The Modified Manson–Coffin Curve Method to estimate fatigue lifetime under complex constant and variable amplitude multiaxial fatigue loading

This paper investigates the accuracy of the so-called Modified Manson–Coffin Curve Method (MMCCM) in estimating fatigue lifetime of metallic materials subjected to complex constant and variable amplitude multiaxial load histories. The MMCCM postulates that fatigue damage is maximised on that material plane experiencing the maximum shear strain amplitude. In the present investigation, the orientation of the critical plane was determined through that direction along which the variance of the resolved shear strain reaches it maximum value. Under variable amplitude complex load histories, this direction was also used to count the resolved shear strain cycles via the classic Rain-Flow method. Further, the degree of multiaxiality and non-proportionality of the time-variable stress states at the assumed critical locations was directly quantified through a suitable stress ratio which accounts for (i) the mean value and the variance of the stress perpendicular to the critical plane as well as for (ii) the variance of the shear stress resolved along the direction experiencing the maximum variance of the resolved shear strain. The accuracy and reliability of the proposed approach was checked against approximately 650 experimental data taken from the literature and generated by testing un-notched metallic materials under complex constant and variable amplitude multiaxial load histories. The sound agreement between estimates and experimental results which was obtained strongly supports the idea that the proposed design technique is a powerful engineering tool allowing metallic materials to be designed against constant and variable amplitude multiaxial fatigue by always reaching a remarkable level of accuracy. This approach offers a complete solution to the strain based multiaxial fatigue problem.     

Estimating fatigue damage under variable amplitude multiaxial fatigue loading

The present paper is concerned with the use of the modified Wöhler curve method (MWCM) to estimate both lifetime and high‐cycle fatigue strength of plain engineering materials subjected to complex load histories resulting, at critical locations, in variable amplitude (VA) multiaxial stress states. In more detail, when employed to address the constant amplitude (CA) problem, the MWCM postulates that fatigue damage reaches its maximum value on that material plane (i.e. the so‐called critical plane) experiencing the maximum shear stress amplitude, fatigue strength depending on the ratio between the normal and shear stress components relative to the critical plane itself. To extend the use of the above criterion to those situations involving VA loadings, the MWCM is suggested here as being applied by defining the critical plane through that direction experiencing the maximum variance of the resolved shear stress. Such a direction is used also to perform the cycle counting: because the resolved shear stress is a monodimensional quantity, stress cycles are directly counted by the classical rain‐flow method. The degree of multiaxiality and non‐proportionality of the time‐variable stress state at the assumed critical sites instead is suggested as being measured through a suitable stress ratio which accounts for the mean value and the variance of the stress perpendicular to the critical plane as well as for the variance of the shear stress resolved along the direction experiencing the maximum variance of the resolved shear stress. Accuracy and reliability of the proposed approach was checked by using several experimental results taken from the literature. The performed validation exercise seems to strongly support the idea that the approach formalized in the present paper is a powerful engineering tool suitable for estimating fatigue damage under VA multiaxial fatigue loading, and this holds true not only in the medium‐cycle, but also in the high‐cycle fatigue regime.     

An improved critical plane and cycle counting method to assess damage under variable amplitude multiaxial fatigue loading

The plane with the maximum variance of the resolved shear stress is taken as the critical plane. Two algorithms are used along with the maximum variance method (MVM) to determine the orientation of the critical plane. The maximum variance of the normal stress on the potential critical planes is calculated to determine the one experiencing the maximum extent of fatigue damage. A new multiaxial cycle counting method is proposed to count cycles on the critical plane. The modified Wöhler curve method is used to assess fatigue damage. About 200 experimental results were collected from the technical literature to validate the approaches being proposed. The results show that the improved design technique being proposed is successful in assessing fatigue damage under variable amplitude multiaxial cyclic loading.     

A novel engineering method based on the critical plane concept to estimate the lifetime of weldments subjected to variable amplitude multiaxial fatigue loading

