A review on coupled modes in V-notched plates of finite thickness: A generalized approach to the problem

Mode 1, 2 and 3 cannot exist in isolation. One mode provokes the existence of a coupled mode which, in some conditions, can be more dangerous than the generating mode itself. This means that three-dimensional problems are automatically at least dual scale. While for a crack this effect was known to exist for a long period of time, it was largely ignored in theoretical studies of V-shaped notches subjected to in-plane and out-of-plane loading as well as in practical fracture problems associated with such geometries. Only recently, some numerical investigations confirmed that highly localized coupled modes do exist in the close vicinity of the notch tip. The present paper is aimed to briefly review important features of these recently identified singular coupled modes. The most significant results from a comprehensive three-dimensional numerical study are presented here to describe the contribution of these modes into the overall stress state in the close vicinity of the notch tip and discuss the implementation of these new results to the failure and integrity assessment of plate structures with sharp notches.     

A successful combination of the equivalent material concept and the averaged strain energy density criterion for predicting crack initiation from blunt V-notches in ductile aluminum plates under mixed mode loading

Crack initiation from blunt V-notch borders in ductile A16061-T6 plates is investigated experimentally and theoretically under mixed mode I/II loading. Experimental observations with naked eye during loading indicated large plastic deformations around the notch tip at the onset of crack initiation, demonstrating large-scale yielding failure regime for the aluminum plates. To theoretically predict the experimentally obtained value of the maximum load that each plate could sustain, i.e. the load-carrying capacity, without performing elastic-plastic failure analyses, the equivalent material concept (EMC) is combined with a well-known brittle fracture criterion, namely the averaged strain energy density (ASED) criterion. It is shown that the combined EMC-ASED criterion could successfully predict the experimental results for various V-notch angles and radii.     

Mixed mode I/II brittle fracture in V-notched Brazilian disk specimens under negative mode I conditions

The well-known round-tip V-notched Brazilian disk specimen is utilized for conducting mixed mode I/II fracture tests on PMMA under negative mode I conditions for different notch angles and various notch radii with the aim to measure experimentally the fracture load and the fracture initiation angle. It is shown by the finite element analysis that although the notch is under negative mode I loading, one side of the notch border still experiences tensile tangential stresses suggesting that fracture would take place from the same side of notch border. Experimental observations also indicated that fracture occurs from the tensile side of the notch border confirming the finite element results. The experimental results are then theoretically estimated by means of two stress-based brittle fracture criteria, namely the round-tip V-notch maximum tangential stress and the mean-stress criteria. It is shown that both criteria provide very good predictions to the experimental results obtained under negative mode I conditions.     

Fatigue strength of Al7075 notched plates based on the local SED averaged over a control volume

When pointed V-notches weaken structural components,local stresses are singular and their intensities are expressed in terms of the notch stress intensity factors(NSIFs).These parameters have been widely used for fatigue assessments of welded structures under high cycle fatigue and sharp notches in plates made of brittle materials subjected to static loading.Fine meshes are required to capture the asymptotic stress distributions ahead of the notch tip and evaluate the relevant NSIFs.On the other hand,when the aim is to determine the local Strain Energy Density(SED)averaged in a control volume embracing the point of stress singularity,refined meshes are,not at all,necessary.The SED can be evaluated from nodal displacements and regular coarse meshes provide accurate values for the averaged local SED.In the present contribution,the link between the SED and the NSIFs is discussed by considering some typical welded joints and sharp V-notches.The procedure based on the SED has been also proofed to be useful for determining theoretical stress concentration factors of blunt notches and holes.In the second part of this work an application of the strain energy density to the fatigue assessment of Al7075 notched plates is presented.The experimental data are taken from the recent literature and refer to notched specimens subjected to different shot peening treatments aimed to increase the notch fatigue strength with respect to the parent material.     

Elastic-plastic fracture analysis of notched Al 7075-T6 plates by means of the local energy combined with the equivalent material concept

The main goal of the present research is to analyze tensile fracture in Al 7075-T6 thin plates weakened by blunt V-notches. For this purpose, first, 27 fracture tests are carried out on rectangular plates containing a central rhombic hole with two blunt V-shaped corners horizontally located. The experimental observations indicated that a plastic region initiates from the notch tip and grows as the tensile load monotonically increases, and finally, fracture happens suddenly with a significant opening of the notch tip. By showing significant plastic deformations around the notch tip and also inclined fracture planes, the specimens after fracture confirm well the ductile rupture in V-notched Al 7075-T6 plates. As the main experimental result, the load-carrying capacity of the notched plates corresponding to the onset of crack initiation from the notch tip is recorded. To theoretically predict the experimental results, the equivalent material concept is utilized together with the well-known brittle fracture criterion, namely the averaged strain energy density criterion. Without requiring elastic-plastic finite element analysis, it is shown that the combination of the averaged strain energy density and equivalent material concept is successful in predicting the load-carrying capacity of the V-notched Al 7075-T6 plates that fail by moderate-scale yielding regime.     

