Scattering of Plane Waves by a Propagating Crack

1975 ◽  
Vol 42 (3) ◽  
pp. 705-711 ◽  
Author(s):  
E. P. Chen ◽  
G. C. Sih
Keyword(s):  
Integral Equations ◽  
Fourier Transforms ◽  
Mixed Boundary ◽  
Crack Opening ◽  
Crack Problem ◽  
Fredholm Integral Equations ◽  
Angle Of Incidence ◽  
Dual Integral Equations ◽  
Dynamic Stress Intensity Factors ◽  
Incident Wavelength

Scattering of plane harmonic waves by a running crack of finite length is investigated. Fourier transforms were used to formulate the mixed boundary-value problem which reduces to pairs of dual integral equations. These dual integral equations are further reduced to a pair of Fredholm integral equations of the second kind. The dynamic stress-intensity factors and crack opening displacements are obtained as functions of the incident wavelength, angle of incidence, Poisson’s ratio of the elastic solid and speed of crack propagation. Unlike the semi-infinite running crack problem, which does not have a static limit, the solution for the finite crack problem can be used to compare with its static counterpart, thus showing the effect of dynamic amplification.

scholarly journals A mixed boundary value problem of a cracked elastic medium under torsion

2021 ◽  
pp. 10-10
Author(s):  
Belkacem Kebli ◽  
Fateh Madani
Keyword(s):  
Boundary Value Problem ◽  
Elastic Medium ◽  
Integral Equations ◽  
Integral Transformation ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Crack Problem ◽  
Fredholm Integral Equations ◽  
Mixed Boundary Value Problem ◽  
Penny Shaped Crack

The present work aims to investigate a penny-shaped crack problem in the interior of a homogeneous elastic material under axisymmetric torsion by a circular rigid inclusion embedded in the elastic medium. With the use of the Hankel integral transformation method, the mixed boundary value problem is reduced to a system of dual integral equations. The latter is converted into a regular system of Fredholm integral equations of the second kind which is then solved by quadrature rule. Numerical results for the displacement, stress and stress intensity factor are presented graphically in some particular cases of the problem.

The solution of Bessel function dual integral equations by converting to the first kind Fredholm Integral Equation: an extension of Noble’s solution

2018 ◽  
Vol 24 (8) ◽  
pp. 2536-2557
Author(s):  
S Cheshmehkani ◽  
M Eskandari-Ghadi
Keyword(s):  
Boundary Value Problems ◽  
Integral Equations ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Integral Transforms ◽  
Original Method ◽  
Fredholm Integral Equations ◽  
Dual Integral Equations ◽  
Mixed Boundary Value Problems ◽  
Factor Method

In certain mixed boundary value problems, Hankel integral transforms are applied and subsequently dual integral equations involving Bessel functions have to be solved. In the literature, if possible by employing the Noble’s multiplying factor method, these dual integral equations are usually converted to the second kind Fredholm Integral Equations (FIEs) and solved either analytically or numerically, respectively, for simple or complicated kernels. In this study, the multiplying factor method is extended to convert the dual integral equations both to the first and the second kind FIEs, and the conditions for converting to each kind of FIE are discussed. Furthermore, it is shown that under some simple circumstances, many mixed boundary value problems arising from either elastostatics or elastodynamics can be converted to the well-posed first kind FIE, which may be solved analytically or numerically. Main criteria for well-posedness of FIEs of the first kind in such problems are also presented. Noble’s original method is restricted to some limited conditions, which are extended here for both first and second kind FIEs to cover a wider range of dual integral equations encountered in engineering mixed boundary value problems.

scholarly journals Dynamic Behavior of an Embedded Foundation under Horizontal Vibration in a Poroelastic Half-Space

