Modification of the fracture criterion for V-shaped notches (plane problem). Relationship between toughness and strength and structural parameters

A fracture criterion of the type of the Neuber-Novozhilov criterion is proposed to describe the fracture in the vicinity of the tip of a V-shaped notch under tensile and shear loading. In the proposed criterion, the limits of averaging of the stresses along the notch axis depend on the presence, location, and size of the initial defects in the material. The crystal lattice parameter of the initial material is chosen for the characteristic linear size. For a V-shaped notch subjected to tension and shear, simple equations are obtained that relate the stress intensity factors for the modified singularity coefficients, the singularity coefficients themselves, and the theoretical tensile and shear strengths of a single crystal of the material taking into account the damage to the material in the vicinity of the notch tip. The equations obtained allow a passage to the limit from a notch to a crack. It is shown that the classical critical stress intensity factor used in the strength analysis of cracked solids is not a material constant.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 1, pp. 106–115, January–February, 2005.     

Determination of stress intensity factors in three-dimensional problems of fracture mechanics

Dependences of displacements of the surface of a notch on the corresponding stress intensity factors were obtained for axisymmetric bodies with internal and external notches under different deformations (tensile, shear, bending, and torsion). An algorithm is proposed to determine the stress intensity factors of three types (opening mode, longitudinal shear, and transverse shear) from displacements of the notch surface near its tip. The effectiveness of the algorithm is shown, as an example, for numerical analysis of various three-dimensional problems of fracture mechanics.     

Asymptotic analysis of transient curved crack in functionally graded materials

Transient mixed-mode elastodynamic crack growth along arbitrary smoothly varying paths in functionally graded materials (FGMs) is considered. The property gradation in FGMs is considered by varying shear modulus and mass density exponentially along the gradation direction. Crack tip out of plane displacement fields and their gradients are developed for propagating curved cracks of arbitrary velocity using asymptotic approach. The mode-mixity due to the inclination of curved crack with respect to property gradient is accommodated in the analysis through superposition of the opening and shear modes. The expansion of the displacement fields and their gradients around the crack-tip are derived in powers of radial coordinates with the coefficients of expansion depending on the instantaneous value of the local curvature of the crack path, time derivatives of crack-tip speed, and time derivative of mode-I and mode-II stress intensity factors. The effect of the transient terms instantaneous local curvature, crack-tip speed, time derivatives of crack-tip speed, and time derivative of mode-I and mode-II stress intensity factors on the contours of constant out of plane displacement are also discussed.     

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.     

Cracks’ closure in 3-D fracture dynamics: the effect of relative location of two coplanar cracks

In this paper, the problem of two equal coplanar cracks with allowance for the crack faces contact interaction was investigated. The problem of the cracks located in homogeneous, isotropic, and linearly elastic solid subjected to normally incident tension–compression wave is solved by the boundary integral equations method. The influence of the distance between two cracks on the stress intensity factors (opening mode and transverse shear mode) is studied for a range of wave numbers. The results are compared with those obtained neglecting cracks’ closure.     

Asymptotic elastic solution for two straight cracks of arbitrary length and location

An infinite elastic plane containing two straight cracks of arbitrary length and location is analyzed within the framework of elastostatics. The mathematical formulation is based on the stress solution for a single crack and leads to a system of singular integral equations that govern the crack surface displacement densities. The solution series in terms of the reciprocal of the crack centre distance is not suitable for cracks that are spaced too closely. It is shown by way of examples that the method of asymptotic solution is convenient for developing approximation expressions of the stress and displacement field with certain characteristics. The formulas for the stress intensity factors and crack opening are given for the case of a constant tensile load. Graphical results are given for the variations of the stress intensity factors with parameters depending on the relative positions of the cracks.     

International conference on role of fracture mechanics in modern technology : Fukuoka, Japan, June 2–6, 1986

An infinite elastic plane containing two straight cracks of arbitrary length and location is analyzed within the framework of elastostatics. The mathematical formulation is based on the stress solution for a single crack and leads to a system of singular integral equations that govern the crack surface displacement densities. The solution series in terms of the reciprocal of the crack centre distance is not suitable for cracks that are spaced too closely. It is shown by way of examples that the method of asymptotic solution is convenient for developing approximation expressions of the stress and displacement field with certain characteristics. The formulas for the stress intensity factors and crack opening are given for the case of a constant tensile load. Graphical results are given for the variations of the stress intensity factors with parameters depending on the relative positions of the cracks.     

垂直于界面V型缺口尖端的应力奇异性及其应用

基于双材料垂直于界面V型缺口理论,给出了单一材料和双材料裂纹问题、V型缺口问题应力强度因子的统一定义,得到了应力外推法计算双材料K_I的公式,数值算例验证了本文方法的有效性.以双材料单向拉伸和三点弯曲模型为对象,深入研究了双材料中弹性模量、泊松比、缺口深度、缺口张角对缺口尖端奇异应力场的影响,获得了一定范围内各种参数变化对缺口尖端奇异应力场的影响规律,为异体材料形成的V型缺口在应力断料中的应用提供了必要的参考依据.     

