# 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.

## 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

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