Stress intensity factor equations for branched crack growth

Overload-induced fatigue crack branching is a well-known crack growth retardation or arrest mechanism, which can quantitatively explain such effects even when arguments based on plasticity induced crack closure cannot be applied, e.g. in high R-ratio or in plane strain controlled fatigue crack growth. However, the few results available for branched cracks cannot be used to predict the subsequent crack growth nor account for the delays observed in practice. In this work, specialized finite element (FE) and fatigue life assessment software are used to solve this problem. The crack path and associated stress intensity factors (SIF) of kinked and bifurcated cracks are numerically obtained by the FE program for several angles and branch lengths, and the companion life assessment program is used to estimate the number of delay cycles associated with them. From these results, crack retardation equations are proposed to model the number of delay cycles and the retardation factor along the crack path, allowing for a better understanding of the influence of crack deflection in the propagation life of structural components.     

The effect of residual stress due to interference fit on the fatigue behavior of a fastener hole with edge cracks

The fatigue property of riveted lap joint is greatly related to the riveting-induced residual stress. However, an accurate study of the fatigue property considering the influence of residual stress quantitatively can be very difficult. A 3D interface element based on the virtual crack closure technology (VCCT) was developed to calculate the stress intensity factor (SIF) for through cracks at the hole edge. The riveting process was analyzed prior to the tensile test in the finite element code, so that the residual stress can be taken into account to get the eventual value of SIF. The result shows that, with the presence of fatigue cracks, the initial stress-strain state in the structure would change, especially near the crack tip, where great compressive stress can be found. The eventual residual stress cannot be derived by simply superimposing the riveting-induced residual stress with the crack-induced residual stress. The 3D-VCCT interface element shows strong ability to solve the SIFs. The FE analysis results agree well with the reported models both in the fatigue crack growth rate (FCGR) and in the shape of the crack front. However, when the crack is extremely short, not only the reported models, but also the present numerical model would fail. Besides, unlike Elber's model and Schijve's model, this study shows that the crack opening stress should not be a function of the stress ratio solely, but also with the crack length included.     

Influence of material microvoids and heterogeneities on fatigue crack propagation

The evaluation of crack initiation, short-crack growth as well as crack path at microscopic scale is a crucial issue for the safety assessment of macroscopically fracture-free structural components. In the present paper, the crack propagation at the material microscale is modeled by taking into account the spatial variability of mechanical characteristics of the material as well as the local multiaxial stress field disturbance induced by inhomogeneities (inclusions or voids). By adopting some crack extension criteria under mixed mode, the short-crack path is determined. A microstructure dependence of the crack path arises in the short-crack regime, while the microstructure of the material does not influence the crack propagation for sufficiently long cracks. A mean weighted equivalent stress-intensity factor (SIF) is computed for kinked short cracks, where the range of such a SIF can be used as a key parameter dictating their fatigue crack growth rate.     

Effects of CFRP bond locations on the Mode I stress intensity factor of centre‐cracked tensile steel plates

The fatigue life of cracked steel members can be greatly extended by externally attached carbon fibre reinforced plastics (CFRP), which reduces the stress intensity factors (SIFs) at the crack tip. Access to cracks is sometimes limited and the CFRP has to be attached away from the cracks. There is a lack of knowledge on SIFs for such strengthening scheme. This paper presents the effects of CFRP bond locations on the Mode I SIF of centre‐cracked tensile (CCT) steel plate. The Mode I SIF at the crack tip is calculated using the finite element (FE) models. A correction factor is introduced as a function of CFRP bond location and crack length. The FE results are compared and agree well with experimental tests conducted by the authors. By combining with another two factors (one considering CFRP mechanical properties and the other considering CFRP bond width) derived previously by the authors, SIF formulae are proposed for CFRP reinforced CCT steel plates.     

Near-threshold fatigue propagation of physically through-thickness short and long cracks in a low alloy steel

In this paper, the near-threshold fatigue behavior of physically through-thickness short cracks and of long cracks in a low alloy steel is investigated by experiments in ambient air. Physically through-thickness short fatigue cracks are created by gradually removing the plastic wake of long cracks in compact tension specimens. The crack closure is systematically measured using the compliance variation technique with numerical data acquisition and filtering for accurate detection of the stress intensity factor (SIF) at the crack opening. Based on the experimental results, the nominal threshold SIF range is shown to be dependent on the crack length and the characteristic of the crack wake which is strongly dependent on the loading history. The effective threshold SIF range and the relation between the crack propagation rate and the effective SIF range after the crack closure correction are shown to be independent on crack length and loading history. The shielding effect of the crack closure is shown to be related to the wake length and load history. The effective threshold SIF range and the relationship between the crack growth rate and the effective SIF range appear to be unique for this material in ambient air. These properties can be considered as specific fatigue properties of the couple material/ambient air environment.     

