Axisymmetric Singular Stresses in a Thick-Walled Spherical Shell With an Internal Edge Crack

1983 ◽  
Vol 50 (1) ◽  
pp. 37-42 ◽  
Author(s):  
A. Atsumi ◽  
Y. Shindo
Keyword(s):  
Integral Equation ◽  
Spherical Shell ◽  
Edge Crack ◽  
Integral Transform ◽  
Crack Opening ◽  
Cauchy Kernel ◽  
Opening Displacement ◽  
Inner Surface ◽  
Dominant Part ◽  
Integral Transform Technique

The paper considers the elastostatic axisymmetric problem for a thick-walled spherical shell containing a circumferential edge crack on the inner surface. The ring-shaped edge crack and the inner surface of the spherical shell are subjected to internal pressure. Using an integral transform technique we obtain a singular integral equation of the first kind which has a generalized Cauchy kernel as the dominant part. The integral equation is solved numerically, and the influence of the geometrical configuration on the stress-intensity factor and the crack-opening displacement is shown graphically in detail.

Diffraction of Antiplane Shear Waves by an Edge Crack

1980 ◽  
Vol 47 (2) ◽  
pp. 359-362 ◽  
Author(s):  
S. F. Stone ◽  
M. L. Ghosh ◽  
A. K. Mal
Keyword(s):  
Integral Equation ◽  
Crack Opening Displacement ◽  
Edge Crack ◽  
Incident Angle ◽  
Crack Opening ◽  
Shear Waves ◽  
Antiplane Shear ◽  
Opening Displacement ◽  
Low Frequencies ◽  
High Frequencies

The diffraction of time harmonic antiplane shear waves by an edge crack normal to the free surface of a homogeneous half space is considered. The problem is formulated in terms of a singular integral equation with the unknown crack opening displacement as the density function. A numerical scheme is utilized to solve the integral equation at any given finite frequency. Asymptotic solutions valid at low and high frequencies are obtained. The accuracy of the numerical solution at high frequencies and of the high frequency asymptotic solution at intermediate frequencies are examined. Graphical results are presented for the crack opening displacement and the stress intensity factor as functions of frequency and the incident angle, Expressions for the far-field displacements at high and low frequencies are also presented.

Circumferential Edge Crack in a Cylindrical Cavity

1977 ◽  
Vol 44 (2) ◽  
pp. 250-254 ◽  
Author(s):  
L. M. Keer ◽  
V. K. Luk ◽  
J. M. Freedman
Keyword(s):  
Integral Equation ◽  
Stress Intensity ◽  
Cylindrical Cavity ◽  
Edge Crack ◽  
Numerical Scheme ◽  
Crack Opening ◽  
Integral Transforms ◽  
Intensity Factors ◽  
Physical Quantities ◽  
Elastostatic Problem

The elastostatic problem of a circumferential edge crack in a cylindrical cavity is investigated. The problem is formulated by means of integral transforms and reduced to a singular integral equation. The numerical scheme of Erdogan, Gupta, and Cook is used to obtain the relevant physical quantities and the stress-intensity factors, and crack opening displacements are computed for several values of crack length.

An Integral Equation Approach to the Semi-Infinite Strip Problem

1973 ◽  
Vol 40 (4) ◽  
pp. 948-954 ◽  
Author(s):  
G. D. Gupta
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Singular Integral ◽  
Laminate Composite ◽  
Integral Transform ◽  
Stress Singularity ◽  
Infinite Strip ◽  
Integral Equation Approach ◽  
Fixed End ◽  
Integral Transform Technique

A semi-infinite strip held rigidly on its short end is considered. Loads in the strip at infinity (far away from the fixed end) are prescribed. Integral transform technique is used to provide an exact formulation of the problem in terms of a singular integral equation. Stress singularity at the strip corner is obtained from the singular integral equation which is then solved numerically. Stresses along the rigid end are determined and the effect of the material properties on the stress-intensity factor is presented. The method can also be applied to the problem of a laminate composite with a flat inclusion normal to the interfaces.

Review of hypersingular integral equation method for crack scattering and application to modeling of ultrasonic nondestructive evaluation

2003 ◽  
Vol 56 (4) ◽  
pp. 383-405 ◽  
Author(s):  
Anders Bostro¨m
Keyword(s):  
Integral Equation ◽  
Nondestructive Testing ◽  
Crack Opening ◽  
Integral Equation Method ◽  
Boundary Element Methods ◽  
Hypersingular Integral Equation ◽  
Ultrasonic Nondestructive Testing ◽  
Opening Displacement ◽  
Hypersingular Integral ◽  
Integral Equation Methods

