On interface crack models with contact zones situated in an anisotropic bimaterial

Summary A boundary value problem for two semi-infinite anisotropic spaces with mixed boundary conditions at the interface is considered. Assuming that the displacements are independent of the coordinate x ₃, stresses and derivatives of displacement jumps are expressed via a sectionally holomorphic vector function. By means of these relations the problem for an interface crack with an artificial contact zone in an orthotropic bimaterial is reduced to a combined Dirichlet-Riemann problem which is solved analytically. As a particular case of this solution, the contact zone model (in Comninou's sense) is derived. A simple transcendental equation and an asymptotic formula for the determination of the real contact zone length are obtained. The classical interface crack model with oscillating singularities at the crack tips is derived from the obtained solution as well. Analytical relations between fracture mechanical parameters of different models are found, and recommendations concerning their implementation are given. The dependencies of the contact zone lengths on material properties and external load coefficients are illustrated in graphical form. The practical applicability of the obtained results is demonstrated by means of a FEM analysis of a finite-sized orthotropic bimaterial with an interface crack. Received 19 October 1998; accepted for publication 13 November 1998     

Model of the fracture process zone at the tip of a crack reaching the nonsmooth interface between elastic media

The paper is concerned with the fracture process zone at the tip of a crack at the nonsmooth interface between isotropic elastic media. A plane symmetric problem is formulated. The zone is modeled by lines of discontinuity of the normal displacement at the interface. The exact solution of the elastic problem is found by the Wiener-Hopf method Translated from Prikladnaya Mekhanika, Vol. 44, No. 10, pp. 13–22, October 2008.     

A transverse crack tip field in bimaterial

The asymptotic field near an interface crack tip is analyzed with the fully nonlinear theory. By dividing the crack tip field into narrowing sectors and an expanding sector, the asymptotic equations for the crack tip field are derived and solved. The singular characters of stress and strain near the crack tip are revealed.     

裂纹垂直于双相介质界面时的应力强度因子

本文利用Ｊ积分与应力强度因子的关系,采用有限元数值方法研究了当裂纹与双相介质的界面垂直时,其裂纹的近界面端和远界面端的应力强度因子随双相介质参数和裂纹端部到界面的距离的变化规律,同时还分析了当边裂纹逐渐扩展时,应力强度因子的变化特征。     

夹杂对两相材料界面裂纹的影响

根据界面上应力和位移的连续条件,得到了单向拉伸状态下,含有椭圆夹杂的无限大双材料组合板的复势解。进一步通过求解Hilbert问题,得到了含有夹杂和半无限界面裂纹的无限大板的应力场,并由此给出了裂尖的应力强度因子K。计算了夹杂的形状、夹杂的位置、夹杂的材料选取以及上、下半平面材料与夹杂材料的不同组合对裂尖应力强度的影响。计算结果表明夹杂到裂尖的距离和夹杂材料的性质对K影响较大,对于不同材料组合,该影响有较大差异。夹杂距裂尖较近时,会对K产生明显屏蔽作用,随着夹杂远离裂尖,对K的影响也逐渐减小。另外,软夹杂对K有屏蔽作用,硬夹杂对K有反屏蔽作用,而夹杂形状对K几乎没有影响。     

A crack perpendicular to and terminating at a bimaterial interface

Using dislocation simulation approach, the basic equation for a finite crack perpendicular to and terminating at a bimaterial interface is formulated. A novel expansion method is proposed for solving the problem. The complete solution to the problem, including the explicit formulae for theT stresses ahead of the crack tip and the stress intensity factors are presented. The stress field characteristics are analysed in detail. It is found that normal stresses {ie27-1} and {ie27-2} ahead of the crack tip, are characterised byQ fields if the crack is within a stiff material and the parameters |p _T | and |q _T | are very small, whereQ is a generalised stress intensity factor for a crack normal to and terminating at the interface. If the crack is within a weak material, the normal stresses {ie27-3} and {ie27-4} are dominated by theQ field plusT stress. This work was supported by the Swedish Research Council for Engineering Sciences.     

Stress field near interface crack tip of double dissimilar orthotropic composite materials

In this paper, double dissimilar orthotropic composite materials interfacial crack is studied by constructing new stress functions and employing the method of composite material complex. When the characteristic equations＇ discriminants △1 〉 0 and △2 〉0, the theoretical formula of the stress field and the displacement field near the mode I interface crack tip are derived, indicating that there is no oscillation and interembedding between the interfaces of the crack.     

