes). Let X $ Ck(t) and to be ®xed. Also, let ddd(X, t),[F(X, t)] and hF(X. t)i be ®xed but arbitrary elements in the domain of the functions cc, sssc, SSSc

and DSSSc. Then, given an arbitrary vector a$R3 and arbitrary tensors A, B $ Lin(R3, R3), there

exists a motion www*k(X*, t*) such that for X* = X and t* = t:

ddd*�X*; t*� � �www*k�X*; t*�� ddd�X; t�; �50�

�F*�X*; t*�� GRADS�www*k�X*; t*�� F�X; t��; �51�

hF*�X*; t*�i � hGRADS�www*k�X*; t*�i � hF�X; t�i; �52�and

_ddd*�x*; t*� � a; �53�

_�F*�x*; t*�� A�F�X; t��; �54�

_hF*�x*; t*�i � AhF�X; t�i � BI: �55�

Proof. Let Xo$Bk(t)[Ck(t) and X$Ck(t) be given together with an arbitrary b $ R3 such that

b�ddd(X, t)$0. Then, chosen a$R3 and A, B $ Lin(R3, R3) arbitrarily, let the function www*k(X*, t*)

be de®ned as follows:

www*k�X*; t*� � 1� �t*ÿ t�b � ddd �a b�

� �wwwk�X0; t� � �F�X0; t� � �t*ÿ t��AF�X0; t� � B��X*ÿ X0�:

�56�Hence, letting Xo=X2xn(X, t) (x$R), for X* = Xo and t* = t, Eq. (56) yields:

www*k�X*; t*� � wwwk�X0; t� � wwwk�X2xn�X; t�; t�: �57�By letting x 4 0+, from Eq. (57) one then obtains:

ddd*�X*; t*�j�x*�x;t*�t� � �www*k�X*; t*��j�X*�X;t*�t� � �wwwk�X; t�� ddd�X; t�: �58�

Furthermore, the gradient of the motion www*k(X*, t*) takes on the form:

F*�X*; t*� � F�X0; t� � �t*ÿ t��AF�X0; t� � B�: �59�Again, letting Xo=X2xn(X, t), for X* = Xo and t* = t, F*(X*, t*) becomes:


F*�X*; t*� � F�X0; t� � F�X2xn�X; t�; t�: �60�Therefore, by letting x4 0+ and making use of the inclusion mapping, from Eqs. (22) and(60) one obtains:

�F*�X*; t*��jX*�X;t*�t� � �F�X; t��; �61�hF*�X*; t*�ij�X*�X;t*�t� � hF�X*; t�i: �62�

With reference to Eqs. (56) and (59), di�erentiation with respect to t*, yields the followingresults:

_www*k�X*; t*� � 1

b � ddd�X; t� �a b�wwwk�X0; t� � �AF�X0; t� � B��X*ÿ X0�; �63�

and

_F*�X*; t*� � AF�X0; t� � B: �64�

Following the same strategy used for the derivation of Eqs. (58) and (61), from Eqs. (63) and(64) one therefore obtains:

_ddd*�X*; t*�j�x*�X;t*�t� �1

b � ddd�X; t� �a b�ddd�X; t� � a; �65�_�F*�X*; t*��j�X*;t*�t� � A�F�X; t��; �66�

_hF*�X*; t*�ij�X*�X;t*�t� � AhF�X; t�i � BI; �67�

where, regarding a, b, A and B as uniform ®elds, use was made of the following relations:

�a b�2 � a b; A2 � A; �B� � 0; hBi � B: �68�

qGoing back to the discussion of the c.z. constitutive response, recall that in order for a set of

constitutive equations to be admissible, one must require that the response functions at handsatisfy the entropy production inequality for any admissible thermodynamic process. The latteris any kinematically admissible transformation which satis®es the momentum balance laws andthe second law of thermodynamics. Both the conservation laws and thermodynamics' secondlaw have been accounted for in deriving the form of the entropy production inequality given inEq. (45), with the exception of the balance of angular momentum which must be enforced, atleast for now, separately. Hence, by substituting Eqs. (47) and (49) into Eq. (45), one sees thatin order for the postulated constitutive relations to be admissible, at any point X $ CZk(t) theymust satisfy the following (variational) inequality:


