Ordered rate constitutive theories in Lagrangian description for thermoviscoelastic solids without memory

This paper presents ordered rate constitutive theories in Lagrangian description for compressible as well as incompressible homogeneous, isotropic thermoviscoelastic solid matter with memory in which the material derivative of order m of the deviatoric stress tensor and heat vector are functions of temperature, temperature gradient, time derivatives of the conjugate strain tensor up to any desired order n, and the material derivatives of up to order m−1 of the stress tensor. The thermoviscoelastic solids described by these theories are called ordered thermoviscoelastic solids with memory due to the fact that the constitutive theories are dependent on orders m and n of the material derivatives of the conjugate stress and strain tensors. The highest orders of the material derivative of the conjugate stress and strain tensors define the order of the thermoviscoelastic solid. The constitutive theories derived here show that the material for which these theories are applicable have fading memory. As is well known, the second law of thermodynamics must form the basis for deriving constitutive theories for all deforming matter (to ensure thermodynamic equilibrium during evolution), since the other conservation and balance laws are independent of the constitution of the matter. The entropy inequality expressed in terms of Helmholtz free energy density \({\Phi}\) does not provide a mechanism to derive a constitutive theory for the stress tensor when its argument tensors are stress and strain rates in addition to others. With the decomposition of the stress tensor into equilibrium and deviatoric stress tensors, the constitutive theory for the equilibrium stress tensor is deterministic from the entropy inequality. However, for the deviatoric stress tensor, the entropy inequality requires a set of inequalities to be satisfied but does not provide a mechanism for deriving a constitutive theory. In the present work, we utilize the theory of generators and invariants to derive rate constitutive theories for thermoviscoelastic solids with memory. This is based on axioms and principles of continuum mechanics. However, we keep in mind that these constitutive theories must satisfy the inequalities resulting from the second law of thermodynamics. The constitutive theories for heat vector q are derived: (i) strictly using conditions resulting from the entropy inequality; (ii) using the theory of generators and invariants with admissible argument tensors that are consistent with the stress tensor as well as the theories in which simplifying assumptions are employed which yield much simplified theories. It is shown that the rate theories presented here describe thermoviscoelastic solids with memory. Mechanisms of dissipation and memory are demonstrated and discussed, and the derivation of memory modulus is presented. It is shown that simplified forms of the general theories presented here result in constitutive models that may resemble currently used constitutive models but are not the same. The work presented here is not to be viewed as extension of the current constitutive models; rather, it is a general framework for rate constitutive theories for thermoviscoelastic solids with memory based on the physics and derivations that are consistent within the framework of continuum mechanics and thermodynamics. The purpose of the simplified theories presented in the paper is to illustrate possible simplest theories within the consistent framework presented here.