Axisymmetric crack terminating at the interface of transversely isotropic dissimilar media

In this paper, the axisymmetric elasticity problem of an infinitely long transversely isotropic solid cylinder imbedded in a transversely isotropic medium is considered. The cylinder contains an annular or a penny shaped crack subjected to uniform pressure on its surfaces. It is assumed that the cylinder is perfectly bonded to the medium. A singular integral equation of the first kind (whose unknown is the derivative of crack surface displacement) is derived by using Fourier and Hankel transforms. By performing an asymptotic analysis of the Fredholm kernel, the generalized Cauchy kernel associated with the case of `crack terminating at the interface' is derived. The stress singularity associated with this case is obtained. The singular integral equation is solved numerically for sample cases. Stress intensity factors are given for various crack geometries (internal annular and penny-shaped cracks, annular cracks and penny-shaped cracks terminating at the interface) for sample material pairs.     

含任意方向裂纹功能梯度材料的应力分析研究

功能梯度材料是在航空航天领域的需求背景下发展起来的,但由于生产技术及工作环境等方面的原因,功能梯度材料内部常常产生各种形式的裂纹并最终导致材料破坏,因此研究含任意方向裂纹功能梯度材料的断裂问题具有重要意义。以含有任意方向裂纹的功能梯度材料为对象,运用积分变换方法,给出了相应材料平面问题的位移场的形式解。通过引入辅助函数并利用相关条件,可将问题转化为求解一组带有Cauchy核的奇异积分方程,继而采用Lobatto-Chebyshev方法对奇异积分方程进行数值求解。最后分析了裂纹方向、材料非均匀指数、载荷条件对混合型应力强度因子的影响。      

Temperature distribution and heat flow around a crack of arbitrary orientation in a functionally graded medium

This paper investigates the heat transfer problem of an infinite functionally graded medium containing an arbitrarily oriented crack under uniform remote heat flux. In the mathematical treatment the crack is approximated as a perfectly insulating cut. By using Fourier transformation, the mixed boundary value problem is reduced to a Cauchy-type singular integral equation for an unknown density function. The singular integral equation is then solved by representing the density function with a Chebyshev polynomial-based series and solving the resulting linear equation using a collocation technique. The temperature field in the vicinity of the crack and the crack-tip heat flux intensity factor are presented to quantify the effect of crack orientation and grading inhomogeneity on the heat flow around the crack.     

Axisymmetric finite cylinder with one end clamped and the other under uniform tension containing a penny-shaped crack

This study considers the axisymmetric analysis of a finite cylinder containing a penny-shaped transverse crack. Material of the cylinder is assumed to be linearly elastic and isotropic. One end of the cylinder is bonded to a fixed support while the other end is subjected to uniform axial tension. Solution is obtained by superposing the solutions for an infinite cylinder loaded at infinity and an infinite cylinder containing four cracks and a rigid inclusion loaded along the cracks and the inclusion. When the radius of the inclusion approaches the radius of the cylinder, its mid-plane becomes fixed and when the radius of the distant cracks approach the radius of the cylinder, the ends become cut and subject to uniform tensile loads. General expressions for the perturbation problem are obtained by solving Navier equations with Fourier and Hankel transforms. Formulation of the problem is reduced to a system of five singular integral equations. By using Gauss-Lobatto and Gauss-Jacobi integration formulas, these five singular integral equations are converted to a system of linear algebraic equations which is solved numerically. Stress distributions along the rigid support, stress intensity factors at the edges of the rigid support and the crack are calculated.     

Elastic equilibrium of an interface crack between rigid stamp and orthotropic strip

Rupture of the interface between an absolutely rigid stamp and an orthotropic infinite strip is investigated. A plane elasticity problem for an interface crack formally leads to oscillatory singularities at the crack tip. In order to overcome this nonphysical solution, a model of an interface crack with frictionless contact zones near the crack tips and the corners of the stamp is developed. By using the method of integral Fourier transforms the problem is reduced to a system of three singular integral equations. The system is solved by the method of collocations with the points of collocation chosen at zeros of the Chebyshev polynomials. The stress intensity factors at the crack tips and the stamp corner points are evaluated.     

Two coplanar Griffith cracks in an infinite elastic layer under torsion

Summary We consider the problem of determining the stress distribution in an infinitely long isotropic homogeneous elastic layer containing two coplanar Griffith cracks which are opened by internal shear stress acting along the lengths of the cracks. The faces of the layer are assumed to be stress free. The cracks are located in the middle plane of the layer parallel to its faces. By using Fourier transforms, we reduce the problem to the solution of a set of triple integral equations with a cosine kernel and a weight function. These equations are solved exactly by using finite Hilbert transform techniques. Finally we derive the closed form expressions for the stress intensity factors and the crack energy. Solutions to the following problems are derived as particular cases: (i) a single crack in an infinite layer under torsion, (ii) two coplanar cracks in an infinite space under torsion, (iii) a single crack in an infinite space under torsion.     