An approximation for the cyclic state of stress ahead of cracks and its implications under fatigue crack growth

Numerical methods are mostly used in the field of fatigue to derive the stress intensity factor (SIF) or J-integral solutions to be employed in damage tolerance analysis of cracked components. In this frame, simple assumptions about material properties are taken into account.More refined approaches try to describe the plasticity-induced crack closure in order to account for retardation effects under variable amplitude loading. In these approaches, the cyclic plasticity is used and cyclic finite element analyses are carried out.In the present work, a novel strategy is presented for the calculation of the relevant parameters to the fatigue crack growth, based on the evaluation of local field parameters (J-integral, T-stress) and cyclic material properties. It is demonstrated that, in case of mild steels and under the assumption of a stress ratio R = −1, the global constraint factor α_g widely employed in fatigue crack growth algorithms such as the strip-yield model, can be calculated in a closed-form on the basis of the expression of the crack-tip fields. Moreover, α_g provides a reasonable explanation of the fatigue crack growth behaviour of the A1N steel for different geometrical and loading configurations. Further investigations carried out on different medium and high strength steel grades show that the plastic radius ahead of small and long cracks at their fatigue limits can be considered as a constant for the material.     

On the through-thickness crack with a curve front in center-cracked tension specimens

Three-dimensional finite element analyses are performed on through-thickness cracks with slightly wavy front in center-cracked plates. Considering there is an inherent relationship between the crack shape and the corresponding stress intensity factor (SIF) distribution of a crack, the curved configuration of the crack is determined using a heuristically derived iterative procedure if the SIF distribution function is known. Several simple SIF distribution functions, for instance the constant SIF distribution along the crack front, are assumed to determine the crack shape. Under the assumption that the rate of fatigue crack growth depends on the SIF range or the effective SIF range, possible effects of plate thickness, crack length and crack closure level gradient on the behaviour of crack tunneling are investigated. The stability of the curved shape of a through-thickness crack in fatigue is also discussed, i.e. whether a crack can maintain its shape satisfying the conditions of constant SIF distribution or other distribution along the crack front during fatigue growth. This study will be useful for a better understanding of the behaviour of crack tunneling and help to evaluate the validity of the two-dimensional linear elastic fracture mechanics in cracked plates.     

Comparison of the simulation and experimental fatigue crack behaviors in the nanoseconds laser shocked aluminum alloy

This investigation was performed to compare the simulation and experimental results of the fatigue crack growth rates and behaviors of the 7050-T7451 aluminum alloy by nanoseconds laser shock processing (LSP). Forman–Newman–deKoning (FNK) model embedded in the Franc2D/L software was utilized to predict fatigue crack growth rate, which was conducted to weigh the stress intensity factor (SIF) changing on the surface cracks. LSP induced high compressive residual stresses that served to enhance fatigue properties by improving the resistance against fatigue crack initiation and propagation. The circulating times of crack growth obtained from the simulation and experimental values indicated a slower fatigue crack growth rates after LSP. The relationships between the elastic–plastic materials crack growth rates and the SIF changing after LSP are resolved.     

Comparison of DBEM and FEM crack path predictions in a notched shaft under torsion

The rather complex 3D fatigue crack growth behaviour of two anti-symmetric “bird wing” cracks, initiated from the two crack front corner points of a notched shaft undergoing torsion, is investigated by the Dual Boundary Element Method (DBEM) and by the Finite Element Method (FEM). Different criteria for the crack path assessment (Minimum Strain Energy Density, Maximum Principal Stress and Approximate Energy Release Rate) and for the Stress Intensity Factor (SIF) evaluation (COD and J-integral) are adopted. The SIF’s and the crack path, calculated by such different approaches, turn out to be well consistent with each other. Moreover the simulated crack path qualitatively agrees with experimental findings available from literature.     

Condition for stable growth of branched cracks

In order to clarify the reason why the stable growth of branched cracks occurs in delayed failure, while not in other subcritical crack propagation process such as fatigue, the stress intensity factor after crack branching in delayed failure was dropped to various values, and the propagation behavior of both cracks was investigated.The well balanced growth of branched cracks in delayed failure occurs only when the crack propagation velocity after crack branching belongs to the region II where the crack propagation velocity is constant independently of $\text{K}$. The fatigue cracks at the tips of artificially branched cracks, on the other hand, can not propagate stably, and only either crack propagates preferentially.The exponent in the crack propagation law ($\text{d}\text{a/}\text{d}\text{t =}\text{c}{}_1\text{K}{}^{\mathrm{<mtext>m</mtext>}}\text{or}\text{d}\text{a/}\text{d}\text{N =}\text{c}{}_2\text{(ΔK)}{}^{\mathrm{<mtext>m</mtext>}}$) expresses the degree of unbalance growth of branched cracks. The stable growth of branched cracks occurs only when the crack propagation velocity is constant independently of $\text{K}$ or $\text{ΔK}$, i.e. $\text{m}\text{= 0}$.     

