Mode II dynamic fields near a crack tip growing in an elastic-perfectly-plastic solid

An asymptotic solution is given for Mode II dynamic fields in the neighborhood of the tip of a steadily advancing crack in an incompressible elastic—perfectly-plastic solid (plane strain). It is shown that, like for Modes I and III (Gao and Nemat-Nasser, 1983), the complete dynamic solution for Mode II predicts a logarithmic singularity for the strain field, but unlike for those modes which involve no elastic unloading, the pure Mode II solution includes two elastic sectors next to the stress-free crack surfaces. This is in contradiction to the quasi-static solution which predicts a small central plastic zone, followed by two large elastic zones, and then two very small plastic zones adjacent to the stress-free crack faces. The stress field for the complete dynamic solution varies throughout the entire crack tip neighborhood, admitting finite jumps at two shock fronts within the central plastic sector. This dynamic stress field is consistent with that of the stationary crack solution, and indeed reduces to it as the crack growth speed becomes zero.     

A transverse crack tip field in bimaterial

The asymptotic field near an interface crack tip is analyzed with the fully nonlinear theory. By dividing the crack tip field into narrowing sectors and an expanding sector, the asymptotic equations for the crack tip field are derived and solved. The singular characters of stress and strain near the crack tip are revealed.     

Role of elasticity in elastic–plastic fracture: Analytical bi-linear crack-tip fields and finite element analysis

A solution for Model-I plane strain crack tip fields in a bi-linear elastic–plastic material is presented. The elastic–plastic Poisson's ratio is introduced to characterize the influence of elastic deformation on the near tip constraint. Attention is focused on the distribution of elastic/plastic strain energy in the sensitive region of the forward sector ahead of a crack tip. The present study shows that the elastic strain energy can be higher than the plastic strain energy in this sensitive sector while large amount of the plastic strain energy develops outside this sector around the crack tip. The effect of elastic deformation in this sensitive region on the structure of crack-tip fields is considerable and the assumption in some important solutions for crack-tip fields reported in literature that the elastic deformation is small and can be ignored is therefore not physically reasonable. Besides, finite element analysis is carried out to validate the analytical solution and good agreement between them is found. It is seen that the present solution with T-stress can properly describe the crack-tip fields under various constraints for different specimens and an analytical relation is established between the critical value of J-integral, J_c, and T-stress for elastic–plastic fracture.     

静止裂纹顶端塑性区内应力场

本文对不可压缩的理想塑性材料裂纹顶端塑性区内的应力场进行了数学分析,证明了当塑性区包围着裂纹顶端而应力函数可用分离变匱型的级数展开且该级数展开的首项与第一类渐近解相同时,第一类渐近解即是塑性区内应力场的精确解。本文又提出了第二类渐近解,说明应力场的渐近解不是唯一的。     

Stress and strain field near tip of mode III growing crack in materials with creep behavior

Using a proposed constitutive relation for materials with creep behavior, the stress and strain distribution near the tip of a Mode III growing crack is examined. Asymptotic equations of the crack tip field are derived and solved numerically. The stresses remain finite at the crack tip. Obtained qualitatively is the crack tip velocity and the local autonomy of the near tip field solution is discussed.     

扩展边界元法分析切口和裂纹结构应力场的准确性

在线弹性理论中,切口/裂纹结构尖端区域存在奇异应力场,数值方法不易求解。本文建立的扩展边界元法(XBEM)对围绕尖端区域位移函数采用自尖端径向距离 的渐近级数展开式表达,其级数项的幅值系数作为基本未知量,而外部区域采用常规边界元法离散方程。两者方程联立求解可获得切口和裂纹结构完整的位移和应力场。扩展边界元法具有半解析法特征,适用于一般的切口和裂纹结构应力场分析,其解可精细描述从尖端区域到整体结构区域的应力场。作者研制了扩展边界元法程序,文中给出了两个算例,通过计算结果分析,表明扩展边界元法求解切口和裂纹结构应力场的准确性和有效性。     

Field structure at mode III dynamically propagating crack tip in elastic-viscoplastic materials

An elastic-viscoplastic mechanics model is used to investigate asymptotically the mode Ⅲ dynamically propagating crack tip field in elastic-viscoplastic materials. The stress and strain fields at the crack tip possess the same power-law singularity under a linear-hardening condition. The singularity exponent is uniquely determined by the viscosity coefficient of the material. Numerical results indicate that the motion parameter of the crack propagating speed has little effect on the zone structure at the crack tip. The hardening coefficient dominates the structure of the crack-tip field. However, the secondary plastic zone has little influence on the field. The viscosity of the material dominates the strength of stress and strain fields at the crack tip while it does have certain influence on the crack-tip field structure. The dynamic crack-tip field degenerates into the relevant quasi-static solution when the crack moving speed is zero. The corresponding perfectly-plastic solution is recovered from the linear-hardening solution when the hardening coefficient becomes zero.     

