The interaction of doubly periodic cracks

In this paper, an extremely accurate and efficient method for computing the interaction of a set of or multiple sets of general doubly periodic cracks has been presented on the basis of superposition principle, pseudo-traction method, and isolating analysis technique. A great number of typical examples are given in this paper. The stress intensity factors (SIF), the minimum strain energy density factors (SED) of crack tip and the critical stress (CRS) of crack growth are calculated with the accuracy of six significant digits for the rectangularly distributed periodic cracks and five significant digits for the general doubly periodic cracks. The relation of the interaction effect of the double periodic cracks with the periods and the ratio of crack length to crack spacing is analyzed. Also in this paper, the key technique problems for this method are discussed.     

THE CONSTITUTIVE RELATION OF CRACK-WEAKENED ROCK MASSES UNDER AXIAL-DIMENSIONAL UNLOADING

An accurate and efficient numerical method for solving the crack-crack interaction problem is presented. The method is mainly by means of the dislocation model, stress superposition principle and Chebyshev polynomial expansion of the pseudo-traction. This method can be applied to compute the stress intensity factors of multiple kinked cracks and multiple rows of periodic cracks as well as the overall strains of rock masses containing multiple kinked cracks under complex loads. Many complex computational examples are given. The dependence of the crack-crack interaction on the crack configuration, the geometrical and physical parameters, and loads pattern, is investigated. By comparison with numerical results under confining pressure unloading, it is shown that the crack-crack interaction under axial-dimensional unloading is weaker than those under confining pressure unloading. Numerical results for single faults and crossed faults show that the single faults are more unstable than the crossed faults. It is found from numerical results for different crack lengths and different crack spacing that the interaction among kinked cracks decreases with an increase in length of the kinked cracks and the crack spacing under axial-dimensional unloading.     

爆炸荷载作用下两平行裂纹对扩展中裂纹的影响规律

用实验和数值模拟方法,研究在爆炸载荷下岩体内部一对平行裂纹对扩展主裂纹的影响规律。实验中,采用带有中心装药孔及预制裂纹的砂岩圆盘试件,利用由示波器、超动态应变仪及裂纹扩展计所组成的测试系统,监测主裂纹扩展速度和扩展距离;数值模拟中,采用了AUTODYN软件进行,模拟了主裂纹及两平行裂纹的扩展规律,对岩石材料,采用线性状态方程及最大拉应力失效准则,并在两平行裂纹间设置相应的观测点记录应力曲线。通过实验与数值模拟分析,得到:爆炸载荷下,紧随冲击波后的稀疏波经过两平行裂纹面反射后变成压缩波,并在两平行裂纹间产生垂直于主裂纹扩展方向的压应力,对裂纹的扩展有压制、止裂作用;而且,这种压应力的大小与两平行裂纹的间距有关,进而导致了不同的止裂效果,影响裂纹的扩展速度及最终扩展长度。     

Bifurcation condition of crack pattern in the periodic rectangular array

Bifurcation condition of crack pattern in the periodic rectangular array plays an important role in determining the final failure pattern of rock material. An approximation for the critical crack size/spacing ratio is established for a uniformly growing periodic rectangular array yields to a non-uniform growing pattern of crack growth. Numerical results show that the critical crack size/spacing ratio λ_cr depends on the number of cracks, the crack spacing, the perpendicular distance between two adjacent rows, as well as the loading conditions. In general, λ_cr increases with the number of lines. It is observed that the critical crack size/spacing ratio λ_cr for the periodic rectangular array decreases with an increase in the perpendicular distance between two adjacent rows. It is clear that the critical crack size/spacing ratio λ_cr for the periodic rectangular array under shear stress increases with increasing the crack spacing.     

