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A stochastic thermoelastic diffusion interaction in an infinitely long annular cylinder

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Abstract

In this work, we study the effect of a stochastic heating on an elastic solid in the form of an infinitely long circular cylinder under the existence of a permeating substance in contact with the outer surface. The stochastic heating is driven by an additive Gaussian white noise. Random elastic, thermal and diffusive properties are obtained in terms of the induced noise. The model is considered in the context of the generalized thermoelastic diffusion theory with one relaxation time. The Laplace transform technique is used to obtain the analytical solution in the transformed domain, while a numerical inversion method for Laplace transform is used to obtain an approximate solution for the temperature, displacement, stress, concentration of the diffusive material and chemical potential. Statistically, first and second moments of the physical properties are derived analytically and discussed.

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References

  1. Allam A.A.: A stochastic half-space problem in the theory of generalized thermoelastic diffusion including heat source. Appl. Math. Model. 38, 4995–5021 (2014)

    Article  MathSciNet  Google Scholar 

  2. Allam A.A., Omar M.A., Ramadan K.T.: A thermoelastic diffusion interaction in an infinitely long annular cylinder. Arch. Appl. Mech. 84, 953–965 (2014)

    Article  Google Scholar 

  3. Aouadi M.: A generalized thermoelastic diffusion problem for an infinitely long solid cylinder. Int. J. Math. Math. Sci. 2006, 1–15 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Asnafi A., Abedi M.: A complete analogical study on the dynamic stability analysis of isotropic functionally graded plates subjected to lateral stochastic loads. Acta Mech. 226, 2347–2363 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Biot M.: Thermoelasticity and irreversible thermo-dynamics. J. Appl. Phys. 27, 240–253 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chester M.: Second sound in solids. Phys. Rev. 131, 2013–2015 (1963)

    Article  Google Scholar 

  7. Dhaliwal R.S., Sherief H.H.: Generalized thermoelasticity for anisotropic media. Q. Appl. Math. 33, 1–8 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  8. de Groot S.R., Mazur P.: Non-Equilibrium Thermodynamics, vol. 140. Wiley, New York (1962)

    Google Scholar 

  9. Graham C., Talay D.: Stochastic Simulation and Monte Carlo Methods. Springer-Verlag, Berlin (2013)

    Book  MATH  Google Scholar 

  10. Honig G., Hirdes U.: A method for the numerical inversion of the Laplace transform. J. Comput. Appl. Math. 10, 113–132 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  11. Klebaner F.C.: introduction to stochastic calculus with applications, 2 edn. Imperial College Press, London (2005)

    Book  MATH  Google Scholar 

  12. Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer, Berlin (1992)

  13. Lawler G.F.: Introduction to stochastic processes. Chapman and Hall/CRC, USA (2006)

    MATH  Google Scholar 

  14. Lord H., Shulman Y.: A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids 15, 299–309 (1967)

    Article  MATH  Google Scholar 

  15. Nowacki W.: Dynamical problems of thermoelastic diffusion in solids i. Bulletin de l’Académie Polonaise des Sciences, Série des Sciences Techniques 22, 55–64 (1974a)

    MathSciNet  Google Scholar 

  16. Nowacki W.: Dynamical problems of thermoelastic diffusion in solids ii. Bulletin de l’Académie Polonaise des Sciences, Série des Sciences Techniques 22, 129–135 (1974b)

    Google Scholar 

  17. Nowacki W.: Dynamic problems of diffusion in solids. Eng. Fract. Mech. 8, 261–266 (1976)

    Article  MATH  Google Scholar 

  18. Nowinski, J.L.: Theory of thermoelasticity with applications, vol. 122 (2000)

  19. Oksendal B.: Stochastic differential equations :An introduction with applications. Springer, Berlin (1995)

    Book  MATH  Google Scholar 

  20. Omar M.A., Aboul-Hassan A., Rabia S.I.: The composite Milstein methods for the numerical solution of Ito stochastic differential equations. J. Comput. Appl. Math. 235, 2277–2299 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Oriani R.A.: Thermomigration in solid metals. J. Phys. Chem. Solids 30, 339–351 (1969)

    Article  Google Scholar 

  22. Philibert J.: One and a half century of diffusion: Fick, Einstein, before and beyond. Diffus. Fundam. 2, 1.1–1.10 (2005)

    Google Scholar 

  23. Sherief H., Anwar M.: State space approach to two-dimensional generalized thermoelasticity problems. J. Therm. Stress. 17, 567–590 (1994)

    Article  MathSciNet  Google Scholar 

  24. Sherief H., Ezzat M.: Solution of the generalized problem of thermoelasticity in the form of series of functions. J Therm. Stress. 17, 75–95 (1994)

    Article  MathSciNet  Google Scholar 

  25. Sherief H.H., El-Maghraby N.M., Allam A.A.: Stochastic thermal shock problem in generalized thermoelasticity. Appl. Math. Model. 37, 762–775 (2013)

    Article  MathSciNet  Google Scholar 

  26. Sherief H.H., Hamza F.A., Salah H.A.: The theory of generalized thermoelastic diffusion. Int. J. Eng. Sci. 42, 591–608 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  27. Sherief H.H., Saleh H.A.: A half-space problem in the theory of generalized thermoelastic diffusion. Int. J. Solids Struct. 42, 4484–4493 (2005)

    Article  MATH  Google Scholar 

  28. Tripathi J.J., Kedar G.D., Deshmukh K.: Generalized thermoelastic diffusion problem in a thick circular plate with axisymmetric heat supply. Acta Mech. 226, 2121–2134 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  29. Tripathi J.J., Kedar G.D., Deshmukh K.: Two-dimensional generalized thermoelastic diffusion in a half-space under axisymmetric distributions. Acta Mech. 226, 3263–3274 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  30. Wang C., Qiu Z., Chen X.: Uncertainty analysis for heat convection-diffusion problem with large uncertain-but-bounded parameters. Acta Mech. 226, 3831–3844 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Basic and Applied Science, College of Engineering and Technology, Arab Academy for Science, Technology and Maritime Transport, Abu Quir Campus, P.O. BOX 1029, Alexandria, Egypt

    Allam A. Allam, Khaled T. Ramadan & M. A. Omar

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  1. Allam A. Allam
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  2. Khaled T. Ramadan
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  3. M. A. Omar
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Correspondence to Allam A. Allam.

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Allam, A.A., Ramadan, K.T. & Omar, M.A. A stochastic thermoelastic diffusion interaction in an infinitely long annular cylinder. Acta Mech 227, 1429–1443 (2016). https://doi.org/10.1007/s00707-016-1569-6

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  • DOI: https://doi.org/10.1007/s00707-016-1569-6

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