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 freedo m can be related to corresponding macroscopic kinematic continuum variables . where the resulting gradients of displacement are selected up to the four th-order and the gradients of rotation up to the third order. The elastic m icro-structural properties For an individual particle are used to derive th e macro-level behavior for a fabric of equal-sized spherical particles, lea ding to a second-gradient micro-polar formulation, In this model, all coeff icients 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-gradien t model and the Cosserat model. In the accompanying paper these reduced for ms are treated in more detail by analyzing the corresponding dispersion rel ations for plane body wave propagation.