Localisation de Déformations et Chaînons de Contraintes en Mécanique des Milieux Granulaires
E. Frossard. Expert, Bureau d’ingénieurs-conseils Coyne et Bellier, firstname.lastname@example.org
Abstract: A former paper has shown recently that a wide set of essential properties of mechanical behavior in granular media, result explicitly of statistical physics which rules the energy dissipation by friction at interparticle contacts, within the moving granular mass (ref.). This new approach ,which includes as key feature a minimum energy-dissipation rule , has been called the Energy Approach (“l’Approche Energétique”).
The present paper is aimed at applying this new approach to the phenomenon of strain localization and stress concentrations, considered here as resulting of the progressive growth of heterogeneities, trigged by statistical fluctuations within the moving granular mass.
The first part focus on the relations, defined by the the framework of the Energy Approach between local and average variables within heterogeneous movements. For usual movements under regular conditions, the heterogeneities of stresses and strain rates, are shown to comply with a set of two compatibility conditions. These conditions, related to thermodynamic consistency, occur to be an extension of the classical Hill-Mandel macro-homogeneity condition. Under these compatibility conditions, the equations of Energy Approach, when verified locally, result to hold also for the average variables, in spite of the presence of strong heterogeneities. including eventual localization of strains in shear bands, together with stress concentrations.
On this basis, are then analyzed the main following features:
- the orientation of shear bands, and stress concentration lines, where classical values are found for shear band orientation, however with the result that the relation between shear band orientation and dilatancy is found to depend strongly upon the type of micro-mechanism producing dilatancy;
- the internal structure of macroscopic shear bands, where the structure defined by the Energy Approach, matches fairly well with the structure described by experimental micrometric measurements recently
published by Nemat-Nasser and Okada(ref.); and where the confrontation with these experimental results leads to detail the features of these shear structures in function of key material parameters, in particular average diameter of particles, e.g. the shear structure width concentrating 99% of dissipation, is found to be about 12 d ;
- the internal structure of eventual macroscopic stress concentration lines, for which specific limitations (static equilibrium conditions, macroscopic Failure Criterion) are shown to limit the macroscopic effects to weak structures as compared with shear bands, keeping the strong stress concentration lines within the microscopic scale of contact forces;
- the criterion of localization development, considered as a particular kind of heterogeneity growth, induced by the minimum dissipation rule, the resulting criterion is found based on non-convexity of specific energy dissipation rate, towards correlated fluctuations in density and strain; the resulting features of localization development correspond quite well to experimental results;
- the evolution of parallel shear bands systems ,where it is shown explicitly that under monotonous strain conditions, the evolution is leading to the concentration of one set of shear bands into a unique shear band, itself concentrating the movements up to the maximum allowable: this results to be a complete micromechanism for generation of failure lines.
poudres & grains NS 3 , 1-119 (Octobre 2004)