应变率对含裂隙红砂岩裂纹扩展模式及破碎特征的影响

以含不同倾角预制裂纹的长方形板状红砂岩为研究对象,采用分离式霍普金森压杆沿试样宽度方向施加冲击荷载,使用高速摄像机记录裂纹扩展过程,获得试样的裂纹路径特征以及动态压缩强度和动态弹性模量,利用筛分统计法分析试样碎块分布特征,结合分形理论定量描述试样破碎程度及特点,探讨中应变率条件下含裂隙试样裂纹扩展模式与动态力学性质和破碎程度的相互关系。研究结果表明,应变率较高时试样会更多地出现远场裂纹和离层裂纹,并且相比相关低应变率实验结果,中应变率范围内试样破坏模式及裂纹分布情况随应变率的变化规律是不同的。随着应变率的提高,试样大体上从1条拉伸裂纹的临界破坏演变成X形剪切裂纹为主的复杂裂隙网络,并且不同角度预制裂隙对于这种裂纹扩展模式的演变有重要影响。在预制裂纹倾角一定的情况下,岩样动态压缩强度和动态弹性模量表现出明显的应变率效应,不同角度预制裂纹对于试样的应变率敏感性有显著影响。随裂纹倾角的增大,试样的动态强度、动态弹性模量和分形维数表现出的变化趋势具有一定的相似性,大体呈现先减小后增大的趋势,裂纹倾角为45°的试样的动态压缩强度、动态弹性模量和分形维数均为最小。随应变率的升高,不同预制裂纹倾角的试样碎块分布更加分散,应变率越高,预制裂纹倾角对于岩石冲击破碎程度、分形维数的影响越显著。

     

MULTIPLE PARALLEL SYMMETRIC PERMEABLE MODEL-Ⅲ CRACKS IN A PIEZOELECTRIC/PIEZOMAGNETIC COMPOSITE MATERIAL PLANE

In this paper, the interactions of multiple parallel symmetric and permeable finite length cracks in a piezoelectric/piezomagnetic material plane subjected to anti-plane shear stress loading are studied by the Schmidt method.The problem is formulated through Fourier transform into dual integral equations, in which the unknown variables are the displacement jumps across the crack surfaces.To solve the dual integral equations, the displacement jumps across the crack surfaces are directly expanded as a series of Jacobi polynomials.Finally, the relation between the electric field, the magnetic flux field and the stress field near the crack tips is obtained.The results show that the stress, the electric displacement and the magnetic flux intensity factors at the crack tips depend on the length and spacing of the cracks.It is also revealed that the crack shielding effect presents in piezoelectric/piezomagnetic materials.     

Interaction between collinear periodic cracks in an infinite piezoelectric body

The problem of collinear periodic cracks in an infinite piezoelectric body is studied. Effect of saturation strips at the crack-tips is taken into account. By means of the Stroh formalism and the conformal mapping technique, the general periodic solutions for collinear cracks are obtained. The stress intensity factors and the size of saturation strips are derived analytically, and their dependencies on the ratio of the periodicity on the half-length of the crack are analyzed in detail. Numerical results show the following two facts. (1) When h/l>4.0, the stress intensity factors become almost identical to those of a single crack in an infinite piezoelectric body. This indicates that the interaction between cracks can be ignored in establishing the criterion for the crack initiation in this case. (2) The speed of the saturation strip size of periodic cracks approaching that of a single crack depends on the electric load applied at infinity. In general, a large electric load at infinity is associated with a slow approaching speed.     

THERMAL FRACTURE OF FUNCTIONALLY GRADED PLATE WITH PARALLEL SURFACE CRACKS

This work examines the fracture behavior of a functionally graded material （FGM） plate containing parallel surface cracks with alternating lengths subjected to a thermal shock. The thermal stress intensity factors （TSIFs） at the tips of long and short cracks are calculated using a singular integral equation technique. The critical thermal shock △Tc that causes crack initiation is calculated using a stress intensity factor criterion. Numerical examples of TSIFs and △Tc for an Al2O3/Si3N4 FGM plate are presented to illustrate the effects of thermal property gradation, crack spacing and crack length ratio on the TSIFs and △Tc. It is found that for a given crack length ratio, the TSIFs at the tips of both long and short cracks can be reduced significantly and △Tc can be enhanced by introducing appropriate material gradation. The TSIFs also decrease dramatically with a decrease in crack spacing. The TSIF at the tips of short cracks may be higher than that for the long cracks under certain crack geometry conditions. Hence, the short cracks instead of long cracks may first start to grow under the thermal shock loading.     

