Simulated caustic patterns for dynamic crack growth in a plate based on a three-dimensional numerical analysis

The rapid growth of a through-the-thickness crack in a nominally elastic plate under Mode I conditions is considered, with emphasis on the three-dimensional aspects of the elastodynamic crack growth process. For points near the crack edge, the assumptions on stress state consistent with plane stress analysis are not valid, and the possibility of an inertial effect associated with rapid lateral contraction must be considered. The steady crack growth process is analyzed here by means of the finite element method, and the degree to which the local fields are described by plane stress analysis is established. In addition, synthetic shadow spot patterns are generated from the numerical three-dimensional solutions. The inferred values of stress intensity factor are compared with the values of the corresponding plane stress dynamic stress intensity factor.     

Plastic zone boundary and Poisson’s ratio depending on plastic loosening

A more accurate determination of the plastic zone boundary for plane strain and plane stress state is proposed. The plastic zone boundary is determined with regard to plastic loosening, given exact stress distribution and Schleicher yield criterion. The presence of mean normal stress in the Schleicher criterion ensures uniform expansion of the plastic zone. The dependence of Poisson’s ratio and constraint ratio for plastic strain on plastic loosening of material is examined. These parameters peak at the tip or in the immediate vicinity of a stress concentrator and decrease with distance from it. In a small neighborhood of the crack tip, a region is found in which Poisson’s ratio is impossible to determine from the Schleicher criterion. The size of this region is identified with the size of the region of exhausted plasticity.     

Optical method of caustics applied in viscoelastic fracture analysis

The optical method of caustics is developed here to study the fracture of viscoelastic materials. By adopting a distribution of viscoelastic stress fields near the crack tip, the method of caustics is used to determine the viscoelastic fracture parameters from the caustic patterns near the crack tip. Two viscoelastic materials are studied. These are PMMA and ternary composites of HDPE/POE-g-MA/CaCO₃. The transmitted and reflective methods of caustics are performed separately to investigate viscoelastic fracture behaviors. The stress intensity factors (SIFs) versus time is determined by a series of shadow spot patterns combined with viscoelastic parameters evaluated by creep tests. In order to understand the viscoelastic fracture mechanisms of HDPE/POE-g-MA/CaCO₃ composites, their fracture surfaces are observed by a Scanning Electron Microscope (SEM). The results indicate that the method of caustics can be used to characterize the fracture behaviors of viscoelastic materials and further to optimize the design of polymer composites.     

The influence of singular field dominance on the caustic methods

The caustic method as applied to fracture mechanics for measuring the stress intensity factor, contains several approximations which limit its applicability and accuracy. The accuracy of the caustic method for the determination of the stress intensity factor is based on the restriction that the K-dominant singular field is fully established at which the experimental data are taken. In this paper, we attempt to identify the regions in which experimental measurements by caustic methods can be performed with confidence. The reliability of the method of caustics applied to the static problem and the case of a stationary crack subjected to dynamic loading are investigated in detail. The results demonstrate that significant deviation can occur in the determination of the stress intensity factor from shadow spot measurements. Conclusions regarding the method of caustics in the determination of the stress intensity factor are extracted from these results.     

Influence of low temperatures on the fracture micromechanism of materials with bcc and fcc lattice structures under single loading conditions

General regularities of the influence of the local stress state of a material on the plastic zone formation at the crack tip and the fracture micromechanism of materials with bcc and fcc lattice structures under single loading (static, impact, high-speed pulsed) conditions have been established. Schemes of plastic zone formation under plane strain, plane stress, and in the transient region from plane strain to plane stress are proposed.     

