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

Abstract

The paper presents rate constitutive theories for finite deformation of homogeneous, isotropic, compressible, and incompressible thermoviscoelastic solids without memory in Lagrangian description derived using the second law of thermodynamics expressed in terms of Gibbs potential Ψ. To ensure thermodynamic equilibrium during evolution, the rate constitutive theories must be derived using entropy inequality [as other three conservation and balance laws are do not provide a mechanism for deriving constitutive theories for the deforming matter (Surana in Advanced mechanics of continuua. in preparation, 2014)]. The two forms of the entropy inequality in Ψ derived using conjugate pairs \({\mathbf{\sigma}^*}\), \({[\dot{J}]}\) : first Piola–Kirchhoff stress tensor and material derivative of the Jacobian of deformation and \({\mathbf{\sigma}^{[0]}}\), \({\dot{\mathbf{\varepsilon}}_{[0]}}\) ; second Piola–Kirchhoff stress tensor and material derivative of Green’s strain tensor are precisely equivalent as the conjugate pairs \({\mathbf{\sigma}^*}\), \({[\dot{J}]}\) and \({\mathbf{\sigma}^{[0]}}\), \({\dot{\mathbf{\varepsilon}}_{[0]}}\) are transformable from each other. In the present work, we use \({\mathbf{\sigma}^{[0]}}\), \({\dot{\mathbf{\varepsilon}}_{[0]}}\) as conjugate pair. Two possible choices of dependent variables in the constitutive theories: Ψ, η, \({\mathbf{\sigma}^{[0]}}\), \({\mathbf{q}}\) and Ψ, η, \({\mathbf{\varepsilon}_{[0]}}\), \({\mathbf{q}}\) (in which η is entropy density and \({\mathbf{q}}\) is heat vector) are explored based on conservation and balance laws. It is shown that the choice of Ψ, η, \({\mathbf{\varepsilon}_{[0]}}\), \({\mathbf{q}}\) is essential when the entropy inequality is expressed in terms of Ψ. The arguments of these dependent variables are decided based on desired physics. Viscoelastic behavior requires considerations of at least \({\mathbf{\varepsilon}_{[0]}}\) and \({\dot{\mathbf{\varepsilon}}_{[0]}}\) (or \({\mathbf{\varepsilon}_{[1]}}\)) in the constitutive theories. We generalize and consider strain rates \({\mathbf{\varepsilon}_{[i]}}\); i = 0, 1, …, n−1 as arguments of the dependent variables in the derivations of the ordered rate theories of up to orders n. At the onset, \({\mathbf{\sigma}^{[0]}}\), \({\mathbf{\varepsilon}_{[i]}}\) ; i = 0, 1, …, n−1, θ and \({\mathbf{g}}\) are considered as arguments of Ψ, η, \({\mathbf{\varepsilon}_{[n]}}\) and \({\mathbf{q}}\). When \({\dot{\Psi}}\) is substituted in the entropy inequality, the resulting conditions eliminate η as a dependent variable, reduce arguments of some of the dependent variables in the constitutive theory etc. but do not provide a mechanism to derive constitutive theories for \({\mathbf{\varepsilon}_{[i]}}\) and \({\mathbf{q}}\). The stress tensor \({\mathbf{\sigma}^{[0]}}\) is decomposed into equilibrium stress \({{}_e \mathbf{\sigma}^{[0]}}\) and deviatoric stress \({{}_d \mathbf{\sigma}^{[0]}}\). Upon substituting this in the entropy inequality, we finally arrive at the inequality that must be satisfied by \({{}_e \mathbf{\sigma}^{[0]}}\), \({{}_d \mathbf{\sigma}^{[0]}}\) and \({\mathbf{q}}\). Derivations of the constitutive theory for \({{}_e \mathbf{\sigma}^{[0]}}\) follow directly from \({{}_e \mathbf{\sigma}^{(0)}}\), equilibrium Cauchy stress tensor, and the constitutive theory for \({\mathbf{\varepsilon}_{[n]}}\) is derived using the theory of generators and invariants. Constitutive theories for the heat vector \({\mathbf{q}}\) of up to orders n that are consistent (in terms of the argument tensors) with the constitutive theories for \({\mathbf{\varepsilon}_{[n]}}\) are also derived. Many simplified forms of the rate theories of orders n are presented. Material coefficients are derived by considering Taylor series expansions of the coefficients in the linear combinations representing \({\mathbf{\varepsilon}_{[n]}}\) and \({\mathbf{q}}\) using the combined generators of the argument tensors about a known configuration \({\underline{\Omega}}\) in the combined invariants of the argument tensors and temperature. It is shown that the rate constitutive theories of order one (n = 1) when further simplified results in constitutive theories that resemble currently used theories but are in fact different. The solid materials characterized by these theories have mechanisms of elasticity and dissipation but have no memory, i.e., no relaxation behavior or rheology. Fourier heat conduction law is shown to be an over-simplified case of the rate theory of order one for \({\mathbf{q}}\). The paper establishes when there is equivalence between the constitutive theories derived here using Ψ and those presented in Surana et al. (Acta Mech 224(11):2785—2816, 2013), that are derived using Helmholtz free energy density Φ.