Constitutive Theories for Linear Micromorphic Thermoviscoelastic Solids

Abstract

This paper presents constitutive theories for linear micromorphic microcontinuum thermoviscoelastic solids in which elasticity and dissipation are considered for the microconstituents, the solid medium and the interaction between the microconstituents and the solid medium. The conservation and the balance laws derived by Surana et al. in a recent paper in which the derivations is initiated for micro deformation using the conservation and balance laws of classical continuum mechanics followed by “integral-average” definitions valid at macro level permitting derivation of conservation and balance laws at macro level are utilized in the present work. Significant aspects of this theory are: 1) Microconstituent rigid rotation physics is treated identically in all 3M theories; 2) The balance of moment of moments balance law, essential in all 3M theories, is used in the present work; 3) Only the symmetric part of nonsymmetric macro Cauchy tensor can be a constitutive tensor; 4) The smoothing weighting function ϕ ( α ) used by Eringen is neither needed nor used in present work; 5) Constitutive tensors of rank two are always symmetric; 6) All constitutive theories are derived using theory of isotropic tensors in conjunction with entropy inequality, and are therefore always thermodynamically and mathematically consistent 7) Conservation of microinertia, as advocated by Eringen, is neither needed nor used in the present work. All three dissipation mechanisms are based on higher order rates up to a desired order of the strain tensors, and hence represent a comprehensive ordered mechanism yielding three ordered spectra of dissipation coefficients. Constitutive theories are first derived using integrity, the complete basis of the constitutive tensor space, and the representation theorem. These are then followed by simplified yet general forms of the constitutive theories, in which physical meaning of the material coefficients can be clearly established. The linear micromorphic theory presented here for thermoviscoelastic solids is compared with Eringen’s theory to identify differences, evaluate their validity based on thermodynamic and mathematical principles, and ultimately determine the thermodynamic and mathematical consistency of the published micromorphic theories.