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.     

Prediction methods of fatigue critical point for notched components under multiaxial fatigue loading

Two methods based on local stress responses are proposed to locate fatigue critical point of metallic notched components under non‐proportional loading. The points on the notch edge maintain a state of uniaxial stress even when the far‐field fatigue loading is multiaxial. The point bearing the maximum stress amplitude is recognized as fatigue critical point under the condition of non‐mean stress; otherwise, the Goodman's empirical formula is adopted to amend mean stress effect prior to the determination of fatigue critical point. Furthermore, the uniaxial stress state can be treated as a special multiaxial stress state. The Susmel's fatigue damage parameter is employed to evaluate the fatigue damage of these points on the notch edge. Multiaxial fatigue tests on thin‐walled round tube notched specimens made of GH4169 nickel‐base alloy and 2297 aluminium‐lithium alloy are carried out to verify the two methods. The prediction results show that both the stress amplitude method and the Susmel's parameter method can accurately locate the fatigue critical point of metallic notched components under multiaxial fatigue loading.     

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.     

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.     

A coupled plasticity correction approach to estimating notch root strains under multiaxial cyclic loading

Based on pseudo strain approach, a coupled plasticity correction approach is proposed to calculate conveniently local strains for blunt notches under multiaxial cyclic loading. A detailed algorithm of the method is presented. Compared with conventional pseudo strain based approach, the proposed method does not require the plasticity model twice. The applicability of the method was evaluated by the measured notch root strains for a notched round shaft subjected to combined axial–torsion multiaxial cyclic loading. The results showed that the proposed method was acceptable and the calculated axial strain range was more accurate than the calculated shear strain range.     

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.     

A weight function-critical plane approach for low-cycle fatigue under variable amplitude multiaxial loading

Low‐cycle fatigue data of type 304 stainless steel obtained under axial‐torsional loading of variable amplitudes are analyzed using four multiaxial fatigue parameters: SWT, KBM, FS and LKN. Rainflow cycle counting and Morrow's plastic work interaction rule are used to calculate fatigue damage. The performance of a fatigue model is dependent on the fatigue parameter, the critical plane and the damage accumulation rule employed in the model. The conservatism and non‐conservatism of predicted lives are examined for some combinations of these variables. A new critical plane called the weight function‐critical plane is introduced for variable amplitude loading. This approach is found to improve the KBM‐based life predictions.     

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.     

A notch critical plane approach of multiaxial fatigue life prediction for metallic notched specimens

An approach based on the local stress response is proposed to locate the fatigue critical point for metallic blunt notched specimens under multiaxial fatigue loading. According to the stress analysis, both stress gradient and gradient of loading nonproportionality exist at notch root. The plane in the vicinity of the notch that passes through the fatigue critical point and experiences the maximum shear stress amplitude is defined as the critical plane for notch specimens (CPN). Furthermore, the Susmel's fatigue damage parameter is modified to assess fatigue life of notched components by combining CPN and the theory of critical distance (TCD). The multiaxial fatigue test of the thin‐walled round tube specimens made of Ni‐base alloy GH4169 is carried out to verify the above approaches. In addition, test data of two kinds of materials are collected. The results show that the maximum absolute error of the fatigue critical point is 9.6° and the majority of the predicted life falls within the three‐time scatter band.     

A new path‐dependent multiaxial fatigue model for metals under different paths

Based on the critical plane approach, a new path‐dependent multiaxial fatigue model in low‐cycle fatigue is proposed. The proposed model includes damage contribution from four sources: the normal strain amplitude, the shear strain amplitude on the critical plane, the hydrostatic mean strain and a new path‐dependent factor. The effect of mean strain is considered by the hydrostatic mean strain. The experimental data of 11 kinds of materials are used to demonstrate the effectiveness of this new model under both zero and non‐zero mean strain multiaxial loading path.     

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.     

