Continuum Thermodynamic and Rate Variational Formulation of Models for Extended Continua

The purpose of this work is the formulation of models for selected extended or generalized continua with the help of continuum thermodynamic and rate variational methods. The current approach is based on energy balance, the dissipation principle, as well as frame-indifference (i. e., Euclidean and material). Energetics and kinetics are based on the free energy density and a dissipation potential, respectively. More specifically, attention is focused here on the class of generalized continua whose energetic behavior depends on (i) the first- and second-order gradients of the standard deformation field, (ii) a microstructure field and its gradient, (iii) local inelastic internal variables. This is sufficiently general to include well-known cases such as second-order Mindlin, director, micropolar (Cosserat), microstretch, or micromorphic, continua, as well as gradient inelasticity. Two types of models are identified depending on whether or not the microstructure field involved is modeled as spatial or non-spatial (e. g., intermediate or material) in nature. In particular, this constitutive assumption influences the form of the evolution-field relation for the microstructure field as well as its coupling to standard momentum balance. Given the resulting continuum thermodynamic model relations, the corresponding initial-boundary-value problem is then formulated in rate-variational form. This is based on bulk and surface rate potentials determining a rate functional whose stationarity conditions yield the corresponding evolution-field relations and flux boundary conditions of the model.