Deformation for a Rectangle by a Finite Fourier Transform

Abstract

In this paper, we introduce a simple method to solve a static, plane boundary value problem in elasticity for an isotropic rectangular region. The method depends on finite Fourier transform to transfer the biharmonic equation to a nonhomogeneous ordinary differential equation of the fourth order. Also, by transfering the boundary conditions, one can find the general solution for the nonhomogeneous ordinary differential equation. Finally, the inverse Fourier transfer allows to get the analytical solution for the biharmonic equation. Using expressions for displacements proved by two of the authors [MSA and AFG], one can obtain the displacements for the rectangular domain.

Exact solution for certain regular thermoelasticity problems in a rectangular long rod by complex harmonics

“…The boundary conditions are given in terms of the second partial derivatives of the stress function as follows [3] and, the following condition at the origin [3] An addition simplifying conditions are apply at the origin [3]…”

“…The method of boundary integral representation is applied in its analytical and computational aspects in [1], [2]. In [3] and [4] the finite Fourier transform technique is used to tackle such problems. In paper [5], a semi-inverse method is used to deal with boundary value problem for a rectangular domain in second gradient elasticity.…”

The stress state of a rectangular elastic domain

“…Here ( ( )) = ′ ( ) + ( ), 1 = 0, 2 = To solve this vector boundary problem the fundamental solution matrix ( ) is constructed. To found it firstly the matrix (where the unit matrix ) must be substituted into the equation (9). From the equality 2 ( ) = ( ) one can derive the ( ) matrix…”

“…In paper [9] a simple method to solve a static, plane boundary value problem of elasticity for an isotropic rectangular region was introduced. The method is based on finite Fourier transform transfering the biharmonic equation to a nonhomogeneous ordinary differential equation of the fourth order.…”

Semi-inverse method à la Saint-Venant for two-dimensional linear isotropic homogeneous second-gradient elasticity

“…In other words, we use a semi-inverse method of the same kind as that used by Saint-Venant. For the sake of simplicity, we choose a squared two-dimensional body, impose a displacement solution in terms of exponential functions (see also [68][69][70][71][72][73][74]), with an arbitrary number of coefficients, for example 16, and assume boundary conditions in an integral form: that is, we prescribe the integral on each side of the necessary boundary conditions. In first-gradient materials we have, for the same problem, only one condition (the kinematical displacement condition or its dual, the force per unit line) per each dimension.…”