The penny-shaped crack at a bonded plane with localized elastic non-homogeneity

This paper examines the axisymmetric problem pertaining to a penny-shaped crack which is located at the bonded plane of two similar elastic halfspace regions which exhibit localized axial variations in the linear elastic shear modulus, which has the form G(z)=G₁+G₂e^±ζz. The equations of elasticity governing this type of non-homogeneity are solved by employing a Hankel transform technique. The resulting mixed boundary value problem associated with the penny-shaped crack is reduced to a Fredholm integral equation of the second kind which is solved in a numerical fashion to generate the crack opening mode stress intensity factor at the tip.     

On stress analysis for a penny-shaped crack interacting with inclusions and voids

An analytical approach to calculate the stress of an arbitrary located penny-shaped crack interacting with inclusions and voids is presented. First, the interaction between a penny-shaped crack and two spherical inclusions is analyzed by considering the three-dimensional problem of an infinite solid, composed of an elastic matrix, a penny-shaped crack and two spherical inclusions, under tension. Based on Eshelby’s equivalent inclusion method, superposition theory of elasticity and an approximation according to the Saint–Venant principle, the interaction between the crack and the inclusions is systematically analyzed. The stress intensity factor for the crack is evaluated to investigate the effect of the existence of inclusions and the crack–inclusions interaction on the crack propagation. To validate the current framework, the present predictions are compared with a noninteracting solution, an interacting solution for one spherical inclusion, and other theoretical approximations. Finally, the proposed analytical approach is extended to study the interaction of a crack with two voids and the interaction of a crack with an inclusion and a void.     

EXACT SOLUTION TO THE PROBLEM OF A HALF-PLANE CRACK IN A TRANSVERSELY ISOTROPIC PIEZOELECTRIC BODY SUBJECTED TO ANTISYMMETRIC TANGENTIAL POINT FORCES

An exact and complete solution of the problem of a half-plane crack in an infinite transversely isotropic piezoelectric body is presented. The upper and lower crack faces are assumed to be loaded antisymmetrically by a couple of tangential point forces in opposite directions. The solution is derived through a limiting procedure from that of a penny-shaped crack. The expressions for the electroelastic field are given in terms of elementary functions. Finally, the numerical results of the second and third mode stress intensity factorsk ₂ andk ₃ of piezoelectric materials and elastic materials are compared in figures. Project supported by the National Natural Science Foundation of China (No. 19872060 and 69982009) and the Postdoctoral Foundation of China.     

Interaction between curved crack and elastic inclusion in an infinite plate

Summary In this paper, the curved-crack problem for an infinite plate containing an elastic inclusion is considered. A fundamental solution is proposed, which corresponds to the stress field caused by a point dislocation in an infinite plate containing an elastic inclusion. By placing the distributed dislocation along the prospective site of the crack, and by using the resultant force function as the right-hand term in the equation, a weaker singular integral equation is obtainable. The equation is solved numerically, and the stress intensity factors at the crack tips are evaluated. Interaction between the curved crack and the elastic inclusion is analyzed. Received 8 October 1996; accepted for publication 27 March 1997     

Two non-elliptical inhomogeneities with internal uniform stresses interacting with a mode-III crack

We use the method of Green's functions to analyze an inverse problem in which we aim to identify the shapes of two non-elliptical elastic inhomogeneities, embedded in an infinite matrix subjected to uniform remote stress, which enclose uniform stress distributions despite their interaction with a finite mode-III crack. The problem is reduced to an equivalent Cauchy singular integral equation, which is solved numerically using the Gauss–Chebyshev integration formula. The shapes of the two inhomogeneities and the corresponding location of the crack can then be determined by identifying a conformal mapping composed in part of a real density function obtained from the solution of the aforementioned singular integral equation. Several examples are given to demonstrate the solution.     

Axisymmetric displacement boundary value problem for a penny-shaped crack

Summary A solution is derived from equations of equilibrium in an infinite isotropic elastic solid containing a penny-shaped crack where displacements are given. Abel transforms of the second kind stress and displacement components at an arbitrary point of the solid are known in the literature in terms of jumps of stress and displacement components at a crack plane. Limiting values of these expressions at the crack plane together with the boundary conditions lead to Abel-type integral equations, which admit a closed form solution. Explicit expressions for stress and displacement components on the crack plane are obtained in terms of prescribed face displacements of crack surfaces. Some special cases of the crack surface shape functions have been given in the paper.     

