Torsional Impact Response of a Thick-Walled Cylinder With a Circumferential Edge Crack

1990 ◽  
Vol 112 (4) ◽  
pp. 367-373 ◽  
Author(s):  
Y. Shindo ◽  
W. Li
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Edge Crack ◽  
Dynamic Stress Intensity Factor ◽  
Impact Response ◽  
Cauchy Kernel ◽  
Dual Integral Equations ◽  
Singular Stress ◽  
Singular Stress Field ◽  
Walled Cylinder

This paper considers the torsional impact response of a long thick-walled cylinder containing an internal or external circumferential edge crack. Laplace and Hankel transforms are used to reduce the elastodynamic problem to a pair of dual integral equations. The dual integral equations are solved by using the standard transform technique, and the result is expressed in terms of an integral equation which has a generalized Cauchy kernel as the dominant part. The kernel of the integral equation is improved in order that the calculation may be made easy. A numerical Laplace inversion technique is used to recover the time dependence of the solution. The dynamic singular stress field is determined, and the numerical results on the dynamic stress intensity factor are obtained to show the influence of inertia, geometry, and their interactions.

Dynamic Singular Stresses for a Griffith Crack in a Soft Ferromagnetic Elastic Solid Subjected to a Uniform Magnetic Field

1983 ◽  
Vol 50 (1) ◽  
pp. 50-56 ◽  
Author(s):  
Y. Shindo
Keyword(s):  
Magnetic Field ◽  
Integral Equations ◽  
Fourier Transforms ◽  
Dynamic Stress Intensity Factor ◽  
Elastic Solid ◽  
Griffith Crack ◽  
Dual Integral Equations ◽  
Singular Stress ◽  
Magnetostatic Field ◽  
Soft Ferromagnetic

The problem of the diffraction of normally incident longitudinal waves on a Griffith crack located in an infinite soft ferromagnetic elastic solid is considered. It is assumed that the solid is a homogeneous and isotropic one and is permeated by a uniform magnetostatic field normal to the crack surfaces. Fourier transforms are used to reduce the problem to two simultaneous dual integral equations. The solution to the integral equations is expressed in terms of a Fredholm integral equation of the second kind having the kernel that is a finite integral. The dynamic singular stress field near the crack tip is obtained and the influence of the magnetic field on the dynamic stress intensity factor is shown graphically in detail. Approximate analytical expressions valid at low frequencies are also obtained and the range of validity of these expressions is examined.

Impact Response of a Single Edge Crack in an Orthotropic Strip

1992 ◽  
Vol 59 (2S) ◽  
pp. S152-S157 ◽  
Author(s):  
Yasuhide Shindo ◽  
Hiroaki Higaki ◽  
Hideaki Nozaki
Keyword(s):  
Integral Equations ◽  
Edge Crack ◽  
Fourier Transforms ◽  
Dynamic Stress Intensity Factor ◽  
Single Edge ◽  
Dual Integral Equations ◽  
Orthotropic Strip ◽  
Transient Problem ◽  
Reinforced Composite ◽  
The Laplace Transform

The elastodynamic response of a single edge crack in an orthotropic strip under normal impact is considered in this study. The edge crack is oriented in a direction normal to the edge of the strip. Laplace and Fourier transforms are used to reduce the transient problem to the solution of a pair of dual integral equations in the Laplace transform plane. The solution to the dual integral equations is then expressed in terms of a Fredholm integral equation of the second kind. A numerical Laplace inversion routine is used to recover the time dependence of the solution. Numerical values on the dynamic stress intensity factor for several fiber-reinforced composite materials are obtained, and the results are graphed to display the influence of the material orthotropy.

A method for the exact solution of a class of two-dimensional diffraction problems

1951 ◽  
Vol 205 (1081) ◽  
pp. 286-308 ◽  
Keyword(s):  
Integral Equation ◽  
Plane Wave ◽  
Integral Equations ◽  
Scattered Field ◽  
Plane Waves ◽  
Angular Spectrum ◽  
Diffraction Theory ◽  
General Interest ◽  
Two Dimensional ◽  
Dual Integral Equations

