Nano-ductile crack propagation in glasses under stress corrosion: spatiotemporal evolution of damage in the vicinity of the crack tip

Recent experiments have evidenced the existence of a ductile fracture mode at the nanometer scale in Aluminosilicate glass. The present study is designed to check whether such a ductile mode is inherent to the amorphous nature of glass. Therefore, the slow crack advance is observed in real time via an Atomic Force Microscope in a minimal glass, amorphous Silica, under stress corrosion. In this case, the Crack propagation proceeds by the nucleation, growth and coalescence of damage cavities as in the Aluminosilicate glass, but the cavity size is significantly larger. We focus here on the kinematics of crack propagation by looking at the spatio-temporal evolution of both the tip of the main crack and the cavity ahead. It is shown that the velocity of the main crack tip is significantly lower than the one of the cavity edge toward the main crack tip, like in metallic alloys. Moreover, the velocities of the different fronts (main crack, frontward and backward cavity tips) at these nanometric scales is one order of magnitude smaller than the crack tip velocity at the continuum scale. This has important consequences for the modelling of stress corrosion, especially at ultra-slow crack propagation.     

Dynamic behaviors of mode III interfacial crack under a constant loading rate

In an earlier study on intersonic crack propagation, Gao et al. (J. Mech. Phys. Solids 49: 2113–2132, 2001) described molecular dynamics simulations and continuum analysis of the dynamic behaviors of a mode II dominated crack moving along a weak plane under a constant loading rate. The crack was observed to initiate its motion at a critical time after the onset of loading, at which it is rapidly accelerated to the Rayleigh wave speed and propagates at this speed for a finite time interval until an intersonic daughter crack is nucleated at a peak stress at a finite distance ahead of the original crack tip. The present article aims to analyze this behavior for a mode III crack moving along a bi-material interface subject to a constant loading rate. We begin with a crack in an initially stress-free bi-material subject to a steadily increasing stress. The crack initiates its motion at a critical time governed by the Griffith criterion. After crack initiation, two scenarios of crack propagation are investigated: the first one is that the crack moves at a constant subsonic velocity; the second one is that the crack moves at the lower shear wave speed of the two materials. In the first scenario, the shear stress ahead of the crack tip is singular with exponent ?1/2, as expected; in the second scenario, the stress singularity vanishes but a peak stress is found to emerge at a distance ahead of the moving crack tip. In the latter case, a daughter crack supersonic with respect to the softer medium can be expected to emerge ahead of the initial crack once the peak stress reaches the cohesive strength of the interface.     

Multiscale modeling of crack initiation and propagation at the nanoscale

Fracture occurs on multiple interacting length scales; atoms separate on the atomic scale while plasticity develops on the microscale. A dynamic multiscale approach (CADD: coupled atomistics and discrete dislocations) is employed to investigate an edge-cracked specimen of single-crystal nickel, Ni, (brittle failure) and aluminum, Al, (ductile failure) subjected to mode-I loading. The dynamic model couples continuum finite elements to a fully atomistic region, with key advantages such as the ability to accommodate discrete dislocations in the continuum region and an algorithm for automatically detecting dislocations as they move from the atomistic region to the continuum region and then correctly “converting” the atomistic dislocations into discrete dislocations, or vice-versa. An ad hoc computational technique is also applied to dissipate localized waves formed during crack advance in the atomistic zone, whereby an embedded damping zone at the atomistic/continuum interface effectively eliminates the spurious reflection of high-frequency phonons, while allowing low-frequency phonons to pass into the continuum region.The simulations accurately capture the essential physics of the crack propagation in a Ni specimen at different temperatures, including the formation of nano-voids and the sudden acceleration of the crack tip to a velocity close to the material Rayleigh wave speed. The nanoscale brittle fracture happens through the crack growth in the form of nano-void nucleation, growth and coalescence ahead of the crack tip, and as such resembles fracture at the microscale. When the crack tip behaves in a ductile manner, the crack does not advance rapidly after the pre-opening process but is blunted by dislocation generation from its tip. The effect of temperature on crack speed is found to be perceptible in both ductile and brittle specimens.     

