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.     

Interim fatigue design recommendations for fillet welded joints under complex loading

Current fatigue design methods for assessing welded steel structures under complex combined or multiaxial loading are known to be potentially unsafe. This has led to a number of research projects over the past 10 years. Some progress has been made in developing better methods, but they are not yet suitable for general design. This paper presents an interim solution based on a review and analysis of relevant published data; all referring to fatigue failure from a fillet weld toe. These indicate that Eurocode 3/IIW S – N curve FAT80/3 (negative inverse slope of 3) is suitable for combined normal and shear stresses acting in phase, and possibly for out-of-phase (i.e. non-proportional loading) bending and shear if the shear stress is not due to torsion. However, a shallower curve FAT80/5 is necessary for out-of-phase torsion and bending or tension. Both curves are used in conjunction with the nominal maximum principal stress range occurring during the loading cycle.     

On the interaction of normal and shear stresses in multiaxial fatigue damage

One of the important issues in assessing multiaxial fatigue damage is interactions between different components of stress such as normal and shear stresses. The present study investigated this interaction effect on the fatigue behavior of materials with shear failure mode when subjected to multiaxial loading conditions. A method is introduced to model this interaction based on the idea that two types of influence are caused by the normal stress acting on the critical plane orientation. These two types of influence are affecting roughness induced closure, as well as fluctuating normal stress which affects the growth of small cracks in mode II. Shear‐based critical plane damage models which use normal stress as a secondary input, such as FS damage model, could then use the summation of these terms. In order to investigate the effect of the method, constant amplitude load paths with different levels of interaction between the normal and shear stresses, as well as variable amplitude tests with histories both taken from service loading conditions and generated using random numbers were designed for an experimental program. The proposed method was observed to result in improved fatigue life estimations where significant interactions between normal and shear stresses exist.     

Multiaxial notch fatigue life prediction based on the dominated loading control mode under variable amplitude loading

A new calculation approach is suggested to the fatigue life evaluation of notched specimens under multiaxial variable amplitude loading. Within this suggested approach, if the computed uniaxial fatigue damage by the pure torsional loading path is larger than that by the axial tension–compression loading path, a shear strain‐based multiaxial fatigue damage parameter is assigned to calculate multiaxial fatigue damage; otherwise, an axial strain‐based multiaxial fatigue damage parameter is assigned to calculate multiaxial fatigue damage. Furthermore, the presented method employs shear strain‐based and axial strain‐based multiaxial fatigue damage parameters in substitution of equivalent strain amplitude to consider the influence of nonproportional additional hardening. The experimental data of GH4169 superalloy and 7050‐T7451 aluminium alloy notched components are used to illustrate the presented multiaxial fatigue lifetime estimation approach for notched components, and the results reveal that estimations are accurate.     

Fatigue life assessment under a complex multiaxial load history: an approach based on damage mechanics

In order to assess the fatigue behaviour of structural components under a complex (cyclic or random) multiaxial stress history, methods based on damage mechanics concepts can be employed. In this paper, a model for fatigue damage evaluation in the case of an arbitrary multiaxial loading history is proposed by using an endurance function which allows us to determine the damage accumulation up to the final failure of the material. By introducing an evolution equation for the endurance function, the final collapse can be assumed to occur when the damage D is complete, that is when D reaches the unity. The parameters of this model, which adopts the stress invariants and the deviatoric stress invariants to quantify the damage phenomenon, are determined through a Genetic Algorithm once experimental data on the fatigue behaviour of the material being examined are known for some complex stress histories. With respect to traditional approaches to multiaxial fatigue assessment, the proposed model presents the following advantages: (1) the evaluation of a critical plane is not necessary; (2) no cycle counting algorithm to determine the fatigue life is required, because it considers the progressive damage process during the fatigue load history; (3) the model can be applied to any kind of stress history (uniaxial cyclic loading, multiaxial in‐phase or out‐of‐phase cyclic loading, uniaxial or multiaxial random loading).     