@c0�ddd; �F�; hFi�@ddd

ÿ sssc�ddd; �F�; hFi��

� _ddd

� @cc�ddd; �F�; hFi�@�F� ÿ 1

2DSSSc�ddd; �F�; hFi�

� �: _�F�

� @cc�ddd; �F�; hFi�@hFi ÿ SSSC�ddd; �F�; hFi�

� �� _hFiR0; �69�

for any admissible motion wwwk(X, t). To determine the general solution of the above inequalityone must ®rst conveniently characterize the ``set of all admissible processes.'' As it turns out, aclass of thought experiments large enough to ®nd the general solution of inequality in Eq. (69)has been already identi®ed in proposition 2. Hence, proposition 2 can now be used to provethe following claim:

Proposition 3 (thermodynamic constraints). Necessary and su�cient conditions for the constitutiverelations in Eqs. (47) and (49) to be thermodynamically admissible are that

sssc � @cc�ddd; �F�; hFi�@ddd

;1

2DSSSc � @cc�ddd; �F�; hFi�

@�F� ;SSSc � @cc�ddd; �F�; F�@hFi : �70�

Proof. For a selected point X $ CZk(t), if one evaluates inequality (69) using the deformationfunction www*k(X*, t*) de®ned in Eq. (56) and lets X* 4 Xo=X2xn(X, t), t* 4 t and x 4 0+,the entropy production inequality takes on the form:

@cc�ddd; �F�; hFi�@ddd

ÿ sssc�ddd; �F�; hFi��

� a� @cc�ddd; �F�; hFi�@�F� ÿ 1

2DSSSc�ddd; �F�; hFi�

� �� A�F�

� @cc�ddd; �F�; hFi�@hFi ÿ SSSc�ddd; �F�; hFi�

� �� AhFi � BI�R0; �71�

where 8a$R3 and 8A, B $ Lin(R3).

Since the choice of a, A and B is completely arbitrary, it immediately follows that the aboveinequality is satis®ed only if the relations in Eq. (70) are satis®ed. qProposition 3 provides a formal proof of a conjecture discussed by Gurtin in Ref. [4]. More

importantly, the above discussion shows that standard methods [39] for the formulation of c.z.constitutive functions can be applied to theories such as that presented herein, provided that aconvenient characterization of the admissible crack motions is found.

A further step in this development consists in deriving the additional constrains placed onthe c.z. constitutive equations by the axiom of material frame indi�erence. The latter requiresthe response functions to be invariant under a change of frame of reference, the latter beingde®ned by the relations

www*k�X; t*� � c�t� �Q�t��wwwk�X; t� ÿ o�; �72�

t* � tÿ a; �73�where a$R, c(t)$R3, Q(t) $ Orth+ (Orth+={QvQÿ1=QT, det(Q)>0}) and a$R. In essence, the


function www*k(X, t*) represents the motion of the system at hand as observed by the movingframe identi®ed by the translation c(t) and rotation Q(t). The crack motion kinematicdescriptors associated with the function www*k(X, t*) are ddd*(X, t*) = [www*k(X, t*)], [F*(X,t*)] = [GRADS(www*k(X, t*))], and hF*(X, t*)i= hGRADS(www*k(X, t*))i. Using Eqs. (72) and (73),it follows that

ddd*�X; t*� � Q�t�ddd�X; t�; �74��F*�X; t*�� Q�t��F�X; t��; �75�hF*�X; t*�i � Q�t�hF�X; t�i: �76�

