Linear dynamical stability in constrained thermoelasticity I. Deformation-temperature constraints

Equations are derived governing an infinitesimal disturbance of a uniform equilibrium state Be of an unbounded body composed of a heat-conducting elastic material subject to a constraint linking the deformation and the temperature. No restriction is placed on the symmetry of the material and the freedom of choice of Be allows the presence of an arbitrary homogeneous prestrain. The stability of Be, in the context of linearized dynamics, is examined by studying the nature of plane-harmonic-wave solutions of the governing equations. It is found that, under very mild restrictions on the relevant material constants in Be, at least one of the four modes of wave propagation is always unstable, a conclusion which seriously undermines the legitimacy of the assumed type of constraint. The particular case of incompressibility at uniform temperature has been discussed by previous authors and is recapitulated here. Finally, an explanation of the inherent instability is provided by treating the constraint as a limit.