Calculus of Variations and Geometric Measure Theory

M. Negri - C. Zanini

From finite to linear elastic fracture mechanics by scaling

created by negri on 13 Sep 2012
modified on 05 Aug 2014


Published Paper

Inserted: 13 sep 2012
Last Updated: 5 aug 2014

Journal: Calc. Var. Partial Differential Equations
Volume: 50
Number: 3-4
Pages: 525-548
Year: 2014
Doi: 10.1007/s00526-013-0645-1
Links: pdf file


In the setting of finite elasticity we study the asymptotic behaviour of a crack that propagates quasi-statically in a brittle material. With a natural scaling of size and boundary conditions we prove that for large domains the evolution (with finite elasticity) converges to the evolution obtained with linearized elasticity. In the proof the crucial step is the (locally uniform) convergence of the non-linear to the linear energy release rate, which follows from the combination of several ingredients: the $\Gamma$-convergence of re-scaled energies, the strong convergence of minimizers, the Euler-Lagrange equation for non-linear elasticity and the volume integral representation of the energy release.