Direct integration method in three-dimensional elasticity and thermoelasticity problems for inhomogeneous transversely isotropic solids: governing equations in terms of stresses

Authors

  • R. M. Kushnir Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, NAS of Ukraine, 79060, Lviv, 3-B Naukova Str.
  • Y. V. Tokovyy Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, NAS of Ukraine, 79060, Lviv, 3-B Naukova Str. https://orcid.org/0000-0003-1610-0113
  • D. S. Boiko Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, NAS of Ukraine, 79060, Lviv, 3-B Naukova Str.

DOI:

https://doi.org/10.17721/1812-5409.2019/1.23

Abstract

An efficient technique for thermoelastic analysis of inhomogeneous anisotropic solids is suggested within the framework of three-dimensional formulation. By making use of the direct integration method, a system of governing equations is derived in order to solve three-dimensional problems of elasticity and thermoelasticity for transversely isotropic inhomogeneous solids with elastic and thermo-physical properties represented by differentiable functions of the variable in the direction that is transversal to the plane of isotropy. By implementing the relevant separation of variables, the obtained equations can be uncoupled and reduced to second-kind integral equations for individual stress-tensor components and the total stress, which represents the trace of the stress tensor. The latter equations can be attempted by any of the numerical, analyticalnumerical, or analytical means available for the solution of the second-kind integral equations. In order to construct the solutions in an explicit form, an advanced solution technique can be developed on the basis of the resolvent-kernel method implying the series representation by the recurring kernels, computed iteratively by the original kernel of an integral equation.

Key words: transversally isotropic inhomogeneous solids, direct integration method, three-dimensional problem, governing equations.

Pages of the article in the issue: 102-105

Language of the article: Ukrainian

References

KALYNYAK, B.М., TOKOVYY, Y.V., and YASINSKYY, А.V. (2019) Pryami ta oberneni zadachi termomekhaniki stosovno optimizatsiyi ta identifikatsiyi termonapruzhenoho stanu deformivnikh tverdykh til. Matematichni metody ta fizykomekhanichni polya. 236(1). p. 21–34.

KOLCHIN, G.B. (1971) Raschet elementov konstrukcij iz uprugikh neodnorodnykh materialov. Chisinau: Kartya Moldoveniaske.

LEKHNITSKII, S.G. (1977). Teoriya uprugosti anizotropnogo tela. Moscow: Nauka.

DING, H., CHEN, W., and ZHANG, L. (2006). Elasticity of transversely isotropic materials. Dordrecht: Springer.

TANIGAWA, Y. (1995) Some basic thermoelastic problems for nonhomogeneous structural materials. Applied Mechanics Reviews. 48(6). p. 287–300.

TOKOVYY, Y.V. (2014). Direct integration method. In R.B. Hetnarski (ed.) Encyclopedia of Thermal Stresses, Doredecht: Springer. 2. p. 951–960.

TOKOVYY, Y.V., KALYNYAK, B.M., and MA, C.-C. (2014). Nonhomogeneous solids: Integral equations approach In R. B. Hetnarski (ed.) Encyclopedia of Thermal Stresses, Doredecht: Springer. 7. p. 3350–3356.

TOKOVYY, Y. (2019) Direct integration of three-dimensional thermoelasticity equations for a transversely isotropic layer. Journal of Thermal Stresses. 42(1). p. 49–64.

No 1 (2019)

Downloads

How to Cite

Kushnir, R. M., Tokovyy, Y. V., & Boiko, D. S. (2019). Direct integration method in three-dimensional elasticity and thermoelasticity problems for inhomogeneous transversely isotropic solids: governing equations in terms of stresses. Bulletin of Taras Shevchenko National University of Kyiv. Physics and Mathematics, (1), 102–105. https://doi.org/10.17721/1812-5409.2019/1.23

Issue

No. 1 (2019)

Section

Differential equations, mathematical physics and mechanics

License

Authors who publish with this journal agree to the following terms:

  1. Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
  2. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
  3. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).