Finite-element analysis of fatigue crack closure under plane strain conditions: stabilization behaviour and mesh size effect

An elastic–plastic finite‐element analysis of fatigue crack closure is performed for plane strain conditions. The stabilization behaviour of crack opening level and the effect of mesh size on the crack opening stress are investigated. It has been well reported that the crack opening level under plane stress conditions becomes stable after the crack advances beyond the initial monotonic plastic zone. In order to obtain a stabilized crack opening level for plane strain conditions, the crack must be advanced through approximately four times the initial monotonic plastic zone. The crack opening load tends to increase with the decrease of mesh size. The mesh size nearly equal to the theoretical plane strain cyclic plastic zone size may provide reasonable numerical results comparable with experimental crack opening data. The crack opening behaviour is influenced by the crack growth increment and discontinuous opening behaviour is observed.     

An interface crack in a coating-substrate composite with mixed-mode Dugdale corrections

Considering both plane stress and plane strain conditions, the plastic zone size and the crack tip opening displacement of an interface crack between a coating and a semi-infinite substrate under a normal load on the crack surfaces are investigated by the mixed-mode Dugdale model. In the model, stresses applied in the plastic zones satisfy the Von Mises yield criterion. The plastic zone size can be calculated by satisfying the condition that the complex stress intensity factors vanish. After the plastic zone size is solved, the crack tip opening displacement can be obtained by dislocation theories. In numerical examples, a uniform load is considered, and the effects of the normalized elastic modulus (the ratio of the elastic modulus of the coating to the elastic modulus of the substrate) and the normalized crack depth (the ratio of the coating thickness to the interface crack length) on the normalized plastic zone size and the normalized crack tip opening displacement are examined. Numerical examples show in the case of thin coatings, the value of the normalized plastic zone size decreases with increasing the normalized elastic modulus.     

COHESIVE MODELLING OF PROCESS REGIONS FOR CRACKS IN LINEAR ELASTIC STRUCTURES—FUNDAMENTAL ASPECTS

The process region at the tip of a crack in a linear elastic structure has been modelled by a cohesive zone. Growth of the front end of the cohesive zone is governed by a critical stress intensity factor criterion, and advance of the original traction free crack is determined by a critical crack opening at the rear end of the cohesive zone. Damage resistance curves relating the applied stress intensity factor to the growth of the cohesive zone have been calculated for an idealized structure containing two characteristic dimensions. Instability resulting in failure of the structure is found to occur either by unstable growth of the front end of the cohesive zone, without a fully developed cohesive zone, or by unstable growth of the original flaw, when the crack opening displacement at the rear end of the cohesive zone reaches a critical value. The influence of the size of the structure compared to the length of the cohesive zone is investigated, and conditions for the limits of validity of the small scale yielding assumption are discussed. Comparisons are made between the maximum load and the length of the cohesive zone at instability resulting from the present analysis, and the values predicted by linear elastic fracture mechanics.     

PROPAGATION AND CLOSURE OF SHORT FATIGUE CRACKS AT NOTCHES UNDER COMPRESSIVE MEAN STRESSES

Abstract— Single-edge-notched specimens of a low-carbon steel were fatigued under cyclic in-plane bending with compressive mean stresses. The development of crack closure with crack growth was studied both experimentally and theoretically. The relation between the crack opening stress and the crack length was a function of the minimum (compressive) applied stress, irrespective of the maximum stress. The effective stress intensity range was a unique parameter in correlating the crack growth rate, even if the crack was embedded in the compressive plastic zone. Under a constant minimum stress, the length of nonpropagating cracks became longer with increasing maximum applied stress. A theoretical model was proposed for predicting the crack opening stress on the basis of the compressive stress distribution at the minimum applied stress. The predicted value agreed fairly well with the experimental result. The model gave upper bounds of the crack growth rate and the length of nonpropagating fatigue cracks within the plastic zone.     

On elastic–plastic fracture behavior of a bi-layered composite plate with a sub-interface crack under mixed mode loading

A generalized Irwin model is proposed to investigate elastic–plastic fracture behavior of a bi-layered composite plate with a sub-interface crack under combined tension and shear loading. The dependence of the stress intensity factors, the plastic zone size, the effective stress intensity factor and the crack tip opening displacement on the crack depth h, the Dundurs’ parameters and the phase angle θ is discussed in detail. Numerical results show that in most cases, if the crack is embedded in a stiffer material, when the crack is close to the interface, the plastic zone size and the crack tip opening displacement will increase. On the contrary, if the crack is embedded in a softer material, when the crack is close to the interface, the plastic zone size and the crack tip opening displacement will decrease.     