Edge cracks in a transversely isotropic hollow cylinder

The analytical solution for the linear elastic, axisymmetric problem of inner and outer edge cracks in a transversely isotropic infinitely long hollow cylinder is considered. The z = 0 plane on which the crack lies is a plane of symmetry. The loading is uniform crack surface pressure. The mixed boundary value problem is reduced to a singular integral equation where the unknown is the derivative of the crack surface displacement. An asymptotic analysis is done to derive the generalized Cauchy kernel associated with edge cracks. It is shown that the stress intensity factor is a function of three material parameters. The singular integral equation is solved numerically. Stress intensity factors are presented for various values of material and geometric parameters.     

The general problem for an arbitrarily oriented crack in a FGM layer1

In this paper, the plane elasticity problem for an unconstrained FGM layer containing an arbitrarily oriented crack is considered. It is assumed that the elastic properties of the material are exponential functions of the thickness coordinate. The problem is formulated in terms of a system of Cauchy-type singular integral equations, which can be solved numerically. The stress intensity factors at the crack tips are computed for mechanical loads. A complete parametric study, by varying both the geometric and material parameters is conducted.     

Incremental elastic compliance and electric resistance of a cylinder with partial loss in the cross-sectional area

Motivated by material science applications, the paper focuses on quantitative characterization and comparison of two microstructural elements typical for lamellar materials - crack and contacting area - in the context of their effect on macroscopic elastic and conductive (thermal or electrical) properties of a body of finite size. The problem is solved in axisymmetric formulation - axial load or axial heat flux is applied to a circular cylinder containing a centered crack, either internal or external. The latter case corresponds to the welding of two halves of the cylinder at the center. The changes in the elastic and conductive properties of the cylinder due to these types of cracks are obtained in explicit analytical form. It is shown that the contributions of internal and external cracks into elastic and conductive properties are similar if the relative loss in the cross-sectional between two parts of the cylinder is up to 70% for elasticity problem and up to 85% for conductivity problem. We also show that the changes in elastic compliance and conductive properties generated by both microstructural elements are interrelated by cross-property connection identical to one obtained for an unbounded material.     

Transient thermal stress intensity factors for an internal longitudinal semi-elliptical crack in a thick-walled cylinder

In this paper, the stress intensity factors are derived for an internal semi-elliptical crack in a thick-walled cylinder subjected to transient thermal stresses. First, the problem of transient thermal stresses in a thick-walled cylinder is solved analytically. Thermal and mechanical boundary conditions are assumed to act on the inner and outer surfaces of the cylinder. The quasi-static solution of the thermoelasticity problem is derived analytically using the finite Hankel transform and then, the stress intensity factors are extracted for the deepest point and the surface points of the semi-elliptical crack using the weight function method. The results show to be in accordance with those cited in the literature in the special case of steady-state problem. Using the closed-form relations extracted for the transient thermal stress intensity factors, some conclusive results are drawn.     

The torsion problem for a circular cylinder with radial edge cracks

Summary A finite Mellin transform technique reduces the torsion problem for a circular cylinder with radial edge cracks to that of solving some integral equations. Expressions are found for the stress intensity factors and crack formation energy. Three particular cases are considered in detail and numerical results given.     

A boundary integral approach for plane analysis of electrically semi-permeable planar cracks in a piezoelectric solid

A plane electro-elastostatic problem involving arbitrarily located planar stress free cracks which are electrically semi-permeable is considered. Through the use of the numerical Green's function for impermeable cracks, the problem is formulated in terms of boundary integral equations which are solved numerically by a boundary element procedure together with a predictor–corrector method. The crack tip stress and electric displacement intensity factors can be easily computed once the boundary integral equations are properly solved.     

Parametric study of oblique edge cracks under cyclic contact loading

The problem of a two-dimensional elastic body, carrying an inclined edge crack and loaded by a cylinder rolling on the surface, is solved by the weight function method. The load induced by the cylinder on the cracked body was represented by the Hertzian pressure distribution, and the nominal stress distribution in the uncracked body was numerically evaluated by the superposition principle. The crack opening displacement components were obtained by an analytical Green's function. The partial crack closure was considered and the influence of the mutual forces between the crack faces included in the analysis, by which the effective stress intensity factors K _I and K _II could be evaluated. By considering different friction conditions between the crack surfaces and several crack inclinations, the evolutions of the effective K _I and K _II during typical loading cycles were analysed.     

