Calculus of Variations and Geometric Measure Theory

S. Almi - E. Davoli - M. Friedrich

Non-interpenetration conditions in the passage from nonlinear to linearized Griffith fracture

created by davoli on 22 Apr 2022
modified on 05 May 2023


Accepted Paper

Inserted: 22 apr 2022
Last Updated: 5 may 2023

Journal: J. Math Pure Appl.
Year: 2022


We characterize the passage from nonlinear to linearized Griffith-fracture theories under non-interpenetration constraints. In particular, sequences of deformations satisfying a Ciarlet-Ne\v{c}as condition in SBV2 and for which a convergence of the energies is ensured, are shown to admit asymptotic representations in GSBD2 satisfying a suitable contact condition. With an explicit counterexample, we prove that this result fails if convergence of the energies does not hold. We further prove that each limiting displacement satisfying the contact condition can be approximated by an energy-convergent sequence of deformations fulfilling a Ciarlet-Ne\v{c}as condition. The proof relies on a piecewise Korn-Poincaré inequality in GSBD2, on a careful blow-up analysis around jump points, as well as on a refined GSBD2-density result guaranteeing enhanced contact conditions for the approximants.

Keywords: non-interpenetration, Griffith fracture, linearization, Ciarlet-Ne\v{c}as, contact condition