17. Large Deformation (Finite Strain) Analysis: Theory

This chapter presents the finite strain formulation of a strain space multiple mechanism model for granular materials. Since the strain space multiple mechanism model has an appropriate micromechanical background in which the branch and complementary vectors are defined in the material (or referential) coordinate, the finite strain formulation is carried out by following the change in these vectors, both in direction and magnitude, associated with deformation in the material. By applying the methodology for compressible materials established in the finite strain continuum mechanics, decoupled formulation that decomposes the kinematic mechanisms into volumetric and isochoric components is adopted. Material (Lagrangian) description of the integrated form is given by a relation between the second Piola-Kirchhoff effective stress and the Green-Lagrange strain tensors; spatial (Eulerian) description by a relation between the Cauchy effective stress and the Euler-Almansi strain tensors. Material description of the incremental form is derived through the material time derivative of the integrated form. The counterpart in the spatial description is derived through the Lie time derivative, given as a relation between the Oldroyd stress rate of Kirchhoff stress and the rate of deformation tensor.