Influence of the root radius of crack-like notches on the fracture load of brittle components

Narrow notches often cause damage that can lead to the destruction of components. The stress field in the vicinity of such crack-like notches in two-dimensional (2D) structures is similar to the stress field around equivalent cracks. Therefore similar investigations are necessary to predict the fracture load for components with cracks or narrow notches. Thus, the asymptotical stress field for a narrow notch with a rounded notch root is deduced from an Airy’s stress function. Based on this stress field a fracture criterion is developed. Comparing the theoretical fracture limit curves derived from the fracture criterion with experimental results it can be shown that for brittle material the local stress state at the fracture initiation point is the same for mode I, mixed-mode and mode II loading.     

Matched asymptotic expansions in nonlinear fracture mechanics—I. longitudinal shear of an elastic perfectly-plastic specimen

A method of analysis based upon matched asymptotic expansions is proposed for a cracked specimen which is subjected to longitudinal shear (mode III) loading. This gives the small-scale yielding estimate of linear fracture mechanics as a first approximation, and provides systematic refinements which take account of the nonlinear interaction between the elastic and the plastic regions. Explicit solutions can be generated for any specimen which is amenable to a linear elastic analysis. Fracture parameters, such as crack opening displacement and the Jintegral, are expressed as power series in the ratio of applied stress to yield stress, and three terms are given explicitly. These are defined from linear elastic solutions alone. The edge-cracked strip and cracking from a semi-circular notch are studied as examples. Comparison with an exact solution for the former geometry suggests that the three-term expansions give useful results up to 75 % of limit load. The latter example is new and shows the effect of a notch on a crack at loads beyond the normal range of validity of linear elastic fracture mechanics.     

An investigation of the losipescu and asymmetrical four-point bend tests

The Iosipescu shear test, utilizing a notched specimen in bending and a modification—the asymmetrical four-point bend (AFPB) test—were evaluated as shear tests for composites. This paper summarizes the results of an extensive numerical and experimental investigation of the Iosipescu and AFPB test methods. Finite-element analyses were conducted to assess the influence of notch parameters and load locations on the stress state in the specimen. The shear moduli and the shear strengths were experimentally measured for a quasiisotropic graphite/epoxy laminate using both the Iosipescu and the AFPB test methods. The tests were conducted for various combinations of notch parameters and load locations. The test results indicate that changes in the notch geometry and load locations aimed at improving the stress distribution in the test section resulted in unexpected changes in the failure mode. Paper was presented at the 1985 SEM Spring Conference on Experimental Mechanics held in Las Vegas, NV on June 9–14.     

Analytical representation of the non-square-root singular stress field at a finite angle sharp notch

The stress field near the tip of a finite angle sharp notch is singular. However, unlike a crack, the order of the singularity at the notch tip is less than one-half. Under tensile loading, such a singularity is characterized by a generalized stress intensity factor which is analogous to the mode I stress intensity factor used in fracture mechanics, but which has order less than one-half. By using a cohesive zone model for a notional crack emanating from the notch tip, we relate the critical value of the generalized stress intensity factor to the fracture toughness. The results show that this relation depends not only on the notch angle, but also on the maximum stress of the cohesive zone model. As expected the dependence on that maximum stress vanishes as the notch angle approaches zero. The results of this analysis compare very well with a numerical (finite element) analysis in the literature. For mixed-mode loading the limits of applicability of using a mode I failure criterion are explored.     

A generalized stress intensity factor to be applied to rounded V-shaped notches

In the presence of sharp (zero radius) V-shaped notches the notch stress intensity factors (N-SIFs) quantify the intensities of the asymptotic linear elastic stress distributions. They are proportional to the limit of the mode I or II stress components multiplied by the distance powered 1 − λ_i from the notch tip, λ_i being Williams’ eigenvalues. When the notch tip radius is different from zero, the definition is no longer valid from a theoretical point of view and the characteristic, singular, sharp-notch field diverges from the rounded-notch solution very next to the notch. Nevertheless, N-SIFs continue to be used as parameters governing fracture if the notch root radius is sufficiently small with respect to the notch depth.Taking advantage of a recent analytical formulation able to describe stress distributions ahead of rounded V-notches, the paper gives a generalized form for the notch stress intensity factors, in which not only the opening angle but also the tip radius dimension is explicitly involved. Such parameters quantify the stress redistribution due to the root radius with respect to the sharp notch case.     