FATIGUE LIFE PREDICTION OF NOTCHED COMPOSITE COMPONENTS

Abstract— The local stress/strain approach has been used to predict the fatigue lives of notched composite components. The method was based on a microstress analysis and the application of a multiaxial fatigue parameter incorporating the alternating strain components on the critical plane. This parameter was able to correlate the fatigue lives obtained under a variety of multiaxial loading and geometrical configurations, enabling a generalized fatigue life curve to be determined on the basis of limited experimental data.
The ability of the multiaxial fatigue parameter to relate the fatigue behaviour of composites was illustrated by predicting the locations of crack initiation sites in a unidirectional silicon carbide fibre reinforced titanium plate containing a circular hole tested under constant amplitude cyclic loading. The same approach was also successfully employed to predict the fatigue lives of graphite reinforced epoxy composite tubes with circular holes tested under several combinations of cyclic tension and torsion.     

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.     

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 high-cycle fatigue life model for variable amplitude multiaxial loading

A high‐cycle fatigue life model for structures subjected to variable amplitude multiaxial loading is presented in this paper. It treats any kind of repeated blocks of variable amplitude multiaxial loading without using a cycle counting method. This model based on a mesoscopic approach is characterized by the following features: (i) the choice of a damage factor related to the accumulated mesoscopic plastic strain per stabilised cycle; (ii) the use of a mesoscopic mechanical behaviour taking into account the fatigue mechanisms such as plasticity and void growth. This behaviour is a von Mises elastoplastic model with linear kinematic hardening and hydrostatic stress dependent yield stress. The fatigue life model has six parameters identified with one SN curve and two fatigue limits. In‐phase and out‐of‐phase experimental tests from the literature are simulated. The predicted fatigue lives are compared to experimental ones.     

Proportional/nonproportional constant/variable amplitude multiaxial notch fatigue: cyclic plasticity,non‐zero mean stresses,and critical distance/plane

This paper deals with the formulation and experimental validation of a novel fatigue lifetime estimation technique suitable for assessing the extent of damage in notched metallic materials subjected to in‐service proportional/nonproportional constant/variable amplitude multiaxial load histories. The methodology being formulated makes use of the Modified Manson‐Coffin Curve Method, the Shear Strain–Maximum Variance Method, and the elasto‐plastic Theory of Critical Distances, with the latter theory being applied in the form of the Point Method. The accuracy and reliability of our novel fatigue lifetime estimation technique were checked against a large number of experimental results we generated by testing, under proportional/nonproportional constant/variable amplitude axial‐torsional loading, V‐notched cylindrical specimens made of unalloyed medium‐carbon steel En8 (080M40). Specific experimental trials were run to investigate also the effect of non‐zero mean stresses as well as of different frequencies between the axial and torsional stress/strain components. This systematic validation exercise allowed us to demonstrate that our novel multiaxial fatigue assessment methodology is remarkably accurate, with the estimates falling within an error factor of 2. By modelling the cyclic elasto‐plastic behaviour of metals explicitly, the design methodology being formulated and validated in the present paper offers a complete solution to the problem of estimating multiaxial fatigue lifetime of notched metallic materials, with this holding true independently of sharpness of the stress/strain raiser and complexity of the load history.     

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.     

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.     

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).     

Multiaxial fatigue and life prediction of elastomeric components

Elastomeric components have wide usage in many industries. The typical service loading for most of these components is variable amplitude and multiaxial. In this study a general methodology for life prediction of elastomeric components under these typical loading conditions was developed and illustrated for a passenger vehicle cradle mount. Crack initiation life prediction was performed using different damage criteria. The methodology was validated with component testing under different loading conditions including constant and variable amplitude in-phase and out-of-phase axial–torsion experiments. The optimum method for crack initiation life prediction for complex multiaxial variable amplitude loading was found to be a critical plane approach based on maximum normal strain plane and damage quantification by cracking energy density on that plane. Rainflow cycle counting method and Miner’s linear damage rule were used for predicting fatigue life under variable amplitude loadings. The fracture mechanics approach was used for total fatigue life prediction of the component based on specimen crack growth data and FE simulation results. Total fatigue life prediction results showed good agreement with experiments for all of the loading conditions considered.