Modelling of complex elastic crystals by means of multi-spin micromorphic media

The aim of the paper is to model essentially magnetic crystals having complicated structure. These crystals have interesting magnetoacoustic properties. Each cell (a point-body) of such a crystal consists of many atoms, combined in clusters (subparticles) responsible for the magnetic properties of a crystal. The magnetic moment of clusters is related to the angular momentum (including moment of momentum and spin). Subparticles are considered as rotating point-bodies, interacting with their neighbours, possessing rotational symmetry. We apply two approaches: phenomenological and microstructural. The phenomenological approach is based on the statement of balance laws. We obtain strain tensors for an inhomogeneous elastic polar medium (Kelvin medium) and the associated constitutive equations. To write down the stress and couple tensor for a point-body corresponding to a cell we have to integrate the laws of balance of momentum and angular momentum over a cell. Thus, the strain energy has a complicated structure depending on various deformations and various internal parameters such as the relative rotation of subparticles in the natural configuration. For the microstructural consideration we model the interaction between subparticles as a potential interaction of sufficiently general kind between rigid bodies both of force and torque nature, and calculate the force and the torque acting from the neighbourhood upon a subparticle. We sum the laws of the balance of linear and angular momenta over all subparticles of a cell. To pass from the discrete to the continuum consideration we expand these laws in two space co-ordinates: a micro-gradient corresponds to the translations in the neighbourhood of the subparticle, and a macro-gradient corresponds to the translation by a basic vector of a crystal lattice. (c) 2005 Elsevier Ltd. All rights reserved.