Hence, with regard to the c.z. constitutive equations, the principle of material frameindi�erence expressed in terms of quantities de®ned in the reference con®guration,7 requiresthat for all orthogonal transformations Q $ Orth+

c*�X; t*� � cc�ddd*�X; t*�; �F*�X; t*��; hF*�X; t*�i�; �77�sss*k�X; t*� � sssc�ddd*�X; t*�; �F*�X; t*��; hF*�X; t*�i�; �78�

DSSS*k�X; t*� � DSSSc�ddd*�X; t*�; �F*�X; t*��; hF*�X; t*�i�; �79�SSS*k�X; t*� � SSSc�ddd*�X; t*�; �F*�X; t*��; hF*�X; t*�i�; �80�

where

c*�X; t*� � c�X; t� � cc�ddd; �F�; hFi�; �81�sss*k�X; t*� � Q�t�sssk�X; t� � Q�t�sssc�ddd; �F�; hFi�; �82�

DSSS*k�X; t*� � Q�t�DSSSk�X; t� � Q�t�DSSSc�ddd; �F�; hFi�; �83�SSS*k�X; t*� � Q�t�SSSk�X; t� � Q�t�SSSc�ddd; �F�; hFi�; �84�

and the arguments (X, t) of the functions ddd, [F] and hFi have been omitted for simplicity.As for the case of an ordinary hyperelastic bulk material, it is possible to show that because

of the thermodynamic restrictions in Eq. (70), Eqs. (78)±(80) are ``automatically'' satis®ed aslong as Eq. (77) is satis®ed. In fact, using the ®rst part of Eq. (70), Eq. (70) and Eq. (81) onehas:

sssc�ddd; �F�; hFi� � @cc�ddd; �F�; hFi�@ddd

� @cc�Qddd;Q�F�;QhFi�@ddd

: �85�

Now, recalling that if f(a) is a smooth function of a$R3 then

@f�Qb�@b

� QT @f�a�@a

��a�Qb�

; �86�

where b$R3, and using the ®rst part of Eq. (70) again to rewrite the last term in Eq. (85), thelatter becomes

7 For a discussion of the material frame indi�erence axiom using quantities de®ned in the reference con®gurationsee Ref. [40].


sssc�ddd; �F�; hFi� � QTsssc�Qddd;Q�F�;QhFi�: �87�Left-multiplying both sides of the above equation by Q and using Eq. (82) one recovers

Eq. (78). The proof that Eqs. (79) and (80) also hold proceeds in a manner analogous to that

used in the case of Eq. (78).

The above discussion implies that, at least for the particular constitutive relations at hand,

the enforcement of the material frame indi�erence axiom reduces to the solution of the

following equation:

cc�ddd; �F�; hFi� � cc�Qddd;Q�F�;Q�F�;QhFi�; 8Q 2 Orth�: �88�Traditionally the general solution to equations such as this is found by setting Q equal to a

``convenient'' element in Orth+ so as to generate a necessary condition for the satisfaction of

the equation at hand. Then, one proceeds to show that the necessary condition obtained is also

a su�cient condition. In the development of a c.z. constitutive theory, the strategy just

outlined does not seem to yield useful results because of the fact that no obvious or physically

motivated8 choice for Q seems to be available. However, it is possible to state a su�cient

condition for the satisfaction of Eq. (88) which is of some interest, because it automatically

yields constitutive relations which satisfy the balance of angular momentum:

Proposition 4 (material frame indi�erence). A su�cient condition for the satisfaction of the axiom

of material frame indi�erence as stated in Eq. (88) and of the balance of angular momentum as

stated in proposition 1 is that:

cc�ddd; �F�; hFi� � cc�d; �C�; hCi�; �89�where cc is a scalar function, d = kdddk, [C]=C+ÿCÿ, hCi= (1/2)(C++Cÿ),

