A continuous Galerkin finite element method for thermoelasticity without energy dissipation

Abstract

The classical thermal theory based on Fourier’s law leads to a diffusive regime. Contrary to that Green and Naghdi [3] developed a theory of thermoelasticity without energy dissipation whose temperature evolution equation is hyperbolic. Among others, the introduction of a new internal variable, i.e. the thermal displacement a with α = T, leads to the theory without energy dissipation, or also called theory of type II, which does not involve energy dissipation. It incorporates thermal wave propagation in a very consistent way and is capable of modeling the second sound phenomenon. The governing equations of the dynamic, linear theory of isotropic and homogeneous thermoelasticity without energy dissipation are the temperature equation

$$ \rho cb\dot T = \rho r + \rho \frac{a} {b}\kappa \Delta \alpha - \rho bT_0 3wKI:\dot \varepsilon $$

(1)

and the balance of linear momentum

$$ \left[ {\mathop {\rho v}\limits^ \cdot } \right] = div\sigma + b, $$

(2)

where the entropy flux $p = - \frac{{\rho k}} {b}\nabla \alpha$ is determined by the same potential which determines the mechanical stresses σ.

This contribution concentrates on numerical aspects of the Green-Naghdi theory of type II. In order to perpetuate the consistency of their theory to the numerical setting we resort to a Galerkin finite element method in space and in time. As the theory itself does not admit energy dissipation, conserving time integration schemes that inherit the underlying conservation principles are of great interest. Customary implicit time-stepping schemes fail to conserve major invariants, for example the total energy. The coupled dynamic system of equations is discretized in time within the framework of finite element methods using a continuous Galerkin (cG) method. In general, the cG-method has proven to qualify well for hyperbolic problems. This fact also holds true for hyperbolic heat conduction and linear thermoelastostatics [1, 2]. The cG(k)- method approximates trial functions piecewise and continuously with polynomials of degree k and test functions piecewise and discontinuously across the element boundaries with polynomials of degree k — 1.

The coupled system is solved monolithically. A numerical example is investigated in order to evaluate the performance of the proposed method.