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

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.

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Transient Response of a Finite Bimaterial Plate Containing a Crack Perpendicular to and Terminating at the Interface

Journal of Applied Mechanics ◽

10.1115/1.2130734 ◽

2005 ◽

Vol 73 (4) ◽

pp. 544-554 ◽

Cited By ~ 1

Author(s):

Xian-Fang Li ◽

L. Roy Xu

Keyword(s):

Integral Equation ◽

Stress Intensity ◽

Singular Integral Equation ◽

Transient Response ◽

Singular Integral ◽

Stress Intensity Factors ◽

Dynamic Stress ◽

Cauchy Kernel ◽

Intensity Factors ◽

Dynamic Stress Intensity Factors

The transient response of a finite bimaterial plate with a crack perpendicular to and terminating at the interface is analyzed for two types of boundaries (free-free and clamped-clamped). The crack surface is loaded by arbitrary time-dependent antiplane shear impact. The mixed initial-boundary value problem is reduced to a singular integral equation of a generalized Cauchy kernel for the crack tearing displacement density or screw dislocation density. The Gauss-Jacobi quadrature technique is employed to numerically solve the singular integral equation, and then the dynamic stress intensity factors are determined by implementing a numerical inversion of the Laplace transform. As an example, numerical calculations are carried out for a cracked bimaterial plate composed of aluminum (material I) and epoxy or steel (material II). The effects of material properties, geometry, and boundary types on the variations of dynamic stress intensity factors are discussed in detail. Results indicate that an overshoot of the normalized stress intensity factor of the crack tip at the interface decreases for a cracked bimaterial plate, and the occurrence of which is delayed for a cracked aluminum/epoxy plate compared to a pure aluminum plate with the same crack.

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Diffraction of Torsional Waves by a Flat Annular Crack in an Infinite Elastic Medium

Journal of Applied Mechanics ◽

10.1115/1.3424662 ◽

1979 ◽

Vol 46 (4) ◽

pp. 827-831 ◽

Cited By ~ 3

Author(s):

Y. Shindo

Keyword(s):

Integral Equation ◽

Stress Intensity ◽

Singular Integral Equation ◽

Elastic Medium ◽

Singular Integral ◽

Dynamic Stress ◽

Dynamic Stress Intensity Factor ◽

Weight Functions ◽

Annular Crack ◽

Torsional Waves

The problem of diffraction of normally incident torsional waves by a flat annular crack embedded in an infinite, isotropic, and homogeneous elastic medium is considered. Using an integral transform technique, the problem is reduced to that of solving a singular integral equation. The solution of the singular integral equation is obtained in the form of the product of the series of Chebyshev polynomials of the first kind and their weight functions. Thus the essential feature of the dynamic singular stress field near the crack is preserved and the crack tip dynamic stress-intensity factor is easily evaluated. Numerical calculations are also carried out and the variations of dynamic stress-intensity factors with the normalized frequency for the ratio of the inner radius to the outer one are shown graphically.

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Approximate Solution Of A Singular Integral Equation With A Multiplicative Cauchy Kernel In The Half-Plane

Computational Methods in Applied Mathematics ◽

10.2478/cmam-2008-0010 ◽

2008 ◽

Vol 8 (2) ◽

pp. 143-154 ◽

Cited By ~ 1

Author(s):

P. KARCZMAREK

Keyword(s):

Integral Equation ◽

Approximate Solution ◽

Singular Integral Equation ◽

Half Plane ◽

Singular Integral ◽

Trigonometric Polynomials ◽

Cauchy Kernel

AbstractIn this paper, Jacobi and trigonometric polynomials are used to con-struct the approximate solution of a singular integral equation with multiplicative Cauchy kernel in the half-plane.

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Impact Response of a Cracked Orthotropic Medium

Journal of Applied Mechanics ◽

10.1115/1.3167101 ◽

1983 ◽

Vol 50 (3) ◽

pp. 630-636 ◽

Cited By ~ 49

Author(s):

M. K. Kassir ◽

K. K. Bandyopadhyay

Keyword(s):

Stress Intensity ◽

Dynamic Stress ◽

Fourier Transforms ◽

Dynamic Stress Intensity Factor ◽

Orthotropic Materials ◽

Dynamic Stress Intensity Factors ◽

Transient Problem ◽

Numerical Laplace Inversion ◽

The Laplace Transform ◽

Applied Stresses

A solution is given for the problem of an infinite orthotropic solid containing a central crack deformed by the action of suddenly applied stresses to its surfaces. Laplace and Fourier transforms are employed to reduce the transient problem to the solution of standard integral equations in the Laplace transform plane. A numerical Laplace inversion technique is used to compute the values of the dynamic stress-intensity factors, k1 (t) and k2 (t), for several orthotropic materials, and the results are compared to the corresponding elastostatic values to reveal the influence of material orthotropy on the magnitude and duration of the overshoot in the dynamic stress-intensity factor.