C2 2 Lin�n?; n?�; C2 � PC2I � �F2�T�F2�; and C � FTF: �90�Proof. For any arbitrary Q, using Eq. (89) in conjunction with the right hand side of Eq. (88),

the latter becomes

cc�Qddd;Q�F�;QhFi� � cc�jjQdddjj; �FTQTQF�; hFTQTQFi�: �91�Recalling that Q is an orthogonal transformation, one obtains

jjQdddjj � jjdddjj � d and QTQ � 1: �92�Substituting these relationships into Eq. (91) and using Eq. (89), one obtains:

cc�Qddd;Q�F�;QhFi� � cc�d; �C�; hCi� � cc�ddd; �F�; hFi�; �93�

8 In the development of constitutive theories for bulk materials, Q is set equal to the transpose of the rotation R

(i.e. Q = RT) derived via the polar decomposition of the deformation gradient F. In the present theory, this choicedoes not provide a useful result in that the deformation gradient is in general discontinuous across the crack sur-face.


that is, Eq. (88). This concludes the part of the proof concerning the axiom of material frameindi�erence.Now, substituting Eq. (89) into the ®rst part of Eq. (70) and recalling that

@jjdddjj@ddd� dddjjdddjj ; �94�

one obtains:

sssc � s�d; �C�; hCi� dddjjdddjj ; �95�

where

s�d; �C�; hCi� � @cc�d; �C�; hCi�@d

: �96�

Similarly, substituting Eq. (88) into the second and third part of Eq. (70) and using Eqs. (A.16)and (A.17) in the appendix, one obtains:

1

2DSSSc � @cc

@�F� �@cc

@�F� � 2hFi @cc

@�C� �1

2�F� @cc

@hCi �97�

and

SSSc � @cc

@hFi �@cc

@hFi � 2�F� @cc

@�C� � 2hFi @cc

@hCi : �98�

Now, expressing SSS2k in terms of SSSk, and F2 in terms of hFi and [F], the conditions in Eq. (31)

can be rewritten as

12 �SSSk � DSSSk��hFi � 1

2 �F��T � 12 �hFi � 1

2 �F��SSSk � DSSSk�T �99�and

12 �SSSk ÿ DSSSk��hFi ÿ 1

2 �F��T � 12 �hFi ÿ 1

2 �F��SSSk ÿ DSSSk�T: �100�Substituting Eqs. (97) and (98) into the left hand sides of Eqs. (99) and (100), one ®nallyobtains:

12 �SSSk � DSSSk��hFi � 1

2 �F��T � �hFi � 12 �F��S1�hFi � 1

2 �F��T �101�and

12 �SSSk ÿ DSSSk��hFi ÿ 1

2 �F��T � �hFi ÿ 12 �F��S2�hFi ÿ 1

2 �F��T �102�where S1 and S2, de®ned as:

S1 � @cc

@hCi � 2@cc

@�C� and S2 � @cc

@hCi ÿ 2@cc

�C� ; �103�


are symmetric tensors in Lin(n_, n_). It is now obvious to see that the right hand sides of Eqs.(101) and (102) form symmetric tensors, thus satisfying Eq. (31). qProposition 4 yields the following corollary.

Corollary 1. A necessary and su�cient condition that cohesive zone constitutive equations of theform

c�X; t� � cc�ddd�X; t��; �104�

sssk�X; t��sssc�ddd�X; t��; �105�be material frame indi�erent is that

cc�ddd� � cc�d�: �106�Proof. The su�cient condition is obtained as a special case of proposition 4. The necessarycondition is easily proven recalling that for a cohesive zone with ddd as the only thermodynamicstate variable, the principle of material frame indi�erence takes on the form:

cc�ddd� � cc�Qddd� 8Q 2 Orth�: �107�Interpreting ddd as the position vector of a point A relative to a ®xed origin O, it is immediate tosee that Eq. (107) demands that the value of cc at A be the same as that for any other point Bon a sphere centered at O with radius d = kdddk. Clearly, this is possible if and only if cc is afunction of the radius d of said sphere. qProposition 4 and corollary (4.1.5) are two conceptually important results. In fact, corollary

