Solving the three-dimensional equations of the linear theory of elasticity

Abstract

The three-dimensional Lamé equations are solved using Cartesian and curvilinear orthogonal coordinates. It is proved that the solution includes only three independent harmonic functions. The general solution of equations of elasticity for stresses is found. The stress tensor is expressed in both coordinate systems in terms of three harmonic functions. The general solution of the problem of elasticity in cylindrical coordinates is presented as an example. The three-dimensional stress-strain state of an elastic cylinder subjected, on the lateral surface, to arbitrary forces represented by a series of eigenfunctions is determined. An axisymmetric problem for a finite cylinder is solved numerically

The study of mathematical models of the linear theory of elasticity by presenting the fundamental solution in harmonic potentials

“…To construct a solution to the boundary-value problem of linear elasticity theory (1), (2), we consider the representation of the displacement vector u in the Revenko form [4] in terms of spatial independent scalar harmonic functions R(r, ϕ, z), Ψ(r, ϕ, z), Q(r, ϕ, z) in the form:…”

“…The general solution based on the scalar and vector spatial harmonic functions has been for the first time proposed by P. F. Papkovich [2] and by G. Neiber [3]. Recent studies of the abovementioned issues have been published in the scientific papers of V. P. Revenko [4][5][6], where the general solution of homogeneous equations of equilibrium in terms of displacements was presented with the help of only three spatial harmonic functions and the volumetric expansion is expressed in terms of one of the above-mentioned functions. Relevant results are presented in Section 3 of this article.…”

The method of developing the complex stress tensor by basic states for the construction solutions of spatial boundary problems of the linear elasticity theory

“…Proving the completeness of general The method of developing the complex stress tensor by basic states for the construction solutions of … 67 solutions, the existence of relationships between them, and the construction of solutions of boundary value problems were discussed by Eubanks and Sternberg, Timoshenko and Goodier, and others [1][2][3][4][5][6]. In particular, [7,8] created a universal design scheme for the development of general solutions and assessment of their completeness and non-unity within the framework of the classical theory of elasticity. The issue of optimizing the number of harmonic potentials in the Papkovich--Neuber representation using the variational approach was analyzed in [9].…”

Dispersion of generalized Rayleigh waves in the half-plane covered with pre-stretched two layers under complete contact

“…With this theory, there is no need for any additional hypotheses, considering the condition that the pre-stresses are much larger than the stresses due to wave propagation, the precise non-linear 3-D equations of continuous media mechanics are --------------* Author's, e-mail: ozisik@yildiz.edu.tr linearized and wave propagation models are made. The 3-DLTE has been elaborated in many investigations such as [5][6][7][8][9] and the others.…”