Ordered Rate Constitutive Theories for Non-Classical Thermoviscoelastic Fluids Incorporating Internal and Cosserat Rotation Rates

Abstract

This paper considers conservation and balance laws for non-classical fluent continua in the presence of internal rotation rates due to the velocity gradient tensor and the rotation rates due to Cosserat rotations. In these balance laws, the internal rotation rates are completely defined as functions of the velocity gradient tensor, but the Cosserat rotation rates are additional three degrees of freedom at each material point. When these rotation rates are resisted by the deforming continua, conjugate moments are created. For thermoviscoelastic fluent continua, these result in additional dissipation mechanism as well as rheology. This paper presents a thermodynamically consistent derivation of constitutive theories for such fluids based on the entropy inequality in conjunction with the representation theorem using integrity, i.e., complete basis. Material coefficients are derived and discussed. The constitutive theories are presented in the absence as well as presence of the balance of moments of moments as a balance law and are compared with the classical continuum theories. Retardation moduli corresponding to the Cauchy stress tensor, both symmetric and antisymmetric, as well as the symmetric and antisymmetric Cauchy moment tensors are derived. The constitutive theories presented in this paper are ordered rate theories. These incorporate convected time derivatives of the strain tensor up to order [Formula: see text] and the convected time derivatives of the stress and moment tensors up to certain orders. Simplified forms of the constitutive theories are also derived. It is shown and concluded that the constitutive theories for thermoviscoelastic fluids based on classical continuum theory such as Maxwell model, Oldroyd-B model, Giesekus model, etc., are all a subset of the constitutive theories presented in this paper for non-classical physics. Both compressible and incompressible thermoviscoelastic fluids are considered.