Ordered rate constitutive theories for thermoviscoelastic solids with memory in Lagrangian description using Gibbs potential

This paper presents ordered rate constitutive theories of orders m and n , i.e., ( m , n ) for finite deformation of homogeneous, isotropic, compressible and incompressible thermoviscoelastic solids with memory in Lagrangian description using entropy inequality in Gibbs potential Ψ as an alternate approach of deriving constitutive theories using entropy inequality in terms of Helmholtz free energy density Φ . Second Piola-Kirchhoff stress σ [0] and Green’s strain tensor ε [0] are used as conjugate pair. We consider Ψ , heat vector q , entropy density η and rates of upto orders m and n of σ [0] and ε [0] , i.e., σ [ i ] ; i = 0, 1, . . . , m and ε [ j ] ; j = 0, 1, . . . , n . We choose Ψ , ε [ n ] , q and η as dependent variables in the constitutive theories with ε [ j ] ; j = 0, 1, . . . , n − 1, σ [ i ] ; i = 0, 1, . . . , m , temperature gradient g and temperature θ as their argument tensors. Rationale for this choice is explained in the paper. Entropy inequality, decomposition of σ [0] into equilibrium and deviatoric stresses, the conditions resulting from entropy inequality and the theory of generators and invariants are used in the derivations of ordered rate constitutive theories of orders m and n in stress and strain tensors. Constitutive theories for the heat vector q (of up to orders m and n − 1) that are consistent (in terms of the argument tensors) with the constitutive theories for ε [ n ] (of up to orders m and n ) are also derived. Many simplified forms of the rate theories of orders ( m , n ) are presented. Material coefficients are derived by considering Taylor series expansions of the coefficients in the linear combinations representing ε [ n ] and q using the combined generators of the argument tensors about a known configuration in the combined invariants of the argument tensors and temperature. It is shown that the rate constitutive theories of order one ( m = 1, n = 1) when further simplified result in constitutive theories that resemble currently used theories but are in fact different. The solid continua characterized by these theories have mechanisms of elasticity, dissipation and memory, i.e., relaxation behavior or rheology. Fourier heat conduction law is shown to be an over simplified case of the rate theory of order one ( m = 1, n = 1) for q . The paper establishes when there is equivalence between the constitutive theories derived here using Ψ and those presented in reference Surana et al. (Acta Mech. doi: 10.1007/s00707-014-1173-6 , 2014 ) that are derived using Helmholtz free energy density Φ . The fundamental differences between the two constitutive theories in terms of physics and their explicit forms using Φ and Ψ are difficult to distinguish from the ordered theories of orders ( m , n ) due to complexity of expressions. However, by choosing lower ordered theories, the difference between the two approaches can be clearly seen.