Mixed-mode I/II fields around a crack with a cohesive zone ahead of the crack tip

Considering a cohesive zone ahead of the crack tip, the linear elastic crack problem under Mode I, Mode II or mixed-mode conditions is formulated in an elliptic coordinate system, so that the cohesive surfaces are conveniently represented by straight line segments. It is shown that the displacement and stress fields around the crack tip and the cohesive zone, expressed in terms of elliptic coordinates, have a simple mathematical form, which does not contain a stress singularity at the crack tip due to the existence of the cohesive zone.     

内聚力模型裂纹问题分析的解析奇异单元

内聚力模型已经被广泛应用于需要考虑断裂过程区的裂纹问题当中,然而常用的数值方法应用于分析内聚力模型裂纹问题时还存在着一些不足,比如不能准确的给出断裂过程区的长度、需要网格加密等。为了克服这些缺点,论文构造了一个新型的解析奇异单元,并将之应用于基于内聚力模型的裂纹分析当中。首先将虚拟裂纹表面处的内聚力用拉格拉日插值的方法近似表示为多项式的形式,而多项式表示的内聚力所对应的特解可以被解析地给出。然后利用一个简单的迭代分析,基于内聚力模型的裂纹问题就可以被模拟出来了。最后,给出二个数值算例来证明本文方法的有效性。     

Fracture mechanics of plates and shells applied to fail-safe analysis of fuselage Part II: Computational results

In this paper, the problem of the fracture of a fuselage stiffened by longitudinal longerons and circumferential frames is analyzed by means of the finite element method. Our research is motivated by the fail-safety design concept of fuselage for civil aircraft. In this study, the total energy release rate are evaluated for five types of basic loading, namely, axial extension, pure bending, twisting, transverse shearing, and radial expansion due to internal pressure. The crack is located either at the mid-point or near the end of the fuselage. It extends in two bays with the stiffener at its center. The stiffener which bisects the crack is assumed to be broken at the location of the crack. Computational results indicate that the total energy release rate G_t increases with the increasing crack length. However, when the crack tip approaches the stiffener, the value of G_t decreases as a result of the reinforcement from the stiffener. For a crack near the end of the fuselage, as a result of boundary effect, the value of G_t is larger in comparison with the case of the crack at the mid-point of the fuselage. We also find that the effect of geometrical nonlinearity can reduce the value of G_t for the fuselage under axial tension or pure bending. For the fractured fuselage under pure bending, shell buckling can occur at the concave side of the fuselage prior to crack growth. The maximum tensile stress in the stiffener in front of the crack tip is also investigated.     

相变滞后对裂纹屏蔽效应的数值分析

本文针对奥氏体-马氏体双相材料,研究裂纹尖端区弥散分布的奥氏体颗粒在应变诱发时发生的相变对裂纹的屏蔽效应。鉴于实验中已发现的不同相变滞后对裂纹屏蔽效应的不同影响,本研究通过在裂纹尖端区不同位置嵌入相变颗粒,考虑到裂纹尖端区应力应变场的奇异分布及其诱发的相变,将裂纹尖端区相变滞后问题转化为相变颗粒在裂纹尖端区的位置问题。计及奥氏体-马氏体相变的体积膨胀效应进行了平面应力裂纹问题数值模拟,得到了单个相变夹杂对裂纹屏蔽效应的影响规律。结果表明：裂纹尖端区相变夹杂的位置对裂纹的屏蔽效应在距裂尖2倍夹杂直径以内影响极大,且以裂尖86度方向为界。其影响规律与McMeeking 和 Evans理论预言的60度方向不同。     

Dynamic mode I perturbation solution for a moving crack unsteadily

Rice et al. (Journal of Mechanics and Physics of Solids42, 813–843) analyze the propagation of a planar crack with a nominally straight front in a model elastic solid with a single displacement component. Using the form of Willis et al. (Journal of the Mechanics and Physics of Solids43, 319–341), of dynamic mode I weight functions for a moving crack, we address that problem solved by Rice et al. in the 3D context of elastodynamic theory. Oscillatory crack tip motion results from constructive-destructive interference of stress intensity waves. Those waves, including system of the dilatational, shear and Rayleigh waves, interact on each other and with moving edge of crack, can lead to continuing fluctuations of the crack front and propagation velocity.     

