On three-constant potentials in polar continua

В работе рассматривается важная с точки зрения прикладных задач проблема моделирования процессов деформирования материалов со сложным откликом на механические воздействия. Приводятся основные уравнения трехконстантной микроструктурной модели для случая одного полярного директора. Выписана точная форма упругого потенциала, из которой, следуя стандартной схеме, выводятся все необходимые уравнения и граничные условия. The paper deals with the problem of modeling the processes of deformation of materials with a complex response to mechanical stress. The governing equations of the three-constant microstructural model for the case of one polar director are presented. The exact form of the Helmholtz free energy is written out. The governing equations and boundary conditions are derived by the standard scheme.

Constitutive relations for polar continua based on statistical mechanics and spatial averaging

The purpose of the current work is the formulation of macroscopic constitutive relations, and in particular continuum flux densities, for polar continua from the underlying mass point dynamics. To this end, generic microscopic continuum field and balance relations are derived from phase space transport relations for expectation values of point fields related to additive mass point quantities. Given these, microscopic energy, linear momentum and angular momentum, balance relations are obtained in the context of the split of system forces into non-conservative and conservative parts. In addition, divergence–flux relations are formulated for the conservative part of microscopic supply-rate densities. For the case of angular momentum, two such relations are obtained. One of these is force-based, and the other is torque-based. With the help of physical and material theoretic restrictions (e.g. material frame-indifference), reduced forms of the conservative flux densities are obtained. In the last part of the work, formulation of macroscopic constitutive relations from their microscopic counterparts is investigated in the context of different spatial averaging approaches. In particular, these include (weighted) volume-averaging based on a localization function, surface averaging of normal flux densities based on Cauchy flux theory and volume averaging with respect to centre of mass.