A review of multiaxial low-cycle fatigue criteria for life prediction of metals

Abstract

Most of real-world structural components that undergo cyclic loading feature multiaxial fatigue. When the cyclic loading involves also significant plastic deformation, multiaxial low-cycle fatigue takes place. Applications where multiaxial low-cycle fatigue can be observed very often involve metal components. To predict their lives multiple criteria and models have been proposed, but their development has not followed a regular path. Multiple reviews are available in literature. However, many of them are outdated, they often employ different classification methods to categorize available criteria, many focus on specific families of criteria, and others do not include sufficient theoretical background. Moreover, none of the available reviews is based on a systematic literature search method. As a result, approaching the topic can result arduous and chaotic, especially for first timers. This work aims at providing a clear, comprehensive, and definitive review of available criteria for multiaxial low-cycle fatigue. First, the basic theoretical background is explained. Secondly, a systematic approach is described and employed to identify all major currently available criteria. Then, they are classified and commentary about different classification styles that can be found in literature is added. Eventually they are described, together with their latest proposed variations. In this way this review can be employed as a guiding reference, especially for engineers approaching the topic for the first time.

Keywords

Multiaxial fatigue fracture mechanics low-cycle fatigue criteria model crack initiation failure prediction

Introduction

Fatigue is the set of phenomena that cause a structural element which experiences cyclic loading to feature a lower strength than that obtained through monotonic tests. The reason why such phenomena were assigned w the name “fatigue” is due to their repetitive nature, which cycle after cycle “tires” the material, which eventually fails under loads that in static conditions would be withstood. The phenomenological explanation lays in the mechanism of crack nucleation and propagation. Indeed, fatigue occurs in microscopic areas of undesired high stress concentration, such as surface defects, where the intensification of stresses is enough to cause a localized microscopic plastic deformation, even if the applied stresses are in the elastic regime (Bolotin, 1999). Such zone of microscopic plastic deformation takes the name of crack tip, and if loads are cyclically repeated it can propagate deep in the material, leading to the growth of an entire crack. This often happens without notice, until suddenly the resistant area is no longer enough to withstand the loads and failure abruptly occurs (Bolotin, 1999). For this reason, components must be carefully designed taking fatigue into account, as due to its inconspicuous nature it could lead to catastrophic consequences. Besides that, fatigue also represents one of the most common, if not the most common, causes of mechanical failure (Campbell, 2012), as it could take place in any kind of application where dynamic loading conditions can be observed.

Fatigue phenomena can be classified according to the number of cycles that are required to lead a material to failure (fatigue life). Two main types of mechanical fatigue are thus identified: High-Cycle Fatigue (HCF) and Low-Cycle Fatigue (LCF). HCF takes place when the cyclic stresses never exceed the material yield strength (i.e., when the component operates always in the elastic region). It is characterized by a number of cycles that canonically ranges from 10⁴ up to 10⁷ cycles (Farhat, 2021), especially in the case of metals, which represent the family of materials that are most often subject to fatigue in industrial applications. Conversely, LCF takes place when the cyclic stresses exceed the material yield strength, i.e., when they induce plastic deformation not only at the crack tip, but in macroscopic regions (Xin, 2013). As a consequence, given a material, LCF conditions result to be more harmful than HCF conditions and the LCF life is shorter than the HCF life. LCF lives of metals usually extend from 10² up to 10³–10⁴ cycles (Concli et al., 2022, 2023; Dufailly and Lemaitre, 1995). Further categories can be identified, although sometimes their ranges overlap, depending on the literature reference. As far as fatigue lives higher than 10⁷ cycles are concerned, Very High-Cycle Fatigue (VHCF) takes place for rages from 10⁷ up to 10¹⁰ cycles, while Ultra High-Cycle Fatigue (UHCF) takes places for ranges from 10⁹ to 10¹² cycles (Wang et al., 2012). With regard to fatigue lives lower than 10² cycles, in literature both Very Low-cycle Fatigue (VLCF) and Ultra Low-cycle Fatigue (ULCF) are definitions employed to refer to fatigue operating in the range between 10° and 10² cycles (Dufailly and Lemaitre, 1995; Pereira et al., 2014). Fatigue phenomena can be classified according to the type of state of stress too. If it is characterized by stresses acting along one single axis, uniaxial fatigue takes place, obtaining a configuration whose analysis is simple and straightforward. Conversely, if multiple axes are simultaneously loaded, multiaxial fatigue takes place, obtaining a configuration whose analysis can result extremely challenging (Socie and Marquis, 1999). MLCF identifies the set of LCF phenomena where stresses act on multiple axis. This means that loads producing plastic strains and that vary over time can be observed acting in different directions during the same load cycle. As a result, the number of parameters that have to be taken into account to predict the life of components subject to MLCF is high. The main ones are: value of the amplitude of the applied strains (or stresses), variability over time of the amplitude of the applied strains (or stresses), loading frequency, elasto-plastic behavior of the material, surface finishing, shape, and size of the structural element, value of the mean stress, proportionality of the applied loads, phase difference between the applied loads, shape of the loading path. All these parameters are related to mechanical loads only. If also thermal or chemical phenomena take place, then the fatigue analysis becomes even more complex. Nevertheless, despite the effort that it requires, being able to predict the lives of components that experience MLCF is of vital importance, as most of real-world applications involve multiaxial states of stresses (Shamsaei and Fatemi, 2009). Note that even under uniaxial loading, a multiaxial state of stress could be obtained anyway, as a result of any geometrical discontinuity (e.g. notches), or due to the presence of a residual stresses.

