Minimum principles, convexity, and thermodynamics in viscoelasticity

Abstract

A linear viscoelastic solid is considered along with the complete set of thermodynamic restrictions on the relaxation function. It is shown that such reslrictions imply the validity of a dissipativity condition, so far regarded as unrelated to the second law. Next it is remarked that the thermodynamic restrictions imply the convexity of a commonly used bilinear functional and the stationarity only if the class of displacement field is appropriate. Then it is proved that the Laplace transform of the solution to the mixed problem gives the strict minimum of a bilinear functional and vice versa. Finally, a bilinear functional with a weight function is considered and it is shown that the solution to the mixed problem gives the strict minimum and vice versa.