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Within the framework of Saint-Venant's compatibility conditions, plane problems of the theory of plasticity with respect to strains and displacements are formulated in this paper. Grid equations for displacements and strains for a rectangular plate are compiled using the finite-difference method. Difference equations for displacements and strains were solved, respectively, by the iterative method and the method of alternating directions. By comparing the numerical results for a rectangular plate, the validity of the formulated plastic problem with respect to deformations is shown.

In the work, within the framework of the Saint-Venant compatibility conditions, two plane problems of thermoelasticity with respect to deformations are formulated. The closedness of boundary value problems is achieved by considering equilibrium equations on the boundary of a given region. Grid equations of thermoelastic problems are compiled using the finite-difference method and solved by the alternative method. The problem of a free thermoelastic rectangle located in a given temperature field is solved numerically. The validity of the formulated boundary value problems and the reliability of the solution are justified by comparing the numerical results of the formulated boundary value problems regarding displacements and stresses.

The article is devoted to the construction of model equations of the theory of elasticity with respect to deformations. Classical and new versions of boundary value problems of the theory of elasticity in strains are considered. In the classical version, model equations in strains are constructed within the framework of the Beltrami-Michell equations. A new version of model equations in strains is based on a new formulation of boundary value problems of the theory of elasticity in stresses. Discrete equations are constructed using the finite-difference method for two-dimensional problems. The well-known problem of tension a rectangular plate with a parabolic load applied on opposite sides has been solved. By comparing the numerical results of boundary value problems in classical and new formulations, as well as the Timoshenko-Goodier solution, the validity of the formulated model equations in strains and the reliability of the obtained numerical results are ensured.

In the work, within the framework of the strain compatibility conditions of Saint-Venant, two equivalent dynamic boundary value problems of thermoelasticity with respect to strains are formulated. In the case of the first boundary value problem, the dynamic equations of thermoelasticity are obtained from the compatibility conditions, in the second case, instead of the first three equations of thermoelasticity, the equations of motion expressed with respect to deformations are considered. Discrete analogues of boundary value problems are constructed using the finite-difference method in the form of explicit and implicit schemes. The solution of explicit schemes is reduced to recurrent relations with respect to deformations and temperature. Implicit schemes are solved by sequential application of the elimination method. The validity of the formulated thermoelastic boundary value problems is substantiated by comparing the numerical results of the problem of a thermoelastic parallelepiped obtained by different methods, as well as solving a similar problem in displacements.

It is known that the plane problems of the theory of elasticity and thermoelasticity are usually reduced to solving the biharmonic equation with respect to the Airy stress function. Solutions to spatial problems can also be defined using the Maxwell and Morera stress functions, etc. In this work, for solving a spatial thermoelastic problem, in the framework of the Beltrami-Michell equations, Poisson-type differential equations are found for each component of the stress tensor. In this case, the right-hand sides of these equations depend on temperature and elastic constants. The boundary conditions consist of the usual surface forces and "additional" boundary conditions derived from the equilibrium equations. The differential equations and boundary conditions of the plane problem of thermoelasticity in stresses are discussed in more detail. Grid equations for plane and spatial problems of thermoelasticity were compiled by the finite-difference method and solved by the iterative method. The problems of a thermoelastic rectangle and a parallelepiped located in a given temperature field are solved numerically. The validity of the formulated boundary value problems and the reliability of the results obtained are substantiated by comparing the numerical results with the solutions of the problems of a thermoelastic plate and a parallelepiped known in the literature. [ABSTRACT FROM AUTHOR]

Usually, planar and spatial problems of the theory of elasticity in stresses are solved using the Airy or Maxwell stress function, respectively. Solving a plane problem usually comes down to solving a biharmonic equation. In the spatial case, it can be solved by the Castigliano variational method. This work is devoted to the mathematical and numerical solution of the spatial problem of elasticity theory directly concerning stresses. In this case, the boundary value problem consists of three equilibrium equations and three Beltrami-Michell equations with the corresponding boundary conditions. At the same time, equilibrium equations are considered as missing boundary conditions. Symmetric finite-difference equations are constructed, and the problems of equilibrium of a parallelepiped under the action of domed and uniformly distributed loads are solved numerically. The reliability of the results obtained is substantiated by comparing the numerical results with the wellknown Filonenko-Borodich solution and with the results of solving the boundary value problem of the equilibrium of the parallelepiped with respect to displacements.

This article is devoted to the formulation and numerical solution of boundary-value problems in the theory of elasticity with respect to deformations. Similar to the well-known Beltrami–Michell stress equations, the Saint-Venant compatibility conditions are written in the form of differential equations for strains. A new version of plane boundary-value problems in strains is formulated. It is shown that for the correctness of plane boundary value problems, in addition to the usual conditions, one more special boundary condition is required using the equilibrium equation. To discretize additional boundary conditions and differential equations, it is convenient to use the finite difference method. By resolving grid equations and additional boundary conditions with respect to the desired quantities at the diagonal nodal points, we obtained convergent iterative relations for the internal and boundary nodes. To solve grid equations, the elimination method was also used. By comparing with the Timoshenko–Goodyear solution on the tension of a rectangular plate with a parabolic load, the validity of the formulated boundary value problems in strains and the reliability of the numerical results are shown. The accuracy of the results has been increased by an average of 15%. [ABSTRACT FROM AUTHOR]