Two phase models for elastic membranes with soft inclusions

Abstract

We derive an effective membrane theory in the thin film limit within a two phase material model for a specimen consisting of an elastic matrix and soft inclusions. The soft inclusions may lead to the formation of cracks within the elastic matrix and the corresponding limiting models are described by Griffith type fracture energy functionals. We also provide simplified proofs of relaxation results for bulk materials.

A Blake-Zisserman-Kirchhoff theory for plates with soft inclusions

“…We state here a version of a relaxation result for bulk materials from [13,54] adapted to our needs. Suppose W : R 3×3 → R is Borel function that satisfies the growth condition c|X| We refer to [54,Theorem 3.1 and Remark 3.3]…”

“…Such materials occur in a variety of quite different applications ranging from biology to geophysics to material science to medicine. We refer to [54] for a more detailed account of applications including references to the literature. Our focus in this contribution lies on thin structures which are of particular interest in mechanical applications.…”

“…Furthermore there are two key steps to establish the lower bound estimates of our main theorem: For estimating the surface contribution we design an auxiliary functional which essentially tracks the surface energies along a given sequence and to which we apply a bulk relaxation result, cf. [13,54]. For the bulk part the main difficulty lies in the identification of the limiting strain, more precisely, in showing that the limiting strain is asymptotically linear in the out-of-plane direction.…”

From atomistic systems to linearized continuum models for elastic materials with voids

“…This corresponds to the investigation of energies that are driven by the competition between elastic and surface energies of perimeter type. Models of this form are gathered under the term stress driven rearrangement instabilities (SDRI), see [13,15,21,27,42,44,45,49,50,63,68,69] for some mathematical and physical literature on the subject.…”