Conical diffraction by multilayer gratings: a recursive integral equation approach

A mixed mode crack problem in functionally graded materials is formulated to a system of Cauchy singular Fredholm integral equations, then the system is solved by the singular integral equation method (SIEM). This specific crack problem has already been solved by N. Konda and F. Erdogan (Konda & Erdogan 1994). However, many mathematical details have been left out. In this paper we provide a detailed derivation, both analytical and numerical, on the formulation as well as the solution to the system of singular Fredholm integral equations. The research results include crack displacement profiles and stress intensity factors for both mode I and mode II, and the outcomes are consistent with the paper by Konda & Erdogan (Konda & Erdogan 1994).

A half-space containing a surface-breaking crack of uniform depth is subjected to three-dimensional dynamic loading. The elastodynamic stress-analysis problem has been decomposed into two problems, which are symmetric and antisymmetric, respectively, relative to the plane of the crack. The formulation of each problem has been reduced to a system of singular integral equations of the first kind. The symmetric problem is governed by a single integral equation for the opening-mode dislocation density. A pair of coupled integral equations for the two sliding-mode dislocation densities govern the antisymmetric problem. The systems of integral equations are solved numerically. The stress-intensity factors are obtained directly from the dislocation densities. The formulation is valid for arbitrary 3-D loading of the half-space. As an example, an applied stress field corresponding to an incident Rayleigh surface wave has been considered. The dependence of the stress-intensity factors on the frequency, and on the angle of incidence, is displayed in a set of figures.

The processes of reflection of three-dimensional electromagnetic waves by locally irregular media interfaces are investigated. The boundary value problem for the system of Maxwell equations in an infinite half-space is reduced to the solution of two systems of singular integral equations. To construct a numerical algorithm for solving these systems of singular integral equations, the approximation and collocation method is used. Special attention is focused on calculation of the kernels of these equations. Solutions of systems of singular integral equations are used to calculated the far-field pattern for the scattered field in the far zone.

Axisymmetric dynamic contact problem for a viscoelastic half-space in the presence of adhesion under the stamp: PMM vol. 37, n≗4, 1973, pp. 731–738

A system of singular integral equations with monotone and concave nonlinearity in the subcritical case is investigated. The specified system and its scalar analogy have direct applications in various areas of physics and biology. In particular, scalar and vector equations of this nature are encountered in the dynamic theory of p-adic strings, in the theory of radiative transfer, in the kinetic theory of gases, and in the mathematical theory of the spread of epidemic diseases. A constructive theorem of existence in the space of continuous and bounded vector functions is proved. The integral asymptotic of the constructed solution is studied. An effective iterative process for constructing an approximate solution to this system is proposed. In a certain class of bounded vector functions, the uniqueness of the solution is proved. In the class of bounded vector functions (with non-negative coordinates) with a zero limit at infinity, the absence of an identically zero solution is proved. Examples of the matrix kernel and nonlinearities are given to illustrate the importance of the obtained results. Some of the examples have applications in the above-mentioned branches of natural science.

In this article, we discuss the solvability of infinite systems of singular integral equations of two variables in the Banach sequence spaces C(I ? I, ?p) with I = [0, T], T > 0 and 1 < p < ? with the help of Meir-Keeler condensing operators and Hausdorff measure of noncompactness. With an example, we illustrate our findings.

In this paper, a monotonicity property of the superposition operator in higher dimensions will be proved. Then by using the concept of measure of noncompactness, we will establish the existence of nondecreasing positive solutions for a system of singular integral equations. Furthermore, the results will be used to investigate the solvability of the system of k th -order initial value problems.

Transient response of an annular crack in a plate of a functionally graded piezoelectric material (FGPM) is studied under thermal shock loading conditions. It is assumed that the thermoelectroelastic properties of the strip vary continuously along the thickness of the strip, and that the crack faces are completely insulated. Some material properties are assumed to be exponentially dependent on the distance from the crack line paralel to the boundaries of the plate. By using both the Laplace and Hankel transforms, the thermal and electromechanical problems are reduced to a singular integral equation and a system of singular integral equations. The singular integral equations are solved numerically, and a numerical method is then employed to obtain the time dependent solutions by way of a Laplace inversion technique. The intensity factors versus time are presented for various values of dimensionless parameters representing the material nonhomogeneity and the crack position.

An exact solution of the stationary thermoelasticity problem about interfacial circular absolutely rigid inclusion, which is in the smooth contact conditions in a piecewise homogeneous transversely homogeneous space, is constructed. The task with the help of the constructed discontinuous solution, by the method of singular integral relations, is reduced to a system of singular integral equations (SIE). An exact solution has been built for the specified SIE, as a result, dependences of translational displacement of the inclusion on temperature, the resultant load, main moment and thermomechanical characteristics of transversely isotropic materials have been obtained.

