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

Abstract

This paper presents ordered rate constitutive theories in Lagrangian description for compressible as well as incompressible homogeneous, isotropic thermoviscoelastic solids without memory in which the deviatoric stress tensor and heat vector as dependent variables in the constitutive theories are functions of temperature, temperature gradient, and the material derivatives of the conjugate strain tensor up to a desired order. The thermoelastic solids described by these theories are called ordered thermoelastic solids due to the fact that the deviatoric stress tensor and heat vector are dependent on the material derivative of the conjugate strain tensor up to a desired order. The highest order of the material derivative of the strain tensor defines the order of the thermoelastic solid or the order of the rate constitutive theory. For thermodynamic equilibrium during the evolution, the constitutive theories must be derived using (or must satisfy) the second law of thermodynamics as conservation of mass, balance of momenta, and energy balance are independent of the constitution of the matter. In this study, we consider the entropy inequality resulting from the second law of thermodynamics in Helmholtz free energy density Φ and conjugate pairs: second Piola–Kirchhoff stress tensor σ ^[0] and Green’s strain tensor ɛ. It is shown that when Φ is a function of the material derivatives of ɛ, the entropy inequality necessitates decomposition of σ ^[0] into equilibrium and deviatoric parts: ${_e{\mathbf{\sigma}}^{[0]}}$ and ${_d{\mathbf{\sigma}}^{[0]}}$. The equilibrium stress tensor is deterministic using the conditions resulting from the entropy inequality, but ${_d{\mathbf{\sigma}}^{[0]}}$ is not. The entropy inequality only requires that work expended due to ${_d{\mathbf{\sigma}}^{[0]}}$ be positive, but provides no mechanism for deriving a constitutive theory for it. In the present work, we use the theory of generators and invariants for deriving a constitutive theory for ${_d{\mathbf{\sigma}}^{[0]}}$. The constitutive theories for the heat vector q are derived using (i) conditions resulting from the entropy inequality. In the simplest case, this yields Fourier heat conduction law, and (ii) the theory of generators and invariants with at least two alternate choices of the argument tensors for q. Merits and shortcomings of the resulting theories are discussed. It is shown that the rate theories presented here describe thermoviscoelastic solids without memory. Simplified cases of the general theory are considered to demonstrate that many of the currently used models for such solids (like Kelvin–Voigt model) resemble the theory presented here, but are quite different, and the theories provide a rationale for modeling the mechanism of dissipation in thermoelastic solids that is consistent with principles of continuum mechanics and thermodynamics. One-dimensional numerical studies using the proposed rate theories and comparisons with current theories are also presented.