DAMAGE TENSORS AND THE CRACK DENSITY DISTRIBUTION

hr. J. Sohdr Swucrures Vol. 30, No. 20, pp. 285%2877,Pnnted in Great BritainDAMAGE19930TENSORS AND THE CRACKDISTRIBUTION0020-7683/93S6.OOf .oO1993 Pergamon Press LtdDENSITYV. A. LUBARDA and D. KRAJCINOVICDepartment of Mechanical and Aerospace Engineering, Arizona State University,Tempe, AZ 85287-6106, U.S.A.(Receivedinfinal form 17 December 1992)Abstract-The paper presents an algorithm for the derivation of damage tensors emphasizing itsrelationship with the actual and approximate crack density distributions. The proposed model isillustrated using scalar, second and fourth order continuous tensor approximations of some typicaltwo and three dimensional crack distributions. It is also shown that the occurrence of regions withnegative crack density (anticrack regions) is in many cases a common and as yet unexplored featureof the approximate solutions.1. INTRODUCTIONBrittle deformation processes in materials with inferior tensile strength, such as rocks,concrete, ceramics and some glassy polymers, are of continuing interest for practitionersand theorists alike. In a great majority of cases the inelastic deformation of these materialsis, at room temperature, attributable to microcracking. Some of the early continuumdamage theories made an attempt to model the macro response of brittle processes modifying existing plasticity theory. However, the constraints which are placed by microcrackson the displacement field are radically different from those imposed by crystalline slips. Thedifference between brittle and ductile deformation is further emphasized by the fact thatthe microcrack growth in a fundamental manner depends on the sign of normal stresses. Itis, therefore, not surprising that the plasticity based phenomenological models were notparticularly successful in replicating salient aspects of the brittle deformation caused bynucleation and growth of a large number of microcracks.The rapid development of the continuum damage mechanics in the last two decadesproduced various contrasting and even contradictory phenomenological models. Usingclever artifices various authors suggested a host of different mathematical representationsfor the damage (internal, hidden) variable. The list runs from scalars (Kachanov, 1958;Lemaitre and Chaboche, 1978 ; Lemaitre, 1987 ; etc.), axial vectors (Davison and Stevens,1973 ; Krajcinovic and Fonseka, 198 1; Talreja, 1985 ; etc.), second order tensors (Vakulenkoand Kachanov, 1971; Dragon and Mroz, 1979 ; Kachanov, 1980, 1992 ; Cordebois andSidoroff, 1982 ; Murakami, 1988 ; Karihaloo and Fu, 1989 ; etc.), fourth order tensors(Chaboche, 1982 ; Simo and Ju, 1987 ; Chow and Wang, 1988 ; Krajcinovic, 1989 ; Lubardaand Krajcinovic, 1993 ; etc.), to a series containing all even order tensors (Onat and Leckie,1988).The primary objective of this paper is to examine the relationship between a given,experimentally determined, distribution of cracks and the scalar, second order and fourthorder tensor damage parameters. The experimentally measured microcrack densities inplanes with different inclinations are typically represented in form of the rosette histogram.This rosette histogram is subsequently approximated by a distribution function defined ona unit sphere and centered in a material point. This distribution function can be furtherexpanded into a series of spherical functions containing dyadic products of unit vectors andthe Kronecker delta tensor. The ensuing series is typically truncated at a desired tensorialrank of these dyadic products. In this sense each of the above mentioned damage representations is an approximation which may or may not be sufficiently accurate in eachparticular case. The objective of this study is to present an algorithm for the derivation of thedamage parameters and to examine their ability to approximate the microcrack distributionfunction in several important cases. A common feature that occurs in many cases, when2859

V. A. LUBARDAand D. KRAJCINOVIC2860either second or fourth order damage tensors are utilized to approximate the actual crackdistribution, are the regions of negative crack density.2. DAMAGETENSORSANDTHE CRACKDENSITYDISTRIBUTIONConsider a solid specimen containing a certain distribution of microcracks accumulatedduring a specific loading program from some initial state. Various damage variables wereintroduced in the literature to adequately represent the degraded state of the material. Ifthe current crack pattern in the representative volume element is such that cracks areuniformly distributed in all planes, regardless of their orientation, a scalar damage variablebecomes a natural choice. The corresponding distribution of damage is referred to asisotropic. If cracks are nonuniformly distributed over differently oriented planes, the damage distribution and correspondingly the material response are anisotropic. A distributionfunction p(n) (defined on a unit sphere) must in this case be introduced to define thedirectional dependence of the crack density. This function can be expanded in a Fouriertype series of certain families of spherical functions (Kanatani, 1984 ; Onat and Leckie,1988), containing dyadic products of the unit vector and the Kronecker delta tensor. Inaddition to the scalar (isotropic) term, the second, fourth and higher even order symmetrictensors appear in this representation. Therefore, the accurate analytical description ofdamage by even higher order tensors is a complicated task, which has generated a lot ofcontroversy in the last two decades. The following derivation in Section 2 is based on moregeneral development, presented by Kanatani (1984).2.1. Scalar damage variableIf the crack distribution is isotropic, the crack density does not depend on the orientation of the normal n to the plane through a material point, i.e. :p(n) P.(1)Expression (1) can also be used to approximate a nearly isotropic distribution, in whichcase p(n) is not a constant, but varies weakly with the orientation of the unit vector n. Thevalue of the average crack density p is then obtained by integrating eqn (1) over all directionsspanning the entire solid angle R 471s4np(n) dR 411 .(2)From eqn (2) the average crack density is :P &’PO(3)wherePO s4np(n)dfh(4)is the density of all cracks within a representative unit volume. Consequently, the damageis characterized by a single scalar parameter po, which does not change the existing symmetries of the original matrix. As a result of its simplicity, the scalar damage variable wasextensively utilized in the literature (Lemaitre, 1987, 1992).2.2. Second order damage tensorIn a general caSe of loading of initially anisotropic rocks the planes containing extremedensities of damage are not mutually perpendicular. Consequently, both the damage itselfand its effect on the material effective stiffness are anisotropic. However, in the case of