With regard to applications where MLCF can be observed, they cover a wide and diversified spectrum, with almost all of them involving metal structural components. As far as the automotive sector is concerned, failure provoked by MLCF occur in internal combustion engines due to the high thermal stresses which take place every time the engine is turned on, employed enough to warm it up, switched off and cooled down. The areas that are typically affected the most are the valve bridge areas of cylinder heads, due to the overlapping of press fit stresses and maximum temperature cyclic loading (Fontanesi and Giacopini, 2013), and the exhaust manifold (Xin, 2013; Xue et al., 2007). Similarly to internal combustion engines, also turbine engines experience MLCF as a consequence of turn on-operation-turn off cycles. The loads that produce it can be thermal loads and centrifugal loads (Dileep et al., 2015; Hu and Wang, 2013). Both of them are applied every time the engine is employed and are removed when it is switched off (in the first case because the engine cools down, and in the latter case because the blades slow down). Centrifugal loads mainly affect blades (Dileep et al., 2015), while thermal loads mainly affect the impeller (Hu and Wang, 2013). In the construction sector, MLCF can take place during earthquakes, whose waves can cause cyclic plastic deformation of metal components that comprise buildings. The most sensitive components are welded joints connecting beams and other structural elements (Huang et al., 2020). Other components that undergo MLCF in buildings are elements which are designed to plastically deform during earthquakes, in order to dissipate the energy transmitted by the earthquake (Nagel et al., 2017). The same elements are employed during similar events, such as hurricane winds and tidal waves (de Castro e Sousa and Nussbaumer, 2019). A fourth field of application of MLCF is represented by piping components of chemical, power and nuclear plants. In this case, MLCF analysis is carried out because water or other fluids flowing at high pressure and high temperature can cause cyclic inelastic strain (Ahn et al., 2002). The components that are analyzed the most are elbow pipes, as they may be subject to wall thinning, due to flow accelerated corrosion (Ahn et al., 2002; Takahashi et al., 2009). In addition, in the case of nuclear power plants also potential seismic loads have to be taken into account (Takahashi et al., 2009; 2014), since for such delicate applications failures could lead to catastrophic consequences and must be avoided at any cost. MLCF failures can be observed also in the field of oil extraction, where steel tubing is run into wells. During this operation, the tubing experiences high cyclic straightening-bending-straitening loads applied by above-surface deployment hardware, which uncoils and coils it again around reels for transportation and storage (Rolovic and Tipton, 1999; Ryu et al., 2018). Due to the large strains that are involved (up to 3%), the service lives of these components can be even as low as less than 100 cycles (Macha and Sonsino, 1999). Other examples of applications involving MLCF are mooring chains and welds of offshore structures (Zarandi and Skallerud, 2020), as well as hull plates of large-scale ships (Deng et al., 2022), which could be cyclically deformed due to severe sea conditions.

Starting from the 1950s multiple criteria have been developed in order to characterize multiaxial high cycle fatigue, especially to predict life under such loads. Starting from the 70 s, the analysis has been extended to MLCF. In this regard one of the first surveys ever produced dates back to 1981, with the work of Garud (Garud et al., 1981), followed by the work of Beaver (Beaver, 1985) in 1985, which however is limited to biaxial fatigue. Two complete reviews of the available criteria for multiaxial fatigue, both in the high-cycle and in the low-cycle regime, were carried out in 1995 and 1996, by Radhakrishnan (Radhakrishnan, 1995) and by You and Lee (RBong-Ryul and Soon-Bok, 1996), respectively. Similar but more circumscribed and at the same time detailed works were performed by Macha and Sonsino (Macha and Sonsino, 1999) in 1999, who reviewed multiaxial fatigue criteria focusing on those based on energy-related parameters and by Karloczuk and Macha (Karolczuk and Macha, 2005) in 2005, who focused on those based on the critical plane theory. In 2011 Papuga (Papuga, 2011) produced a survey listing and evaluating the soundness of seventeen main multiaxial fatigue life prediction models. The same year, Fatemi and Shamsaei (Fatemi and Shamsaei, 2011) carried out an overview paper about multiaxial fatigue, discussing multiple aspects, among which also some of the available criteria at the time. Even though the reviews listed up to now are of great scientific relevance and interest, and even though they discuss criteria for the life prediction of metals subject to multiaxial fatigue, none of them is devoted only to those that are employed specifically for MLCF. As a result, they mention only some of them, while missing many others. Multiple MLCF life prediction criteria are missing also due to the age of the works, with the latest one missing more than one decade of recent advancements in the field. Similarly, also more recent reviews treat the topic of multiaxial fatigue prediction criteria, without mentioning all of those related to MLCF or without describing them in enough detail. Foti et al. (Foti et al., 2023) developed a thorough and all-round review of the failure mechanisms of multiaxial fatigue of additively manufactured metallic components, treating a remarkable number of aspects, though missing various MLCF life prediction criteria. Luo et al. (Luo et al., 2017), focused on the damage parameters suitable for non-proportional loading, and Carpinteri et al. (Carpinteri et al., 2017), focused on those suitable for variable amplitude loading, with both works addressing criteria both for the low and the high-cycle fatigue regime. The work of Zhong and Lu (Zhong and Lu, 2020), reviews many of the multiaxial fatigue prediction criteria based on energy parameters, including those based on the critical plane theory that contain energy parameters, which are generally suitable for MLCF. Eventually, also the review of Deng et al. was carried out for multiaxial fatigue in general, but among those available in literature it is probably the one that mentions and describes the highest number of MLCF life prediction criteria.

In light of the absence of a survey of the available criteria for multiaxial fatigue specifically in the low-cycle regime, in light of the large variety of applications where MLCF can be encountered and of the high relevance that such applications have in the industrial sector, investigating the topic of MLCF of metals results of the utmost scientific interest and importance. As a result, the objective that is set for this work is to investigate and assess the current state of the art regarding available MLCF criteria and models for metals, producing a comprehensive, thorough, and up to date overview which can assist scholars when approaching such challenging and captivating subject. The paper will be organized as follows. In Section ‘Theoretical background’ the basic theoretical background required to understand the characteristics of MLCF criteria will be provided. In Section ‘MLCF criteria’ the systematic literature search strategy that has been employed to identify the currently available criteria is described. Furthermore, criteria are categorized and thoroughly described, together with their variants proposed in the years following their introduction. Eventually, in the final section conclusions will be drawn.

Theoretical background

After the introductive overview about MLCF discussed above, some further notions are discussed in this section, with the goal of providing readers with the required knowledge to better understand the working principles of MLCF criteria, which will be explained in Section ‘MLCF criteria’. The fundamentals of LCF will be explained at first, then those of multiaxial states of stress and strain will follow, collecting all those notions at the basis of the definition of MLCF. Eventually, additional useful considerations about loading conditions will be discussed.

