Determination of stress field in an elastic solid weakened by parallel penny-shaped cracks

Summary This paper investigates the stress intensity factors of two penny-shaped cracks with different sizes in a three-dimensional elastic solid under uniaxial tension. The two cracks are symmetrically parallel located in the isotropic solid. Based on Eshelby's equivalent inclusion method and the superposition pronciple of the elasticity theory, a closed-form analytical elastic solution for the stress intensity factors (SIFs) on the boundaries of the cracks is obtained when the center distance between the two cracks is much larger than the crack sizes. A numerical method is employed to extract the solution for small center distance case. It is found that, due to the interaction between the two cracks, the first and second kinds of SIFs exist at the same time even if the applied stress is pure tension. Numerical examples are given for different configurations and it is clearly shown that the SIFs are strongly determined by the distance between the centers of the two cracks.     

Stress intensity factors for a bounded plate containing two collinear cracks

This paper presents a method of numerical analysis for the problem of two collinear cracks in a finite, linearly elastic, isotropic plate and subjected to in plane forces.

The problem is treated imagining the plate with the two cracks draws in an unbounded region. Using the analytical solution of a point force applied to an infinite plate with two collinear cracks of equal length, the boundary conditions are written by superimposing the effect of interior loading upon the effect of concentrated loads applied on a curve parallel to the outer boundary. The boundary condition are satisfied in a least square sense.

Numerical example are given for square plates with inner or edge cracks. The accuracy is discussed.     

Dynamic stresses around two cracks placed symmetrically to a large crack

Dynamic stresses around three cracks in an infinite elastic plate have been solved. Two cracks, which are small and equal, are situated ahead of a large crack so as to allow for geometrical symmetry. Time-harmonic normal traction acts on each surface of these cracks. To solve the problem, two solutions are combined. One of them is a solution for a crack in an infinite plate and another is that for two collinear cracks in an infinite plate. The Schmidt method is used to satisfy the boundary conditions on the cracks' surfaces with use of the combined solutions. Stress intensity factors are calculated numerically for some of these crack configurations.     

Multiple holes,cracks, and inclusions in anisotropic viscoelastic solids

By using the elastic–viscoelastic correspondence principle, the problems with multiple holes, cracks, and inclusions in two-dimensional anisotropic viscoelastic solids are solved for the cases with time-invariant boundaries. Based upon this principle and the existing methods for the problems with anisotropic elastic materials, two different approaches are proposed in this paper. One is concerned with an analytical solution for certain specific cases such as two collinear cracks, collinear periodic cracks, and interaction between inclusion and crack, and the other is a boundary-based finite element method for the general cases with multiple holes, cracks, and inclusions. The former considers only specific cases in infinite domain and can be used as a reference for any other numerical methods, and the latter is applicable to any combination of holes, cracks and inclusions in finite domain, whose number, size and orientation are not restricted. Unlike the conventional finite element method or boundary element method which usually needs very fine meshes to get convergence solutions, in the proposed boundary-based finite element method no meshes are needed along the boundaries of holes, cracks and inclusions. To show the accuracy and efficiency of these two proposed approaches, several representative examples are implemented analytically and numerically, and they are compared with each other or with the solutions obtained by the finite element method.     

Distributed cracks and kinked cracks in bonded dissimilar half planes with an interface crack

This paper is concerned with the interactions between an interface crack and other arbitrarily distributed cracks in two bonded dissimilar half planes. Special emphasis is placed on the cracks kinked at a tip of the interface crack, which remain unsolved as far as the authors are concerned. For the present, we pay attention to the stress intensity factors at the tips of the kinks or the distributed cracks, and not to those at the tips of the interface crack. The analysis is based on continuous distributions of the body forces along the cracks, and their densities are determined with a new procedure in order to get highly accurate results. The present analysis for distributed line cracks applies to kinked cracks, branched cracks and those piercing the interface just by joining some of the line cracks. Numerical calculations are performed for various important problems, and the effects of geometric and mechanical parameters on the stress intensity factors are examined.     

