The interaction between heat and dynamic response of viscoelastic bodies with temperature-dependent properties is studied. First, equations governing the small deformation thermomechanical response to sinusoidal loading are shown to be equivalent to a set of two variational principles. Viscoelastic properties of a solid propellant are characterized and then used in numerical examples dealing with sinusoidal shear loading of slabs and cylinders. As the first problem, an approximate variational method is used to calculate one-dimensional transient and steady-state temperature distributions in a massless slab. An exact steady-state solution is obtained for the thermomechanical behavior of a slab with concentrated mass and is then used to deduce the solution for a similarly loaded cylinder. Finally, the influence of distributed mass in a cylinder is studied using a variational method. It is found that without inertia a large temperature rise may occur when the applied stress amplitude is above a certain critical value which depends on thermal and mechanical properties, geometry, and frequency. Moreover, the combination of temperature-dependent properties and inertia leads to temperature and displacement jump instabilities that are similar to those existing in a nonlinear spring-mass system with a spring that softens with increasing displacement.

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