# Statistical Determination of Crack Influence Function

The basic characteristic of the new formulation is the crack influence function Л, whose rate of decay is determined by a certain characteristic length f This function represents the stress field due to pressurizing a single crack in the given elastic structure, all other cracks being absent. In practice, the structure is always finite, and thus the values of Лд„ should, in principle, be calculated taking into account the geometry of the structure. However, the crack is often very small compared to the dimensions of the structure. Then the present formulation has the advantage that one can use, as a very good approximation, the stress field for a single crack in an infinite body, which is well known and calculated easily. This is, of course, not possible for cracks very near the boundary of the structure.

The cracks in structures are distributed randomly and their number is vast. Thus, on the macro­continuum level, function Л„„ cannot characterize the stress fields of the individual cracks. Rather, it should characterize the stress field of a representative crack obtained by a suitable statistical averaging of the random situation on the microstructure level.

A method of rigorous mathematical formulation of the macroscopic continuum crack influence function A was briefly proposed in the addendum to Bazant (1994b) and was developed in detail in Bazant and Jirasek (1994a). This method will now be described.

The crack that is pressurized by unit pressure, as specified in the definition of Л, will be called the source crack, and the frozen crack in the structure on which the influence is to be found will be called
the target crack (Fig. 13.3.4a). For the purpose of calculations, the target crack is, of course, closed and glued, as if it did not exist, and the stresses transmitted across the target crack are calculated assuming the body to be continuous, in the following, the global axes will be denoted with capital letters and the position vectors of the target and source cracks by X and H (Fig. 13.3.4a). We take axes (x, y) to be, respectively, parallel and perpendicular to the source crack with origin at its center, and call £ the vector from the source to the target crack (Fig. 13.3.4a). Then function Л(0,£) represents the influence of a source crack centered at x = 0 on a target crack centered at £ (Fig. 13.3.4a).

At the given macro-continuum point, there may or may not be a crack in the microstructure. Function Л corresponding to that point must reflect the smeared statistical properties of all the possible microcracks occurring near that point. To do this, we must idealize the random crack arrangements in some suitable manner.

We will suppose that the center of the source crack can occur randomly anywhere within a square of size s centered at point x = 0; see Fig. 13.3.4b, c, where various possible cracks are shown by the dashed curves, but only one of these, the crack showed by the solid lines, is actually realized. The value of s is imagined to represent the typical spacing of the dominant cracks. In a material such as concrete, approximately s ~ mda where da — spacing of the largest aggregate pieces and m = coefficient larger than 1 but close to 1 (m would equal da if the aggregates were arranged at the densest ideal packing and if there were no mortar layers within the contact zones).

To simplify the statistical structure of the system of dominant cracks, one may imagine the material to be subdivided by a square mesh of size s as shown in Fig. 13.3.4c, with one and only one crack center occurring within each square of the mesh. This is, of course, a simplification of reality because the underlying square mesh introduces a certain directional bias (as is well known from finite element analysis of fracture). It would be more realistic to assume that the possible zone of occurrence of the center of each crack is not a square but has a random shape and area about s x s, and that all these areas are randomly arranged. But this would be too difficult for statistical purposes, and probably unimportant with respect to the other simplifications of the model.

Let us now center coordinates x and у in the center of the square s x s, as shown in Fig. 13.3.4b, and consider the influence of a source crack within this square on a target crack at coordinates £ = (£, ?/). The macroscopic crack influence function should describe the influence of any possible source crack within the given square in the average, smeared macroscopic sense. Therefore, Л(0,£) is defined as the mathematical expectation £ with regard to all the possible random realizations of the source crack center within the given square s x s, that is  (13.3.34)

The vector (£ — x, Г) — y) — r = vector from the center x s (x, y) of a source crack to the center

£ = (£, 77) of the target crack. In detail, Ц0,О = [ f to(x, y)a{‘](£-x, ri-y)dxdy

S J-S/2J-S/2

Here crO is the field of the maximum principal stresses caused by applying a unit pressure on the faces of the source crack, and the integrals represent the statistical averaging over the square sxs (Fig. 13.3.4b). Certain specified weights w(x, y) have been introduced for this averaging. At first, one might think that uniform weights w might be appropriate, but that would not be realistic near the boundaries of the square because a crack cannot intersect a crack centered in the adjacent square, and, in practice, cannot even lie too close. Rigorously, one would have to consider the joint probability of the occurrences of the crack center locations in the adjacent squares, but this would be too complicated. We prefer to simply reduce the probability of occurrence of the source crack as the boundary of the square is approached. For numerical computations we choose a bell-shaped function in both the x and у directions, given as  225 U’° = ’64“

(13.3.36)

for x < s/2,y < s/2, and w(x, y) — 0 otherwise; constant wq is selected so the integral ofw(x, y) over the square s x, s be equal to 1. It may be added that there is also a practical reason for introducing this weight function. If the weights were uniform over the square, function Л would not have a smooth shape, which would be inconvenient and probably also unrealistic for a continuum model.

The stress field to be substituted into (13.3.36) is given for two dimensions by the well-known Westergaard’s solution (see Chapter 4). However, the integral in Ecp (13.3.35) is difficult to evaluate analytically, and it is better to use numerical integration to obtain Л.

The asymptotic properties of function Л for large r can nevertheless be determined easily (BaZant 1992b, 1994b) by considering the lines of influence from various possible source cracks to the given target crack as shown in Fig. 13.3.4b. If the target crack is very far from the square in which the source crack is centered, all the possible rays of influence are nearly equally long and come from nearly the same direction. Therefore, the integral in Eq. (13.3.35) should exactly preserve the long range asymptotic field

o-O.