On the continuity of the trace operator inGSBV(Ω) andGSBD(Ω)

Abstract

In this paper, we present a new result of continuity for the trace operator acting on functions that might jump on a prescribed (n − 1)-dimensional set Γ, with the only hypothesis of being rectifiable and of finite measure. We also show an application of our result in relation to the variational model of elasticity with cracks, when the associated minimum problems are coupled with Dirichlet and Neumann boundary conditions.

Weak formulation of elastodynamics in domains with growing cracks

“…In the case has also finite perimeter, the trace operator T r(•) can be extended to the space G S B D( ; ), using the notion of approximate limit on the point of the reduced boundary F (see [15,Definition 3.9]). Moreover, the following theorem holds true.…”

“…Our choice of Neumann forces, in some sense, is natural. In fact looking at the construction of made in [15], roughly speaking, it turns out that measures the "closeness" of to the boundary. From a physical point of view, this might be interpreted as the fact that, when the elastic material between the Neumann boundary and the crack is infinitesimally small, then its elastic reaction can only balance traction forces which decrease their intensity (proportionally to ).…”

“…From a mathematical point of view, the difficulty is due to the lack of continuity of the trace operator acting on functions having jump sets close to the boundary. In order to solve this problem, we make use of the results obtained in [15], which allow us to restrict our attention to a suitable space of traction forces F.…”

A new proof of compactness in G(S)BD

“…where e(u) denotes the approximate symmetric gradient of u. In this respect, we mention results on compactness and lower semicontinuity [5,12,16,15,27,30,43], Ambrosio-Tortorelli approximations [11,13,14,23,33], dimension reduction, homogenization, atomistic derivation, and nonlocal approximations [1,3,6,10,26,28,31,38,40,41,42], linearization in elasticity [2,24,25], and modeling of fracture, epitaxially strained films, and stress-driven rearragnement instabilities [19,22,29,34]. The common feature of the above mentioned works is that the underlying ambient space is of euclidean type.…”