A Nonlinear Microdilation Microcontinuum Theory with Nonlinear and Linear Microconstituent Kinematics for Thermoelastic Solids

Abstract

This paper presents the conservation and balance laws for nonlinear microdilation microcontinuum theory for thermoelastic solid medium in which microconstituents can only have pure volumetric deformation without distortion of shape. We consider nonlinear elasticity for microconstituents, for the medium as well as for interaction of the microconstituents with the solid medium. The theory is based on classical rotations as rigid body rotations of the microconstituents and use of balance of moment of moments balance law for thermodynamic equilibrium of the deforming matter. A check on the closure of the mathematical model consisting of conservation and the balance laws and the constitutive theories reveals that additional four equations are needed for closure of the mathematical model. It is shown that one of the four equations can be extracted from balance of angular momenta. We present two alternatives for the remaining three equations. In the first case, one could use Eringen’s conservation of microinertia conservation law to obtain three equations, hence we have closure of the mathematical model. In the second alternative if we only consider linear volumetric deformation of the microconstituents, then we will only require one additional equation, hence the mathematical model has closure with additional one equation extracted from the balance of angular momenta. Pros and cons of both approaches are discussed in this paper from the point of view of thermodynamic and mathematical consistency of the resulting theory. Since the microconstituents are deformable, we begin derivation of the conservation and balance laws for the microconstituents followed by integral-average definitions that facilitate the derivation of macro conservation balance laws incorporating microconstituent kinematics. Constitutive theories are initiated using conjugate pairs in the entropy inequality in conjunction with axiom of causality and are derived using representation theorem, hence ensuring their thermodynamic and mathematical consistency. Since the classical rotations and the conjugate moments is a new kinematically conjugate pair in the theory, the balance of moment of moments balance law is necessitated by classical thermodynamics for thermodynamic equilibrium of the microdilation solid medium. In the derivation of the conservation and the balance laws for the microdilation theory, we ensure that the modification of the conservation and the balance laws of classical continuum mechanics are supported by classical thermodynamics. The thermodynamically and mathematically consistent microdilation theory presented here is compared with Eringen’s microstretch theory.