Fracture of bicrystal metal/ceramic interfaces: A study via the mechanism-based strain gradient crystal plasticity theory

Two continuum mechanical models of crystal plasticity theory namely, conventional crystal plasticity theory and mechanism-based crystal plasticity theory, are used to perform a comparative study of stresses that are reached at and ahead of the crack tip of a bicrystal niobium/alumina specimen. Finite element analyses are done for a stationary crack tip and growing cracks using a cohesive modelling approach. Using mechanism-based strain gradient crystal plasticity theory the stresses reached ahead of the crack tip are found to be two times larger than the stresses obtained from conventional crystal plasticity theory. Results also show that strain gradient effects strongly depend on the intrinsic material length to the size of plastic zone ratio (l/R₀). It is found that the larger the (l/R₀) ratio, the higher the stresses reached using mechanism-based strain gradient crystal plasticity theory. An insight into the role of cohesive strength and work of adhesion in macroscopic fracture is also presented which can be used by experimentalists to design better bimaterials by varying cohesive strength and work of adhesion.     

Continuing crack-tip deformation and fracture for plane-strain crack growth in elastic-plastic solids

Analysis of the deformation field consistent with a Prandtl stress distribution travelling with an advancing plane-strain crack reveals the functional form of the near tip crack profile in an elastic-plastic solid. The crack opening δ is shown to have the form δ ～ r In (const./r) at a distance r from the tip. This observation coupled with data generated from finite element investigations of growing cracks in small-scale yielding permits the construction of a relation characterizing the deformation at an extending crack tip. A ductile crack-growth criterion consisting of the attainment of a critical opening at a small characteristic material distance from the tip is adopted. Predictions of the stability of a growing crack for both small-scale yielding specimens and those subject to general yielding are discussed.     

Stress and energy density fields near a blunted crack front in power hardening material

Using Jaumann and Dienes rates of Euler stress in elastic-plastic constitutive equations of finite deformation, plane strain finite element analysis for a compact tension specimen with a blunted crack front is made. The Euler stress, Kirchhoff stress and volume strain energy density near a blunted crack tip are computed. Constitutive relations with different deformation rates affect the the near crack tip solution in a region within an order of magnitude of the crack opening displacement. The results differed from the corresponding solution of deformation plasticity (or nonlinear elasticity) with increasing deformation. They are smaller in a local region of about 2 to 10 times of the crack opening distance.The volume energy density near the crack tip is computed, the stationary values of which determine the locations of extensive yielding and possible sites of crack initiation. It remained nearly constant with increasing deformation. Such a character tends to support the volume energy density criterion as a means for quantifying the ductile fracture behavior of metals.     

Effect of crack-tip shape on the near-tip field in glassy polymer

The physical nature of a crack tip is not absolutely sharp but blunt with finite curvature. In this paper, the effects of crack-tip shape on the stress and deformation fields ahead of blunted cracks in glassy polymers are numerically investigated under Mode I loading and small scale yielding conditions. An elastic–viscoplastic constitutive model accounting for the strain softening upon yield and then the subsequently strain hardening is adopted and two typical glassy polymers, one with strain hardening and the other with strain softening–rehardening are considered in analysis. It is shown that the profile of crack tip has obvious effect on the near-tip plastic field. The size of near-tip plastic zone reduces with the increase of curvature radius of crack tip, while the plastic strain rate and the stresses near crack tip enhance obviously for two typical polymers. Also, the plastic energy dissipation behavior near cracks with different curvatures is discussed for both materials.     

Fatigue crack propagation life analysis of solder joints under thermal cyclic loading

Fatigue crack propagation life analysis of solder joints under thermal cyclic loadings was investigated by the strain energy release rate method using finite element analysis. A relationship between the crack-growth rate and the strain energy release rate was derived. Finite element simulations were carried out to investigate the effect of crack growth along the interface of solder and lead in a solder joint assembly. The crack propagation life of the solder joint with an interface crack was predicted from the derived relationship between crack growth rate and values of the strain energy release rate. It was found that crack propagation life is much higher than the crack initiation life.     

