Singular Integral Equations. By N.I. Muskhelishvili. Translated fromthe 2nd Russian edition by J.R.M. Radok. Pp. 447. F1. 28.50. 1953. (Noordhoff, Groningen)

Modeling and forecasting of Turkey's energy consumption using socio-economic and demographic variables

: One encounters difficulties in the solution of integral equations, namely, the occurrence of singularities in the kernel, and the occurence of unknown-type singularities in the solution of the integral equation. Standard methods of approximation based on exactness for polynomials up to a certain degree are very poor, and frequently fail for functions having such singularities. The purpose of this contract was to develop methods for solving integral equations which work well in spite of the presence of singularities. This goal has been accomplished, in that the new approximation methods which were developed do work well in the presence of singularities. (Author)

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Numerical Solution of Singular Integral Equations.

A receding contact plane problem between a functionally graded layer and a homogeneous substrate

The static bending and buckling behaviors of bi-directional functionally graded (BFG) plates with porosity are investigated in this paper. An improved first-order shear deformation theory with an assuming parabolic distribution shear stresses is developed to describe the displacement, strain, and stress fields of the plates. The significant novelty of the proposed theory is that the transverse shear stresses equal to zero at two free surfaces of the BFG plates. Therefore, no shear correction factor is required as in other first-order shear deformation theory. A four-node quadrilateral plate element (IMQ4) is developed based on the improved first-order shear deformation theory, mixed finite element method (FEM) and Hamilton's principle for analysis of BFG plates. Several comparison studies are provided to demonstrate the precision and robustness of the proposed plate element IMQ4. Then the proposed plate element, IMQ4, is employed to analyze the bending and buckling responses of the BFG plates. Some new numerical results on the flexural and buckling behaviors of BFG plates are achieved via a deep parametric study. 

Static bending and buckling analysis of bi-directional functionally graded porous plates using an improved first-order shear deformation theory and FEM

The plane symmetric double receding contact problem for a rigid stamp and two elastic layers having different elastic constants and heights is considered. The external load is applied to the upper elastic layer by means of a rigid stamp and the lower elastic layer is bonded to a rigid support. The problem is solved under the assumptions that the contact between two elastic layers, and between the rigid stamp and the upper elastic layer are frictionless, the effect of gravity force is neglected and only compressive normal tractions can be transmitted through the interfaces. The problem is reduced to a system of singular integral equations in which the contact pressures are the unknown functions by using integral transform technique and boundary conditions of the problem. The system of singular integral equations is solved numerically by making use of appropriate Gauss-Chebyshev integration formulas for two stamp geometries. Numerical results for the contact pressures, the contact areas, the normal stresses σ x , σ y , and the shear stress τ xy are given for various dimensionless quantities.

Double receding contact problem for a rigid stamp and two elastic layers

A receding contact problem for an anisotropic elastic medium consisting of a layer and a half plane

This paper considers a receding contact problem for two elastic layers (with different elastic constants and heights) supported by two elastic quarter planes. The lower layer is supported by two elastic quarter planes, and the upper elastic layer is subjected to a symmetrical distributed load whose length is 2a on its top surface. It is assumed that contact between all surfaces is frictionless, and the effect of gravity force is neglected. First, the problem is formulated and solved using the theory of elasticity and integral transform technique. Using the integral transform technique and boundary conditions of the problem, the problem is reduced to a system of singular integral equations in which contact pressures and contact areas are unknown. The system of singular integral equations is solved numerically by using the Gauss-Jacobi integration formulation. Second, the receding contact problem has been developed based on the FEM ANSYS software. Two-dimensional analysis of the problem is carried o...

Comparison between Analytical and ANSYS Calculations for a Receding Contact Problem

In this paper, the plane problem of a frictionless receding contact between an elastic functionally graded layer and two homogeneous quarter planes is considered when the graded layer is pressed against the quarter planes. The top of the layer is subjected to normal tractions over a finite segment. The graded layer is modeled as a non-homogeneous medium with a constant Poisson’s ratio and exponentially varying shear modules. The problem is converted into the solution of a Cauchy-type singular integral equation in which the contact pressure and the receding contact half-length are the unknowns using integral transforms. The singular integral equation is solved numerically using Gauss–Jacobi integration. The corresponding receding contact half-length that satisfies the global equilibrium condition is obtained using an iterative procedure. The effect of the material non-homogeneity parameter on the contact pressure and on the length of the receding contact is investigated.

A receding contact problem between a functionally graded layer and two homogeneous quarter planes

The receding contact problem for two elastic layers whose elastic constants and heights are different supported by two elastic quarter planes is considered. The lower layer is supported by two elastic quarter planes and the upper elastic layer is subjected to symmetrical distributed load whose heights are 2a on its top surface. It is assumed that the contact between all surfaces is frictionless and the effect of gravity force is neglected. The problem is formulated and solved by using Theory of Elasticity and Integral Transform Technique. The problem is reduced to a system of singular integral equations in which contact pressures are the unknown functions by using integral transform technique and boundary conditions of the problem. Stresses and displacements are expressed depending on the contact pressures using Fourier and Mellin formula technique. The singular integral equation is solved numerically by using Gauss-Jacobi integration formulation. Numerical results are obtained for various dimensionless quantities for the contact pressures and the contact areas are presented in graphics and tables.

Ahmet Birinci

Papers

Double receding contact problem for a rigid stamp and two elastic layers

A receding contact problem for an anisotropic elastic medium consisting of a layer and a half plane

Comparison between Analytical and ANSYS Calculations for a Receding Contact Problem

A receding contact problem between a functionally graded layer and two homogeneous quarter planes

The receding contact problem of two elastic layers supported by two elastic quarter planes