Theoretical fatigue–effective notch stresses at spot welds

Notch stress formulae are derived for the application of a notch stress approach to the fatigue assessment of spot welds. A keyhole notch is assumed to describe the edge of the weld spot between the overlapping plates. The stress fields at the keyhole notch under 'singular' and 'non-singular' in-plane loading modes inclusive of the stress concentration factors K _t are derived from the relevant Airy stress functions. The formulae are applied to typical loading cases of spot welds and compared with finite element solutions. Fatigue-effective notch stresses inclusive of fatigue notch factors K _f are calculated by applying the microstructural support hypothesis of Neuber. The notch stresses at the keyhole are also derived for out-of-plane shear loading based on the relevant harmonic stress functions. The multiaxial notch stresses at the weld spot edge are thus completely described.     

Selection of the critical plane orientation in two-parameter multiaxial fatigue failure criterion under combined bending and torsion

The present paper is focused on engineering application of the algorithm of fatigue life calculation under multiaxial fatigue loading. For that reason, simple two-parameter multiaxial fatigue failure criterion is proposed. The criterion is based on the normal and shear stresses on the critical plane. Experimental results obtained under multiaxial proportional, non-proportional cyclic loading and variable-amplitude bending and torsion were used to verify the proposed two-parameter criterion and other well-known multiaxial fatigue criteria. Elastic–plastic behaviour of the bulk material was taken into account in calculation of the stress/strain distribution across the specimen cross-section. It is shown that the proposed two-parameter multiaxial fatigue failure criterion gives the best correlation between the experimental and calculated fatigue lives.     

A multiaxial fretting fatigue test for spline coupling contact

A novel fretting fatigue experimental methodology is presented for mimicking the salient fretting variables for arbitrary axial locations within a complex spline coupling geometry, under combined torque, axial loading and rotating bending moment. The approach permits the simulation, in a simplified test arrangement, of the superimposed multiaxial fretting conditions between the spline teeth. This is achieved via the combination of a low frequency in-plane cyclic normal clamping load and a higher frequency out-of-plane cyclic fatigue stress. The latter is known from spline fatigue tests to play a critical role, along with torque and axial loads, in fretting fatigue cracking of splines.     

Multiaxial fatigue life assessment of welded structures

Welded structures, such as welded pressure vessel components subjected to multiaxial cyclic loading, are particularly susceptible to fatigue damage. In this paper, a new path-length-based effective stress range is proposed to assess the fatigue life of weld joints under multiaxial fatigue loading. The path-length measure, a function of both normal and shear components on a critical crack plane, has a solid root in classic fracture mechanics and its application is validated by correlating nominal fatigue data including pure-bending, pure-torsion, in-phase, and out-of-phase loading. Path-Dependent Maximum Range (PDMR), a unique general-purpose fatigue life assessment package for multiaxial variable-amplitude loading, is introduced in this paper. Finally, the application of PDMR to multiaxial fatigue life assessment of complex loading cases is also discussed.     

Invarianten‐basierte Mehrachsigkeitshypothese zur Anwendung bei Schwingbeanspruchung

Multiaxial hypothesis based on invariants for the application of fatigue loading Increasing deviations of experimentally determined and calculated fatigue lives can be observed for multiaxially loaded specimens with increasing contribution of shear stresses. An improvement of this situation can be gained by linking the calculation procedure to both constant amplitude life curves for pure push/pull and shear loading. A hypothesis is presented in this paper which is formulated strictly using only the invariants of the stress tensor to interpolate between the border cases. A modification of this hypothesis is able to take nonproportional loading due to a phase shift between the stress components into account.     

Biaxial fatigue for proportional and non‐proportional loading paths

In order to study the use of a local approach to predict crack‐initiation life on notches in mechanical components under multiaxial fatigue conditions, the study of the local cyclic elasto‐plastic behaviour and the selection of an appropriate multiaxial fatigue model are essential steps in fatigue‐life prediction. The evolution of stress–strain fields from the initial state to the stabilized state depends on the material type, loading amplitude and loading paths. A series of biaxial tension–compression tests with static or cyclic torsion were carried out on a biaxial servo‐hydraulic testing machine. Specimens were made of an alloy steel 42CrMo4 quenched and tempered. The shear stress relaxations of the cyclic tension–compression with a steady torsion angle were observed for various loading levels. Finite element analyses were used to simulate the cyclic behaviour and good agreement was found. Based on the local stabilized cyclic elastic–plastic stress–strain responses, the strain‐based multiaxial fatigue damage parameters were applied and correlated with the experimentally obtained lives. As a comparison, a stress‐invariant‐based approach with the minimum circumscribed ellipse (MCE) approach for evaluating the effective shear stress amplitude was also applied for fatigue life prediction. The comparison showed that both the equivalent strain range and the stress‐invariant parameter with non‐proportional factors correlated well with the experimental results obtained in this study.     

