Solution of the gradient thermoelasticity problem for a cylinder with a heat-protected coating
DOI:
https://doi.org/10.7242/1999-6691/2021.14.3.21Keywords:
gradient thermoelasticity, hollow cylinder, Cauchy problems, shooting method, WKB method, thermal protection coating, inhomogeneous materialsAbstract
The investigation of the stress-strain state of an infinitely long thermoelastic cylinder is carried out taking into account the scale effects. A thermal protective coating is applied to the outer side surface of the cylinder, the thermomechanical characteristics of which are functions of the radial coordinate. Thermal boundary conditions of the first kind are set on the stress-free side surfaces of the cylinder. To take into account the scale effects, the one-parameter gradient theory of thermoelasticity proposed by Aifantis is used. Additional boundary conditions and conjugation conditions for couple stresses are specified. The displacements and stresses are represented as the sum of the solutions of the thermoelasticity problem in the classical formulation and the gradient parts. After finding the radial temperature distribution, the thermoelasticity problem in the classical formulation with respect to radial displacements and stresses is solved numerically by the shooting method. Additional boundary layer terms for radial displacements at small values of the gradient parameter are found using the asymptotic method for solving linear differential equations with spatially varying coefficients (the Wentzel-Kramers-Brillouin method - WKB method). Calculations of radial displacements, Cauchy stresses, couple stresses and total stresses in the case of both homogeneous and inhomogeneous coatings are carried out using specific examples. It has been found that the Cauchy stresses and total stresses experience a jump at the boundary between the cylinder and the coating. The couple stresses at small gradient parameters are much less than the total stresses. An increase of the scale parameter reduces the values of radial displacements and total stresses. Deformations are continuous in both cases of coating (homogeneous and inhomogeneous). A comparative study of the influence of the value of the inhomogeneity parameter on the distribution of displacements and total stresses is carried out.
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Padture N.R., Gell M., Jordan E.H. Thermal barrier coatings for gas-turbine engine applications. Science, 2002, vol. 296, pp. 280-284. https://doi.org/10.1126/science.1068609">https://doi.org/10.1126/science.1068609
Bialas M. Finite element analysis of stress distribution in thermal barrier coatings. Coating Tech., 2008, vol. 202, pp. 6002-6010. https://doi.org/10.1016/j.surfcoat.2008.06.178">https://doi.org/10.1016/j.surfcoat.2008.06.178
Vatulyan A., Nesterov S., Nedin R. Regarding some thermoelastic models of «coating-substrate» system deformation. Continuum Mech. Thermodyn., 2020, vol. 32, pp. 1173-1186. https://doi.org/10.1007/s00161-019-00824-9">https://doi.org/10.1007/s00161-019-00824-9
Toupin R.A. Elastic materials with couple-stresses. Rational Mech. Anal., 1962, vol. 11, pp. 385-414. https://doi.org/10.1007/BF00253945">https://doi.org/10.1007/BF00253945
Mindlin R.D. Micro-structure in linear elasticity. Rational Mech. Anal., 1964, vol. 16, pp. 51-78. https://doi.org/10.1007/BF00248490">https://doi.org/10.1007/BF00248490
Ahmadi G., Firoozbakhsh K. First strain gradient theory of thermoelasticity. J. Solid Struct., 1975, vol. 11, pp. 339-345. https://doi.org/10.1016/0020-7683(75)90073-6">https://doi.org/10.1016/0020-7683(75)90073-6
Lur'ye M.V. Zadachi Lame v gradiyentnoy teorii uprugosti [Lame problems in the gradient theory of elasticity]. DAN SSSR – Akad. Nauk SSSR, 1968, vol. 181, no. 5, pp. 1087-1089.
Altan B.S., Aifantis E.C. On some aspects in the special theory of gradient elasticity. JMBM, 1997, vol. 8, pp. 231- https://doi.org/10.1515/JMBM.1997.8.3.231">https://doi.org/10.1515/JMBM.1997.8.3.231
Askes H., Aifantis E.C. Numerical modeling of size effects with gradient elasticity – Formulation, meshless discretization and examples. J. Fruct., 2002, vol. 117, pp. 347-358. https://doi.org/10.1023/A:1022225526483">https://doi.org/10.1023/A:1022225526483
Askes H., Aifantis E.C. Gradient elasticity in statics and dynamics: An overview of formulations, length scale identification procedures, finite element implementations and new results. J. Solid. Struct. 2011. Vol. 48. P. 1962-1990. https://doi.org/10.1016/j.ijsolstr.2011.03.006">https://doi.org/10.1016/j.ijsolstr.2011.03.006
Aifantis E.C. Gradient effects at the macro, micro and nano scales. JMBM, 1994, vol. 5, pp. 335-353. https://doi.org/10.1515/JMBM.1994.5.3.355">https://doi.org/10.1515/JMBM.1994.5.3.355
Lur’ye S.A., Belov P.A., Rabinskiy L.N., Zhavoronok S.I. Masshtabnyye effekty v mekhanike sploshnykh sred. Materialy s mikro- i nanostrukturoy [Scale effects in continuum mechanics. Materials from micro- and nanostructures]. Moscow, Izd-vo MAI, 2011. 160 p.
