In the articles Thermoelasticity with second sound: A review1 and Hyperbolic thermoelasticity: A review of recent literature2, Chandrasekharaiah has presented an in-depth look at nonconventional (a.k.a. generalized or non-Fourier) theories of thermoelasticity. The motivation driving the formulation of these theories is the desire to overcome the infinite propagation speed of thermal signals predicted by conventional thermoelasticity (CTE), the so-called “paradox of heat conduction.”
In 1, two of these nonconventional theories are examined in the context of the Danilovskaya problem (DP). (In the DP, the homogeneous and isotropic thermoelastic half-space under a stress free boundary condition (BC) at is subjected to a Heaviside, or step, temperature BC at time The first he refers to as extended thermoelasticity (ETE) and the second as temperature-rate dependent thermoelasticity (TRDTE). In both ETE and TRDTE, the parabolic diffusion equation of CTE is replaced with a hyperbolic heat transport equation. As a result, both theories predict thermal waves (ie, second sound) propagating with finite speeds.
In ETE, a single relaxation time appears and second sound propagates with speed where κ is used here to denote the thermal diffusivity. It is noted that ETE reduces to CTE in the limit TRDTE was presented in 1972 by Green and Lindsay 3. This theory involves the two relaxation times and α, where and, in the case of homogeneous and isotropic materials, reduces to CTE in the limit (While it has been postulated that is actually non-negative, it must be noted that TRDTE admits second sound only when 1.) An important aspect of TRDTE is that Fourier’s heat law is not violated in materials that have a center of symmetry at each point 2,3.
Although it was not pointed out in 1, Chandrasekharaiah did note in 2 several physically unrealistic results associated with TRDTE, in particular the fact that the displacement suffers jump discontinuities in the presence of a step temperature BC. A natural question that arises is why was this problem with TRDTE not reported in 1, especially since the author of that paper derived parts of the small-time solution to the DP for a TRDTE medium? (See 2 and the references therein for a discussion of the problems with TRDTE.)
The intent of the present Letter is the following: Show that the small-time expression given in 1 for the normal stress corresponding to the DP for a TRDTE medium is incorrect; Show how this erroneous expression could have lead to the aforementioned shortcoming of TRDTE being missed in 1; and Give for the record the correct small-time expressions for the normal stress, displacement, and strain corresponding to the DP for a TRDTE medium. Lastly, all quantities below are dimensionless, unless stated otherwise the same notation employed in 1 is used here, and the reader is referred to 1 for the definition of all undefined symbols.
Figure 1 shows a comparison of the inverse of Eq. (6) with the small-time solution given in Eq. (8). The inverse of Eq. (6) was computed numerically using Tzou’s Riemann sum inversion algorithm (TRSIA) 5 and the values of the material parameters were obtained from Table II of 1. As shown in Fig. 1, Eq. (8) is a very good/excellent approximation to for In addition, the two propagating jumps are clearly visible, with and it is noted that are the elastic (trailing) and thermal (leading) wavefronts, respectively.
It must be pointed out that the presence of propagating jumps in violates the continuity of displacements requirement [6, p. 142], and thus indicates that TRDTE is inconsistent with the continuum theory of matter under a step (actually any discontinuous) temperature BC (see 2 and the references therein). These jumps, which occur in both the coupled and uncoupled cases, vanish only in the limit (For a treatment of the uncoupled, spherically symmetric case for a shell, see 7.)
Finally, it should be mentioned that an error similar to the one corrected here, in which all and terms are missing from the Laplace inverse, occurs in the expression for the strain (ie, Eq. (48)) in 8. (It is of interest to note that had the correct expression for the strain been obtained in 8, the drawbacks with TRDTE could have been uncovered in 1980.) However, while Eq. (5.58) of 1 is incorrect, and this error appears to have directly resulted in the primary physically objectionable feature of TRDTE being overlooked in 1 as well as to the mistaken claim (1, p 371) that the TRDTE expression for reduces to its ETE counterpart (ie, Eq. (4.39) of 1) when Chandrasekharaiah’s two articles 1,2 nevertheless provide an excellent review of the literature on nonconventional thermoelasticity and contain a wealth of information on the subject.
PM Jordan was supported by CORE/ONR/NRL funding (PE 602435N).