A crack of arbitrary shape inside an infinite transversely isotropic cylinder under arbitrary load

Summary  In the first part of the article an infinite circular cylinder is considered, made of transversely isotropic elastic material and weakened by a plane crack perpendicular to its axis O z. The crack is opened by an arbitrary normal stress. The second part is devoted to the same crack loaded by an arbitrary tangential stress. The complete solution in both cases is presented as a sum of the solution of a similar problem of a crack in an infinite space and an integral transform term, the parameters of which are determined from a set of linear algebraic equations derived from the boundary conditions. Governing integral equations with respect to the yet unknown crack displacement discontinuities are obtained. In the case of a circular crack, these equations can be inverted and solved by the method of consecutive interations. Received 30 November 2000; accepted for publication 3 May 2001     

A penny-shaped crack in a filament-reinforced matrix—II. The crack problem

Using the filament model developed in the previous paper, the elastostatic interaction problem between a penny-shaped crack and a slender inclusion or filament in an elastic matrix is formulated. For a single filament as well as multiple identical filaments located symmetrically around the crack the problem is shown to reduce to a singular integral equation. The solution of the problem is obtained for various geometries and filament to-matrix stiffness ratios, and the results relating to the angular variation of the stress intensity factor and the maximum filament stress are presented.     

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. The project supported by the National Natural Science Foundation of China (19872002) and Climbing Foundation of Northern Jiaotong University     

Interaction of elastic waves with a penny-shaped crack in an infinitely long cylinder

This paper contains an analysis of the interaction of longitudinal waves with a penny-shaped crack located in an infinitely long elastic cyclinder. The problem is reduced to a Fredholm integral equation of the second kind which is solved numerically for a range of values of the frequency of the incident waves and the radius of the cylinder. Numerical values of the dynamic stress intensity factor at the rim of the crack have been calculated.     

Electroelastic analysis of a piezoelectric cylindrical fiber with a penny-shaped crack embedded in a matrix

The electroelastic response of a penny-shaped crack in a piezoelectric cylindrical fiber embedded in an elastic matrix is investigated in this study. Fourier and Hankel transforms are used to reduce the problem to the solution of a pair of dual integral equations. They are then reduced to a Fredholm integral equation of the second kind. Numerical values on the stress intensity factor, energy release rate and energy density factor for piezoelectric composites are obtained to show the influence of applied electric fields.     

Thermal stress analysis of a three-dimensional anticrack in a transversely isotropic solid

A three-dimensional analysis is performed for an infinite transversely isotropic elastic body containing an insulated rigid sheet-like inclusion (an anticrack) in the isotropy plane under a remote perpendicularly uniform heat flow. A general solution scheme is presented for the resulting boundary-value problems. Accurate results are obtained by constructing suitable potential solutions and reducing the thermal problem to a mechanical analog for the corresponding isotropic problem. The governing boundary integral equation for a planar anticrack of arbitrary shape is obtained in terms of a normal stress discontinuity. As an illustration, a complete solution for a rigid circular inclusion is obtained in terms of elementary functions and analyzed. This solution is compared with that corresponding to a penny-shaped crack problem.     

The approximate solution of penny-shaped cracks periodically distributed in infinite elastic body

The penny-shaped cracks periodically distributed in infinite elastic body are studied. The problem is approximately simplified to that of a single crack embedded in finite length cylinder and the stress intensity factor is obtained by solving a Fredholm integral equation. Numerical results are given and the effects of crack interaction on the stress intensity factor are discussed. The project suppoted by National Natural Science Foundation of China     

Interaction between rigid-disc inclusion and penny-shaped crack under elastic time-harmonic wave incidence

Influence of a rigid-disc massive inclusion on a neighboring penny-shaped crack induced by the time-harmonic wave propagation in an infinite elastic matrix is investigated by the numerical solution of associated 3D elastodynamic problem. No restrictions on the mutual orientation of interacting objects and direction of wave incidence are assumed. The inclusion is perfectly bonded with a matrix and supposes the translations and rotations, the crack faces are load-free. Frequency-domain problem is reduced to a system of boundary integral equations (BIEs) relative to the interfacial stress jumps (ISJs) on the inclusion and the crack opening displacements (CODs). The subtraction technique in conjunction with mapping technique, under taking into account the structure of solution at the fronts of inclusion and crack, is applied for regularization of BIEs obtained. A discrete analogue of equations is constructed by using the collocation scheme. Numerical calculations are carried out for the grazing incidence of a plane P-wave on the crack, where the interacting inclusion is coplanar and perpendicular to the crack, and has the same radius. The shielding and amplification effects of inclusion are assessed by the analysis of mode-I stress intensity factor (SIF) in the crack vicinity depending on the wave number, incident wave direction, position of the crack front point, inclusion mass, crack-inclusion orientation and distance.     