This paper summarizes an attempt at proposing a new engineering method suitable for estimating the fatigue lifetime of steel‐ and aluminium‐welded connections subjected to variable amplitude multiaxial fatigue loading. In particular, the proposed approach is based on the use of the so‐called Modified Wöhler Curve Method (MWCM), i.e. a bi‐parametrical critical plane approach, whose accuracy has been checked so far solely in addressing the constant amplitude multiaxial fatigue problem. In order to extend the use of our criterion to variable amplitude situations, the critical plane is suggested here as being determined by taking full advantage of the maximum variance concept, that is, such a plane is assumed to be the one containing the direction along which the variance of the resolved shear stress reaches its maximum value. The main advantage of such a strategy is that the cycle counting can directly be performed by considering the shear stress resolved along the maximum variance direction: by so doing, the problem is greatly simplified, allowing those well‐established cycle counting methods specifically devised to address the uniaxial variable amplitude problem to be extended to those situations involving multiaxial fatigue loading. The validity of the proposed methodology was checked by using two different datasets taken from the literature and generated by testing both steel and aluminium tube‐to‐plate welded connections subjected to in‐phase and 90° out‐of‐phase variable amplitude bending and torsion. This new fatigue life assessment technique was seen to be highly accurate allowing the estimates to fall within the calibration scatter bands not only when the constants in the governing equations were calculated by using the experimental uniaxial and torsional fully reversed fatigue curves, but also when they were determined by using the reference curves supplied, for the investigated geometry, by the available standard codes. These results seem to strongly support the idea that, thanks to its peculiar features, our method can be considered as an effective engineering approach capable of performing multiaxial fatigue assessment under variable amplitude loading which fully complies with the recommendations of the available standard codes.     

A Review of Critical Plane Orientations in Multiaxial Fatigue Failure Criteria of Metallic Materials

The paper presents a review of multiaxial fatigue failure criteria based on the critical plane concept. The criteria have been divided into three groups, according to the fatigue damage parameter used in the criterion, i.e. (i) stress, (ii) strain and (iii) strain energy density criteria. Each criterion was described mainly by the critical plane orientation. Multiaxial fatigue criteria based on the critical plane concept usually apply different loading parameters in the critical plane whose orientation is determined by (a) only shear loading parameters (crack Mode II or III), (b) only normal loading parameters (crack Mode I) or sometimes (c) mixed loading parameters (mixed crack Mode). There are also criteria based on few critical plane orientations and criteria based on critical plane orientations determined by a weighted averaging process of rotating principal stress axes.     

High-cycle fatigue mechanisms in 1045 steel under non-proportional axial-torsional loading

Detailed microscopic analyses have been made on the high-cycle mechanisms in 1045 steel under various stress-controlled axial-torsional loadings. A special attention has been paid to a critical example of non-proportional loading, i.e., 90° out-of-phase loading with different stress ratios. The replica technique has been used to monitor crack initiation and propagation from the microstructure scale. The orientations of persistent slip bands and Stage I cracks are in good agreement with the critical plane concept. The evolutions of crack length with cycle life as well as the crack aspect ratios depend on the loading condition. However at a given life, the data are consolidated in terms of crack depth versus cycle life. The McDiarmid parameter correlates stress-life data under proportional loadings. However, it underestimates fatigue lives under out-of-phase loading at high stress ratio and it overestimates them in the case where all planes experience the same shear stress amplitude (stress ratio = 0.5). More damaging mechanisms are involved in crack initiation and crack propagation. It is recommended to test the fatigue performance of materials in this last condition that involves the worst damage mechanisms.     

Notch and mean stress effect in fatigue as phenomena of elasto-plastic inherent multiaxiality

The present paper summarises an attempt of estimating fatigue lifetime of notched metallic materials by directly accounting for the degree of multiaxiality of the local elasto-plastic stress/strain-fields acting on the fatigue process zone. In more detail, the proposed approach takes as its starting point the assumption that Stage I is the most important stage to be modelled to accurately estimate fatigue damage, and this holds true independently of the sharpness of the assessed geometrical feature. According to this initial idea, and by taking full advantage of the so-called Modified Manson-Coffin Curve Method (MMCCM), the hypothesis is then formed that the crack initiation plane is always coincident with that material plane experiencing the maximum shear strain amplitude. Subsequently, to devise an efficient design method capable of taking into account the detrimental effect of stress/strain gradients arising also from severe stress/strain concentration phenomena, the MMCCM is suggested here as being applied in terms of the Theory of Critical Distances (TCD), the latter being used in the form of the Point Method (PM). Further, in light of the well-known fact that the value of the mean stress/strain components in the vicinity of the stress/strain raisers’ apices can be different from the corresponding nominal values due to the actual elasto-plastic behaviour of the material being assessed, it is shown, through the MMCCM itself, that also the mean stress effect can directly and accurately be treated as a problem of inherent multiaxiality. Finally, as a preliminary validation, the accuracy and reliability of the proposed approach is checked through several experimental results taken from the literature and generated by testing, under uniaxial fatigue loading, samples containing a variety of geometrical features, the effect of different nominal load ratios being investigated as well.     