2019 ◽  
Vol 9 (4) ◽  
pp. 740 ◽  
Author(s):  
Yang Chen ◽  
Wen Zhao ◽  
Pengjiao Jia ◽  
Jianyong Han ◽  
Yongping Guan
Keyword(s):  
Integral Equations ◽  
Half Space ◽  
Mixed Boundary ◽  
Fredholm Integral Equations ◽  
Wave Forces ◽  
Dual Integral Equations ◽  
Horizontal Vibration ◽  
Embedded Foundation ◽  
Embedded Foundations ◽  
Caisson Foundations

More and more huge embedded foundations are used in large-span bridges, such as caisson foundations and anchorage open caisson foundations. Most of the embedded foundations are undergoing horizontal vibration forces, that is, wind and wave forces or other types of dynamic forces. The embedded foundations are regarded as rigid due to its high stiffness and small deformation during the forcing process. The performance of a rigid, massive, cylindrical foundation embedded in a poroelastic half-space is investigated by an analytical method developed in this paper. The mixed boundary problem is solved by reducing the dual integral equations to a pair of Fredholm integral equations of the second kind. The numerical results are compared with existing solutions in order to assess the accuracy of the presented method. To further demonstrate the applicability of this method, parametric studies are performed to evaluate the dynamic response of the embedded foundation under horizontal vibration. The horizontal dynamic impedance and response factor of the embedded foundation are examined based on different embedment ratio, foundation mass ratio, relative stiffness, and poroelastic material properties versus nondimensional frequency. The results of this study can be adapted to investigate the horizontal vibration responses of a foundation embedded in poroelastic half-space.

Antiplane Crack Problem in Functionally Graded Piezoelectric Materials

2002 ◽  
Vol 69 (4) ◽  
pp. 481-488 ◽  
Author(s):  
Chunyu Li ◽  
G. J. Weng
Keyword(s):  
Integral Equations ◽  
Piezoelectric Material ◽  
Electric Fields ◽  
Crack Problem ◽  
Functionally Graded ◽  
Fredholm Integral Equations ◽  
Dual Integral Equations ◽  
Functionally Graded Piezoelectric Material ◽  
Functionally Graded Piezoelectric Materials ◽  
Functionally Graded Piezoelectric

In this paper the problem of a finite crack in a strip of functionally graded piezoelectric material (FGPM) is studied. It is assumed that the elastic stiffness, piezoelectric constant, and dielectric permitivity of the FGPM vary continuously along the thickness of the strip, and that the strip is under an antiplane mechanical loading and in-plane electric loading. By using the Fourier transform, the problem is first reduced to two pairs of dual integral equations and then into Fredholm integral equations of the second kind. The near-tip singular stress and electric fields are obtained from the asymptotic expansion of the stresses and electric fields around the crack tip. It is found that the singular stresses and electric displacements at the tip of the crack in the functionally graded piezoelectric material carry the same forms as those in a homogeneous piezoelectric material but that the magnitudes of the intensity factors are dependent upon the gradient of the FGPM properties. The investigation on the influences of the FGPM graded properties shows that an increase in the gradient of the material properties can reduce the magnitude of the stress intensity factor.

Dynamic Response of Circular Plates Resting on Viscoelastic Half Space

1978 ◽  
Vol 45 (2) ◽  
pp. 379-384 ◽  
Author(s):  
Y. J. Lin
Keyword(s):  
Integral Equations ◽  
Half Space ◽  
Mixed Boundary ◽  
Dynamic Responses ◽  
Thin Plates ◽  
Fredholm Integral Equations ◽  
Circular Plates ◽  
Dual Integral Equations ◽  
Impedance Function ◽  
Vertical Excitation

Dynamic responses of circular thin plates resting on viscoelastic half space subject to harmonic vertical and rocking excitations are studied. The analysis is based on the assumption that the contact between the plate and the surface of the half space is frictionless. This dynamic mixed boundary-value problem leads to sets of dual integral equations which are reduced to Fredholm integral equations of the second kind and solved by numerical procedures. The numerical results show that the rocking impedance function is independent of the plate flexibility, but the vertical excitation is not.