Finite strain stress fields near the tip of an interface crack between a soft incompressible elastic material and a rigid substrate

We present a numerical study of finite strain stress fields near the tip of an interface crack between a rigid substrate and an incompressible hyperelastic solid using the finite element method (FEM). The finite element (FE) simulations make use of a remeshing scheme to overcome mesh distortion. Analyses are carried out by assuming that the crack tip is either pinned, i.e., the elastic material is perfectly bonded (no slip) to the rigid substrate, or the crack lies on a frictionless interface. We focus on a material which hardens exponentially. To explore the effect of geometric constraint on the near tip stress fields, simulations are carried out under plane stress and plane strain conditions. For both the frictionless interface and the pinned crack under plane stress deformation, we found that the true stress field directly ahead of the crack tip is dominated by the normal opening stress and the crack face opens up smoothly. This is also true for an interface crack along a frictionless boundary in plane strain deformation. However, for a pinned interface crack under plane strain deformation, the true opening normal stress is found to be lower than the shear stress and the transverse normal stress. Also, the crack opening profile for a pinned crack under plane strain deformation is completely different from those seen in plane stress and in plane strain (frictionless interface). The crack face flips over and the tip angle is almost tangential to the interface. Our results suggest that interface friction can play a very important role in interfacial fracture of soft materials on hard substrates.     

Caustics,pseudocaustics and the related illuminated and dark regions with the computational method of quantifier elimination

The method of caustics is a powerful experimental method in elasticity and particularly in fracture mechanics for crack problems. The related method of pseudocaustics is also of interest. Here we apply the computational method of quantifier elimination implemented in the computer algebra system Mathematica in order to determine (i) the non-parametric equation and two properties of the caustic at a crack tip and especially (ii) the illuminated and the dark regions related to caustics and pseudocaustics in plane elasticity and plate problems. The present computations concern: (i) The derivation of the non-parametric equation of the classical caustic about a crack tip through the elimination of the parameter involved (here the polar angle) as well as two geometrical properties of this caustic. (ii) The derivation of the inequalities defining the illuminated region on the screen in the problem of an elastic half-plane loaded normally by a concentrated load with the boundary of this illuminated region related to some extent to the caustic formed. (iii) Similarly for the problem of a clamped circular plate under a uniform loading with respect to the caustic and the pseudocaustic formed. (iv) Analogously for the problem of an equilateral triangular plate loaded by uniformly distributed moments along its whole boundary, which defines the related pseudocaustic. (v) The determination of quantities of interest in mechanics from the obtained caustics or pseudocaustics. The kind of computations in the applications (ii) to (iv), i.e. the derivation of inequalities defining the illuminated region on the screen, seems to be completely new independently of the use here of the method of quantifier elimination. Additional applications are also possible, but some of them require the expansion of the present somewhat limited power of the quantifier elimination algorithms in Mathematica. This is expected to take place in the future.     

The study of crack tip stress and strain field in elasto-plastic materials

Making use of the elastic-plastic finite element program and the small scale yielding crack tip solution, three point bend and compact tension specimens are analysed at various levels of work hardening. Special attention is given to the singular stress distribution at a crack tip, to the fracture behaviour characterized by path independent integral and by crack tip opening displacement. Comparison is made between experimental and theoretical results for a structural low alloy steel. Correlation between the two is satisfactory, the experimental data are bounded by plane strain and plane stress solutions. It is indicated that the uncertainty in the true stress-strain curve in the region of initial yield may strongly influence the theoretical calculations.The authors would like to thank Professor Dr. H.Bethge, director of the Institute of Solid State Physics and Electron Microscopy in Halle (GDR) and Professor Dr. P.Ry, director of the Institute of Physical Metallurgy, Czechoslovak Academy of Sciences in Brno for their support of this cooperative research project.     

