Brittle fracture dynamics with arbitrary paths. II. Dynamic crack branching under general antiplane loading

The dynamic propagation of a bifurcated crack under antiplane loading is considered. The dependence of the stress intensity factor just after branching is given as a function of the stress intensity factor just before branching, the branching angle and the instantaneous velocity of the crack tip. The jump in the dynamic energy release rate due to the branching process is also computed. Similar to the single crack case, a growth criterion for a branched crack is applied. It is based on the equality between the energy flux into each propagating tip and the surface energy which is added as a result of this propagation. It is shown that the minimum speed of the initial single crack which allows branching is equal to 0.39c, where c is the shear wave speed. At the branching threshold, the corresponding bifurcated cracks start their propagation at a vanishing speed with a branching angle of approximately 40°.     

Dynamic crack tip fields and dynamic crack propagation characteristics of anisotropic material

Based on mechanics of anisotropic material, the dynamic crack propagation problem of I/II mixed mode crack in an infinite anisotropic body is investigated. Expressions of dynamic stress intensity factors for modes I and II crack are obtained. Components of dynamic stress and dynamic displacements around the crack tip are derived. The strain energy density theory is used to predict the dynamic crack extension angle. The critical strain energy density is determined by the strength parameters of anisotropic materials. The obtained dynamic crack tip fields are unified and applicable to the analysis of the crack tip fields of anisotropic material, orthotropic material and isotropic material under dynamic or static load. The obtained results show Crack propagation characteristics are represented by the mechanical properties of anisotropic material, i.e., crack propagation velocity M and fiber direction α. In particular, the fiber direction α and the crack propagation velocity M give greater influence on the variations of the stress fields and displacement fields. Fracture angle is found to depend not only on the crack propagation but also on the anisotropic character of the material.     

Brittle fracture dynamics with arbitrary paths I. Kinking of a dynamic crack in general antiplane loading

We present a new method for determining the elasto-dynamic stress fields associated with the propagation of anti-plane kinked or branched cracks. Our approach allows the exact calculation of the corresponding dynamic stress intensity factors. The latter are very important quantities in dynamic brittle fracture mechanics, since they determine the crack path and eventual branching instabilities. As a first illustration, we consider a semi-infinite anti-plane straight crack, initially propagating at a given time-dependent velocity, that changes instantaneously both its direction and its speed of propagation. We will give the explicit dependence of the stress intensity factor just after kinking as a function of the stress intensity factor just before kinking, the kinking angle and the instantaneous velocity of the crack tip.     

Damage growth by debonding in a single fibre metal matrix composite: Elastoplasticity and strain energy density criterion

A displacement-based finite element-based numerical approach has been employed to study the damage growth in a unidirectional SiC/Al composite containing a pre-existing crack along the fibre/matrix interface. The composite is modeled as a two-material cylinder subjected to uniform displacement. A detailed analysis is made for the stress field in the vicinity of the debond crack tip. This approach incorporates an elastic-plastic analysis combined with a strain energy density criterion to predict debonded crack growth direction, extended stable growth and final termination. The influence of contact taking place between the debonded surfaces is also considered. It is shown that such surface contact leads to reduced stress and strain fields around the crack tip, while the extent of reduction is increased with debonding length. By combining the reduced stress field with the strain energy density criterion, a limiting value for the debonding extension can be calculated for the critical applied displacement that led to fibre fracture.     

Stress-strain field near crack tip and calculation of critical stress of crack propagation in a pure bending beam of rectangular section with one-sided mode i crack

This paper studies the stress-strain field near crack tip in a pure bending beam of rectangular section with one-sided mode I crack by the analytic method of Ref. [1], then it gives the stress and strain components at the crack tip when the crack propagates and further it obtains the formulas of calculating the elastic deformed area width, the deformed intensity area width and the equation groups of calculating the critical stress of crack propagation, last the equation group of calculating critical stress of crack propagation is verified by calculating instance. The maximum error is 0.18%. First Received May 7, 1994.     