Fatigue damage evolution in silicon films for micromechanical applications

In this paper we examine the conditions for surface topography evolution and crack growth/fracture during the cyclic actuation of polysilicon microelectromechanical systems (MEMS) structures. The surface topography evolution that occurs during cyclic fatigue is shown to be stressassisted and may be predicted by linear perturbation analyses. The conditions for crack growth (due to pre-existing or nucleated cracks) are also examined within the framework of linear elastic fracture mechanics. Within this framework, we consider pre-existing cracks in the topical SiO₂ layer that forms on the Si substrate in the absence of passivation. The thickening of the SiO₂ that is normally observed during cyclic actuation of Si MEMS structures is shown to increase the possibility of stable crack growth by stress corrosion cracking prior to the onset of unstable crack growth in the SiO₂ and Si layers. Finally, the implications of the results are discussed for the prediction of fatigue damage in silicon MEMS structures.     

不同倾角预制裂纹缺陷与运动裂纹的相互作用

为了研究运动裂纹与不同倾角预制裂纹缺陷之间的作用机制,采用数字激光动态焦散线方法对含不同倾角预制裂纹缺陷的三点弯曲梁进行冲击实验。研究结果表明,在冲击载荷作用下,预制裂纹缺陷尖端均产生了次生裂纹;当运动主裂纹与预制裂纹缺陷贯通时,次生裂纹不会立即起裂,而是经过0~10 μs的能量积蓄后,次生裂纹才起裂;运动主裂纹应力强度因子峰值与次裂纹起裂时的应力强度因子值均随预制裂纹倾角的增大而增大。     

Quasi-brittle fracture diagram of structured bodies in the presence of edge cracks

The Neuber-Novozhilov approach is used to obtain necessary and sufficient fracture criteria. Using a modified Leonov-Panasyuk-Dugdale model, simple relations for the critical fracture parameters are derived for opening mode edge cracks for the case where the diameter of the prefracture zone coincides with the diameter of the plasticity zone. These relations are suitable for studying fracture where the crack length is negligibly small. A fracture diagram using critical stresses under both criteria is proposed for a wide range of crack length. At a certain level of loading, three regions are identified, in the first of which the crack is stable, in the second, the crack extends but remains stable, and in the third, the crack is unstable. Experimental data on the fracture of specimens with edge cracks are obtained. It is established that the theoretical critical fracture curves are in good agreement with the obtained critical parameters for flat tensile specimens with two collinear edge cracks.     

带有两个共线裂纹载流薄板温度场的复变函数解

采用复变函数方法对带有两个共线穿透裂纹的导电薄板，在瞬间电流作用下的温度场进行了计算。针对带有两个共线裂纹或夹杂的载流薄板形成了边值问题，给出封闭形式的解，得到了电流在裂纹尖端的奇异特征。在形成了点热源的基础上，导出裂纹端附近区域的温度场表达式，为计算裂纹尖端附近应力场打下了基础。     

STUDY ON THE SURFACE CRACK GROWTH BEHAVIOR IN 14MnNbq BRIDGE STEEL

Three-dimensional crack closure correction methods are investigated in this paper.The fatigue crack growth tests of surface cracks in 14MnNbq steel for bridge plate subjected to tensile and bending loadings are systematically conducted.The experimentally measured fatigue crack growth rates of surface cracks are compared with those of through-thickness cracks in detail.It is found that the crack growth rates of surface cracks are lower than those of through-thickness cracks.In order to correct their differences in fatigue crack growth rates, a dimensionless crack closure correction model is proposed.Although this correction model is determined only by the experimental data of surface cracks under tensile loading with a constant ratio R=0.05, it can correlate the surface crack growth rates with reasonable accuracy under tensile and bending loadings with various stress ratios ranging from 0 to 0.5.Furthermore, predictions of fatigue life and crack aspect ratio for surface cracks are discussed, and the predicted results are also compared with those obtained from other prediction approaches.Comparison results show that the proposed crack closure correction model gives better prediction of fatigue life than other models.     

