Thermoelasticity that uses fractional heat conduction equation

Journal of Mathematical Sciences

Volumn 162, Issue 2, 2009, Pages 296-305

  Povstenko, Y Z ^a,b  

^a PIDSTRYHACH INSTITUTE FOR APPLIED PROBLEMS OF MECHANICS AND MATHEMATICS   (Ukraine)

^b JAN DLUGOSZ UNIVERSITY   (Poland)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 70350315751     PISSN: 10723374     EISSN: 15738795     Source Type: Journal    
DOI: 10.1007/s10958-009-9636-3     Document Type: Article

References (67)

1. N. Petrov and J. Brankov, Contemporary Problems of Thermodynamics [Russian translation], Mir, Moscow (1986).
2. Ya. S. Pidstryhach and Yu. M. Kolyano, Generalized Thermomechanics [in Russian], Naukova Dumka, Kiev (1976).
3. Yu. N. Rabotnov, Creep of Structural Elements [in Russian], Nauka, Moscow (1966).
4. Yu. N. Rabotnov, Elements of Hereditary Mechanics of Solids [in Russian], Nauka, Moscow (1977).
5. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Integrals and Derivatives of Fractional Order and Their Applications [in Russian], Nauka i Tekhnika, Minsk (1987)
6. English translation: S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Newark (1993).
7. R. L. Bagley and P. J. Torvik, On the appearance of the fractional derivative in the behavior of real materials, J. Appl. Mech., 51, No. 2, 294-298 (1984).
8. P. L. Butzer and U. Westphal, An introduction to fractional calculus, in: R. Hilfer (editor), Applications of Fractional Calculus in Physics, World Scientific, Singapore (2000), pp. 1-85.
9. M. Caputo, Linear models of dissipation whose Q is almost frequency independent, Part II, Geophys. J. Roy. Astron. Soc., 13, No. 5, 529-539 (1967).
10. A. Carpinteri and P. Cornetti, A fractional calculus approach to the description of stress and strain localization, Chaos, Solitons Fractals, 13, No. 1, 85-94 (2002).
11. A. Carpinteri, P. Cornetti, and K. M. Kolwankar, Calculation of the tensile and flexural strength of disordered materials using fractional calculus, Chaos, Solitons Fractals, 21, No. 3, 623-632 (2004).
12. A. Carpinteri, P. Cornetti, and S. Puzzi, Scaling laws and multiscale approach in the mechanics of heterogeneous and disordered materials, Appl. Mech. Rev., 59, No. 5, 283-305 (2006).
13. C. Cattaneo, Sulla conduzione del calore, Atti Sem. Mat. Fis. Univ. Modena, 3, 83-101 (1948).
14. C. Cattaneo, Sur une forme de l'équation de la chaleur éliminant le paradoxe d'une propagation instantanée, C. R. Acad. Sci., 247, No. 4, 431-433 (1958).
15. D. S. Chandrasekharaiah, Thermoelasticity with second sound: a review, Appl. Mech. Rev., 39, No. 3, 355-376 (1986).
16. D. S. Chandrasekharaiah, Hyperbolic thermoelasticity: a review of recent literature, Appl. Mech. Rev., 51, No. 12, 705-729 (1998).
17. A. S. Chaves, A fractional diffusion equation to describe Lévy flights, Phys. Lett. A, 239, No. 1-2, 13-16 (1998).
18. P. J. Chen and M. E. Gurtin, On second sound in materials with memory, Z. Angew. Math. Phys., 21, No. 2, 232-241 (1970).
19. W. Day, The Thermodynamics of Simple Materials with Fading Memory, Springer, Berlin (1972).
20. H. Demiray and A. C. Eringen, On nonlocal diffusion of gases, Arch. Mech., 30, No. 1, 65-77 (1978).
21. A. C. Eringen, Theory of nonlocal thermoelasticity, Int. J. Eng. Sci., 12, No. 12, 1063-1077 (1974).
22. Y. Fujita, Integrodifferential equation which interpolates the heat equation and the wave equation, Osaka J. Math., 27, No. 2, 309-321 (1990).
23. R. Gorenflo, A. Iskenderov, and Y. Luchko, Mapping between solutions of fractional diffusion-wave equations, Fractional Calculus Appl. Anal., 3, No. 1, 75-86 (2000).
24. R. Gorenflo and F. Mainardi, Fractional calculus: integral and differential equations of fractional order, in: A. Carpinetti and F. Mainardi (editors), Fractals and Fractional Calculus in Continuum Mechanics, Springer, Wien (1997), pp. 223-276.
25. R. Gorenflo and F. Mainardi, Random walk models for space-fractional diffusion processes, Fractional Calculus Appl. Anal., 1, No. 2, 167-191 (1998).
26. R. Gorenflo and F. Mainardi, Fractional calculus and stable probability distributions, Arch. Mech., 50, No. 3, 377-388 (1998).
27. R. Gorenflo, F. Mainardi, D. Moretti, and P. Paradisi, Time fractional diffusion: a discrete random walk approach, Nonlin. Dynamics, 29, Nos. 1-4, 129-143 (2002).
28. A. E. Green and P. M. Naghdi, Thermoelasticity without energy dissipation, J. Elasticity, 31, No. 3, 189-208 (1993).
29. M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal., 31, No. 2, 113-126 (1968).
30. A. Hanyga, Multidimensional solutions of space-time-fractional diffusion equations, Proc. Roy. Soc. London A, 458, No. 2018, 429-450 (2002).
31. R. B. Hetnarski and J. Ignaczak, Generalized thermoelasticity, J. Thermal Stresses, 22, No. 4-5, 451-476 (1999).
32. R. B. Hetnarski and J. Ignaczak, Nonclassical dynamical thermoelasticity, Int. J. Solids Struct., 37, No. 1-2, 215-224 (2000).