Dynamic caustics and its applications

For the study of elastodynamic problems of propagating cracks it is necessary to evaluate the dynamic stress intensity factor K^d_I which depends on the form of expressions for the stress components existing at the running crack tip at any instant of the propagation of the crack and the corresponding dynamic mechanical and optical properties of the material of the specimen under identical loading conditions. In this paper the distortion of the form of the corresponding reflected caustic from the lateral faces of a dynamically loaded transparent and optically inert specimen containing a transverse crack running under constant velocity was studied on the basis of complex potential elasticity theory and the influence of this form on the value of the dynamic stress intensity factor was given. The method was applied to the study of a propagating Mode I crack in a PMMA specimen under various propagation velocities and the corresponding dynamic stress intensity factor K^d_I evaluated. Also, crack propagation behaviour of notched composites in dynamic loading modes are reviewed and evaluated. A relatively large data base using metal-epoxy particulates, rubber-toughened poly(methyl methacrylate), and Sandwich plates are given. In all cases, a combination of high-speed photography and the optical method of dynamic caustics has been used. Results on the dynamic crack propagation mode, fracture toughness and crack propagation velocities of several rubber-modified composite models are presented. The composite models studied include specimens with one and/or two ‘complex’ two-stage inclusions, i.e. PMMA round inclusions surrounded by concentric rubber rings, one and/or press-fifting inclusions without rubber interface, all under dynamic loading. In all cases both qualitative and quantitative results were obtained. Also, results on crack propagation mode, crack propagation velocity, stress intensity factors and on the influence of the sandwich phases on crack propagation mode are presented.     

The effect of the boundary conditions on in-plane and out-of-plane stress field in three dimensional plates weakened by free-clamped V-notches

Dealing with the material microstructure an analytical multiscale model has recently been developed by Sih. Physically, the different orders of the stress singularities are related to the different constraints associated with the defect thought as a microscopic V-notch at the tip of the main crack. Irregularities of the material microstructure tend to curl the crack tip being the clamped-free boundary conditions the more realistic and general representation of what occurs on the microscopic V-notch. As a result, mixed mode conditions are always present along the V-notch bisector line.It is known for a long time that at the antisymmetric (mode II) stress distribution ahead of the crack tip generates a coupled out-of-plane singular mode. Recent theoretical and numerical analyses have demonstrated that this out-of-plane mode due to three-dimensional effects occurs also in the case of large V-notches where the mode II stress field is no longer singular. In addition, when the notch opening angle is non-zero, the three-dimensional singular stress state is strongly influenced by the plate thickness.The aim of this paper is to investigate the effect of free-fixed boundary conditions along the notch edges in three dimensional plates weakened by pointed V-notches and quantify the intensity of the out-of-plane singularity occurring under this constraint configuration. For the sake of simplicity a macronotch is considered rather than a micronotch. A synthesis of the magnitude of the stress state through the plate thickness is carried out by using the mean value of the strain energy density over a given control volume embracing the notch tip. The capability of the strain energy density to capture all the combined effects due to the out-of-plane mode make it a powerful parameter at every scale levels.     

Fracture behaviors induced by thermal stress in an anisotropic half plane superconductor

This Letter presents a theoretical analysis to the fracture behavior of a large single domain YBCO superconductor under thermal stress based on the two-dimensional theory of anisotropic thermoelasticity, the coupled finite element and infinite element numerical method. The thermal stress intensity factors are obtained due to a uniform heat flux by a line crack in a generally half plane superconductor. It is found that the thermal stress intensity factors decrease with the decrease of temperature, and while the longer the crack length is, the larger the stress intensity factors. Additionally, the J-integral at the crack tip is also investigated, a similar behavior to the thermal stress intensity factors is found. These results are benefit for us to understand the fracture mechanism of superconductor both in theory and application.     

The caustic approach to Rayleigh-waves and their interaction with surface irregularities

During the past 25 years the method of caustics has matured to the stage where it offers a highly potential tool for qualitative as well as quantitative solutions of engineering problems in many fields. In conjunction with high-speed recording techniques such as high-speed cinematography, the method of caustics serves as a means for visualizing the highly complex interaction processes that occur when elastic stress waves interact with material inhomogeneities or discontinuities such as inclusions, surface or embedded cracks and contacting surfaces during impact.In this contribution, the method of caustics is applied to a number of dynamic problems all of which are associated with the propagation and interaction of Rayleigh-waves:

• the pseudo-caustic of a Rayleigh-wave;

• Rayleigh-wave generation during dynamic impact of two elastic bodies;

• Rayleigh-wave interaction with a shallow embedded crack;

• Rayleigh-wave interaction with various configurations of sets of surface cracks.

