The quasi-adiabatic approximation for solving dynamic problems of non-hyperboring thermoelasticity
Dr. Aleksey Pichugin (Brunel)
Thursday 15th December, 2011 14:00-15:00 325
It is known that propagation of ultrasound in the thermoelastic media results in motions that are predominantly adiabatic, except when the sound frequency is exceptionally low or exceptionally high. One can re-formulate the governing equations of thermoelasticity in the non-dimensional form that emphasises this observation and exposes the presence of 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 asymptotically equivalent â€œquasi-adiabaticâ€ system of the equations of ordinary elasticity with frequency-dependent modulae, characterising the effect of thermoelastic dissipation. Solutions resulting from the use of such â€œquasi-adiabaticâ€ approximation assume a specific dependence between the displacement and temperature fields and, subsequently, can satisfy the necessary boundary conditions for temperature only in degenerate cases. This difficulty is resolved by constructing a complementary boundary layer solution, which can be used to derive the consistent boundary conditions for the quasiadiabatic governing equations. Importantly, we demonstrate that contributions from the thermoelastic boundary layer may dominate the thermoelastic dissipation. The described methodology enables elementary solution of a number of initial and boundary value problems of thermoelasticity, which is illustrated by a selection of model examples.