Abstract
We use the Cagniard-De Hoop method to develop the displacement and temperature fields in a half-space subjected on its free surface to an instantaneously applied heat source. We include in our analysis the thermal relaxation time of heat conduction, which insures that the termal waves propagate with a finite signal speed. We express our solution in terms of a small thermo-elastic coupling coefficient, and obtain explicit expressions for the wave-speeds and wave-amplitudes. Due to the existence of the thermal damping, we give only the short-time solution. We then present numerical results for the dilatation and the temperature so as to illustrate the salient features of the problem.
Zusammenfassung
Die Autoren benutzen die Methode von Cagniard-De Hoop zur Ermittlung des Verschiebungs-und des Temperaturfeldes in einem elastischen Halbraum unter dem Einfluß einer an der freien Oberfläche plötzlich angebrachten Wärmequelle. Sie berücksichtigen dabei die Relaxationszeit der Wärmeleitung, welche eine endliche Fortpflanzungsgeschwindigkeit der Wärmewellen garantiert. Die Lösung wird in einem kleinen thermoelastischen Koppelungskoeffizienten ausgedrückt, und es werden explizite Ausdrücke für die Fortpflanzungsgeschwindigkeiten und Wellenamplituden gegeben. Mit Rücksicht auf die thermische Dämpfung werden nur kurzfristige Lösungen betrachtet. Schließlich werden numerische Resultate für die Dilatation und die Temperatur angegeben, um die wesentlichen Eigenschaften des Problems zu illustrieren.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
H. Lamb,On the Propagation of Tremors Over the Surface of an Elastic Solid, Phil. Trans. R. Soc. London,203, 1–42 (1904).
A.T. de Hoop,A Modification of Cagniard's Method for Solving Seismic Pulse Problems, Appl. Sci. Res.8, 349–356 (1959).
L. Cagniard,Reflection and Refraction of Progressive Seismic Waves (trans. by E. Flinn and C. Dix. McGraw-Hill, New York 1962).
E.A. Kraut,Advances in the Theory of Anisotropic Elastic Wave Propagation, Rev. Geoph.3, 401–448 (1963).
T. Karlsson andJ.F. Hook,Lamb's Problem for an Inhomogeneous Medium with Constant Velocities of Propagation, Bull. seism. Soc. Am.53, 1007–1022 (1963).
D.C. Gakenheimer andJ. Miklowitz,Transient Excitation of an Elastic Half-Space by a Point Load Traveling on the Surface, J. Appl. Mech.12, 1–11 (1969).
A. Nayfeh andS. Nemat-Nasser,Thermoelastic Waves in Solids with Thermal Relaxation, Acta Mechanica.12, 53–69 (1971).
C.C. Ackerman, B. Bertman, H.A. Fairbank andR.A. Guyer.Second Sound in Solid Helium, Phys. Rev. Letters16, 789–791 (1966).
C.C. Ackerman andW.C. Overton, Jr.,Second Sound in Solid Helium-3, Phys. Rev. Letters22, 764–766 (1969).
T. McNelly, S. Rogers, D. Channin, R. Rollefson, W. Goubau, G. Schmidt, J. Krumhansl andR. Pohl,Heat Pulses in NaF: Onset of Second Sound, Phys. Rev. Letters24, 100–102 (1970).
H. Jackson andC. Walker,Second Sound in NaF, Phys. Rev. Letters25, 26–28 (1970).
E.M. Lifshitz,Superfluidity, Sci. Am.198, 30–36 (1958).
L.D. Landau,The Theory of Superfluidity of Helium II, J. Phys. U.S.S.R.5, 71–90 (1941).
J.C. Ward andJ. Wilks,The Velocity of Second Sound in Liquid Helium near the Absolute Zero, Phil. Mag.42, 314–316 (1951).
J.C. Ward andJ. Wilks,On the Second Sound and the Thermo-Mechanical Effect, Phil. Mag.43, 48–50 (1952).
R.B. Dingle,Derivation of the Velocity of Second Sound from Maxwell's Equation of Transfer, Phil. Mag.42, 374–376 (1952).
K.R. Atkins andD.V. Osborne,The Velocity of Second Sound Below 1 o K, Phil. Mag.41, 1078–1081 (1950).
M. Chester,Second Sound in Solids, Phys. Rev.131, 2013–2015 (1963).
H.W. Lord andY. Shulman,A Generalized Dynamical Theory of Thermoelasticity, J. Mech. Phys. Solids15, 299–309 (1967).
F.R. Norwood andW.E. Warren,Wave Propagation in the Generalized Dynamical Theory of Thermoelasticity, Q. J. Mech. Appl. Math.22, 283–290 (1969).
S. Kaliski,Wave Equations of Thermo-Electric-Magneto-Elasticity, Proc. Vibr. Probl.3, 231–265 (1965).
M.E. Gurtin andA.C. Pipkin,A General Theory of Heat Conduction, Arch. ration. Mech. Analysis31, 113–126 (1968).
J.D. Achenbach,The Influence of Heat Conduction on Propagating Stress Jumps, J. Mech. Phys. Solids16, 273–282 (1968).
C.C. Ackerman andR.A. Guyer,Temperature Pulses in Dielectric Solids, Ann. Phys.50, 128–185 (1968).
Y.C. Fung,Foundations of Solid Mechanics (Prentice-Hall, Englewood Cliffs, N.J. 1965).
Author information
Authors and Affiliations
Additional information
This work was in part (A.H.N.) supported by the National Science Foundation under Grant GK-277 at the University of California, San Diego.
Rights and permissions
About this article
Cite this article
Nayfeh, A.H., Nemat-Nasser, S. Transient thermo-elastic waves in a half-space with thermal relaxation. Journal of Applied Mathematics and Physics (ZAMP) 23, 50–68 (1972). https://doi.org/10.1007/BF01593202
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01593202