Kurt Frischmuth, Witold Kosinski
Thermomechanical coupled waves in a nonlinear medium
Preprint series: Preprints aus dem Fachbereich Mathematik, Universität Rostock
65M06 Finite difference methods
80A20 Heat and mass transfer, heat flow
ZDM: M50
PACS: 66.70
CR: G.1.8.
Abstract: Several technological situations at moderate and high temperatures as
well as many physical experiments at low temperatures show the necessity of
taking into account the wave structure of heat transport.
Motivated by experimental evidence a thermo-mechanical framework for a
deformable heat-conducting continuum has been developed. In this
framework a history effect is present, it is introduced via a proportionality law
between the heat flux vector and the gradient of a scalar thermal variable.
The theory leads to a modified model of thermoelasticity with an extra
thermal stress effect and wave--type heat conduction.
The model is governed by a system of quasi--linear hyperbolic equations.
All essential material functions are examined, and the
impact of their nonlinearity on the solution to initial-boundary
value problems is studied.

A numerical scheme is developed to solve initial-boundary
value problems on finite domains by
a modification of {\sc Lax-Wendroff}'s scheme.
It turns out that the previously studied case of NaF crystals considered
as {\em rigid heat conductors}
was quantitatively quite acceptable: while it does not
show any elastic waves, it still gives correct speeds and amplitudes of
the pure thermal waves. However, this feature of {\em upward compatibility}
of theories is lost for materials with higher thermal expansion
Hence, for new applications, it is of the essence to develop numerical
methods for the fully coupled theory in 2D and 3D cases.
Keywords: heat conduction, balance laws, hyperbolic systems