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.     

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.     

带有中间裂纹载流薄板放电瞬间耦合场的数值模拟

应用耦合场理论采用数值分析的方法计算了具有中间裂纹的导电薄板在放电瞬间裂纹端区域附近的温度场、温度梯度场的分布状态。计算结果表明：由于电流产生的焦耳热源的作用，裂纹尖端处温度瞬时急剧升高，能够在很小的范围内熔化形成微小的口，裂纹前缘的曲率半径增加了几个数量级，显著地减少了应力集中，阻止了干线裂纹源的形成，有效地遏制了裂纹的扩展，数值模拟采用了热－电耦合的分析方法，考虑了材料电阻随温度变化和电流流过生成的内热源间的相互作用，同时考虑了导热系数和比热随温度变化所产生的影响，对于具有中间裂纹的载流反，根据结构、材质、边界条件及通电电流的对称性，计算得到了两个裂纹尖端完全对称的结果。     

弹性T项对裂尖参数和裂纹扩展路径稳定性的作用

研究了弹性T项在主裂纹与近裂尖微空洞干涉问题中对裂尖参数和裂纹扩展路径稳定性的影响作用.利用“伪力”法,并通过解决主裂纹与近裂尖微空洞干涉问题,对远场纯1型载荷和远场弹性T项载荷下,该干涉问题中弹性T项的影响作用进行了分析从数值结果可以看出：由于空洞的存在;释放了弹性T项所引起的应力,弹性T项对裂尖参数;应力强度因子和J积分都有直接显著的影响,因而,它对该载荷下的裂纹扩展路径的稳定性有控制作用。     

Two mechanisms of ductile fracture: void by void growth versus multiple void interaction

Two distinct mechanisms of crack initiation and advance by void growth have been identified in the literature on the mechanics of ductile fracture. One is the interaction a single void with the crack tip characterizing initiation and the subsequent void by void advance of the tip. This mechanism is represented by the early model of Rice and Johnson and the subsequent more detailed numerical computations of McMeeking and coworkers on a single void interacting with a crack tip. The second mechanism involves the simultaneous interaction of multiple voids on the plane ahead of the crack tip both during initiation and in subsequent crack growth. This mechanism is revealed by models with an embedded fracture process zone, such as those developed by Tvergaard and Hutchinson. While both mechanisms are based on void nucleation, growth and coalescence, the inferences from them with regard to crack growth initiation and growth are quantitatively different. The present paper provides a formulation and numerical analysis of a two-dimensional plane strain model with multiple discrete voids located ahead of a pre-existing crack tip. At initial void volume fractions that are sufficiently low, initiation and growth is approximately represented by the void by void mechanism. At somewhat higher initial void volume fractions, a transition in behavior occurs whereby many voids ahead of the tip grow at comparable rates and their interaction determines initiation toughness and crack growth resistance. The study demonstrates that improvements to be expected in fracture toughness by reducing the population of second phase particles responsible for nucleating voids cannot be understood in terms of trends of one mechanism alone. The transition from one mechanism to the other must be taken into account.     

Interfacial cracks between piezoelectric and elastic materials under in-plane electric loading

A new experimental technique for accelerated fatigue crack growth tests was recently developed (Du et al., 2001). The technique, which uses piezoelectric actuators, enables application of cyclic loading at frequencies several orders higher than that by mechanical loading. However, the validity of this technique relies on the equivalence between piezoelectric and mechanical loading. In this paper, the behavior of an interfacial crack between a piezoelectric material and an elastic material under in-plane electric loading is studied. The displacement mismatch along a bonded interface due to electric potential loading on the piezoelectric material is modeled by inserting an array of uniformly distributed dislocations along the interface. By means of Fourier transformation methods, the governing equations are converted to an integral equation, which is then converted to a standard Hilbert problem. A closed form solution for stresses, electric field, and electric displacements along the bonded interface is obtained. The results agree very well with those obtained from numerical simulations. The results show that the closed form solution is accurate not only for far field distributions of stresses and electric variables, but also for the asymptotic distributions near the crack tip. The solution also suggests the likelihood of domain switching in the piezoelectric material near the crack tip, a process that may influence the interfacial fracture resistance.     

