A SINGULAR PERTURBATION APPROACH TO VARIATIONAL PROBLEMS IN FRACTURE MECHANICS

Abstract

We consider functionals of the form I(u, S)=∫Ω\S W(Du)dx+∫S ɸ(u+, u−)dℋn−1, with ɸ(u,v)~|u–v| for small values of |u–v|, which are related to the variational formulation of static or quasi-static phenomena in damage and fracture mechanics. Here ℋn–1 denotes the (n−1)-dimensional Hausdorff measure, Ω is the reference configuration, the function u represents the displacement, which is differentiable outside the “discontinuity surface” S, and u+, u− are the traces of u on both sides of S. The latter can be interpreted as a crack or a plasticity surface. The functions W and ϕ represent the bulk and surface energy densities respectively. These functionals in general are not lower semicontinuous in their natural topology. Hence we may have minimizing sequences with unbounded discontinuity surfaces, and in the limit we could obtain in general a diffuse zone of “non-differentiability.” In order to assure that we obtain solutions whose “fracture” remain confined only on a surface at most, we propose a singular perturbation approach. We approximate the functional I with a sequence of functionals of the form Iε(u, S)=∫Ω\S W(Du)dx+∫S ɸε(u+, u−)dℋn−1. We show that in the model case of ɸ(u, v)=|u−v|, if ɸε(u, v)~|u−v|+εϕ1(u, v) the limits of the minimizers of Iε not only minimize the corresponding problems for I, but also minimize a “first order” problem involving only an appropriate “surface energy density.”