A minimum principle for the quasi-static problem in linear viscoelasticity

A minimum principle is set up for the quasi-static boundary-value problem (QSP) in linear viscoelasticity. A linear homogeneous and isotropic viscoelastic solid under unidimensional displacements is considered along with the complete set of thermodynamic restrictions on the relaxation function. It is assumed that boundary conditions are of Dirichlet type and initial history data are not given. The variational formulation of QSP is set up through a convex functional based on a "weighted" $L^2$ inner product as the bilinear form and is strictly related to the thermodynamic restrictions on the relaxation function. As an aside, the same technique is proved to be applicable to analogous physical problems such as the quasi-static heat flux equation.