Crack Initiation at V-Notch Tip under In-Plane Mixed Mode Loading: A Review of the Fictitious Notch Rounding Concept

The fictitious notch rounding concept has been recently applied for the first time to V-notches with root hole subjected to in-plane mixed mode loading. Out-of-bisector crack propagation is taken into account. The fictitious notch radius is determined as a function of the real notch radius (the microstructural support length) and the notch opening angle. Due to the complexity of the problem, a method based on the simple normal stress failure criterion has been used. It is combined with the maximum tangential stress criterion to determine the crack propagation angle. An analytical method based on Neuber’s procedure has been developed. The method provides the values of the microstructural support factor as a function of the mode ratio and the notch opening angle. The support factor is considered to be independent of the microstructural support length. Finally, for comparison, the support factor is determined on a purely numerical basis by iterative analysis of finite element models. The present paper is aimed to give a brief overview of the recent findings on this challenging topic making clear the state of the art.     

Emission of stress waves during fracture

Emission of both longitudinal and surface (Rayleigh) waves during fracture of plates under conditions of plane stress and plane strain were studied experimentally. The non-equilibrated tensile stress in the fractured section of the plate creates an elastic wave, which travels radially along the plate at the sound speed. Moreover, the high surface deformation around the crack tip, due to the high stress concentration there, propagates as a surface wave following fracture of this zone, at the respective Rayleigh wave speed with a circular wavefront. The influence of the thickness of the plate and the type of fracture (brittle or ductile) was examined and interesting results were derived, by utilizing a high speed photography technique.     

Three-dimensional vibrations of cylindrical elastic solids with V-notches and sharp radial cracks

This paper provides free vibration data for cylindrical elastic solids, specifically thick circular plates and cylinders with V-notches and sharp radial cracks, for which no extensive previously published database is known to exist. Bending moment and shear force singularities are known to exist at the sharp reentrant corner of a thick V-notched plate under transverse vibratory motion, and three-dimensional (3-D) normal and transverse shear stresses are known to exist at the sharp reentrant terminus edge of a V-notched cylindrical elastic solid under 3-D free vibration. A theoretical analysis is done in this work utilizing a variational Ritz procedure including these essential singularity effects. The procedure incorporates a complete set of admissible algebraic-trigonometric polynomials in conjunction with an admissible set of “edge functions” that explicitly model the 3-D stress singularities which exist along a reentrant terminus edge (i.e., α>180°) of the V-notch. The first set of polynomials guarantees convergence to exact frequencies, as sufficient terms are retained. The second set of edge functions—in addition to representing the corner stress singularities—substantially accelerates the convergence of frequency solutions. This is demonstrated through extensive convergence studies that have been carried out by the investigators. Numerical analysis has been carried out and the results have been given for cylindrical elastic solids with various V-notch angles and depths. The relative depth of the V-notch is defined as (1−c/a), and the notch angle is defined as (360°−α). For a very small notch angle (1° or less), the notch may be regarded as a “sharp radial crack.” Accurate (four significant figure) frequencies are presented for a wide spectrum of notch angles (360°−α), depths (1−c/a), and thickness ratios (a/h for plates and h/a for cylinders). An extended database of frequencies for completely free thick sectorial, semi-circular, and segmented plates and cylinders are also reported herein as interesting special cases. A generalization of the elasticity-based Ritz analysis and findings applicable here is an arbitrarily shaped V-notched cylindrical solid, being a surface traced out by a family of generatrix, which pass through the circumference of an arbitrarily shaped V-notched directrix curve, r(θ), several of which are described for future investigations and close extensions of this work.     

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.     

Mode II brittle fracture assessment using an energy based criterion

In this paper the minimum strain energy density criterion is modified to predict the values of mode II fracture toughness reported in the literature for several brittle and quasi-brittle materials. The experimental results are all related to mode II fracture tests performed on the semicircular bend specimen. The modified mode II fracture criterion takes into account the effect of T-stress (in addition to the singular terms of stresses/strains) when calculating the strain energy density factor at a very small critical distance from the crack tip. It is shown that the proposed criterion provides significantly better predictions for mode II fracture toughness compared with the classical minimum strain energy density criterion.     

