Simultaneous Singular Integral Equations Research Articles

Singular integral method based closed form solution for moving EMPS model in semipermeable two‐equal‐collinear cracked MEE material

AbstractThis paper presents the analytical closed‐form solutions of moving two equal collinear semipermeable cracks in magneto‐electro‐elastic material considering the strip electric‐magnetic polarization saturation (EMPS) model. To simulate the dynamic/moving cracks problem, two equal collinear Yoffe type cracks moving with a constant subsonic velocity along the plane of the cracks is considered. The problem is solved by considering semipermeable crack face conditions and under the in‐plane electro‐magnetic‐mechanical loading. Applying distributed dislocation technique and symmetry of the collinear cracks, the problem is reduced into simultaneous singular integral equations which have been solved in closed form expressions using finite Hilbert approach. The explicit expressions for dynamic distributed dislocation densities, crack opening displacement (COD), crack opening potential (COP), crack opening induction (COI), and local stress intensity factors (LSIFs) are evaluated for both the cases of the EMPS model (magnetic zone is smaller or greater than electric zone). For evaluation of inner and outer electric and magnetic zone lengths (MZLs), an iterative approach is applied to solve the developed two non‐linear equations. Also, to implement the semipermeable crack‐face conditions, a bisection iterative method has been applied. Based on the developed closed form solutions, numerical studies are presented for standard fracture parameters with respect to crack propagation velocity, electrical loading and inter‐crack space distance. This study shows that electric displacement and magnetic induction defined over the crack‐surface for semipermeable conditions decrease whereas the numerical values of saturated zone lengths, COD, COP, COI, and LSIFs increase with increasing crack propagation velocity. Moreover, for both static and dynamic cracks problems, the significant crack‐interaction effects have been observed on all the studied fracture parameters except the crack‐face conditions.

Analysis of a mode-I crack in a one-dimensional orthorhombic quasicrystal strip

A mode-I crack in a one-dimensional (1D) orthorhombic quasicrystal (QC) strip under in-plane phonon and phason stress loading is considered. Fourier transforms are applied to reduce the mixed boundary value problem of the mode-I crack to solving a system of simultaneous singular integral equations. Asymptotic expressions of the phonon and phason stresses and displacement fields near the crack tips have been obtained in an explicit form. The crack-tip singularities of the mode-I crack have been investigated and the intensity factors of the stresses in the phonon and phason fields are derived explicitly. The stress intensity factors (SIFs) and the hoop stress intensity factors (HSIFs) have been determined to investigate the effect of the geometric size and the crack kinking phenomenon. The effect of the thickness ratio of the cracked strip on the SIFs and energy release rates has been investigated. When the thickness of the cracked strip becomes infinite large, the results obtained for the crack problem can be reduced to the analytic solution for a mode-I crack in an infinite 1D orthorhombic QC media.

Analysis of an anti-plane crack in a one-dimensional orthorhombic quasicrystal strip

In this paper, a central crack in a one-dimensional (1D) orthorhombic quasicrystal strip under anti-plane loadings has been studied. Fourier transform is applied to reduce the mixed boundary value problem of the crack to solving a system of simultaneous singular integral equations. Asymptotic expressions of the phonon and the phason fields near the crack tips have been obtained. The crack-tip singularities of the anti-plane crack have been investigated, and the intensity factors of the stresses in the phonon and the phason fields are derived explicitly. The stress intensity factors (SIFs) and the hoop stress intensity factors (HSIFs) have been determined to investigate the effect of the geometric size and the crack skewing phenomenon. This crack skewing phenomenon is different from the mode-III crack problems in isotropic elastic materials and transversely isotropic elastic materials where the crack propagates along the original crack plane.

Analysis on the propagation of Griffith crack in a magnetoelastic self-reinforced strip subjected to moving punch of constant load

The present study analysed the characteristics of a moving Griffith crack in a self-reinforced strip of finite thickness and infinite extent with the moving parallel punches of constant load acting on the boundaries of the strip at both sides due to the propagation of magnetoelastic plane waves under mechanical point loading. With the aid of integral transform technique, problem has been reduced to the pair of simultaneous singular integral equations with Cauchy-type singularities. The point load at the edge of the moving crack is considered in terms of Dirac delta function, and the expression of stress intensity factor (SIF) at the crack tip with constant point loading has been established in closed form by using the well-known properties of Hilbert transformation. Moreover, some of the special cases have been deduced from the obtained expression of SIF for the force of constant intensity, without punch pressure and anisotropy in the considered strip. Numerical computations and graphical demonstrations have been carried out to observe the profound effect of magnetoelastic coupling parameter, punch pressure, crack length, distinct positions of point load and the velocity of crack associated with magnetoelastic plane wave on SIF for self-reinforced materials and isotropic material strip. A comparative study of SIF at the tip of moving crack has been made for the self-reinforced and isotropic materials to highlight some of the important peculiarities of the problem.