In the last few years Copson, Schwinger and others have obtained exact solutions of a number of diffraction problems by expressing these problems in terms of an integral equation which can be solved by the method of Wiener and Hopf. A simpler approach is given, based on a representation of the scattered field as an angular spectrum of plane waves, such a representation leading directly to a pair of ‘dual’ integral equations, which replaces the single integral equation of Schwinger’s method. The unknown function in each of these dual integral equations is that defining the angular spectrum, and when this function is known the scattered field is presented in the form of a definite integral. As far as the ‘radiation’ field is concerned, this integral is of the type which may be approximately evaluated by the method of steepest descents, though it is necessary to generalize the usual procedure in certain circumstances. The method is appropriate to two-dimensional problems in which a plane wave (of arbitrary polarization) is incident on plane, perfectly conducting structures, and for certain configurations the dual integral equations can be solved by the application of Cauchy’s residue theorem. The technique was originally developed in connexion with the theory of radio propagation over a non-homogeneous earth, but this aspect is not discussed. The three problems considered are those for which the diffracting plates, situated in free space, are, respectively, a half-plane, two parallel half-planes and an infinite set of parallel half-planes; the second of these is illustrated by a numerical example. Several points of general interest in diffraction theory are discussed, including the question of the nature of the singularity at a sharp edge, and it is shown that the solution for an arbitrary (three-dimensional) incident field can be derived from the corresponding solution for a two-dimensional incident plane wave.

scholarly journals On the Fredholm integral equation associated with pairs of dual integral equations

1969 ◽  
Vol 16 (3) ◽  
pp. 185-194 ◽  
Author(s):  
V. Hutson
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Bessel Function ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Unknown Function ◽  
Sufficient Conditions ◽  
Neumann Series ◽  
Dual Integral Equations ◽  
Image Position

Consider the Fredholm equation of the second kindwhereand Jv is the Bessel function of the first kind. Here ka(t) and h(x) are given, the unknown function is f(x), and the solution is required for large values of the real parameter a. Under reasonable conditions the solution of (1.1) is given by its Neumann series (a set of sufficient conditions on ka(t) for the convergence of this series is given in Section 4, Lemma 2). However, in many applications the convergence of the series becomes too slow as a→∞ for any useful results to be obtained from it, and it may even happen that f(x)→∞ as a→∞. It is the aim of the present investigation to consider this case, and to show how under fairly general conditions on ka(t) an approximate solution may be obtained for large a, the approximation being valid in the norm of L2(0, 1). The exact conditions on ka(t) and the main result are given in Section 4. Roughly, it is required that 1 -ka(at) should behave like tp(p>0) as t→0. For example, ka(at) might be exp ⌈-(t/ap)⌉.

scholarly journals A convergence theorem for singular integral equations

1981 ◽  
Vol 22 (4) ◽  
pp. 539-552 ◽  
Author(s):  
David Elliott
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Integral Equations ◽  
Singular Integral ◽  
Sufficient Conditions ◽  
Singular Integral Equations ◽  
Convergence Theory ◽  
Cauchy Kernel ◽  
Algebraic Equations ◽  
Linear Algebraic Equations

AbstractThe principal result of this paper states sufficient conditions for the convergence of the solutions of certain linear algebraic equations to the solution of a (linear) singular integral equation with Cauchy kernel. The motivation for this study has been the need to provide a convergence theory for a collocation method applied to the singular integral equation taken over the arc (−1, 1). However, much of the analysis will be applicable both to other approximation methods and to singular integral equations taken over other arcs or contours. An estimate for the rate of convergence is also given.

scholarly journals Torsional Impact Response of a Penny-Shaped Interface Crack in Bonded Materials With a Graded Material Interlayer

2002 ◽  
Vol 69 (3) ◽  
pp. 303-308 ◽  
Author(s):  
C. Li ◽  
Z. Duan ◽  
Z. Zou
Keyword(s):  
Integral Equation ◽  
Stress Intensity ◽  
Singular Integral Equation ◽  
Interface Crack ◽  
Singular Integral ◽  
Dynamic Stress ◽  
Mixed Boundary ◽  
Dynamic Stress Intensity Factor ◽  
Cauchy Kernel ◽  
Dynamic Stress Intensity Factors

In this paper, the dynamic response of a penny-shaped interface crack in bonded dissimilar homogeneous half-spaces is studied. It is assumed that the two materials are bonded together with such a inhomogeneous interlayer that makes the elastic modulus in the direction perpendicular to the crack surface is continuous throughout the space. The crack surfaces are assumed to be subjected to torsional impact loading. Laplace and Hankel integral transforms are applied combining with a dislocation density function to reduce the mixed boundary value problem into a singular integral equation with a generalized Cauchy kernel in Laplace domain. By solving the singular integral equation numerically and using a numerical Laplace inversion technique, the dynamic stress intensity factors are obtained. The influences of material properties and interlayer thickness on the dynamic stress intensity factor are investigated.