Sequences of high-speed cinematographic recordings of the evolution of a dynamic process provide the raw data for data and image analysis for the determination of stress intensity factors, contact forces, load-time traces or other quantities of physical interest.The paper also discusses some of the advantages and disadvantages of the method of caustics over other methods currently used in photomechanics such as dynamic photoelasticity. Data reduction and analysis aspects will also be highlighted.     

碳纤维布补强梁裂尖奇异性的光学焦散线实验研究

通过光学焦散线方法对碳纤维布加固含裂纹梁进行三点弯荷载下的静荷载和冲击实验,研究了其断裂特性。在静态实验研究中,记录了不同载荷状态下裂纹的焦散斑光学图像,计算了与之对应的裂纹尖端的应力强度因子,分析了试件在粘贴加固不同长度的碳纤维布情况下焦散斑和应力强度因子演化规律。在动态实验中,记录了裂纹扩展的过程和在此过程中不同时刻的焦散斑和裂纹扩展的路线。通过各瞬间的焦散斑图像,分析裂纹的扩展长度及速度,并得到各时刻的动态应力强度因子。     

On the plastic zone size and crack tip opening displacement of a sub-interface crack in an infinite bi-material plate

Elastic–plastic stress analysis has been carried out for the plastic zone size and crack tip opening displacement of a sub-interface crack with small scale yielding. In our study, the shape of plastic zone is assumed as a long, slim strip at both crack tips. In the plastic zone, both normal stress and shear stress exist and are considered due to the bi-material interface. The values of the plastic zone size, normal stress and shear stress are determined by satisfying the conditions where both Modes I and II stress intensity factors vanish and Von Mises yield criterion is met. In the present paper, the sub-interface crack is simulated by continuously distributed dislocations which will result in singular integral equations. Those singular integral equations can be solved by reducing them to a set of linear equations. The values of the plastic zone size and crack tip opening displacement are obtained through an iterative procedure. Finally, the effect of normalized loading, normalized crack depth (distance to the interface) and Dundurs’ parameters on the normalized plastic zone size and the normalized crack tip opening displacement is discussed.     

晶体相场法研究应力状态及晶体取向对微裂纹尖端扩展行为的影响

采用晶体相场模拟研究了单向拉伸作用下初始应力状态、晶体取向角度对单晶材料内部微裂纹尖端扩展行为的影响, 以(111)晶面上的预制中心裂纹为研究对象探讨了微裂纹尖端扩展行为的纳观机理, 结果表明: 微裂纹的扩展行为主要发生在<011>(111)滑移系上, 扩展行为与扩展方向与材料所处的初始应力状态及晶体取向紧密相关. 预拉伸应力状态将首先诱发微裂纹尖端生成滑移位错, 进而导致晶面解理而实现微裂纹尖端沿[011]晶向扩展, 扩展到一定程度后由于位错塞积, 应力集中, 使裂纹扩展方向沿另一滑移方向[101], 并形成锯齿形边缘; 预剪切应力状态下, 微裂纹尖端首先在[101]晶向解理扩展, 并诱发位错产生, 形成空洞聚集型长大的二次裂纹, 形成了明显的剪切带; 预偏变形状态下微裂纹尖端则直接以晶面解理形式[101]在上进行扩展, 直至断裂失效; 微裂纹尖端扩展行为随晶体取向不同而不同, 较小的取向角度会在裂纹尖端形成滑移位错, 诱发空位而形成二次裂纹, 而较大的取向角下的裂纹尖端则以直接解理扩展为主, 扩展方向与拉伸方向几近垂直.     