Elastodynamic analysis of nonplanar cracks in a half-space

Summary The interaction of time harmonic antiplane shear waves with nonplanar cracks embedded in an elastic half-space is studied. Based on the qualitatively similar features of crack and dislocation, with the aid of image method, the problem can be formulated in terms of a system of singular integral equations for the density functions and phase lags of vibrating screw dislocations. The integral equations, with the dominant singular part of Hadamard's type, can be solved by Galerkin's numerical scheme. Resonance vibrations of the layer between the cracks and the free surface are observed, which substantially give rise to high elevation of local stresses. The calculations show that near-field stresses due to scattering by a single crack and two cracks are quite different. The interaction between two cracks is discussed in detail. Furthermore, by assuming one of the crack tips to be nearly in contact with the free surface, the problem can be regarded as the diffraction of elastic waves by edge cracks. Numerical results are presented for the elastodynamic stress intensity factors as a function of the wave number, the incident angle, and the relative position of the cracks and the free surface.     

Three collinear cracks in an infinite elastic medium

The plane strain problem of determining the distribution of stress in the vicinity of three cracks embedded in an infinite isotropic elastic medium is considered. The cracks are collinear, the two side cracks are equal in length and located symmetrically with respect to the middle crack. The surface tractions acting on the cracks are completely arbitrary. Some special cases of the loading are discussed in detail.     

Basic solutions of two parallel mode-I cracks or four parallel mode-I cracks in the piezoelectric materials

In this paper, the stress and the electric intensity factors of two parallel mode-I cracks or four parallel mode-I cracks in the piezoelectric materials were examined by means of the Schmidt method for the permeable electric boundary conditions. The present problem can be solved by using the Fourier transform and the technique of dual integral equation, in which the unknown variables are the jumps of displacements across the crack surfaces, not the dislocation density functions. To solve the dual integral equations, the displacement jumps are directly expanded in a series of Jacobi polynomials. Finally, the effects of the distance between two parallel cracks and the distance between two collinear cracks on the stress and the electric intensity factors in the piezoelectric materials are analyzed. These results can be used for the strength evaluation of the piezoelectric materials with multi-cracks.     

Interaction of two unequal cracks in a prestressed fiber reinforced composite

We consider an elastic orthotropic material representing a fiber reinforced composite. The composite is prestressed and contains two collinear cracks having different lengths. The faces of the cracks are acted by symmetrically distributed constant normal incremental stresses. We determine the critical values of the applied incremental stresses for which the cracks tips start to propagate and we analyse the interaction of the cracks as function of their lengths and of the distance between the cracks. This revised version was published online in July 2006 with corrections to the Cover Date.     

Dynamic stress intensity factors around three cracks in an infinite elastic plane subjected to time-harmonic stress waves

The time-harmonic problem for an infinite elastic plane weakened by three parallel cracks has been solved. In this problem, two cracks are situated symmetrically on either side of a central crack and incident stresses impinge perpendicular to the cracks. Using the Fourier transform technique, the boundary conditions are reduced to four simultaneous integral equations. To solve the equations, the differences of displacements inside the cracks are expanded in a series. The unknown coefficients in the series are solved by the Schmidt method. The dynamic stress intensity factors are calculated numerically for several crack configurations. This revised version was published online in August 2006 with corrections to the Cover Date.     

两条断续预制裂纹粗晶大理岩强度参数的研究

基于在岩石力学伺服试验机上得到的两条断续预制裂纹粗晶大理岩常规三轴压缩试验结果,采用Coulomb和Hoek-Brown强度准则,获得了断续预制裂纹粗晶大理岩的强度参数,评价了Coulomb和Hoek-Brown准则用于断续预制裂纹岩样强度分析的适用性。研究表明:完整岩样强度和围压之间的关系用Coulomb和Hoek-Brown准则均可以很好地表征,而断续预制裂纹岩样强度更好地符合Hoek-Brown强度准则。利用Hoek-Brown准则确定的粘聚力和内摩擦角也可以表征岩石材料真实的力学特性。断续预制裂纹岩样强度对围压的敏感程度高于完整岩样,计算得到的粘聚力显著低于完整岩样,而内摩擦角高于完整岩样。断续预制裂纹岩样的粘聚力以及内摩擦角与裂纹倾角关系复杂,单轴压缩强度是否参与回归影响较大。由Hoek-Brown准则得到的粗晶大理岩强度参数T(m,s)(联合回归均值)对完整岩样和断续预制裂纹岩样分别为T(6.542,0.99)和T(6.625,0.156),可知断续预制裂纹对岩石强度参数m(软硬程度)影响不大,而对参数s(破碎程度)存在较大影响。     

