Linearization of quasistatic fracture evolution in brittle materials

Abstract

We prove a linearization result for quasistatic fracture evolution in nonlinear elasticity. As the stiffness of the material tends to infinity, we show that rescaled displacement fields and their associated crack sets converge to a solution of quasistatic crack growth in linear elasticity without any a priori assumptions on the geometry of the crack set. This result corresponds to the evolutionary counterpart of the static linearization result [Friedrich, Math. Eng. 2 (2020), 75–100], where a Griffith model for nonsimple brittle materials has been considered featuring an elastic energy which also depends suitably on the second gradient of the deformations. The proof relies on a careful study of unilateral global minimality, as determined by the nonlinear evolutionary problem, and its linearization together with a variant of the jump transfer lemma in GSBD [Friedrich and Solombrino, Ann. Inst. H. Poincaré C Anal. Non Linéaire 35 (2018), 27–64].