非均匀弹性材料中反平面的运动裂纹

用解析方法研究了非均匀弹性材料中反平面运动裂纹问题。首先采用余弦变换求解非均匀材料的基本方程，然后根据混合边值条件建立裂纹运动的对偶积分方程，再把对偶积分方程化为第二类Fredholm积分方程。给出了数值算例，计算结果表明材料的非均匀性对动应力强度因子有较大的影响。     

Solution of a flat elliptical crack in an electrostrictive solid

Based on the assumption that the elastic strain of electrostrictive materials is a higher-order small quantity, this paper studies the 3D problem of an infinite electrostrictive solid with a flat elliptical crack which is electrically permeable. According to existing solutions of similar problems in pure elastic materials, with the displacement function method, we first derived explicit expression for displacement potential function and obtained stress field near the crack and open displacement of crack surface. Then, the general solution for the stress intensity factor was derived, and the corresponding solutions were also presented for a penny-shaped crack and a permeable line-crack as two special cases of the present problem. Finally, numerical results were given to discuss the effect of environment at infinity and electric field inside the crack on the stress-intensity factors.     

Scattering of monochromatic elastic waves on a planar crack of arbitrary shape

Scattering of monochromatic elastic waves on an isolated planar crack of arbitrary shape is considered. The 2D-integral equation for the crack opening vector is discretized by Gaussian approximating functions. For such functions, the elements of the matrix of the discretized problem have forms of standard one-dimensional integrals that can be tabulated. For regular grids of approximating nodes, the matrix of the discretized problem has the Toeplitz structure, and the corresponding matrix–vector products can be calculated by the fast Fourier transform technique. The latter strongly accelerates the process of iterative solution of the discretized problem. Examples of calculations of crack opening vectors, dynamic stress-intensity factors, and differential cross-sections of circular (penny-shaped) and non-circular cracks for various incident wave fields are presented. For a penny-shaped crack and longitudinal incident waves normal to the crack plane, an efficient semi-analytical method of the solution of the scattering problem is developed. The results of both methods are compared in a wide frequency region of the incident field.     

The discontinuity in the elastostatic displacement vector across a penny-shaped crack under arbitrary loads

Consider an infinite elastic solid containing a penny-shaped crack. A familiar problem in linear elastostatics is the determination of the displacement in the solid when the crack is subjected to an arbitrarily prescribed loading; denote the corresponding boundary-value problem by P. We construct the exact Green's function for P, using a method due to Guidera and Lardner [4]. We then use our Green's function to solve P, by obtaining expressions for the discontinuity in the displacement vector across the crack. Finally, we compare our solutions for P with those obtained recently by Krenk [6].     

The failure of a strain-softening solid containing an internal flaw

The paper presents results from a theoretical analysis of the effect of a penny-shaped crack on the failure of an infinite solid that is subjected to tension loadings normal to the crack plane. The material is strain-softening and the fully developed softening zone size and the crack tip stress intensity K associated with the attainment of this state are shown to be very dependent on the initial crack size. For load control conditions, it is shown that complete failure can occur prior to the full development of a softening zone, the failure stress depending on the initial crack size. Particular emphasis is focused on the limiting situation where the failure stress approaches the limit stress, i.e. the maximum stress that the strain-softening material can sustain, and the effect of the strain-softening law on this limiting situation is determined.     

横观各向同性材料的三维断裂力学问题

从三维横观各向同性材料弹性力学理论出发, 使用Hadamard有限部积分概念, 导出了三维状态下单位位移间断(位错)集度的基 本解. 在此基础上, 进一步运用极限理论, 将任意载荷作用下, 三维无限大横观各向 同性材料弹性体中, 含有一个位于弹性对称面内的任意形状的片状裂纹问题, 归结为求 解一组超奇异积分方程的问题. 通过二维超奇异积分的主部分析方法, 精确地求得了裂纹前沿光滑点附近的应力奇异指数和奇异应力场, 从而找到了以裂纹表面位移间断表示的应力强度因子表达式及裂纹局部扩展所提供 的能量释放率. 作为以上理论的实际应用,最后给出了一个圆形片状裂纹问题 的精确解例和一个正方形片状裂纹问题的数值解例. 对受轴对称法向均布载荷作用下圆形片状裂纹问题, 讨论了超奇异积分方程的精确求解方法, 并获得了位移间断和应力强度因子的封闭解, 此结果与现有理论解完全一致.