Evaluation of crack initiation angle under mixed mode loading at diverse strain rates

Crack initiation angle, under mixed mode loading at several strain rates, is analysed using an experimental–numerical approach. The physical phenomenon for the problem at hand is influenced by the local and global conditions. One of such factors is the strain rate at the crack tip. For this purpose, PMMA plates with centred angled cracks under mixed mode loading were tested. The strain rate at the neighbourhood of the crack tip before crack propagation was evaluated. Considering that this material is strain rate sensitive, the numerical models were calibrated with the modulus of elasticity measured in tension tests at the observed strain rates. Numerical evaluations were performed with the finite element method in conjunction with the volume energy density criterion. An improvement in the evaluation of the crack propagation angle was observed. In order to complete the analysis, the crack initiation angle was also evaluated with the strain energy density factor S, considering the mechanical properties of PMMA, as evaluated at the observed strain rates, and the stress intensity factors k₁ and k₂. Results are in agreement with those observed experimentally.     

考虑材料循环塑性的疲劳裂纹扩展模拟

提出了一种考虑材料循环塑性性能的研究疲劳裂纹扩展与闭合行为的有限元模拟方法．对所选用的循环塑性本构关系进行了基本实验检验．探讨了在疲劳裂纹扩展有限元分析中网格尺寸的影响，给出了网格优化准则．研究了在循环硬化条件下考虑裂纹合效应时裂纹面张开廓形、裂纹尖端应力、应变场和正反向塑性区的演变规律．对于循环硬化和不同循环应力比Ｒ等因素对裂纹张开应力水平的影响也作了考察     

Effect of constraint induced by crack depth on creep crack-tip stress field in CT specimens

In this paper, the effect of constraint induced by the crack depth on creep crack-tip stress field in compact tension (CT) specimens is examined by finite element analysis, and the effect of creep deformation and damage on the Hutchinson–Rice–Rosengren (HRR) singularity stress field are discussed. The results show the constraint induced by crack depth causes the difference in crack-tip opening stress distributions between the specimens with different crack depth at the same C*. The maximum opening stress appears at a distance from crack tips, and the stress singularity near the crack tips does not exist due to the crack-tip blunting caused by the large creep deformation in the vicinity of the crack tips. The actual stress calculated by the finite element method (FEM) in front of crack tip is significantly lower than that predicted by the HRR field. Based on the reference stress field in the deep crack CT specimen with high constraint, a new constraint parameter R is defined and the constraint effect in the shallow crack specimen is examined at different distances ahead of the crack tip from transient to steady-state creep conditions. During the early stages of creep constraint increases with time, and then approaches a steady state value as time increases. With increasing the distance from crack tips and applied load, the negative R increases and the constraint decreases.     

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.     

Stochastic modelling of crack growth based on damage accumulation

The growth rate of a fatigue crack is modelled from a damage accumulation standpoint. The material ahead of the crack-tip is considered to be composed of assembly of uniaxial fatigue elements which accumulate damage per load cycle. Each element is subjected to increased levels of stress and strain ranges as the crack propagates. A linearly accumulated damage criterion is assumed, and failure of an element indicates a void initiation at its position. Both deterministic and stochastic analyses are included. The historical damage of the material before it reaches the crack tip vicinity is quantified and is shown to be significant for the first few elements. The predicted results agree fairly well with the experimental data.     