基于奇异强度因子的V型切口应力场和N-SIF评估方法

根据线弹性断裂力学理论，V形切口处的应力场具有奇异性，应力值趋于无穷大，峰值应力不能直接用于评定疲劳强度。通过引入了奇异强度因子“as”，单边缺口应力分布和缺口应力强度因子(N-SIF)的半解析公式被推导。考虑张开角和几何尺寸等因素，基于奇异强度因子拟合得到了切口应力评估的简易公式，可用于切口应力场和N-SIF值的快速评估。将简易公式评估结果与有限元结果以及传统文献结果进行对比分析，结果表明，本文简易公式可以准确地预报拉伸载荷下单边V型切口角平分线上的应力场和N-SIF值，实现了切口试样应力场的快速评估。     

Fracture analysis of U-notched disc-type graphite specimens under mixed mode loading

Fracture phenomenon was investigated both experimentally and theoretically for a type of coarse-grained polycrystalline graphite weakened by a U-shaped notch under mixed mode loading. First, 36 disc-type graphite specimens containing a central U-notch, so called in literature as the U-notched Brazilian disc (UNBD), were prepared for four different notch tip radii and the fracture tests were performed under mode I and mixed mode I/II loading conditions. Then, the experimentally obtained fracture loads and the fracture initiation angles were predicted by using the U-notched maximum tangential stress (UMTS) and the newly formulated U-notched mean stress (UMS) fracture criteria. Both the criteria were developed in the form of the fracture curves and the curves of fracture initiation angle, in terms of the notch stress intensity factors (NSIFs). The results showed that while the criteria could predict successfully the experimental notch fracture toughness values, the UMS criterion provides slightly better predictions than the UMTS criterion, particularly for shear-dominant deformations. Also, found in this research was that the curves of fracture initiation angle were almost identical for the two criteria which both could predict well the experimental results.     

Fracture investigation at V-notch tip using coherent gradient sensing (CGS)

Local deformation field and fracture characterization of mode I V-notch tip are studied using coherent gradient sensing (CGS). First, the governing equations that relate to the CGS measurements and the elastic solution at mode I V-notch tip are derived in terms of the stress intensity factor, material constant, notch angle and fringe order. Then, a series of CGS fringe patterns of mode I V-notch are simulated, and the effects of the notch angle on the shape and size of CGS fringe pattern are analyzed. Finally, the local deformation field and fracture characterization of mode I V-notch tip with different V-notch angles are experimentally investigated using three-point-bending specimen via CGS method. The CGS interference fringe patterns obtained from experiments and simulations show a good agreement. The stress intensity factor obtained from CGS measurements shows a good agreement with finite element results under K-dominant assumption.     

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.     

The dynamic behaviour of sharp V-notches under impact loading

The dynamic behaviour of sharp V-notches which are either symmetric or oblique to the longitudinal boundary of a homogeneous elastic and isotropic strip subjected to an impact plane pulse was studied by the method of caustics. The stress pulse impinging on the flanks of the notch reflects and diffracts in different ways depending on the geometry of the notch relative to the coming pulse. For compressive stress pulses a stress concentration at the bottom of the notch does not create a crack propagation phenomenon, whereas for tensile pulses there is a possibility for an incubation, nucleation and eventual propagation of a crack. A complete experimental study of the incubation nucleation and propagation of cracks from the bottoms of notches in thin strips under tensile stress pulses was undertaken, whereas for compressive stress pulses the stress concentration at the bottom of the notch was evaluated. Interesting results were disclosed concerning the reinforcement of pulses by reflection and caging in, the evolution of stress concentration at the notch and the mode of crack propagation inside the plate. Dynamic stress intensity factors were evaluated all over the paths of crack propagation indicating a close intimacy between crack velocity and values of SIFs.     

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.     

