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We developed a technique for the construction of solutions of axisymmetric problems of elasticity and thermoelasticity in stresses for a space and a half space whose elastic properties are arbitrary functions of the coordinate z. By using the direct integration method and the Hankel integral transformation, the problems are reduced to governing integral equations accompanied by a local boundary condition and an integral condition in the case of the half space. The solutions of the deduced equations are constructed in the explicit form by using the resolvent-kernel method.

A method for reducing the three-dimensional problem of the theory of elasticity for a homogeneous transversely isotropic half-space to key integral equations of the second kind for individual components of the stress tensor is proposed. Using the resolvent kernel method, we obtain explicit solutions of these equations in the space of the double integral Fourier transform. The shape of the solution constructed in this way does not depend on the relationships between the elastic modules of the material, which allows, in particular, to ensure the attenuation of the components of the stress tensor at infinitely distant points of the half-space for various types of transversely isotropic materials. Cite as: Yu. V. Tokovyy, D. S. Boiko, “Integral equations of a three-dimensional elasticity problem for a homogeneous transversely isotropic half-space,” Prykl. Probl. Mekh. Mat. , Issue 18, 83–92 (2020) (in Ukrainian), https://doi.org/10.15407/apmm2020.18.83-92

We describe the main stages of the development of investigations originated by Prof. Vihak in the fields of the theory and methods of optimal control over the thermal processes and thermal stressed state of deformed solids, theory of inverse problems of heat conduction and thermomechanics, and the mathematical methods in the mechanics of deformable solids. We indicate the existence of close relationship between the obtained theoretical results and the applied problems of thermal power engineering motivating investigations of this kind. We describe a significant contribution made by the researchers of the Pidstryhach Institute for Applied Problems in Mechanics and Mathematics of the Ukrainian National Academy of Sciences to the development of investigations in these fields. The prospects of further development of these studies are analyzed and some facts from the life and creative activity of Prof. V. М. Vihak are presented.

An algorithm for the computation and analysis of the Cosserat spectrum for an axisymmetric elasticity boundary-value problem in a finite-length solid cylinder with boundary conditions in terms of stresses is proposed. By making use of the cross-wise superposition method, the spectral problem is reduced to systems of linear algebraic equations. A solution method for the mentioned systems is presented and the asymptotic behavior of the Cosserat eigenvalues is established. On this basis, the key features of the Cosserat spectrum for the mentioned problem are analyzed with special attention given to the effect of the cylinder aspect ratio.

An exact analytical solution to the three-dimensional elasticity problem for a transversely-isotropic composite layer is constructed by making use of the direct integration method along with the Fourier double-integral transform. The original problem is reduced to a system of governing partial-differential equations for separate stress-tensor components. The governing equations are accompanied with corresponding local and integral boundary conditions, obtained on the basis of the original local boundary conditions imposing the normal and shearing forces on the limiting planes of the layer. The numerical analysis of the obtained solution is presented for certain transversely-isotropic composite materials.

This article presents an analytical technique for solving the axisymmetric elasticity problem for a finite solid cylinder subjected to discontinuous normal loadings on its ends and lateral surface. This technique is based on application of the method of crosswise superposition by representing the solution for stresses in the form of decompositions into Fourier and Bessel–Dini series. For determination of the coefficients in these series, the infinite systems of linear algebraic equations are obtained and solved by means of a modified algorithm of advanced reduction. The technique is numerically validated for typical cases of discontinuous loading. It is shown that the solution procedure is efficient for determination of the stresses in the cylinder including its edges and discontinuity points of normal loadings.

We consider an axially symmetric problem of the thermostressed state of a solid cylinder of finite length with a load-free surface. Using the method of superposition, we have constructed the complete analytical solution of this problem, which is reduced to the solution of a system of linear algebraic equations. We have proposed a method for determining the asymptotic behavior of coefficients in these systems, which enables us to develop an efficient algorithm for the calculation of stresses in the cylinder, including regions near its end-face circles. Typical examples are considered.

Results of an analysis of the plane stress state of a thin annular disk subjected to uniform pressure on two diametrically opposite segments of the rim are presented. Some features of the stress distribution are investigated by means of both theoretical technique and experimental testing. Analytical expressions for the components of the displacement vector that correspond to the stress state of a ring under the action of diametral compression are established.

An analytical approach to solve plane static non-axisymmetric elasticity and thermoelasticity problems for radially inhomogeneous hollow cylinders is presented. This approach is based upon the direct integration method proposed by Vihak (Vigak). The essence of the method mentioned is in the integration of the original differential equilibrium equations, which are independent of the stress–strain relations. This gives the opportunity to deduce the relations, which are invariant with respect to various properties of the material, for the stress-tensor components. From these relations each of the stress-tensor components have been expressed in terms of the governing one. A solution of the equation for the governing stress in the form of Fourier series is presented. To determine the Fourier coefficients, an integral Volterra-type equation is derived and solved by a simple iteration method with rapid convergence. Other stress-tensor components are expressed through the obtained governing stress in the form of an explicit functional dependence on force and thermal loadings.

We have developed a method for analytical solving of the plane thermoelasticity problem in terms of stresses for a strip, which is infinite with respect to width. To derive the governing equations, we have used a method of direct integration of differential equilibrium and compatibility equations. Reducing the governing equations to the integral Volterra type equation of the second kind, we have solved it in Fourier transforms by applying a method of simple iteration. Nehomogeninio strypo termoelastiškumo uždavinio suvedimas į Volterra tipo integralinę lygtį Straipsnyje vystomas analizinio sprendinio metodas nehomogeninio strypo termoelastiškumo uždaviniui strypo įtempimams rasti, kai strypo ilgis yra begalinis pločio atžvilgiu. Pagrindinės lygtys išvedamos naudojant diferencialines pusiausvyros ir suderinamumo lygtis ir tiesioginį integravimą. Suvedus pagrindines lygtis į antrojo tipo Volterra integralinę lygtį, naudojant Furje transformaciją, ji sprendžiama paprastosios iteracijos metodu. First Published Online: 14 Oct 2010