A new finite element method for strain gradient theories and applications to fracture analyses

A new compatible finite element method for strain gradient theories is presented. In the new finite element method, pure displacement derivatives are taken as the fundamental variables. The new numerical method is successfully used to analyze the simple strain gradient problems – the fundamental fracture problems. Through comparing the numerical solutions with the existed exact solutions, the effectiveness of the new finite element method is tested and confirmed. Additionally, an application of the Zienkiewicz–Taylor C1 finite element method to the strain gradient problem is discussed. By using the new finite element method, plane-strain mode I and mode II crack tip fields are calculated based on a constitutive law which is a simple generalization of the conventional J₂ deformation plasticity theory to include strain gradient effects. Three new constitutive parameters enter to characterize the scale over which strain gradient effects become important. During the analysis the general compressible version of Fleck–Hutchinson strain gradient plasticity is adopted. Crack tip solutions, the traction distributions along the plane ahead of the crack tip are calculated. The solutions display the considerable elevation of traction within the zone near the crack tip.     

Interface crack between a compressible elastomer and a rigid substrate with finite slippage

We study the deformation of a crack between a soft elastomer and a rigid substrate with finite interfacial slippage. It is assumed that slippage occurs when the interfacial shear traction exceeds a threshold. This leads to a slip zone ahead of the crack tip where the shear traction is assumed to be equal to the constant threshold. We perform asymptotic analysis and determine closed-form solutions describing the near-tip crack opening displacement and the corresponding stress distributions. These solutions are consistent with numerical results based on finite element analysis. Our results reveal that slippage can significantly affect the deformation and stress fields near the tip of the interface crack. Specifically, depending on the direction of slippage, the crack opening profile may appear more blunted or sharpened than the parabola arising from for the case of zero interfacial shear traction or free slippage. The detailed crack opening profile is determined by the constant shear traction in the slip zone. More importantly, we find that the normal stress perpendicular to the interface can increase or decrease when slippage occurs, depending on the direction of slippage and the shear traction in the slip zone.     

基于数字散斑相关方法测定Ⅰ型裂纹应力强度因子

提出了一种通过数字散斑相关方法测定金属材料Ⅰ型裂纹尖端位置和应力强度因子的实验方法.实验采用疲劳试验机对含Ⅰ型缺口的Cr12MoV钢试件预制裂纹,通过数字散斑相关方法测试试件在三点弯曲加载条件下裂纹的扩展过程及裂尖区域的位移场.将位移场数据代入裂尖位移场方程组,采用牛顿-拉普森方法求解含未知参量的裂尖非线性位移场方程组,计算裂尖位置和应力强度因子.实验结果表明,采用该方法可以准确地测定金属材料Ⅰ型裂纹应力强度因子、裂尖位置及裂纹扩展长度,解决了以往研究中因不能准确测定裂纹尖端位置,而无法准确计算Ⅰ型裂纹裂尖断裂参数的难题,揭示了金属材料裂纹扩展过程中应力强度因子演化特征.     

Computational crack path prediction

A computer program has been developed for the numerical prediction of curved crack growth paths under proportional loading conditions. The numerical prediction is performed by the step-by-step method in cooperation with the stress analysis ahead of the crack tip and the determination of the curved increment of the crack growth. The stress analysis is performed by the method of superposition of analytical and finite-element solutions, and the results are then utilized to determine the coefficients of the analytical expression of the curved crack path obtained by the first order perturbation method.The first numerical example is given for the crack path prediction in DCB-type specimen, where we often observe abrupt crack curving. Computational prediction is performed by introducing slight and small initial branching at the original crack tip. Within few steps of numerical calculations unstable crack curving is obtained, and the predicted path shows extremely good agreement with the experimentally measured path. The second numerical prediction is made for an edge crack approaching a circular hole, which may be considered as a crack arrester. In the present case the effect of the initially introduced slight kink diminishes with increasing crack length. The crack turns back to the original direction, resulting arrest at the hole.     

Logarithm strain singularity at tip of mode I growing crack in elasticand perfectly-plastic material

Reanalyzed in detail is the stress and strain distribution near the tip of a Mode I steadily growing crack in an elastic and perfectly-plastic material. The crack tip region is divided into five angular sectors, one of which is singular in character and represents a rapid transition zone that becomes a line of strain discontinuity in the limit as crack tip is approached. It is shown for an incompressible material (ν=0.5) under plane strain that the local strain in all the angular sectors possesses the same logarithm singularity, i.e., In r where r is the radial distance measured from the crack tip. This result also prevails for the compressible material ( v < 0.5) and resolves a long standing controversy concerning the strain singularity in the sector just ahead of the crack tip.     

