On the continuum modeling of materials with kinematic structure

Summary

In this work, a phenomenological field-based approach to the formulation of models for structured materials or continua is presented. The corresponding results are in particular relevant to that class of such materials for which thedistribution of (micro)structure at each material point and the evolution of this distribution may influence the behavior of the material as a whole (e.g., in nematic liquid crystals with variable orientation). Essential to the underlying approach is the assumption or idealization that the structured continuum in question is characterized kinematically by additional degrees of freedom in comparison to standard continua. On this basis, the kinematics and balance relations for the structure continuum are formulated with respect to a (generalized) kinematic space via direct generalization of standard kinematics and balance relations. In particular, the formation of the latter for the structured continuum is based on the corresponding (total) energy balance. Indeed, analogous to the standard case, the assumed Euclidean frame-indifference of this balance, together with the transformation properties of the fields appearing in it, determine the forms of the remaining balance relations for the structured continuum. With these general results in hand, account is next taken of the fact that the degrees of freedom of the standard continuum constitute a subset of those of the structured continuum. As already established in previous work, this fact can be represented in a mathematically-precise fashion as a fibre bundle, with base space the standard kinematic space, i.e., three-dimensional Euclidean point space, and total space the kinematic space for the structured continuum. In this context, the kinematic space for the structure itself is represented by the typical fibre of the fibre bundle. Among other results, one obtains on this basis a split of the momentum balance for the structured continuum into momentum balances for the standard continuum and for the structure. Further, the fibre bundle representation induces naturally forms of all fields and balance relations with respect to standard kinematic space, i.e., forms averaged over the degrees of freedom of the structure. In the last part of the work, the results of the current approach are applied to the special cases of rigid-rod- and rigid-body-like structure, and compared with previous work.