A STUDY ON FATIGUE CRACK GROWTH UNDER OUT-OF-PHASE COMBINED LOADINGS

Abstract— Fatigue tests were performed on thin-walled tubular specimens of S45C steel under tension-compression, pure torsion, in-phase and out-of-phase axial-torsional loadings. The relationship between cracking behaviour and stress components on the crack plane was investigated. Measurement of microcrack density showed that microcracking was governed predominantly by the shear stress amplitude acting on the crack plane for all loading conditions. The failure crack was formed by coalescence of many cracks initiated near the maximum shear planes. The cracks grew turning their orientation to the direction perpendicular to the maximum normal stress. The transition of crack orientation occurred at relatively longer crack lengths at a higher stress ratio. The crack growth behaviour for all loading modes can be correlated using an equivalent strain intensity parameter based on shear and normal strains on the crack plane.     

A critical distance/plane method to estimate finite life of notched components under variable amplitude uniaxial/multiaxial fatigue loading

The present paper summarises the main features of a design technique we have devised to specifically perform, by post-processing the linear-stress fields in the vicinity of the assumed crack initiation sites, the fatigue assessment of notched components subjected to in-service variable amplitude (VA) uniaxial/multiaxial fatigue loading. In more detail, fatigue damage is estimated through the Modified Wöhler Curve Method (MWCM) applied along with the Theory of Critical Distances (TCDs), the latter being used in the form of the Point Method (PM). According to the philosophy on which the linear-elastic TCD is based, the adopted critical distance is treated as a material property whose length increases as the number of cycles to failure decreases. To correctly apply the MWCM, the orientation of the critical plane is suggested here as being calculated through that direction experiencing the maximum variance of the resolved shear stress. Further, the above direction is used also to perform the cycle counting: since, by definition, the resolved shear stress is a monodimensional stress quantity, fatigue cycles are counted by taking full advantage of the classical three-point Rain Flow method. From a philosophical point of view, the real novelty contained in the present paper is that eventually all the different pieces of theoretical work we have done over the last 15 years by investigating different aspects of the uniaxial/multiaxial fatigue issue are consistently brought together by formalising a design methodology of general validity. The accuracy and reliability of the proposed fatigue assessment technique was checked by using 124 experimental results generated by testing notched cylindrical samples of carbon steel C40. The above tests were run under three different load spectra, by exploring uniaxial as well as in- and out-of-phase biaxial situations, in the latter case the axial and torsional load signals being not only characterised by non-zero mean values, but also by different frequencies. To conclude it can be said that such a systematic validation exercise allowed us to prove that the proposed approach is highly accurate, resulting in estimates falling within the constant amplitude (CA) fully-reversed uniaxial and torsional scatter bands used to calibrate the method itself (this holding true independently of both complexity of the applied VA loading path and sharpness of the tested notch).     

A method for determining the fatigue critical plane for biaxial random vibration in the frequency domain

This paper provides a direct method to determine the critical plane orientation for biaxial random vibration. The critical plane is obtained by finding the direction that maximizes the normal stress variance. It is found that the shear stress is uncorrelated with the normal stress in this orientation. Furthermore, the direction of maximal normal stress is shown to coincide with the principal direction in the case of proportional stress components. Spectral fatigue damage methods proposed in recent literature involve a Monte Carlo enumeration step to find the critical plane orientation. By using the proposed technique, computationally expensive enumeration methods are avoided and greater accuracy in the fatigue damage estimate may result.     

Experimental and numerical investigations on the influence of the loading direction on the fatigue crack growth

During a service loading fatigue cracks can be subjected to a mixed mode loading if, due to the alteration of the loading direction, the basic crack modes (Modes I, II and III) are combined. An alteration of the loading direction, e.g. can occur either occasionally paired with an overload (mixed mode overload) or permanently in terms of a mixed mode block loading as a combination of normal and shear stresses.Within the scope of this paper, experimental investigations on both mixed mode overloads, which are interspersed into a Mode I baseline level loading, and mixed mode block loadings are presented. The experimental investigations show that the retardation effect decreases with an increasing amount of Mode II of the overload. Due to the block loading, the fatigue crack growth rate is retarded as well, and the crack is also deflected. The kinking angle depends on the fraction of shear stresses. Furthermore, a detailed elastic–plastic finite element analysis of the fatigue crack growth after mixed mode overloads is presented in order to understand the mechanism of the load interaction effects. By such numerical simulations, it can be shown that, due to mixed mode overloads, plastic deformations occur, which on the one hand reduce the near-tip closure and on the other hand cause a far-field closure. Also the stress distribution before and after the crack tip changes. A mixed mode overload causes lower closure and the crack tip deformations become asymmetrical, which is a reason for the smaller retardation effect of a mixed mode overload.     

