[Paper Review] On the Constitutive Relations in Thermo-Electroelasticity

Adriano Montanaro|arXiv (Cornell University)|Oct 7, 2009

Thermoelastic and Magnetoelastic Phenomena3 references3 citations

TL;DR

This paper derives thermodynamically consistent constitutive relations for thermo-electroelastic materials using the Coleman-Noll approach within a second-sound theory. It establishes that the free energy function must be independent of the temperature gradient, and the heat flux must satisfy a reduced dissipation inequality, ensuring thermodynamic compatibility and proving that static heat flux vanishes in thermal equilibrium.

ABSTRACT

We give a derivation of the thermodynamic restrictions on the constitutive relations of an electrically polarizable and finitely deformable heat conducting elastic continuum, interacting with the electric field. This is made following the method of Coleman-Noll in a thermodynamic theory with the Clausius-Duhem inequality.

Motivation & Objective

• To extend the thermodynamic framework of thermoelasticity to thermo-electroelasticity by incorporating finite deformation, electric fields, and heat conduction.
• To derive thermodynamic restrictions on constitutive relations for electrically polarizable, heat-conducting, finitely deformable continua.
• To ensure compatibility with the second law of thermodynamics by enforcing the dissipation inequality via the Coleman-Noll method.
• To generalize Tiersten’s foundational work on electro-thermoelasticity by embedding it within a thermodynamically consistent, field-based framework.
• To establish that the heat flux must vanish in thermal equilibrium, a direct consequence of the reduced dissipation inequality.

Proposed method

• Applies the Coleman-Noll method to a continuum with finite deformation, temperature, electric potential, and heat flux as state variables.
• Uses the Clausius-Duhem inequality as the thermodynamic restriction, leading to a dissipation inequality involving free energy, stress, polarization, and heat flux.
• Derives constitutive relations via the chain rule on the free energy function ψ(F, θ, W, G), where F is the deformation gradient, θ is temperature, W is the electric field, and G is the temperature gradient.
• Imposes thermodynamic compatibility by requiring the dissipation inequality to hold for arbitrary time rates of state variables, leading to constraints on the functional dependence of ψ.
• Derives the entropy, first Piola-Kirchhoff stress, and polarization vector per unit mass from the partial derivatives of ψ with respect to F, θ, and W.
• Establishes the reduced dissipation inequality Q·G ≤ 0, which governs the behavior of the heat flux vector Q in response to the temperature gradient G.

Experimental results

Research questions

• RQ1 How can the Coleman-Noll method be extended to derive thermodynamically consistent constitutive relations for thermo-electroelastic materials?
• RQ2 What are the necessary conditions on the free energy function ψ to ensure thermodynamic compatibility in the presence of finite deformation, electric fields, and heat conduction?
• RQ3 How does the dependence of the heat flux on the temperature gradient affect the dissipation inequality in this framework?
• RQ4 Under what conditions does the heat flux vanish in thermal equilibrium, and how is this derived from the thermodynamic constraints?
• RQ5 How does this formulation generalize Tiersten’s earlier work on electro-thermoelasticity within a thermodynamically consistent framework?

Key findings

• The free energy function ψ is independent of the temperature gradient G, which is a necessary condition for thermodynamic consistency.
• The entropy η is determined by the negative partial derivative of ψ with respect to temperature: η = −∂ψ/∂θ.
• The first Piola-Kirchhoff stress tensor S is derived from the gradient of ψ with respect to the deformation gradient F: S = ρR ∂ψ/∂F.
• The polarization vector per unit mass Π is derived from the negative gradient of ψ with respect to the electric field W: Π = −∂ψ/∂W.
• The reduced dissipation inequality Q·G ≤ 0 must be satisfied, which ensures that the heat flux does not generate energy in the system.
• In thermal equilibrium (G = 0), the heat flux Q vanishes identically: Q(F, θ, W, 0) = 0, proving that no heat flux exists in steady-state thermal conditions.