FATIGUE CRACK GROWTH FROM A CIRCULAR NOTCH UNDER HIGH LEVELS OF BIAXIAL STRESS

Abstract— A series of experiments have been conducted on cruciform specimens to investigate fatigue crack growth from circular notches under high levels of biaxial stress. Two stress levels (Δσ₁= 380 and 560 MPa) and five stress biaxialities (λ=+1.0, +0.5, 0, −0.5 and −1.0; where λ=σ₂/σ₁ were adopted in the fatigue tests in type 316 stainless steel having a monotonic yield strength of 243 MPa. The results reveal that fatigue crack growth rates are markedly influenced by both the stress amplitude and the stress biaxiality. A modified model has been developed to describe fatigue crack growth under high levels of biaxial stress.     

THE COMBINED EFFECTS OF STRESS RATIO AND YIELD STRENGTH ON THE LEFM FATIGUE THRESHOLD CONDITION

Abstract— Long-crack (LEFM) fatigue threshold, Δ, K ₀ values are predicted which include the commonly observed effects of stress ratio, R , and yield strength, σ _y . It is assumed that the yield strength effect on threshold is indirectly related to grain size and so is not an independent variable. Two intrinsic thresholds of a material are invoked to explain the observations of higher Δ K ₀ values and a higher R -ratio sensitivity of Δ K ₀ in low strength materials compared to high strength materials. The paper shows that Δ K ₀ is almost independent of both yield strength and stress ratio at high values of the stress ratio. Quantitative relations are developed to estimate curves of (i) Δ K ₀ versus R and (ii) Δ K ₀ versus σ _y . These curves show good agreement with experimental data for steels and aluminium alloys. A method is presented that may be used as an alternative procedure for obtaining quick and conservative estimates of Δ K ₀ for design applications.     

FINITE ELEMENT ANALYSIS OF SPECIMEN GEOMETRY EFFECTS ON FATIGUE CRACK CLOSURE

Abstract— Elastic-plastic finite element analysis is used to study fatigue crack closure at three different crack length to width ratios for three plane stress specimen geometries: center-cracked plate, single-edge-cracked plate (tension), and single-edge-cracked plate (bend). The maximum stress to flow stress ratio, S_max/σ_O, which successfully describes closure results in many center-cracked plate configurations, does not correlate the effect of different geometries on the normalized opening stress, S _open/ S _max. Crack opening stresses for different geometries and crack lengths are successfully correlated by a normalized stress intensity parameter, K _max/ K ₀, where K ₀=σ₀φa. The quality of the correlation is very high at small K _max/ K ₀, and gradually deteriorates as K _max/ K ₀ increases beyond the small-scale yielding regime.     

FATIGUE LIFE PREDICTION OF HEAT-TREATED CARBON STEELS AND LOW ALLOY STEELS BASED ON A SMALL CRACK GROWTH LAW

Abstract— Since heat-treated high strength steels are often used as materials for machines and structures that operate under severe service conditions, it is important to evaluate their fatigue life. Hence the growth law of a small fatigue crack must be known in order to estimate the fatigue life of machines and structures since the life of such members is controlled mainly by the behaviour of a small crack. The growth rate of a small crack can not be predicted usually by linear elastic fracture mechanics, but can be determined uniquely by the term σⁿ_a l , where σ_a is stress amplitude, l is crack length and n is a material constant. In this paper, the small-crack growth law of heat-treated carbon steels and low alloy steels was studied. An effective and convenient method based on a small-crack growth law, d l /d N = C ₃(σ_a/σ_a)ⁿ l is proposed, where σ_u is the ultimate tensile strength, for predicting the small crack propagation life of heat- treated steels with different tensile strength levels, together with a method for determining the fatigue life of plain members.     

EFFECT OF WAVEFORM ON CORROSION FATIGUE CRACK GROWTH

Abstract A study was made on the effects of stress rise time T ₁, maximum stress holding time T ₂, stress decreasing time T ₃ and minimum stress holding time T ₄ of a cycle on fatigue crack growth for a low alloy carbon steel in 3% NaCl solution. Measurements of the effective stress intensity range ratio U and observations of crack tip response were performed to clarify the causes of waveform effects.
The results were summarized as follows; T ₁ had a strong accelerating effect due to corrosive dissolution of the fresh surfaces of the crack which were formed during T ₁. The crack growth rate was enhanced as T ₁ increased and reached a constant value (about 3 times that in air) after T ₁= 10s. The crack growth rate at low Δ K , however, decreased as T ₁ increased more than T ₁= 1 s. T ₂, T ₃ and T ₄ decreased the crack growth rate. The extent of decrease not only depended on the period of T ₂ (or T ₃, T ₄), but also on Δ K and T ₁. A previously derived crack growth law which considered waveform and frequency effects, is also valid to a first approximation for the present results.     