A shear stress-based parameter for fretting fatigue crack initiation

The purpose of this study was to investigate the fretting fatigue crack initiation behaviour of titanium alloy, Ti–6Al–4V. Fretting contact conditions were varied by using different geometries of the fretting pad. Applied forces were also varied to obtain fretting fatigue crack initiation lives in both the low- and high-cycle fatigue regimes. Fretting fatigue specimens were examined to determine the crack location and the crack angle orientation along the contact surface. Salient features of fretting fatigue experiments were modelled and analysed with finite element analysis. Computed results of the finite element analyses were used to formulate a shear stress-based parameter to predict the fretting fatigue crack initiation life, location and orientation. Comparison of the analytical and experimental results showed that fretting fatigue crack initiation was governed by the maximum shear stress, and therefore a parameter involving the maximum shear stress range on the critical plane with the correction factor for the local mean stress or stress ratio effect was found to be effective in characterizing the fretting fatigue crack initiation behaviour in titanium alloy, Ti–6Al–4V.     

SHEAR FATIGUE CRACK GROWTH: A LITERATURE SURVEY

A fatigue crack is often initiated by a localized cyclic plastic deformation in a crystal where the active slip plane coincides with the plane of maximum shear stress. Once a crack is initiated, the crack will propagate on the maximum shear plane for a while and, in the majority of the cases, will eventually change to the plane of the applied tensile stress. The “shear” and “tensile” modes of fatigue crack propagation are termed stage I and stage II fatigue crack growth. They are also known as mode II and mode I fatigue crack growth. However, the mechanism of the tensile mode fatigue crack propagation is shear in nature. Considerable progress has been made recently in the understanding of mode II fatigue crack growth. This paper reviews the various test methods and related data analyses. The combined mode I and mode II elastic crack tip stress field is reviewed. The development and the design of the compact shear specimen are described and the results of fatigue crack growth tests using the compact shear specimens are reviewed. The fatigue crack growth tests and the results of inclined cracks in tensile panels, center cracks in plates under biaxial loading, cracked beam specimens with combined bending and shear loading, center cracked panels and the double edge cracked plates under cyclic shear loading are reviewed and analyzed in detail.     

A bi-parametric Wöhler curve for high cycle multiaxial fatigue assessment

This paper presents a method for estimating high-cycle fatigue strength under multiaxial loading conditions. The physical interpretation of the fatigue damage is based on the theory of cyclic deformation in single crystals. Such a theory is also used to single out those stress components which can be considered significant for crack nucleation and growth in the so-called Stage I regime. Fatigue life estimates are carried out by means of a modified Wöhler curve which can be applied to both smooth and blunt notched components, subjected to either in-phase or out-of-phase loads. The modified Wöhler curve plots the fatigue strength in terms of the maximum macroscopic shear stress amplitudes, the reference plane − where such amplitudes have to be evaluated − being thought of as coincident with the fatigue microcrack initiation plane. The position of the fatigue strength curve also depends on the stress component normal to such a plane and the phase angle as well. About 450 experimental data taken from the literature are used to check the accuracy of the method under multiaxial fatigue conditions.     

EFFECT OF BIAXIAL MEAN STRESS ON CYCLIC STRESS-STRAIN RESPONSE AND BEHAVIOUR OF SHORT FATIGUE CRACKS IN A HIGH STRENGTH SPRING STEEL

Abstract— Biaxial fatigue tests were conducted on a high strength spring steel using hour-glass shaped smooth specimens. Four types of loading system were employed, i.e. (a) fully reversed cyclic torsion, (b) uniaxial push—pull, (c) fully reversed torsion with a superimposed axial static tension or compression stress, and (d) uniaxial push—pull with a superimposed static torque, to evaluate the effects of mean stress on the cyclic stress—strain response and short fatigue crack growth behaviour. Experimental results indicate that a biaxial mean stress has no apparent influence on the stress—strain response in torsion, however a superimposed tensile mean stress was detrimental to torsional fatigue strength. Similarly a superimposed static shear stress reduced the push—pull fatigue lifetime. A compressive mean stress was seen to be beneficial to torsion fatigue life. The role of mean stress on fatigue lifetime, under mixed mode loading, was investigated through experimental observations and theoretical analyses of short crack initiation and propagation. Using a plastic replication technique the effects of biaxial mean stress on both Stage I (mode II) and Stage II (mode I) short cracks were evaluated and analysed in detail. A two stage biaxial short fatigue crack growth model incorporating the influence of mean stress was subsequently developed and applied to correlate data of crack growth rate and fatigue life.     