Analysis of crack extension in anisotropic materials based on local normal stress

The normal stress ratio theory is applied to predict crack extension behavior in center-notched unidirectional graphite-epoxy of arbitrary fiber axis orientation, subjected to arbitrary far-field planar loading. The theory is applied within analytical solutions for two infinite plate geometries: a plate with a sharp center crack, and a plate with an elliptical center flaw. A critical analytical case is identified suggesting that application of the theory within a stress solution modelling crack tip shape may increase the accuracy of crack growth direction predictions. Crack extension direction, location of crack extension, and critical stress predictions of the theory are compared to those obtained from experiments on specimens subjected to tensile, shear, and mixed-mode far-field loading. The comparison shows that, applied within each analytical solution, the normal stress ratio theory provides verifiable predictions of crack growth behavior. By modelling actual notch tip shape, the elliptical notch solution is able to provide accurate qualitative predictions of the origin of crack extension along the periphery of a cut notch tip in a way that the sharp crack analysis cannot. The sharp notch solution appears to provide slightly more accurate crack growth direction predictions, however. Also, in predicting critical applied far-field stresses, the sharp crack solution appears to exhibit a stronger ability to model subtle experimental trends.     

DETERMINATION OF CRACK SURFACE DISPLACEMENTS FOR A RADIAL CRACK EMANATING FROM A SEMI-CIRCULAR NOTCH USING WEIGHT FUNCTION METHOD

A radial crack emanating from a semi-circular notch is of significant engineering importance. Accurate determination of key fracture mechanics parameters is essential for damage tolerance design and fatigue crack growth life predictions. The purpose of this paper is to provide an efficient and accurate closed-form weight function approach to the calculation of crack surface displacements for a radial crack emanating from a semi-circular notch in a semi-infinite plate.Results are presented for two load conditions: remote applied stress and uniform stress segment applied to crack surfaces. Based on a correction of stress intensity factor ratio, highly accurate analytical equations of crack surface displacements under the two load conditions are developed by fitting the data obtained with the weight function method. It is demonstrated that the WuCarlsson closed-form weight functions are very efficient, accurate and easy-to-use for calculating crack surface displacements for arbitrary load conditions. The method will facilitate fatigue crack closure and other fracture mechanics analyses where accurate crack surface displacements are required.     

Inverse notch sensitivity: Cracks can make nonwoven fabrics stronger

The strength of materials is always reduced in the presence of notches and cracks and this phenomenon – known as notch sensitivity – is critical in structural design. Good structural materials (ductile metals, elastomers) tend to be notch insensitive, which was considered to be the optimum behavior. Here, we report that inverse notch insensitivity (where the failure stress of the notched specimen is higher than that of the unnotched counterpart) can be achieved in polypropylene nonwoven fabrics. This behavior is only possible because of the peculiar microstructure of nonwoven fabrics, in which fracture of interfiber bonds provides a source of non-linear deformation and leads to a change in the network topology. The former facilitates crack tip blunting, spreading damage in the ligament, while the re-orientation of the fibers perpendicular to the notch plane strengthens the material and improves the maximum load bearing capability.     

A new version of the Neuber rule accounting for the influence of the notch opening angle for out-of-plane shear loads

The paper deals with nonlinear stress and strain distributions at the root of sharp and rounded notches with different opening angles under antiplane shear loading and small scale yielding. In order to make an easier comparison with the Neuber rule, the material is thought of as obeying the particular nonlinear law used in the past just by Neuber.By solving the linear differential equation resulting from the use of the hodograph transformation, a new relationship linking linear and nonlinear stress and strain concentrations is found. The relationship is written also in terms of the relevant notch stress intensity factors. In contrast with the Neuber rule, this relationship strictly depends on the notch opening angle. Even when the notch opening angle is zero, it does not match the Neuber Rule, but results in an additional factor 2 which is in agreement with Hult and McClintock’s solution when the notch tip radius tends to zero and the notch becomes a crack.     