Stress intensity factors for curved and kinked cracks in plane extension

A singular integral equation containing the crack opening displacement (COD) is developed for solving plane elasticity problems. The crack may contain any number of kinks at different intervals and orientations, such as a saw-tooth shape. Cracks in the form of a sine wave can also be treated. The crack tip stress intensity factors are evaluated for a variety of crack shapes and the results are displayed graphically. The distance between the crack tips is found to be a dominant factor on the crack tip stress intensity while the angle between the tangent to the crack tip and load direction determines the proportion of Mode I and II stress intensity factors.     

Fatigue damage in polycrystals – Part 2: Intrinsic scatter of fatigue life

Part 2 deals with the evolution of plastic flow resistance with crack growth from its minimum value (fatigue limit) towards its saturated bulk value (cyclic yield stress). The far-field stress level, the geometry of the crack and the grain size distribution of the material are those parameters that control the area of crack tip plasticity and hence the rate towards saturation. The implication of the far-field stress is held responsible for the violation of the similitude concept and the failure of the stress intensity factor to describe conditions of short cracking. However, an engineering tool based on the stress intensity factor and being able to predict the fatigue life of short cracks can be constructed, considering that the distribution of crack growth rates is intrinsically defined by the material itself. The above allows the development of a set of equations able to construct the fatigue life scatter of the material.     

皮质骨断裂力学行为的实验研究

采用稳态裂纹扩展和疲劳裂纹扩展的实验, 研究了牛皮质骨横向和纵向裂纹扩展的断裂力学行为. 沿着两个方向制备了紧凑拉伸(CT)试件. 由于试件尺寸的限制,采用数值计算方法确定了裂纹尖端应力强度因子与裂纹长度的关系. 在实验中, 采用数字图像相关法精确测定裂纹尖端的位置. 由于裂纹沿横向扩展时有较大的偏斜, 将采用$J$积分测量其断裂韧性. 实验结果表明, 在裂纹扩展的一定范围内, 皮质骨的断裂韧度随着裂纹不断扩展而增大, 即表现出上升的阻力曲线(R-curve)性质.而皮质骨的横向裂纹扩展的断裂韧度要远远大于纵向裂纹扩展的断裂韧度, 表现出各向异性的阻力曲线行为. 在疲劳裂纹扩展中, 纵向疲劳裂纹扩展率要大于横向疲劳裂纹扩展率, 这说明皮质骨具有各向异性的疲劳裂纹扩展性质.      

基于DIC的爆炸加载下脆性材料裂纹扩展规律的试验研究

通过数字图像相关法（DIC），应用PMMA对爆炸加载条件下脆性材料的裂纹扩展规律进行了试验研究。基于对称性试验模型，实现了裂纹尖端位置和应变场信息的同步记录。以此为基础，通过对比分析获知，主应变场应变值最大点不能作为裂纹尖端的判断依据。并以动态裂纹扩展速度为参量，应用断裂动力学和最小二乘牛顿迭代法，计算出了考虑惯性效应的Ⅰ-Ⅱ混合型裂纹的应力强度因子:K_Ⅰ和K_Ⅱ值会随着裂纹扩展方向改变而发生突变；K_Ⅰ最大值为2.63 MPa·m1/2，最小值为0.89 MPa·m1/2；其整体变化趋势表明，爆炸加载条件下脆性材料裂纹扩展随能量积聚和释放呈循环阶梯式递减发展。     

Torsional impact of transversely isotropic solid with functionally graded shear moduli and a penny-shaped crack

The torsional impact response of a penny-shaped crack in an unbounded transversely isotropic solid is considered. The shear moduli are assumed to be functionally graded such that the mathematics is tractable. Laplace transform and Hankel transform are used to reduce the problem to solving a Fredholm integral equation. The crack tip stress fields are obtained. Investigated are the influence of material nonhomogeneity and orthotropy on the dynamic stress intensity factor. The peak value of the dynamic stress intensity factor can be suppressed by increasing the shear moduli's gradient and/or increasing the shear modulus in a direction perpendicular to the crack surface.     

