On large deformation granular strain measures for generating stress-strain relations based upon three-dimensional discrete element simulations

Generating stress-strain relations based upon three-dimensional discrete element simulations, for hierarchical multiscale constitutive modeling of granular materials at finite strain, requires measures of stress and strain in the reference and current configurations. The Cauchy stress tensor calculated from interparticle forces and branch vectors resulting from discrete element simulations, is well-established, but the various finite strain definitions existing in the literature are not as well studied or accepted. Thus, the paper develops seven finite strain measures in three-dimensions (Cartesian coordinates assumed): two are calculated in the reference configuration, and five are calculated in the current configuration. In the process, a time-integrated deformation gradient is formulated, with which the Cauchy stress tensor can be mapped to either the Kirchhoff stress, first Piola-Kirchhoff stress, or second Piola-Kirchhoff stress. The difficulty in calculating strain measures for granular materials is the discrete particulate nature of the materials. In the paper, strain measures are motivated by the equivalent continuum method, but are extended to finite strain in three-dimensions. Simulations are conducted to test the finite strain measures, and it is found that these strain measures are independent of rigid body rotation of a particle assembly. Granular Cauchy stress is also calculated for these discrete element simulations to be able to plot stress-strain curves in the current configuration. Finally, two numerical examples with large deformation are simulated: (1) cavity expansion, and (2) pile penetration; and their results are analyzed.