# Tangential Stiffness Tensor Via Solution of a Body with Many Growing Cracks

The power of the microplane model is limited by the assumption of a kinematic (or static) constraint, which is a simplification of reality. To avoid this simplification, one needs to tackle the boundary value problem of the growth of many statistically unifonnly distributed cracks in an infinite elastic body. This problem is not as difficult as it seems. It appears possible that an approximate solution of this problem might once supersede the microplane model as the most realistic predictive approach to cracking damage.

The problem of calculating the macroscopic stiffness tensor of elastic materials intersected by various types of random or periodic crack systems has been systematically explored during the last two decades and effective methods such as the self-consistent scheme (Budianski and O’Conell 1976; Iloenig 1978), the differential scheme (Roscoc 1952; Hashin 1988), or the Mori-Tanaka method (Mori and Tanaka 1973) have been developed. A serious limitation of these studies was that they dealt with cracks that neither propagate nor shorten (Fig. 14.5.1a). This means that, in the context of response of a material with growing damage illustrated by the curve in Fig. 14.5.1a, these formulations predict only the secant elastic moduli (such as Es in Fig. 14.5.1a). Such information does not suffice for calculating the response of a body with progressing damage due to cracking.

To calculate the response of a material with cracks that can grow or shorten, it is also necessary to determine the tangential moduli, exemplified by Et in Fig. 14.5.1b. Knowledge of such moduli makes it possible, for a given strain increment, to determine the inelastic stress drop 6ocr (Fig. 14.5.1b). This problem has recently been studied by Bazant and Prat (1995, 1997) and Prat and Bazant (1997). Its solution will now be briefly reviewed.

We consider a representative volume V of an elastic material containing on the microscale many cracks (microcracks). On the macroscale, we imagine the cracks to be smeared and the material to be represented by an approximately equivalent homogeneous continuum whose local deformation within the representative volume can be considered approximately homogeneous over the distance of several average crack sizes. Let є and a be the average strain tensor and average stress tensor within this representative volume. To obtain a simple formulation, we consider only circular cracks of effective radius a.

Consider the material to be intersected by N families of random cracks, labeled by subscripts /t — 1,2,..N. Each crack family may be characterized by its spatial orientation?7)(, its effective crack radius «,,, and the number of cracks in family ц per unit volume of the material. The compliance tensor may be considered as the function C — С(а;, аг,…, ajy; щ, гі2,… ,rijv). Approximate estimation of this function has been reviewed by Kachanov and co-workers (Kachanov 1992, 1993; Sayers and Kachanov 1991; Kachanov, Tsukrov and Shafiro 1994).  To obtain the tangential compliance tensor of the material on the macroscale, the cracks must be allowed to grow during the prescribed strain increment Ьє. This means that the energy release rate per unit length

Figure 14.5.1 Stress-strain curves and moduli: (a) effective secant moduli; (b) tangential moduli and inelastic stress decrement due to crack growth.

of the front edge of one crack must be equal to the given critical value R(atl) (or to the fracture energy Gy, in the case of LEFM). For the sake of simplicity, we will enforce the condition of criticality of cracks only in an overall (weak) sense, by assuming that the average overall energy release rate of all the cracks of one family within the representative volume equals their combined energy dissipation rate.

Our analysis will be restricted to the case when the number of cracks in each family is not allowed to change (бпц = 0), i. e., no new cracks are created and no existing cracks are allowed to close. This does not seem an overly restrictive assumption because a small enough crack has a negligible effect on the response and can always be assumed at the outset. Besides, concrete is full of microscopic cracks (or flaws) to begin with, and no macroscopic crack nucleates from a homogeneous material.   The incremental constitutive relation can be obtained by differentiation of Hooke’s law. It reads 6e — CSa + (dC/da^cr 6aM from which

U=1

where 6 denotes infinitesimal variations and C(a(I) is the fourth-order macroscopic secant compliance tensor of the material with the cracks, and E(aJt) is the fourth-order secant stiffness tensor, whose 6×6 matrix is inverse of the matrix of C(a;i).

