Inertia effects as a possible missing link between micro and macro second-order work in granular media

Abstract

International audienceThis paper is concerned with a theoretical question as to the definition of instabilities in a granular assembly and its proper formulation at the microscopic level. Recently, this question has taken up much prominence with the emergence of intriguing failure modes such as diffuse failure associated to unstable plasticity of granular materials and microstructural instabilities. An analysis of the second-order work as a general and necessary criterion to detect instabilities is conducted both at the macroscopic and microscopic levels including large deformations. On the basis of a micromechanical analysis of a body consisting of arbitrary interacting particles in a representative element volume (REV), a general formula is derived to quantify the microscopic second-order work involving local variables on the grain scale. The latter emerges as a sum of a configurational term that involves contact forces between neighboring grains, plus a kinetic part consisting of the mechanical unbalance of intergranular forces under dynamics at incipient failure. The present analysis is thought to serve as a clarification of the question of failure in geomaterials typified by a transition from static to a dynamic regime with release of kinetic energy originating from microstructural interactions

Failure in granular media from an energy viewpoint

“…However, in the case of localized failure, the energy analyses distinguished the shear band zone from the remaining part for which the various quantities were computed separately. Furthermore, and towards deeper insights into the link between energetic sources and failure micromechanisms, the same energetic analyses can be performed at the mesoscopic level by considering well-defined subsets of particles such as the minimal polygonal loops [21][22][23], tessellated particle systems [24], or clusters of particles [20,25] exhibiting negative values of second order work [18,26]. For simplicity, none of these later were included in this paper.…”

Microstructural self-organization in granular materials during failure

“…This approach was shown to be a powerful mechanical framework that describes most of the failure modes observed before the conventional plastic limit is reached. Furthermore, taking advantage of a micromechanical approach, a microscopic formulation of the second-order work can be proposed [17] to bridge the macroscopic scale to relevant local scales. In particular, the formulation enables us to identify specific contacts associated with a negative local second-order work.…”

“…Based on the second-order work criterion [12], it has been shown [17] that a link between the macroscopic and microscopic scales in terms of instability within a granular media can be established. The second-order work W 2 , which is generally expressed in terms of the tensorial variables as [9] …”

A multiscale description of failure in granular materials

“…The Lagrangian formulation given in Eq. (1) can be readily differentiated, then providing the following expression of the second-order work (see [6] for more details):…”

“…Stress response path for an undrained biaxial compression (top), and incremental deviatoric strain fields computed at states numbered from 1 to 3 (bottom) contacting particles, and p f denotes the resultant force applied to the particle 'p' of position p x . As specified in [6], the creation or the deletion of contacts is accounted for in this approach. The symbol and 'q', c f is set to zero.…”