Fictitious Notch Rounding approach of pointed V-notch under in-plane shear

The Fictitious Notch Rounding approach (FNR) is applied here for the first time to V-shaped notches under in-plane shear loading. The fictitious radius is evaluated for different opening angles as a function of the Microstructural Characteristic Length (MCL), the actual radius and the failure hypothesis. A multiaxiality factor is introduced and found to be very sensitive to the opening angle. Under mode II loading, the problem is more complex than under mode I and mode III, mainly because the maximum elastic stress is outside the notch bisector line. The main problem is the choice of the expected crack initiation angle, which defines the direction where the relevant stress has to be integrated. This integration carried out over the MCL gives the effective stress value for the pointed V-notch. To this end, two different criteria are used, the Maximum Tangential Stress (MTS) criterion and the Minimum Strain Energy Density criterion (MSED). A large number of finite element analyses have been carried out to determine the multiaxiality factor. This factor has been evaluated by comparing the theoretical stress concentration factor (SCF) obtained from fictitiously rounded notches to the effective stress concentration factor obtained by integrating the relevant stress over the Microstructural Characteristic Length.     

Micro-crack tip fracture of commercial grade aluminum under mixed mode loading

In situ tensile tests were made in a scanning electron microscope (SEM) to investigate the deformation and micro-fracture in the immediate vicinity of a micro-crack tip in commercial pure aluminum with large-size crystal. Examined are the slip line field, stress intensity factor, strain energy density factor and crack tip opening displacement (CTOD) for mixed mode loading. Blunting and sharpening effects are observed. The latter is controlled by localized slip while the former by uniformed slip of the operating slip system with the highest crack tip Schmid factor. The operating slip system depends on the crystallographic orientation of crystal containing micro-cracks.The damage and fracture take place in the blunted region and depend on the coarsening and spacing of uniformed slip lines. The mixed mode micro-crack propagates along the direction where the voids grow and coalesce into the micro-crack. The direction also depends on the orientation of the applied loading. This suggests that the formation of macro-fracture mechanics could be applied. In particular, the minimum strain energy density criterion is suitable for determining the direction of micro-crack instability in the mixed mode. The in situ data were used to yield a nearly constant critical, minimum strain energy density factor for onset of micro-cracking.     

Quasicontinuum simulation of single crystal nano-plate with a mixed-mode crack

The quasicontinuum (QC) method is employed to simulate a nickel single crystal nano-plate with a mixed-mode crack. Atomic stresses near the crack tip are fitted according to the elastoplastic fracture mechanics equations. It is found that the atomic stress fields neighboring the crack tip are also singular and controlled by the atomic stress intensity factors. And then the critical energy release rates for brittle and ductile fracture are computed and compared in order to predict crack propagation or dislocation emission. Four possible slip directions at the crack tip are pointed out. Finally, the slip direction around the crack tip is determined by the shear stress and it is well consistent with the atomic pictures from the QC simulation.     

Meshless simulation of crack propagation in multiphase materials

The material body considered in this work consists of multiphases. Digital imaging data are taken as the input to specify the configuration and composition of the specimen. Meshless method is demonstrated as a superior numerical tool to analyze crack initiation and propagation in multiphase material. A fracture criterion, based on the ratio of the opening stress over the material toughness distributed in front of the crack tip, is proposed to determine the direction of crack propagation of mixed mode fracture problem in multiphase material. Numerical results are presented and discussed.     

Numerical analysis of quasi-static crack branching in brittle solids by a modified displacement discontinuity method

Mechanism of quasi-static crack branching in brittle solids has been analyzed by a modified displacement discontinuity method. It has been assumed that the pre-existing cracks in brittle solids may propagate at the crack tips due to the initiation and propagation of the kink (or wing) cracks. The originated wing cracks will act as new cracks and can be further propagated from their tips according to the linear elastic fracture mechanics (LEFM) theory. The kink displacement discontinuity formulations (considering the linear and quadratic interpolation functions) are specially developed to calculate the displacement discontinuities for the left and right sides of a kink point so that the first and second mode kink stress intensity factors can be estimated. The crack tips are also treated by boundary displacement collocation technique considering the singularity variation of the displacements and stresses near the crack tip. The propagating direction of the secondary cracks can be predicted by using the maximum tangential stress criterion. An iterative algorithm is used to predict the crack propagating path assuming an incremental increase of the crack length in the predicted direction (straight and curved cracks have been treated). The same approach has been used for estimating the crack propagating direction and path of the original and wing cracks considering the special crack tip elements. Some example problems are numerically solved assuming quasi-static conditions. These results are compared with the corresponding experimental and numerical results given in the literature. This comparison validates the accuracy and applicability of the proposed method.     

