The effect of an elliptical inclusion on a crack

The interaction between an elliptical inclusion and a crack is analyzed by body force method. The investigated stress field is simulated by superposing the fundamental solutions for a point force applied at a point in an infinite plate containing an elliptical inclusion. Based on numerical results, effects of the inclusion shape on the crack tip stress intensity factor are discussed. It is found that for small cracks emanating from a stress-higher point on the inclusion interface the stress intensity factors are mainly determined by the stresses, occurring at the crack starting point before the crack initiation, and the inclusion root radius, besides the crack length. However, for the cracks occurring in a stress-lower region around the inclusion, it is difficult to characterize the effect of the inclusion geometry on the stress intensity factors of small cracks by the inclusion root radius alone. This revised version was published online in July 2006 with corrections to the Cover Date.     

On the numerical evaluation of stress intensity factors for an interface crack of a general shape

A numerical algorithm is presented for the problem of a crack along the interface of an elastic inclusion embedded in an elastic plane subjected to uniform stress at infinity. The algorithm is based on a Fredholm integral equation of the second kind and allows for fast and accurate solutions to geometries of great complexity. In an example crack opening displacement and stress intensity factors are computed for a crack in the interface of an inclusion with nineteen protruding arms. Copyright © 1999 John Wiley & Sons, Ltd.     

Fracture mechanics of brittle matrix ductile fiber composites

A model to predict the increase in critical flaw size or stable crack growth potential which can occur by the inclusion of ductile fibers in a brittle matrix is considered. The model is based upon the super-position of two known stress intensity solutions; one for the crack opening mode resulting from a remotely applied stress and the second, an opposing stress intensity that results from a crack closing force exerted by unbroken fibers spanning the crack surfaces. The extent of stable growth possible is computed at the ultimate stress of the brittle phase as functions of fiber strength and of volume fraction for various amounts of fiber rupture. A hot pressed beryllium matrix is used as an example. The crack surface displacement over which a given fiber is capable of deforming without rupture is found to be sensitive to the fiber-matrix interface strength. The factors leading to maximum crack surface displacement without rupture are a high strain hardening capability of the fiber and an interface designed to fail at fiber stresses between yield and ultimate strengths.     

The role of inclusions in the fracture of ceramic materials

The stress concentrations that occur at inclusions due to thermal expansion and elastic modulus mismatch are discussed and the stress intensity factors at interface cracks that result from these stresses are calculated. It is shown that conservative failure prediction based on an equivalence between inclusion size and crack size is usually acceptable if the shear modulusμ or thermal expansion coefficientα for the inclusion is larger than the matrix values. If, however,μ andα are smaller for the inclusion than the matrix, extensive cracking can develop at the inclusions which may lead to premature failure. For this case the only effective methods for failure prediction are techniques which give directly the maximum stress intensity factor, i.e., proof testing and/or acoustic emission.     

Bonded elastic half-planes with an interface crack and a perpendicular intersecting crack that extends into the adjacent material—I

The solution is given for two bonded isotropic linearly elastic half-planes of different elastic properties having a crack along the interface as welt as a perpendicular crack in one of the half-planes which may intersect the interface crack. The appropriate integral equations are developed using displacement dislocations on the crack surfaces. Numerical results are presented for the stress intensity factors, strain energy release rate, stresses and displacements.     

Determination of stress intensity factors for bimaterial interface stationary rigid line inclusions by boundary element method

The stress intensity factors for a rigid line inclusion lying along a bimaterial interface are calculated by the boundary element method with the multiregion and the discontinuous traction singular elements. The relationships between the stress intensity factors and the inclusion surface stresses are derived. The numerically computed stress intensity factors for the bimaterial interface rigid line inclusion in the infinite body are proved to be in good agreement within 3% when compared with the previous exact solutions. In the finite bimaterial models, the stress intensity factors for the center and edge rigid line inclusions at the interface are computed with the variation of the rigid line inclusion length and the shear modulus ratio under the uniaxial and biaxial loading conditions.     

Crack–inclusion problem for a long rectangular slab of superconductor under an electromagnetic force

The interaction between a crack and an inclusion in a type-II superconductor is investigated in this paper. Using the finite element method, the crack–inclusion problem can be solved. Numerical results are presented to illustrate fracture behavior of superconductor under electromagnetic force. The magnetic behavior of the superconductor is described by the critical-state Bean model. The stress intensity factors at the crack tip are obtained and discussed for decreasing field after zero-field cooling. Numerical results show that the stress intensity factors at crack tip are always larger with an elastic inclusion than for a rigid inclusion. Because of the barrier effect of the rigid inclusion, the values of the stress intensity factors decrease when the crack approaches the inclusion. Relative to rigid inclusion and no inclusion cases, elastic inclusion leads to the largest value of the stress intensity factor at crack tip. Thus, the crack propagation is easier near an elastic inclusion and the rigid inclusion is helpful for crack arrest.     