THE INFLUENCE OF STRENGTH AND STRESS RATIO ON SHORT-CRACK THRESHOLDS AND NON-PROPAGATING FATIGUE CRACKS

Quantitative predictions of the influence of yield strength and stress ratio, R , on the physically small crack fatigue threshold stress intensity, Δ K _0(s), are presented. It is shown that at R = 0 to -1, although the threshold stress Δ₀ increases, the threshold stress intensity, Δ K _0(s), decreases with increasing yield strength. Moreover, a lower bound value, Δ K _0(s)(min) is shown to have a constant value, irrespective of the strength and stress ratio. For a given strength, Δ K _0(s), decreases with increasing R in the range -1 R 0.6 and attains a constant low value for R > 0.6. Predicted values of Δ K _0(s) are in good agreement with experimental data for steels. The formation and length of non-propagating fatigue cracks, a _np, are also discussed. The methods suggested for estimating Δ K _0(s) and a _np may be found useful in design procedures.     

Material fatigue properties for assessing mechanical components weakened by notches and defects

This paper addresses the problem of estimating the material fatigue properties for assessing real mechanical components. Initially, some practical rules are proposed to estimate the fully reversed plain fatigue limit, Δσ₀, using the material tensile stress. These rules are obtained by subdividing materials into five different groups: carbon steels, low-alloy steels, high-alloy steels, aluminium alloys and cast irons. Subsequently, using a large database of fatigue data found in the literature, it is demonstrated that the fully reversed torsional plain fatigue limit can be directly estimated from the fully reversed uniaxial plain fatigue limit by simply using Von Mises' formula. Finally, some empirical equations are proposed to estimate El Haddad's short crack constant, a ₀. These equations are based on the assumption that this material property can be derived from the plain fatigue limit determined at a given load ratio, R . Since the a ₀ values depend on the load ratio, so a ₀ versus Δσ₀ relationships can directly account for the R influence. The aim of this paper is to provide engineers engaged in assessing real structural components with empirical rules to estimate the material fatigue properties. All these pieces of information are needed to apply the most modern methods suitable for assessing components weakened by any kind of geometrical feature and subjected to any kind of fatigue loading.     

INVERSION OF THE STRAIN-LIFE AND STRAIN-STRESS RELATIONSHIPS FOR USE IN METAL FATIGUE ANALYSIS

Abstract— Two methods are described for inverting the strainrange/life and strainrange/stressrange equations commonly used in fatigue analysis in order to obtain closed-form expressions for life and stressrange in terms of strainrange. In the Collocation approach the form used is N _f= A (Δε—Δε₀)γ or N _f= A (Δε)Ψ(Δε—Δε₀)γ. In the Spline-Function approach the curve is divided into two regions. At strainranges above where the elastic and plastic lines intersect the equation is N _f= N _T R ^1/c exp δ R α; at lower strainranges it is N _f= N _T R ^1/b exp δ R β, where N _T is transition life, R is strainrange normalised to transition strainrange, and b, c , α, β, δ are constants determinable from the constants of the equation to be inverted. Similar expressions are derived for the cyclic stress/strain curve in terms of the same constants. The methods are illustrated by an example, and found to have close conformity to the basic equations to be inverted.     

A SHEAR STRESS BASED CRITICAL-PLANE CRITERION OF MULTIAXIAL FATIGUE FAILURE FOR DESIGN AND LIFE PREDICTION

Abstract— The use of a previously presented general criterion of failure for high cycle multiaxial fatigue, τ_a/ t _A,B+σ_n.max/2σ_T= 1, is extended to cases where the shear and normal stress on the critical plane are non-proportional and also to give life predictions in the range of 10⁴ to 10⁶ cycles. The criterion takes account of whether case A cracks, growing along the surface, or case B cracks, growing in from the surface, occur.     

CLOSURE AND GROWTH OF MODE I CRACKS IN BIAXIAL FATIGUE

Abstract— —The closure behavior of mode I fatigue cracks under biaxial loading is studied with an elastic-plastic plane stress finite element model. Biaxial stresses are shown to have a significant impact on crack closure behavior at higher maximum stresses. In general, normalized crack opening stresses are highest for equibiaxial loading and lowest for pure shear loading. The differences are apparently negligible for maximum applied stresses less than about 0.4 σ₀. Experimental crack growth data are quantitatively consistent with these trends. Correlations of the experimental data with a simple Δ K _eff were successful as first-order engineering estimates. Changes in forward and reversed plastic zone sizes with biaxiality are not entirely consistent with trends in crack growth rates.     