The scattering of elastic waves by cracks in isotropic and anisotropic solids has important applications in various areas of mechanical engineering and geophysics, in particular in ultrasonic nondestructive testing and evaluation. The scattering by cracks can be investigated by integral equation methods, eg, boundary element methods, but here we are particularly concerned with more analytically oriented hypersingular integral equation methods. In these methods, which are only applicable to very simple crack shapes, the unknown crack opening displacement in the integral equation is expanded in a set of Chebyshev functions, or the like, and the integral equation is projected onto the same set of functions. This procedure automatically takes care of the hypersingularity in the integral equation. The methods can be applied to cracks in 2D and 3D, and to isotropic or anisotropic media. The crack can be situated in an unbounded space or in a layered structure, including the case with an interface crack. Also, problems with more than one crack can be treated. We show how the crack scattering procedures can be combined with models for transmitting and receiving ultrasonic probes to yield a complete model of ultrasonic nondestructive testing. We give a few numerical examples showing typical results that can be obtained, also comparing with some experimental results. This review article cites 78 references.    

scholarly journals Periodic Array of Cracks in a Half-Plane Subjected to Arbitrary Loading

1987 ◽  
Vol 54 (3) ◽  
pp. 642-648 ◽  
Author(s):  
H. F. Nied
Keyword(s):  
Integral Equation ◽  
Half Plane ◽  
Crack Surface ◽  
Mixed Boundary ◽  
Crack Opening ◽  
Periodic Array ◽  
Hypersingular Integral Equation ◽  
Opening Displacement ◽  
Surface Displacements ◽  
Displacement Based

The plane elastic problem for a periodic array of cracks in a half-plane subjected to equal, but otherwise arbitrary normal crack surface tractions is examined. The mixed boundary value problem, which is formulated directly in terms of the crack surface displacements, results in a hypersingular integral equation in which the unknown function is the crack opening displacement. Based on the theory of finite part integrals, a least squares numerical algorithm is employed to efficiently solve the singular integral equation. Numerical results include crack opening displacements, stress intensity factors, and Green’s functions for the entire range of possible periodic crack spacing.

Nature of Stress Singularities at the Reentrant Wedge Corner for a Branched Crack Problem

1999 ◽  
Vol 66 (1) ◽  
pp. 278-280 ◽  
Author(s):  
A. S. Selvarathinam and ◽  
J. G. Goree
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Isotropic Material ◽  
Main Crack ◽  
Crack Opening ◽  
Crack Problem ◽  
Stress Singularity ◽  
Stress Singularities ◽  
Opening Displacement ◽  
Reentrant Corner

The solution of the branched crack problem for an isotropic material, employing the dislocation method as developed by Lo (1978), results in a singular integral equation in which the slope of the crack-opening displacement is the unknown. In this brief note, using the function-theoretic method, the behavior of this unknown function is investigated at the corner where the branched and main crack meet and it is shown that the order of stress singularity obtained at the reentrant corner of the branched crack is given by the Williams’ (1952) characteristic equation for the isotropic wedge.

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.

scholarly journals On a Method in Dynamic Elasticity Problems for Heterogeneous Wedge-Shaped Medium

2021 ◽  
Vol 273 ◽  
pp. 04002
Author(s):  
Vyacheslav Berkovich ◽  
Viсtor Poltinnikov
Keyword(s):  
Integral Equation ◽  
Wave Field ◽  
Boundary Integral Equation ◽  
Integral Transform ◽  
Boundary Integral ◽  
Geometric Characteristics ◽  
Dynamic Elasticity ◽  
Elasticity Problems ◽  
Steady Oscillations ◽  
Integral Transform Technique

The method of analysis of steady oscillations arising in the piecewise homogeneous wedge-shaped medium composed by two homogeneous elastic wedges with different mechanical and geometric characteristics is presented. Method is based on the distributions’ integral transform technique and allows reconstructing the wave field in the whole medium by displacement oscillations given in the domain on the boundary of the medium. The problem in question is reduced to a boundary integral equation (BIA). Solvability problems of the BIA are examined and the structure of its solution is established.

scholarly journals Interaction of a finite crack with shear waves in an infinite magnetoelastic medium

2021 ◽  
Vol 15 (1) ◽  
Author(s):  
Sourav Kumar Panja ◽  
Subhas Chandra Mandal
Keyword(s):  
Integral Equation ◽  
Integral Transformation ◽  
Crack Opening ◽  
Shear Waves ◽  
Fourier Integral ◽  
Opening Displacement ◽  
Field Interaction ◽  
Low Frequencies ◽  
Finite Crack ◽  
Magnetoelastic Medium

The aim of this paper is to investigate the interaction of a finite crack with shear waves in an infinite magnetoelastic medium. Fourier integral transformation is applied to convert the boundary value problem for a homogeneous, isotropic elastic material to the Fredholm integral equation of second kind. The integral equation is solved by the perturbation method and the effect of magnetic field interaction on the crack is discussed. The stress intensity factor at the crack tip is determined numerically and plotted for low frequencies. Moreover, shear stress outside the crack, crack opening displacement, and crack energy are evaluated and shown by means of graphs.

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