Shape sensitivity analysis for non-differentiable performance measures in multiscale crack propagation simulation using regression hybrid method

In this paper, we present a regression hybrid method that calculates shape sensitivity coe?cients for multiscale crack propagation problems with performance measures that are non-differentiable in numerical implementation. These measures are crack propagation speed (or crack speed) defined at atomistic level obtained by solving coupled atomistic/continuum structures using the bridging scale method (BSM). The major contributions of this paper are: first, by analyzing the characteristics of the performance measures of crack speed in design space, this paper verifies for the first time that these measures are theoretically continuous and differentiable with respect to design variables, and as a result, the sensitivity coe?cients exist in theory; second, to overcome the non-differentiability of the performance measures in numerical computation due to the finite size of integration time step, this paper proposes a regression hybrid method that calculates the shape sensitivity coe?cients of crack speed through polynomial regression analysis based on the sensitivity of atomic responses, which is calculated through analytical shape design sensitivity analysis (DSA). And finally, the proposed method supports for 3D crack propagation problems with periodic boundary condition in one direction. A nano-beam example is used to demonstrate numerically the feasibility, accuracy, and e?ciency of the proposed method.     

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.     

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°.     

Remarks on the energy release rate for an antiplane moving crack in couple stress elasticity

This paper is concerned with the steady-state propagation of an antiplane semi-infinite crack in couple stress elastic materials. A distributed loading applied at the crack faces and moving with the same velocity of the crack tip is considered, and the influence of the loading profile variations and microstructural effects on the dynamic energy release rate is investigated. The behavior of both energy release rate and maximum total shear stress when the crack tip speed approaches the critical speed (either that of the shear waves or that of the localized surface waves) is studied. The limit case corresponding to vanishing characteristic scale lengths is addressed both numerically and analytically by means of a comparison with classical elasticity results.     

Transition of mode II cracks from sub-Rayleigh to intersonic speeds in the presence of favorable heterogeneity

Understanding sub-Rayleigh-to-intersonic transition of mode II cracks is a fundamental problem in fracture mechanics with important practical implications for earthquake dynamics and seismic radiation. In the Burridge-Andrews mechanism, an intersonic daughter crack nucleates, for sufficiently high prestress, at the shear stress peak traveling with the shear wave speed in front of the main crack. We find that sub-Rayleigh-to-intersonic transition and sustained intersonic propagation occurs in a number of other models that subject developing cracks to intersonic loading fields. We consider a spontaneously expanding sub-Rayleigh crack (or main crack) which advances, along a planar interface with linear slip-weakening friction, towards a place of favorable heterogeneity, such as a preexisting subcritical crack or a small patch of higher prestress (similar behavior is expected for a small patch of lower static strength). For a range of model parameters, a secondary dynamic crack nucleates at the heterogeneity and acquires intersonic speeds due to the intersonic stress field propagating in front of the main crack. Transition to intersonic speeds occurs directly at the tip of the secondary crack, with the tip accelerating rapidly to values numerically equal to the Rayleigh wave speed and then abruptly jumping to an intersonic speed. Models with favorable heterogeneity achieve intersonic transition and propagation for much lower prestress levels than the ones implied by the Burridge-Andrews mechanism and have transition distances that depend on the position of heterogeneity. We investigate the dependence of intersonic transition and subsequent crack propagation on model parameters and discuss implications for earthquake dynamics.     

Analysis of dynamic growth of a tensile crack in an elastic-plastic material

The influence of inertia on the stress and deformation fields near the tip of a crack growing in an elastic-plastic material is studied. The material is characterized by the von Mises yield criterion and J₂ flow theory of plasticity. The crack grows steadily under plane strain conditions in the tensile opening mode. Features of the stress and deformation state at points near the moving crack tip are described for elastic-perfectly plastic response and for several crack propagation speeds. It is found that inertia has a significant effect on the elastic-plastic response of material particles near the crack tip, and that elastic unloading may occur behind the crack tip for higher speeds. The relationship between the applied crack driving force, represented by a remote stress intensity factor, and the crack tip speed is examined on the basis of a critical crack tip opening angle growth criterion. The calculated result is compared with dynamic fracture toughness versus crack speed data for a 4340 steel.     

