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 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.     

Comparison of HCF life prediction methods based on different critical planes under multiaxial variable amplitude loading

High‐cycle fatigue life prediction methods based on different critical planes, including the maximum shear stress (MSS) plane, the weighted average shear stress plane and the Maximum Variance shear stress plane, are compared by two multiaxial cycle counting methods, i.e. the main and auxiliary channels (MAC) counting and the relative equivalent stress counting. A modified damage model is used to calculate the multiaxial fatigue damage. Compared with the experimental lives for 7075‐T651 aluminium alloy, the predicted results show that the MSS method together with MAC counting is suitable for the multiaxial fatigue life prediction.     

Evaluation of fatigue life for titanium alloy TC4 under variable amplitude multiaxial loading

Fatigue tests under variable amplitude multiaxial loading were conducted on titanium alloy TC4 tubular specimens. A method to estimate the fatigue life under variable amplitude multiaxial loading has been proposed. Multiaxial fatigue parameter based on Wu–Hu–Song approach and rainflow cycle counting and Miner–Palmgren rule were applied in this method. The capability of fatigue life prediction for the proposed method was checked against the test data of TC4 alloy under variable amplitude multiaxial loading. The prediction results are all within a factor of two scatter band of the test results.     

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.     

Fatigue assessment of metallic components under uniaxial and multiaxial variable amplitude loading

In the present paper, the fatigue lifetime of metallic structural components subjected to variable amplitude loading is evaluated by applying 2 different multiaxial high‐cycle fatigue criteria. Such criteria, proposed by some of the present authors, are based on the critical plane approach and aim at reducing a given multiaxial stress state to an equivalent uniaxial stress condition. In particular, the procedure employed by both criteria consists of the following 3 steps: (1) definition of the critical plane; (2) counting of loading cycles; and (3) estimation of fatigue damage. Finally, the previous criteria are validated by comparing the theoretical results with experimental data related to smooth metallic specimens subjected to uniaxial and multiaxial variable amplitude loading.     

An improved multiaxial high‐cycle fatigue criterion based on critical plane approach

Several groups of fatigue damage parameters are discussed and then an improved multiaxial high‐cycle fatigue criterion based on critical plane defined by the plane of maximum shear stress range is presented in this paper. A compromising solution to consider the mean normal stress acting on the critical plane is also proposed. The new fatigue criterion extends the range of metallic materials which is valid for the ratio 1.25 < f_?1/t_?1 < 2. The predictions based on the presented model show a good agreement with test data.     

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.     

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.     

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.     

Fatigue analysis of ductile and brittle behaving steels under variable amplitude multiaxial loading

The analysis of fatigue behavior under multiaxial variable amplitude stress states, despite its wide applicability, has not been fully studied. Issues such as varying degrees of nonproportionality of the load history, cycle counting, damage accumulation, failure behavior of the material, and mean stress fluctuations which can significantly affect the results of these analyses have not been well understood. In this study, a methodology for the analysis of fatigue behavior under multiaxial variable amplitude loading conditions is employed which accounts for the aforementioned issues. At its core, the applied methodology uses critical plane analysis based on the failure behavior of each material to assess the fatigue damage. In order to evaluate the performance of the analysis method, axial, torsional, and combined axial‐torsional variable amplitude tests were performed on one ductile and one brittle behaving steel. The applied methodology resulted in close estimation of the experimental fatigue life for both ductile and brittle behaving steels.     

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.     

Use of an energy‐based/critical plane model to assess fatigue life under low‐cycle multiaxial cycles

For engineering components subjected to multiaxial loading, fatigue life prediction is crucial for guaranteeing their structural security and economic feasibility. In this respect, energy‐based models, integrating the stress and strain components, are widely used because of their availability in fatigue prediction. Through employing the plastic strain energy concept and critical plane approach, a new energy‐based model is proposed in this paper to evaluate the low‐cycle fatigue life, in which the critical plane is defined as the maximum damage plane. In the proposed model, a newly defined NP factor κ^*  is used to quantify the nonproportional (NP) effect so that the damage parameter can be conveniently calculated. Moreover, a simple estimation method of weight coefficient is developed, which can reflect different contributions of shear and normal plastic strain energy on total fatigue damage. Experimental data of 10 kinds of materials are employed to assess the effectiveness of this model as well as three other energy‐based models.     

