Cracks Near Boundary
When the boundary is near, the crack influence function should be obtained by solving the stress field of a pressurized crack located at a certain distance d from the boundary; Fig. 13.3.5d-g. Obviously, the function will depend on d as a parameter, i. e., Л(x,£,d). Functions Л will be different for a free boundary, fixed boundary, sliding boundary, and elastically supported boundary or interface with another solid (Fig. 13.3.5d-g). When the crack is near a boundary corner (Fig. 13.3.50, Л represents the solution of the stress field of a pressurized crack in the wedge, and will depend on the distances from both boundary planes of the wedge. These solutions will be much more complicated than fora crack in infinite body, and
simplifications will be needed. On the oilier hand, because of the statistical nature of the crack system, exact solutions of these problems are not needed. Only their essential features arc.
A crude but simple approach to the boundary effect is to consider the same weight function as for an infinite solid, protruding outside the given finite body. In the previous nonlocal formulations, based on the idea of spatial averaging, the same weight function as for the infinite solid has been used in the spatial integral and the weight function has simply been scaled up (renormalized), so that the integral of the weight function over the reduced domain would remain 1. In the present formulation, such scaling would have to be applied to all the weight functions whose integral should be 1, i. e., a, ф, В, C. For those weight functions whose integral should vanish, a different scaling would be needed to take the proximity of the boundary into account; for example, the values at the boundary should be scaled up so that the spatial integral would always vanish, as indicated in (13.3.22). As a reasonable simplification, this might perhaps be done by replacing the к, ш values for the integration points £„ of the boundary finite elements by where the multiplicative factor кь is determined from the condition that, Л;і„ =- 0 (with
the summation carried over all the points in the given finite body);