Calculus of Variations and Geometric Measure Theory

S. Conti - M. Focardi - F. Iurlano

Phase field approximation of cohesive fracture models

created by focardi on 27 May 2014
modified on 17 Dec 2018


Published Paper

Inserted: 27 may 2014
Last Updated: 17 dec 2018

Journal: Ann. I. H. Poincaré
Volume: 33
Pages: 1033--1067
Year: 2016


We obtain a cohesive fracture model as a $\Gamma$-limit of scalar damage models in which the elastic coefficient is computed from the damage variable $v$ through a function $f_k$ of the form $f_k(v)=\mathrm{min}\{1,\varepsilon_k^{1/2} f(v)\}$, with $f$ diverging for $v$ close to the value describing undamaged material. The resulting fracture energy can be determined by solving a one-dimensional vectorial optimal profile problem. It is linear in the opening $s$ at small values of $s$ and has a finite limit as $s\to+\infty$. If the function $f$ is allowed to depend on the index $k$, for specific choices we recover in the limit Dugdale's and Griffith's fracture models, and models with surface energy density having a power-law growth at small openings.

Keywords: $\Gamma$-convergence, Damage problems, cohesive fracture, phase field models