Field structure at mode Ⅲ dynamically propagating crack tip in elastic-viscoplastic materials

An elastic-viscoplastic mechanics model is used to investigate asymptotically the mode Ⅲ dynamically propagating crack tip field in elastic-viscoplastic materials. The stress and strain fields at the crack tip possess the same power-law singularity under a linear-hardening condition. The singularity exponent is uniquely determined by the viscosity coefficient of the material. Numerical results indicate that the motion parameter of the crack propagating speed has little effect on the zone structure at the crack tip. The hardening coefficient dominates the structure of the crack-tip field. However, the secondary plastic zone has little influence on the field. The viscosity of the material dominates the strength of stress and strain fields at the crack tip while it does have certain influence on the crack-tip field structure. The dynamic crack-tip field degenerates into the relevant quasi-static solution when the crack moving speed is zero. The corresponding perfectly-plastic solution is recovered from the linear-hardening solution when the hardening coefficient becomes zero.     

Measurement of the strain field near a crack tip in polymethylmethacrylate by holographic interferometry

An experimental method has been developed for the detailed, real-time study of the strain field near a crack tip in any transparent material. This method combines holography and interferometry to measure the field of averaged transverse strain in a sheet specimen under inplane loading. It is shown that the solution for an edge crack in a semi-infinite sheet based on two-dimensional linear elastic theory can be applied to the finite-width polymethylmethacrylate specimen. This solution is observed to become less valid as the crack tip is approached due to the breakdown of the plane-stress assumption. This effect is evaluated for a range of thicknesses from much less to much more than the crack length.     

A weighted residual method for elastic-plastic analysis near a crack tip and the calculation of the plastic stress intensity factors

In this paper,a weighted residual method for the elastic-plastic analysis near a crack tip is systematically given by taking the model of power-law hardening under plane strain condition as a sample.The elastic-plastic solutions of the crack tip field and an approach based on the superposition of the nonlinear finite element method on the complete solution in the whole crack body field,to calculate the plastic stress intensity factors,are also developed.Therefore,a complete analysis based on the calculation both for the crack tip field and for the whole crack body field is provided.     

On the fracture toughness of ferroelastic materials

The toughness enhancement due to domain switching near a steadily growing crack in a ferroelastic material is analyzed. The constitutive response of the material is taken to be characteristic of a polycrystalline sample assembled from randomly oriented tetragonal single crystal grains. The constitutive law accounts for the strain saturation, asymmetry in tension versus compression, Bauschinger effects, reverse switching, and strain reorientation that can occur in these materials due to the non-proportional loading that arises near a propagating crack. Crack growth is assumed to proceed at a critical level of the crack tip energy release rate. Detailed finite element calculations are carried out to determine the stress and strain fields near the growing tip, and the ratio of the far field applied energy release rate to the crack tip energy release rate. The results of the finite element calculations are then compared to analytical models that assume the linear isotropic K-field solution holds for either the near tip stress or strain field. Ultimately, the model is able to account for the experimentally observed toughness enhancement in ferroelastic ceramics.     

Wedge indentation into elastic-plastic single crystals, 1: Asymptotic fields for nearly-flat wedge

Asymptotic stress and deformation fields under the contact point singularities of a nearly-flat wedge indenter and of a flat punch are derived for elastic ideally-plastic single crystals with three effective in-plane slip systems that admit a plane strain deformation state. Face-centered cubic (FCC), body-centered cubic (BCC), and hexagonal-close packed (HCP) crystals are considered. The asymptotic fields for the flat punch are analogous to those at the tip of a stationary crack, so a potential solution is that the deformation field consists entirely of angular constant stress plastic sectors separated by rays of plastic deformation across which stresses change discontinuously. The asymptotic fields for a nearly-flat wedge indenter are analogous to those of a quasistatically growing crack tip fields in that stress discontinuities can not exist across sector boundaries. Hence, the asymptotic fields under the contact point singularities of a nearly-flat wedge indenter are significantly different than those under a flat punch. A family of solutions is derived that consists entirely of elastically deforming angular sectors separated by rays of plastic deformation across which the stress state is continuous. Such a solution can be found for FCC and BCC crystals, but it is shown that the asymptotic fields for HCP crystals must include at least one angular constant stress plastic sector. The structure of such fields is important because they play a significant role in the establishment of the overall fields under a wedge indenter in a single crystal. Numerical simulations—discussed in detail in a companion paper—of the stress and deformation fields under the contact point singularity of a wedge indenter for a FCC crystal possess the salient features of the analytical solution.     