复合材料承受剪切时双裂纹周围的应力强度因子

本文研究了由各向同性和各向异性半无限接合而成的复合材料中的应力强度因子问题，在复合材料的接合面附近处具有与接合面平行且共线的两个Ｇｒｉｆｆｉｔｈ裂纹，裂纹面上作用有剪应力，本文利用付利叶变换将混合边值问题归毕为求解奇异积分方程问题，为求解这些方程，将裂纹面上，下的位移差展成级数，并满足理解纹面外侧边界条件，级数中的待定系数利用裂纹面内的边界条件和施密特方法求得，本文对硼纤维塑料和铝板接合的复合材料     

On quadruple integral equations related to a certain crack problem

An exact solution of a four part mixed boundary value problem representing a three colinear crack system connected with specified crack opening displacements between the cracks is obtained. The three cracks thus become one with pressure and/or opening displacement prescribed on the crack face. From considerations of dual symmetry and a formulation based on Papkovich-Neuber harmonic functions, the boundary value problem is reduced to solving a quadruple set of integral equations. An exact solution of these equations is derived using a modified finite Hilbert transform technique. The closed form results for the stress distributions and the crack-tip stress intensity factors are presented. Limiting cases of the solution yield results which agree with well known solutions.     

Mixed-mode stress intensity factors and critical angles of cracks in bolted joints by weight function method

Summary   Mechanical joints, such as bolted or riveted joints, are widely used in structural components. Reliable determination of stress intensity factors for cracks in bolted joints is required to evaluate their safety and fatigue life. The weight function method is an efficient technique to calculate stress intensity factors for various loading conditions by the stress analysis of an uncracked model. In this paper, the mixed-mode stress intensity factors for cracks in bolted joints are analyzed by the weight function method, and coefficients included in the weight function are determined by finite element analysis for reference loadings. The critical angle at which mode I stress intensity factor becomes maximum is determined, and the effects of the amount of clearance and crack length on the critical angle are investigated. Received 28 February 2001; accepted for publication 22 June 2001 RID=" ID=" The authors are grateful for the support provided by a grant from the Korea Science & Engineering Foundation (KOSEF) and Safety and Structural Integrity Research Center at the Sungkyunkwan University.     

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.     

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。     

THE BEHAVIOR OF TWO PARALLEL SYMMETRIC PERMEABLE CRACKS IN PIEZOELECTRIC MATERIALS

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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.     

The stress intensity factors near the tips of two collinear cracks in composite under in-plane shear

In this paper, the stress-intensity factors for two collinear cracks in a composite bonded by an isotropic and an anisotropic half-plane were calculated. The cracks are paralell to the interface, and the crack surfaces are loaded by uniform shear stresses. By using Fourier transform, the mixed boundary value problem is reduced to a set of singular integral equations. For solving the integral equations, the crack surface displacements are expanded in triangular series and the unknown coefficients in the series are determined by the Schmidt method. The stress intensity factors for the cracks in the boron-fibre plastics and aluminium joined composite and in carbon-fibre reinforced plastics were calculated numerically.     

Two parallel penny-shaped cracks under the action of dynamic impacts

Summary The three-dimensional elastodynamic response of two parallel penny-shaped cracks embedded in an infinite elastic solid under the action of impact loading is investigated. A time-domain boundary integral equation method is used for calculating the time-dependent crack opening displacements and subsequently the dynamic stress intensity factors. Numerical computations are carried out for various geometry parameters. The results are presented in graphical form and discussed. The effect of the locations of the cracks on the dynamic stress intensity factors is presented. Received 8 May 1996; accepted for publication 25 September 1996     

On the higher order asymptotic analysis of a non-uniformly propagating dynamic crack along an arbitrary path

Transient mixed-mode elastodynamic crack growth along arbitrary smoothly varying paths is considered. Asymptotically, the crack tip stress field is square root singular with the angular variation of the singular term depending weakly on the instantaneous values of the crack tip speed and on the mode-I and mode-II stress intensity factors. However, for a material particle at a small distance away from the moving crack tip, the local stress field will depend not only on the instantaneous values of the crack tip speed and stress intensity factors, but also on the past history of these time dependent quantities. In addition, for cracks propagating along curved paths the stress field is also expected to depend on the nature of the curved crack path. Here, a representation of the crack tip fields in the form of an expansion about the crack tip is obtained in powers of radial distance from the tip. The higher order coefficients of this expansion are found to depend on the time derivative of crack tip speed, the time derivatives of the two stress intensity factors as well as on the instantaneous value of the local curvature of the crack path. It is also demonstrated that even if cracks follow a curved path dictated by the criterion K ₁₁ ^d =0, the stress field may still retain higher order asymmetric components related to non-zero local curvature of the crack path.     

Analysis of stress intensity factor in orthotropic bi-material mixed interface crack

Adopting the complex function approach, the paper studies the stress intensity factor in orthotropic bi-material interface cracks under mixed loads. With consideration of the boundary conditions, a new stress function is introduced to transform the problem of bi-material interface crack into a boundary value problem of partial differential equations. Two sets of non-homogeneous linear equations with 16 unknowns are constructed. By solving the equations, the expressions for the real bi-material elastic constant εt and the real stress singularity exponents λt are obtained with the bi-material engineering parameters satisfying certain conditions. By the uniqueness theorem of limit,undetermined coefficients are determined, and thus the bi-material stress intensity factor in mixed cracks is obtained. The bi-material stress intensity factor characterizes features of mixed cracks. When orthotropic bi-materials are of the same material, the degenerate solution to the stress intensity factor in mixed bi-material interface cracks is in complete agreement with the present classic conclusion. The relationship between the bi-material stress intensity factor and the ratio of bi-material shear modulus and the relationship between the bi-material stress intensity factor and the ratio of bi-material Young's modulus are given in the numerical analysis.     

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.