断裂过程区应力应变场的数学分析

本文将条状分离区看作是位错连续分布区域,利用复变函数理论及切比雪夫多项式级数展开提出了一种统一的数学分析的方法,用以对均匀材料裂纹前方分离过程区的非线性效应进行有效的,精确的解析分析。避免以前各种模型一般都需要借助于有限元法或其他离散方法才能得到数值解答的难点。本文利用这一新方法,对断裂过程区是内聚力模型,条状屈服区模型的裂纹问题,受约束金属薄膜的裂纹问题,进行了分析计算,并与已有的结果进行了比较,得到了满意的结果。     

An ESPI experimental study on the phenomenon of fracture in glass. Is it brittle or plastic?

The crack opening displacement (COD) in annealed soda-lime (float) glass has been measured with an electronic speckle pattern interferometry (ESPI) apparatus using coherent laser light. Specimens, naturally pre-cracked with a particular technique, were loaded under strain-driven bending until crack propagated; at regular intervals loading was paused to let the crack reach subcritical equilibrium and the COD measured. By using a post-processing algorithm comparing four images lighted with phase-shifted laser beams, surface displacements could be measured at a resolution of .Glass transparency has allowed to see through that the propagating crack front is not sharp but curved, jagged and merged in an opaque neighborhood. Numerical simulations show that the measured CODs cannot be reproduced if cohesive surface forces à la Barenblatt-Dugdale bridge the crack lips; instead a plastic-like region must form in a bulk neighborhood of the tip, where inelastic strains are associated with volume increase rather than deviatoric distortion. For this, a Gurson-Tvergaard model of porous plasticity, accounting for the formation of microvoids/microcracks, has been found more efficient than classical von Mises plasticity. This study confirms the formation at the crack tip of a process zone, whose occurrence in brittle materials like glass is still a subject of controversy.     

Cohesive-zone laws for void growth — I. Experimental field projection of crack-tip crazing in glassy polymers

A hybrid framework for inverse analysis of crack-tip cohesive-zone model is developed in this two-part paper to measure cohesive-zone laws of void growth in polymers by combining analytical, experimental, and numerical approaches. This paper focuses on experimental measurements of the cohesive-zone laws for two nonlinear fracture processes in glassy polymers, namely multiple crazing in crack-growth toughening of rubber-toughened high-impact polystyrene (HIPS) and crazing of steady-state crack growth in polymethylmethacrylate (PMMA) under a methanol environment. To this end, electronic speckle pattern interferometry (ESPI) is first applied to measure the crack-tip displacement fields surrounding the fracture process zones in these polymers. These fields are subsequently equilibrium smoothed and used in the extraction of the cohesive-zone laws via an analytical solution method of the inverse problem, the planar field projection method (P-FPM) [Hong, S., Kim, K.-S., 2003. Extraction of cohesive-zone laws from elastic far-fields of a cohesive crack tip: a field projection method. Journal of the Mechanics and Physics of Solids 51, 1267-1286]. Results show that the proposed framework of the P-FPM could provide a systematic way of finding the shape of the cohesive-zone laws governed by the different micro-mechanisms in the fracture processes. In HIPS, inter-particle multiple crazing develops and the craze zone broadens ahead of a crack-tip under mechanical loading. The corresponding cohesive-zone relationship of the multiple-craze zone is found to be highly convex, which indicates effectiveness of rubber particle toughening. It is also observed that the effective peak traction, 7 MPa, in the crack-tip cohesive zone of HIPS (30% rubber content) is lower than the uniaxial yield stress of 9 MPa, presumably due to stress multi-axiality effects. In contrast, in PMMA, methanol localizes the crack-tip craze, weakening the craze traction for craze-void initiation to about 9 MPa and the fibril pull-out stress to less than 6 MPa. This reduction in cohesive traction, coupled with a strongly concave traction-separation cohesive-zone relationship, signifies environmental embrittlement of PMMA. These experimentally determined cohesive-zone laws are compared with detailed numerical analyses of effective microscale-void growth ahead of a crack tip in Part II.