Application of distributed dislocation method to curved crack moving near a press-fitted inclusion in a two-dimensional infinite plate

This paper describes the application of the distributed dislocation method to the problem of a curved crack moving near a press-fitted inclusion in a two-dimensional infinite plate which is subjected to a tensile loading. The formulation is based upon the superposition of the following three fundamental problems: a continuous distribution of edge dislocations along the crack line in an infinite plate with an inclusion, residual stress generated by the press-fitted inclusion without the crack, and stress field in the infinite elastic plate containing the circular inclusion, but without the crack, which is subjected to a uniform tensile loading. The superposition leads to simultaneous singular integral equations, which are introduced by relating the traction-free condition along the curved crack line to the dislocation density functions as unknown functions in the integral equations. The stress intensity factors can be derived directly from those dislocation density functions. In this study, the crack-propagation program, which is considered that the crack tip is automatically moved in a direction satisfying the restriction that the stress intensity factor KII is zero, was developed. The developed program is applied to the elastic problem including the press-fitted inclusion and crack, and the influence of the initial crack locations and residual stress generated by inserting the inclusion into the hole on the geometry of the crack path is discussed.

A study of the double edge cracks and a central crack in an orthotropic strip under the tensile loads

In this article, An elastostatic problem with two symmetrically located edge cracks and a centre crack in an orthotropic elastic material under normal loadings is considered. Two harmonic functions are considered which are expressed in terms of displacement of stress and component of axes. The problem is reduced into a pair of simultaneous singular integral equations of the first kind with Cauchy-type singularities, which are solved using Chebyshev polynomials. The normalized stress intensity factors are found against ratios of cracks’ length are computed for different particular cases for the orthotropic material Steel-Mylar and the results are depicted through graphs.

Biaxial Loading of a Plate Containing a Hole and Two Co-Axial Through Cracks

Abstract The paper presents the solution linear elasticity problem for an isotropic plate weakened by a hole and two co-axial cracks. The plate is exerted by uniform traction at infinity. The corresponding 2D problem is solved by the method of Kolosova-Muskhelishvili complex potentials. The method implies reduction of the problem to simultaneous singular integral equations (SIE) for the functions defined the region of the cracks and hole. For particular case the solution the SIE is obtained analytically in a closed form. A thorough analysis of the stress intensity factors (SIF) is carried out for various cases of the hole shape: penny-shaped, elliptical and rectangular.

Interaction of Three Interfacial Cracks between an Orthotropic Half-Plane Bonded to a Dissimilar Orthotropic Layer with Punch

Abstract This article deals with the interactions between a central crack and a pair of outer cracks situated at the interface of an orthotropic elastic half-plane bonded to a dissimilar orthotropic layer with a punch. The problem is reduced to the solution of three simultaneous singular integral equations that are finally solved using Jacobi polynomials. The phenomena of crack shielding and crack amplification have been depicted through graphs for different particular cases.

Interaction between interfacial and sub‐interfacial cracks in a composite media – Revisited

The plane strain problem of determining stress intensity factors and stress magnification factors for an interfacial Griffith crack situated at the interface of two bonded dissimilar orthotropic media having sub‐interfacial Griffith crack is considered. The problem is reduced to the solution of two pair of simultaneous singular integral equations which are finally been solved by using Jacobi polynomials. The propagation of interfacial crack through amplification and shielding factors are shown graphically for different particular cases.

Interaction of three interfacial Griffith cracks between bonded dissimilar orthotropic half planes

The paper deals with the interaction of a pair of outer cracks on a central crack situated at the interface of two dissimilar orthotropic half-planes. The mixed boundary value problem is reduced to solving a pair of simultaneous singular integral equations which have finally been solved numerically by using Jacobi polynomials. The analytical expressions for stress intensity factors at the central crack tip and the expression of the strain energy release rate have been derived for general loading. Numerical values of the interaction effects of the outer cracks on the central crack have been calculated through stress magnification factors. It is seen that the interaction effects are either shielding or amplification depending on the size of the outer cracks and their spacing from the central crack.

Moving Griffith crack in an orthotropic strip with punches at boundary faces

Integral transform technique is employed to solve the elastodynamic problem of steady‐state propagation of a Griffith crack centrally situated along the midplane of orthotropic strip of finite thickness 2h and subjected to point loading with centrally situated moving punches under constant pressure along the boundaries of the layer. The problem is reduced to the solution of a pair of simultaneous singular integral equations with Cauchy‐type singularities which have finally been solved through the finite Hilbert transform technique. For large h, analytical expression for the stress intensity factor at the crack tip is obtained. Graphical plots of the numerical results are also presented.

Interference Effects of Multiple Surface Cracks in a Layered Material due to Hot Rolling Contact

This paper deals with the mutual interference of multiple two-dimensional surface cracks in a surface coating layered material under thermal stresses due to rolling contact with heat input. Contact loading is simulated as a contact pressure load with both the normal and shear components having parabolic distribution. In the present crack analysis, the surface cracks are replaced by the distributed edge dislocations, and the crack face friction is not considered. The problem is reduced to simultaneous singular integral equations for dislocation densities. The integral equations can be solved numerically by considering the nature of the singularities at the crack tips. The numerical results of the stress intensity factors showing the effects of the mutual interference of a pair of surface cracks are given for some tribological material coatings on a steel substrate. The effects of the frictional coefficient, the heat input strength, the crack length and the coating thickness upon the magnitude of the stress intensity factors and it's mutual interference are considered numerically.