Higher order fields for damaged nonlinear antiplane shear notch, crack and inclusion problems

Damaged nonlinear antiplane shear problems with a variety of singularities are studied analytically. A deformation plasticity theory coupled with damage is employed in analysis. The effect of microscopic damage is considered in terms of continuum damage mechanics approach. An exact solution for the general damaged nonlinear singular antiplane shear problem is derived in the stress plane by means of a hodograph transformation, then corresponding higher order asymptotic solutions are obtained by reversing the stress plane solution to the physical plane. As example, traction free sharp notch and crack, rigid sharp wedge and flat inclusion, and mixed boundary sharp notch problems are investigated, respectively. Consequently, higher order fields are obtained, in which analytical expressions of the dominant and second order singularity exponents and angular distribution functions of the near tip fields are derived. Effects of the damage and hardening exponents of materials and the geometric angle of notch/wedge on the near tip quantities are discussed in detail. It is found that damage leads to a weaker dominant singularity of stress, but to little stronger singularities of the dominant and second order terms of strain compared to that for undamaged material. It is also seen that damage has important effect on the angular distribution functions of the near tip stress and strain fields. As special cases, higher order analytical solutions of the crack and rigid flat inclusion tip fields are obtained, respectively, by reducing the notch/wedge tip solutions. Effects of damage and hardening exponents on the dominant and second order terms in the solutions of the crack and inclusion tip fields are discussed.     

Analytical solution for a strained reinforcement layer bonded to lip-shaped crack under remote mode III uniform load and concentrated load

In this paper, the analytical solution of stress field for a strained reinforcement layer bonded to a lip-shaped crack under a remote mode III uniform load and a concentrated load is obtained explicitly in the series form by using the technical of conformal mapping and the method of analytic continuation. The effects of material combinations, bond of interface and geometric configurations on interfaciai stresses generated by eigenstrain, remote load and concentrated load are studied. The results show that the stress concentration and interfaciai stresses can be reduced by rational material combinations and geometric configurations designs for different load forms.     

Glass Fracture in the Region of Its Contact with Steel Balls

Results of experiments on indentation of steel balls of various diameters into glass specimens shaped as rectangular parallelepipeds are reported. The limiting load of formation of an annular crack near the contact area and the radius of this crack are experimentally determined. The field of the contact stress in the fracture region is found by using the Huber solution of the Hertz problem of ball indentation into an elastic half-space. Modeling fracture caused by contact interaction is based on using the local criterion of the maximum stress and several nonlocal criteria: average stress criterion, Nuismer criterion, and gradient criterion. For calculating the parameter that has the length dimension and is included into the nonlocal fracture criteria, the ultimate tensile stress and the critical stress intensity factor of glass are experimentally determined for beams without and with a notch. It is demonstrated that the experimental data on the annular crack radius are most adequately predicted by the gradient criterion among all criteria considered in the study. However, this criterion overpredicts the ultimate tensile stress in the course of beam bending, which is caused by the scale factor.     

A finite element investigation of the effect of crack tip constraint on hole growth under mode i and mixed mode loading

In this work, the effect of constraint on hole growth near a notch tip in a ductile material under mode I and mixed mode loading (involving modes I and II) is investigated. To this end, a 2-D plane strain, modified boundary layer formulation is employed in which the mixed mode elastic K–T field is prescribed as remote boundary conditions. A finite element procedure that accounts for finite deformations and rotations is used along with an appropriate version of J₂ flow theory of plasticity with small elastic strains. Several analyses are carried out corresponding to different values of T-stress and remote elastic mode-mixity. The interaction between the notch and hole is studied by examining the distribution of hydrostatic stress and equivalent plastic strain in the ligament between the notch tip and the hole, as well as the growth of the hole. The implications of the above results on ductile fracture initiation due to micro-void coalescence are discussed.     

Photoelastic determination of stress-intensity factors

The increasing number of analytical and numerical solutions for the crack-tip stress-intensity factor has greatly widened the scope of application of linear elastic fracture-mechanics technology. Experimental verification of a particular solution by elastic stress analysis is often a necessary supplement to provide the criteria for proper application to actual design problems. In this paper, it is shown that the photoelastic technique can be used to obtain rather good estimates of the stress-intensity factor for various specimen geometries and loading conditions. Treated are the following cases: wedge-opening load specimen, several notched rotating-disk configurations, and a notched pressure vessel. A sharp crack is simulated by a relatively narrow notch terminating in a root radius of 0.010 in or less. Stress distributions along the section of symmetry ahead of the notch tip are obtained using three-dimensional frozen-stress photoelasticity. The results are used to determine the stress-intensity factor, cK _I , by three methods. Two of these are based on Irwin's expressions for the elastic stress field at the tip cf a crack, and the other is a result of Neuber's hyperbolic-notch analysis. Agreement, with available analytical solutions is good.