Fractional heat conduction equation and associated thermal stresses in an infinite solid with spherical cavity

Quarterly Journal of Mechanics and Applied Mathematics

Volumn 61, Issue 4, 2008, Pages 523-547

  Povstenko, Y Z ^a  

^a JAN DLUGOSZ UNIVERSITY   (Poland)

Author keywords

[No Author keywords available]

Indexed keywords

ELASTICITY; ENERGY DISSIPATION; IMAGE CODING; INTEGRAL EQUATIONS; INTERPOLATION; SPHERES; STRENGTH OF MATERIALS; THERMAL STRESS; THERMOELASTICITY; WAVE EQUATIONS;

CAPUTO FRACTIONAL DERIVATIVES; FRACTIONAL; FRACTIONAL DERIVATIVES; HEAT CONDUCTION EQUATIONS; INTEGRAL TRANSFORM TECHNIQUES; NUMERICAL RESULTS; SPHERICAL CAVITIES; THERMOELASTICITY WITHOUT ENERGY DISSIPATIONS;

HEAT CONDUCTION;

EID: 54949108246     PISSN: 00335614     EISSN: 14643855     Source Type: Journal    
DOI: 10.1093/qjmam/hbn016     Document Type: Article

References (42)

1. Ya. S. Podstrigach and Yu. M. Kolyano, Generalized Thermomechanics (Naukova Dumka, Kiev 1976) (in Russian).
2. D. S. Chandrasekharaiah, Thermoelasticity with second sound: A review, Appl. Mech. Rev. 39 (1986) 355-376.
3. D. S. Chandrasekharaiah, Hyperbolic thermoelasticity: A review of recent literature, ibid. 51 (1998) 705-729.
4. K. K. Tamma and X. Zhou, Macroscale and microscale thermal transport and thermomechanical interactions: Some noteworthy perspectives, J. Thermal Stresses 21 (1998) 405-449.
5. R. B. Hetnarski and J. Ignaczak, Generalized thermoelasticity, ibid. 22 (1999) 451-476.
6. R. B. Hetnarski and J. Ignaczak, Nonclassical dynamical thermoelasticity, Int. J. Solids Struct. 37 (2000) 215-224.
7. M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speed, Arch. Rat. Mech. Anal. 31 (1968) 113-126.
8. P. J. Chen and M. E. Gurtin, A second sound in materials with memory, Z. Angew. Math. Phys. 21 (1970) 232-241.
9. C. Cattaneo, On the conduction of heat, Atti. Semin. Fis. Univ. Modena 3 (1948) 3-21.
10. C. Cattaneo, Sur une forme de l'équation de la chaleur éliminant le paradoxe d'une propagation instantanée, C. R. Acad. Sci. 247 (1958) 431-433.
11. P. Vernotte, Les paradoxes de la théorie continue de l'équation de la chaleur, ibid. 246 (1958) 3154-3155.
12. S. Kaliski, Wave equation of thermoelasticity, Bull. Acad. Polon. Sci. Sér. Sci. Techn. 13 (1965) 253-260.
13. H.W. Lord and Y. Shulman, A generalized dynamical theory of thermoelasticity, J. Mech. Phys. Solids 15 (1967) 299-309.
14. A. E. Green and P. M. Naghdi, Thermoelasticity without energy dissipation, J. Elast. 31 (1993) 189-208.
15. R. Gorenflo, F. Mainardi, D. Moretti and P. Paradisi, Time fractional diffusion: A discrete random walk approach, Nonlinear Dyn. 29 (2002) 129-143.
16. Y. Z. Povstenko, Fractional heat conduction equation and associated thermal stress, J. Thermal Stresses 28 (2005) 83-102.
17. Y. Z. Povstenko, Stresses exerted by a source of diffusion in a case of a non-parabolic diffusion equation, Int. J. Engng Sci. 43 2005) 977-991.
18. Y. Z. Povstenko, Two-dimensional axisymmentric stresses exerted by instantaneous pulses and sources of diffusion in an infinite space in a case of time-fractional diffusion equation, Int. J. Solids Struct. 44 (2007) 2324-2348.
19. Y. Z. Povstenko, Fundamental solution to three-dimensional diffusion-wave equation and associated diffusive stresses, Chaos Solitons Fractals 36 (2008) 961-972.
