Analytical and experimental studies on fatigue crack path under complex multi-axial loading

In real engineering components and structures, many accidental failures are due to unexpected or additional loadings, such as additional bending or torsion, etc. Fractographical analyses of the failure surface and the crack orientation are helpful for identifying the effects of the non‐proportional multi‐axial loading. There are many factors that influence fatigue crack paths. This paper studies the effects of multi‐axial loading path on the crack path. Two kinds of materials were studied and compared in this paper: AISI 303 stainless steel and 42CrMo4 steel. Experiments were conducted in a biaxial testing machine INSTRON 8800. Six different biaxial loading paths were selected and applied in the tests to observe the effects of multi‐axial loading paths on the additional hardening, fatigue life and the crack propagation orientation. Fractographic analyses of the plane orientations of crack initiation and propagation were carried out by optical microscope and SEM approaches. It was shown that the two materials studied had different crack orientations under the same loading path, due to their different cyclic plasticity behaviour and different sensitivity to non‐proportional loading. Theoretical predictions of the damage plane were made using the critical plane approaches such as the Brown–Miller, the Findley, the Wang–Brown, the Fatemi–Socie, the Smith–Watson–Topper and the Liu's criteria. Comparisons of the predicted orientation of the damage plane with the experimental observations show that the critical plane models give satisfactory predictions for the orientations of early crack growth of the 42CrMo4 steel, but less accurate predictions were obtained for the AISI 303 stainless steel. This observation appears to show that the applicability of the fatigue models is dependent on the material type and multi‐axial microstructure characteristics.     

Crack initiation and growth path under multiaxial fatigue loading in structural steels

In real engineering components and structures many accidental failures occur due to unexpected or additional loadings, such as additional bending or torsion. There are many factors influencing the fatigue crack paths, such as the material type (microstructure), structural geometry and loading path. It is widely believed that fatigue crack nucleation and early crack growth are caused by cyclic plasticity. This paper studies the effects of multiaxial loading paths on the cyclic deformation behaviour, crack initiation and crack path. Three types of structural steels are studied: Ck45, medium carbon steel, 42CrMo4, low alloy steel and the AISI 303 stainless steel. Four biaxial loading paths were applied in the tests to observe the effects of multiaxial loading paths on the additional hardening, fatigue crack initiation and crack propagation orientation. Fractographic analyses of the plane orientations of crack initiation and propagation were carried out by optical microscope and SEM approaches. It is shown that these materials have different crack orientations under the same loading path, due to their different cyclic plasticity behaviour and different sensitivity to non-proportional loading. Theoretical predictions of the damage plane were conducted using the critical plane approaches, either based on stress analysis or strain analysis (Findley, Smith–Watson–Topper, Fatemi–Socie, Wang–Brown–Miller, etc). Comparisons of the predicted crack orientation based on the critical plane approaches with the experimental observations for the wide range of loading paths and the three structural materials are shown and discussed. Results show the applicability of the critical plane approaches to predict the fatigue life and crack initial orientation in structural steels.     

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.     

A novel accumulative fatigue damage model for multiaxial step spectrum considering the variations of loading amplitude and loading path

In this paper, based on the process of the fatigue crack initiation and the critical plane theory, a continuous stress parameter was proposed to quantify the driving force of the fatigue crack initiation for the fully reversed multiaxial fatigue loading. In this stress parameter, the shear stress amplitude and normal stress amplitude on the critical plane were combined with the variable coefficients which were affected by the normalized fatigue life and the loading non‐proportionality. Owing to these coefficients, for the multiaxial loadings with different non‐proportionalities, the driving force of the fatigue crack initiation during the whole life could be described. After that, a novel accumulative fatigue damage model was established for the multiaxial two‐stage step spectrum. In this model, the accumulative damage was calculated according to the variation of the proposed stress parameter on the critical plane. Considering the directionality of the multiaxial fatigue damage, for the spectrum in which the loading path was variable, the damage accumulation was carried out on the critical planes of the both loadings, and the larger one was chosen as the final accumulative fatigue damage. In order to verify the new model, up to 41 different multiaxial two‐stage step spectrum loading tests on 2024‐T4 aluminium alloy were collected. The new model, as well as other five commonly used models, was applied to calculate the accumulative fatigue damage. The final results showed that, compared with other commonly used models, the new model had the most accurate results with the smallest scatters.     

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.     

