Finite element methods for micropolar models of granular materials

We present a thermodynamically based finite element scheme for rate-independent materials and demonstrate its application in modelling the rheological behaviour of granular materials. Starting from the laws of thermodynamics, we have recently developed a new class of micropolar-type constitutive relations for two-dimensional densely packed granular media. This class of constitutive laws is expressed in terms of particle-scale properties, thus providing a direct link between observed macroscopic behaviour and the underlying particle–particle interactions. Here, we demonstrate how the connection to the underlying physics can be maintained and carried through to the finite element implementation phase of the modelling process via the same thermodynamical principles used to construct the constitutive laws. Notably, the study indicates that while the traditional Galerkin-FEM method admits a range of weighting functions, the proposed formulation provides an additional constraint that narrows the choice of admissible weighting functions via the second law of thermodynamics. Additionally, this paper presents insights into the finite element implementation of micropolar models deemed to be appropriate for modelling several classes of heterogeneous media (e.g. granular materials, cellular composites and biological materials). As the kinematics and kinetics of micropolar continua are enriched by the addition of rotational degrees of freedom to each material point, the equations governing boundary value problems for such materials differ from those of other continuum models both from the viewpoint of the constitutive law and the governing conservation laws. Analysis of elastoplastic deformation of micropolar continua is presented.