A method for assessing critical plane‐based multiaxial fatigue damage models

Fatigue failure is a complex phenomenon. Therefore, development of a fatigue damage model that considers all associated complexities resulting from the application of different cyclic loading types, geometries, materials, and environmental conditions is a challenging task. Nevertheless, fatigue damage models such as critical plane‐based models are popular because of their capability to estimate life mostly within ±2 and ±3 factors of life for smooth specimens. In this study, a method is proposed for assessing the fatigue life estimation capability of different critical plane‐based models. In this method, a subroutine was developed and used to search for best estimated life regardless of critical plane assumption. Therefore, different fatigue damage models were evaluated at all possible planes to search for the best life. Smith‐Watson‐Topper (normal strain‐based), Fatemi‐Socie (shear strain‐based), and Jahed‐Varvani (total strain energy density‐based) models are compared by using the proposed assessment method. The assessment is done on smooth specimen level by using the experimental multiaxial fatigue data of 3 alloys, namely, AZ31B and AZ61A extruded magnesium alloys and S460N structural steel alloy. Using the proposed assessment method, it was found that the examined models may not be able to reproduce the experimental lives even if they were evaluated at all physical planes.     

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.     

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.     

Criteria of multiaxial random fatigue based on stress,strain and energy parameters of damage in the critical plane

In this paper generalized criteria of multiaxial random fatigue based on stress, strain and strain energy density parameters in the critical plane have been discussed. The proposed criteria reduce multiaxial state of stress to the equivalent uniaxial tension–compression or alternating bending. Relations between the coefficients occurring in the considered criteria have been derived. Thus, it is possible to take into account fatigue properties of materials under simple loading states during determination of the multiaxial fatigue life. Presented models have successfully correlated fatigue lives of cast iron GGG40 and steel 18G2A specimens under constant amplitude in‐phase and out‐of‐phase loadings including different frequencies.     

Investigation of the multiaxial fatigue behaviour of 316 stainless steel based on critical plane method

In this work, the multiaxial behaviour of 316 stainless steel is studied under the lens of critical plane approach. A series of experiments were developed on dog bone–shaped hollow cylindrical specimens made of type 316 stainless steel. Five different loading conditions were assessed with (a) only tensile axial stress, (b) only hoop stress, (c) combination of axial and hoop stresses with square shape, (d) combination of tensile axial and hoop stresses with L shape, and (e) combination of compressive axial and hoop stresses with L shape. The fatigue analysis is performed with four different critical plane theories, namely, Wang‐Brown, Fatemi‐Socie, Liu I, and Liu II. The efficiency of all four theories is studied in terms of the accuracy of their life predictions and crack failure plane angle. The best fatigue life predictions were obtained with Liu II model, and the best predictions of the failure plane were obtained with Liu I model.     

A structural stress‐based critical plane method for multiaxial fatigue life estimation in welded joints

This paper aims at proposing a new fatigue life estimation model that is preferably adapted to welded joints subjected to multiaxial loading. First, a mesh‐size insensitive structural stress is defined that enables to characterize the stress concentration effect appropriately. Second, the multiaxial stress state and loading path influence are taken into account in the lifetime prediction model by adopting a suitable critical plane method, originally proposed by Carpinteri and co‐authors. Experimental verification is conducted for a given welded joint geometry under different loading conditions, including uniaxial, torsional and multiaxial loads. The reliability and effectiveness of the new method are validated through substantive fatigue testing data.     

A review of multiaxial fatigue criteria for random variable amplitude loads

Nowadays, the estimation of fatigue life under multiaxial random loading is still an extremely complex task. In this paper, a comprehensive review of the multiaxial random fatigue criteria available in the literature is presented. Such a review is mainly devoted to stress‐based criteria for the evaluation of fatigue life in high‐cycle regime. Time and frequency domain approaches are examined. The focus of this paper is related to uniform stress/strain distribution, but also the effect of stress/strain gradient is tangentially addressed. More than 200 references are cited.