Origin of anomalous inverse notch effect in bulk metallic glasses

Understanding notch-related failure is crucial for the design of reliable engineering structures. However, substantial controversies exist in the literature on the notch effect in bulk metallic glasses (BMGs), and the underlying physical mechanism responsible for the apparent confusion is still poorly understood. Here we investigate the physical origin of an inverse notch effect in a Zr-based metallic glass, where the tensile strength of the material is dramatically enhanced, rather than decreased (as expected from the stress concentration point of view), by introduction of a notch. Our experiments and molecular dynamics simulations show that the seemingly anomalous inverse notch effect is in fact caused by a transition in failure mechanism from shear banding at the notch tip to cavitation and void coalescence. Based on our theoretical analysis, the transition occurs as the stress triaxiality in the notched sample exceeds a material-dependent threshold value. Our results fill the gap in the current understanding of BMG strength and failure mechanism by resolving the conflicts on notch effects and may inspire re-interpretation of previous reports on BMG fracture toughness where pre-existing notches were routinely adopted.     

Crack initiation at the rectangular step notch surface under combined loading

Summary  Under external forces acting on the face of a notch, cracks originate at corners, and the system is liable to fail. An analysis is presented of the stress field in the neighborhood of the notch tips, based on the integral representation of the biharmonic solution and on numerical methods. Computations were performed for constant loading or constant displacement distributed along one face of the notch. The coefficients in the principal terms of the asymptotic formulae for the circumferential and shear stresses depend on the angle and height of the notch face and on the boundary conditions. The maximal values of these coefficients determine the stress intensity factors for the opening and shear modes. The angles corresponding to the maximal values of the intensity factors indicate the directions of initiation of opening and sliding cracks. Received 30 May 2000; accepted for publication 3 April 2001     

Applicability of the crack faces thermoelectric boundary conditions for thermopiezoelectric materials

The applicability and effect of the crack surfaces thermoelectric boundary conditions in thermopiezoelectric fracture mechanics problem are discussed by using the finite thickness notch approach. The stress and electric displacement intensity factors at the notch tips, and thermal flux and electric displacement inside the notch are derived in closed-form. The numerical results are compared with the ideal crack solutions. It is found that the electrically impermeable crack boundary condition assumption is reasonable if the flaw in the material is a notch with finite width, and the thermal conductivity of air or vacuum inside the crack must be considered.     

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 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.     

Reliability assessment of a bi-material notch: Strain energy density factor approach

Joints of different materials have many applications in structural engineering and microelectronics. In the present contribution the joint is modelled as a bi-material notch. The singular stress field near the notch tip is investigated. Depending on the notch geometry and materials, the stress field can have one or two singularities. It is shown that to study the problem of a crack onset at the notch, both terms have to be taken into account. Criteria for the direction and for crack nucleation are formulated. The approach utilizes the knowledge of the strain energy density factor distribution in a bi-material notch vicinity.     

Fictitious Notch Rounding approach of pointed V-notch under in-plane shear

The Fictitious Notch Rounding approach (FNR) is applied here for the first time to V-shaped notches under in-plane shear loading. The fictitious radius is evaluated for different opening angles as a function of the Microstructural Characteristic Length (MCL), the actual radius and the failure hypothesis. A multiaxiality factor is introduced and found to be very sensitive to the opening angle. Under mode II loading, the problem is more complex than under mode I and mode III, mainly because the maximum elastic stress is outside the notch bisector line. The main problem is the choice of the expected crack initiation angle, which defines the direction where the relevant stress has to be integrated. This integration carried out over the MCL gives the effective stress value for the pointed V-notch. To this end, two different criteria are used, the Maximum Tangential Stress (MTS) criterion and the Minimum Strain Energy Density criterion (MSED). A large number of finite element analyses have been carried out to determine the multiaxiality factor. This factor has been evaluated by comparing the theoretical stress concentration factor (SCF) obtained from fictitiously rounded notches to the effective stress concentration factor obtained by integrating the relevant stress over the Microstructural Characteristic Length.     

