Micro-mechanical modelling of granular material. Part 1: Derivation of a second-gradient micro-polar constitutive theory

This contribution is one in a series of two papers. In the current paper a constitutive law is developed that includes the micro-structural effects by particle displacement as well as particle rotation. Both degrees of freedom can be related to corresponding macroscopic kinematic continuum variables. where the resulting gradients of displacement are selected up to the fourth-order and the gradients of rotation up to the third order. The elastic micro-structural properties for an individual particle are used to derive the macro-level behavior for a fabric of equal-sized spherical particles, leading to a second-gradient micro-polar formulation. In this model, all coefficients are expressed in terms of particle stiffness and particle structure. It is shown that the second-gradient micro-polar model can be reduced to simpler forms, such as the classic linear elastic model, the second-gradient model and the Cosserat model. In the accompanying paper these reduced forms are treated in more detail by analyzing the corresponding dispersion relations for plane body wave propagation.