A new approach to numerical simulation of boundary value problems of the theory of elasticity in stresses and strains

Abstract

The main parameters characterizing the process of deformation of solids are displacements, strain and stress tensors. From the point of view of the strength and reliability of the structure and its elements, researchers and engineers are mainly interested in the distribution of stresses in the objects under study. Unfortunately, all boundary value problems are formulated and solved in solid mechanics mainly with respect to displacements, or an additional stress functions. And the required stresses are calculated from known displacements or stress functions. In this case, the accuracy of stress calculation is strongly affected by the error of numerical differentiation, as well as the approximation order of the boundary conditions. The formulation of boundary value problems directly with respect to stresses or strains allows to increase the accuracy of stress calculation by bypassing the process of numerical differentiation.
 Therefore, the present work is devoted to the formulation and numerical solution of boundary value problems directly with respect to stresses and strains. Using the well-known Beltrami-Miеchell equation, and considering the equilibrium equation as ah additional boundary condition, a boundary value problem(BVP) is formulated directly with respect to stresses. In a similar way, using the strain compatibility condition, the Beltrami-Mitchell type equations for strains are written.
 The finite difference equations for two-dimensional BVP are constructed and written in convenient a form for the use of iterative method. A number of problems on the equilibrium of a rectangular plate under the action of various loads applied on opposite sides are numerically solved. The reliability of the results is ensured by comparing the numerical results of the 2D elasticity problems in stresses and strains, and with the exact solution, as well as with the known solutions of the plate tension problem under parabolic and uniformly distributed loads

New Coupled Thermoelasticity Boundary-Value Problems in Strains

“…can be considered instead of equations (10) in problem A. Thus, equations (15)(16) and (11)(12)(13)(14) represent a new version of the related problem of thermoelasticity under deformation (problem B).…”

“…We will consider the boundary value problems of thermoelasticity A and B, formulated in the previous paragraph, in the case of plane deformation. Problem A (10)(11)(12)(13)(14)…”

“…From the point of view of differential equations, boundary conditions regarding deformations do not raise questions. In [11,12,13], the static problem of a rectangular plate with boundary conditions regarding deformations was solved numerically and compared with the exact solution.…”

“…In this case, considering the equilibrium equation on the boundary of a given region, three additional boundary conditions are obtained that ensure the closedness of the system of differential equations. Similar to the static problem, in the related boundary value problem (10)(11)(12)(13)(14), instead of the first three differential equations for the compatibility of deformations, three equations of motion expressed with respect to deformations are considered. To find the equations of motion expressed with respect to deformations, we differentiate the equation of motion (1) taking into account relation (2) with respect to 1 2 , xx and 3…”

“…Usually, when formulating static problems of the theory of elasticity in stresses, the six Beltrami-Michell equations are considered in combination with three equilibrium equations, and an overdetermination of the number of equations is observed. But, as shown in [11,12,13], to formulate a closed boundary value problem it is enough to consider the first or second group of Beltrami-Michell equations in combination with three equilibrium equations.…”