The explicit solution of some axialsymmetric scalar screen problems in Sobolev spaces is presented. The well-posedness of the boundary integral equations, which are formulated as dual integral equations, is proved. A reduction for the mixed boundary value problem to a system of singular integral equations with a symbol from the Wiener-algebra is given.The approach of dual integral equations has a long history [Mixed Boundary Value Problems in Potential Theory, North-Holland, Amsterdam, 1966] whereas the new developments reviewed in [E. Meister and F. O. Speck, Modern Wiener–Hopf methods in diffraction theory, Proc. Conf. Dundee, 1988, in Ordinary and Partial Differential Equations, B. Sleeman and R. Jarvis, eds., 1989, pp. 130–171] make it possible to handle these equations in Sobolev spaces.

A flat annular crack in a magnetoelectroelastic layer subjected to mechanical, electric and magnetic loadings is investigated under magnetoelectrically impermeable boundary condition on the crack surface. Using Hankel transform technique, the mixed boundary value problem is reduced to a system of singular integral equations. With the aid of Gauss-Chebyshev integration technique, the integral equations are further reduced to a system of algebraic equations. The field intensity factor and energy release rate are determined. Numerical results reveal the effects of electric and magnetic loadings and crack configuration on crack propagation and growth.

Plane-deformation problems of thermal stresses around an elastic ribbonlike conductor in an infinite, homogeneous, isotropic medium under uniform heat flow are analyzed by the method of continuous distribution of singular points. The line heat sources and concentrated forces are distributed continuously along the ribbonlike conductor. The boundary conditions yield a system of singular integral equations for density functions with Cauchy kernels and logarithmic singular kernel. A solution is assumed in the form of the product of a series of Chebyshev polynomials of the first kind and their weight function and is determined by transforming the singular integral equations into linear algebraic equations. One of the advantages of this method is that the strength of the thermal-stress singularity at the tips of the ribbonlike conductor can be easily evaluated. Numerical results for the strength of the thermal-stress singularity are plotted in terms of nondimensional parameters, consisting of the dimensions of the ribbonlike conductor and the material constants of the conductor and matrix.

This paper deals with numerical solutions of singular integral equations in interaction problems of diamond-shaped inclusions with angular corners under various loading conditions. The problems are formulated as a system of singular integral equations with Cauchy-type or logarithmic-type singularities, where the unknown functions are the densities of body forces distributed in infinite plates having the same elastic constants as those of the matrix and inclusions. In order to analyze the problems accurately, the unknown functions of the body force densities are expressed as a linear combination of two types of fundamental density functions and power series, where the fundamental density functions are chosen to represent the symmetric stress singularity of 1/r1-λ1 and the skew-symmetric stress singularity of 1/r1-λ2. Then, newly defined stress intensity factors of angular corners are systematically calculated for various shapes and spacings of two diamond-shaped inclusions in a plate subjected to uniaxial tension, biaxial tension and in-plane shear. The present method is found to yield rapidly converging numerical results for the interaction of diamond-shaped inclusions.

This paper describes numerical solutions of singular integral equations of the body force method in interaction problems of ellipsoidal inclusions under uniform tension. The problems are formulated as a system of singular integral equations with Cauchy-type or logarithmic-type singularities, where the densities of body forces distributed in the r-and z-directions of infinite bodies having the same elastic constants of the matrix and inclusions are unknown functions. In order to satisfy the boundary conditions along the inclusions, eight fundamental density functions proposed in our previous paper are used. Then the body force densities are approximated by a linear combination of the fundamental density functions and polynomials. The present method is found to give rapidly converging numerical results for stress distribution along the boundaries of both the matrix and inclusions.

This paper is concerned with a method of decreasing stress concentration due to a notch and a hole by providing additional holes in the region of the principal notch or hole. A singular integral equation method that is useful for this optimization problem is discussed. To formulate the problem the idea of the body force method is applied using the Green's function for a point force as a fundamental solution. Then, the interaction problem between the principal notch and the additional holes is expressed as a system of singular integral equations with a Cauchy-type singular kernel, where densities of the body force distribution in the x- and y-directions are to be unknown functions. In solving the integral equations, eight kinds of fundamental density functions are applied; then, the continuously varying unknown functions of body force densities are approximated by a linear combination of products of the fundamental density functions and polynomials. In the searching process of the optimum conditions, the direction search of Hooke and Jeeves is employed. The calculation shows that the present method gives the stress distribution along the boundary of a hole very accurately with a short CPU time. The optimum position and the optimum size of the auxiliary hole are also determined efficiently with high accuracy.