Low-cycle fatigue

As previously mentioned, LCF is that type of fatigue than occurs when the loads withstood by the material exceed its yield strength, producing macroscopic plastic deformations that are cyclically repeated. Materials typically respond to plastic deformation in three different ways: with a cyclic strain hardening behavior, with a cyclic strain softening behavior or remaining stable (Campbell, 2012). Different behaviors produce different curves (called hysteresis loops) on the stress-strain diagram. Figure 1 displays them for a case where it is supposed to work in uniaxial tension-compression and strain control conditions (i.e. controlling the values of the applied upper and lower strains). In Figure 1(a) the behavior of a strain-hardening material is represented, showing that cycle after cycle the stresses that are required to produce the imposed deformations gradually increase, i.e., the material gains hardness. This behavior is due to the disordered orientations characterizing the grains and sub-grains of strain hardening materials. When dislocation move due to plastic deformation they twist and interact with each other, making their displacement more difficult. Such disordered orientations cause the twisting and interaction of dislocations when they move due to plastic deformation dislocations are moved due to plastic deformation (Zhang, 2013). Figure 1(b) displays the behavior of a strain-softening material, showing that cycle after cycle the stresses that are required to obtain the imposed deformations decrease, i.e., the material loses hardness. This behavior is due to the fact that strain-softening materials feature intrinsic dislocation intertwists and obstacles in the grains, which are removed by plastic deformation, making dislocation displacement easier (Zhang, 2013). Such difference in the response to plastic deformation is the reason why LCF tests are typically carried out in strain-controlled conditions, as opposed to HCF tests, which are typically carried out in stress-controlled conditions (Concli et al., 2022; 2023). Yet, independently of the material behavior, after a certain amount of cycles materials typically reach a stabilization, showing no further hardening or softening, and producing always the same hysteresis loop on the stress strain diagram. This is often referred as the cyclic hysteresis loop. The areas defined by a hysteresis loop on the stress-strain diagram can be related to the energy employed in the deformation process, as shown in Figure 2. ΔW^(P) is the energy related to the plastic strain, and corresponds to the area inside the hysteresis loop, while ΔW^(E+) and ΔW^(E−) are the energies related to the positive and the negative elastic strain respectively.

For a given material, by applying different values of deformations (or loads) it is possible to obtain different stabilized cycles. The vertexes of every stabilized hysteresis curve can be extracted and put together, obtaining a curve of on the stress-strain diagram (Figure 3) that can be described through the Ramberg-Osgood equation (39):

ε = \frac{σ}{E} + K {(\frac{σ}{E})}^{n}

(1)

where ε is the strain, σ is the stress, E is the Young’s modulus, and K and n are constants that depend on the material under analysis. By doing the opposite procedure, through the Ramberg-Osgood curve of a material and the Masing hypothesis (which assumes that the material features the same behaviour both in tension and in compression conditions (Masing, 1926)) it is possible to reconstruct the stabilized hysteresis loop for a given stress-strain couple. Reconstructing the Ramberg-Osgood curve of a material is the first step in characterizing its LCF behaviour. The second one is to identify the relationship between the magnitude of applied strains and the number of cycles characterizing the fatigue life. In the case of high-strain LCF conditions, where the effect of elastic strain on the fatigue life is negligible, such relationship is described through the Coffin-Manson equation (41):

ε_{p} = ε_{f}' {(2 N_{f})}^{c}

(2)

where ε_p is the applied plastic strain,

ε'_{f}

is the fatigue ductility coefficient, N_f is the fatigue cycles to failure, and c is the fatigue ductility exponent. Conversely, in case of HCF, tests are performed in the elastic region and in stress-controlled conditions, and the relationship that is found is that between applied stresses and fatigue life, through the Basquin equation (42):

σ_{e} = σ'_{f} {(2 N_{f})}^{b}

(3)

where σ_e is the applied elastic stress amplitude,

σ'_{f}

is the fatigue strength coefficient, and b is the fatigue strength exponent. For LCF conditions where both elastic (ε_e) and plastic (ε_p) strains are to be taken into account (the vast majority of LCF applications), the Basquin and the Coffin-Manson equations are combined to define the relationship between total strain amplitude (ε_a) and fatigue life:

ε_{a} = ε_{e} + ε_{p} = \frac{σ_{e}}{E} + ε_{p} = \frac{σ'_{f}}{E} {(2 N_{f})}^{b} + ε_{f}' {(2 N_{f})}^{c}

(4)

As mentioned above, this relationship is valid for uniaxial conditions only, and in the case of MLCF it cannot be employed in its original form. However, what MLCF criteria often do is to calculate an equivalent strain, which can be used instead of ε_a in a strain-life equation similar to the Basquin-Coffin-Manson one.

Multiaxial state of stress and strain

Given a structural component subject to an undefined load, the state of stress and strain in any point can be described in a Cartesian system through tensors comprised of nine stress and nine strain components respectively:

[σ] = [\begin{array}{l} σ_{x} & τ_{x y} & τ_{x z} \\ τ_{y x} & σ_{y} & τ_{y z} \\ τ_{z x} & τ_{z y} & σ_{z} \end{array}]

(5)

[ε] = [\begin{array}{l} ε_{x} & \frac{1}{2} γ_{x y} & \frac{1}{2} γ_{x z} \\ \frac{1}{2} γ_{y x} & ε_{y} & \frac{1}{2} γ_{y z} \\ \frac{1}{2} γ_{z x} & \frac{1}{2} γ_{z y} & ε_{z} \end{array}]

(6)

In Figure 4 it is represented how the components of the stress tensor are distributed on an infinitesimal element taken at the point under analysis. Note that to grant equilibrium, the shear stresses τ_xy = τ_yx, τ_xz = τ_zx and τ_yz = τ_zy, as well as the shear strains γ_xy = γ_yx, γ_xz = γ_zx and γ_yz = γ_zy, meaning that the tensors are symmetrical. As a result, from this point on, τ_xy, τ_xz, τ_yz, γ_xy, γ_xz, γ_yz will be employed instead of τ_yx, τ_zx, τ_zy, γ_yx, γ_zx, γ_zy respectively (also the contrary is possible). In addition, it follows also that only six components are required to define the stress and the strain tensor, σ_x, σ_y, σ_z, τ_xy, τ_xz, τ_yz and ε_x, ε_y, ε_z, γ_xy, γ_xz, γ_yz respectively. Given a state of stress and strain, there are infinite couples of stress and strain tensors that describe them, each one with different values of their components, depending on the reference system that is considered. Yet, they all represent the same state of stress and strain, only from different point of views. In uniaxial fatigue the magnitude of the stresses changes over time, but their direction is always aligned with one axis only. This means that there will be a reference system (the one with the X axis parallel to the stresses direction) where σ′_x is the only stress tensor component that is not zero. The same can be observed for the Y and the Z direction, producing stress tensors where σ′_y and σ′_z are the only components that are not zero respectively. This does not apply to the strain tensor, due to the effect of the Poisson ratio (ν), which, even if one of the reference system axes is aligned with the stresses direction, produces a secondary deformation in the other two directions. In the case of multiaxial fatigue (and therefore in the case of MLCF as well), the stresses inside the material change their magnitude over time and their directions are aligned with two or more different axes. Moreover, it might be that also the stresses direction can change over time. This means that under multiaxial fatigue, during the life of the component the stress tensor features multiple components that are different from zero and whose value changes over time.