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.     

Elastic analysis of a mode II crack in a decagonal quasi—crystal

The elastic analysis of a mode II Griffith crack penetrating through a decagonal quasi-crystal along the periodic axis is made within the context of the continuum theory. By using a general solution obtained previously, the problem in the case of uniform shear stress at infinity is solved, and the analytical expressions for the entire stress field disturbed by an internal crack are derived in an explicit form. The asymptotic fields of the displacement and stress around a crack tip in both phonon and phason fields indicate that the stresses near a crack tip exhibit the square-root singularity. The formula for evaluating the energy release rate is also given. If imposing that the phason field is absent, the well-known results of a mode II crack in a conventional material are recovered from the present results.     

Atomic scale investigation of grain boundary structure role on intergranular deformation in aluminium

The role that grain boundary (GB) structure plays on the directional asymmetry of an intergranular crack (i.e. cleavage behaviour is favoured along one direction, while ductile behaviour along the other direction of the interface) was investigated using atomistic simulations for aluminium 〈1 1 0〉 symmetric tilt GBs. Middle-tension (M(T)) and Mode-I crack propagation specimens were used to evaluate the predictive capability of the Rice criterion. The stress–strain response of the GBs for the M(T) specimens highlighted the importance of the GB structure. The observed crack tip behaviour for certain GBs (Σ9 (2 2 1), Σ11 (3 3 2) and Σ33 (4 4 1)) with the M(T) specimen displayed an absence of directional asymmetry which is in disagreement with the Rice criterion. Moreover, in these GBs with the M(T) specimen, the dislocation emission from a GB source at a finite distance ahead of the crack tip was observed rather than from the crack tip, as suggested by the Rice criterion. In an attempt to understand discrepancy between the theoretical predictions and atomistic observations, the effect of boundary conditions (M(T), Mode-I and the edge crack) on the crack tip events was examined and it was concluded that the incipient plastic events observed were strongly influenced by the boundary conditions (i.e. activation of dislocation sources along the GB, in contrast to dislocation nucleation directly from the crack tip). In summary, these findings provide new insights into crack growth behaviour along GB interfaces and provide a physical basis for examining the role of the GB character on incipient event ahead of a crack tip and interface properties, as an input to higher scale models.     

Effect of stress state on deformation and fracture of nanocrystalline copper:Molecular dynamics simulation

Deformation in a microcomponent is often constrained by surrounding joined material making the component under mixed loading and multiple stress states. In this study, molecular dynamics(MD) simulation are conducted to probe the effect of stress states on the deformation and fracture of nanocrystalline Cu. Tensile strain is applied on a Cu single crystal,bicrystal and polycrystal respectively, under two different tension boundary conditions. Simulations are first conducted on the bicrystal and polycrystal models without lattice imperfection. The results reveal that, compared with the performance of simulation models under free boundary condition, the transverse stress caused by the constrained boundary condition leads to a much higher tensile stress and can severely limit the plastic deformation, which in return promotes cleavage fracture in the model. Simulations are then performed on Cu single crystal and polycrystal with an initial crack. Under constrained boundary condition, the crack tip propagates rapidly in the single crystal in a cleavage manner while the crack becomes blunting and extends along the grain boundaries in the polycrystal. Under free boundary condition, massive dislocation activities dominate the deformation mechanisms and the crack plays a little role in both single crystals and polycrystals.     

Fracture Assessment of Notched Bainitic Functionally Graded Steels under Mixed Mode (I + II) Loading

The averaged value of the strain-energy density over a well-defined volume is one of the powerful criteria to assess the static strength of U- and V-notched specimens. This contribution is the first to investigate the effect of notch parameters (notch radius, notch depth and notch opening angle) for fracture assessment of specimens weakened by blunt V-notches made of bainitic functionally graded steels under mixed mode loading (I + II). A numerical method has been used to evaluate the boundary of the control volume, the mean value of the strain-energy density and the critical fracture load. Different values of the notch radius (0.5, 1.0, 1.5 and 2.0 mm), notch depth (5.5, 6.0 and 6.5 mm), notch opening angle (30°, 60° and 90°) and distance of the applied load from the notch bisector line (5 and 10 mm) have been considered. Moreover, this contribution shows that the mean value of the strain-energy density over the control volume can also be accurately determined from a coarse mesh for functionally graded steels.     