Singularity of dynamic stress fields around an interface crack between viscoelastic bodies

The singular nature of the dynamic stress fields around an interface crack located between two dissimilar isotropic linearly viscoelastic bodies is studied. A harmonic load is imposed on the surfaces of the interface crack. The dynamic stress fields around the crack are obtained by solving a set of simultaneous singular integral equations in terms of the normal and tangent crack dislocation densities. The singularity of the dynamic stress fields near the crack tips is embodied in the fundamental solutions of the singular integral equations. The investigation of the fundamental solutions indicates that the singularity and oscillation indices of the stress fields are both dependent upon the material constants and the frequency of the harmonic load. This observation is different from the well-known −1/2 oscillating singularity for elastic bi-materials. The explanation for the differences between viscoelastic and elastic bi-materials can be given by the additional viscosity mismatch in the case of viscoelastic bi-materials. As an example, the standard linear solid model of a viscoelastic material is used. The effects of the frequency and the material constants (short-term modulus, long-term modulus and relaxation time) on the singularity and the oscillation indices are studied numerically.

On elastodynamical problem of interfacial Griffith cracks in composite media

The plane strain problem of determining stress intensity factors for two equal and parallel moving interfacial Griffith cracks in composite media consisting of an orthotropic layer bonded two dissimilar orthotropic half planes is considered. The problem is reduced to solution of two pair of simultaneous singular integral equations containing Cauchy kernels. Expressions for stress intensity factor are obtained for the case of a general loading distribution.

Interaction of general plane P wave and cylindrical inclusion partially debonded from its viscoelastic matrix

The interaction of a general plane P wave and an elastic cylindrical inclusion of infinite length partially debonded from its surrounding viscoelastic matrix of infinite extension is investigated. The debonded region is modeled as an arc-shaped interface crack between inclusion and matrix with non-contacting faces. With wave functions expansion and singular integral equation technique, the interaction problem is reduced to a set of simultaneous singular integral equations of crack dislocation density function. By analysis of the fundamental solution of the singular integral equation, it is found that dynamic stress field at the crack tip is oscillatory singular, which is related to the frequency of incident wave. The singular integral equations are solved numerically, and the crack open displacement and dynamic stress intensity factor are evaluated for various incident angles and frequencies.

Stress intensity factor around a Griffith carck in an orthotropic punched layer

A study is made of the plane strain problem to determine the stress intensity factor for a Griffith crack in an orthotropic elastic layer with punches located along the boundaries of the layer. The problem is reduced to the solution of a pair of simultaneous singular integral equations with the Cauchy type singularities. These integral equations are then solved approximately by using the Chebyshev polynomials. An expression for the stress intensity factor (SIF) for the case of a general loading is obtained. Numerical results for the SIF are presented graphically for some particular cases of interest.

On a Moving Griffith Crack at the Interface of two Bonded Dissimilar Orthotropic Half-Planes

The plane strain problem of determining strain energy release rate and crack energy for an interfacial moving Griffith crack situated at the interface of two bonded dissimilar orthotropic half-planes is considered. The problem is reduced to solution of a pair of simultaneous singular integral equations which are finally solved by using Jacobi polynomials. Expressions for strain energy release rate and crack energy are obtained for some particular cases and the results are presented graphically.

Study of a static cruciform crack problem in an orthotropic elastic plane

This paper deals with the plane strain problem of determining stress intensity factors and crack energies of a static cruciform crack in an infinite orthotropic elastic medium. The problem is reduced to the solution of two simultaneous singular integral equations with Cauchy-type singularities. Expressions for stress intensity factors (SIF) and crack energies for the case general loading distribution are obtained. Numerical results for some special cases are presented for the Boron-Epoxy composite material.

Interaction of elastic waves by an interfacial Griffith crack

A study is made of the problem of finding the distribution of stress field in the vicinity of a Griffith crack located at the interface of two dissimilar orthotropic half-planes. The crack is opened by the interaction of a plane harmonic elastic wave incident normally on the crack. The Fredholm integral equation is derived for the determination of diffraction wave field. The problem is then reduced to a pair of simultaneous singular integral equations which are solved by using the Jacobi polynomials. An expression for the stress intensity factors is obtained in the end.

Stress intensity factors for a moving interfacial Griffith crack in composite media

This paper deals with two plane strain problems for determining stress intensity factors (SIF) involved in a moving interfacial Griffith crack in composite orthotropic media. The problems are reduced to solution of a pair of simultaneous singular integral equations with Cauchy type singularities. Expressions for SIF for general and particular cases of the problems are obtained. Several special and limiting cases of these problems are discussed. Numerical results for some particular cases are also presented by graphical representations.