Interaction of a screw dislocation with a notch in a piezoelectric bi-material

Summary The electro-elastic interaction of a screw dislocation and a notch in a piezoelectric bi-material is analyzed. The electro-elastic fields induced by the dislocation are derived using the conformal mapping and the image-dislocation approach, where the solution for a piezoelectric bi-material without a notch is used as a base. The stress and the electric displacement intensity factors of the notch and the image force on the dislocation are given explicitly. We find that intensity factors are expressed in terms of the effective material constants, while the radial component of the image force is independent of the notch angle and the angular position of the dislocation in the polar coordinate system. Numerical results for the image force are provided for the use when one of the two media is purely elastic. They illustrate the behavior of the dislocation in the neighborhood of the notch.     

Out-of-plane interface cracks in dissimilar piezoelectric materials

Summary This paper investigates the problem of an anti-plane interfacial crack between two dissimilar piezoelectric material layers. A single crack is first considered. The effect of interaction of two collinear cracks in the medium on the field intensity factors is investigated. The solutions of several particular cases, including an infinite piezoelectric bi-material and a piezoelectric material bonded to an elastic medium, are given. The bi-material constants governing the behavior of the crack tip fields are identified. By considering the crack as a notch of finite thickness, it is shown that the thickness of the notch has a pronounced influence on the crack tip field. The results for the assumption of a permeable crack represent the limit case where the notch thickness is reduced to zero.BLW would like to thank the National Science Foundation of China (#10102004) and the City University of Hong Kong (DAG #7100219) for the support of this work. YGS also thanks the Multidiscipline Scientific Research Foundation Project (HIT. MD 2001. 39) of the Harbin Institute of Technology and the SRF for ROCS, SEM.accepted for publication 3 April 2003     

分子动力学模拟金属玻璃板拉伸时剪切带的形成和演化过程

金属玻璃在低温高应力条件下容易生成剪切带而导致结构的破坏,大大限制了它的推广应用。本文采用分子动力学模拟研究了三种Cu64Zr36（不带缺口、一侧带缺口、两侧带缺口）金属玻璃板试样在拉伸过程中剪切带的形成和演化过程及其力学性能。结果表明：不带缺口金属玻璃板试样在低温高应力的拉伸过程中会自发出现局部剪切转变区,发生剪切局部化,继续拉伸会在与加载轴大约成45°方向上形成剪切带。剪切带的形成与剪切转变区的分布和局部化有关,带缺口比不带缺口的试样会更早出现应变局部化,即在较低的拉伸应变下便形成剪切带,其拉伸强度也相应较低。相同条件下,一侧带缺口与两侧带缺口的试样在拉伸强度上几乎相同,但两侧带缺口试样的应变局部化程度稍低,主要是两侧缺口处均出现了剪切转变区,导致其分布和局部化不够集中,这也是形成主剪切带和次剪切带的主要原因。以上结果为进一步从原子尺度理解金属玻璃剪切带的形成和演化特征提供了重要的信息。     

分子动力学模拟金属玻璃板拉伸时剪切带的形成和演化过程

金属玻璃在低温高应力条件下容易生成剪切带而导致结构的破坏,大大限制了它的推广应用。本文采用分子动力学模拟研究了三种Cu64Zr36（不带缺口、一侧带缺口、两侧带缺口）金属玻璃板试样在拉伸过程中剪切带的形成和演化过程及其力学性能。结果表明：不带缺口金属玻璃板试样在低温高应力的拉伸过程中会自发出现局部剪切转变区,发生剪切局部化,继续拉伸会在与加载轴大约成45°方向上形成剪切带。剪切带的形成与剪切转变区的分布和局部化有关,带缺口比不带缺口的试样会更早出现应变局部化,即在较低的拉伸应变下便形成剪切带,其拉伸强度也相应较低。相同条件下,一侧带缺口与两侧带缺口的试样在拉伸强度上几乎相同,但两侧带缺口试样的应变局部化程度稍低,主要是两侧缺口处均出现了剪切转变区,导致其分布和局部化不够集中,这也是形成主剪切带和次剪切带的主要原因。以上结果为进一步从原子尺度理解金属玻璃剪切带的形成和演化特征提供了重要的信息。     

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.     

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.     

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.     