正交各向异性材料切口尖端热弹奇性特征分析

研究正交各向异性平面V形切口,计算其热弹奇性特征.通过引入切口尖端物理场的渐近级数展开式,将应力和热流平衡方程转化为关于奇性指数的特征常微分方程组,采用插值矩阵法求解,获取切口尖端的热流、应力奇性指数和对应的特征角函数.算例表明,该法精度高适应性强.     

Cohesive solutions of intersonic moving dislocations

A class of cohesive solutions of moving glide dislocations with intersonic speeds has been derived on the basis of the fundamental equation of a moving dislocation introduced by Weertman in conjunction with a proposed generalized Bilby–Cottrell–Swinden–Dugdale model. In this model we assume a straight weak path within an infinite elastic plate. Two length scales, namely the width (thickness) of the weak path and the material intrinsic length, which scales strain-gradient-induced hardening and energy dissipation, are taken into account by applying the traction–separation law for the decohesion of the weak path. Dislocations propagate along this weak path with a speed higher than the shear wave speed. The accumulation of these moving dislocations forms a macroscale crack growth with a cohesive zone ahead of the crack tip. Similar to the Bilby–Cottrell–Swinden–Dugdale model, the remote enforced stress and/or stress-rate boundary conditions are represented as an equivalent crack surface traction associated with the dislocation distribution. The involved Cauchy integral and corresponding eigenvalue problem are solved using the algorithms introduced by Muskhelishvili and by Weertman. The problems associated with three types of decohesion law are constant traction, traction linearly dependent on separation, and separation- and separation-rate-dependent traction. These problems are solved using three different solution strategies: the direct-integration method, the iteration method and the Jacobi polynomial expansion respectively. The derived solutions provide explicit relations between the remote load propagation speed, the material intrinsic length, the weak path thickness and the strain-rate-hardening parameter. The solutions demonstrate that the intersonic speed region can be divided into two subdomains; steady-state propagation occurs within the subdomain where the propagation speeds are equal to or greater than the Eshelby speed (c _s × 2^1/2, where c _s is the shear wave speed). For a weak path with a finite width and the corresponding decohesion law scaled by material intrinsic length, an intersonic crack propagation will not take place if only a constant remote stress is imposed. A ‘steady-state’ crack surface load and/or remote stress-rate boundary condition, which can be considered as a point force or a distributed force with a constant distance to the moving crack tip, is required to maintain steady-state intersonic crack propagation.     

相干梯度敏感干涉测量技术及在静态断裂力学实验中的应用

介绍了相干梯度敏感(CGS)干涉测量技术的基本原理及其在静态Ⅰ型断裂实验中的应用,验证了该方法对裂纹尖端局部变形场和断裂特性进行定量研究的可行性。给出了代表静态Ⅰ型裂纹尖端奇异场光力学信息的CGS控制方程,模拟并分析了Ⅰ型裂纹尖端的CGS条纹模式,对静态Ⅰ型裂纹尖端变形场和断裂特性进行了三点弯曲的CGS试验,并提取了应力强度因子KⅠ。结果表明,试验结果与理论分析结果相吻合。     

复合材料反平面切口端部奇性研究

基于复合材料切口尖端位移场的渐近展开,将切口的反平面平衡控制方程转化为关于切口奇性指数的特征微分方程,采用一种变换将其化为线性特征微分方程组,引入插值矩阵法计算相应边界条件下方程组的特征值以获取切口尖端的应力奇性指数.研究单相材料切口、双相材料切口及止于异质界面切口的奇异性,算例表明该方法可以一次性计算出多阶奇性指数.对所取得的非奇异指数尽管切口不表现出奇性状态,但却是描述切口尖端完整应力场必不可少的参量.     

Fracture assessment of polymethyl methacrylate using sharp notched disc bend specimens under mixed mode I + III loading

Mixed mode I/III behavior of Perspex (polymethyl methacrylate (PMMA)) is studied experimentally and theoretically in this research using a new and simple laboratory test configuration. The specimen is a circular disc containing a sharp V-notch along the diameter that is loaded by the conventional three-point bend fixture. The critical values of notch stress intensity factors (K _I ^V and K _III ^V ) were obtained for the whole combinations of modes I and III simply by changing the notch inclination angle relative to the loading rollers. The value of notch fracture toughness under pure or dominantly tension loads was greater than its corresponding value under mode III or dominantly torsion loads. The experimental results were also predicted very well by employing the local strain energy density (SED) criterion.