1 states that a hyperelastic cohesive zone model with ddd as the only kinematic state variable isnecessarily isotropic in the sense discussed in remark 1. Hence, elastic c.z. models based onlyon ddd, which by virtue of the second law of thermodynamics are characterized by SSS+

k =0,cannot ``handle'' anisotropy. A similar result can be proven even for inelastic c.z. models. Forexample, for a rate dependent model of the type

sssk � sssc�ddd; _ddd�; �108�the following proposition can be proven:

Proposition 5 (rate dependent c.z. model: objectivity). A necessary and su�cient condition that thecohesive zone constitutive law in Eq. (108) be material frame indi�erent is that

sssc�ddd; _ddd� � s�ddd; _ddd� dddjjdddjj ; �109�

where

s�ddd; _ddd� � s�Qddd; _Qddd�Q _ddd� 8Q 2 Orth� and 8 _Q 2 Lin�R3;R3�: �110�Proof. Recalling that the statement of the principle of material frame indi�erence in this casereads

Qsssc�ddd; _ddd� � sssc�Qddd; _Qddd�Q _ddd� 8Q 2 Orth� and 8 _Q 2 Lin�R3;R3�; �111�


the proof of the su�cient condition is immediate. To prove the necessary condition let Q= Rd

where the latter is any rotation whose axis contains ddd [34]. Then

Rdddd � ddd and _Rdddd� Rd_ddd � _ddd: �112�

Hence, using Rd in Eq. (111) in place of Q, one obtains:

Rdsssc�ddd; _ddd� � sssc�ddd; _ddd�: �113�By de®nition, this result indicates that sssc must belong to the axis of Rd and that therefore dddkmust be parallel to the opening displacement ddd. qWithout proving similar results for all possible cohesive zone models, and using the language

of strength of materials, one can safely say that any cohesive zone model whose kinematicsrelies only on the opening displacement can be always interpreted to be physically equivalentto a continuous distribution over the cohesive zone of ``two-force members'' and, therefore,intrinsically incapable of describing any anisotropic microstructure. This result is of practicalimportance in view of the widespread use of simple models such as this, particularly innumerical solution schemes [15, 35±37]. By contrast, proposition (4.1.4) being only a su�cientcondition for the satisfaction of the material frame indi�erence principle, implies that theinclusion of the kinematic parameters [F] and hFi allows one to describe a rather wider rangeof c.z. models, both isotropic and anisotropic.

5. Conclusions

A theory of crack behavior under deformation has been presented where a crack is viewed asa special type of interphase with its own constitutive equations. As indicated in theintroduction, the theory presented is fully three-dimensional and independent of the crackpropagation regime. These were some of the main objectives of the work. Other features thatmake this analysis new include the use of kinematic descriptors such as [F] and hFi which areintended to account for the deformation state tangent to the crack plane. This inclusion wasconsidered necessary in view of some recent investigations in dynamic crack propagation whichhave demonstrated the importance of o�-crack-plane deformation in the triggering ofinstabilities [29] in the crack front velocity. It was shown that the new parameters [F] and hFiprovide a possible strategy for the description of anisotropic cohesive zone models. Although adirect comparison has been omitted, this theory is a generalization of that proposed byCostanzo and Allen [14]. The latter was developed under the assumption of quasi-staticevolution and was conjectured to be applicable even in the solution of dynamic crackpropagation problems by Costanzo and Walton [15]. Hence, the results presented hereinprovide a formal proof that the aforementioned conjecture was indeed correct. Future workwill involve the completion of the general theory by formulating a theory of crack evolution.


Acknowledgements

The author wishes to thank Jay R. Walton (Department of Mathematics, Texas A&MUniversity), Lev Truskinovski (Aerospace Engineering & Mechanics Department, University ofMinnesota) and Maria-Carme Calderer (Department of Mathematics, Penn State University)for several valuable discussions and advice.