层状磁电复合材料界面共线裂纹平面问题分析

本文研究了面内电磁势载荷作用下双层压电压磁复合材料中共线界面裂纹问题．考虑了压电材料的导磁性质和压磁材料的介电性质,引入了界面电位移和磁感强度的连续性条件．利用Fourier 变换得到一组第二类Cauchy 型奇异积分方程．进一步导出了相应问题的应力强度因子、电位移强度因子和磁感强度强度因子的表达式,给出了应力强度因子的数值结果．结果表明电磁载荷会导致界面裂纹尖端I、II 混合型应力奇异性,同时还伴随着电位移和磁感强度的奇异性．比较了双裂纹左右端的应力强度因子,发现在面内极化方向上施加面内磁势载荷时共线裂纹内侧尖端区域的两个法向应力场发生互相干涉增强．     

A novel method for residual strength prediction for sheets with multiple site damage: Methodology and experimental validation

A unified method for solving the strip yield model for collinear cracks in finite and infinite sheet is proposed. The method is based on the weight function of a single crack. Two collinear cracks in finite and infinite sheets are used to apply and verify this method. The plastic zone size, crack opening displacement and stress distribution along the ligament between cracks obtained by using the present method are extensively compared with existing available results and finite element solutions, and very good agreements are observed. Combined with the Crack Tip Opening Angle (CTOA) criterion, the unified method is used to predict the crack growth behavior and residual strength for 2024-T3 aluminum alloy sheet with Multiple Site Damage (MSD). Thirty-two sheets with four types of MSD are designed and tested to verify this method. It is shown that the present method is able to predict various crack growth behaviors observed in experiment. The predicted residual strengths are within 9% of the corresponding test results. Compared to the elastic–plastic finite element method, the present method is much more efficient.     

On safe crack shapes in elastic bodies

According to the Griffith criterion, a crack propagation occurs, provided that the derivative of the energy functional with respect to the crack length reaches some critical value. We consider a generalization of this criterion to the case of nonlinear cracks satisfying a nonpenetration condition and investigate the dependence of the shape derivative of the energy functional on the crack shape. In the paper, we find the crack shape which gives the maximal deviation of the energy functional derivative from a given critical value and, in particular, prove that this optimality problem admits a solution.     

基于局域分析的疲劳短裂纹群体演化随机模型

采用局域裂纹数密度描述金属材料中不同局部区域的疲劳短裂纹群体损伤的发展情况通过考虑在不同局域存在的材料性质的随机涨落及局部损伤对损伤总量发展的影响，建立了局域裂纹数密度演化随机方程对方程数值求解从而模拟了材料的疲劳短裂纹损伤过程结果显示出主裂纹出现的随机性，并讨论了裂纹总数与最大裂纹尺度在统计意义上的演化特征     

Interaction of microcracks with a macrocrack yielded in a narrow strip

Plastic zone growth of collinear cracks has had a longstanding interest in ductile fracture. This work further considers yield zone growth in an isotropic, homogeneous elastic–perfectly plastic infinite plate containing a macrocrack with several neighboring microcracks. Normal loading is considered at distances far away from the cracks. The strip yield is adopted where the plastic zone is assumed to be confined to two narrow strips extending from the ends of a finite length crack while the microcracks are assumed to be elastic. The plastic zone length and crack opening displacement are found from asymptotic solution and compared with finite element solution.     

两种各向异性材料界面共线裂纹的反平面问题

本文研究两种各向异性材料界面共线裂纹的反平面剪切问题。利用复变函数方法，提出了一般问题公式和某些实际重要问题的封闭形式解。考察了裂纹尖端附近的应力分布并给出了应力强度因子公式。从本文解签的特殊情形，可以直接导出两种各向同性材料界面裂纹，均匀各向异性材料共线裂纹以及均匀各向同性材料共线裂纹的相应问题公式，其中包括已有的经典结果。     

半无限横观各向同性体中共面双裂纹边界元分析

基于双横观各向同性材料基本解的对偶边界元方法,分析了半无限横观各向同性材料中两条共面裂纹的相互作用问题。裂纹面上分别作用着法向和切向两种均布荷载,材料的各向同性面及裂纹面平行于半无限域自由面。数值计算得到了该类裂纹的应力强度因子(SIF)值和两条共面裂纹相互作用的SIF值影响系数。根据数值结果分析了自由面对该类裂纹SIF值的影响,以及裂纹间距与形状、自由面对SIF值影响系数的影响。结果表明,自由面对作用法向力时的该类裂纹SIF值有明显的影响,裂纹间距与形状对SIF值影响系数影响较大,但自由面对SIF值影响系数基本无影响。