Penny-shaped crack in a long circular cylinder subjected to a uniform shearing stress

This paper contains an analysis of the stress distribution in a long circular cylinder of elastic material containing a penny-shaped crack when it is deformed by the application of a uniform shearing stress. The crack with its center on the axis of the cylinder lies on the plane perpendicular to that axis, and the cylindrical surface is stress-free. By making a suitable representation of the stress function for the problem, the problem is reduced to the solution of a pair of Fredholm integral equations of second kind. These are solved numerically, and the percentage increase in the stress intensity factor due to the effect of the finite radius of the cylinder is presented in graphical form for various proximity ratios.     

Torsion of a long circular cylinder having a spherical cavity with a peripheral edge crack

The singular stress problem of a peripheral edge crack around a spherical cavity in a long circular cylinder under torsion is investigated. The problem is solved by using integral transforms and is reduced to the solution of two integral equations. The solution of these equations is obtained numerically by the method due to Erdogan, Gupta, and Cook, and the stress intensity factors, and crack opening displacements are displayed graphically.     

Tension of a long circular cylinder having a spherical cavity with a peripheral edge crack

The singular stress problem of a peripheral edge crack around a spherical cavity in a long circular cylinder under tension is investigated. The problem is solved by using integral transforms and is reduced to the solution of three integral equations. The solution of these equations is obtained numerically by the method due to Erdogan, Gupta, and Cook, and the stress intensity factors are displayed graphically.     

Axisymmetric circumferential internal crack problem of a thick-walled cylinder with inner and outer claddings

The elastostatic axisymmetric problem for a long thick-walled cylinder containing an axisymmetric circumferential internal crack with two claddings is considered. The claddings having different elastic properties than the hollow cylinder are assumed to be bonded to inner and outer wall of the hollow cylinder. The problem is formulated in terms of a singular integral equation of a well known type, the derivative of the crack surface displacement being the density function, using the standard transform technique. By using appropriate quadrature formulas, the integral equation is reduced to a system of linear algebraic equations. This system is solved numerically and the related stress-intensity factors are calculated for the cases of hollow cylinder with two claddings bonded to inner and outer wall of the cylinder, a cladding bonded to inner wall of the cylinder, a cladding bonded to outer wall of the cylinder and no cladding under axial tensile load. The influence of the geometrical configuration, the claddings and internal crack length on the stress-intensity factors is shown graphically.     

Interaction of a crack with a pair of thin, elastic inclusions

The influence of a pair of thin, elastic inclusions on the stress intensity at a crack tip is considered. The interaction problem is formulated as a set of coupled singular integral equations and the solution is obtained by the Gauss-Chebyshev numerical technique. The stress intensity is computed as a function of certain geometric and elastic parameters of the inclusions, and the toughening effect of these parameters is evaluated.     

Interfacial crack problem in layered soft ferromagnetic materials in uniform magnetic field

The magnetoelastic plane strain problem of an interfacial Griffith crack between two dissimilar soft ferromagnetic elastic materials subjected to a uniform magnetostatic field is considered within the framework of linear magnetoelasticity. By making use of the Fourier integral transform technique, the mixed boundary problem is then reduced to a pair of singular integral equations of the second kind. Solutions of the singular integral equations are obtained numerically by means of a Jacobi polynomial expansion method. Effects of the magnetic field, the combinations of the magnetic properties of materials and the geometric parameters on the magnetoelastic stress intensity factors in the vicinity of crack tip are shown graphically.     

Cylindrical interface crack in a hollow layered functionally graded cylinder under static torsion

A hollow functionally graded composite cylinder under static torsion, which consists of an inner and outer elastic circular tube with a cylindrical interface crack, is studied in this work. By utilizing Fourier integral transform method, the mixed boundary value problem is reduced to a Cauchy singular integral equation, from which the numerical results of the stress intensity factor are obtained by the Lobatto–Chebyshev quadrature technique. Numerical results demonstrate the coupled effects of geometrical, physical, and functionally graded parameters on the interfacial fracture behavior.     