Analysis of the Interface Crack for Rubber-like Materials

The interface crack between two dissimilar rubber materials under mixed load is asymptotically analyzed in this paper. It is shown that the crack tip field is composed of one expanding sector and two shrinking sectors. For the case considered, the interface is located in the expanding sector. Analytical solutions are obtained for both the expanding sector and the shrinking sectors. The structures of the crack tip field obtained in this paper is compared with those in previous works for different constitutive equations.     

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.     

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。     

Study on fracture behavior of ferroelectric ceramics under combined electromechanical loading by using a moiré interferometry technique

This paper discusses an in situ observation of fracture behavior around a crack tip in ferroelectric ceramics under combined electromechanical loading by use of a moiré interferometry technique. The deformation field induced by the electric field and the stress concentration near the crack tip in three-points bending experiments was measured. By analysis of the moiré images it is found that under a constant mechanical load, the electric field almost has no effect on the crack extension in the case that the directions of the poling, electric field and crack extension are perpendicular to each other. When the poling direction is parallel to the crack extension direction and perpendicular to the electric field, the strain decreases faster than that calculated by FEM with and without electrical loading as one goes away from the crack tip. In addition, as the electric field intensity increases, the strain near the crack tip increases, and the strain concentration becomes more significant. The project supported by the National Natural Science Foundation of China (10132010, 10025209, 10232023)     

新型各向异性核工业石墨断裂力学行为的实验研究和数值分析

本文分别采用激光和白光DSCM(数字散斑相关测量)方法对一种新型各向异性核工业石墨的裂纹尖端位移、应变场进行了实验研究，两种方法都取得了较好的结果。考虑核工业石墨制备过程形成的各向异性特点，本文构建了三维各向异性有限元模型，采用奇异单元，计算模拟石墨裂纹尖端的变形和应力场，通过对实验结果和有限元计算结果的比较可以发现两者具有相近的趋势。     

VISCOPLASTIC SOLUTION TO FIELD AT STEADILY PROPAGATING CRACK TIP IN LINEAR- HARDENING MATERIALS

An elastic-viscoplastic constitutive model was adopted to analyze asymptotically the tip-field of moving crack in linear-hardening materials under plane strain condition. Under the assumption that the artificial viscosity coefficient was in inverse proportion to power law of the rate of effective plastic strain, it is obtained that stress and strain both possess power law singularity and the singularity exponent is uniquely determined by the power law exponent of the rate of effective plastic strain. Variations of zoning structure according to each material parameter were discussed by means of numerical computation for the tip-field of mode Ⅱ dynamic propagating crack, which show that the structure of crack tip field is dominated by hardening coefficient rather than viscosity coefficient. The secondary plastic zone can be ignored for weak hardening materials while the secondary plastic zone and the secondary elastic zone both have important influence on crack tip field for strong hardening materials. The dynamic solution approaches to the corresponding quasi-static solution when the crack moving speed goes to zero, and further approaches to the HR (Hui-Riedel) solution when the hardening coefficient is equal to zero.     

AN INVESTIGATION ON ASYMPTOTIC NEAR-TIP FIELDS OF DYNAMIC CRACK IN AN ELASTIC-PERFECTLY PLASTIC COMPRESSIBLE MATERIAL

The stress and deformation fields near the tip of a mode-I dynamic crack steadilypropagating in an elastic-perfectly plastic compressible material are considered under plane strain condi-tions. Within the framework of infinitesimal displacement gradient theory, the material is character-ized by the Von Mises yield criterion and the associated J_2 flow theory of plasticity. Through rigorousmathematical analysis, this paper eliminates the possibilities of elastic unloading and continuousasymptotic fields with singular deformation, and then constructs a fully continuous and boundedasymptotic stress and strain field. It is found that in this solution there exists a parameter (?)_0 whichcannot be determined by asymptotic analysis but may characterize the effect of the far field. Lastly thevariations of continuous stresses, velocities and strains around the crack tip are given numerically fordifferent values of (?)_0.     

一种基于应变梯度监测Ⅰ型裂纹的方法

基于线弹性断裂力学中I型裂纹的欧文解答,解析推导了在单向拉伸作用下无限大平板中I型裂纹尖端应变梯度场,建立了应变梯度与裂纹扩展之间的关联;基于挠曲电效应建立了电极化强度与应变梯度之间的力电耦合关系,提出了一种利用应变梯度传感器监测I型裂纹的方法,获知裂纹尖端坐标和裂纹扩展长度.本研究拟为应用应变梯度传感器对工程结构中裂纹扩展的实时监测提供初步的理论依据及方法.挠曲电感应技术在结构健康监测领域前景广阔.