Two-scale models of polycrystals: Macroscale motion decomposition

The paper considers the application of multiscale mathematical models to description of severe inelastic deformation of single- and polycrystals for which geometrically and physically nonlinear constitutive relations are required. For definiteness, the discussion is given on the example of two-scale models. One of the unsolved problems in nonlinear solid mechanics (and in construction of multiscale models in particular) is how to single out the component of material motion responsible for geometric nonlinearity (for example, when rigid rotations are imposed on strain-induced motion) or, in other words, how to decompose the macroscale motion into quasi-rigid motion and strain-induced motion. In multiscale constitutive models, this problem is added complexity by the necessity to match the constitutive relations with “affined” parameters of different scales due to different definitions of quasi-rigid motion on different scales. We consider three possible methods of representing the quasi-rigid motion on the macroscale. For each method, consistency conditions are determined; from these conditions also follows a relation between macroscale quasi-rigid rotation and mesoscale rotation of elements. It is shown that determination of rotations on the scales “from the top down” makes it impossible to select an arbitrary rotation model on the lower scales, and this considerably limits the applicability of physically substantiated rotation models on the lower scales. It is shown that the proposed method of defining the macroscale quasi-rigid motion as an averaged mesoscale spin makes it possible to obtain consistent constitutive relations of neighbor scales with no limitations on the choice of a lower scale rotation model.