Crack initiation, propagation and coalescence from frictional flaws in uniaxial compression

An extensive experimental program has been conducted on pre-cracked specimens of a rock-model material to investigate crack propagation and coalescence from frictional discontinuities. Prismatic gypsum specimens have been prepared with three pre-existing closed cracks (flaws). The flaws all have a constant length of 12.7 mm and are parallel to each other. Different geometries are obtained by changing the angle of the flaws with respect to the direction of loading, the spacing, and the continuity of the flaws. In the experiments, three different types of cracks have been observed: wing cracks, coplanar shear, and oblique shear cracks. These are the same types of cracks observed with open flaws. Crack initiation occurs simultaneously at all the tips of the flaws for wing and shear cracks. Mean crack initiation stress is higher for secondary cracks than for wing cracks. The differences however decrease as the flaws are oriented at smaller angles with the direction of loading. The types of coalescence (i.e. the type of cracks and crack pattern that link two flaws) from closed flaws are similar to those from open flaws. However, the type of coalescence observed in a specimen with open flaws is not necessarily produced when using the same geometry but with closed flaws. The most important conclusion reached in this research is that the fracturing processes in open and closed flaws are similar. Friction along the flaws increases the initiation and coalescence stress and favors linkage through shear cracks.     

On the problem of two coplanar cracks inside an infinite isotropic elastic solid

A method is proposed for the approximate evaluation of normal displacements and normal stresses on the plane of two coplanar cracks located inside an infinite isotropic elastic solid and subjected to normal internal pressure. The formulation results in a single integral equation for the unknown normal stresses on the plane of the cracks. Numerical results are given for the stress intensity factor K_I of two coplanar circular cracks and two coplanar elliptical cracks opened up under a uniform internal pressure.     

Solution of multiple crack problem in a finite plate using an alternating method based on two kinds of integral equation

This paper investigates a solution of multiple crack problem in a finite plate using an alternating method. The finite plate with cracks is an overlapping region of two regions: namely the infinite region exterior to the cracks and the finite region interior to finite plate without cracks. It is assumed that the cracks are applied by some loading and edges of the finite plate are of traction free. Governing equations for the problem and an alternating method are suggested. In the iteration, we need to solve two boundary value problems. One is the multiple crack problem in an infinite plate, and the other is the boundary value problem for the finite plate without crack. Several numerical examples are provided to prove the effectiveness of the suggested method.     

Crack coalescence morphology in rock-like material under compression

This paper uses peridynamic simulations to determine the extent of coalescing damage and identify the underlying causes. The basic crack types and crack coalescence patterns in specimens with a flaw pair under uniaxial compression are systematically investigated. Various crack types including horsetail cracks, anti-wing cracks, and tensile wing cracks are successfully observed and the coalescence sequences are identified. By varying angles, six crack coalescence categories with respect to the overlapping ratios provide insightful information of different crack growths and indicate various cracking modes underlying various coalescence patterns. The arrangement of the flaw pair strongly influences the crack initiation position and trajectories, allowing for different coalescence morphologies. Coalescence formed by two internal tensile wing cracks, or transfixion, shows unbroken crack segments with a further loading, along with growing shear cracks until failure. In contrast, after the coalescence is formed through two horsetail cracks, the interior of the rhombic shape gets deformed with further loading. The peridynamic code adopted in this research can provide realistic simulation results and help researchers to conduct expanded tests as well as to enhance understanding the fracture of rock-like material.     

Interaction between a spherical inhomogeneity and two symmetrically placed penny-shaped cracks

Stress analysis is carried out for a three-dimensional elastic solid containing an elastic spherical inhomogeneity and two coplanar penny-shaped cracks. Each of the two cracks is located on either side of the elastic spherical inhomogeneity and the geometry is subjected to uniform tensile stress at infinity. The interaction between the inhomogeneity and the cracks is tackled by the superposition principle of elasticity theory and Eshelby's equivalent inclusion method. Analytical solutions for the stress intensity factors on the boundaries of the cracks and the stress field inside the inhomogeneity are evaluated in series form. Numerical calculations are reported for several special cases, and show the variations of the stress intensity factors and stress field inside the inhomogeneity with the configuration and elastic properties of the solid and the inhomogeneity.     

