N. I. Ioakimidis

Application of finite-part integrals to the singular integral equations of crack problems in plane and three-dimensional elasticity

SummaryA modification of the form of the singular integral equation for the problem of a plane crack of arbitrary shape in a three-dimensional isotropic elastic medium is proposed. This modification consists in the incorporation of the Laplace operator δ into the integrand. The integral must now be interpreted as a finite-part integral. The new singular integral equation is equivalent to the original one, but simpler in form. Moreovet, its form suggests a new approach for its numerical solution, based on quadrature rules for one-dimensional finite part integrals with a singularity of order two. A very simple application to the problem of a penny-shaped crack under constant pressure is also made. Moreover, the case of straight crack problems in plane isotropic elasticity is also considered in detail and the corresponding results for this special case are also derived.

On the uniform convergence of Gaussian quadrature rules for Cauchy principal value integrals and their derivatives

The convergence of the aforementioned quadrature rules for integrands possessing H6lder-continuous derivatives of an appropriate order is proved to be uniform and not only pointwise. The rate of convergence is also established and an application to the numerical solution of singular integral equations is made.

A natural approach to the introduction of finite-part integrals into crack problems of three-dimensional elasticity

Abstract The classical singular integral equation for the problem of a plane crack inside an infinite isotropic elastic medium and under an arbitrary normal pressure distribution was recently modified and written without the use of the Laplace operator Δ or the derivatives of the unknown function, but with the use of a finite-part integral. In this paper, a second complete derivation of the same equation is made (not based on previous forms of this equation) by using a limiting procedure, which makes it clear why the finite-part integral results in this equation. It is believed that the present results will be used in future for the introduction of finite-part integrals into a lot of crack problems in the theory of three-dimensional elasticity.

On the natural interpolation formula for cauchy type singular integral equations of the first kind

A Cauchy type singular integral equation of the first kind can be numerically solved either directly, through the use of a Gaussian numerical integration rule, or by reduction to an equivalent Fredholm integral equation of the second kind, where the Nyström method is applicable. In this note it is proved that under appropriate but reasonable conditions the expressions of the unknown function of the integral equation, resulting from the natural interpolation formulae of the direct method, as well as of the Nyström method, are identical along the whole integration interval.ZusammenfassungEine singuläre Integralgleichung erster Art vom Cauchy-Typ kann entweder direkt, mittels einer Gaußschen numerischen Integrationsformel, oder durch Reduktion auf eine äquivalente Fredholmsche Integralgleichung zweiter Art, wo die Nyström-Methode anwendbar ist, gelöst werden. In dieser Arbeit wird bewiesen, daß unter geeigneten und sinnvollen Bedingungen die Ausdrücke der unbekannten Funktion der Integralgleichung, die einerseits bei den natürlichen Integrationsformeln der direkten Methode und anderseits bei der Nyström-Methode entstehen, im ganzen Integrationsintervall gleich sind.

On the numerical evaluation of derivatives of Cauchy principal value integrals

Quadrature rules for the approximate evaluation of derivatives of Cauchy principal value integrals (with respect to the free variable inside the integral) can be obtained by formal differentiations of the right sides of the corresponding quadrature rules (without derivatives). The justification of this method, under appropriate conditions, is presented in this short communication. The results of this short communication are also applicable to the numerical evaluation of a class of finite-part integrals and to the numerical solution of singular integrodifferential equations.ZusammenfassungQuadraturformeln für die approximative Berechnung der Ableitungen Cauchyscher Hauptwertintegrale (bezüglich der freien Variablen der Integrale) können durch formale Differentiationen der rechten Seiten der verwendeten Integrationsformeln (ohne Ableitungen) genommen werden. Die Rechtfertigung dieser Methode unter geeigneten Bedingungen erscheint in dieser kurzen Mitteilung. Die Ergebnisse dieser kurzen Mitteilung sind auch anwendbar für die numerische Berechnung der endlichen Bestandteile gewisser divergenter Integrale und für die numerische Lösung singulärer Integrodifferentialgleichungen.

The numerical solution of crack problems in plane elasticity in the case of loading discontinuities

Abstract The direct quadrature method of numerical solution of Cauchy type singular integral equations encountered in plane elasticity crack problems is applied to the case where the loading distribution along the crack edges presents jump discontinuities. This is made by using a well-known modification of the quadrature method which is free of undesirable errors due to the loading discontinuities. Hence, the method is ideal to treat the aforementioned class of crack problems and, particularly, crack problems where the Dugdale-Barenblatt elastic-perfectly plastic model is adopted. Finally, a numerical application of the method to the problem of a periodic array of cracks with a loading distribution presenting a jump discontinuity is made. The numerical results obtained in this problem compare favorably with the corresponding theoretical results available in this special problem.

A remark on the numerical solution of singular integral equations and the determination of stress-intensity factors

SummaryAs is well-known, an efficient numerical technique for the solution of Cauchy-type singular integral equations along an open interval consists in approximating the integrals by using appropriate numerical integration rules and appropriately selected collocation points. Without any alterations in this technique, it is proposed that the estimation of the unknown function of the integral equation is further achieved by using the Hermite interpolation formula instead of the Lagrange interpolation formula. Alternatively, the unknown function can be estimated from the error term of the numerical integration rule used for Cauchy-type integrals. Both these techniques permit a significant increase in the accuracy of the numerical results obtained with an insignificant increase in the additional computations required and no change in the system of linear equations solved. Finally, the Gauss-Chebyshev method is considered in its original and modified form and applied to two crack problems in plane isotropic elasticity. The numerical results obtained illustrate the powerfulness of the method.

Quadrature Methods for the Determination of Zeros of Transcendental Functions - A Review

A review of quadrature methods for the numerical determination of zeros of algebraic or transcendental functions is presented. Most of these methods are based on the classical theory of analytic functions, but, recently, relevant methods based on the elementary theory of real functions were also developed. On the other hand, purely numerical methods were also recently proposed. The common point of these methods is the use of numerical integration rules for the determination of the aforementioned zeros. This makes these methods completely different from the classical (usually iterative) competitive methods. Emphasis is placed on the description of the fundamental principles of each method and, further, on making appropriate reference to the existing relevant literature. Some generalizations of these methods (e.g. to the cases of poles or systems of two equations in two unknowns) are also presented.

On convergence of two direct methods for solution of Cauchy type singular integral equations of the first kind

The methods for direct numerical solution of Cauchy type singular integral equations of the first kind based on Gauss-Chebyshev or Lobatto-Chebyshev numerical integration and the reduction of such an integral equation to a system of linear equations are proved to converge under appropriate conditions.

Elementary applications of MATHEMATICA to the solution of elasticity problems by the finite element method

Abstract The computer algebra system MATHEMATICA is used for the SAN (semi-analytical/numerical) solution of two simple elasticity problems, having been reduced to systems of linear algebraic equations by the finite element method. In both cases, one parameter was left in symbolic form and Taylor series expansions with respect to this parameter (either a material constant or a geometric parameter) were used in the SAN results. The Gauss-Seidel iterative method for systems of linear equations and its SOR variant were used for the solution of the systems of linear equations always in the SAN environment offered by MATHEMATICA. The SAN results were compared with the corresponding numerical results and they were found to be equally acceptable. The present approach, which can be generalized in a variety of ways, offers the advantage (over purely numerical techniques) that it permits the incorporation of symbolic parameters into the results of the finite element method.