Gao X.-L., Park S.K. Variational formulation of a simplified strain gradient elasticity theory and its application to a pressurized thick-walled cylinder problem. J. Solid. Struct., 2007, vol. 44, pp. 7486-7499. https://doi.org/10.1016/j.ijsolstr.2007.04.022">https://doi.org/10.1016/j.ijsolstr.2007.04.022
Chu L., Dui G. Exact solutions for functionally graded micro-cylinders in first gradient elasticity. J. Mech. Sci., 2018, vol. 148, pp. 366-373. https://doi.org/10.1016/j.ijmecsci.2018.09.011">https://doi.org/10.1016/j.ijmecsci.2018.09.011
Sadeghi H., Baghani M., Naghdabadi R. Strain gradient thermoelasticity of functionally graded cylinders. Scientia IranicaB, 2014, vol. 21, pp. 1415-1423.
Hosseini M., Dini A., Eftekhari M. Strain gradient effects on the thermoelastic analysis of a functionally graded micro-rotating cylinder using generalized differential quadrature method. Acta Mech., 2017, vol. 228, pp. 1563-1580. https://doi.org/10.1007/s00707-016-1780-5">https://doi.org/10.1007/s00707-016-1780-5
Lurie S.A., Solyaev Yu.O., Rabinsky L.N., Kondratova Yu.N., Volov M.I. Simulation of the stress-strain state of thin composite coating based on solutions of the plane problem of strain-gradient elasticity for layer. Vestnik PNIPU. Mekhanika – PNRPU Mechanics Bulletin, 2013, no. 1, pp. 161-181.
Li A., Zhou S., Zhou S., Wang B. A size-dependent bilayered microbeam model based on strain gradient elasticity theory. Struct., 2014, vol. 108, pp. 259-266. https://doi.org/10.1016/j.compstruct.2013.09.020">https://doi.org/10.1016/j.compstruct.2013.09.020
Li A., Zhou S., Zhou S., Wang B. A size-dependent model for bi-layered Kirchhoff micro-plate based on strain gradient elasticity theory. Struct., 2014, vol. 113, pp. 272-280. https://doi.org/10.1016/j.compstruct.2014.03.028">https://doi.org/10.1016/j.compstruct.2014.03.028
Fu , Zhou S., Qi L. The size-dependent static bending of a partially covered laminated microbeam. Int. J. Mech. Sci., 2019, vol. 152, pp. 411-419. https://doi.org/10.1016/j.ijmecsci.2018.12.037">https://doi.org/10.1016/j.ijmecsci.2018.12.037
Lurie S.A., Pham T., Soliaev J.O. Gradient model of thermoelasticity and its application for the modeling of thin layered composite structures. MKMK – Journal of Composite Mechanics and Design, 2012, vol. 18, no. 3, pp. 440-449.
Vatulyan А.О., Nesterov S.А. On the deformation of a composite rod in the framework of gradient thermoelasticity. Materials Physics Mechanics, 2020, vol. 46, pp. 27-41. https://doi.org/10.18149/MPM.4612020_3">https://doi.org/10.18149/MPM.4612020_3
Stoer J., Bulirsch R. Introduction to numerical analysis. Springer, 2002. 746 p. https://doi.org/10.1007/978-0-387-21738-3">https://doi.org/10.1007/978-0-387-21738-3
Maslov V.P. Kompleksnyy metod VKB v nelineynykh uravneniyakh [Complex WKB method in nonlinear equation]. Moscow, Nauka, 1977. 384 p.
Vatulyan A.O., Nesterov S.A. On the identification problem of the thermomechanical characteristics of the finite functionally graded cylinder. Sarat. Un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika – Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2021, vol. 21, no. 1, pp. 35-47. https://doi.org/10.18500/1816-9791-2021-21-1-35-47">https://doi.org/10.18500/1816-9791-2021-21-1-35-47
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