The role of air in fatigue load interaction

Natural fatigue crack formation and growth were studied in notched Al–Cu alloy coupons through high‐resolution SEM fractography. The experiments were conducted under programmed loading conditions designed to induce microscopic marking of the crack formation and growth process under varying stress ratio and closure‐free crack tip conditions. Control experiments were performed by switching between an air and vacuum environment. In air, varying the stress ratio from 0.74 down to 0.64 retards crack growth by up to a factor of five. This ‘closure‐free’ stress ratio history effect totally disappears in vacuum, suggesting a significant environmental influence on stress ratio and its history. Crack‐tip stress state appears to moderate environmental action, revealing a potential mechanism sensitive to residual stress. Consequently, crack closure, residual stress and crack front and plane orientation are identified as major load interaction mechanisms whose synergistic action controls fatigue under variable amplitude loading. The study also appears to suggest that as a consequence of the crack seeking the path of least resistance, load‐sequence sensitive crack plane and front orientation may only induce retardation effects.     

FATIGUE CRACK GROWTH UNDER MIXED-MODE I AND II LOADING

Abstract— Mixed-mode fatigue crack growth has been studied using four point bend specimens under asymmetric loads. A detailed finite element analysis provides the stress intensity factors for curved cracks under different mixed-mode load conditions. Both fatigue crack growth direction and crack growth rate are studied. The maximum tangential stress and the minimum strain energy density criteria were found to provide satisfactory predictions of the crack growth directions. An effective stress intensity factor was used to correlate the fatigue crack growth rates successfully. It is found that the use of mode I fatigue crack growth rate properties results in a conservative crack growth rate prediction for mixed-mode load conditions.     

Fatigue of engineering structures under combined nonproportional loads: An overview

Nonproportionality in fatigue is a matter of loading sequences acting on structures. In such loading cases, a life assessment‐based search of the initiation site is required. This means that for a variety of potential sites, their stress‐strain sequences must be determined and the corresponding fatigue lives have to be calculated. For the determination of local stress‐strain sequences, a superposition of the action of loads at all times for all potential sites must be performed. The superposition has not only to be performed at a particular time of a load reversal but also for a sufficiently large number of intermediate points in time. For the assessment of crack initiation under local nonproportional stress‐strain sequences, a multiaxial fatigue damage hypothesis must be specified. Due to the superiority with respect to accuracy, critical plane approaches are the favourite ones when dealing with nonproportional fatigue of ductile metals. For describing fatigue crack growth, the time sequences for the stress intensity factors corresponding to the 3 opening modes must be obtained by superposition. A mixed‐mode hypothesis must be applied to predict both the direction and the increment of crack growth at each crack front point. A first prediction trial should be based on an appropriate extension of the maximum tangential stress hypothesis. An application of a variant of the maximum shear stress hypothesis might be an amendment. Especially in the case of crack turning, the consideration of crack face contact is mandatory.     

Low‐cycle fatigue life prediction of various metallic materials under multiaxial loading

This paper proposed a simple life prediction model for assessing fatigue lives of metallic materials subjected to multiaxial low‐cycle fatigue (LCF) loading. This proposed model consists of the maximum shear strain range, the normal strain range and the maximum normal stress on the maximum shear strain range plane. Additional cyclic hardening developed during non‐proportional loading is included in the normal stress and strain terms. A computer‐based procedure for multiaxial fatigue life prediction incorporating critical plane damage parameters is presented as well. The accuracy and reliability of the proposed model are systematically checked by using about 300 test data through testing nine kinds of material under both zero and non‐zero mean stress multiaxial loading paths.     

Computational modeling of mixed-mode fatigue crack growth using extended finite element methods

The extended finite element method (XFEM) combined with a cyclic cohesive zone model (CCZM) is discussed and implemented for analysis of fatigue crack propagation under mixed-mode loading conditions. Fatigue damage in elastic-plastic materials is described by a damage evolution equation in the cohesive zone model. Both the computational implementation and the CCZM are investigated based on the modified boundary layer formulation under mixed-mode loading conditions. Computational results confirm that the maximum principal stress criterion gives accurate predictions of crack direction in comparison with known experiments. Further popular multi-axial fatigue criteria are compared and discussed. Computations show that the Findley criterion agrees with tensile stress dominant failure and deviates from experiments for shear failure. Furthermore, the crack propagation rate under mixed mode loading has been investigated systematically. It is confirmed that the CCZM can agree with experiments.