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.

scholarly journals Hydromechanics of low-Reynolds-number flow. Part 2. Singularity method for Stokes flows

1975 ◽  
Vol 67 (4) ◽  
pp. 787-815 ◽  
Author(s):  
Allen T. Chwang ◽  
T. Yao-Tsu Wu
Keyword(s):  
Exact Solutions ◽  
Stokes Flow ◽  
Fundamental Solutions ◽  
The Body ◽  
Circular Cylinders ◽  
Geometrical Parameters ◽  
Flow Problems ◽  
Singularity Method ◽  
Wide Range ◽  
Body Shapes

The present study further explores the fundamental singular solutions for Stokes flow that can be useful for constructing solutions over a wide range of free-stream profiles and body shapes. The primary singularity is the Stokeslet, which is associated with a singular point force embedded in a Stokes flow. From its derivatives other fundamental singularities can be obtained, including rotlets, stresslets, potential doublets and higher-order poles derived from them. For treating interior Stokes-flow problems new fundamental solutions are introduced; they include the Stokeson and its derivatives, called the roton and stresson.These fundamental singularities are employed here to construct exact solutions to a number of exterior and interior Stokes-flow problems for several specific body shapes translating and rotating in a viscous fluid which may itself be providing a primary flow. The different primary flows considered here include the uniform stream, shear flows, parabolic profiles and extensional flows (hyper-bolic profiles), while the body shapes cover prolate spheroids, spheres and circular cylinders. The salient features of these exact solutions (all obtained in closed form) regarding the types of singularities required for the construction of a solution in each specific case, their distribution densities and the range of validity of the solution, which may depend on the characteristic Reynolds numbers and governing geometrical parameters, are discussed.

An inhomogeneous problem for an elastic half-strip: An exact solution

2021 ◽  
pp. 108128652199641
Author(s):  
Mikhail D Kovalenko ◽  
Irina V Menshova ◽  
Alexander P Kerzhaev ◽  
Guangming Yu
Keyword(s):  
Exact Solution ◽  
Boundary Conditions ◽  
Exact Solutions ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Theory Of Elasticity ◽  
Orthogonality Relation ◽  
Infinite Strip ◽  
Inhomogeneous Problem ◽  
Half Strip

We construct exact solutions of two inhomogeneous boundary value problems in the theory of elasticity for a half-strip with free long sides in the form of series in Papkovich–Fadle eigenfunctions: (a) the half-strip end is free and (b) the half-strip end is firmly clamped. Initially, we construct a solution of the inhomogeneous problem for an infinite strip. Subsequently, the corresponding solutions for a half-strip are added to this solution, whereby the boundary conditions at the end are satisfied. The Papkovich orthogonality relation is used to solve the inhomogeneous problem in a strip.

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.

Hele-Shaw flows and water waves

2000 ◽  
Vol 409 ◽  
pp. 223-242 ◽  
Author(s):  
DARREN G. CROWDY
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Exact Solutions ◽  
Water Waves ◽  
Free Boundary Problems ◽  
Capillary Water ◽  
Mathematical Transformation ◽  
Boundary Problems ◽  
Physical Problems ◽  
Euler Flows

By adapting a new mathematical approach to the problem of steady free-surface Euler flows with surface tension recently devised by the present author, it is demonstrated that exact solutions for steady, free-surface multipole-driven Hele-Shaw flows with surface tension can be constructed using similar methods. Moreover, a (one-way) mathematical transformation between exact solutions to the two distinct free-boundary problems is identified: known exact solutions for free-surface Euler flows with surface tension are shown to automatically generate steady quadrupolar-driven Hele-Shaw flows (with non-zero surface tension) existing in exactly the same domain with the same free surface. This correspondence highlights the essential dynamical differences between the two physical problems. Using the transformation, the exact Hele-Shaw analogues of all known exact solutions for free-surface Euler flows (including Crapper's classic capillary water wave solution) are catalogued thereby producing many previously unknown exact solutions for steady Hele-Shaw flows with capillarity. In particular, this paper reports what are believed to be the first known exact solutions for Hele-Shaw flows with surface tension in a doubly-connected fluid region.

scholarly journals Approximate Solutions of the Generalized Abel’s Integral Equations Using the Extension Khan’s Homotopy Analysis Transformation Method

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Mohamed S. Mohamed ◽  
Khaled A. Gepreel ◽  
Faisal A. Al-Malki ◽  
Maha Al-Humyani
Keyword(s):  
Exact Solutions ◽  
Integral Equations ◽  
Method Comparison ◽  
Approximate Solutions ◽  
Transformation Method ◽  
Theory Of Elasticity ◽  
Transform Method ◽  
Numerical Examples ◽  
Homotopy Analysis ◽  
User Friendly

User friendly algorithm based on the optimal homotopy analysis transform method (OHATM) is proposed to find the approximate solutions to generalized Abel’s integral equations. The classical theory of elasticity of material is modeled by the system of Abel integral equations. It is observed that the approximate solutions converge rapidly to the exact solutions. Illustrative numerical examples are given to demonstrate the efficiency and simplicity of the proposed method. Finally, several numerical examples are given to illustrate the accuracy and stability of this method. Comparison of the approximate solution with the exact solutions shows that the proposed method is very efficient and computationally attractive. We can use this method for solving more complicated integral equations in mathematical physical.

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