Abstract
We obtain the fundamental solution of thermoelasticity with two relaxation times for an infinite space which is acted upon by a spherically symmetric instantaneous point source of heat. The operational method and the Laplace transform technique are used to derive the solution in the form of a series of functions. Numerical results are given and represented graphically.
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Abbreviations
- \(\lambda , \mu \) :
-
Lame’s constants
- \(\rho \) :
-
Density
- \(c_E\) :
-
Specific heat at constant strain
- k :
-
Thermal conductivity
- t :
-
Time
- \(r, \theta , \varphi \) :
-
Spherical polar coordinates
- \(u_{i}\) :
-
Components of displacement vector
- \(\sigma _{ij}\) :
-
Components of stress tensor
- \(e_{ij}\) :
-
Components of strain tensor
- T :
-
Absolute temperature
- Q :
-
Intensity of the heat source per unit mass
- \(\alpha _t\) :
-
Coefficient of linear thermal expansion
- \(\tau , \upsilon \) :
-
Relaxation times
- H(.):
-
Heaviside unit step function
- \(\delta (.)\) :
-
Dirac’s delta function
- \(\varepsilon =\) :
-
\(\frac{\gamma ^{2} T_0}{\rho c_E (\lambda +2\mu )}\)
- \(\gamma \) :
-
\(=(3\lambda +2\mu )\alpha _t\)
- \(b=\) :
-
\(\frac{\gamma T_0}{\mu }\)
- \(T_{0}\) :
-
reference temperature chosen so that \(\frac{\left| {T-T_0} \right| }{T_0} \ll 1\)
- \(W=W(r,t)=\) :
-
\(\hbox {e}^{t-r} H(t-r)\)
- \(U=U(r,t)=\) :
-
\(\frac{1}{2}\left[ {\hbox {e}^{t-r} { erfc}\left( {\frac{r}{2\sqrt{t}}-\sqrt{t}}\right) +\hbox {e}^{t+r} { erfc}\left( {\frac{r}{2\sqrt{t}}+\sqrt{t}}\right) }\right] \)
- \(V=V(r,t)=\) :
-
\(\frac{1}{2}\left[ {\hbox {e}^{t-r} { erfc}\left( {\frac{r}{2\sqrt{t}}-\sqrt{t}}\right) -\hbox {e}^{t+r} { erfc}\left( {\frac{r}{2\sqrt{t}}+\sqrt{t}}\right) }\right] \)
- \({ erfc}(x) =\) :
-
\(\frac{2}{\sqrt{\pi }}\int \limits _\mathrm{x}^\infty {\hbox {e}^{-\mathrm{y}^{2}}} \hbox {d}y=\) Complementary error function
References
Biot, M.: Thermoelasticity and irreversible thermodynamics. J. Appl. Phys. 27, 240–253 (1956)
Lord, H., Shulman, Y.: A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids 15, 299–309 (1967)
Sherief, H.H., Ezzat, M.: Solution of the generalized problem of thermoelasticity in the form of series of functions. J. Therm. Stress. 17, 75–95 (1994)
Anwar, M.N., Sherief, H.H.: A problem in generalized thermoelasticity for an infinitely long annular cylinder. Int. J. Eng. Sci. 26, 301–306 (1988)
Sherief, H.H., Anwar, M.N.: A two dimensional generalized thermoelasticity problem for an infinitely long cylinder. J. Therm. Stress. 17, 213–227 (1994)
Sherief, H.H., El-Maghraby, N.: Effect of body forces on a 2D generalized thermoelastic long cylinder. Comput. Math. Appl. 66, 1181–1191 (2013)
Sharma, J., Pathania, V.: Thermoelastic waves in coated homogeneous anisotropic materials. Int. J. Mech. Sci. 48, 526–535 (2006)
Green, A., Lindsay, K.: Thermoelasticity. J. Elast. 2, 1–7 (1972)
Sherief, H.H.: State space approach to thermoelasticity with two relaxation times. Int. J. Eng. Sci. 31, 1177–1189 (1993)
Sherief, H.H.: A thermo-mechanical shock problem for thermoelasticity with two relaxation times Int. J. Eng. Sci. 32, 315–324 (1994)
Sherief, H.H., Saleh, H.: A problem for an infinite thermoelastic body with a spherical cavity. Int. J. Eng. Sci. 36, 473–487 (1998)
Sherief, H.H., Megahed, F.: A two-dimensional thermoelasticity problem for a half- space subjected to heat sources. Int. J. Solids Struct. 36, 1369–1382 (1999)
Sherief, H.H.: Fundamental solution for thermoelasticity with two relaxation times. Int. J. Eng. Sci. 30, 861–870 (1992)
Sherief, H.H.: Fundamental solution of the generalized thermoelastic problem for short times. J. Therm. Stress. 9, 151–164 (1986)
Sherief, H.H., Anwar, M.N.: Problem in generalized thermoelasticity. J. Therm. Stress. 9, 165–181 (1986)
Kothari, S., Kumar, R., Mukhopadhyay, S.: On the fundamental solutions of generalized thermoelasticity with three phase-lags. J. Therm. Stress. 33, 1035–1048 (2010)
Ezzat, M.: Fundamental solution in thermoelasticity with two relaxation times for cylindrical region. Int. J. Eng. Sci. 33, 2011–2020 (1995)
Sladek, V., Sladek, J.: Boundary integral equation method in thermoelasticity part i general analysis. Appl. Math. Model. 7, 241–253 (1983)
Anwar, M.N., Sherief, H.H.: Boundary integral equation formulation for generalized thermoelasticity in a Laplace transform domain. Appl. Math. Model. 12, 161–166 (1988)
Anwar, M.N., Sherief, H.H.: Boundary integral equation formulation for thermoelasticity with two relaxation times. J. Therm. Stress. 17, 257–270 (1994)
El-Karamany, A., Ezzat, M.: Analytical aspects in boundary integral equation formulation for the generalized linear micropolar thermoelasticity. Int. J. Mech. Sci. 46, 389–409 (2004)
Sherief, H.H., Abd El-Latief, A.: Boundary element method in generalized thermoelasticity. Encycl. Therm. Stress. 9, 407–415 (2014)
Nowacki, W.: A dynamical problem of thermoelasticity. Arch. Mech. Stos. 9, 325–334 (1957)
Hetnarski, R.: Solution of the coupled problem of thermoelasticity in the form of series of functions. Arch. Mech. Stos. 16, 919–941 (1964)
Sherief, H.H., Ezzat, M.: Solution of the generalized problem of thermoelasticity in the form of series of functions. J. Therm. Stress. 17, 75–95 (1994)
Churchill, R.V.: Operational Mathematics, 3rd edn. McGraw-Hill, New York (1972)
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Sherief, H.H., Hussein, E.M. Fundamental solution of thermoelasticity with two relaxation times for an infinite spherically symmetric space. Z. Angew. Math. Phys. 68, 50 (2017). https://doi.org/10.1007/s00033-017-0794-8
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DOI: https://doi.org/10.1007/s00033-017-0794-8