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Investigation on mixed-mode dynamic stress intensity factors of an interface crack in bi-materials with an inclusion

Composite Structures ◽

10.1016/j.compstruct.2018.02.078 ◽

2018 ◽

Vol 202 ◽

pp. 491-499 ◽

Cited By ~ 3

Author(s):

Kai Huang ◽

Licheng Guo ◽

Hongjun Yu

Keyword(s):

Stress Intensity ◽

Interface Crack ◽

Stress Intensity Factors ◽

Mixed Mode ◽

Dynamic Stress ◽

Intensity Factors ◽

Dynamic Stress Intensity Factors

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Interface Stress Analysis of the Bending Cylinder Containing Inclusion

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.201-203.951 ◽

2011 ◽

Vol 201-203 ◽

pp. 951-955

Author(s):

Xin Yan Tang

Keyword(s):

Integral Equation ◽

Stress Intensity ◽

Singular Integral Equation ◽

Stress Analysis ◽

Singular Integral ◽

Stress Intensity Factors ◽

Integral Equation Method ◽

Equation Method ◽

Intensity Factors ◽

Engineering Structures

Using the elasticity and the singular integral equation method, an analysis of a bending cylinder containing inclusions is carried out. The disturbing interface stresses on the inclusion sides and the stress intensity factors at the inclusion tips are obtained. The results given in this paper are useful for the strength design of the engineering structures or mechanical components containing inclusions.

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Approximate Solution of a Singular Integral Equation with a Cauchy Kernel on the Euclidean Plane

Computational Methods in Applied Mathematics ◽

10.1515/cmam-2017-0052 ◽

2018 ◽

Vol 18 (4) ◽

pp. 741-752

Author(s):

Dorota Pylak ◽

Paweł Karczmarek ◽

Paweł Wójcik

Keyword(s):

Integral Equation ◽

Approximate Solution ◽

Singular Integral Equation ◽

Singular Integral ◽

Euclidean Plane ◽

Singular Integral Equations ◽

Cauchy Kernel ◽

Challenging Problem ◽

One Dimensional ◽

Interpolating Polynomials

AbstractMultidimensional singular integral equations (SIEs) play a key role in many areas of applied science such as aerodynamics, fluid mechanics, etc. Solving an equation with a singular kernel can be a challenging problem. Therefore, a plethora of methods have been proposed in the theory so far. However, many of them are discussed in the simplest cases of one–dimensional equations defined on the finite intervals. In this study, a very efficient method based on trigonometric interpolating polynomials is proposed to derive an approximate solution of a SIE with a multiplicative Cauchy kernel defined on the Euclidean plane. Moreover, an estimation of the error of the approximated solution is presented and proved. This assessment and an illustrating example show the effectiveness of our proposal.

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A convergence theorem for singular integral equations

The Journal of the Australian Mathematical Society Series B Applied Mathematics ◽

10.1017/s033427000000285x ◽

1981 ◽

Vol 22 (4) ◽

pp. 539-552 ◽

Cited By ~ 10

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.

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Dynamic Stress Intensity Factor for a Radial Crack on a Circular Cavity in a Piezoelectric Medium

Key Engineering Materials ◽

10.4028/www.scientific.net/kem.452-453.273 ◽

2010 ◽

Vol 452-453 ◽

pp. 273-276

Author(s):

Tian Shu Song ◽

Dong Li ◽

Ming Yue Lv ◽

Ming Ju Zhang

Keyword(s):

Stress Intensity Factor ◽

Stress Intensity ◽

Intensity Factor ◽

Dynamic Stress ◽

Radial Crack ◽

Dynamic Stress Intensity Factor ◽

Crack Tip ◽

Piezoelectric Medium ◽

Circular Cavity ◽

Dynamic Stress Intensity Factors

The problem of dynamic stress intensity factor is investigated theoretically in present paper for a radial crack on a circular cavity in an infinite piezoelectric medium, which is subjected to time-harmonic incident anti-plane shearing. First, a pair of electromechanically coupled Green’s functions are constructed which indicate the basic solutions for a semi-infinite piezoelectric medium with a semi-circular cavity. Second, based on the crack-division technique and conjunction technique, integral equations for the unknown stresses’ solution on the conjunction surface is established, which are related to the dynamic stress intensity factor at the crack tip. Third, the analytical expression on dynamic stress intensity factor at the crack tip is obtained. At last, some calculating cases are plotted to show how the frequencies of incident wave, the piezoelectric characteristic parameters of the material and the geometry of the crack and the circular cavity influence upon the dynamic stress intensity factors. While some of the calculating results are compared with the same situation about a straight crack and with static solutions.

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