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Computational Mathematics and Mathematical Physics

, Volume 59, Issue 9, pp 1470–1474 | Cite as

Effect of Heat on Deformations in Material with a Defect

  • E. V. AstakhovaEmail author
  • A. V. GlushkoEmail author
  • E. A. LoginovaEmail author
Article
  • 22 Downloads

Abstract

A system of thermoelasticity equations is considered. Boundary transmission conditions are specified by the differences in temperature, heat fluxes, deformations, and their first derivatives on the boundary. The stationary case is studied. The boundary (crack) is represented by the interval \([ - 1;1]\) of the \(O{{x}_{1}}\) axis. The given problem is investigated, its solution is found, and the well-posedness of its formulation is proved. The results of previous works are generalized. The subject of greatest interest is the asymptotic behavior, as \({{x}_{1}} \to \pm 1,\;{{x}_{2}} \to 0\), of the displacements \(u({{x}_{1}},{{x}_{2}}),\)\(v({{x}_{1}},{{x}_{2}})\) of a point \(({{x}_{1}},{{x}_{2}})\) under material deformations and the asymptotic behavior of their derivatives. Here, the functions \(u({{x}_{1}},{{x}_{2}}),\)\(v({{x}_{1}},{{x}_{2}})\) are assumed to depend on the material temperature \(T({{x}_{1}},{{x}_{2}})\) at the point \(({{x}_{1}},{{x}_{2}})\).

Keywords:

transmission problems asymptotics with respect to smoothness system of thermoelasticity equations heat conduction deformation boundary conditions 

Notes

REFERENCES

  1. 1.
    R. C. Koeller, “Applications of fractional calculus to the theory of viscoelasticity,” J. Appl. Mech. 51, 299–307 (1984).MathSciNetCrossRefGoogle Scholar
  2. 2.
    P. J. Torvik and R. L. Bagley, “On the appearance of the fractional derivative in the behavior of real materials,” J. Appl. Mech. 51, 294–298 (1984).CrossRefGoogle Scholar
  3. 3.
    S. Suresh and A. Mortensen, “Fundamentals of functionally graded materials,” Compr. Struct. Integr. 2, 607–644 (2003).Google Scholar
  4. 4.
    G. S. Mishuris and G. Kuhn, “Asymptotic behavior of the elastic solution near the tip of a crack situated at a nonideal interface,” Z. Angew. Math. Mech. 81, 811–826 (2001).CrossRefGoogle Scholar
  5. 5.
    V. Sladek, J. Sladek, and C. Zhang, “Transient heat conduction in anisotropic and functionally graded media by local integral equations,” Eng. Anal. Boundary Elem. 29 (11), 1047–1065 (2005).CrossRefGoogle Scholar
  6. 6.
    S. Krahulec, J. Sladek, V. Sladek, and Y.-Ch. Hon, “Meshless analyses for time-fractional heat diffusion in functionally graded materials,” Eng. Anal. Boundary Elem. 62, 57–64 (2016).MathSciNetCrossRefGoogle Scholar
  7. 7.
    G. Vitucci and G. Mishuris, “Analysis of residual stresses in thermoelastic multilayer cylinders,” J. Eur. Ceram. Soc. 36, 2411–2417 (2016).CrossRefGoogle Scholar
  8. 8.
    W.-S. Lei, “Non-oscillatory and non-singular asymptotic solutions to stress fields at interface cracks,” Fatigue Fract. Eng. Mater. Struct., June, 1–18 (2017).Google Scholar
  9. 9.
    A. S. Ryabenko and A. S. Chernikova, “On the uniqueness of the solution to a problem modeling the heat distribution in a plane with a crack at the interface between two materials,” Vestn. Voronezh. Gos. Univ. Ser. Fiz. Mat., No. 4, 124–133 (2017).Google Scholar
  10. 10.
    S. El-Borgi, F. Erdogan, and L. Hidri, “A partially insulated embedded crack in an infinite functionally graded medium under thermo-mechanical loading,” Int. J. Eng. Sci. 42, 371–393 (2004).CrossRefGoogle Scholar
  11. 11.
    A. V. Glushko and E. A. Loginova, “Asymptotic properties of the solution to the steady heat distribution problem in an inhomogeneous plane with a crack,” Vestn. Voronezh. Gos. Univ. Ser. Fiz. Mat., No. 2, 47–50 (2010).Google Scholar
  12. 12.
    E. A. Loginova, “Construction of the solution to the heat distribution problem in an inhomogeneous material with a crack,” Vestn. S.-Peterb. Univ., Ser. 1: Mat. Mekh. Astron., No. 1, 40–47 (2012).Google Scholar
  13. 13.
    A. V. Glushko, E. A. Loginova, V. E. Petrova, and A. S. Ryabenko, “Study of steady-state heat distribution in a plane with a crack in the case of variable internal thermal conductivity,” Comput. Math. Math. Phys. 55 (4), 690–698 (2015).MathSciNetCrossRefGoogle Scholar
  14. 14.
    A. V. Glushko, E. A. Loginova, and S. V. Pronina, “Solution to the problem of deformations in an inhomogeneous material with a crack under loading,” in Proceedings of the International Science-Practical Conference on Topical Issues of Natural and Mathematical Sciences under Present-Day Conditions of National Development (ITsRON, St. Petersburg, 2017), Vol. 4, pp. 11–15.Google Scholar
  15. 15.
    A. V. Glushko, E. A. Loginova, and S. V. Pronina, “Asymptotic behavior of the solution derivatives in the problem of elastic deformations in an inhomogeneous material under mechanical loading,” Vestn. Voronezh. Gos. Univ. Ser. Fiz. Mat., No. 4, 70–87 (2017).Google Scholar
  16. 16.
    S. M. Nikol’skii, Approximation of Functions of Several Variables and Imbedding Theorems (Springer-Verlag, New York, 1975; Nauka, Moscow, 1977).Google Scholar
  17. 17.
    G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge Univ. Press, Cambridge, 1995).zbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Voronezh State UniversityVoronezhRussia

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