Out of phase, non-proportional and variable amplitude loading

How multiaxial loading is applied strongly influences the lives of components. The same cyclic loads applied to the same component in different ways may produce lives that are completely different. For this reason, it is important to describe the main loading characteristics and how they influence the complexity of MLCF life analysis. Firstly, when multiple cyclic loads are combined, it is important to evaluate the phase angle between them. If cyclic loads are applied so that they vary synchronously, reaching the highest and the lowest values contemporarily, then they are in phase. Vice versa, if they do not vary synchronously, they are out of phase. Figure 5 features two sinusoidal loads, one in phase (Figure 5(a)) and one out of phase (Figure 5(b)).

When loads are out of phase, then in most cases they are also non-proportional. As anticipated above, if loads are applied in a way that the direction of the principal stresses and their ratio do not change over time, then the loading is said to be proportional. Conversely, if the loads are applied so that the principal stress directions, or ratios, or both, change, then the loading is said to be non-proportional. As a result, if two or more loads are out of phase, then typically they are also non-proportional. However, it is not said that if two loads are in phase (and synchronous) then they are necessarily proportional. For example, in the case depicted by Figure 6, the two loads are in phase, but they are not proportional. This is because even though they reach highest and lowest values contemporarily, one load keeps its amplitude constant, while the other one varies it. This means that the ratio between the two stresses (and therefore between the principal stresses) changes over time, making the load non-proportional. Indeed, what can change over time when dealing with multiaxial fatigue is also the amplitude of the applied loads. Together with non-proportionality and out of phase loading, variable amplitude is a condition that further complicates the characterization of the states of stress and strains of structural components under multiaxial loading and the variability over time of the P_n vector. Under non-proportional loading, some materials feature additional cyclic hardening that is not observed in proportional loading conditions (Socie and Marquis, 1999). Such phenomenon in called non-proportional hardening and in these cases the Ramberg-Osgood equation is no longer valid. Moreover, the lives of materials are often much shorter that under equivalent proportional loading (Zhao et al., 2022). For this reason, it is important to evaluate also the hardening behavior of the material under analysis. Typically, the maximum additional hardening is obtained when 90 degrees out of phase loading is applied (Kanazawa et al., 1979).

In addition to complicate loading conditions, in case of LCF the effect of plastic deformation has to be taken into account as well, making MLCF analysis a challenge that not rarely reaches extreme complicacy. Multiple models, each one with a different goal, are required. Broadly speaking, the main ones are:

A constitutive model of the material under analysis. Constitutive models define the response of materials to the mechanical and/or thermal conditions they experience. In MLCF analysis they play a fundamental role as not only the elastic, but also the plastic deformation response it to be described. In case of non-proportional loading, models that analyze non-proportional cyclic hardening should be included;

Damage criteria, which are reviewed in this work, which have the task to process the stresses and strains (which are bounded by the constitutive model that is employed) to assess the damage that the material experiences after a cycle;

Damage accumulation models, which define the rules to use to determine how damage accumulates and sums up cycle after cycle. Miner’s rule is one of the most known damage accumulation models (Miner, 1945);

Cycle counting models, which are particularly useful for variable amplitude loading. They are methodologies to count and process the cycles corresponding to each of the loads applied. The cycle counting model that is probably most employed in the academic and industrial sectors is the rain-flow counting (Standard, 2011).

MLCF criteria

This section represents the core of this work. It is divided into three parts. The first one describes the literature search strategy that has been employed to identify all the major criteria for MLCF life prediction that are available in literature. The second one describes how currently available criteria have been classified in this work. According to the classification that is made, the third one presents all major currently available criteria for MLCF, together with their variants proposed in the years following their introduction.

Systematic literature search

The strategy that has been employed to identify the literature mentioning all currently available MLCF criteria is based on a systematic approach. Unlike traditional (non-systematic) reviews, systematic reviews grant results which are replicable, objective and are not biased by the authors past experience on the topic. The following steps are applied (Figure 7):

the scientific database to be employed for researching papers concerning MLCF criteria and models is chosen. Scopus has been selected by the authors, as they evaluated it as the most reliable, recognized and accessible database available in the academic world.

the search terms to be employed for researching papers concerning MLCF criteria and models for metals are defined. They are combined through Boolean operators, obtaining the string that is entered in Scopus: (TITLE-ABS-KEY (“low cycle fatigue” OR “lcf” OR “low-cycle fatigue” OR “low-cycle-fatigue”) AND TITLE-ABS-KEY (“predict*” OR “criteri*” OR “prevision*”) AND TITLE-ABS-KEY (“model” OR “fem” OR “finite element” OR “fea”) AND TITLE-ABS-KEY (“metal*” OR “alloy*” OR “alum*” OR “titanium” OR “inconel” OR “steel”) AND TITLE-ABS-KEY (“multiaxial”)). The output generated by Scopus at the date of search (23.02.2023) is comprised of 172 scientific papers;

the results of the Scopus search undergo a first screening: papers in languages other than English are excluded. No discriminating criterion is set on the date of publication, as the first MLCF criteria have been proposed back in the 70’s. The remaining papers after the first screening are 124;

the results of the first screening undergo a second screening: after reading the abstracts, papers that are not pertinent to the research scope are excluded. The remaining papers after the first screening are 109;

Multiaxial fatigue criteria classification

Classifying the multiaxial fatigue criteria that have been proposed up to now is no easy task at all. Different authors provide different classifications, resulting in no uniform and standard subdivision available in literature. As a consequence, different references assign the same criteria to different subcategories. This in the case for example of criteria that are based on the theory of critical plane and that exploit energy-related parameters, which sometimes are classified as energy criteria (Macha and Sonsino, 1999; Yu et al., 2017), while in other cases are classified as critical plane criteria (Li et al., 2011). Besides, multiple references employ too few and wide families to provide a clear understanding, especially for engineers approaching the subject for the first time. This is the case when criteria are subdivided according to the physical nature of their failure parameter into stress-based criteria, strain-based criteria and energy-based criteria, which is a commonly employed type of categorization. This subdivision is for sure correct from a logical point of view, but the authors of this work believe that criteria such as those based on the critical plane theory should be classified as separate families. For sure, they could be redistributed between the three families mentioned above, but they have received such a high scientific relevance and have been developed to such an extent, that they can represent a category themselves and should receive a dedicated description. Moreover, to avoid any potential source of misunderstanding, after describing the families among which multiaxial fatigue criteria are subdivided in this work, a commentary will be provided also for those families that are not employed in this work but that can be found in literature references that use other categorization systems. It will also be explained how the criteria belonging to such categories fall among the ones employed in this work. In this way, this review paper aims at providing readers with a clear, comprehensive and definitive categorization of criteria for multiaxial fatigue and at being a guiding reference (especially for scholars approaching the subject for the first time) to understand the large number of papers of MLCF available in literature. Furthermore, in doing so, a clarifying overview is given before discussing MLCF criteria in detail. Indeed, the categorization described in this section will be also employed to organize the description of the criteria contained in Section ‘MLCF criteria’.