Crack kinking from an interface crack with initial contact between the crack lips

The authors recently theoretically studied crack kinking and opening from an initially closed crack (without friction) in some homogeneous medium. The same problem, but for an interface crack, is considered here. Comninou has shown that the asymptotic stress field prior to kinking is governed by a single, mode II stress intensity factor (SIF). Using this result, plus a homogeneity property of the problems of elastic fracture mechanics with unilateral contact envisaged, a change of scale, and two reasonable hypotheses, we establish the expression of the SIF at the tip of the small, open crack extension. It is shown that whatever the geometry of the external boundary and the crack and whatever the loading, these SIF depend solely upon the initial (mode II) SIF (in a linear way), the kink angle and Dundurs' parameters α and β. Using this result and Goldstein and Salganik's “principle of local symmetry” to predict the kink angle, one finds that it is independent of the loading but does depend upon Dundurs' parameters α and β. This contrasts with the case of an ordinary (initially closed) crack in some homogeneous medium, for which the kink angle was not only independent of the loading but an absolute constant. However, it is numerically found that the influence of the mismatch of elastic properties upon the kink angle is rather weak.     

Analysis of stress intensity factor in orthotropic bi-material mixed interface crack

Adopting the complex function approach, the paper studies the stress intensity factor in orthotropic bi-material interface cracks under mixed loads. With consideration of the boundary conditions, a new stress function is introduced to transform the problem of bi-material interface crack into a boundary value problem of partial differential equations. Two sets of non-homogeneous linear equations with 16 unknowns are constructed. By solving the equations, the expressions for the real bi-material elastic constant εt and the real stress singularity exponents λt are obtained with the bi-material engineering parameters satisfying certain conditions. By the uniqueness theorem of limit,undetermined coefficients are determined, and thus the bi-material stress intensity factor in mixed cracks is obtained. The bi-material stress intensity factor characterizes features of mixed cracks. When orthotropic bi-materials are of the same material, the degenerate solution to the stress intensity factor in mixed bi-material interface cracks is in complete agreement with the present classic conclusion. The relationship between the bi-material stress intensity factor and the ratio of bi-material shear modulus and the relationship between the bi-material stress intensity factor and the ratio of bi-material Young's modulus are given in the numerical analysis.     

Fracture analysis of mode-II crack perpendicular to imperfect bimaterial interface

The problem of a mode-II crack close to and perpendicular to an imperfect interface of two bonded dissimilar materials is investigated.The imperfect interface is modelled by a linear spring with the vanishing thickness.The Fourier transform is used to solve the boundary-value problem and to derive a singular integral equation with the Cauchy kernel.The stress intensity factors near the left and right crack tips are evaluated by numerically solving the resulting equation.Several special cases of the mode-II crack problem with an imperfect interface are studied in detail.The effects of the interfacial imperfection on the stress intensity factors for a bimaterial system of aluminum and steel are shown graphically.The obtained observation reveals that the stress intensity factors are dependent on the interface parameters and vary between those with a fully debonded interface and those with a perfect interface.     

功能梯度双材料弱/微间断界面的冲击断裂分析

提出强间断、弱间断、微间断和全连续界面的概念与分类,建立功能梯度弹性双材料弱间断 界面冲击断裂问题的力学模型,采用积分变换法推导问题的Cauchy奇异积分方程,并用配 点法求得数值解. 分析表明,弱/微间断性对于FGMs界面裂纹应力强度因子有着重要影响, 而且微间断性是优于弱间断性的一种界面力学性能连接关系. 以FGMs界面某一侧 的力学性能函数在界面处的Taylor展开式的低阶项作为界面另一侧的力学性能函数,便可 以使FGMs界面成为``微间断'界面. 界面的一阶微间断对应力强度因子的减小作用较为明 显,而高阶(二阶及以上)微间断对应力强度因子的影响较小. 减小界面的弱间断程度或使 FGMs界面具备``微间断性',都将利于提高功能梯度双材料界面抗冲击断裂能力,在一定 程度上达到界面增韧的目的.     

Fracture-mechanical assessment of electrically permeable interface cracks in piezoelectric bimaterials by consideration of various contact zone models

Summary An interface crack with an artificial contact zone at the right-hand side crack tip between two piezoelectric semi-infinite half-planes is considered under remote mixed-mode loading. Assuming the stresses, strains and displacements are independent of the coordinate x ₂, the expression for the displacement jumps and stresses along the interface are found via a sectionally holomorphic vector function. For piezoceramics of the symmetry class 6 mm and for electrically permeable crack faces, the problem is reduced to a combined Dirichlet-Riemann boundary value problem which can be solved analytically. Further, analytical expressions for the stresses, electrical displacements, derivatives of elastic displacement jumps, stress and electrical intensity factors are found at the interface. Real contact zone lengths and the well-known oscillating solution are derived from the obtained solution as well. Analytical relationships between the fracture-mechanical parameters of various models are found, and recommendations are suggested concerning the application of numerical methods to the problem of an interface crack in the discontinuity area of a piezoelectric bimaterial. Received 16 March 1999; accepted for publication 31 May 1999     