20. Y. Z. Povstenko, Fractional radial diffusion in a cylinder, J. Mol. Liq. 137 (2008) 46-50.
21. Y. Z. Povstenko, Fundamental solution to central symmetric problems for fractional heat conduction equation and associated thermal stresses, J. Thermal Stresses 31 (2008) 127-148.
22. H. Parkus, Instationäre Wärmespannungen (Springer, Wien 1959).
23. N. Noda, R. Hetnarski and Y. Tanigawa, Thermal Stresses, 2nd edn. (Taylor and Francis, New York 2003).
24. B. Mukhopadhyay, R. Bera and L. Debnath, On generalized thermoelastic disturbances in an elastic solid with a spherical cavity, J. Appl. Math. Stochastic Anal. 4(1991) 225-240.
25. S. B. Sinha and K. A. Elsibai, Thermal stresses for an infinite body with a spherical cavity with two relaxation times, J. Thermal Stresses 19 (1996) 495-510.
26. H. H. Sherif and H. A. Saleh, A problem for an infinite thermoelastic body with a spherical cavity, Int. J. Engng Sci. 36 (1998) 473-487.
27. A. M. Abd-All, A. N. Abd-alla and N. A. Zeidan, Transient thermal stresses in a spherical orthotropic elastic medium with spherical cavity, Appl. Math. Comput. 105 (1999) 231-252.
28. P. M. Jordan and P. Puri, Thermal stresses in a spherical shell under three thermoelastic models, J. Thermal Stresses 24 (2001) 47-70.
29. S. Mukhopadhyay, Thermoelastic interaction without energy dissipation in an unbounded medium with a spherical cavity due to a thermal shock at the boundary, ibid. 25 (2002) 877-887.
30. A. Kar and M. Kanoria, Thermoelastic interaction with energy dissipation in an unbounded body with a spherical hole, Int. J. Solids Struct. 44 (2007) 2961-2971.
31. R. Gorenflo and F. Mainardi, Fractional calculus: Integral and differential equations of fractional order, Fractals and Fractional Calculus in Continuum Mechanics (ed. A. Carpinetti and F. Mainardi; Springer, Wien 1997) 223-276.
32. A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations (Elsevier, Amsterdam 2006).
33. Ya. S. Podstrigach and R. N. Shvets, The quasi-static problem in coupled thermoelasticity, Int. Appl. Mech. 5(1969) 33-39.
34. B. A. Boley and J. H. Weiner, Theory of Thermal Stresses (John Wiley, New York 1960).
35. W. Nowacki, Thermoelasticity (Polish Scientific Publishers, Warszawa 1986).
36. A. Erdélyi, W. Magnus, F. Oberhettinger and F. Tricomi, Higher Transcendental Functions, Vol. 3 (McGraw-Hill, New York 1955).
37. E. Sternberg and J. G. Chakravorty, Thermal shock in an elastic body with a spherical cavity, Q. Appl. Math. 17 (1959) 205-218.
38. Y. Fujita, Integrodifferential equation which interpolates the heat equation and the wave equation, Osaka J. Math. 27 (1990) 309-321.
39. A. Erdélyi, W. Magnus, F. Oberhettinger and F. Tricomi, Tables of Integral Transforms, Vol. 1 (McGraw-Hill, New York 1954).
40. A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, Integrals and Series: Elementary Functions (Nauka, Moscow 1981) (in Russian).
41. A. S. Galitsyn and A. N. Zhukovsky, Integral Transforms and Special Functions in Heat Conduction Problems (Naukova Dumka, Kiev 1976) (in Russian).
42. R. Gorenflo and F. Mainardi, Fractional oscillations and Mittag-Leffler functions, Preprint PR-A-96-14. Fachbereich Mathematik und Informatik (Freie Universität Berlin, Berlin 1996) 1-22.