On the solutions of a class of dual integral equations occurring in diffraction problems

1990 ◽  
Vol 429 (1877) ◽  
pp. 399-427 ◽  
Keyword(s):  
Integral Equations ◽  
Electromagnetic Waves ◽  
Near Field ◽  
Standard Form ◽  
Crack Problem ◽  
Sufficient Conditions ◽  
Bench Mark ◽  
Angle Of Incidence ◽  
Dual Integral Equations ◽  
High Frequencies

It is shown that there exists a category of two-dimensional diffraction problems, which can be put into a ‘standard form‘ of dual integral equations. These diffraction problems include: diffraction of electromagnetic waves by a finite strip, a finite slit, the diffraction of scalar or vector elastic waves by a rigid strip or crack, etc. A general method for solving such dual integral equations is given by the artifice of constructing a set of functions of compact support biorthogonal to another given set of functions. The sufficient conditions for a given dual integral equations to be solvable in this manner are also determined. Hence, the method forms a complement to the Weiner-Hopf method. To illustrate the method solutions are obtained for a bench-mark problem : the diffraction of light by a finite perfectly conducting strip (or equivalently the diffraction of SH waves by a crack). Comparison with results obtained by others for low, intermediate and high frequencies show the utility and accuracy of the method for the entire range of frequencies. Both the near field and the far field are obtained, the latter is shown to correspond to the Fraunhoffer diffraction pattern for high frequency. It is also shown that for the equivalent crack problem the stress intensity factor (SIF) fluctuates rapidly with changes in the angle of incidence for high frequencies, thus making the SIF especially sensitive to angle of incidence at high frequencies.

scholarly journals The solution of some integral equations

1987 ◽  
Vol 29 (2) ◽  
pp. 203-215
Author(s):  
John F. Ahner ◽  
John S. Lowndes
Keyword(s):  
Wave Equation ◽  
Boundary Value Problems ◽  
Integral Equations ◽  
Laplace Equation ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Fredholm Integral Equations ◽  
Mixed Boundary Value Problems ◽  
Triple Integral Equations

AbstractAlgorithms are developed by means of which certain connected pairs of Fredholm integral equations of the first and second kinds can be converted into Fredholm integral equations of the second kind. The methods are then used to obtain the solutions of two different sets of triple integral equations tht occur in mixed boundary value problems involving Laplace' equation and the wave equation respectively.

Dual integral equations associated with Fourier transforms

1986 ◽  
Vol 38 (2) ◽  
pp. 165-171
Author(s):  
Nguyen Van Ngok ◽  
G. Ya. Popov
Keyword(s):  
Integral Equations ◽  
Fourier Transforms ◽  
Dual Integral Equations

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.

Three-Dimensional Analysis of Cracking in a Multilayered Composite

1995 ◽  
Vol 62 (2) ◽  
pp. 273-281 ◽  
Author(s):  
C. H. Kuo ◽  
L. M. Keer
Keyword(s):  
Integral Equations ◽  
Stiffness Matrix ◽  
Crack Opening ◽  
Crack Problem ◽  
Three Dimensional ◽  
Single Layer ◽  
Singular Integral Equations ◽  
Material System ◽  
Multilayered Composite ◽  
Penny Shaped Crack

The three-dimensional problem of a multilayered composite containing an arbitrarily oriented crack is considered in this paper. The crack problem is analyzed by the equivalent body force method, which reduces the problem to a set of singular integral equations. To compute the kernels of the integral equations, the stiffness matrix for the layered medium is formulated in the Hankel transformed domain. The transformed components of the Green’s functions and derivatives are determined by solving the stiffness matrix equations, and the kernels are evaluated by performing the inverse Hankel transform. The crack-opening displacements and the three modes of the stress intensity factor at the crack front are obtained by numerically solving the integral equations. Examples are given for a penny-shaped crack in a bimaterial and a three-material system, and for a semicircular crack in a single layer adhered to an elastic half-space.

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