Crack path evaluation on HC and BCC microstructures under multiaxial cyclic loading

In this paper the multiaxial loading path effect on the fatigue crack initiation, fatigue life and fracture surface topology are evaluated for two different crystallographic microstructures (bcc and hc): high strength low-alloy 42CrMo4 steel and the extruded Mg alloy AZ31B-F, respectively.A series of multiaxial loading paths were carried out in load control, smooth specimens were used. Experimental fatigue life and fractographic results were analyzed to depict the mechanical behavior regarding the different microstructures.A theoretical analysis was performed with various critical plane models such as the Fatemi–Socie, SWT and Liu in order to correlate the theoretical estimations with the experimental data. A new approach based on maximum stress concentration factors is proposed to estimate the crack initiation plane, estimations from this new approach were compared with the measured ones with acceptable results. To implement this new approach a virtual micro-notch was considered using FEM. Moreover, the multiaxial loading path effect on stress concentration factors is also studied. The obtained results clearly show the effect of the applied load conditions on local microstructures response.     

Life prediction based on weight‐averaged maximum shear strain range plane under multiaxial variable amplitude loading

Based on Wang and Brown's reversal counting method, a new approach to the determination of the critical plane is proposed by the defined plane with a weight‐averaged maximum shear strain range under multiaxial variable amplitude loading. According to the determined critical plane, a detailed procedure of multiaxial fatigue life prediction is introduced to predict lives in the low‐cycle multiaxial fatigue regime. The proposed approach is verified by two multiaxial fatigue damage models and Miner's linear cumulative damage law. The results showed that the proposed approach can effectively predict the orientation of the failure plane under multiaxial variable amplitude loading and give a satisfactory life prediction.     

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.     

Corrosion fatigue behaviour of 304 stainless steel under proportional and non‐proportional multiaxial loading condition

Corrosion fatigue and electrochemical tests under proportional loading and non‐proportional loading were conducted on 304 stainless steel in 0.63 mol L^?1 NaCl solution at room temperature. Two biaxial loading paths were applied to study the effect of proportional loading and non‐proportional loading on corrosion fatigue behaviour. Surface and fractographic observations of multiaxial corrosion fatigue specimens were carried out by scanning electron microscopy. It was shown that proportional loading had a more significant effect on the occurrence of local corrosion compared with non‐proportional loading because the continuous rotation of the principal stress plane under non‐proportional loading inhibits the pit formation.     

Estimating the orientation of Stage I crack paths through the direction of maximum variance of the resolved shear stress

Our formalisation of the Shear Stress-Maximum Variance Method takes as a starting point the hypothesis that, in ductile materials subjected to fatigue loading, the crack initiation planes, i.e. the so-called Stage I planes, are those containing the direction experiencing the maximum variance of the resolved shear stress. From a computational point of view, the most remarkable implication of the above assumption is that, as soon as the variance and covariance terms characterising the considered load history are known, the effective time needed to estimate the orientation of the critical plane does not depend on the length of the load history itself. Further, such a computational efficiency is seen to be associated with an high-level of accuracy in estimating fatigue lifetime of both plain and notched engineering components, this holding true under constant as well as under variable amplitude uniaxial/multiaxial fatigue loading. In this scenario, by assuming that the orientation of Stage I planes can directly be determined through the orientation of Stage II crack paths, the present paper investigates whether, independently from the degree of multiaxiality and non-proportionality of the applied loading history, the direction of maximum variance of the resolved shear stress is also capable of accurately estimating the orientation of Stage I crack paths.     

Comparative study on biaxial low-cycle fatigue behaviour of three structural steels

In this study the uniaxial/biaxial low‐cycle fatigue behaviour of three structural steels (Ck45 normalized steel, 42CrMo4 quenched and tempered steel and AISI 303 stainless steel) are studied, evaluated and compared. Two parameters are considered for estimating non‐proportional fatigue lives: the coefficient of additional hardening and the factor of non‐proportionality. A series of tests of uniaxial/biaxial low‐cycle fatigue composed of tension/compression with cyclic torsion were carried out on a biaxial servo‐hydraulic testing machine. Several loading paths were carried out, including proportional and non‐proportional ones, in order to verify the additional hardening caused by different loading paths. The experiments showed that the three materials studied have very different additional hardening behaviour. Generally, the transient process from the initial loading cycle to stabilized loading cycle occurs in a few cycles. The stabilized cyclic stress/strain parameters are controlling parameters for fatigue damage. A factor of non‐proportionality of the loading paths is evaluated based on the Minimum Circumscribed Ellipse approach. It is shown that the microstructure has a great influence on the additional hardening and the hardening effect is dependent on the loading path and also the intensity of the loading.     