多层材料结构的界面裂纹尖端复应力强度因子

本文建立了一种多层材料复合结构的界面裂纹问题分析模型。当两种材料之间插入第三种薄层弹性材料,裂纹位于第三种材料与第一或第二种弹性材料的界面上,且插页材料3~#的厚度相对于裂纹尺寸或平面内其他尺寸很小时,可以得到该问题裂纹尖端的复应力强度因子通式。本文用有限元法对结果进行了数值验证,并进行了有关问题的讨论。     

Initial kinking of an interface crack between two elastic media

The initial kinking of a thin fracture process zone near the crack tip under plane strain is studied using the Wiener-Hopf method. The crack is located at the interface between dissimilar elastic media. The fracture process zone is modeled by a straight line of normal displacement discontinuity emerging from the crack tip at an angle to the interface. The angle between the process zone and the interface is determined from the condition of strain energy maximum in the process zone. The dependences of the length and angle of the process zone on the external load and other parameters of the problem are studied. The results are compared with theoretical and experimental data obtained by other researchers __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 10, pp. 28–41, October 2007.     

The stress-strain fields at crack tip in axially cracked cylindrical shells and the calculation of stress intensity factors

A perturbation solution for stress-strain fields (including modes I, II, III) at crack tip in axially cracked cylindrical shells is given. The analysis, using 10th-order differential equations which take the transverse shear deformations into account, involves perturbation in a curvature parameter λ², (λ²=[12(1-v ²)]^1/2 a ²/Rh). Stress intensity factors for finite size cylindrical shells under bending and internal pressure loading are evaluated. A good accuracy can be obtained without using fine meshes in a region near the crack tip. Besides, the influence of the finite size and the shearing stiffness on bulging factors, which are commonly used in engineering, are analyzed.     

A study of dynamic caustics around running interface crack tip

Based on the first invariant of stress singular field in the vicinity of running tip of an interface crack, mapping equations of the caustic curve on the reference plane and the initial curve on the specimen plane are developed. The dynamic caustics are analyzed for the crack propagating along the interface between two bonded dissimilar materials. The variation of the caustic configurations is shown with the velocity change of the running crack and the ratio change of the stress intensity factors. Two characteristic dimensions are proposed that are not only practically measurable from optical caustic contours but also suitable to represent the behavior of transient caustics. The project supported by the National Natural Science Foundation of China and the Scientific Commission of Yunnan Province of China     

DYNAMIC STRESS FIELD AROUND THE MODE Ⅲ CRACK TIP IN AN ORTHOTROPIC FUNCTIONALLY GRADED MATERIAL

The problem of a Griffith crack in an unbounded orthotropic functionally graded material subjected to antipole shear impact was studied. The shear moduli in two directions of the functionally graded material were assumed to vary proportionately as definite gradient. By using integral transforms and dual integral equations, the local dynamic stress field was obtained. The results of dynamic stress intensity factor show that increasing shear moduli’s gradient of FGM or increasing the shear modulus in direction perpendicular to crack surface can restrain the magnitude of dynamic stress intensity factor.     

Effects of electric/magnetic impact on the transient fracture of interface crack in piezoelectric-piezomagnetic sandwich structure: anti-plane case

Due to the incompatibility of the interlaminar deformations,the interface debonding or cracking usually happens in a layered magnetoelectric(ME)structure under an applied load.In this paper,the transient responses of the anti-plane interface cracks in piezoelectric(PE)-piezomagnetic(PM)sandwich structures are studied by the standard methods of the integral transform and singular integral equation.Discussion on the numerical examples indicates that the PE-PM-PE structure under electric impact is more likely to fracture than the PM-PE-PM structure under a magnetic impact.The dynamic stress intensity factors(DSIFs)are more sensitive to the variation of the active layer thickness.The effects of the material constants on the DSIFs are dependent on the roles played by PE and PM media during the deformation process.     

Contact zone assessment for a fast growing interface crack in an anisotropic bimaterial

Summary A plane strain problem for a crack with a frictionless contact zone at the leading crack tip expanding stationary along the interface of two anisotropic half-spaces with a subsonic speed under the action of various loadings is considered. The cases of finite and infinite-length interface cracks under the action of a moving concentrated loading at its faces are considered. A finite-length crack for a uniform mixed-mode loading at infinity is considered as well. The associated combined Dirichlet-Riemann boundary value problems are formulated and solved exactly for all above-mentioned cases. The expressions for stresses and the derivatives of the displacement jumps at the interface are presented in a closed analytical form for an arbitrary contact zone length. Transcendental equations are obtained for the determination of the real contact zone length, and the associated closed form asymptotic formulas are found for small values of this parameter. It is found that independently of the types of the crack and loading, an increase of the crack tip speed leads to an increase of the real contact zone length and the correspondent stress intensity factor. The latter increase significantly for an interface crack tip speed approaching the Ragleigh wave speed.