The quasi-adiabatic approximation for coupled thermoelasticity

The equations of coupled thermoelasticity are considered in the case of stationary vibrations. The dimensional and order-of-magnitude analysis of the parameters occurring within these equations prompts the introduction of the new non-dimensionalisation scheme, highlighting the nearly-adiabatic nature of the resulting motions. The departure from the purely adiabatic regime is characterised by a natural small parameter, proportional to the ratio of the mean free path of the thermal phonons to the vibration wavelength. When the governing equations are expanded in terms of the small parameter, one can formulate an equivalent “quasi-adiabatic” system of the equations of ordinary elasticity with frequency-dependent modulae, characterising the thermoelastic dissipation. Unfortunately, this model lacks the degrees of freedom necessary to satisfy boundary condition(s) for the temperature. Thus, we also derive a complementary boundary layer solution and show that to the leading order it is described by thermoelastic equations in the quasi-static approximation. Further simplifications are possible for purely dilatational motions; we illustrate this point by solving a model thermoelastic problem in 1D.