scholarly journals Discussion: “Application of the Fourier Method to the Solution of Certain Boundary Problems in the Theory of Elasticity” (Pickett, Gerald, 1944, ASME J. Appl. Mech., 11, pp. A176–A182)

1945 ◽  
Vol 12 (3) ◽  
pp. A185
Author(s):  
Eric Reissner
Keyword(s):  
Fourier Method ◽  
Theory Of Elasticity ◽  
Boundary Problems

Application of the Fourier Method to the Solution of Certain Boundary Problems in the Theory of Elasticity

1944 ◽  
Vol 11 (3) ◽  
pp. A176-A182
Author(s):  
Gerald Pickett
Keyword(s):  
Boundary Condition ◽  
Exact Solutions ◽  
Fourier Method ◽  
Theory Of Elasticity ◽  
Circular Cylinders ◽  
Boundary Problems

Abstract The paper shows how the Fourier method may be used to obtain exact solutions for stresses in rectangular prisms or circular cylinders for any boundary condition.

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.

scholarly journals Stability of mathematical models of the main problems of the anisotropic theory of elasticity

2020 ◽  
Vol 30 (1) ◽  
pp. 112-124
Author(s):  
A.V. Yudenkov ◽  
A.M. Volodchenkov
Keyword(s):  
Boundary Value Problems ◽  
Analytic Functions ◽  
Boundary Value ◽  
Research Result ◽  
Theory Of Elasticity ◽  
Boundary Problems ◽  
Main Research ◽  
Contour Shape ◽  
Variable Function ◽  
Anisotropic Bodies

The boundary problems of the complex-variable function theory are effectively used while investigating equilibrium of homogeneous elastic mediums. The most complicated systems of the boundary value problems correspond to the case when an elastic body exhibits anisotropic properties. Anisotropy of the medium results in the drift of boundary conditions of the function that in general disrupts analyticity of the functions of interest. The paper studies systems of the boundary value problems with drift for analytic vectors corresponding to the primal elastic problems (first, second and mixed problems). Systems of analytic vectors with drift are reduced to equivalent systems of Hilbert boundary value problems for analytic functions with weak singularity integrators. The obtained general solution of the primal boundary value problems for the anisotropic theory of elasticity allows us to check the above problems for stability with respect to perturbations of boundary value conditions and contour shape. The research is relevant as there is necessity to apply approximate numerical methods to the boundary value problems with drift. The main research result comes to be a proof of stability of the systems of the vector boundary value problems with drift for analytic functions on the Hölder space corresponding to the primal problems of the elastic theory for anisotropic bodies in the case of change in the boundary value conditions and contour shape.

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.

Determination of the Stress State of the Layer with a Cylindrical Elastic Inclusion

2019 ◽  
Vol 968 ◽  
pp. 413-420
Author(s):  
Vitaly Yu. Miroshnikov ◽  
Alla V. Medvedeva ◽  
Sergei V. Oleshkevich
Keyword(s):  
Stress State ◽  
Reduction Method ◽  
Elastic Inclusion ◽  
Fourier Method ◽  
Theory Of Elasticity ◽  
Geometrical Parameters ◽  
Spatial Problem ◽  
Algebraic Equations ◽  
Linear Algebraic Equations

A spatial problem of the theory of elasticity for the layer with an infinite round cylindrical inclusion is investigated. At the boundaries of the layer, displacements are given. The cylindrical elastic inclusion is rigidly coupled with the layer and their boundary surfaces do not intersect. The solution to the spatial problem is obtained by the generalized Fourier method, with regard to the Lamé system of equations. The obtained infinite systems of linear algebraic equations are solved by a reduction method. As a result, the values ​​of displacements and stresses in the elastic body are determined. A comparative analysis of the stress state for different geometrical parameters is carried out, and a comparison is made with the stress state in the layer with a cylindrical cavity.

Use of the Generalized Impulse Momentum Equations in Analysis of Wave Propagation

1991 ◽  
Vol 113 (4) ◽  
pp. 532-542 ◽  
Author(s):  
Wei-Hsin Gau ◽  
A. A. Shabana
Keyword(s):  
Wave Equations ◽  
Initial Conditions ◽  
Fourier Method ◽  
Coefficient Of Restitution ◽  
Theory Of Elasticity ◽  
Jump Discontinuity ◽  
Mechanical Joints ◽  
Dynamic Formulation ◽  
Propagation Of Elastic Waves ◽  
Algebraic Constraint

A procedure is developed in this paper to study the propagation of impact-induced axial waves in constrained beams that undergo large rigid body displacements. The solution of the wave equations is obtained using the Fourier method. Kinematic conditions that describe mechanical joints in the system are formulated using a set of nonlinear algebraic constraint equations that are introduced to the dynamic formulation using the vector of Lagrange multipliers. The initial conditions which represent the jump discontinuity in the elastic coordinates as the result of impact are predicted using the generalized impulse momentum equations that involve the coefficient of restitution as well as the Jacobian matrix of the kinematic constraints. The convergence of the series solutions presented in this paper is examined and the analytical and numerical results are found to be consistent with the solutions obtained by the use of the classical theory of elasticity in the case of plastic impact. The cases in which the coefficient of restitution is different from zero are also examined and it is shown that the generalized impulse momentum equations can be used with confidence to study the propagation of elastic waves in applications related to multibody dynamics.

scholarly journals Investigation of the second main problem of elasticity for a layer with n cylindrical inclusions

2021 ◽  
pp. 156-166
Author(s):  
Vitaly Miroshnikov ◽  
Tetiana Denisova
Keyword(s):  
Boundary Conditions ◽  
Reduction Method ◽  
Fourier Method ◽  
High Accuracy ◽  
Theory Of Elasticity ◽  
System Of Equations ◽  
Spatial Problem ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Cylindrical Inclusions

When designing structures in the form of a reinforced layer, one has to deal with the situation when the reinforcement bars are located close to each other. In this case, their influence on each other increases. In order to obtain the stress-strain state in the contact zone of the layer and the inclusion, it is necessary to have a method that would allow obtaining a result with high accuracy. In this work, an analytical-numerical approach to solving the spatial problem of the theory of elasticity for a layer with a given number of longitudinal cylindrical inclusions and displacements given at the boundaries of the layer has been proposed. The solution of the problem has been obtained by the generalized Fourier method with respect to the system of Lame's equation in local cylindrical coordinates associated with inclusions and Cartesian coordinates associated with layer boundaries. Infinite systems of linear algebraic equations obtained by satisfying the boundary conditions and conjugation conditions of a layer with inclusions have been solved by the reduction method. As a result, displacements and stresses have been obtained at different points of the considered medium. When the order of the system of equations is 6, the accuracy of fulfilling the boundary conditions was 10-2 for values from 0 to 1. Numerical studies of the algebraic system of equations give grounds to assert that its solution can be found with any degree of accuracy by the reduction method, which is confirmed by the high accuracy of fulfilling the boundary conditions. In the numerical analysis, variants of the layer with 1 and 3 inclusions have been compared. The result has shown that close placement of reinforcement bars increases stresses  on the surface of these inclusions. The values of stresses on the layer contact surfaces with inclusions have also been obtained. The proposed solution algorithm can be used in the design of structures, the computational scheme of which is the layer with longitudinal cylindrical inclusions and displacements specified at the layer boundaries.

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