A constitutive model of coupled thermoelasticity with plasticity

It is well-known that the classical linear theory of heat conduction, based on Fourier's law for the thermal flux, predicts that a thermal effects at a point of a body is felt instantly at other points of the body. Therefore in past years several alternative theories of heat conduction have been proposed. In this paper we consider the non-classical theory of thermoelasticity developed by Green and Naghdi (1977) which incorporates the approach based on Fourier's law (referred to as type I), the theory without energy dissipation (type II) and a theory which allows finite wave propagation as well as energy dissipation (type III). Contributions on the Green and Naghdi (GN) approach can be found, among others, in Bargmann and Steinmann (2006) and references therein. An analytical treatment of coupled thermoelastic problems is complex so that the development of alternative methods of analysis turns out to be important. Accordingly variational formulations of such problems can be of great interest from a theoretical point of view and from a computational standpoint since they are the foundation to develop mixed finite elements. In the present paper we treat the non-classical thermoelastic problem following the GN model of type II. The purpose of the contribution is to provide a constitutive model of thermoelasticity coupled with plasticity suitable to formulate variational formulations (see e.g. Marotti de Sciarra, 2009 for nonlocal problems) in order to consistently derive a finite element approach and the related algorithmic procedure. The coupled thermoelastic model with plasticity is formulated in a geometrically linear range and is based on the internal variable theories of inelastic behaviours of associative type.