Effect of overload on fatigue crack retardation of aerospace Al-alloy laser welds using crack-tip plasticity analysis

Investigations on fatigue crack growth retardation due to single tensile and periodic multiple over load in strength undermatched laser beam welded 3.2 mm thick aerospace grade aluminium alloy 2139-T8 sheets are conducted. The effect of overload on the fatigue crack propagation behaviours of the homogenous base metal and welded panels (200 mm wide, centre cracked) was compared using experimental and FE analysis methods. The effective crack tip plasticity has been determined in homogeneous M(T) specimens using Irwin’s method and in both homogeneous and laser welded specimen by calculating crack tip plastic strain using FE analysis for single tensile overload. The crack retardation due to the overload in welded specimens is described by the Wheeler Model. The crack tip plastic zone size in the welded specimen was determined by FE analysis using maximum plastic zone extension at the mid sheet thickness. The results show that the Wheeler Model can be implemented to the highly heterogeneous undermatched weld to describe the crack retardation in fatigue following single tensile overload. Fatigue crack growth retardation due to single overload is found to be larger than the base metal. However, after periodic multiple overload, shorter crack retardation has occurred for undermatched welds than the base metal.     

Crack retardation mechanism due to overload in base material and laser welds of Al alloys

To determine the retardation mechanisms due to overload and to predict the subsequent evolution of crack growth rate, investigations are conducted on crack retardation caused by single tensile overloads in base material and laser-welded sheets of AA6056-T6 Al alloy. The effect of the overload ratio on the fatigue crack propagation behaviour of the C(T) 100 specimens was analysed by using experimental and Finite Element (FE) methods. The crack growth rate and fracture surface features were investigated for both base material and laser-welded sheets. The retardation due to overload is described in terms of the affected regions in front of the crack tip. The size and shape of the crack-tip plastic zone and the damage profile induced during the application of the overload in the base material are predicted by FE analysis in conjunction with a porous-metal plasticity model. The results show that the mechanisms of retardation in under-matched welds are substantially different from that of the homogenous base material. More significant crack retardation due to overload has been observed in the laser weld of AA6056-T6. Based on SEM observations of the fracture surfaces and the damage profiles predicted by the proposed FE model, the shape of the crack front formed during the overload application can be predicted. During the overload, the crack front extends into a new shape, which can be predicted by the ductile damage model; a higher load results in a more curved crack front. These outcomes are used to determine the dominant retardation mechanisms and the significance of retardation observed in each region ahead of the crack tip and finally to define the minimum crack growth rate after overload.     

Estimation of fatigue crack growth retardation due to crack branching

Quantitative analysis is provided to estimate the reduction of fatigue crack growth rate due to overload crack branching. A recent mixed-mode fatigue crack growth model based on the dilatational component of the accumulated strain energy density near the crack tip is modified to quantify the retardation factor of crack growth rate following an overload. It is found that crack branching due to an overload results in considerable reduction of fatigue crack growth rate. The retardation factor estimated by the proposed methodology is correlated with test results for the 2090-T8E41 aluminum–lithium alloy indicating encouraging agreement.     

Threshold and growth mechanism of fatigue cracks under mode II and III loadings

ABSTRACT The fatigue crack growth behaviour of 0.47% carbon steel was studied under mode II and III loadings. Mode II fatigue crack growth tests were carried out using specially designed double cantilever (DC) type specimens in order to measure the mode II threshold stress intensity factor range, ΔK_IIth. The relationship ΔK_IIth > ΔK_Ith caused crack branching from mode II to I after a crack reached the mode II threshold. Torsion fatigue tests on circumferentially cracked specimens were carried out to study the mechanisms of both mode III crack growth and of the formation of the factory‐roof crack surface morphology. A change in microstructure occurred at a crack tip during crack growth in both mode II and mode III shear cracks. It is presumed that the crack growth mechanisms in mode II and in mode III are essentially the same. Detailed fractographic investigation showed that factory‐roofs were formed by crack branching into mode I. Crack branching started from small semi‐elliptical cracks nucleated by shear at the tip of the original circumferential crack.     

