Computational Mathematics and Mathematical Physics

, Volume 59, Issue 9, pp 1470–1474

# Effect of Heat on Deformations in Material with a Defect

Article

### 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

## REFERENCES

1. 1.
R. C. Koeller, “Applications of fractional calculus to the theory of viscoelasticity,” J. Appl. Mech. 51, 299–307 (1984).
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).
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).
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).
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).
7. 7.
G. Vitucci and G. Mishuris, “Analysis of residual stresses in thermoelastic multilayer cylinders,” J. Eur. Ceram. Soc. 36, 2411–2417 (2016).
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).
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).
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).