Fracture problems of rubbers: J-integral estimation based upon η factors and an investigation on the strain energy density distribution as a local criterion

The plane elasticity problem studied is of a circular inclusion having a circular arc-crack along the interface and a crack of arbitrary shape in an infinite matrix of different material subjected to uniform stresses at infinity. The solution of the problem is given using Muskhelishvili's complex variable method with sectionally holomorphic functions. First, the solution to the (auxiliary) problem of a dislocation (or force) applied at a point in the matrix with the circular inclusion partially bonded is derived fully in its general form by solving the appropriate Rieman-Hilbert problem. It is subsequently used as the Green's function for the initial problem by introducing an unknown density function associated with a distribution of dislocations along the crack in the matrix. The initial problem is then reduced to a singular integral equation (SIE) over the crack in the matrix only. The SIE is solved numerically by appropriate quadratures and the stress intensity factors reported for the arc-cut and a straight crack in the matrix for a range of values of the geometrical parameters.     

Analysis of dynamic interaction between an inclusion and a nearby moving crack by BEM

In this paper, the dynamic interaction between an inclusion and a nearby moving crack embedded in an elastic medium is studied by the boundary element method (BEM). To deal with this problem, the multi-region technique and two kinds of time-domain boundary integral equations (BIEs) are introduced. The system is divided into two parts along the interface between the inclusion and the matrix medium. Each part is linear, elastic, homogeneous and isotropic. The non-hypersingular traction boundary integral equation is applied on the crack surfaces; while the traditional displacement boundary integral equation is used on the interface and external boundaries. In the numerical solution procedure, square root shape functions are adopted as to describe the proper asymptotic behavior in the vicinity of the crack-tips. The crack growth is modeled by adding new elements of constant length to the moving crack tip, which is controlled by the fracture criterion based on the maximum circumferential stress. In each time step, the direction and the speed of the crack advance are evaluated. The numerical results of the crack growth path, speed, dynamic stress intensity factors (DSIFs) and dynamic interface tractions for various material combinations and geometries are presented. The effect of the inclusion on the moving crack is discussed.     

Antiplane shear crack terminating at and going through a bimaterial interface

The antiplane shear problem of two bonded elastic half planes containing a crack perpendicular to the interface is considered. The cases of a semi-infinite crack terminating at the interface, a finite crack away from and terminating at the interface, two cracks one on each side of the interface, and a finite crack crossing the interface are separately investigated. The nature of the stress singularity for the crack terminating at and going through the interface is studied, and it is shown that at the irregular point on the interface, for the former the power of singularity is not -1/2 and for the latter the stresses are bounded. For a material pair of aluminum-epoxy some numerical results giving the stress intensity factors, the density functions, and the crack opening displacements are presented.     

Anti-plane shear stress problem of two bonded dissimilar half spaces with an elliptical hole or rigid inclusion on the interface

The problem of anti-plane shear stress of two bonded dissimilar half spaces with an elliptical hole or a rigid inclusion at the interface and having interfacial cracks is presented. Uniform anti-plane shear stresses and the stress free or zero displacement boundary conditions on the elliptical hole are considered. The two cases are reduced to Riemann–Hilbert problems and closed form solutions are obtained by use of the complex stress function and the conformal mapping approaches. Stress distributions, as well as stress intensity factor, are shown. When the elliptical hole collapses, the known solutions of the interfacial crack and thin rigid fiber can be obtained. If the coordinates in the Plemelj function are changed, a debonding length can be determined.     

The inclusion problem with a crack crossing the boundary

The problem of an elastic plane containing an elastic inclusion is considered. It is assumed that both the plane and the inclusion contain a radial crack and the two cracks are collinear. The problem is formulated in terms of a system of singular integral equations. In the interesting limiting cases in which the crack tips approach the interface from either one or both sides, the dominant parts of the kernels become generalized Cauchy kernels giving rise to stress singularities of other than 13-1 power. For these unusual cases of a crack terminating at or crossing the interface stress intensity factors are defined and some detailed results are given for various crack-inclusion geometries and material combinations.     

Analysis of an edge crack in a semi-infinite composite with a long reinforced phase

For the purpose of clarifying the micro fracture of continuous fiber unidirectionally reinforced composite materials, the problem of an edge crack perpendicular to a long reinforced phase is considered on the basis of the plane strain theory of elasticity. The stress intensity factor at the tip of the crack, and the stresses on the interface between the matrix and the reinforced phase and in the reinforced phase are discussed. In the analysis, the method of continuous distributions of dislocations is used. Then, a singular integral equation is derived and is solved by the technique developed by Erdogan and Gupta. From the numerical results it was concluded that:

1. The stress intensity factor decreases monotonically as the crack tip approaches the reinforced phase. That is, the presence of the reinforced phase can result in crack arrest.
2. When the crack tip exists near the reinforced phase, the normal stress on the interface between the matrix and the reinforced phase has a maximum at the intersection of the extension of the edge crack and the reinforced phase, while the shear stress on the interface and the normal stress in the reinforced phase take, respectively, maxima at symmetric points with respect to the crack surface in the immediate vicinity of the intersection.
3. The maximum values of the stresses on the interface and in the reinforced phase increase monotonically as the crack tip approaches the reinforced phase.