33. R. Hilfer (editor), Applications of Fractional Calculus in Physics, World Scientific, Singapore (2000).
34. J. Ignaczak, Generalized thermoelasticity and its applications, in: R. B. Hetnarski (editor), Thermal Stresses III, Elsevier, New York (1989), pp. 279-354.
35. D. D. Joseph and L. Preziosi, Heat waves, Rev. Mod. Phys., 61, No. 1, 41-73 (1989).
36. D. D. Joseph and L. Preziosi, Addendum to the paper 'Heat waves,' Rev. Mod. Phys., 62, No. 2, 375-391 (1990).
37. S. Kaliski, Wave equations of thermoelasticity, Bull. Acad. Pol. Sci. Sér. Sci. Techn., 13, No. 4, 253-260 (1965).
38. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam (2006).
39. V. S. Kiryakova, Generalized Fractional Calculus and Applications, Longman, Harlow (1994).
40. H. W. Lord and Y. Shulman, A generalized dynamical theory of thermoelasticity, J. Mech. Phys. Solids, 15, No. 5, 299-309 (1967).
41. F. Mainardi, Applications of fractional calculus in mechanics, in: P. Rusev, I. Dimovski, and V. Kiryakova (editors), Transform Methods and Special Functions (Varna 96), Bulgarian Academy of Sciences, Sofia (1998), pp. 309-334.
42. F. Mainardi and R. Gorenflo, On Mittag-Leffler-type functions in fractional evolution processes, J. Comput. Appl. Math., 118, No. 1-2, 283-299 (2000).
43. F. Mainardi, Y. Luchko, and G. Pagnini, The fundamental solution of the space-time fractional diffusion equation, Fractional Calculus Appl. Anal., 4, No. 2, 153-192 (2001).
44. R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339, No. 1, 1-77 (2000).
45. R. Metzler and J. Klafter, The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A: Math. Gen., 37, No. 31, R161-R208 (2004).
46. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York (1993).
47. T. B. Moodi and R. J. Tait, On thermal transients with finite wave speeds, Acta Mech., 50, No. 1/2, 97-104 (1983).
48. R. R. Nigmatullin, To the theoretical explanation of the 'universal response,' Phys. Stat. Sol. (b), 123, No. 2, 739-745 (1984).
49. R. R. Nigmatullin, On the theory of relaxation with 'remnant' temperature, Phys. Stat. Sol. (b), 124, No. 1, 389-393 (1984).
50. F. R. Norwood, Transient thermal waves in the general theory of heat conduction with finite wave speeds, J. Appl. Mech., 39, No. 3, 673-676 (1972).
51. J. W. Nunziato, On heat conduction in materials with memory, Quart. Appl. Math., 29, No. 2, 187-204 (1971).
52. K. B. Oldham and J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Academic Press, New York (1974).
53. P. Paradisi, R. Cesari, F. Mainardi, A. Maurizi, and F. Tampieri, A generalized Fick's law to describe non-local transport effects, Phys. Chem. Earth (B), 26, No. 4, 275-279 (2001).
54. P. Paradisi, R. Cesari, F. Mainardi, and F. Tampieri, The fractional Fick's law for non-local transport processes, Physica A, 293, No. 1-2, 130-142 (2001).
55. I. Podlubny, Fractional Differential Equations, Academic Press, New York (1999).
56. Y. Z. Povstenko, Fractional heat conduction equation and associated thermal stresses, J. Thermal Stresses, 28, No. 1, 83-102 (2005).
57. Y. Z. Povstenko, Stresses exerted by a source of diffusion in a case of a non-parabolic diffusion equation, Int. J. Eng. Sci., 43, No. 11-12, 977-991 (2005).
58. Y. Z. Povstenko, Thermoelasticity based on fractional heat conduction equation, in: F. Ziegler, R. Heuer, and C. Adam (editors), Proceedings of the 6th International Congress on Thermal Stresses (May 26-29, 2005, Vienna, Austria), Vol. 2, Vienna University of Technology, Vienna (2005), pp. 501-504.
59. Y. Z. Povstenko, Two-dimensional axisymmetric 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, No. 7-8, 2324-2348 (2007).
60. Y. Z. Povstenko, Fundamental solution to three-dimensional diffusion-wave equation and associated diffusive stresses, Chaos, Solitons Fractals, 36, No. 4, 961-972 (2008).
61. Y. Z. Povstenko, Fractional radial diffusion in a cylinder, J. Mol. Liquids, 137, No. 1-3, 46-50 (2008).
62. Y. Z. Povstenko, Fundamental solutions to central symmetric problems for fractional heat conduction equation and associated thermal stresses, J. Thermal Stresses, 31, No. 2, 127-148 (2008).
63. Yu. A. Rossikhin and M. V. Shitikova, Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids, Appl. Mech. Rev., 50, No. 1, 15-67 (1997).
64. A. I. Saichev and G. M. Zaslavsky, Fractional kinetic equations: solutions and applications, Chaos, 7, No. 4, 753-764 (1997).
65. K. K. Tamma and X. Zhou, Macroscale and microscale thermal transport and thermomechanical interactions: some noteworthy perspectives, J. Thermal Stresses, 21, No. 3/4, 405-449 (1998).
66. P. Vernotte, Les paradoxes de la théorie continue de l'équation de la chaleur, C. R. Acad. Sci., 247, No. 22, 3154-3155 (1958).
67. G. M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport, Phys. Rep., 371, No. 6, 461-580 (2002).