Torsion problems for a cylinder with a rectangular hole and a rectangular cylinder with a crack

Using the method of singular integral equation and the crack-cutting technique, the rigorous solutions are obtained for a cylinder with a rectangular hole and a rectangular cylinder with a crack, which exactly satisfy the boundary conditions and the conditions at the corner points. After that the torsional rigidities and the stress intensity factors at the crack tip are determined. Next, for the doubly connected circular cylinder with a rectangular hole the expressions for the singular stresses around the concave corner points are derived and the generalized stress intensity factors are then defined. Since the crack-cutting technique is used in this paper, the solution of the matching rectangular cylinder is also obtained and its numerical results coincide with those in references. Thus the method proposed here is verified. The project supported by National Natural Science Foundation of China     

Stress intensity factors of the eccentric and edge cracked cylinders

Using the single crack solution and the regular solution of harmonic function,thetorsion problem of a cracked cylinder is reduced to solving a set of mixed-type integralequations which can be solved by combining the numerical method of singular integralequation with the boundary element method.Several numerical examples arecalculated and the stress intensity factors are obtained.     

The solution of integral equations with strongly singular kernels applied to the torsion of cracked circular cylinder

In this paper,the functions of warping displacement interruption defined on the crack lines are taken for the fundamental unknown functions.The torsion problem of cracked circular cylinder is reduced to solving a system of integral equations with strongly singular kernels.Using the numerical method of these equations,the torsional rigidities and the stress intensity factors are calculated to solve the torsion problem of circular cylinder with star-type and other different types of cracks.The numerical results are satisfactory.     

Diffraction of torsional elastic waves by a peripheral edge crack around a spherical cavity

This paper deals with the problem of finding the stress distribution in the neighborhood of a peripheral edge crack in a spherical cavity. The crack is excited by a torsional standing wave.The problem is solved by using integral transforms and is reduced to the solution of a singular integral equation. The solution of this equation is obtained numerically by the method due to Erdogan, Gupta, and Cook, and the stress intensity factors are displayed graphically.     

用样条边界元与奇异积分方程法联合求解多裂纹柱体的扭转

通过间解的分离，本文将径向多裂纹柱体的导曲函两个调和函数表示，使问题归为解一组混混合型积分方程。针对方程的特点，本文联合使用三次样条边界法与奇异积分方程的数值方法对所得方程建立了数值法，并对裂纹相交情形作了特殊处理。最后对工程中感兴趣的一些典型的多裂纹柱体的扭转作了例题计算，结果表明，本文方法具有收敛快，精度高的特点。     

Transverse cracking of orthotropic composite laminates: a fracture mechanics approach

This research is concerned with the fracture mechanics of a laminated composite medium, which contains a central layer sandwiched by two outer layers. There is a periodic array of cracks in the central layer along the central axis of the medium. Fourier transform is used to reduce the problem to the solution of a system of dual integral equations, which are solved by the singular integral equation technique. Rigorous fracture mechanics analysis, which exactly satisfies all boundary conditions of the problem, is conducted. Numerical solutions for the crack tip field and the stress in the medium are obtained for various values such as crack length, crack spacing and layer thickness. Results are also given for the reduction of the equivalent Young’s modulus of the laminate due to multiple cracking. The cases of axial extension and residual temperature change of the composite medium are accounted for.     

弹性功能梯度材料板条中周期裂纹的反平面问题

讨论了弹性功能梯度材料板条中裂纹的反平面问题. 用Fourier 变换方法得到了一个基本解. 这个基本解表示了实轴上一点作用有点位错时引起的影响. 利 用此基本解可得单裂纹和周期裂纹问题的奇异积分方程. 在周期裂纹求解时, 远处裂纹对于中央裂纹的影响作了有效的近似处理. 最后, 给出了数值结果, 它表示了材料性质对于裂纹端应力强度因子的影响.     