平面幂律塑性Ⅰ型裂纹尖端应力场全解

在航空航天、船舶、石油管道和核电等领域,服役结构或部件在长期极端条件下运行,不可避免地会产生裂纹,因此,为研究含裂纹结构的准静态断裂行为,必须了解裂纹尖端附近区域的应力应变场特点.对于幂律材料裂纹构元,研究平面应变和平面应力条件下Ⅰ型裂纹尖端应力场的解析分布.基于能量密度等效和量纲分析,推导了能量密度中值点代表性体积单元(representative volume element, RVE)的等效应力解析方程,并定义其为应力因子,进而针对有限平面应变和平面应力紧凑拉伸(compact tension, CT)试样和单边裂纹弯曲(single edge bend, SEB)试样,以应力因子作为应力特征量,并构造用于表征裂尖等效应力等值线的蝶翅轮廓式和扇贝轮廓式三角特殊函数,提出描述幂律塑性条件下平面I型裂纹尖端应力场的半解析模型.该半解析模型形式简单,对CT和SEB试样的裂尖应力场的预测结果与有限元分析的结果比较表明,两者之间均密切吻合,模型公式可直接用于预测Ⅰ型裂纹尖端应力分布,方便于断裂安全评价和理论发展.     

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.     

三点弯曲试样动态应力强度因子计算研究

利用Hopkinson压杆对三点弯曲试样进行冲击加载,采集了垂直裂纹面距裂尖2mm和与裂纹面成60°距裂尖5mm处的应变信号。根据裂尖附近测试的应变信号计算试样的动态应力强度因子,并与有限元计算结果进行比较,结果表明由于裂尖有一段疲劳裂纹区,通过裂尖附近应变信号来计算动态应力强度因子时,如果裂尖位置确定不准及粘贴应变片位置不够准确对计算结果将带来很大影响。因此利用应变片法计算动态应力强度因子时,为了获得更准确的计算结果,在实验后应对试件裂纹面进行分析测量,重新确定裂尖位置,必要时需对应变片至裂尖距离进行修正后再计算动态应力强度因子值。     

Fracture behavior of a nickel-base single crystal superalloy as predicted by the strain energy density criterion

Results from compact tension specimens and scanning electronic microscopy are used to study the fracture characteristics of nickel-base single crystal superalloy DD3. Finite element analyses are also made to better understand the outcome. The experiments showed that the initiation and growth of cracks were dependent on the crystallographic orientation of specimens, and the fracture surfaces coincided with the slip planes. Three-dimensional stresses near the crack tip are obtained. The strain energy density criterion is applied to predict the fracture of the superalloy. Agreement with experiments is good.     

采用显微网格法研究韧性材料平面应力状态下裂尖附近的应变场

用显微网格数字图象处理方法，测量了韧性材料平面应力条件下Ｉ型单边裂纹尖端附近的应变场。实验结果表明：在裂尖两侧存在与裂纹方向夹角约为５０°的大变形带。最大应变位于离裂尖一定距离的大变形带上，随着载荷增加，最大应变的位置逐渐靠近裂纹尖端。应变峰值轨迹线在裂纹顶端呈放射状，载荷增加时，应变峰值的轨迹线位置并不改变。     

U形切口的有限断裂准则

有限断裂力学准则综合了应力和能量参数,假设裂纹或切口端部有限裂纹长度的增长．特别地,该有限裂纹的长度不是材料的基本常数,而是与构件的结构有关．基于U形切口两种形式：点方式和线方式有限断裂准则,对文献中的铝合金U形切口三点弯曲断裂实验进行了分析验证．一方面基于材料的断裂韧度和抗拉强度,预测切口件断裂载荷;另一方面根据几组不同的切口根部半径及其对应的临界切口应力强度因子,同时估算材料的断裂参数：断裂韧度和抗拉强度．将点方式和线方式两种不同形式有限断裂准则的预测结果,与平均周向应力准则、最大周向应力准则以及文献中相关结果进行了比较得出：无论是预测断裂载荷还是估算材料断裂参数,线方式有限断裂准则,与文献中相关结果比较吻合,尤其是估算的断裂韧度精度较高．     

The role of macroscopic hardening and individual length-scales on crack tip stress elevation from phenomenological strain gradient plasticity

