Lyapunov functions for thermomechanics with spatially varying boundary temperatures (Q578965)

In this interesting paper the authors study evolution problems in thermomechanics where the temperature \(\theta\) equals an assigned function of place \(\theta_ 0(X)\) on part of the reference boundary, the complementary part being thermally insulated and the mechanical boundary conditions being standard. They show that, provided that Planck inequality prevails, either one of the following constitutive prescriptions for the referential heat flux vector \(q_ R\) allows construction of a Lyapunov function: (i) \(q_ R=\hat q_ R(X,\theta,\nabla \theta)=-\kappa (\theta)\nabla \theta\), with log \(\kappa\) (\(\theta)\) a concave function of log \(\theta\) ; (ii) \(q_ R=\hat q_ R(X,\theta,\nabla \theta)=- K(X)\nabla \theta\), with K a uniformly positive matrix. This result compares with and partially complements a classical result of Duhem, who in 1911 showed that there is a Lyapunov function, the so- called equilibrium free energy, for the case when \(\theta_ 0(X)=const\). and \(q_ R\) may depend on the deformation.