On the propagation of transients through thermoviscoelastic media

Authors:
M. F. McCarthy, T. B. Moodie and R. P. Sawatzky

Journal:
Quart. Appl. Math. **46** (1988), 539-557

MSC:
Primary 73U05; Secondary 73F15

DOI:
https://doi.org/10.1090/qam/963589

MathSciNet review:
963589

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We examine the propagation of thermal and mechanical transients through general linear thermoviscoelastic media. A linear theory of heat conduction in deformable materials with memory is employed to study the one-dimensional problem of a homogeneous thermoviscoelastic half-space subjected to thermal and mechanical disturbances at its boundary. A ray series approach is used to generate asymptotic wave front expansions for the temperature, strain, and stress response of the medium to the disturbances. General properties of the propagation process are obtained simply and directly. As an example, we specialize the solution to the case of a medium where the conduction of heat obeys Cattaneo's law and the viscoelastic response is that of a standard linear solid. The propagation of transients through this material is depicted graphically for various values of the thermal and mechanical parameters.

**[1]**Carlo Cattaneo,*Sur une forme de l’équation de la chaleur éliminant le paradoxe d’une propagation instantanée*, C. R. Acad. Sci. Paris**247**(1958), 431–433 (French). MR**0095680****[2]**Morton E. Gurtin and A. C. Pipkin,*A general theory of heat conduction with finite wave speeds*, Arch. Rational Mech. Anal.**31**(1968), no. 2, 113–126. MR**1553521**, https://doi.org/10.1007/BF00281373**[3]**V. K. Agarwal,*Some remarks on a generalized heat conduction equation*, Amer. J. Phys.**49**, 503-504 (1981)**[4]**H. W. Lord and Y. Shulman,*A generalized dynamical theory of thermoelasticity*, J. Mech. Phys. Solids**15**, 299-309 (1967)**[5]**P. J. Chen and M. E. Gurtin,*On second sound in materials with memory*, Z. Angew. Math. Phys.**21**, 232-241 (1970)**[6]**J. D. Achenbach,*The influence of heat conduction on propagating stress jumps*, J. Mech. Phys. Solids**16**, 273-282 (1968)**[7]**F. R. Norwood and W. E. Warren,*Wave propagation in the generalized dynamical theory of thermoelasticity*, Quart. J. Mech. Appl. Math.**22**, 283-290 (1969)**[8]**E. B. Popov,*Dynamic coupled problem of thermoelasticity for a half-space taking account of the finiteness of the heat propagation velocity*, J. Appl. Math. Mech. (PMM)**31**, 349-356 (1967)**[9]**H. W. Lord and A. A. Lopez,*Wave propagation in thermoelastic solids at very low temperature*, Acta Mech.**10**, 85-98 (1970)**[10]**Y. H. Pao and D. K. Banerjee,*Thermal pulses in dielectric crystals*, Lett. Appl. Eng. Sci.**1**, 33-41 (1973)**[11]**D. K. Banerjee and Y. H. Pao,*Thermoelastic waves in anisotropic solids*, J. Acoust. Soc. Amer.**56**, 1444-1454 (1974)**[12]**A. Nayfeh and S. Nemat-Nasser,*Thermoelastic waves in solids with thermal relaxation*, Acta Mech.**12**, 53-69 (1971)**[13]**A. H. Nayfeh,*Transient thermo-elastic waves in a half-space with thermal relaxation*, Z. Angew. Math. Phys.**23**, 50-68 (1972)**[14]**A. H. Nayfeh,*Propagation of thermoelastic disturbances in non-Fourier solids*, AIAA J.**15**, 957-960 (1977)**[15]**M. F. McCarthy and P. M. O'Leary,*Wave propagation in linear thermoviscoelastic materials*, J. Thermal Stresses**5**, 347-364 (1982)**[16]**Morton E. Gurtin,*Time-reversal and symmetry in the thermodynamics of materials with memory*, Arch. Rational Mech. Anal.**44**(1971/72), 387–399. MR**0349098**, https://doi.org/10.1007/BF00249968**[17]**F. Mainardi,*On thermal waves in generalized linear theory of heat conduction*, CISM Symposium on thermomechanics in solids, Udine, Italy (1974)**[18]**T. B. Moodie and R. J. Tait,*On thermal transients with finite wave speeds*, Acta Mech.**50**(1983), no. 1-2, 97–104. MR**729370**, https://doi.org/10.1007/BF01170443**[19]**R. P. Sawatzky and T. B. Moodie,*On thermoelastic transients in a general theory of heat conduction with finite wave speeds*, Acta Mech.**56**(1985), no. 3-4, 165–187. MR**820808**, https://doi.org/10.1007/BF01177116**[20]**R. M. Christensen,*Theory of viscoelasticity: an introduction*, 2nd ed., Academic Press, New York, 1982**[21]**Peter J. Chen and Jace W. Nunziato,*Thermodynamic restrictions on the relaxation functions of the theory of heat conduction with finite wave speeds*, Z. Angew. Math. Phys.**25**(1974), 791–798 (English, with German summary). MR**0366242**, https://doi.org/10.1007/BF01590264**[22]**R. Courant and D. Hilbert,*Methods of mathematical physics. Vol. I*, Interscience Publishers, Inc., New York, N.Y., 1953. MR**0065391**

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC:
73U05,
73F15

Retrieve articles in all journals with MSC: 73U05, 73F15

Additional Information

DOI:
https://doi.org/10.1090/qam/963589

Article copyright:
© Copyright 1988
American Mathematical Society