Department of Mathematics Colloquium
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Abstract |
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| This is joint work with Jagan M. Padbidri
(George W. Woodruff School of Mechanical Engineering, Georgia Institute
of Technology) and Balasingam Muhunthan (Department of Civil and
Environmental Engineering, Washington State University) We use numerical simulations to uncover (1) the micromechanism of dilatancy and critical state, (2) the intrinsic length scale that characterizes shear bands in granular materials. Dense granular materials exhibit a peculiar behaviour − dilatancy - volume increase when sheared under constant pressure. More precisely, when sheared under constant pressure, their volume either dilates or decreases, depending on the combination of pressure and porosity. The critical state is the boundary between dilating and compacting states when material shears at constant volume. The set of critical state points in the pressure-porosity space forms the critical state line. The phenomenological Critical state theory, based on such observations, and its modifications, are at the core of modern geomechanics. Yet, current understanding of dilatancy and critical state is purely empirical. The fundamental question: what are the micromechanisms that produce dilatancy and compaction? - has not been answered, except in a vague manner. The classic simplistic answer that nearly rigid particles must climb over each other to accommodate the imposed shear, only brings about other questions: Why other materials don’t dilate as the rigid sphere model of atoms would predict? Why the critical state depends on pressure? We show that the key to this distinct granular behaviour is the presence of intrinsic stress, the existence of which has been postulated earlier, but its physical nature has remained conjectural. We use the graph theory representation of particles assemblies, first to provide the micromechanical definition of the intrinsic stress, then to quantify its effect on the change of volume under shear. Persistent shear bands in granular materials occur at later stages of deformation. Typically, widths of shear bands are about 10-20 particle diameters. What determines this length? Strain localization in the form of shear bands is accompanied by accompanied by massive rolling of particle. On a single contact level, rolling is favored over frictional sliding, as a mechanism for rearrangement of particles. Yet, on a level of an assembly, rolling is constrained by neighbors. The result is a characteristic rolling correlation length. Our numerical simulations, specifically designed for this problem, indicate that the transmission of rotations depends on direction. Specifically, it depends on the strength of the force chain branch in that direction. The maximum propagation distance is comparable to observed widths of shear bands.
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