Appendix A

With reference to Eqs. (89) and (90), this appendix is devoted to the derivation of aconvenient characterization of the derivatives

@cc�d; �C�; hCi�@�F� and

@cc�d; �C�:hCi�@hFi : �A:1�

Before characterizing them, the quantities above need to be properly de®ned. To this end, let[F] = [F(l)] and hFi= hF(Z)i, where l, Z$R are two independent scalar parameters. hence, thequantities in Eq. (A.1) will be de®ned as follows:

@cc�d; �C�; hCi�@�F� : Lin�n?;R3�4R such that

@cc�d; �C�; hCi�@l

� @cc�d; �C�; hCi�@�F� � d�F�

dl; �A:2�

@cc�d; �C�; hCi�@hFi : Lin�n?;R3�4R such that

@cc�d; �C�; hCi�@Z

� @cc�d; �C�; hCi�@hFi � dhFi

dZ: �A:3�

Using a similar strategy, that is, letting [C] = [C(z)] and hCi= hCi(x), with z, x$R, one canalso de®ne the partial derivatives:

@cc�d; �C�; hCi�@�C� : Lin�n?; n?�4R such that

@cc�C�; hCi�@z

� @cc�d; �C�; hCi�@�C� � d�C�

dz; �A:4�

@cc�d; �C�; hCi�@�C� : Lin�n?; n?�4R such that

@cc�C�; hCi�@x

� @cc�d; �C�; hCi�@�C� � d�C�

dx; �A:5�


With reference to Eq. (90), now recall that the quantities [C] and hCi can be expressed as:

�C� � hFi � 1

2�F�

� �T

hFi � 1

2

� �ÿ hFi ÿ 1

2�F�

� �T

hFi ÿ 1

2�F�

� �; �A:6�

2hCi � hFi � 1

2�F�

� �T

hFi � 1

2�F�

� �� hFi ÿ 1

2�F�

� �T

hFi ÿ 1

2�F�

� �; �A:7�

which can be simpli®ed to read

�C� � hFiT�F� � �F�ThFi �A:8�

hCi � hFiThFi � 1

4�F�T�F�: �A:9�

Equations (A.8) and (A.9) can now be used in conjunction with Eqs. (A.2) and (A.3) to obtain:

@cc�d; �C�; hCi�@l

� @cc�d; �C�; hCi�@�C� � @�C�

@l� @cc�d; �C�; hCi�

@hCi � @hCi@l

; �A:10�

@cc�d; �C�; hCi�@Z

� @cc�d; �C�; hCi�@�C� � @�C�

@Z� @cc�d; �C�; hCi�

@hCi � @hCi@Z

; �A:11�

where

@�C�@l� hFiT d�F�

dl� d�F�

dl

� �hFi; @hCi

@l� 1

4

d�F�dl

� �T

�F� � 1

4�F�T d�F�

dl; �A:12�

@�C�@Z� dhFi

dZ

� �T

�F� � �F�T dhFidZ

;@hCi@Z� dhFi

dZ

� �T

hFi � hFiT dhFidZ

: �A:13�

Taking into account the symmetry of the linear operators de®ned in Eqs. (A.4) and (A.5), Eqs.(A.10) and (A.11) can be simpli®ed to read:

@cc

@l� 2

@cc

@�C� � hFiT d�F�dl� 1

2

@cc

@hCi � �F�T d�F�dl

; �A:14�

@cc

@Z� 2

@cc

@�C� � �F�T dhFi

dZ� 2

@cc

@hCi � hFiT dhFi

dZ: �A:15�

Finally, by comparing Eqs. (A.14) and (A.15) with Eqs. (A.2) and (A.3), respectively, one seesthat the operators in Eq. (A.1) admit the representation

@cc�d; �C�; hCi�@�F� � 2hFi @cc�d; �C�; hCi�

@�C� � 1

2�F� @cc�d; �C�; hCi�

@hCi ; �A:16�


@cc�d; �C�; hCi�@hFi � 2�F� @cc�d; �C�; hCi�

@�C� � 2hFi @cc�d; �C�; hCi�@hCi : �A:17�

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a continuum theory of cohesive zone models: deformation and constitutive equations

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