Coupled quantum–continuum analysis of crack tip processes in aluminum

A concurrent multiscale method is presented that couples a quantum mechanically governed atomistic domain to a continuum domain. The approach is general in that it is applicable to a wide range of quantum and continuum material modeling methodologies. It also provides quantifiable and controllable coupling errors via a force-based-coupling strategy. The applications presented here utilize an atomistic region that is governed by Kohn–Sham density functional theory and a continuum region governed by linear elasticity with discrete dislocation capabilities. As a validation we compute the core structure of a screw dislocation in aluminum and compare to previously published results. Then we investigate two crack orientations in aluminum and predict the critical load at which crack propagation and crack tip dislocation nucleation occurs. We compute critical loads with both LDA and GGA exchange correlation functionals and compare our results to popular empirical potentials in the context of classical continuum models. Overall this work aims to lay a foundation for future quantum mechanics-based investigations of crack tip processes involving Al alloys and impurity elements.     

Atomistic/continuum simulation of interfacial fracture Part II: Atomistic/dislocation/continuum simulation

Coupled atomistic/dislocation/continuum simulation of interfacial fracture is performed in this paper. The model consists of a nanoscopic core made by atomistic assembly and a surrounding elastic continuum with discrete dislocations. Atomistic dislocations nucleate from the crack tip and move to the continuum layer where they glide according to the dislocation dynamics curve. An atoms/continuum averlapping belt is devised to facilitate the transition between the two scales. The continuum constraint on the atomic assembly is imposed through the mechanics atmosphere along the overlapping belt. Transmissions of mechanics parameters such as displacements, stresses, masses and momenta across the belt are realized. The present model allows us to explore interfacial fracture processes under different mode mixity. The effect of atomistic zigzag interface on the fracture process is revealed: it hinders dislocation emission from the crack tip, especially under high mode mixity. The project supported by the National Natural Science Foundation of China     

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。     

Study on the mechanisms and quantitative law of mode I supersonic crack propagation

Continuum mechanics predicts that the propagation speed of non-equilibrium information in solids is limited by the longitudinal wave speed, so is crack propagation. However, solids are essentially discrete systems. In this paper, via theoretical analysis and numerical simulations, it is demonstrated in a straightforward way that non-equilibrium disturbance (e.g. force, displacement, energy, and so on) can propagate at a supersonic speed in discrete systems, although the magnitude of the disturbance attenuates very quickly. In dynamic fracture, a cascade of atomic-bond breaking events provides an amplification mechanism to counterbalance the attenuation of the disturbance. Therefore, supersonic crack propagation can be realized in a domino way. Another key factor for supersonic crack propagation is to ensure sufficient energy flowing into the crack tip. Since most energy can only be transferred at a speed limited by the longitudinal wave speed, the conditions for the occurrence of supersonic crack propagation are not easily met in most situations, unless there is high pre-stored energy along the crack path or continuous energy supply from the loading concomitantly moving with the crack tip. A quantitative relation between supersonic crack propagation speed and material properties and parameters is given, which implies that knowing all the classical macroscopic quantities is not enough in determining the supersonic crack propagation speed, and the microstructure does play a role. Moreover, it is interesting to note that fracture toughness affects the crack propagation speed in the subsonic regime, but not in the supersonic regime, because the deformation/stress is uniform in front of a supersonic crack where strength criterion dominates.     

Propagating semi-infinite crack in fluid-saturated porous medium of finite height

Derived in this work are the Mode I stress intensity factor results for a constant velocity semi-infinite crack moving in a fluid-saturated porous medium with finite height. Two limiting cases are discussed; they correspond to a low and high speed crack propagation. To be expected is that the crack front stress intensification would increase as the medium height is reduced in relation to the segment length in which mechanical pressure is applied. Moreover, the stress intensity factor for the high speed crack is larger than the low speed crack, the magnification of which depends on the material. Dissatisfaction of the crack surface and tip boundary condition is found in the present solution which calls possibly for the additional consideration of a local boundary layer as discussed by other authors.     

