Hilbert transform approach to solve a problem of collinear Griffith crack in the mid-plane of an infinite orthotropic strip

In this paper, an integral transformation of the displacement is employed to determine the solution of the elastodynamics problem of two collinear Griffith cracks with constant velocity situated in a mid-plane of an infinite orthotropic strip where the boundaries are assumed to be stress-free. By use of the integral transformation of the displacement, the problem is reduced to a solution of the triple integral equations and the finite Hilbert transformation technique is used to solve it. The expressions of stresses are obtained asymptotically for large values of strip depth. An analytical expression of the stress intensity factor at the crack tip is obtained and represented graphically for two different orthotropic materials.

Contact Problem Of Conducting And Heated Punch On A Multifield Foundation

The solution for a multifield material subjected to temperature loading in a circular region is presented in an explicit analytical form. The study concerns the steady – state thermal loading infinite region (heated embedded inclusion), half – space region and two – constituent magneto – electro – thermo – elastic material region. The new mono – harmonic potential functions, obtained by the author, are used in the analysis of punch problem. The more interested case in which the contact region is annular is analyzed. By using the methods of triple integral equations and series solution technique the solution for an indentured multifield substrate over an annular contact region is given. The sensitivity analysis of obtained indentation parameters shows some interesting points. In particular, it shows that the increasing of the applied electric and magnetic potentials reduces the indentation depth in multifield materials.

Stress Intensity Factor of Peripheral Edge Crack around a Spherical Cavity under Internal Pressure Corrected

The singular stress problem of a flat annular crack around a spherical cavity subjected to internal pressure is investigated. By application of an integral transforms and the theory of triple integral equations, the problem is reduced to the solution of a singular integral equation of the first kind. The equations gotten for the case of peripheral edge crack around a spherical cavity is solved numerically, and the stress intensity factors are shown graphically. The results in this paper are basically consistent with the existing literature in special cases.

The Study of Triple Integral Equations with Generalized Legendre Functions

A method is developed for solutions of two sets of triple integral equations involving associated Legendre functions of imaginary arguments. The solution of each set of triple integral equations involving associated Legendre functions is reduced to a Fredholm integral equation of the second kind which can be solved numerically.

Two-Dimensional Electrostatic Problem in a Plane with Earthed Elliptic Cavity due to One or Two Collinear Charged Electrostatic Strips

A two-dimensional electrostatic problem in a plane with earthed elliptic cavity due to one or two charged electrostatic strips is considered. Using the integral transform technique, each problem is reduced to the solution of triple integral equations with sine kernels and weight functions. Closed-form solutions of the set of triple integral equations are obtained. Also closed-form expressions are obtained for charge density of the strips. Finally, the numerical results for the charge density are given in the form of tables.

Closed-form solution for piezoelectric layer with two collinear cracks parallel to the boundaries

We consider the problem of determining the stress distribution in an infinitely long piezoelectric layer of finite width, with two collinear cracks of equal length and parallel to the layer boundaries. Within the framework of reigning piezoelectric theory under mode III, the cracked piezoelectric layer subjected to combined electromechanical loading is analyzed. The faces of the layers are subjected to electromechanical loading. The collinear cracks are located at the middle plane of the layer parallel to its face. By the use of Fourier transforms we reduce the problem to solving a set of triple integral equations with cosine kernel and a weight function. The triple integral equations are solved exactly. Closed form analytical expressions for stress intensity factors, electric displacement intensity factors, and shape of crack and energy release rate are derived. As the limiting case, the solution of the problem with one crack in the layer is derived. Some numerical results for the physical quantities are obtained and displayed graphically.

Axial Vibration of a Padded Annulus on a Semi-Infinite Viscoelastic Medium

The vibratory punch problem for a viscoelastic half-space indented by a padded annular disk is investigated. By virtue of transform methods, the problem is formulated as a set of triple integral equations which are reducible to a Fredholm integral equation of the second kind. In the formulation, the response of a thin buffer which regularizes the load transfer to the semi-infinite solid is approximated via a plane stress-type solution. A set of numerical results is included to demonstrate the effect of padding characteristics on the dynamic system response. Apart from providing an interpretation tool for the nondestructive testing methods involving buffered annular loading systems, the present solution can be used as an effective approximation to the corresponding rigid-punch problem which has so far eluded a rigorous mathematical treatment.