The effect of the interphase on crack-inclusion interactions

The problem of a radial or circumferential matrix crack interacting with a circular inclusion surrounded by an interphase region is investigated. The problem is formulated using Kolosov-Muskhelishvili complex potentials where the crack is modeled as a distribution of dislocations. The complex potentials for a dislocation interacting with a circular inclusion with an interphase are first rederived and then used in the crack formulation. The corresponding Cauchy singular integral equations are then solved using the Lobatto–Chebyshev quadrature technique. After comparing the current solution with previously published results, the influence of the interphase stiffness and thickness on a radial or circumferential matrix crack is studied for a glass fiber-epoxy composite. From this study it was found that compliant interphases increase the Mode I stress intensity factors for radial cracks while stiff interphases shield these cracks from the inclusion relative to the no-interphase cases. Additionally, the compliant interphases were found to be more affected by the thickness of the interphase. Results for the circumferential cracks were not as straightforward. Compliant interphases decreased the Mode II stress intensity factors but, depending on the interphase thickness and distance from the crack, could either shield or enhance the Mode I stress intensity factors. Stiff interphases increased the Mode II SIF but decreased the Mode I SIF as compared to the no-interphase cases.     

Three-dimensional problem of a running crack

The problem of an uniformly propagating finite crack in an infinite medium is solved using the dynamic equations of elasticity in 3-dimensions. Equal and opposite tractions are applied arbitrarily to the crack surfaces. The problem is reduced to the dual integral equations and solved with the aid of the series method. The numerical examples are presented on some graphs.     

Torsion of an elastic space with a crack on the surface of revolution

Numerical methods for solving integral equations of an axisymmetric problem of torsion of an elastic space with cracks on the surface of revolution are suggested for the cases of cracks crossing the axis of symmetry and cracks that have no common points with this axis. We also present relations for calculating the stress intensity factors at crack tips. Numerical results are obtained for a conic or paraboloidal simply connected crack and for a doubly connected crack lying on a surface formed by the revolution of an arbitrarily oriented straight segment or a parabolic arc. The crack faces are either subjected to a constant load or free of any forces; the body is subjected to torsion at infinity.Karpenko Physico-Mechanical Institute, Ukrainian Academy of Sciences, L'viv. Translated from Fiziko-Khimicheskaya Mekhanika Materialov, Vol. 29, No. 6, pp. 87–93, November–December, 1993.     

Transient analysis of multiple parallel cracks under anti-plane dynamic loading

The problem of a homogeneous linear elastic body containing multiple non-collinear cracks under anti-plane dynamic loading is considered in this work. The cracks are simulated by distributions of dislocations and an integral equation relating tractions on the crack planes and the dislocation densities is derived. The integral equation in the Laplace transform domain is solved by the Gaussian–Chebyshev integration quadrature. The dynamic stress intensity factor associated with each crack tip is calculated by a numerical inverse Laplace scheme. Numerical results are given for one crack and two or three parallel cracks under normal incidence of a plane horizontally shear stress wave.     

Strain-driven corrosion crack growth: A pilot study of intergranular stress corrosion cracking

This work proposes a model for corrosion driven crack growth. The model poses a moving boundary problem, where a chemical attack removes material from the body. The rate of the chemical attack is a function of the strain along the body surface. No crack growth criterion is needed for the analysis. A finite strain formulation is used and the material model is assumed hyperelastic. The problem is stated for a large body, containing a large crack. A low frequency cyclic loading is considered. Thus, corrosion is assumed to dissolve material with a rate approximately proportional to the strain rate. The problem is solved using finite element method based program, enhanced with a procedure handling the moving boundary. Parametric studies are performed for a series of different initial shapes of the near-tip region. Presented results show that the crack growth rate is largely dependent on the initial crack geometry. For a set of initial shapes and load levels steady-state conditions of growth are achieved, while for the others the cracks show tendency to branch.     

Strain-driven corrosion crack growth: A pilot study of intergranular stress corrosion cracking

This work proposes a model for corrosion driven crack growth. The model poses a moving boundary problem, where a chemical attack removes material from the body. The rate of the chemical attack is a function of the strain along the body surface. No crack growth criterion is needed for the analysis. A finite strain formulation is used and the material model is assumed hyperelastic. The problem is stated for a large body, containing a large crack. A low frequency cyclic loading is considered. Thus, corrosion is assumed to dissolve material with a rate approximately proportional to the strain rate. The problem is solved using finite element method based program, enhanced with a procedure handling the moving boundary. Parametric studies are performed for a series of different initial shapes of the near-tip region. Presented results show that the crack growth rate is largely dependent on the initial crack geometry. For a set of initial shapes and load levels steady-state conditions of growth are achieved, while for the others the cracks show tendency to branch.     

裂纹深度及尖端曲率对尖角裂纹周围应力场的影响

对于尖角裂纹,裂纹深度和裂纹尖端曲率对于裂纹附近应力场的影响有可能要大于裂纹张开角度的影响。根据弹性断裂理论和复变函数理论建立了一个计算模型,利用保角映射方法解决了尖角裂纹的边界条件问题,得到了具有不同深度和尖端曲率的裂纹周围应力分布,并对此应力分布规律进行了分析。