A Derivation of Linearized Griffith Energies from Nonlinear Models

Abstract

We derive Griffith functionals in the framework of linearized elasticity from nonlinear and frame indifferent energies in a brittle fracture via -convergence. The convergence is given in terms of rescaled displacement fields measuring the distance of deformations from piecewise rigid motions. The configurations of the limiting model consist of partitions of the material, corresponding piecewise rigid deformations and displacement fields which are defined separately on each component of the cracked body. Apart from the linearized Griffith energy the limiting functional also comprises the segmentation energy, which is necessary to disconnect the parts of the specimen.

Functionals Defined on Piecewise Rigid Functions: Integral Representation and $$\varGamma $$-Convergence

“…However, our primary purpose comes from the study of free-discontinuity problems defined on the space GSBD p , see [38], which has obtained steadily increasing attention over the last years, cf., e.g., [30,31,32,33,34,35,46,47,48,50]. We have indeed already mentioned before how the analysis of partition problems has proved to be a relevant tool in the study of freediscontinuity problems on SBV .…”

Equilibrium Configurations for Epitaxially Strained Films and Material Voids in Three-Dimensional Linear Elasticity

“…We stress that this cannot be obtained by some general existence results, but is achieved through a very careful construction (pp. [28][29][30][31][32][33][34][35][36], that follows only partially the analogous one in [18]. We believe that the construction in [18] could indeed be improved by adopting an approach similar to ours, in order to take also some pathological situations into account.…”

“…To this end, we will need the following measure-theoretical result. (See [34,Lemma 4.1,4.2] and note that the statement in fact holds in arbitrary space dimensions for measurable functions. )…”

Griffith energies as small strain limit of nonlinear models for nonsimple brittle materials

“…Eventually, a result in dimension two without a priori assumptions on the crack paths and the deformations, in the general framework of free discontinuity problems (see [35]), has been derived in [39]. This analysis relies fundamentally on delicate geometric rigidity results in the spirit of [46,22].…”

“…Concerning the passage to the linearized system, the essential step is to establish a compactness result in terms of suitably rescaled displacement fields which measure the distance of the deformations from the identity. Whereas in [39] this is achieved by means of delicate geometric rigidity estimates, the main idea in our approach is to partition the domain into different regions in which the gradient is 'almost constant'. This construction relies on the coarea formula in BV and is the fundamental point where the presence of a second order term in the energy is used to pass rigorously to a linear theory.…”