A porous media fracture model based on homogenization theory

TL;DR

This paper introduces a new homogenization-based regularized fracture model for crack propagation in porous media, derived through asymptotic expansions and cell problem solutions, providing a mathematically rigorous framework.

Contribution

The paper develops a novel homogenized fracture model for porous media using asymptotic expansions and homogenization theory, offering a new mathematical approach.

Findings

Derivation of a homogenized fracture model for porous media.

Separation of minimality conditions for primary and secondary variables.

Balance of homogenized energy relation established.

Abstract

A novel regularized fracture model for crack propagation in porous media is proposed. Our model is obtained through homogenization theory and formal asymptotic expansions. We start with a regularized quasi-static fracture model posed in a periodically perforated domain obtained by periodic extension of a re-scaled unit cell with a hole. This setup allows us to write two separated minimality conditions for the primary (displacement) and secondary variables plus a balance of energy relation. Then we apply the usual asymptotic expansion matching to deduce limit relations when the re-scaling parameter of the unit cells vanishes. By introducing cell problems solutions and a homogenized tensor we can recast the obtained relations into a novel model for crack propagation in porous media. The proposed model can be interpreted as a regularized quasi-static fracture model for porous media. This model yields two separated (homogenized) minimality conditions for the primary and secondary variables and a balance of homogenized energy relation.