Multiaxial notch fatigue life prediction based on pseudo stress correction and finite element analysis under variable amplitude loading

A new computational methodology is proposed for fatigue life prediction of notched components subjected to variable amplitude multiaxial loading. In the proposed methodology, an estimation method of non‐proportionality factor (F) proposed by authors in the case of constant amplitude multiaxial loading is extended and applied to variable amplitude multiaxial loading by using Wang‐Brown's reversal counting approach. The pseudo stress correction method integrated with linear elastic finite element analysis is utilized to calculate the local elastic‐plastic stress and strain responses at the notch root. For whole local strain history, the plane with weight‐averaged maximum shear strain range is defined as the critical plane in this study. Based on the defined critical plane, a multiaxial fatigue damage model combined with Miner's linear cumulative damage law is used to predict fatigue life. The experimentally obtained fatigue data for 7050‐T7451 aluminium alloy notched shaft specimens under constant and variable amplitude multiaxial loadings are used to verify the proposed methodology and equivalent strain‐based methodology. The results show that the proposed methodology is superior to equivalent strain‐based methodology.     

A new approach of fatigue life prediction for metallic materials under multiaxial loading

Based on experimental data found in literatures, four traditionally multiaxial fatigue life criteria are analyzed and verified. It is discovered that these conventional criteria cannot reflect well the combined effect both under tension and torsion loadings for some materials, such as 6082-T6 and AlCu4Mg1, due to lack of enough consideration about the influence of stress amplitude ratio and stress level on fatigue life even under proportional loading. In order to solve this problem, a new approach of fatigue life prediction, based on the equal-life curve, is proposed and it is composed of three parts: the multiaxial fatigue life surface, a new path-dependent factor for multiaxial high-cycle fatigue and a material parameter describing material sensitivity to non-proportional loading. Finally, the precision of the presented approach is systematically checked against the experimental data found in literatures for four different materials under proportional and non-proportional loadings.     

Notch fatigue life prediction considering nonproportionality of local loading path under multiaxial cyclic loading

An innovative numerical methodology is presented for fatigue lifetime estimation of notched bodies experiencing multiaxial cyclic loadings. In the presented methodology, an evaluation approach of the local nonproportionality factor F for notched specimens, which defines F as the ratio of the pseudoshear strain range at 45° to the maximum shear plane and the maximum shear strain range, is proposed and discussed deeply. The proposed evaluation method is incorporated into the material cyclic stress‐strain equation for purpose of describing the nonproportional hardening behavior for some material. The comparison between multiaxial elastic‐plastic finite element analysis (FEA) and experimentally measured strains for S460N steel notched specimens shows that the proposed nonproportionality factor estimation method is effective. Subsequently, the notch stresses and strains calculated utilizing multiaxial elastic‐plastic FEA are used as input data to the critical plane‐based fatigue life prediction methodology. The prediction results are satisfactory for the 7050‐T7451 aluminum alloy and GH4169 superalloy notched specimens under multiaxial cyclic loading.     

Multiaxial fatigue life prediction method based on path‐dependent cycle counting under tension/torsion random loading

A path‐dependent cycle counting method is proposed by applying the distance formula between two points on the tension‐shear equivalent strain plane for the identified half‐cycles first. The Shang–Wang multiaxial fatigue damage model for an identified half‐cycle and Miner's linear accumulation damage rule are used to calculate cumulative fatigue damage. Therefore, a multiaxial fatigue life prediction procedure is presented to predict conveniently fatigue life under a given tension and torsion random loading time history. The proposed method is evaluated by experimental data from tests on cylindrical thin‐walled tubes specimens of En15R steel subjected to combined tension/torsion random loading, and the prediction results of the proposed method are compared with those of the Wang–Brown method. The results showed that both methods provided satisfactory prediction.     