From a local stress approach to fracture mechanics: a comprehensive evaluation of the fatigue strength of welded joints

This paper investigates the possibility of unifying different criteria concerned with the fatigue strength of welded joints. In particular, it compares estimates based on local stress fields due to geometry (evaluated without any crack-like defect) and residual life predictions in the presence of a crack, according to LEFM. Fatigue strength results already reported in the literature for transverse non-load-carrying fillet welds are used as an experimental database. Nominal stress ranges were largely scattered, due to large variations of joint geometrical parameters. The scatter band greatly reduces as soon as a 0.3-mm virtual crack is introduced at the weld toe, and the behaviour of the joints is given in terms of Δ K _I versus total life fatigue. Such calculations, not different from residual life predictions, are easily performed by using the local stress distributions determined near the weld toes in the absence of crack-like defects. More precisely, the analytical expressions for K _I are based on a simple combination of the notch stress intensity factors K ₁^N and K ₂^N for opening and sliding modes. Then, fatigue strength predictions, as accurate as those based on fracture mechanics, are performed by the local stress analysis in a simpler way.     

Weibull stress model for cleavage fracture under high-rate loading

This paper examines the effects of loading rate on the Weibull stress model for prediction of cleavage fracture in a low-strength, A515-70 pressure vessel steel. Interest focuses on low-to-moderate loading rates ( K˙ _I < 2500  MPa √m  s⁻¹ ). Shallow cracked SE(B) specimens were tested at four different loading rates for comparison with previous quasi-static tests on shallow notch SE(B)s and standard C(T)s. To utilize these dynamic experimental data, we assume that the Weibull modulus ( m ) previously calibrated using quasi-static data remains invariant over the loading rates of interest. The effects of dynamic loading on the Weibull stress model enter through the rate-sensitive material flow properties, the scale parameter ( σ _u ) and the threshold Weibull stress ( σ _w-min ). Rate-sensitive flow properties are modelled using a viscoplastic constitutive model with uniaxial, tension stress–plastic strain curves specified at varying plastic strain rates. The analyses examine dependencies of σ _w-min and σ _u on K˙ _I . Present results indicate that σ _w-min and σ _u are weak functions of loading rate K˙ _I for this pressure vessel steel. However, the predicted cumulative probability for cleavage exhibits a strong sensitivity to σ _u and, consequently, the dependency of σ _u on K˙ _I is sufficient to preclude use of the static σ _u value for high loading rates.     

CALCULATION OF THE STRESS INTENSITY FACTOR FOR A PARTIAL CIRCUMFERENTIALLY CRACKED TUBE LOADED IN BENDING BY USING THE SHELL LINE-SPRING MODEL

Abstract— In this paper the line-spring model (LSM) developed by Rice and Levy is used to obtain an approximate solution of the stress intensity factor for a partial circumferential, externally cracked tube under axial tension and four point bending. The calculation is based on the work done by Delale and Erdogan for cylindrical shells containing a circumferential or an axial semi-elliptical, part-through crack. The range of utility of their analysis is enlarged to thicker wall tubes with nonelliptic and longer part-through circumferential cracks. Values of K ₁ calculated by the LSM are compared with those from a finite element analysis for remote tensile loading and bending cases, which shows fairly good agreement. The calculations are also applied to a fatigue crack growth test in a tube in four point bending to correlate the d a /d N vs λ K ₁ data.     

ELASTIC-PLASTIC FRACTURE MECHANICS FOR INITIATION AND PROPAGATION OF NOTCH FATIGUE CRACKS

Abstract Initiation and propagation are considered to be controlled by the extent of total plastic shear deformation φ. Crack initiation and crack propagation occur when φ, exceeds a critical threshold value which can be equated to threshold conditions determined from linear elastic fracture mechanics analyses. When a crack is in a plastically deformed zone φ_t=φ _p +φ _e . where φ _p is the component of φ _t due to notch bulk plasticity and φ _e , is the component of φ _t due to a linear elastic fracture mechanics (LEFM) analysis of the crack tip plasticity field.
When cracks initiate at notch roots φ _t > φ_th. As the crack propagates in the notch plastic zone the rate of decrease of v _p will be different from the rate of increase of φ _e and it is possible for φ _t to decrease to a level below φ_th thereby creating a non-propagating crack.     

A MODEL OF FATIGUE CRACK GROWTH AFTER AN OVERLOAD IN D16T Al-ALLOY SUBJECTED TO BIAXIAL CYCLIC LOADS AT VARIOUS R-RATIOS

Abstract— Fatigue crack growth after a biaxial overload has been investigated. The crack retardation parameters, N _D, and, a _D, do not have monotonous dependencies on the biaxial stress ratio, λ, because the shear stress, τ_III, acting in the perpendicular direction of the specimen face, influenced the values of these parameters.
It has been found that the plastic zone size parameters, r _ab, and Δ, do not increase monotonously with increasing λ ratio. The plastic zone size in the crack growth direction, r _ho= a _D13, was calculated on the basis of newly proposed relations.
Crack growth after an overload was simulated on the basis of the equivalent mode I stress intensity factor, IC_C, invoking a unified kinetic diagram and calculated crack increments, a _D13 and a _D=Δ_c, where Δ_c is the maximum value of the calculated size of plastic zone. The experimental data for crack growth after an overload had good agreement with the calculated data.     