Compensation of stress intensity factors in hollow cylinders containing several cracks under torsion by electro-elastic coating

In this article, a formulation for a hollow cylinder reinforced with an electroelastic layer is investigated. The hollow cylinder and its electro-elastic coating are under the Saint-Venant torsional loading. First, the solution to the problem containing a Volterra-type screw dislocation is obtained by using the Fourier transform. The problem is then reduced to a set of Cauchy singular integral equations by the distributed dislocation method. Finally, several examples are presented to show the effect of the electro-elastic coating on the reduction of the stress intensity factors at the crack tips.     

Stress intensity factors around a cylindrical crack in an interfacial zone in composite materials

Axisymmetric stresses around a cylindrical crack in an interfacial cylindrical layer between an infinite elastic medium with a cylindrical cavity and a circular elastic cylinder made of another material have been determined. The material constants of the layer vary continuously from those of the infinite medium to those of the cylinder. Tension surrounding the cylinder and perpendicular to the axis of the cylinder is applied to the composite materials. To solve this problem, the interfacial layer is divided into several layers with different material properties. The boundary conditions are reduced to dual integral equations. The differences in the crack faces are expanded in a series so as to satisfy the conditions outside the crack. The unknown coefficients in the series are solved using the conditions inside the crack. Numerical calculations are performed for several thicknesses of the interfacial layer. Using these numerical results, the stress intensity factors are evaluated for infinitesimal thickness of the layer.     

An embedded crack in a functionally graded orthotropic coating bonded to a homogeneous substrate under a frictional Hertzian contact

In this paper, we consider the elasto-static problem of an embedded crack in a graded orthotropic coating bonded to a homogeneous substrate subject to statically applied normal and tangential surface loading. The crack direction is parallel to the free surface. The coating is graded in the thickness direction and is orthogonal to the crack direction. This coating is modelled as a non-homogeneous medium with an orthotropic stress–strain law. The equivalent crack surface stresses are first obtained and substituted in the plane elasticity equations. Using integral transforms, the governing equations are converted into singular integral equations which are solved numerically to yield the displacement field as well as the crack-tip stress intensity factors. This study presents a complete theoretical formulation for the problem in the static case. A numerical predictive capability for solving the singular integral equations and computing the crack-tip stress intensity factors is proposed. Since the loading is compressive, a previously developed crack-closure algorithm is applied to avoid interpenetration of the crack faces. The main objective of the paper is to investigate the effects of the material orthotropy and non-homogeneity of the graded coating on the crack-tip stress intensity factors, with and without using the crack-closure algorithm, for the purpose of gaining better understanding on the behavior and design of graded coatings.     

Stress-intensity factors for 3-D dynamic loading of a cracked half-space

A half-space containing a surface-breaking crack of uniform depth is subjected to three-dimensional dynamic loading. The elastodynamic stress-analysis problem has been decomposed into two problems, which are symmetric and antisymmetric, respectively, relative to the plane of the crack. The formulation of each problem has been reduced to a system of singular integral equations of the first kind. The symmetric problem is governed by a single integral equation for the opening-mode dislocation density. A pair of coupled integral equations for the two sliding-mode dislocation densities govern the antisymmetric problem. The systems of integral equations are solved numerically. The stress-intensity factors are obtained directly from the dislocation densities. The formulation is valid for arbitrary 3-D loading of the half-space. As an example, an applied stress field corresponding to an incident Rayleigh surface wave has been considered. The dependence of the stress-intensity factors on the frequency, and on the angle of incidence, is displayed in a set of figures.     

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.     

Multiple cracking of fiber/matrix composites—Analysis of normal extension

This paper is concerned with the fracture of a fiber embedded in a matrix of finite radius. There is a periodic array of cracks in the fiber along the central axis of the medium. The paper accounts for the cases of axial extension and residual temperature change of the composite medium. Fourier and Hankel transforms are used to reduce the problem to the solution of a system of dual integral equations, which are solved by the singular integral equation technique. Rigorous fracture mechanics analysis, which exactly satisfies all boundary conditions of the problem, is conducted. Numerical solutions for the crack tip field and the stress in the fiber are obtained for various values such as crack radius, crack spacing and fiber volume fraction.