石墨烯裂纹扩展行为研究

基于分子动力学方法对含预制裂纹石墨烯进行扶手椅向拉伸断裂模拟。使用连续介质理论结合分子动力学计算石墨烯能量释放率,确定石墨烯能量释放率GIC为10.25 J/m2;应力强度因子KIC为 3.33MPam^1/2。进一步对影响石墨烯裂纹扩展速率的因素-初始裂纹长度与加载速率进行讨论。结果表明：裂纹初始长度与加载率会在一定程度上影响石墨烯中裂纹扩展速率。裂纹扩展速率会随着初始裂纹长度的增加而降低;但随着初始裂纹长度的增加,裂纹扩展速率对其敏感度降低。裂纹扩展速率会随着加载率的升高而增大。 初始裂纹长度与加载率对裂纹扩展速率的影响有一定的关联性,加载率的升高会降低裂纹扩展速率对初始裂纹长度变化的敏感度。在此基础上确定了石墨烯中裂纹扩展极限速率为8350 m/s。关联性,加载率的升高会降低裂纹扩展速率对初始裂纹长度变化的敏感度。在此基础上确定了石墨烯中裂纹扩展极限速率为8350 m/s。     

Molecular-dynamics simulation-based cohesive zone representation of intergranular fracture processes in aluminum

A traction-displacement relationship that may be embedded into a cohesive zone model for microscale problems of intergranular fracture is extracted from atomistic molecular-dynamics (MD) simulations. An MD model for crack propagation under steady-state conditions is developed to analyze intergranular fracture along a flat Σ99 [1 1 0] symmetric tilt grain boundary in aluminum. Under hydrostatic tensile load, the simulation reveals asymmetric crack propagation in the two opposite directions along the grain boundary. In one direction, the crack propagates in a brittle manner by cleavage with very little or no dislocation emission, and in the other direction, the propagation is ductile through the mechanism of deformation twinning. This behavior is consistent with the Rice criterion for cleavage vs. dislocation blunting transition at the crack tip. The preference for twinning to dislocation slip is in agreement with the predictions of the Tadmor and Hai criterion. A comparison with finite element calculations shows that while the stress field around the brittle crack tip follows the expected elastic solution for the given boundary conditions of the model, the stress field around the twinning crack tip has a strong plastic contribution. Through the definition of a Cohesive-Zone-Volume-Element—an atomistic analog to a continuum cohesive zone model element—the results from the MD simulation are recast to obtain an average continuum traction-displacement relationship to represent cohesive zone interaction along a characteristic length of the grain boundary interface for the cases of ductile and brittle decohesion.     

Crack tip superblunting: experiment,theory and numerical simulation

Experimental revelation ofsuperblunting along plane strain predominant crack tip segment is reported here for modified polyproplene. As elucidated by a heuristic model of progressive circumferential cold-draw, the formation of superblunting crack tip depends critically on the ratio between the cold-draw propagation speed and the loading speed, and contributes significantly to the material toughening, especially for the improvement in impact toughness. Detailed numerical calculations are conducted based on a hyperelastic-viscoplastic and anisotropic damage constitutive model at finite deformation. The simulated results recapitulate the essential features of crack tip superblunting. The project supported by the National Natural Science Foundation of China     

关联参照模型和位错发射过程的分子动力学模拟

提出关联参照模型和随位错位置变化的柔性位移边界条件．提供了一个在固定位移边界条件下位错穿越边界的方法．应用三维分子动力学方法研究了体心立方（ＢＣＣ）金属晶体钼裂尖发射位错的力学行为．     

裂尖超钝化:实验、理论与数值模拟

本文报道了在改性聚丙烯裂尖平面应变段发生超钝化的实验发现。利用环向冷拉模型,可说明裂尖超钝化的形成取决于冷拉传播速率与加载率之比。裂尖超钝化可使材料韧性,尤其是冲击韧性的改善,达到一个新的水平。本文在超弹性-粘塑性各向异性损伤的有限变形本构模型下进行了详尽的数值计算。模拟结果成功地再现了裂尖超钝化的基本特征。     

Oscillatory fracture paths in thin elastic sheets

We report a novel mode of quasi-static oscillatory crack propagation when a cutting tip of moderately large width is driven through a thin brittle polymer film. Experiments show that the amplitude and wavelength of the oscillatory crack paths scale linearly with the width of the cutting tip over a wide range of length scales but are independent of the width of the sheet and of the cutting speed. We propose a mechanism for this instability, based on the coupling between crack propagation and out-of-plane deformations of the film. To cite this article: B. Roman et al., C. R. Mecanique 331 (2003).