Analysis of dilatancy relation and shear-band formation in granular materials based on Eshelby-Mandel tensor

The theory of configurational or material forces based on the Eshelby stress tensor (also called energy-momentum tensor) has provided a general and efficient way to describe the motion of material defects and other inhomogeneities within the framework of continuum mechanics. In this paper, we explore how to use the configurational forces to describe the behavior of homogeneous granular materials by considering the material characteristics on both continuum and discrete particle levels. In particular, dissipative driving forces based on the Eshelby-Mandel stress tensor are utilized as the driving force of the configuration variations in the form of shear-induced volume change. The energy dissipation induced by the relative sliding at particle contacts is considered in the configurational forces. To characterize the dilation of a homogeneous granular material with uniform deformation, a virtual plane is introduced to facilitate the analysis and to derive the dilatancy formulation. With the consideration of the shear-band geometry and the requirement of configurational force equilibrium across the boundary of a shear-band, the condition for the onset of a shear band is derived. For granular specimens subjected to biaxial compression, the analyses recover the well-known Rowe's dilatancy formulation and yield the shear-band orientation identical to that obtained from the classical bifurcation analysis within the framework of elasto-plasticity.