A mixed variational principle in quasi-static aging viscoelasticity

The aging viscoelastic problem is usually posed by means of integro-differential equations while a variational formulation would provide many advantages both in the theoretical posing of the problem and the numerical approximate solution. The aim of this paper is to present a variational formulation which is suitable for aging viscoelasticity, by dealing with the problem on infinite temporal domain. Differently from finite domains, the investigation regarding the continuity of the viscoelastic operator becomes interesting and requires a careful choice of the functional spaces and of the restrictions on the viscoelastic kernels. Furthermore, the operator is not symmetric and a functional with a critical point at the solutions can be sought only by special methods. The usual form of the constitutive law makes it convenient to present a mixed formulation in which displacements and stresses are unknown. Finally, an application to one of the most classical problems on this topic is developed.