SIFs of CCT plate repaired with single‐sided composite patch

Constant amplitude load fatigue tests are performed to obtain crack propagation data for LF2‐aluminium centre crack tension (CCT) plates un‐repaired and repaired with single‐sided composite patches. Then, the James–Anderson method, an experimental method, is used to obtain the stress intensity factor (SIF) formula for the repaired CCT plates with carbon–fibre composite patches. At last, crack propagation life prediction and finite element (FE) calculation are carried out to validate the experimental SIF formula. It is shown that the present SIF formula can exactly predict the fatigue‐crack propagation life of the patched CCT plates and is close to the FE results, which implies the effectivity of the experimental SIF formula in the present paper.     

新的估算表面裂纹应力强度因子经验公式

该文给出了新的估算拉伸和纯弯曲载荷下表面裂纹应力强度因子的经验公式。根据疲劳裂纹扩展的数值模拟结果确定强度因子分布函数;利用按已知应力强度因子分布函数求裂纹形状及相应应力强度因子的方法计算给定尺寸的表面裂纹的应力强度因子;通过对数值结果的曲线回归得到估算表面裂纹应力强度因子经验公式。利用该公式对有限厚度和宽度平板内表面裂纹的应力强度因子进行了估算,并与已知的半椭圆形表面裂纹的应力强度因子解进行了比较。该文结果为估算表面裂纹应力强度因子提供了一种新的途径。     

TORSIONAL FATIGUE OF A MEDIUM CARBON STEEL CONTAINING AN INITIAL SMALL SURFACE CRACK INTRODUCED BY TENSION–COMPRESSION FATIGUE: CRACK BRANCHING,NON-PROPAGATION AND FATIGUE LIMIT

In order to examine the threshold condition for the fatigue limit of materials containing a small crack under cyclic torsion, reversed torsional fatigue tests were carried out on 0.47% C steel specimens containing an initial small crack. Initial small semi-elliptical cracks ranging from 200 to 1000 μm in length were introduced by the preliminary tension–compression fatigue tests using specimens containing holes of 40 μm diameter. The threshold condition for the fatigue limit of the specimens containing artificial small defects under rotating bending and cyclic torsion are also reviewed. Crack growth behaviour from an initial crack was investigated. The torsional fatigue limit for a semi-elliptical small crack is determined by the threshold condition for non-propagation of Mode I branched cracks. The torsional fatigue limit of specimens containing an initial small crack can be successfully predicted by the extended application of the √area parameter model in combination with the σ_θmax criterion.     

Retardation and repair of fatigue cracks in a microcapsule toughened epoxy composite—Part II: In situ self-healing

Successful arrest and retardation of fatigue cracks is achieved with an in situ self-healing epoxy matrix composite that incorporates microencapsulated dicyclopentadiene (DCPD) healing agent and Grubbs’ first generation Ru catalyst. Healing agent is released into the crack plane by the propagating crack, where it polymerizes to form a polymer wedge, generating a crack tip shielding mechanism. Due to the complex kinetics of healing a growing crack, the resulting in situ retardation and arrest of fatigue cracks exhibit a strong dependence on the applied range of cyclic stress intensity ΔK_I. Significant crack arrest and life-extension result when the in situ healing rate is faster than the crack growth rate. In loading cases where the crack grows too rapidly (maximum applied stress intensity factor is a significant percentage of the mode-I fracture toughness value), a carefully timed rest period can be used to prolong fatigue life up to 118%. At moderate ΔK_I, in situ healing extends fatigue life by as much as 213%. Further improvements in fatigue life-extension are achieved by employing a rest period, which leads to permanent arrest at this moderate ΔK_I. At lower values of applied stress intensity factor, self-healing yields complete arrest of fatigue cracks providing infinite fatigue life-extension.     

VARIABLE AMPLITUDE LOADING OF SMALL FATIGUE CRACKS IN 6261-T6 ALUMINIUM ALLOY

Abstract— Constant and variable amplitude (VA) loading fatigue studies were carried out on a 6261 aluminium alloy using cylindrical plain hour-glass specimens. Crack growth was monitored via surface replication using cellulose acetate.
Crack growth results at constant amplitude loading show the typical intermittent high and low periods of growth rate associated with crack-microstructure interactions. Acceleration in growth rate during an overload block depends on crack length and stress amplitude ratio. It appears to pass through a maximum at a crack length corresponding to the first microstructural barrier. Microstructural-based modelling is therefore required for small fatigue cracks, rather than solely closure-based modelling. The Navarro-de los Rios model of short fatigue crack growth appears able to provide good indications of crack growth rates under VA block loading, and gives reasonable life predictions.
For short cracks (surface length < 80 μm) and a small overload ratio (6.7%), crack growth may show severe retardation during the overload block. This is ascribed to crack tip blunting being more important than the increase in stresses when closure is low. It appears from a Miner's rule type exercise, that VA block loading has its major effect on growth at a surface crack length of 20 μm. This means that the crack initiation period cannot be ignored in life prediction models for small fatigue cracks.