     

含界面裂纹圆形夹杂的十二次对称二维准晶热应力分析

针对点热源作用下,无限大十二次对称二维准晶基体和圆形弹性夹杂界面之间含多条裂纹的问题进行了研究。基于复变函数分区全纯理论、留数定理、广义 Liouville 定理、Riemann-Schwarz 解析延拓定理及复应力函数奇性主部分析方法,获得了集中热源作用于准晶基体内任意一点时,准晶基体和圆形弹性夹杂内外温度场、声子场热应力的一般复势解。由此获得了含一条界面裂纹和两条界面裂纹时温度场以及声子场热应力的封闭形式解答,将所得结果与已有结果进行了对比,验证了该方法的有效性。最后通过数值算例分析了夹杂半径、点热源强度及裂纹角度对热应力和裂纹尖端热应力强度因子的影响规律。结果表明：随着热源强度的增大,裂纹尖端的声子场热应力也逐渐增大;随着裂纹角度的增大,裂纹尖端的声子场热应力强度因子变大;随着半径的增大,热应力强度因子的变化趋势越来越明显,并且取得的峰值越高,即裂纹角度和夹杂半径的增加,促进了裂纹的扩展。这些结论为准晶材料的结构设计和使用提供了科学依据。     

Stress solution for a crack at an arbitrary angle to an interface

Two-dimensional elasticity solution and the stress intensity factors are determined for a finite crack in one of the materials of a bimaterial composite. The crack has an arbitrary orientation and distance from the straight interface. The solution for general stress boundary conditions on the crack surface is presented in the form of coupled Fredholm integral equations of the second kind. Numerical values of the stress intensity factors are computed for various crack orientations, distances from the interface, and different combinations of material properties when the boundary conditions are uniform pressure and uniform shear stress.     

Kinking of an interface crack in an orthotropic bimaterial

We investigated the asymptotic problem of a kinked interface crack in an orthotropic bimaterial under in‐plane loading conditions. The stress intensity factors at the tip of the kinked interface crack are described in terms of the stress intensity factors of the interface crack prior to the kink combined with a dimensionless matrix function. Using a modified Stroh formalism and an orthotropy rescaling technique, the matrix function was obtained from the solutions of the corresponding problem in transformed bimaterial. The effects of orthotropic and bimaterial parameters on the matrix function were examined. A reduction in the number of dependent material parameters on the matrix function was made using the modified Stroh formalism. Moreover, the explicit dependence of one orthotropic parameter on the matrix function was determined using an orthotropic rescaling technique. The effects of the other material parameters on the matrix function were numerically examined. The energy release rate was obtained for a kinked interface crack in an orthotropic bimaterial.     

The stress intensity factors of slightly undulating interface cracks of bimaterials

A modified interface crack with slightly undulating profile, which has a good agreement with reality and retains the simplicity of a mathematical model, is presented in this paper. This model is utilized to reveal some of the properties of uneven cracks, especially the stress intensity factors. As we know, many failures occurring in the interface are induced by crucial lateral stresses which are parallel to the interface. Hence, when the lateral stresses are much stronger than others, the corresponding solution is also derived for understanding how the lateral stresses affect the stress intensity factors as the crack is uneven. In the present paper, the Hilbert's problem enables different perturbed-interface cracks to be solved in an unified manner. Muskhelishvili's potential formulation is used to derive, by means of a perturbation analysis technique, an homogeneous and general Hilbert's problem.     

Calculation of the stress intensity factor for arbitrary finite cracked body by using the boundary weight function method

The state of thermal stresses for a periodic two-layered elastic space weakened by an interface thermally insulated Griffith crack and loaded by concentrated line heat sources is investigated. The analysis is performed within the framework of the homogenized model with microlocal parameters. The stress intensity factors at the crack tips are determined. Two examples of the application of the results obtained are detailed.     

Boundary element analysis of structures with bridged interfacial cracks

The direct boundary integral equations method has been applied to analyze stresses in a fracture process zone (a crack bridged zone) and to calculate stress intensity factors module for structures with bridged interfacial cracks under mechanical loading. Bridged zones at interfacial cracks are considered as parts of these cracks with assumption that surfaces of interfacial cracks are connected by distributed spring-like bonds with given bond deformation law. For numerical analysis of piecewise structures with bridged interfacial cracks the multi-domain formulation of the boundary elements method is used. The stress intensity factors module evaluation is performed on the basis of displacements and stresses computed at nodal points of special quadratic boundary elements adjoined to a crack tip. The comparative study between the results obtained by the boundary elements method and the results obtained previously by the singular integral–differential equations method is performed and the validity of the presented numerical formulation is demonstrated. The new problem for a bridged circumferential crack between a cylindrical inclusion and a matrix in plate of finite size is also solved. Stresses distributions along the bridged zone and the stress intensity factors modulus dependencies versus the bridged zone length and bonds stiffness are presented and discussed for this problem.