Stability of an advancing crack to small perturbation of its path

In this work we investigate the stability of a nominally straight two-dimensional quasistatically growing crack to a small perturbation of its path. Formulae for perturbations of stress intensity factors induced by slight deviation of the crack trajectory were developed by Movchan et al. (Int. J. Solids Struct. 35, 3419) Their solution is exploited to derive an equation for the perturbation of the crack path on the assumption that the crack advances in pure “opening” mode (i.e. local K_II=0). Various types of loading conditions are considered, including a cracked body loaded by a pair of point body forces and a crack whose faces are subjected to given tractions acting in the direction normal to the crack boundary. The body is also subjected to a remotely maintained uniaxial stress, aligned with the direction of the unperturbed crack. The loading is assumed to advance as the crack advances, to maintain the critical value of Mode I stress intensity factor. Numerical computations of possible crack paths have been performed, extending results on crack stability obtained by Cotterell and Rice (Int. J. Fract. 16, 155). The results show that in the case of loading by point body forces the stability of the crack path depends on the positions of the points of application of the applied forces and the magnitude of the applied stress acting parallel to the crack. There exists a critical value of this stress such that the crack path is stable for values less than critical and unstable otherwise. It is shown that the crack is always unstable in the case of point force tractions applied normal to the crack faces.     

Mixed mode fatigue crack growth with a sudden change in loading direction

With a sudden change in the maximum load level, there will be a corresponding change in the crack driving force regardless of whether the load is applied monotonically or cyclically. The effective strain energy density factor range ΔS_p,eff has been used to correlate mixed mode fatigue crack propagation where the crack growth direction is not known as an a priori. Examined in this work is a sudden change of load direction on fatigue crack growth while the load level remains unchanged. Yielding is assumed to be localized near the crack tip such that the crack growth behavior can be described adequately by the elastic stress field. Under the conditions investigated, minimal change on crack growth rates is observed. No firm conclusion could be drawn on deviation of crack path for the case considered.     

压应力对压剪裂纹扩展的影响研究

本文针对压剪裂纹的启裂及扩展问题,研究了脆性材料中裂纹面压应力变化对其扩展的影响规律.借助双轴加载试验机可自由调节两个轴位移和力的优势,设计了一种单边对称双裂纹压剪试样.试验中,施加裂纹面压应力至不同的预设值后,使剪应力单调增大直至裂纹启裂及扩展,得到不同预设压应力下压剪裂纹启裂及扩展规律. 随着预设压应力增大,启裂角增大,裂纹扩展路径与初始裂纹的偏角也增大.但随着压应力增大,启裂角增大至一定值后趋于稳定,实验发现,依据裂纹是否闭合,压应力对压剪裂纹扩展的作用大致可分为两个阶段:闭合前,压应力对裂纹启裂载荷及启裂角、扩展路径均有影响,预设压应力增大,裂纹启裂载荷增大、启裂角增大,扩展路径愈来愈偏离初始裂纹方向;闭合后,压应力虽然增大,启裂角和临界压剪应力比始终恒定,压应力对临界剪力和扩展路径存在一定影响.研究表明,裂纹启裂角与启裂时的压剪应力比存在一定的对应关系.启裂时的压剪应力比增大,启裂角增大,启裂时的压剪应力比恒定,启裂角不变.      

Impact failure prediction of Mode III crack in orthotropic functionally graded strip

Mode III impact of a crack in an orthotropic functionally graded strip is investigated. The shear moduli in two directions of the material are assumed to vary proportionately with gradient. Laplace transform and Fourier cosine transform are used to reduce the problem to solving a Fredholm integral equation. The crack tip stress field is obtained by considering the asymptotic behavior of Bessel function. Energy density factor criterion is applied to obtain the maximum of minimum energy density and direction of crack initiation. Numerical results are given graphically. The effects of orthotropy, nonhomogeneity and height of the strip on the energy density factor are discussed.