求解双材料界面裂纹应力强度因子的扩展有限元法

基于双材料界面裂纹尖端的基本解,构造扩展有限元法(eXtended Finite Element Methods, XFEM)裂尖单元结点的改进函数。有限元网格剖分不遵从材料界面,考虑3种类型的结点改进函数:弱不连续改进函数、Heaviside改进函数和裂尖改进函数,建立XFEM的位移模式,给出计算双材料界面裂纹应力强度因子(Stress Intensity Factors, SIFs)的相互作用积分方法。数值结果表明:XFEM无需遵从材料界面剖分网格,该文的方法能够准确评价双材料界面裂纹尖端的SIFs。     

The interaction of two inclined cracks with dynamic stress wave loadings

To gain insight into the phenomenon of the interaction of stress waves with material defects and the linkage of two cracks, the transient response of two semi-infinite inclined cracks subjected to dynamic loading is examined. The solutions are obtained by the linear superposition of fundamental solutions in the Laplace transform domain. The fundamental solution is the exponentially distributed traction on crack faces proposed by Tsai and Ma [1]. The exact closed form solutions of stress intensity factor histories for these two inclined cracks subjected to incident plane waves and diffracted waves are obtained explicitly. These solutions are valid for the time interval from initial loading until the first wave scattered at one crack tip returns to the same crack tip after being diffracted by another crack tip. The result shows that the contribution of diffracted waves to stress intensity factors is much less than the incident waves. The probable crack propagation direction is predicted from the fracture criterion of maximum circumferential tensile stress. The linkage of these two cracks is also investigated in detail.     

定向断裂控制爆破爆生裂纹扩展机理的实验研究

采用新型数字激光动态焦散线实验系统,对缺陷介质双孔定向断裂控制爆破裂纹扩展的动态行为进行了研究。结果表明,预制斜裂纹阻断了爆生主裂纹的扩展,最终两条主裂纹分别与翼裂纹形成相互勾连的形状。爆生主裂纹尖端以张拉应力场为主,其断裂为近似I型断裂。当爆生主裂纹运动到预制裂纹附近时,主裂纹端部应力场与预制裂纹尖端奇异应力场相互叠加,在预制裂纹尖端形成较强的拉剪应力场,且受已有主裂纹面的影响,预制裂纹扩展表现为弯向主裂纹面的弯曲断裂。研究结果可为含节理岩体定向断裂控制爆破提供理论依据。     

Efficient solution of multiple cracks in great number using eigen COD boundary integral equations with iteration procedure

A newly developed computational approach is proposed in the paper for the analysis of multiple crack problems based on the eigen crack opening displacement (COD) boundary integral equations. The eigen COD particularly refers to a crack in an infinite domain under fictitious traction acting on the crack surface. With the concept of eigen COD, the multiple cracks in great number can be solved by using the conventional displacement discontinuity boundary integral equations in an iterative fashion with a small size of system matrix to determine all the unknown CODs step by step. To deal with the interactions among cracks for multiple crack problems, all cracks in the problem are divided into two groups, namely the adjacent group and the far-field group, according to the distance to the current crack in consideration. The adjacent group contains cracks with relatively small distances but strong effects to the current crack, while the others, the cracks of far-field group are composed of those with relatively large distances. Correspondingly, the eigen COD of the current crack is computed in two parts. The first part is computed by using the fictitious tractions of adjacent cracks via the local Eshelby matrix derived from the traction boundary integral equations in discretized form, while the second part is computed by using those of far-field cracks so that the high computational efficiency can be achieved in the proposed approach. The numerical results of the proposed approach are compared not only with those using the dual boundary integral equations (D-BIE) and the BIE with numerical Green's functions (NGF) but also with those of the analytical solutions in literature. The effectiveness and the efficiency of the proposed approach is verified. Numerical examples are provided for the stress intensity factors of cracks, up to several thousands in number, in both the finite and infinite plates.