Abstract
An asymptotic model for time-harmonic motion in fully-coupled linear thermoelastic orthorhombic materials is presented. The asymptotic approach takes advantage of the observation that the parameter expressing departure from the purely adiabatic regime is extremely small in practice. Consequently, the leading order bulk response turns out to be non-dissipative, and is governed by the usual equations of elastodynamics with adiabatic material constants. In the case of isothermal stress-free boundary conditions, it is shown that thermoelastic interaction is dominated by a thermoelastic boundary layer. Hence, effective boundary conditions may be constructed, which duly account for the influence of this boundary layer and successfully describe dispersion and dissipation of surface waves to leading order. As an illustration, in the special case of an isotropic half-space with free isothermal boundary conditions, we recover the asymptotic results by Chadwick and Windle (1964). Numerical comparison of the dispersion curves for surface waves in an orthorhombic half-space shows excellent agreement between the exact fully-coupled thermoelastic problem and the corresponding quasi-adiabatic approximation, even for relatively large wavenumbers.
Highlights
Since the pioneering work of Danilovskaya (1950), it is known that disregard of coupling between strain and temperature fields may lead to significant inaccuracy in the prediction of the dynamic response of thermoelastic solids
Thermomechanical interaction is a key factor in understanding dispersion and dissipation at high frequencies, which are typical of modern signal processing and passband filter devices
The fully coupled equations of thermoelasticity pose a formidable obstacle to analytical approaches, while numerical solutions suffer from the usual flaw of lending little insight on the role of the problem’s parameters
Summary
Since the pioneering work of Danilovskaya (1950), it is known that disregard of coupling between strain and temperature fields may lead to significant inaccuracy in the prediction of the dynamic response of thermoelastic solids. We perform dimensional analysis of the steady-state equations of thermoelasticity and identify the natural small parameters appearing in the problem (Sec.2) This allows to highlight the relevant asymptotic regimes and to show that motion is usually dominated by adiabatic interaction between the elastic and the thermal fields. Within this limit, the leading order regular expansion turns out to be formally equivalent to classical orthorhombic elasticity, adiabatic parameters replace isothermal (Sec.). Boundary layer solutions are derived and used to construct effective boundary conditions (Sec.3.2) The latter, coupled with classical elasticity, provide a practical and efficient “quasi-adiabatic” model that incorporates thermoelastic effects without delving into the intricacies of the fully coupled problem. The validity of our “quasi-adiabatic” model is confirmed by comparing the asymptotic dispersion curves versus the dispersion curves of the fully coupled problem (Sec.5)
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