A Second Gradient Theory of Thermoelasticity

This paper is concerned with a linear theory of thermoelasticity without energy dissipation, where the second gradient of displacement and the second gradient of the thermal displacement are included in the set of independent constitutive variables. In particular, in the case of rigid heat conductors the present theory leads to a fourth order equation for temperature. First, the basic equations of the second gradient theory of thermoelasticity are presented. The boundary conditions for thermal displacement are derived. The field equations for homogeneous and isotropic solids are established. Then, a uniqueness result for the basic boundary-initial-value problems is presented. An existence theorem is established for the first boundary value problem. The problem of a concentrated heat source is investigated using a solution of Cauchy-Kowalewski-Somigliana type.