Continuum thermodynamic formulation of models for materials characterized by an evolving microstructural distribution

The current contribution is concerned with the formulation of continuum thermodynamic field models for materials with microstructure for which the distribution of (micro)structure at each material point and its evolution relative to the continuum as a whole influences the macroscopic behaviour.

Examples of such materials include metallic polycrystals (grain size, grain orientation, dislocation orientation, dislocation curvature), granular materials (grain size, grain orientation, contact), polymers (molecular chain length, chain orientation), uniaxial and biaxial nematic liquid crystals (crystallite orientation distribution). To account for the motion and evolution of such microstructure with respect to the material as a whole, the corresponding extended or structured continuum is endowed with additional kinematic degrees of freedom.

Once these are established, one may proceed by analogy with the case of the standard continuum to formulate kinematics, balance relations, and constitutive relations, for the extended one. For example, the formulation of extended balance relations follows in this way in the context of the total energy balance and and invariance arguments. 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 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 this way arises naturally in the current approach a distribution function and its role as mediator of the constitutive influence of microstructural evolution on macroscopic continuum material behaviour fields. In the last part of the work, a number of applications of the current approach are discussed and compared with previous work.