The vast majority of life prediction models for multiaxial fatigue can be subdivided into five main categories (Figure 8): criteria based on empirical formulas, criteria based on equivalent stresses, criteria based on equivalent strains, criteria based on energy parameters, and criteria based on the theory of critical plane.

Criteria based on empirical models rely on the development of experimental stress-life and strain-life curves (Basquin, 1910; Wöhler, 1860). With respect to other criteria, they are much less frequently employed, probably due to the high amount of experimental data that they require and due to the fact that each of them is valid for a specific material. Equivalent stress and equivalent strain criteria combine multiaxial stresses and strains respectively, in order to calculate an equivalent stress/strain that in uniaxial conditions is assumed to produce the same fatigue damage that is produced in multiaxial conditions by the stress/strains that are combined to obtain it (Paul, 2016). Equivalent stresses are typically employed in HCF conditions (as they take into account only elastic deformations), while equivalent strains in LCF conditions (as they take into account plastic deformations) (Stephens et al., 2000).

Energy criteria for multiaxial fatigue exploit the energy required to deform the material during each loading cycle as parameter to assess the fatigue damage and to predict the fatigue life (Filippini et al., 2003; Zhao and Qu, 2016). In particular, they rely on the strain energy density, and depending on the strain energy density they consider, they can be classified into three sub categories: criteria employing elastic strain energy, criteria employing plastic strain energy, criteria employing the sum of elastic and plastic strain energies (Macha and Sonsino, 1999). The first category is suitable to model HCF, the second to model LCF, while the third category can be used for both HCF and LCF (Macha and Sonsino, 1999).

Critical plane criteria are based on the identification of the plane(s) of maximum fatigue damage (hence called the” critical” plane), which allows to predict the orientation of cracks (Concli et al., 2022). More specifically, given a point, the fatigue damage in that point is evaluated for every plane passing through it, and the plane which displays the highest value of a combination of stress and/or strain components is considered the critical plane. The combination of stress/strain components depends on the specific criterion that is employed. Approaches based on critical plane theory are often used together with numerical analysis, to evaluate the damage for every point of the meshes of structural elements (Concli, 2022; Concli et al., 2021a, 2021b, 2022; Pagliari et al., 2023). Just as energy criteria, critical plane criteria can be subdivided, into three sub-categories (Figure 8): criteria that are based only on stress-related parameters, criteria that are based only on strain-related parameters, and criteria that are based on a combination of strain and stress parameters (sometimes also referred to as energy-related parameters) (Yu et al., 2017). Critical plane models employing only stress parameters are suitable to predict the lives of components experiencing HCF, where the deformation is fully elastic. Conversely, critical plane models employing only strain parameters are suitable to predict the lives of components experiencing LCF, since the strain components enable to take the plastic deformation into account. Eventually, critical plane models based on both stress and strain parameters can be employed to predict both HCF and LCF lives (Karolczuk and Macha, 2005). Critical plane criteria based on both stress and strain parameters combine the reliability that is granted by using energy as an indicator of fatigue damage with the capability of critical plane theory of predicting the orientation of crack initiation and propagation (Luo et al., 2017). Critical plane approaches are typically appreciated for their effectiveness in predicting fatigue lives under multiaxial loading, and for the wide range of materials and applications where they can be applied, as well as for their capability of identifying the crack orientation (Fatemi and Shamsaei, 2011; Shamsaei and Fatemi, 2009; Socie and Marquis, 1999; Zhao and Qu, 2016).

As described above, there is no consensus in literature about a common categorization methodology for multiaxial fatigue criteria. Consequently, families different from those that have been just described can be found in literature. To avoid any misunderstanding, here they are mentioned, and it is explained how they redistribute across the categories defined above. To start, it is worth mentioning criteria relying on calculations of space averages, which is sometimes indicated as a separate family (Brighenti et al., 2013; Brighenti and Carpinteri, 2012; Filippini et al., 2003). These criteria are based on the determination of an average parameter to estimate fatigue lives, and in the case of the subdivision of criteria done in this work they redistribute according to the type of average parameter that is calculated. For example, Filippini et al (Filippini et al., 2003) calculates an equivalent strain as an average of the total shear strains acting on all planes passing through a material point. According to the categorization methodology of this work, this model falls into the family of equivalent strain criteria. The same occurs for criteria relying on calculations of stress or strain invariants, which can also be found as described as a separate family (Brighenti et al., 2013; Brighenti and Carpinteri, 2012; Filippini et al., 2003), but that in the current subdivision fall under multiple of the categories described above.

MLCF criteria

In this section all the main currently available criteria for MLCF life prediction of metals will be described in detail. An explanation of their working principles will be given, and modifications that have been proposed in the years following their proposal will be listed together with each of them. Note that as this review focuses on MLCF, only criteria suitable to model it will be described. This means that some of the categories mentioned in Section ‘Multiaxial fatigue criteria classification’ (equivalent stress criteria and stress-based critical plane criteria) will not be found in this section, as they are suitable for multiaxial high-cycle fatigue only.

Equivalent strain criteria

Equivalent strain criteria combine multiaxial strains, in order to calculate an equivalent strain that in uniaxial conditions is assumed to produce the same fatigue damage that is produced in multiaxial conditions by the strains that are combined to obtain it (Paul, 2016). Even though equivalent strain criteria are of simple and convenient application, and even though they can provide accurate fatigue life estimations under proportional loading, they are often not suitable to predict fatigue under non-proportional loading (Li et al., 2011; Luo et al., 2017; Zhao and Qu, 2016; Zhong and Lu, 2020 ).

Octahedral strain range (OCT)

One of the most basic strain-based criteria is the octahedral strain range criterion (hereinafter referred to as OCT criterion), which is sometimes referred in literature also as “von Mieses strain range” criterion. Given a point of the material under analysis, this model calculates an equivalent strain range ( $Δ ε$ _OCT) on that point combining the ranges of the components of the strain tensor ( $Δ ε$ _x, $Δ ε$ _y, $Δ ε$ _z, $Δ γ$ _xy, $Δ γ$ _xz, $Δ γ$ _yz) in the following way (Filippini et al., 2003):

\begin{array}{l} Δ ε_{OCT} = \frac{1}{(1 + ν) \sqrt{2}} ({(Δ ε_{x} - Δ ε_{y})}^{2} + {(Δ ε_{y} - Δ ε_{z})}^{2} + {(Δ ε_{z} - Δ ε_{x})}^{2} \\ + \frac{3}{2} (Δ γ_{x y}^{2} + Δ γ_{y z}^{2} + Δ γ_{x z}^{2}))^{\frac{1}{2}} \end{array}

(7)

where ν is the Poisson’s ratio. The equivalent strain can be substituted to the strain amplitude in the BCM equation, to link it to the fatigue life (Zheng et al., 2022):

Δ ε_{OCT} = \frac{σ_{f}'}{E} {(2 N_{f})}^{b} + ε_{f}' {(2 N_{f})}^{c}

(8)

Unfortunately, the OCT criterion cannot be employed to evaluate non-proportional loading, as it is not able take into account non-proportional hardening (Filippini et al., 2003). To make this clear, let us consider two load states, both characterized by a normal and a shear strain that are applied. In the first state the strains are in phase (i.e., the load is proportional), while in the second one they are out of phase (i.e., the load is non-proportional). The OCT criterion produces the same equivalent strain for both loads, which is contradictory to reality, as in the case of non-proportional load a higher damage parameter should be obtained to represent the higher degree of damage that occurs under such loading conditions.

