Thermoelasticity with thermomechanical constraints

Abstract

Equations are derived governing the behaviour of small disturbances superimposed on an underlying equilibrium con"guration of a thermoelastic body. The body may be materially inhomogeneous, non-homogeneously prestrained and be subjected to a non-uniform temperature resulting in non-constant-coe$cient partial di!erential equations. These equations are generalized to the case where thermomechanical constraints are present, both deformation-temperature and deformation-entropy constraints. It is known that the "rst of these types of constraints leads to material instabilities and the second does not. By examining nearly constrained materials, and taking an appropriate limit, we "nd that the instabilities associated with deformation-temperature constraints arise because the heat capacity at constant deformation becomes negative whilst deformation-entropy constraints are stable because the same heat capacity remains positive, though tending to zero in the limit of the constraint holding exactly. The ten important moduli of thermoelasticity are examined in the limit of each constraint holding exactly. It is found, for example, that for each type of constraint the heat capacity at constant stress remains positive and bounded away from zero. Results on wave propagation are also presented.

On the Müller paradox for thermal-incompressible media

“…To obtain these results, we used temperature as natural thermodynamical variable. Nevertheless, Manacorda [21] first noted (see also [22,23]) that in the case V ≡ V (T ), instabilities occur in wave propagations. The instabilities are due to the chemical potential non-concavity (see Remark in previous section) and the sound velocity c becomes complex.…”

A consistent thermodynamical model of incompressible media as limit case of quasi-thermal-incompressible materials

“…Taking account of Eqs. (6) 1 , (8) 2 , (9), (11) and (12), Eq. (14) yields the sound velocity in the form:…”

“…By using Eq. ( 1), inequality (11) can be written in terms of the thermal expansion coefficient α and the compressibility factor β:…”

“…Let us also note that in the solid case, Dunwoody and Odgen [12] showed that conditions for infinitesimal stability are unattainable if the trace of the strain tensor is dependent on temperature. Starting from Manacorda's observation, other authors such as Scott et al [10,11] proposed an alternative definition of incompressibility: instead of V ≡ V (T ), they assumed V ≡ V (S), where S is the specific entropy; then, in the ideal incompressible case, the Mach number becomes unbounded and the sound velocity becomes infinite. From an experimental point of view, this assumption is unrealistic because the entropy is not an observable and no direct evaluation of V ≡ V (S) is possible.…”

Thermoelastic Waves in a Constrained Isotropic Plate: Incompressibility at Uniform Entropy

“…By developing these ideas further for the case of a nearly constrained material, Scott [11] was able to demonstrate that the heat capacity at constant deformation tends to zero through positive values as the limit of deformation-entropy constraint is achieved, which is anom-THERMOELASTIC PLATE WAVES 387 alous. In a later paper Scott [12] applied similar analysis to investigate the behavior of the heat capacity at constant stress and found that it remains positive and bounded away from zero for either type of constraint.…”