A theorem in thermoelasticity and its application to linear stability

Equations are written down governing the propagation of plane sinusoidal waves of small amplitude through a homogeneously prestrained equilibrium state of a materially homogeneous thermoelastic body of arbitrary elastic and thermal symmetry. The symmetric isothermal and isentropic acoustic tensors are defined in the usual way and it is assumed that the former is positive definite, so that it has three real and positive eigenvalues. It is shown, under the usual assumption that the specific heat at constant deformation is positive, that the three real and positive eigenvalues of the isentropic acoustic tensor are interlaced with those of the isothermal acoustic tensor, the smallest eigenvalue belonging to the isothermal and the largest to the isentropic acoustic tensor. Under the additional assumption that the symmetrized thermal conductivity tensor is positive definite, it is further shown that this result on the interlacing of the eigenvalues is sufficient to guarantee, for all positive values of the frequency of the sinusoidal waves, that the material is linearly stable in the sense that sinusoidal waves may not increase without bound in the direction of propagation. In the final section, the wide diversity in behaviour of the complex squared wave speed as a function of frequency is illustrated graphically. The stability result is extended to negative frequencies as these would be required in any Fourier synthesis of the sinusoidal wave solutions. A connection with Whitham’s wave hierarchy approach is mentioned.