Thermodynamic and rate variational formulation of models for inhomogeneous gradient materials with microstructure and application to phase field modeling

In this work, thermodynamic models for the energetics and kinetics of inhomogeneous gradient materials with microstructure are formulated in the context of continuum thermodynamics and material theory. For simplicity, attention is restricted to isothermal conditions. The materials of interest here are characterized by (1) first- and second-order gradients of the deformation field and (2) a kinematic microstructure field and its gradient (e.g., in the sense of director, micromorphic or Cosserat microstructure). Material inhomogeneity takes the form of multiple phases and chemical constituents, modeled here with the help of corresponding phase fields. Invariance requirements together with the dissipation principle result in the reduced model field and constitutive relations. Special cases of these include the well-known Cahn-Hilliard and Ginzburg-Landau relations. In the last part of the work, initial boundary value problems for this class of materials are formulated with the help of rate variational methods.