The surface area of one circular crack of radius o. lL is A — and the area change when the crack radius increases by 6a is 6AM — 2ттаІІ6а1і. We assume we can replace the actual crack radii by their effective radius aд.

The crack radius increments 6aд must be determined in conformity to the laws of fracture mechanics. Let us assume that the cracks (actually microcracks) can be described by equivalent linear elastic fracture mechanics (LEFM) with an R-curve R(atl). This means that the energy release rates must be equal to (or Gf, in the case of LEFM). For the special case Л(ам) — Gj = fracture energy of the matrix of the material, the cracks follow LEFM.     The complementary energy density of the material is U = U (a, aM) = cr ■ С(aM)cr, where the dot indicates scalar product of two second-order tensors. To make the problem tractable, we impose the energy criterion of fracture mechanics (energy balance condition) only in the overall, weak sense. This leads to the following N conditions of crack growth (Bazant and Prat 1995; Prat and Bazant 1997):

(repetition of subscript/tin this and subsequent equations does not imply summation); — crack growth indicator which is equal to 1 if the. crack is growing (6atl > 0) and 0 if it is closing (Safl < 0), while any value 0 < kil < 1 can be used if 6aд ~ 0.

Differentiation of (14.5.2) provides the incremental energy balance conditions;

dC – A, ( QLC

cr. 6a + ( – jer • Де Qa~a " ^п^КцСуб^и J 6a„ — 2m^atlGy6(/t — 1,2,…N)

(14.5.3)

where бкц — 0 except when the crack growth changes to no growth or to closing, or vice versa. The handling of the large jump in ajL is exact if — s” and atl — a“ld because 6(K, tlatl) — – (ісдам)°и – – І exactly.

Substitution of (14.5.1) into (14.5.3) leads to a system of N equations for N unknowns 6ci….6ajy:

N

J2A = (F=l,-N) (14.5.4)

u=     where

В/г ~ & ■ 7Г—E6e — 27rn/ta°ldG/бкр (14.5.6)

0(1^

After solving (14.5.4), evaluating 6cr from (14.5.1) and incrementing cr, one must check whether the case 6afl > 0 and ONp < 0 (or ejy < 0) occurs for any /t. If it does, the corresponding equation in the system (14.5.4) must be replaced by the equation 6ap = 0. The solution of such a modified equation system must be iterated until the case 6atl > 0 and < 0 (or cfy < 0) would no longer occur for any /л.

The formulation needs to be further supplemented by a check for compression. The reason is that the energy expression is quadratic, which means that (14.5.2) is invariant when cr is replaced by —cr. Thus, a negative stress intensity factor K/lL can occur even when (14.5.4) is satisfied, and so the sign of K/p must be checked for each crack family )i. The case K/p < 0 is inadmissible.

Since the present formulation yields only the values of (Kip)2 ~ ETZ(at,) but not the values of Kip, the sign of Kip must be inferred approximately. It can be considered the same as the sign of the stress ONp — iip ■ сгтір in the direction normal to the cracks of 71-th family («,, = unit vector normal to the cracks). [Alternatively, the sign of Kip could be inferred from the sign of the normal component eiv> = "a ‘ £л ”п cracking strain tensor ecr = CCTcr. J Approximate though such estimation

surely is on the microscale of an individual crack, it nevertheless is fully consistent with the macroscopic approximation of C implied in this model. The reason is that all the composite material models for cracked solids are based on the solution of one crack in an infinite solid, for which the sign of Kip coincides with the sign of ацр (or ).

Usually the six independent components of Se are known or assumed, and then (14.5.4) represents a separate system of only N equations for Sai, …бац (this is a. simplification compared to the formulation in the paper, which led to a system of N -1 6 equations). In each iteration of each loading step, the values of Kp are set according to the sign of 6atl in the preceding step or preceding iteration.