Anti-Plane Moving Crack in Functionally-Graded Material

2010 ◽  
Vol 29-32 ◽  
pp. 549-553
Author(s):  
Qi Liu
Keyword(s):  
Integral Equations ◽  
Functionally Graded Material ◽  
Integral Transform ◽  
Mixed Boundary ◽  
Dynamic Stress Intensity Factor ◽  
Functionally Graded ◽  
Dual Integral Equations ◽  
Moving Crack ◽  
Fourier Cosine ◽  
Graded Material

In this paper the anti-plane moving crack in a functionally-graded material is studied by the analytical method. First the governing equations for a functionally-graded material are obtained using a Fourier cosine integral transform. Then the dual integral equations for moving crack are established according to the mixed boundary value conditions. It is shown that the dual integral equations can be reduced to the Fredholm integral equation of the second kind. Numerical results shown in the present paper indicate that the non-homogeneity of material has an important influence on the dynamic stress intensity factor.

scholarly journals On Certain Dual Integral Equations

1961 ◽  
Vol 5 (1) ◽  
pp. 21-24 ◽  
Author(s):  
E. T. Copson
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Dual Integral Equations ◽  
Mellin Transforms ◽  
Abel Integral Equation ◽  
Image Position ◽  
Fourier Integrals

In his book on Fourier Integrals, Titchmarsh [l] gave the solution of the dual integral equationsfor the case α > 0, by some difficult analysis involving the theory of Mellin transforms. Sneddon [2] has recently shown that, in the cases v = 0, α = ±½, the problem can be reduced to an Abel integral equation by making the substitutionorIt is the purpose of this note to show that the general case can be dealt with just as simply by puttingThe analysis is formal: no attempt is made to supply details of rigour.

A formal solution of certain dual integral equations involving H-functions

1967 ◽  
Vol 63 (1) ◽  
pp. 171-178 ◽  
Author(s):  
R. K. Saxena
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Formal Solution ◽  
Fractional Integration ◽  
Dual Integral Equations ◽  
First Case ◽  
The Common ◽  
The Given ◽  
Fourier Kernel

AbstractThe problem discussed is the formal solution of certain dual integral equations involving H-functions. The method followed is that of fractional integration. The given dual integral equations have been transformed, by the application of fractional integration operators, into two others with a common kernel and the problem is then reduced to solving one integral equation. In the first case the common kernel comes out to be a symmetrical Fourier kernel given earlier by Fox and the formal solution is then immediate. In the second case the common kernel is a generalized Fourier kernel and dual integral equations of this type have recently been studied by Fox.

Intervallic Coiflets for Numerical Calculation of Dynamic Stress Intensity Factor

2012 ◽  
Vol 562-564 ◽  
pp. 668-671
Author(s):  
Jian Ping Xuan ◽  
Yuan Feng Liu ◽  
Tie Lin Shi
Keyword(s):  
Stress Intensity Factor ◽  
Stress Intensity ◽  
Intensity Factor ◽  
Integral Equations ◽  
Dynamic Stress ◽  
Dynamic Stress Intensity Factor ◽  
High Accuracy ◽  
Fredholm Integral Equations ◽  
Wavelet Expansion ◽  
Dual Integral Equations

There are lots of practical problems which are related to the solution of Fredholm integral equations of the second kind. The present work proposes intervallic Coiflets for solving the equations. Illustrative problem involving dynamic stress and electric fields of a cracked piezoelectric excited by anti-plane shear wave is addressed. Permeable boundary condition has been used to obtain a pair of dual integral equations of the symmetric and antisymmetric parts which can be reduced to the solutions of two Fredholm integral equations of the second kind. The dynamic stress intensity factor is expressed in terms of the right-end values of two unknown functions in Fredholm integral equations. The two unknown functions are solved by intervallic Coiflets which have less the endpoints error. And intervallic Coiflets have low calculation cost and high accuracy due to the wavelet expansion coefficients are exactly obtained without calculating the wavelet integrations. The calculation results agree well with the existing method, which show the high accuracy of the estimation and demonstrate validity and applicability of the method.

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