This paper quantifies the effect of strain gradient plasticity (SGP) on crack tip stress elevation for a broad range of applied loading conditions and constitutive model parameters, including both macroscopic hardening parameters and individual material length-scales controlling gradient effects. Finite element simulations incorporating the Fleck-Hutchinson SGP theory are presented for an asymptotically sharp stationary crack. Results identify fundamental scaling relationships describing (i) the physical length-scales over which strain gradients are prominent, and (ii) the degree of stress elevation over conventional Hutchinson-Rice-Rosengren (HRR) fields. Results illustrate that the three length-scale theory predicts much larger SGP effects than the single length-scale theory. Critically, the first length-scale parameter dominates SGP stress elevation: this suggests that SGP effects in fracture can be predicted using the length-scales extracted from nanoindentation, which exhibits similar behavior. Transitional loading/material parameters are identified that establish regimes of SGP relevance: this provides the foundation for the rational application of SGP when developing new micromechanical models of crack tip damage mechanisms and associated subcritical crack propagation behavior in structural alloys.     

Mixed mode (I and II) crack tip fields in bulk metallic glasses

Stationary crack tip fields in bulk metallic glasses under mixed mode (I and II) loading are studied through detailed finite element simulations assuming plane strain, small scale yielding conditions. The influence of internal friction or pressure sensitivity on the plastic zones, notch deformation, stress and plastic strain fields is examined for different mode mixities. Under mixed mode loading, the notch deforms into a shape such that one part of its surface sharpens while the other part blunts. Increase in mode II component of loading dramatically enhances the normalized plastic zone size, lowers the stresses but significantly elevates the plastic strain levels near the notch tip. Higher internal friction reduces the peak tangential stress but increases the plastic strain and stretching near the blunted part of the notch. The simulated shear bands are straight and extend over a long distance ahead of the notch tip under mode II dominant loading. The possible variations of fracture toughness with mode mixity corresponding to failure by brittle micro-cracking and ductile shear banding are predicted employing two simple fracture criteria. The salient results from finite element simulations are validated by comparison with those from mixed mode (I and II) fracture experiments on a Zr-based bulk metallic glass.     

单元类型和尺寸对裂尖疲劳塑性数值模拟的影响研究

本文采用Jiang-Sehitoglu循环塑性模型和多轴疲劳准则对紧凑拉伸式样裂尖的循环塑性变形、裂纹扩展速率和残余应力进行了有限元数值模拟,着重考察了单元的类型和最小单元尺寸对裂尖循环塑性和裂纹扩展速率的影响.紧凑拉伸试样的材料为1070钢,数值模拟采用了线性单元(四节点)和二次单元(八节点)两种单元,裂尖附近有限元单元的最小尺寸从0.007mm到0.24mm不等.文中将裂纹扩展速率的预测值与实验值进行了比较,通过对裂纹扩展速率的比较,确定在疲劳塑性分析时对单元类型和尺寸进行合理选取.     

动载下裂纹应力强度因子计算的改进型扩展有限元法

相较于常规扩展有限元法(extended finite element method, XFEM), 改进型扩展有限元法(improved XFEM) 解决了现有方法线性相关与总体刚度矩阵高度病态问题, 在数量级上提升了总体方程的求解效率, 克服了现有方法在动力学问题中的能量正确传递、动态应力强度因子数值震荡、精度低下问题. 本文基于改进型XFEM, 采用Newmark 隐式时间积分算法, 重点研究了动载荷作用下扩展裂纹尖端应力强度因子的求解方法, 与静力学方法相比, 增加了裂纹扩展速度项与惯性项的贡献. 通过数值算例研究了网格单元尺寸、质量矩阵、时间步长、裂尖加强区域、惯性项、扩展速度项及相互作用积分区域J-domain的网格与单元尺寸对动态应力强度因子求解精度的影响, 验证了改进型XFEM计算动态裂纹应力强度因子方法的有效性. 针对文献中具有挑战性的 "I 型半无限长裂纹先稳定后扩展"问题, 改进型XFEM给出目前为止精度最好的动态应力强度因子数值解.