Atomistic study of dislocation loop emission from a crack tip

We report the first atomistic calculation of the saddle-point configuration and activation energy for the nucleation of a 3D dislocation loop from a stressed crack tip in single crystal Cu. The transition state is found using reaction pathway sampling schemes, the nudged elastic band, and dimer methods. For the (111)[110] crack, loaded typically at 75% of the athermal critical strain energy release rate for spontaneous dislocation nucleation, the calculated activation energy is 1.1 eV, significantly higher than the continuum estimate. Implications concerning homogeneous dislocation nucleation in the presence of a crack-tip stress field are discussed.     

Elliptically polarized white light photoviscoelastic technique and its application to viscoelastic fracture

The present paper demonstrates the successful application of the photoviscoelastic technique using elliptically polarized white light to the stress field evaluation of the crack growth in a viscoelastic strip. Using the proposed technique, which can determine both isochromatic and isoclinic parameters simultaneously from a single color image, the time-dependent stress state around a slowly propagating crack tip in a viscoelastic strip plate is successfully analyzed. Then, the time-dependent stress intensity factor extended for linearly viscoelastic materials is evaluated from the experimental results using a method based on the least squares. The results show that the proposed critical stress intensity factor for fast crack growth may be considered as a characteristic property of the material under monotonically increasing load.     

边界元法分析二维线弹性裂纹扩展

用边界元法研究裂纹扩展过程.首先将尖端区域Williams渐近展开的特征分析法与边界积分方程结合,解出切口尖端附近应力奇异性区域的各应力场渐近展开项系数,获得平面切口/裂纹结构完整的位移和应力场.再基于考虑非奇异应力项贡献的最大周向应力脆性断裂准则,运用边界元法分析边缘含裂纹半圆形弯曲试样在荷载作用下的启裂方向,对裂纹扩展过程给出自动跟踪方法,通过算例证明边界元法模拟裂纹扩展过程的正确性和有效性.     

Actual light deflections in regions of crack tips and their influence on measurements in photomechanics

Crack-tip photomechanics procedures are based on certain simplifying assumptions that are seldom discussed. In a recent paper the theoretical bases of the shadow optical methods of caustics have been analysed and tested using the results obtained by three analytical-experimental procedures, namely classical strain gage techniques, isodynes, and strain-gradient index method. It has been concluded that the straing-radient index method appears to be a suitable tool for analysis of stress states near crack tips and notches and, in particular, for testing the predictive power of the pertinent singular solutions of the linear elastic fracture mechanics and the ranges of their applicability. In the present paper, a more detailed analysis of all results obtained in light deflection experiments allows to quantify the contribution of both involved effects and to determine the distortion of the faces of the investigated plates along their crack planes. The ability of the strain-gradient light bending method to analyse some features of the three-dimensional stress-state is reported. Finally, the presented experimental evidence allows to draw conclusions related to limits of applicability of certain photomechanical measurements near crack tips.^†     

A linear least squares approach for evaluation of crack tip stress field parameters using DIC

In the present work, an experimental study is carried out to estimate the mixed-mode stress intensity factors (SIF) for different cracked specimen configurations using digital image correlation (DIC) technique. For the estimation of mixed-mode SIF׳s using DIC, a new algorithm is proposed for the extraction of crack tip location and coefficients in the multi-parameter displacement field equations. From those estimated coefficients, SIF could be extracted. The required displacement data surrounding the crack tip has been obtained using 2D-DIC technique. An open source 2D DIC software Ncorr is used for the displacement field extraction. The presented methodology has been used to extract mixed-mode SIF׳s for specimen configurations like single edge notch (SEN) specimen and centre slant crack (CSC) specimens made out of Al 2014-T6 alloy. The experimental results have been compared with the analytical values and they are found to be in good agreement, thereby confirming the accuracy of the algorithm being proposed.