Ordered rate constitutive theories for non-classical thermoviscoelastic solids with memory incorporating internal and Cosserat rotations

Abstract

This paper considers conservation and balance laws for non-classical solid continua in the presence of internal rotations ( $${}_i \pmb {\varvec{\varTheta }}$$ ) due to the Jacobian of deformation and Cosserat rotations ( $${}_e \pmb {\varvec{\varTheta }}$$ ) at each material point. In these balance laws, internal rotations are completely defined as functions of the displacement gradient tensor, but Cosserat rotations are additional three degrees of freedom at each material point. When these rotations are resisted by the deforming matter, conjugate moments are created. For thermoviscoelastic solids with memory, these result in additional energy storage, dissipation mechanism, and rheology. This paper presents a thermodynamically consistent derivation of constitutive theories for such solids based on the entropy inequality in conjunction with representation theorem. Material coefficients are derived and discussed. The constitutive theories are presented in the absence as well as in the presence of the balance of moment of moments as additional balance law for non-classical continuum theories, and the resulting theories are compared with classical continuum theories in which this balance law is not needed. Retardation moduli corresponding to the Cauchy stress tensor as well as the Cauchy moment tensor are derived. In this paper we only consider small strain, small deformation physics.