What Happens beyond Drucker’s Proposition in Heterogeneous Media

Abstract

This paper briefly reviews our recent analytical and experimental results on 3 interrelated features beyond the peak load in heterogeneous media: continuous bifurcation, damage localization and catastrophic rupture (CR). Firstly, an Elastic Statistically-Brittle model (ESB) was introduced to formulate the basic features of a kind of heterogeneous media, like rocks and cements. The global mean field approximation (GMF) shows that the measure of heterogeneity, like the Weibull modulus m in the distribution of meso-strength plays a key role to distinguish CR from gradual failure. Then, with the ESB model and corresponding experimental results, continuous bifurcation and damage localization are discussed. In accord with these, regional mean field approximation (RMF) is adopted and it shows that any scale of damage localization can satisfy the conservation laws in continuum mechanics. This implies that catastrophic rupture could appear at any state beyond the peak load, depending on the unknown evolution of damage localization zone. Hence, catastrophic rupture seems to occur stochastically at macroscopic level. On the other hand, both experimental and analytic studies demonstrate that a robust power law singularity (-1/2) appears ahead of CR. Preliminary applications of these ideas are briefly described.