If Sap — 0, or ifi(due to numerical error) |ба^| is nonzero but less than a certain chosen very small positive number 6, Up is arbitrary and can be anywhere between 0 and 1, which makes equation (14.5.3) superfluous. The condition (14.5.2) of energy balance in the of constant crack length becomes meaningless and must be dropped. It needs to be replaced by an equation giving Up (or Kip) as a function of cr), which must be used to check whether Kp indeed remains within the interval (0,1). However, it seems that for proportionally increasing loads the case Sap — 0 is not important and its programming could be skipped, using conveniently the value Kp ~ 0.5.

The foregoing solution was outlined in Bazant and Prat (1995) and worked out in detail in Prat and Bazant (1997) (with Addendum in a later issue).

To be able to use (14.5.5), we must have the means to evaluate the effective secant stiffness C as a function of ttp. Bazant and Prat (1995) and Prat and Bazant (1997) adopted the approximate approach developed by Sayers and Kachanov (1991) using the symmetric second-order crack density tensor

N

OL = nval ® % (14.5.7)

д=1   (Vakulenko and Kachanov 1971; Kachanov 1980, 1987b). In this approach, the effective secant compli­ance C is derived from an elastic potential F which is considered as a function of (he crack density tensor cr (in addition of the stress tensor cr):   The elastic potential F(a, cr) can be expanded into atensorial power series. Sayers and Kachanov (1991) proposed to approximate potential F by a tensor polynomial that is quadratic in a and linear in cr:

in which 1)1 and i]2 are assumed to depend only on a — tr a = ^2 rnap’ l’,c *’rsl invariant of a (Sayers and Kachanov 1991). The strain tensor follows from (14.5.9):  The functions rji (a) and 772(a) can be obtained by taking the particular form of the preceding formu­lation for the case of random isotropically distributed cracks and equating the results to those obtained using, e. g., the differential scheme (Ilashin 1988). To this end, we note that if the orientation, density, and size of cracks is isotropically distributed, then a must be spherical, and since its trace is equal to a, it must be a = (a/3)l. Substituting this into the preceding equation we get for this case:      Noting now that the resulting behavior in (14.5.11) must be elastic with effective elastic modulus Ecff and effective Poisson’s ratio Ум, we get the following relationship between the functions 77;, the effective elastic constants and the a:    where E and у are the elastic constants of the uncracked material (included in C°); here we indicate explicitly that the effective elastic constants depend on ft. This dependence can be obtained, for example, by using the differential scheme which, for quasibritlle materials, gives better predictions than the self – consistent scheme (Bazant and Prat 1997). The resulting relationships (see e. g., Hashin 1988 for the details of the derivation) are as follows:

From Eqs. (14.5.7), (14.5.10), and (14.5.12)—(14.5.14) the effective compliance tensor is obtained as a function of aM. Then Saй is calculated from F. q. 14.5.4 for a given be as indicated before.

The crack density may be characterized as a continuous function nlL of spherical angles ф and 0 (Prat and Bazant 1997). Function ntl is then sampled at spherical angles ф^ and (7fI corresponding to the orientations of the microplanes in the microplane model. For isotropic materials such as concrete, the distribution of is initially uniform, and a very small but nonzero value must be assigned to every as the initial condition because no new cracks are allowed to nucleate. For an initially anisotropic material such a rock with joints, function is assumed to peak at a few specified discrete orientations ф*,в*.

In on-going studies, the //-curves are used by Bazant and co-workers as a means to approximate the effect of plastic strains in the matrix of the material occurring simultaneously with the crack growth. Unlike classical plasticity, the plastic strain cannot be considered here to be smeared in a continuum manner because the cracks cause stress concentrations. Therefore, plasticity of the matrix will get manifested by the formation of a plastic zone at the front edge of each microcrack. As is well known, the effect of such a plastic zone can be approximately described by an Я-curve.