Equivalence of the notch stress intensity factor,tip opening displacement and energy release rate for a sharp V-notch

For an infinite elastic plane with a sharp V-notch under the action of symmetrically loading at infinity, the length of crack initiation ahead of the V-notch’s tip is estimated according to a modified Griffith approach. Applying a new conservation integral to the perfectly plastic strip (Dugdale model) ahead of the V-notch’s tip, the relationship between notch stress intensity factor (NSIF) and notch tip opening displacement (NTOD) is presented. Also, the relationship between NSIF and perfectly plastic strip size (PPSS) is found. Since there are three fracture parameters (NSIF, NTOD, and PPSS) with changeable notch opening angle in two basic relationships, it is necessary to select one critical parameter with changeable notch opening angle or two independent critical parameters, respectively. With the help of a characteristic length, it is found by this new conservation integral that the NSIF, NTOD and energy release rate are equivalent in the case of small-scale yielding. Especially, the characteristic length possesses clear physical meaning and, for example, depends on both the critical NSIF and SIF or both the NTOD and CTOD, respectively, in which SIF and CTOD are from the tip of a crack degenerated from the sharp V-notch. The dependence of NSIF on NTOD and PPSS is presented according to the equivalence, and the critical NSIF depending on the critical NTOD with a notch opening angle is also predicted.     

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.     

Notch sensitivity in nanoscale metallic glass specimens: Insights from continuum simulations

Recent experiments have shown that nano-sized metallic glass (MG) specimens subjected to tensile loading exhibit increased ductility and work hardening. Failure occurs by necking as opposed to shear banding which is seen in bulk samples. Also, the necking is generally observed at shallow notches present on the specimen surface. In this work, continuum finite element analysis of tensile loading of nano-sized notched MG specimens is conducted using a thermodynamically consistent non-local plasticity model to clearly understand the deformation behavior from a mechanics perspective. It is found that plastic zone size in front of the notch attains a saturation level at the stage when a dominant shear band forms extending across the specimen. This size scales with an intrinsic material length associated with the interaction stress between flow defects. A transition in deformation behavior from quasi-brittle to ductile becomes possible when this critical plastic zone size is larger than the uncracked ligament length. These observations corroborate with atomistic simulations and experimental results.     

Adiabatic shear localization in the impact of edge-notched specimens

Impact experiments are performed on edgenotched specimens in the two-dimensional punch geometry. Materials tested include 18Ni(350) maraging steel; S7 tool steel; 4340, 300M, HP 9-4-20 and D-6ac ultra high-strength steels; and Ti6Al4V alloy. These materials have shown a high susceptibility to dynamic shear failure in previous studies. Impact velocity ranged from 25 m/s to 45 m/s, and shear bands were found to form at the notch tip and at the die corner on the back side of the specimen for all materials tested. Metallurgical analysis confirms the existence of adiabatic shear bands followed by a crack propagating through the fully developed shear band. High-speed photography was used to observe the initiation of adiabatic shear bands shortly after impact. Laser-etched lines on the specimen surfaces allowed the determination of the time of impact and the initiation time of shear failure. The elapsed time between the two was used to estimate the stress intensity factor at the time of shear band initiation. Comparisons of shear band initiation stress intensity factors at the notch tip and die corner are made. It is seen that the shear bands initiate at approximately the same stress intensity factor at both the notch tip and die corner. Finite element simulations support the use of a square root singularity for the stress in the plate near the corners of a deformable punch or die.     

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.     

Why does necking ignore notches in dynamic tension?

Recent experimental work has revealed that notched tensile specimens, subjected to dynamic loading, may fail by growing a neck outside of the notched region. This apparent lack of sensitivity to a classical stress concentration case was reported but not explained or modeled.The present paper combines experimental and numerical work to address this issue. Specifically, it is shown that the dynamic tensile failure locus is dictated by both the applied velocity boundary condition and the material mechanical properties, specifically strain-rate sensitivity and strain-rate hardening.It is shown that at sufficiently high impact velocities, the flows stress in the notch vicinity becomes quite higher than in the rest of the specimen, so that while the former resists deformation, it transfers the load to the latter. The result will be the formation of a local neck and failure away from the notch.This effect is shown to be active when the material properties are perturbed only at the local level, as in the case of machining of the notch, which in itself may again be sufficient to stabilize the structure under local failure until a neck forms elsewhere.While the physical observations are quite counterintuitive with respect to the engineering views of stress concentrator's effect, the present work rationalizes those observations and also provides information for the designers of dynamically tensioned structures that may contain notches or similar flaws.