Lebensdauerberechnung festgewalzter Kerbproben unter Axialbelastung auf der Basis eines Rißfortschrittskonzepts

Axial fatigue life calculation of fillet rolled specimens by means of a crack growth model Fillet rolling is a method which significantly improves the fatigue strength of members. Residual compressive stresses induced in the surface layer during the fillet rolling process are able to retard or prevent crack propagation. An elastic‐plastic on the J‐integral based crack growth model considering the crack opening and closure phenomenon in nonhomogeneous plastic stress fields is described. Experimentally determined crack growth curves and fracture fatigue life curves at constant amplitude loading were used to verify the developed model.     

Non-orthogonal stress modes for interfacial fracture based on local plastic dissipation

Interfacial cracks have several features which are different from those of cracks in homogeneous materials. Among those, the loading mode dependency of interfacial toughness has been a main obstacle to the widespread utilization of interfacial fracture mechanics. In this study, plasticity-induced toughening of an interface crack between an elastic–plastic material and an elastic material is studied. A useful relationship between the plastic dissipation and the plastic zone size is derived via an effective crack length model. Non-orthogonal stress modes for interface cracks are proposed on the basis of the plastic dissipation mechanism and a mixed-mode criterion for interfacial crack growth is also proposed using these stress modes. The non-orthogonal stress modes are able to represent the asymmetric behavior, mode-dependent toughening and ε-dependency of interfacial crack growth.     

Effects of non-singular stresses on crack-tip fields for pressure-sensitive materials,Part 1: Plane strain case

Mode I near-tip stress fields for elastic perfectly plastic pressure-sensitive materials under plane strain and small-scale yielding conditions are presented. A Coulomb-type yield criterion described by a linear combination of the effective stress and the hydrostatic stress is adopted in the analysis. The finite element computational results sampled at the distance of a few crack opening displacements from the tip show that, as the pressure sensitivity increases, the magnitudes of the normalized radial and hoop stress ahead of the tip decrease, the total angular span of the singular plastic sectors decreases, and the angular span of the elastic sectors bordering the crack surfaces increases. When non-singular T stresses are considered along the boundary layer of the small-scale yielding model, the near-tip stresses decrease as the T stress decreases. The plastic zone shifts toward the crack surfaces as the T stress increases. When the discontinuities of the radial stress and the out-of-plane normal stress along the border between the plastic sector and the elastic sector are allowed, the angular variations of the asymptotic crack-tip fields agree well with those of the finite element computations. Variation of the Q stresses for pressure-sensitive materials can be found from the asymptotic solutions when the plastic zone size ahead of the tip is relatively larger than the crack opening displacement. In addition the T stress is shown to have strong effects on the plastic zone sizes and shapes which could affect the toughening of pressure-sensitive materials.     

Mechanisms and modeling of low cycle fatigue crack propagation in a pressure vessel steel Q345

The fatigue process near crack is governed by highly concentrated strain and stress in the crack tip region. Based on the theory of elastic–plastic fracture mechanics, we explore the cyclic J-integral as breakthrough point, an analytical model is presented in this paper to determine the CTOD for cracked component subjected to cyclic axial in-plane loading. A simple fracture mechanism based model for fatigue crack growth assumes a linear correlation between the cyclic crack tip opening displacement (ΔCTOD) and the crack growth rate (da/dN). In order to validate the model and to calibrate the model parameters, the low cycle fatigue crack propagation experiment was carried out for CT specimen made of Q345 steel. The effects of stress ratio and crack closure on fatigue crack growth were investigated by elastic–plastic finite element stress–strain analysis of a cracked component. A good comparison has been found between predictions and experimental results, which shows that the crack opening displacement is able to characterize the crack tip state at large scale yielding constant amplitude fatigue crack growth.     

沿直线有多条裂纹的薄板弯曲问题

本文首先用复变函数的虚部构成了一平面上沿y轴有矩形开口的位移函数,并用加权积分法将裂纹尖端交换为光滑接触的形状：然后由此函数构成的重调和函数,导出了沿不同弹性介质界面(y轴)有多条裂纹的薄板弯曲问题的应力函数。用加权积分法将裂纹尖端无穷大的应力集中有限化,并不意味着消除了复变函数中的奇异点。文中图示了将应力奇点移至其他分支,而在XY全平面应力呈有限连续的情形。本分析方法的成功之处在于在裂纹尖端附近构成了一开口位移和截面内力相并存的过渡区,同时消除了过去研究中呈现的界面裂纹尖端附近无穷大应力的剧烈振荡现象。     

