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.

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.    

Elastodynamic Analysis of a Periodic Array of Mode III Cracks in Transversely Isotropic Solids

1992 ◽  
Vol 59 (2) ◽  
pp. 366-371 ◽  
Author(s):  
Ch. Zhang
Keyword(s):  
Integral Equation ◽  
Crack Opening Displacement ◽  
Boundary Integral Equation ◽  
Crack Opening ◽  
Transversely Isotropic ◽  
Boundary Integral ◽  
Crack Spacing ◽  
Periodic Array ◽  
Mode Iii ◽  
Opening Displacement

Time-harmonic elastodynamic analysis is presented for a periodic array of collinear mode III cracks in an infinite transversely isotropic solid. The scattering problem by a single antiplane crack is first formulated, and the scattered displacement field is expressed as Fourier integrals containing the crack opening displacement. By using this representation formula and by considering the periodicity conditions in the crack spacing, a boundary integral equation is obtained for the crack opening displacement of a reference crack. The boundary integral equation is solved numerically by expanding the crack opening displacement into a series of Chebyshev polynomials. Numerical results are given to show the effects of the crack spacing, the wave frequency, the angle of incidence, and the anisotropy parameter on the elastodynamic stress intensity factors.

Vibratory Motion of a Body on an Elastic Half Plane

1968 ◽  
Vol 35 (4) ◽  
pp. 697-705 ◽  
Author(s):  
P. Karasudhi ◽  
L. M. Keer ◽  
S. L. Lee
Keyword(s):  
Integral Equation ◽  
Half Plane ◽  
Good Accuracy ◽  
Coupling Effect ◽  
Mixed Boundary ◽  
Mixed Boundary Value Problem ◽  
Dual Integral Equations ◽  
Entire Surface ◽  
Vibratory Motion ◽  
Rocking Vibration

The vertical, horizontal and rocking vibrations of a body on the surface of an otherwise unloaded half plane are studied. The problems are formulated so that one stress vanishes over the entire surface, and an oscillating displacement is prescribed in the loaded region. The problems are mixed with respect to the prescribed displacement and the remaining stress. Each case leads to a mixed boundary value problem represented by dual integral equations which are reduced to a single Fredholm integral equation. Although numerical methods are used to solve the integral equation, the contact stresses are found to be presentable in closed form to good accuracy. An estimate of the stiffnesses for coupled horizontal and rocking vibration is also suggested and it is found that the coupling effect is significant.

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.

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.

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.

Three-dimensional analysis of cracks in layered transversely isotropic media

1989 ◽  
Vol 424 (1867) ◽  
pp. 307-322 ◽  
Keyword(s):  
Integral Equation ◽  
Green's Functions ◽  
Green’S Functions ◽  
Crack Opening ◽  
Mathematical Formulation ◽  
Transversely Isotropic ◽  
Boundary Integral ◽  
Opening Displacement ◽  
Transversely Isotropic Media ◽  
Isotropic Media

A general mathematical formulation to analyse cracks in layered transversely isotropic media is developed in this paper. By constructing the Green’s functions, an integral equation is obtained to determine crack opening displacements when an applied crack face traction is specified. For the infinite body, the Green’s functions have solutions in a closed form. For layered media, a flexibility matrix in the integral transformed domain is formed that establishes the relation between the traction and the displacement for a single layer; the global matrix is formed by assembling all of the flexibility matrices constructed for each layer. The Green’s functions in the spatial domain are obtained by inversion of the Hankel transform. Finally, the crack opening displacement and the crack-tip opening displacement for a vertical planar crack in a layered transversely isotropic medium are obtained numerically by the boundary integral equation method.

Load Carrying Capacity of Notched CFRP Laminates

2010 ◽  
Vol 430 ◽  
pp. 47-51
Author(s):  
H. Suzuki ◽  
S. Kinugawa ◽  
Hideki Sekine
Keyword(s):  
Carrying Capacity ◽  
Crack Surface ◽  
Crack Opening ◽  
Damage Zone ◽  
Load Carrying Capacity ◽  
Opening Displacement ◽  
Cohesive Stress ◽  
Cfrp Laminates ◽  
Tension Softening ◽  
Load Carrying

On the basis of a micromechanical study, a method for evaluating load carrying capacity of notched CFRP laminates is developed. The damage zone at a notch tip in CFRP laminates is modeled as a fictitious crack with a cohesive stress acting on the crack surface. Then, applying the Weibull weakest link theory to the strength of surviving fiber bundles on the crack surface, we derive the relationship between the cohesive stress and the crack opening displacement, i.e., the tension-softening relation. By incorporating it in a BEM scheme, the load-displacement relationship is simulated. The simulated result for notched CPRP laminates is compared with experimental ones, and it is found that the simulated and experimental results of load carrying capacity are consistent.

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