Mieses equivalent strain by ASME (ASME)

The ASME Boiler and Pressure Vessel Procedure (A. Code, 1988) defines an equivalent strain criterion, based on the OCT criterion, that works for non-proportional loading as well. This model (hereinafter referred to as ASME criterion), calculates an equivalent strain by finding the maximum octahedral shear strain between two generic time instants t_i and t_j (Li et al., 2006; Li et al., 2006):

\begin{array}{l} Δ ε_{ASME} = \max [\frac{1}{(1 + ν) \sqrt{2}} ({(Δ ε_{x, i j} - Δ ε_{y, i j})}^{2} + {(Δ ε_{y, i j} - Δ ε_{z, i j})}^{2} + {(Δ ε_{z, i j} - Δ ε_{x, i j})}^{2} \end{array}

+ \frac{3}{2} (Δ γ_{x y, i j}^{2} + Δ γ_{y z, i j}^{2} + Δ γ_{x z, i j}^{2}))^{\frac{1}{2}}]

(9)

where

Δ ε_{x, i j}

Δ ε_{y, i j}

Δ ε_{z, i j}

Δ γ_{x y, i j}

Δ γ_{x y, i j}

and

Δ γ_{x y, i j}

are the normal and shear strain ranges calculated between t_i and t_j (e.g.

Δ ε_{x, i j}

= [

ε_{x}

(t_i)−

ε_{x}

(t_j)]). As reported in (Filippini et al., 2003; Li et al., 2006a, 2006b), the

δ ε

_ASME can give lower damage parameters in case of non-proportional loading (while it should give higher values to represent the higher degree of damage due to non-proportionality), making it non completely suitable for multiaxial out of phase loading.

For this reason, Li et al. (Li et al., 2006a, 2006b) integrated the strain with two parameters developed by Itoh et al. (Itoh et al., 1995) that enable to account for non-proportional loading, f_NP and α*:

Δ ε_{ASME, L I} = (1 + α^{*} f_{N P}) Δ ε_{ASME}

(10)

where

Δ ε

_ASME,LI is the modified parameter. f_NP and α* are explained in detail as the following criterion is described.

Itoh, Sakane, Ohnami and Socie (ISOS)

Itoh et al. proposed an equivalent strain criterion (ISOS criterion) for both proportional and non-proportional loading conditions (Itoh et al., 1995). They stated that models that employ both stress and strain parameters are able to effectively predict lives under non-proportional loading, but also that estimating the stress amplitude under non-proportional loading can be difficult. For this reason, they developed a criterion that relies only on strain. The model calculates an equivalent strain parameter ( $Δ ε$ _ISOS ):

Δ ε_{ISOS} = (1 + α^{*} f_{N P}) Δ ε_{1}

(11)

where

Δ ε

₁ is the principal strain range under non-proportional loading, α* is a coefficient expressing the additional hardening response of the material under analysis due to non-proportional loading, and f_NP is a parameter that represents the severity of the non-proportional loading. More in detail, α* is calculated as the ratio between the stress amplitude when the loading is 90° out of phase (which as stated in Section ‘Out of phase, non-proportional and variable amplitude loading’ is the condition where the maximum non-proportional hardening is found) and the stress amplitude when the loading is proportional (when no additional hardening takes place). For example, for 304 stainless steel the stress amplitude under 90°degrees out of phase loading is 90% higher than the stress amplitude obtained with proportional loading (and maintaining the same ratio between applied strain and shear strain). As a result, α* is 0.9. Conversely, in the case of aluminum alloys, which feature no non-proportional hardening, α* is close to 0. With regard to f_NP, in case of proportional loading is zero. In case of non-proportional loading, it can be calculated as:

f_{N P} = \frac{1.57}{T ε_{1, \max}} \int_{0}^{T} (| sin (ξ (t)) | ε_{1} (t)) d t

(12)

where T is the time of one cycle, ε₁(t) is the maximum absolute value of principal strains at time t, ε_1,max is the maximum value of ε₁(t), and ξ(t) is the phase angle between ε_1,max and ε₁(t) (Song et al., 2021). Combining the amplitude and the change in direction of the principal strain allows to account for the additional damage induced by non-proportional loading.

Borodii and Strizhalo (Borodii and Strizhalo, 2000) changed the non-proportionality severity factor fNP with a coefficient presented by Borodii in (Borodii, 1996), obtaining satisfactory results for Al6061 alloy, Al, Cu, and 310 SS. However, their model is suitable for biaxial fatigue only, while Karunananda et al. (Karunananda et al., 2011) extended it to triaxial fatigue, by using von Mises equivalent strain (as defined in equation (7) instead of $Δ ε$ ₁, and verified it for 304 L stainless steel and Haynes 188. Liu and Yan (Liu and Yan, 2018) proposed a modified version of ISOS criterion by extending it to a more convenient coordinate system and proving its applicability to both high- and low-cycle fatigue. Morishita and Itoh (Morishita and Itoh, 2016) integrated into ISOS criterion a local strain range parameter to consider the local concentration of stresses caused by notches. They successfully applied the modified criterion to 316 L stainless steel specimens.

Energy criteria

Energy criteria for multiaxial fatigue exploit the energy required to deform the material during each loading cycle as parameter to assess the fatigue damage and to predict the fatigue life (Filippini et al., 2003; Zhao and Qu, 2016). The criteria that are suitable to model MLCF are based either on the plastic strain energy or on the sum of elastic and plastic strain energies (Macha and Sonsino, 1999). Energy criteria are characterized by good reliability, broad applicability, and higher ease of use with respect to other criteria (Zhong and Lu, 2020). As disadvantages, they are very sensitive to the accuracy of the constitutive equation of the material under analysis (Zhao and Qu, 2016), and in some cases they cannot take into account non-proportional loading, especially as far as traditional criteria are concerned (Li et al., 2011; Zhao and Qu, 2016). However, there are also multiple successful examples of energy criteria that work with non-proportional loading conditions (Macha and Sonsino, 1999; Zhao and Qu, 2016). Eventually, being energy a scalar quantity, they cannot estimate the orientation of the planes on which fatigue cracks initiate and propagate (Luo et al., 2017).

