Merits and limits of a variational definition of the effective toughness of heterogeneous materials

The objective of this study is to assess the variational definition of the effective fracture energy of a heterogeneous brittle material recently proposed by Schneider (2020) based on available mathematical homogenization results. First, similarities between the variational problems defining this effective surface energy and variational limit loads problems, are highlighted. These analogies allow us to compute the effective surface energy using the analytical and computational tools developed to determine the extremal surfaces of ideally plastic heterogeneous materials. Second, the same analogy opens the way to the derivation of variational bounds on the effective surface energy which appear to be in good agreement with the numerical simulations of other authors (Lebihain et al., 2021). Third, the status of this effective surface energy with respect to the effective toughness of heterogeneous materials is discussed. It is shown to always be a lower bound to the actual energy dissipated along the propagation of a crack in a heterogeneous material. Examples where both energies are different are exhibited.

Automated summary

This study aims to evaluate the variational definition of the effective fracture energy of a heterogeneous brittle material based on available mathematical homogenization results. The research reveals the similarities between the variational problems defining the effective surface energy and variational limit loads problems, and calculates the effective surface energy using analytical and computational tools. Additionally, the study derives variational bounds on the effective surface energy, which align well with numerical simulations conducted by other researchers. The relationship between this effective surface energy and the effective toughness of heterogeneous materials is also discussed.