Cohesive Fracture Model Based on Necking

A linear hardening model together with a linear elastic background material is first used to discuss some aspects of the mathematical and physical limitations and constraints on cohesive laws. Using an integral equation approach together with the cohesive crack assumption, it is found that in order to remove the stress singularity at the tip of the cohesive zone, the cohesive law must have a nonzero traction at the initial zero opening displacement. A cohesive zone model for ductile metals is then derived based on necking in thin cracked sheets. With this model, the cohesive behavior including peak cohesive traction, cohesive energy density and shape of the cohesive traction–separation curve is discussed. The peak cohesive traction is found to vary from 1.15 times the yield stress for perfectly plastic materials to about 2.5 times the yield stress for modest hardening materials (power hardening exponent of 0.2). The cohesive energy density depends on the critical relative plate thickness reduction at the root of the neck at crack initiation, which needs to be determined by experiments. Finally, an elastic background medium with a center crack is employed to re-examine the shape effect of cohesive traction–separation curve, and the relation between the linear elastic fracture mechanics (LEFM) and cohesive zone models by considering the cohesive zone development and crack growth in the infinite elastic medium. It is shown that the shape of the cohesive curve does affect the cohesive zone size and the apparent energy release rate of LEFM for the crack growth in the elastic background material. The apparent energy release rate of LEFM approaches the cohesive energy density when the crack extends significantly longer than the characteristic length of the cohesive zone.     

基于小波内聚力模型的界面裂纹扩展

利用区间B样条小波良好的局部化性能,将内聚力模型（CZM）引入小波有限元法（WFEM）数值分析中,以区间B样条小波尺度函数作为插值函数,构造小波内聚力界面单元,推导了小波内聚力界面单元刚度矩阵,基于虚拟裂纹闭合技术（VCCT）计算界面裂纹应变能释放率（SERR）,采用β-Κ断裂准则,实现界面裂纹扩展准静态分析。将WFEM和传统有限元法（CFEM） 的SERR数值分析结果与理论解进行比较,结果表明:采用WFEM和CFEM计算的SERR分别为96.60 J/m2 和 101.43 J/m2,2种方法的SERR数值解与理论解相对误差分别为1.85%和3.06%,这明确表明WFEM在计算界面裂纹扩展方面能用较少单元和节点数获得较高的计算精度和效率。在此基础上,探讨了界面裂纹初始长度和双材料弹性模量比对界面裂纹扩展的影响,分析结果表明:界面裂纹尖端等效应力随界面裂纹初始长度的增加而增加;双材料弹性模量比相差越大,界面裂纹越易于扩展,且裂纹扩展长度也越大,因此可通过调节双材料弹性模量比来延缓界面裂纹扩展。      

An elastic-plastic finite element analysis of crack tip fields under biaxial loading conditions

A square plate containing a central crack and subjected to biaxial stresses has been studied by a finite element analysis. An elastic analysis shows that the crack opening displacement and stress of separation ahead of the crack tip are not affected by the mode of biaxial loading and therefore the stress intensity factor adequately describes the crack tip states in an elastic continuum.

An elastic-plastic analysis involving more than localized yielding at the crack tip provides different solutions of crack tip stress fields and crack face displacements for the different modes of biaxial loading.

The equi-biaxial loading mode causes the greatest separation stress but the smallest plastic shear ear and crack displacement. The shear loading system induces the maximum size of shear ear and crack displacement but the smallest value of crack tip separation stress.

     

Stochastic microcrack nucleation in lamellar solids

Growth of a crack across an interface between two grains of an elastic lamellar material having different lamellar orientations is investigated for materials having a heterogeneous spectrum of individual lamellar toughnesses. Numerical analyses carried out using a cohesive zone model and the finite element method show that microcracking in the adjacent lamellae can preferentially occur at low-toughness lamellae spatially remote from the crack tip rather than at higher-toughness lamellae close to the crack tip. An analytic model based on linear elastic fracture mechanics and an initial microcrack is shown to predict the location and macroscopic toughness at which microcrack nucleation and growth occur in good agreement with the numerical analyses, using only the initial microcrack size as a single parameter. These results demonstrate that microcrack nucleation requires a sufficiently high stress over a sufficiently large region and thus that microcrack nucleation ahead of a main crack can be the dominant small-scale damage mechanism in such heterogeneous systems.     

Cyclic plasticity near a cracklike elliptical flaw

An elastic-plastic analysis of a cracklike elliptical flaw under cyclic tensile loading is discussed. A highly efficient numerical approach combining aspects of the finite element and boundary collocation methods was developed to allow accurate solution detail in the root region of the flaw. Conditions of localized yielding at the flaw root is the focus of the work with applied stress levels small relative to yield stress and plastic zone dimensions comparable to the root radius of curvature. The flaw is considered isolated in an infinite sheet under plane strain constraint. Numerical results are given for the stress and strain distributions and the plastic zone changes during a constant amplitude cyclic loading. These elastic-plastic results are compared with the predictions of elastic and fully plastic analysis and also with sharp crack solutions.     