Crack density distribution2861initially isotropic materials subjected to proportional loading, the density of damage ismaximum in the plane perpendicular to the largest principal stress, and minimum in theplane normal to the minimum principal stress. To study this case of damage distribution,it seems reasonable to approximate its density distribution by an oval curve (Fig. 1). Thisclass of damage distribution can be approximated adequately by a second order tensor. Ifpii denotes the components of the second order crack density tensor, the density of cracksembedded in the planes with a normal n is defined by the expression :depicted by the oval shape of Fig. 1. Integrating eqn (5) over the entire solid angle, andusing :Jn,njdQ f 6,,4a(6)where Sfj denotes the Kronecker delta tensor, it follows that the first invariant of the secondorder crack density tensor is :3PoPkk bn’The summation convention is used for the repeated indices, and p. is the total crackdensity, defined by eqn (4).Multiplying eqn (5) with njni, integrating over all directions and using :Jninjnknf dR ‘ Iijk,p(8)4nit follows that :(9)In eqn (8)ii/k/ f(6ijdkI 6ik8j[ ddjk).(10)Substituting eqn (7) into eqn (9), the crack density tensor can be expressed as :pij E(Dijf dij).The symmetric second order tensor :Fig. 1. An oval shape corresponding to the second order tensor dmcription of the crack densityd st bution.(11)

V. A. LUBARDA and2862D. KRAJCINOVICon the right hand side of eqn (11) is referred to as the damage tensor. In view of eqn (1 l),the second order tensor approximation of the crack density distribution (5) can be rewrittenas:p(n) g DijFZ,Rj- g.(13)It is instructive to rewrite eqn (13) using the deviatoric part of the damage tensor Q,,defined as :0; :sp(n)(ninj)’ dfi Dij- 6,,4n(14)where (ninj)’ ninj-@ii is the deviatoric part of the tensor nin,/. Using eqns (11) and (14)the deviatoric part of the second order crack density tensor is :(15)Hence, eqns (5) and (13) can be rewritten as :p(n) ggS,.&ninj. (16)The first term on the right hand side of eqn (16)defined by a single scalar po. The second term in eqn (16)approximation of the deviation of the crack distributionIn the case of two dimensional analysis, the densityPO while the counterpartss2np(n)de,represents the isotropic damage,represents the second order tensorfrom its average value.of all cracks within a unit area is :(17)of eqns (6) and (8) are :s2nninid6’ nS/ie(18)(19)Consequently,becomes :the second order approximationof the crack density distribution(20)where the second order damage tensor Dij is given by :

Crack density distribution2863D, (21)s 2np(n)nini de.Equation (20) can be rewritten in terms of the deviatoric part of the damage tensorD D, - fp,,6,, as :p(n) g zD&z,.(22)Again, the last term on the right hand side of eqn (22) represents the deviation fromthe isotropic damage.2.3. Fourth order damage tensorExperimental observations of microcracks in different materials, subjected to a varietyof conditions, abound in the existing literature. In general, microcracks are embedded inplanes perpendicular to the maximum principal stress. For example, in a material such asrock (Hallbauer et al., 1973; Zheng et al., 1991) subjected to uniaxial compression, theangles subtended by microcrack planes and maximum normal stress are distributed withina range of (10-15)‘. The range of angles can be much larger in the case of internalpressures generated by expansive chemical reactions, corrosion, internal heat sources, etc.Higher order tensor variables are, therefore, often needed to improve the accuracy of theapproximate (smooth) representation of the complicated crack distribution, generated inthe course of arbitrary load programs. The fourth order tensor approximation of the crackdensity distribution is defined as :p(n) Pi&WjWb(23)where P , are the components of the fourth order crack density tensor. Integrating eqn(23) over all directions n and using eqn (8), it follows that :Piijj 5PO471’The symmetry properties of the crack density tensor, giving piijj pljlj pii,!, areutilized in derivation of eqn (24). Furthermore, it can be shown that :sninjnkn,n,ng dR 4; Ir/k,a,j,4n(25)whereThe tensor Zijklin eqn (26) is defined in eqn (9). Multiplying eqn (23) by n,ng andintegrating the product over all directions, leads to : ,rli, It can be similarly shown that :SASmzo-5& (DM!f 8 ).(27)