Critical plane approach with two families of microcracks for modelling of unilateral fatigue damage

New multiaxial fatigue damage model based on the critical plane approach is proposed. Two different physical mechanisms of the fatigue damage development on each potential failure plane (critical plane) are considered. In general, each critical plane contains two families of a parallel microcracks. The proposed model reproduces simultaneously fatigue damage induced anisotropy, the influence of positive and negative mean stresses, unilateral fatigue damage, microcrack closure effect and fatigue behaviour under variable amplitude loading. The expression for the equivalent stress in the damage evolution equation includes the stress intensity for the amplitudes as well as joint invariants for the mean values of the stress tensor and for the vectors associated with the directions of microcracks. The theoretical predictions are compared with experimental data under uniaxial cyclic loading of brass specimens. The influence of positive and negative mean stresses on the fatigue life of brass is investigated.     

A HIGH-CYCLE FATIGUE CRITERION APPLIED IN BIAXIAL AND TRIAXIAL OUT-OF-PHASE STRESS CONDITIONS

A high-cycle fatigue criterion suitable for multiaxial non-proportional stress loading is proposed in this paper. The criterion is based on some microscopic considerations related to the crystalline structure of metals. The purpose of the present paper is mainly the application of this criterion in two loading cases: (a) biaxial loads involving two normal stresses or one normal and one shear stress, and (b) triaxial load with two normal stresses and one shear stress. Stress states of these kinds are very common in piping assemblies. Application of the proposed criterion in the case of triaxial loading, where the three stress components are of the same frequency, but out-of-phase, leads to a simple analytical formula. This formula is the equation of a bounding surface that delimits in the space of the above three stresses the safety domain against fatigue crack initiation. A remarkable theoretical result concerns the phase difference of the shear stress, which does not appear in the derived formula. Consequently, according to our proposal the safety domain (i.e. the limiting fatigue endurance) under combined out-of-phase biaxial normal stress loading and torsion is independent of the phase difference of the torsion. Obviously this result holds also for the simpler case of axial load and torsion. On the contrary the phase difference between the two normal stresses has a strong detrimental effect on the fatigue endurance of a metal. As is shown these theoretical conclusions are in good agreement with fatigue limit test data found in the scientific literature.     

Multiaxial fatigue damage parameter and life prediction for medium-carbon steel based on the critical plane approach

The tension–torsion fatigue characteristics were investigated under proportional and non-proportional loading in this paper. The fatigue cracks on the surface of multiaxial fatigue specimens were observed and analyzed by a scan electron microscope. On the basis of the investigation on the Kindil–Brown–Miller and Fatemi–Socie’s critical plane approaches, a shear strain based multiaxial fatigue damage parameter was proposed by von Mises criterion based on combining the maximum shear strain and the normal strain excursion between adjacent turning points of the maximum shear strain on the critical plane. The proposed multiaxial fatigue damage parameter does not include the weight constants. According to the proposed multiaxial fatigue damage parameter, the multiaxial fatigue life prediction model was established with the Coffin–Manson equation, which is used to predict the multiaxial fatigue life of medium-carbon steel. The results showed that the proposed multiaxial fatigue damage parameter could be used under either multiaxial proportional or non-proportional loading.     

Strain-based multiaxial fatigue damage modelling

A new multiaxial fatigue damage model named characteristic plane approach is proposed in this paper, in which the strain components are used to correlate with the fatigue damage. The characteristic plane is defined as a material plane on which the complex three‐dimensional (3D) fatigue problem can be approximated using the plane strain components. Compared with most available critical plane‐based models for multiaxial fatigue problem, the physical basis of the characteristic plane does not rely on the observations of the fatigue crack in the proposed model. The cracking information is not required for multiaxial fatigue analysis, and the proposed model can automatically adapt for different failure modes, such as shear or tensile‐dominated failure. Mean stress effect is also included in the proposed model by a correction factor. The life predictions of the proposed fatigue damage model under constant amplitude loading are compared with a wide range of metal fatigue results in the literature.     

A fracture plane approach in multiaxial high-cycle fatigue of metals

The high-cycle fatigue behaviour of metals under multiaxial loading is examined. By employing the weight function method, the authors propose to correlate the fatigue fracture plane orientation with the averaged principal stress directions. The results derived by applying such an approach are compared with the experimental data collected from the relevant literature, concerning different types of metals under in-phase or out-of-phase sinusoidal biaxial normal and shear stress states. Theoretical results determined by McDiarmid are also reported.