Finite element and experimental study on multiaxial fatigue analysis of rail clip failures

The rail clip fastening system is an important structural component of railway track systems providing flexibility and turnover resistance for running rails. High replacement frequency of fasteners was observed compared with other components because of fatigue failures of rail clips. In this study, implicit and explicit finite element (FE) models were developed for E‐clip and Fast‐clip with material and fatigue properties obtained from experimental testing. The fatigue loading experiments were conducted to determine the strain‐life relationship. The assessments of the fatigue damage and fatigue life were analysed using the FE results for the rail clip strain/stress components with the Fatemi‐Socie multiaxial fatigue criterion. A time‐efficient smallest enclosing circle algorithm was developed to search the critical plane orientation and the maximum shear strain amplitude for fatigue analysis. This work provides a method for FE and experimental study of multiaxial fatigue analysis of rail clip failures subjected to dynamic loading.     

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.     

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).     

Kurzrisswachstum bei mehrachsig nichtproportionaler Beanspruchung

Short fatigue crack growth under multiaxial nonproportional loading Initiation and short fatigue crack growth have been investigated under nonproportional cyclic loading. A critical plane approach based on fracture mechanics is used for modelling the fatigue process. A Paris‐type crack growth law, formulated using the effective cyclic J‐integral as crack driving force parameter, is integrated to give crack growth curves. Crack opening stresses and strains are calculated with approximation equations. Jiang's plasticity model is used to predict the stress‐strain path. The good agreement between model and real damage evolution is shown comparing experimentally determined crack growth curves, crack orientations, and life curves.     

A critical plane-strain energy density criterion for multiaxial low-cycle fatigue life under non-proportional loading

A series of multiaxial low-cycle fatigue experiments was performed on 45 steel under non-proportional loading. The present evaluations of multiaxial low-cycle fatigue life were systematically analysed. A combined energy density and critical plane concept is proposed that considers different failure mechanisms for a shear-type failure and a tensile-type failure, and from which different damage parameters for the critical plane-strain energy density are proposed. For tensile-type failures in material 45 steel and shear-type failures in material 42CrMo steel, the new damage parameters permit a good prediction for multiaxial low-cycle fatigue failure under non-proportional loading. The currently used critical plane models are a special and simple form of the new model.     

Online multiaxial fatigue damage evaluation method by real‐time cycle counting and energy‐based critical plane criterion

To realize online multiaxial fatigue damage assessment for the mechanical components in service, an online multiaxial cycle counting method is proposed coupled with the segment processing technique and Wang‐Brow's relative equivalent strain concept. Meanwhile, considering all the stress and strain components, which contribute to the fatigue damage on the critical plane, a multiaxial fatigue damage model without any weight coefficients is also proposed in an equivalent form of shear strain energy. Then, an online fatigue damage evaluation method for multiaxial random loading is developed by combining with the proposed damage model and online cycle counting method. The experimental results showed that the proposed online cycle counting method can be successfully applied to the calculation of multiaxial fatigue damage under random loading. Moreover, the proposed online multiaxial fatigue damage evaluation method can provide satisfactory predictions.     

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.     

Fatigue analysis of railway wheel using a multiaxial strain‐based critical‐plane index

A fatigue damage model to assess the development of subsurface fatigue cracks in railway wheels is presented in this paper. A 3‐dimensional finite element model (FEM) is constructed to simulate repeated cycles of contact loading between a railway wheel and a rail. The computational approach includes a hard‐contact over‐closure relationship and an elastoplastic material model with isotropic and kinematic hardening. Results from the simulation are used in a multiaxial critical‐plane fatigue damage analysis. The employed strain‐based critical‐plane fatigue damage approach is based on Fatemi‐Socie fatigue index that takes into account the non‐proportional and out‐of‐phase nature of the multiaxial state of stress occurs when a railway wheel rolls on a rail. It predicts fatigue‐induced micro‐crack nucleation at a depth of about 3.7 mm beneath the wheel tread, as well as the crack plane growth orientation which indicates the possible failure pattern. Additionally, the influence of various factors such as contribution of normal stresses, higher wheel load, and material model have been investigated.     

Fatigue life calculation by means of the cycle counting and spectral methods under multiaxial random loading

The paper contains a new algorithm for estimation of fatigue life in HCF regime under multiaxial random loading using spectral methods. Loading of Gaussian distribution and narrow‐ and broad‐band frequency spectra were assumed. Various characteristic states of multiaxial loading were considered. The equivalent stress history was determined with use of the failure criteria of multiaxial fatigue based on the critical plane. For determination of the critical plane position, the method of variance was applied. During simulation, the authors compared the results obtained by a spectral method in the frequency domain with those from the rain‐flow algorithm in the time domain. The paper also contains the results of fatigue tests for 18G2A structural steel subjected to bending and combined bending with torsion. The tests were performed in order to verify the proposed algorithms for determination of fatigue life. It has been shown that under multiaxial random loading results of fatigue life calculated according to the considered algorithms in frequency and time domains are well correlated with the results of experiments.