PROBABILISTIC ASPECTS OF CLEAVAGE CRACK INITIATION SITES AND FRACTURE TOUGHNESS

Abstract— Fracture toughness tests were performed in the ductile-brittle transition temperature range using 110 specimens of the three-point bend and CT types. Probabilistic characteristics of fracture toughness and cleavage crack initiation sites were analysed in detail, together with the fibrous crack shape, from which the plane strain region in the specimen was deduced. The criterion for obtaining plane strain at the mid plane of the specimen was established as: B ≤ 0.004{ K _c( J )/σ _y }²+ 0.01. The thickness effect of cleavage fracture toughness for the specimen satisfying this equation is mainly caused by the statistical distribution of the weakest points ahead of the crack front (the Weibull volume effect).     

A GENERAL CRITERION FOR HIGH CYCLE MULTIAXIAL FATIGUE FAILURE

Abstract— A new simple general criterion of failure for high cycle multiaxial fatigue, τ_a/ t _A.B+σ_{n. max}/2σ_T= 1 presented. The failure criterion is based on a critical plane approach where fatigue strength is a function of the shear stress amplitude and the maximum normal stress on the critical plane of maximum shear stress amplitude. The criterion takes account of whether case A cracks, growing along the surface, or case B cracks, growing into the surface, occur. It requires knowledge of the material properties, tensile strength, σ_T, and reversed shear fatigue strength for case A, t _A, or case B, t _B, cracking, whichever is relevant. t _A is the reversed torsion fatigue strength and t _B is found from a case B cracking test case. The criterion is applicable in the region, 0.5 t ≤ t _a≤ t , and 0 ≤σ_n.max≤σ_T.     

The analysis of thermoelastic isopachic data from crack tip stress fields

A computer program—FACTUS (fracture analysis of crack tips using SPATE)—has been developed for the efficient analysis of thermoelastic data obtained from around a crack tip. The program is based on earlier work for the determination of stress intensity factors (SIFs), and also includes a novel solution procedure for the derivation of the non-singular stress term σ _{0 x} . The program has been used in the analysis of a series of large plate specimens with central or edge slots/cracks. The derived SIFs are compared with independent values. Issues, e.g. crack closure and the extent and effect of the plastic zone, are discussed.     

THE BOUSSINESQ WEDGE AND THE Jk, L AND M INTEGRALS

Abstract— This paper examines the application of the J _k, L and M integrals, in complex-variable form, to the Boussinesq wedge. The wedge is symmetrical and subjected to a point couple and point forces at the apex of the wedge. In the case of a point couple acting at the wedge apex the J _y, L and M integrals are found to vanish for all wedge angles whereas J _x displays a 1/ r ³ path-dependence; where r is a radial dimension measured from the wedge apex. When the wedge is subjected to point forces at the wedge apex then J _x and J _y are 1/ r path-dependent whereas L and M are path-independent.
The property that the L and M integrals are path-independent for the Boussinesq wedge is applied to the problem of determining the modes I and II stress intensity factors for a corner-loaded edge crack in a half-plane subjected to both normal and parallel point forces to the free surface of the half-plane.     

Scale effects in the initiation of cracking of a scarf joint

The singular stress field at the interface-corner of a bi-material scarf joint is analysed for a strip of finite width, w, under remote tension and bending. The two substrates are taken as linear elastic and isotopic. The intensity of the singular stress field is calculated using a domain integral method, and is plotted as a function of joint geometry and material mismatch parameters. It is envisaged that the intensity of singularity can serve as a valid fracture criterion provided the zone of nonlinearity is fully embedded within the singular elastic field. It is assumed that fracture initiates when the magnitude of the corner singularity attains a critical value; consequently, the fracture strength of the joint depends upon the size of the structure. In addition, the interfacial stress intensity factor and the associated T-stress are determined for an edge interfacial crack. When the crack is short with respect to the width of the strip, the stress intensity factor is dominated by the presence of the corner singularity; a boundary layer formulation is used to determine the coupling between the crack tip field and the interface-corner field. The solution suggests that an interfacial crack grows unstably with a rapidly increasing energy release rate, but with only a small change in mode mix. This revised version was published online in July 2006 with corrections to the Cover Date.