Solution of the problem of gradient thermoelasticity for a coated strip

A.O. Vatulyan ^a,b∗ , S.A. Nesterov ^b∗∗

^a Southern Federal University, Rostov-on-Don, 344006 Russia

^b Southern Mathematical Institute – Brach of the Vladikavkaz Scientific Centre of the Russian Academy of Sciences, Vladikavkaz, 362025 Russia

E-mail: ^∗aovatulyan@sfedu.ru, ^∗∗1079@list.ru

Received November 20, 2020

ORIGINAL ARTICLE

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DOI: 10.26907/2541-7746.2021.2.181-196

For citation: Vatulyan A.O., Nesterov S.A. Solution of the problem of gradient thermoelasticity for a coated strip. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2021, vol. 163, no. 2, pp. 181–196. doi: 10.26907/2541-7746.2021.2.181-

196. (In Russian)

Abstract

The formulation of a one-parameter problem of gradient thermoelasticity for the “thermal protective coating – substrate” system which is modeled by a composite strip is presented. The lower boundary of the strip is rigidly clamped and maintained at zero temperature, and on the upper boundary, free of stresses, a heat flux localized over small segment acts, while the rest of the upper boundary is thermally insulated. First, the Fourier transform in the horizontal coordinate is applied to the equilibrium and heat conduction equations and the boundary conditions. After finding the temperature transformant, the transformants of horizontal and vertical displacement are determined. The Vishik–Lyusternik’s asymptotic approach is used to find the transformants of displacements, taking into account the presence of boundary layer solutions in the vicinity of the strip boundaries. The numerical inversion of the transformants is based on the compound quadrature formula of Philon. A comparison is made of the distribution of Cauchy displacements and stresses obtained on the basis of solving the problem in the classical formulation and in the gradient formulation. It is found that a change in the gradient parameter insignificantly affects the distribution of displacements, but strongly on the distribution of Cauchy stresses and moment stresses. The displacements are continuous, equal to zero in the containment, have certain symmetry when distributed along the horizontal coordinate, and attenuate with distance from the source. Near the termination, the Cauchy stresses decrease exponentially to zero in accordance with the boundary conditions, experience a jump on the mate line. Since displacements and deformations are continuous on the line of conjugation of the strips, due to the jump in thermomechanical characteristics, a Cauchy stress jump occurs in the vicinity of the line of conjugation of the strips. The magnitude of the Cauchy stress jump also depends on the ratio between the gradient parameter and the coating thickness. It is revealed that when the thickness of the coating is less than two gradient parameters, the stress jump changes exponentially and then goes to a stationary value. The moment stresses are continuous and peak at the interface of the materials.

Keywords: strip, coating, gradient thermoelasticity, Cauchy stresses, moment stresses, boundary layer, Vishik–Lyusternik’s method, stress jump

Acknowledgments. The study was supported by the Southern Mathematical Institute – Brach of the Vladikavkaz Scientific Centre of the Russian Academy of Sciences, Vladikavkaz, Russia.

References

1. Toupin R.A. Elastic materials with couple stresses. Arch. Ration. Mech. Anal., 1962, vol. 11, pp. 385–414. doi: 10.1007/BF00253945.
2. Mindlin R.D. Micro-structure in linear elasticity. Arch. Ration. Mech. Anal., 1964, vol. 16, pp. 51–78. doi: 10.1007/BF00248490.
3. Ahmadi G., Firoozbakhsh K. First strain gradient theory of thermoelasticity. Int. J. Solids Struct., 1975, vol. 11, no. 3, pp. 339–345.
4. Altan B.S., Aifantis E.C. On some aspects in the special theory of gradient elasticity. J. Mech. Behav. Mater., 1997, vol. 8, no. 3, pp. 231–282. doi: 10.1515/JMBM.1997.8.3.231.
5. Lurie S.A., Fam T., Solyaev Yu.O. Gradient model of thermoelasticity and its applications to the modeling of thin-layered composite structures. Mekh. Kompoz. Mater. Konstr., 2012, vol. 18, no. 3, pp. 440–449. (In Russian)
6. Aifantis K., Askes H. Gradient elasticity with interfaces as surfaces of discontinuity for the strain gradient. J. Mech. Behav. Mater., 2007, vol. 18, no. 4, pp. 283–306. doi: 10.1515/JMBM.2007.18.4.283.
7. Lurie S.A., Solyaev Yu.O., Rabinskii L.N., Kondratova Yu.N., Volov M.I. simulation of the stress-strain state of thin composite coatings based on solutions of the plane problem of strain-gradient elasticity for a layer. Vestn. Permsk. Nats. Issled. Politekh. Univ. Mekh., 2013, no. 1, pp. 161–181. (In Russian)
8. Zhang N.H., Meng W.L., Aifantis E.C. Elastic bending analysis of bilayered beams containing a gradient layer by an alternative two-variable method. Compos. Struct., 2011, vol. 93, no. 12, pp. 3130–3139. doi: 10.1016/j.compstruct.2011.06.019.
9. Li A., Zhou Sh., Zhou Sh., Wang B. A size-dependent bilayered microbeam model based on strain gradient elasticity theory Compos. Struct., 2014, vol. 108, pp. 259–266. doi: 10.1016/j.compstruct.2013.09.020.
10. Li A., Zhou Sh., Zhou Sh., Wang B. A size-dependent model for bilayered Kirchhoff micro-plate. Compos. Struct., 2014, vol. 113, pp. 272–280. doi: 10.1016/j.compstruct.2014.03.028.
11. Fu G., Zhou Sh., Qi L. The size-dependent static bending of a partially covered laminated microbeam. Int. J. Mech. Sci., 2019, vol. 152, pp. 411–419. doi: 10.1016/j.ijmecsci.2018.12.037.
12. Sadeghi H., Baghani M., Naghdabadi R. Strain gradient thermoelasticity of functionally graded cylinders. Sci. Iran., Trans. B, 2014, vol. 21, no. 4, pp. 1415–1423.
13. Papargyri-Beskou S., Tsinopoulos S. Lame’s strain potential method for plane gradient elasticity problems. Arch. Appl. Mech., 2015, vol. 85, nos. 9–10, pp. 1399–1419. doi: 10.1007/s00419-014-0964-5.
14. Boley B., Weiner J. Teoriya temperaturnykh napryazhenii [Theory of Thermal Stresses]. Moscow, Mir, 1964. 517 p. (In Russian)
15. Kovalenko A.D. Termouprugost’ [Thermoelasticity]. Kiev. Yshch. Shk., 1975. 216 p. (In Russian)
16. Krylov V.I. Priblizhennoe vyshislenie integralov [Approximate Calculation of Integrals]. Moscow, Nauka, 1967. 500 p. (In Russian)
17. Filon L.N.G. III. On a quadrature formula for trigonometric integrals. Proc. R. Soc. Edinburgh, 1930, vol. 49, pp. 38–47.
18. Vishik M.I., Lyusternik L.A. Regular degeneration and boundary layer for linear differential equations with small parameter. Usp. Mat. Nauk, 1957, vol. 12, no. 5, pp. 3–122. (In Russian)

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