Non-orthogonal stress modes for interfacial fracture based on local plastic dissipation

Interfacial cracks have several features which are different from those of cracks in homogeneous materials. Among those, the loading mode dependency of interfacial toughness has been a main obstacle to the widespread utilization of interfacial fracture mechanics. In this study, plasticity-induced toughening of an interface crack between an elastic-plastic material and an elastic material is studied. A useful relationship between the plastic dissipation and the plastic zone size is derived via an effective crack length model. Non-orthogonal stress modes for interface cracks are proposed on the basis of the plastic dissipation mechanism and a mixed-mode criterion for interfacial crack growth is also proposed using these stress modes. The non-orthogonal stress modes are able to represent the asymmetric behavior, mode-dependent toughening and ε-dependency of interfacial crack growth.     

Experimental and finite element study on stable crack growth in three point bending

Experimental and finite element results are presented on mode I and mixed mode (involving I and II only) stable crack growth under static loading through an aircraft grade aluminium alloy (D16AT) in three point bending. The results include load-displacement diagrams, J-integrals, plastic zones, tunneling (or crack front curving), etc. During experiment a substantial amount of tunneling is observed, the extent of which increases as the extension progresses in both mode I and mixed mode. The tunneling reduces as a_o/w increases. The crack extends initially almost along a straight line at an angle with the initial crack in a mixed mode. The maximum load is observed to be as high as 1.6 times the initiation load in the whole range examined. From the finite element study it is seen that, in a mixed mode, the J-integral at the onset of extension is the lowest compared with the values at the later stages. The plastic zone size grows as the stable extension progresses; the growth is approximately the maximum along the crack extension line. The direction of initial crack extension in a mixed mode can be predicted through an elastic finite element analysis and using the criterion of maximum tangential principal stress. The study also indicates that the load-displacement diagram associated with a mixed mode stable crack growth can be predicted reasonably accurately using the criterion of crack opening angle.     

基于断裂力学的高强度钢材梁柱节点受力性能分析

为了研究国产Q460C高强度结构钢材梁柱节点的断裂行为,该文基于断裂力学理论,计算了Q460C高强度钢材焊缝及热影响区材料的断裂韧性,并且采用三维有限元断裂模型,以I型裂纹尖端应力强度因子K_I和J积分为定量的评价指标,分析了焊缝及热影响区不同长度的裂纹对梁柱节点断裂韧性的需求。弹性分析表明,K_I沿梁翼缘宽度方向呈W形分布,最大值出现在翼缘中心,且与名义弯曲应力呈线性关系,而焊根裂纹的断裂韧性需求比热影响区裂纹更高。弹塑性分析表明,J_I最大值出现在翼缘的边缘,热影响区裂纹的断裂韧性需求比焊根裂纹更高。研究结果表明,Q460C高强度钢材梁柱节点的断裂由焊根裂纹控制,断裂承载力与梁全截面塑性承载力相近,临界转角小于0.02rad,因此建议通过改善焊接工艺或局部构造来保证节点拥有足够的转动能力。     

THE VARIATION OF CRACK OPENING STRESS INTENSITY WITH TIME

It was found that crack opening stress intensities are not constant after holding for various times in the unloaded state for iron and a sequence of Fe-Si binary alloys. The crack opening level decreased with time first because of asperity creep and substructure relaxation, and then increased again with time due to reverse plastic zone relaxation allowing a new set of asperities to come into contact. It is demonstrated that the crack closure phenomenon is not only a result of residual plasticity, oxidation and asperities, but also is due to asperity creep and reverse plastic zone relaxation at the crack tip for low strength materials.     

Elastic–plastic fracture behavior of a radial matrix crack interacting with a circle inclusion with generalized Irwin corrections

A generalized Irwin plastic zone model is proposed to investigate the interaction problem for a circular inclusion embedded in an elastic–plastic matrix that contains a radial crack, oriented at an arbitrary angle from a remote load. The distributed dislocation technology is applied to formulate the current problem. The effective stress intensity factors, the plastic zone size, and the crack tip opening displacement are evaluated by solving the formulated singular integral equations. In the numerical examples, the inclusion is taken as a void and a rigid body. The effects of the crack angle and the inclusion–crack distance (the distance from the inclusion center to the crack center) on the effective stress intensity factors, the plastic zone size, and the crack tip opening displacement are discussed in detail. Numerical results show that if the crack angle is not large, the values of the plastic zone size and the crack tip opening displacement are less than the corresponding values in the homogenous case when the inclusion is a rigid body; when the inclusion is a void, these values are larger than the corresponding values in the homogenous case.