Integrodifferential equations of the three-dimensional problem of the theory of elasticity for a body with a system of fine inclusions

1984 ◽  
Vol 20 (1) ◽  
pp. 11-16
Author(s):  
M. M. Stadnik
Keyword(s):  
Three Dimensional ◽  
Theory Of Elasticity ◽  
Integrodifferential Equations ◽  
Dimensional Problem

Mathematical Theory of Transversally Isotropic Shells of Arbitrary Thickness at Static Load

2019 ◽  
Vol 968 ◽  
pp. 496-510
Author(s):  
Anatoly Grigorievich Zelensky
Keyword(s):  
Mathematical Theory ◽  
Three Dimensional ◽  
Nonlinear Problems ◽  
Theory Of Elasticity ◽  
Elastic Plates ◽  
Dimensional Problem ◽  
Boundary Problems ◽  
Complex Boundary ◽  
Plates And Shells ◽  
Basic Equations

Classical and non-classical refined theories of plates and shells, based on various hypotheses [1-7], for a wide class of boundary problems, can not describe with sufficient accuracy the SSS of plates and shells. These are boundary problems in which the plates and shells undergo local and burst loads, have openings, sharp changes in mechanical and geometric parameters (MGP). The problem also applies to such elements of constructions that have a considerable thickness or large gradient of SSS variations. The above theories in such cases yield results that can differ significantly from those obtained in a three-dimensional formulation. According to the logic in such theories, the accuracy of solving boundary problems is limited by accepted hypotheses and it is impossible to improve the accuracy in principle. SSS components are usually depicted in the form of a small number of members. The systems of differential equations (DE) obtained here have basically a low order. On the other hand, the solution of boundary value problems for non-thin elastic plates and shells in a three-dimensional formulation [8] is associated with great mathematical difficulties. Only in limited cases, the three-dimensional problem of the theory of elasticity for plates and shells provides an opportunity to find an analytical solution. The complexity of the solution in the exact three-dimensional formulation is greatly enhanced if complex boundary conditions or physically nonlinear problems are considered. Theories in which hypotheses are not used, and SSS components are depicted in the form of infinite series in transverse coordinates, will be called mathematical. The approximation of the SSS component can be adopted in the form of various lines [9-16], and the construction of a three-dimensional problem to two-dimensional can be accomplished by various methods: projective [9, 14, 16], variational [12, 13, 15, 17]. The effectiveness and accuracy of one or another variant of mathematical theory (MT) depends on the complex methodology for obtaining the basic equations.

On the Axisymmetric Problem of the Theory of Elasticity for an Infinite Region Containing Two Spherical Cavities

1952 ◽  
Vol 19 (1) ◽  
pp. 19-27
Author(s):  
E. Sternberg ◽  
M. A. Sadowsky
Keyword(s):  
Axisymmetric Problem ◽  
Three Dimensional ◽  
Theory Of Elasticity ◽  
Uniform Field ◽  
Dimensional Problem ◽  
Spherical Cavities ◽  
Infinite Region ◽  
In Series ◽  
The Common ◽  
Axis Of Symmetry

Abstract This paper contains a solution in series form for the stress distribution in an infinite elastic medium which possesses two spherical cavities of the same size. The loading consists of tractions applied to the cavities, as well as of a uniform field of tractions at infinity, and both are assumed to be symmetric with respect to the common axis of symmetry of the cavities and with respect to the plane of geometric symmetry perpendicular to this axis. The loading is otherwise unrestricted. The solution is based upon the Boussinesq stress-function approach and apparently constitutes the first application of spherical dipolar co-ordinates in the theory of elasticity. Numerical evaluations are given for the case in which the surfaces of the cavities are free from tractions and the stress field at infinity is hydrostatic. The results illustrate the interference of two sources of stress concentration in a three-dimensional problem. The approach used here may be extended to cope with the general equilibrium problem for a region bounded by two nonconcentric spheres.

scholarly journals Stress Distribution in a Composite Coating by Local Loading of the Free Surface

2014 ◽  
Vol 8 (2) ◽  
pp. 83-87
Author(s):  
Roman Kulchytsky-zhyhailo ◽  
Waldemar Kołodziejczyk
Keyword(s):  
Stress Distribution ◽  
Three Dimensional ◽  
Theory Of Elasticity ◽  
Layered System ◽  
Dimensional Problem ◽  
Elliptical Distribution ◽  
Local Loading ◽  
Integral Fourier Transform ◽  
Periodical Structure ◽  
Homogenized Model

Abstract A three-dimensional problem of the theory of elasticity for halfspace with multilayered coating with periodical structure is considered. The fundamental layer consists of two layers with different thicknesses and different mechanical properties. The coating is described by the homogenized model with microlocal parameters. The solution is derived by using integral Fourier transform. Calculations were conducted with the assumption of elliptical distribution of normal and tangential tractions applied to the surface of the layered system in a cir-cular area. Analysis of the stresses was restricted to the first principal stress distribution.

scholarly journals ГРАНИЧНЫЕ УСЛОВИЯ КОНТАКТНОГО ТИПА В ЗАДАЧЕ О КРУГОВОЙ ЦИЛИНДРИЧЕСКОЙ ПОЛОСТИ В УПРУГОМ ПРОСТРАНСТВЕ

2018 ◽  
pp. 121-128
Author(s):  
В. Ю. Мирошников ◽  
Т. В. Денисова ◽  
В. С. Проценко
Keyword(s):  
Boundary Conditions ◽  
Three Dimensional ◽  
Fourier Method ◽  
Theory Of Elasticity ◽  
Numerical Comparison ◽  
Elastic Space ◽  
Dimensional Problem ◽  
Cylindrical Coordinates ◽  
Contact Type ◽  
Tangential Stresses

A three-dimensional problem of the theory of elasticity is considered, when contact-type conditions (normal displacements and tangential stresses) are given on a cylindrical cavity in elastic space. The solution is obtained on the basis of the Fourier method with respect to the Lame equations in cylindrical coordinates. The solvability and uniqueness of the problem for these boundary conditions is proved. Normal and tangential stresses are found in the elastic body. A numerical comparison is made of the influence of the boundary conditions in the form of displacements and boundary conditions of the contact type on the stressed state of the elastic space.

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