Damage cascade in a softening interface

Arnaud Delaplace,Stéphane Roux,Gilles Pi Jaudier-Cabot

doi:10.1016/s0020-7683(98)00054-7

Abstract

A model describing the damage at an interface which is coupled to an elastic homogeneous block is introduced. Resorting to a real-space renormalization analysis, we show that in the absence of heterogeneity localization proceeds through a cascade of bifurcations which progressively concentrates the damage from the global interface to a narrow region leading to a crack nucleation. The equivalent homogeneous interface behaviour is obtained through this entire cascade, allowing for the analysis of size effects. When random heterogeneities are introduced in the interface, prior to the onset of localization damage proceeds by a sequence of avalanches whose mean size diverges at the first bifurcation point of the homogeneous interface. The large scale features of the bifurcation cascade are preserved, while the details of the late stage are smeared out by the randomness.

Highlights

Where H is the tangent sti}ness operator at the continuum point level and n is the orientation of the localized band[ Another one is the loss of stability at the material level in the sense of the Drucker postulate[ Note that in some well!de_ned cases "associative constitutive laws#\ the loss of uniqueness coincides with the loss of stability in the rheological sense[
For continuous models\ the localization is well!de_ned\ using some criteria based on the loss of uniqueness or the study of the tangent sti}ness operator[ For discrete models\ the localization could not be de_ned in such a manner[ The solution is always unique\ and no tangent could be calculated on the response because of the ~uctuations that are superimposed[
In some well!known cases\ the loss of uniqueness coincides with the loss of stability\ where a bifurcation point is encountered[ In the _rst part\ we show that the study of avalanche statistics allows to detect this point[ Precisely\ the divergence of the avalanche sizes could be directly compared to the loss of stability in a continuous model[

Summary

Introduction

The second approach is discrete random modelling[ It is directed towards the description of the study of the material heterogeneities "i[e[ at a scale lower than the representative volume of the material# "see e[g[ Delaplace et al[\ 0885 ^ Fokwa\ 0881#[ Because of heterogeneity\ the usual employed localization criteria in continuum models cannot be used for two reasons mainly ] the _rst one is that the solution is always unique[ To some extent\ the situation for discrete models is the same as the situation for some rate dependent models where bifurcation is not possible and strain localization cannot be viewed as a loss of uniqueness problem anymore "see e[g[ Dudzinski and Molinari\ 0880 ^ Leroy\ 0880#[ The second one is that no tangent operator can be calculated because of the discrete characteristic of the response\ and because of the ~uctuations that appear all along the curve[ There lies a subtle di.culty ] consider the dimensionless ratio o a:L\ of the microstructure units a of a discrete model over the system size L\ which characterizes the dis! The second approach is discrete random modelling[ It is directed towards the description of the study of the material heterogeneities "i[e[ at a scale lower than the representative volume of the material# "see e[g[ Delaplace et al[\ 0885 ^ Fokwa\ 0881#[ Because of heterogeneity\ the usual employed localization criteria in continuum models cannot be used for two reasons mainly ] the _rst one is that the solution is always unique[ To some extent\ the situation for discrete models is the same as the situation for some rate dependent models where bifurcation is not possible and strain localization cannot be viewed as a loss of uniqueness problem anymore "see e[g[ Dudzinski and Molinari\ 0880 ^ Leroy\ 0880#[ The second one is that no tangent operator can be calculated because of the discrete characteristic of the response\ and because of the ~uctuations that appear all along the curve[ There lies a subtle di.culty ] consider the dimensionless ratio o a:L\ of the microstructure units a of a discrete model over the system size L\ which characterizes the dis! creteness of the medium[ When o tends to 9\ a continuum description is expected to hold[ As we will see later\ the stressÐstrain response of the system converges towards a smooth law whose tangent operator H can be de_ned[ The latter does provide information on the stability of the structure[ \ when stability is analysed using actual responses for non!zero o\ it can be shown that the ~uctuations in the stressÐstrain responses give rise to a non!di}erentiable law\ and this feature brings some useful additional informations on the approach to the loss of stability[ Elaborating over these notions leads to the useful concept of {avalanches|[

Objectives

Conclusion