Ellyin

Ellyin (Ellyin, 1974) proposed to analyze fatigue under proportional loading though a criterion based on cyclic (i.e., in the stabilized phase mentioned in Section ‘Low-cycle fatigue’) distortion energy (ΔW_d), which can be expressed as the integral over a (stabilized) cycle of the product between the deviatoric stress tensor (s_ij) and the increment of the deviatoric strain tensor (e_ij):

Δ W_{d} = \int_{cycle} s_{i j} d ε_{i j}

(13)

Since the components of the deviatoric strain tensor $ε_{i j}$ are obtained as the sum of elastic and plastic strains, ΔW_d represents the total cyclic distortion energy. Ellyn expressed ΔW_d in terms of a stress and strain parameter called effective stress $(\bar{σ})$ and effective strain ( $\bar{ε}$ ) respectively:

Δ W_{d} = K_{E L} Δ \bar{ε} Δ \bar{σ} + Δ \bar{ε} {\bar{σ}}_{m}

(14)

where K_EL is a coefficient that depends on the stress level when the material is in the plastic region,

Δ \bar{ε}

is the range of the effective strain,

Δ \bar{σ}

is the range of the effective stress and

{\bar{σ}}_{m}

is the mean effective stress.

\bar{ε}, \bar{σ}

and

{\bar{σ}}_{m}

can be calculated as:

\bar{ε} = 2 \sqrt{(\frac{1}{3} ε_{i j} ε_{i j})}

(15)

\bar{σ} = \sqrt{(\frac{3}{2} s_{i j} s_{i j})}

(16)

{\bar{σ}}_{m} = \frac{{\bar{σ}}_{U} + {\bar{σ}}_{L}}{2}

(17)

where

{\bar{σ}}_{U}

and

{\bar{σ}}_{L}

are the lower and upper limit of the effective strain respectively. After expressing the distortion energy as in equation (14), Ellyn’s criterion relates it to fatigue life as:

K_{E L} Δ \bar{ε} Δ \bar{σ} + Δ \bar{ε} {\bar{σ}}_{m} = λ β

(18)

where β is a constant and λ is a parameter in the order of:

λ \sim N_{f}^{0.35}

(19)

Ellyin and Kujawski (Ellyin and Kujawski, 1993) developed Ellyin’s criterion so that it could be employed also for non-proportional loading, by integrating it with a special function of multiaxial constraints.

Lefebvre, neale, and ellyn (LNE)

Lefebvre et al. (Lefebvre et al., 1981) developed a criterion (LNE criterion) that employs the energy related to the effective strain. However, differently from Ellyn’s model which uses the total effective strain $(\bar{ε})$ energy, the LNE criterion makes use of the effective plastic strain ( ${\bar{ε}}^{(P)}$ ) energy only. It relates it to fatigue life as:

Δ \bar{σ} Δ {\bar{ε}}^{(P)} = K_{LNE 1} {(N_{f})}^{K_{LNE 2}}

(20)

where

Δ {\bar{ε}}^{(P)}

is the effective plastic strain range, while K_LNE1 and K_LNE2 are constants that depend on the mechanical properties of the material and are a function of the stress ratio (Macha and Sonsino, 1999).

Δ {\bar{ε}}^{(P)}

can be calculated as:

Δ {\bar{ε}}^{(P)} = \sqrt{(\frac{2}{3} ε_{i j}^{(P)} ε_{i j}^{(P)})}

(21)

where

ε_{i j}^{(P)}

is the deviatoric plastic strain tensor. As stated in Zhong and Lu, (2020), LNE criterion lacks the required information to correlate experimental data when variable amplitude and non-proportional loading conditions are involved.

Garud

Garud’s criterion (Garud, 1981) is based on employing only the plastic strain energy as the damage parameter to determine the fatigue life until crack initiation (i.e. up to the point cracks become visible). In this model, the plastic strain energy ( $Δ$ W^(P)) over a cycle is calculated as the sum of the areas of the hysteresis loops of each one of the stress tensor components. Indicating as σ_ij a generic component of the stress tensor, the plastic strain energy for that component is calculated as:

Δ W_{i j}^{(P)} = \int_{cycle} σ_{i j} d ε_{i j}^{(P)}

(22)

Summing the $Δ W_{i j}^{(P)}$ of each one of the nine components of the stress tensor, the plastic strain energy $Δ W^{(P)}$ is obtained. Garud’s criterion relates $Δ W^{(P)}$ to the fatigue life as:

N_{f} = K_{GAR 1} {(Δ W^{(P)})}^{- K_{GAR 2}}

(23)

whereK_GAR1 and K_GAR2 are material constants. As stated in (Macha and Sonsino, 1999), Garud’s criterion lacks the required information to correlate experimental data when variable amplitude and non-proportional loading conditions are involved. Lu et al. (Lu et al., 2019) integrated Garud’s criterion with Itoh’s hardening coefficient α*, Itoh’s non-proportionality severity factor f_NP, obtaining good accuracy for 316 L stainless stell under both proportional and non-proportional loading conditions.

Jahed and varvani

Jahed and Varvani-Farahani proposed a criterion (JV criterion) based on axial and torsional strain energies, which first was suitable only for proportional loading (Jahed and Varvani-Farahani, 2006), and soon after was extended to non-proportional loading as well (Jahed et al., 2007). The former version is here presented. The criterion is expressed as:

N_{f} = \frac{Δ W^{(A)}}{Δ W} N_{A} + \frac{Δ W^{(T)}}{Δ W} N_{T}

(24)

where

Δ W^{(A)}

is the strain energy under purely axial loading,

Δ W^{(T)}

is the strain energy under purely torsional loading,

Δ

W is the total cyclic strain energy, N_A is the fatigue life under purely axial loading, and N_T is the fatigue life under purely torsional loading. To calculate

Δ

W in the case of non-proportional loading, Jahed and Varvani-Farahani suggest employing the modified incremental cyclic plasticity model of Mroz (Mroz, 1967), while to calculate N_A and N_T the Basquin-Coffin-Manson equations expressed in energy terms are employed:

Δ W_{A} = E'_{t} {(N_{A})}^{B} + E'_{f} {(N_{A})}^{C}

(25)

Δ W_{T} = T'_{t} {(N_{T})}^{B_{s}} + T'_{f} {(N_{T})}^{C_{s}}

(26)

where

E'_{t}, T'_{t}, E'_{f}, T'_{f}

, B, B_S, C and C_S are the axial fatigue strength coefficient, the torsional fatigue strength coefficient, the axial fatigue toughness coefficient, the shear fatigue toughness coefficient, the axial fatigue strength exponent, the torsional fatigue strength exponent, the axial fatigue toughness exponent and the shear fatigue toughness exponent, all expressed in energy terms. All eight parameters can be determined through the strain-life curves (on a logarithmic diagram) under purely axial and purely torsional loading conditions.

