A SINGULAR PERTURBATION APPROACH TO VARIATIONAL PROBLEMS IN FRACTURE MECHANICS
Andrea Braides, Alessandra Coscia · Mathematical Models and Methods in Applied Sciences · 1993
We consider functionals of the form I(u, S)=∫ Ω \ S W(Du)dx+∫ S ɸ(u + , u − )dℋ n−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 ℋ n–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 I ε (u, S)=∫ Ω\S W(Du)dx+∫ S ɸ ε (u + , u − )dℋ n−1 . We show that in the model case of ɸ(u, v)=|u−v|, if ɸ ε (u, v)~|u−v|+εϕ 1 (u, v) the limits of the minimizers of I ε not only minimize the corresponding problems for I, but also minimize a “first order” problem involving only an appropriate “surface energy density.”