Critical plane criteria

Critical plane criteria are based on finding the plane where the value of a variable (which is chosen as the most proper indicator of material damage) is maximum. This allows to define the direction that cracks will follow when nucleating and propagating at a very early stage. As stated above, the variable representing the level of damage in each plane typically is a combination of stress and/or strain components, which depends on the criterion. Critical plane criteria based on strain terms only are not able to include information about the material deformation response, such as cyclic hardening caused by non-proportional loading (Shamsaei and Fatemi, 2009). On the contrary, criteria based on both stress and strain terms do. They often include a normal stress term, which takes into account the effect of the mean or residual stress, as well as of cyclic hardening (Shamsaei and Fatemi, 2009), making them suitable to evaluate also the lives of components subject to non-proportional loading.

Most of MLCF critical plane criteria identify the critical plane either as the plane featuring either the maximum normal strain range ( $Δ ε_{n, \max}$ ) or the maximum shear strain range ( $Δ γ_{\max}$ ). As a result, given a point where a crack is expected to nucleate, in order to identify the critical plane it is necessary to calculate the normal or the shear strain ranges (depending on the criterion that is being employed) for every plane passing through that point. This can be better understood by looking at Figure 9, which shows a generic candidate plane passing through an undefined point (called ith plane), having $Φ$ _n and θ_n as spherical coordinates. X′Y′Z′ is the coordinate system integral to the ith plane. $ε'_{x} (t)$ is the normal strain, while $γ'_{x y} (t)$ and $γ'_{x z} (t)$ are the shear strains on that plane. They vary over time since multiaxial fatigue occurs.

As far as the normal strain range is concerned, it is calculated, for every ith candidate plane passing though the point under analysis, as:

Δ ε_{n, i th} = \max_{1 \leq j \leq p} {ε'_{x} (j)} - \min_{1 \leq j \leq p} {ε'_{x} (j)}

(27)

where p is the number of stages in which loading cycles have been subdivided. Figure 10 shows graphically what is expressed by equation (27): for a given candidate plane

ε'_{x} (t)

is calculated for every time instant and the difference between the maximum and the minimum values define the strain range on the ith candidate plane. Such calculation should be performed for every ith plane. The greatest

Δ ε_{n, i_{t h}}

that is found is

Δ ε_{n, \max}

, and the plane featuring is the critical plane, if the criterion identifies the plane of maximum normal strain range as the critical plane. If the point of crack nucleation is unknown, the procedure should be also repeated for every point of the component under analysis.

Similarly, the shear strain range for every ith plane passing through the point under analysis ( $Δ γ_{i}$ _th ) can be calculated as (Yu et al., 2017):

\frac{Δ γ_{i th}}{2} = \begin{array}{c} \max \\ 1 \leq j \leq p \\ j + 1 \leq m \leq p \end{array} {\sqrt{{[γ_{x y}^{'} (m) - γ_{x y}^{'} (j)]}^{2} + {[γ_{x z}^{'} (m) - γ_{x z}^{'} (j)]}^{2}}}

(28)

Such calculation should be performed for every ith plane. The greatest $Δ γ_{i}$ _th that is found is $Δ γ_{\max}$ , and the plane featuring it is the critical plane, if the criterion that is being employed identifies the critical plane as the plane of maximum shear strain range. Also in this case, if the point of crack nucleation is unknown, the procedure should be also repeated for every point of the component under analysis.

Smith-Watson-Topper (SWT)

The Smith, Watson, and Topper (SWT) criterion was introduced in 1970 and was originally proposed as a model applicable to uniaxial fatigue (Smith, 1970). Thanks to the work of Socie (Socie, 1987) it was later extended to multiaxial fatigue and in the years to follow it became one of the most extensively employed criteria for the prediction of lives of structural components under MLCF, becoming one of the pillars of MLCF analysis. As a result, in most of the cases the SWT model is mentioned in literature, what is actually meant is the extension by Socie, which is also what is described here. SWT criterion is a critical plane criterion based on two assumptions. The first one is that cracks initiation occurs along the plane that features the maximum range of normal strain ( $Δ ε_{n, \max}$ ), which is identified as the critical plane. The second one is that crack growth is driven by the stresses normal to the critical plane. As a result, once the critical plane is found through the procedure described above, the SWT parameter can be calculated as:

SWT = \frac{Δ ε_{n, \max}}{2} σ_{n, \max}

(29)

where

σ_{n, \max}

is the maximum stress normal to the critical plane that is experienced during one loading cycle. It is integrated in the parameter since the SWT criterion assumes stresses normal to the critical plane to regulate the crack growth. The SWT parameter can then be inserted into a strain-life relationship of the Basquin-Coffin-Manson type as:

\frac{Δ ε_{n, \max}}{2} σ_{n, \max} = \frac{σ_{f}'^{2}}{E} {(2 N_{f})}^{2 b} + σ_{f}' ε_{f}' {(2 N_{f})}^{b + c}

(30)

Since it is based on the assertion that cracks nucleate on the plane of maximum principal strain range, the SWT model gives accurate results when predicting the lives of materials characterized by tensile fracture (Dileep et al., 2015; Shamsaei and Fatemi, 2009). In addition, SWT criterion is suitable to analyze cases where non-proportional loading is involved, and can take into account the effect of mean stress, since $σ_{n, \max}$ can be expressed as:

σ_{n, \max} = σ_{n, mean} + σ_{n, a}

(31)

where

σ_{n, mean}

and

σ_{n, a}

are the mean stress and the stress amplitude respectively (Dileep et al., 2015).

Ince and Glinka (Ince and Glinka, 2014) adjusted SWT parameter by separating the elastic component of the strain from the plastic one:

\frac{Δ ε_{n, \max}^{(E)}}{2} \frac{σ_{n, \max}}{σ'_{f}} + \frac{Δ ε_{n, \max}^{(P)}}{2} = \frac{σ_{f}'}{E} {(2 N_{f})}^{2 b} + ε_{f}' {(2 N_{f})}^{c}

(32)

where

Δ ε_{n, \max}^{(E)}

and

Δ ε_{n, \max}^{(P)}

are the elastic and the plastic components of the maximum normal strain range. They tested the model on Incoloy 901 superalloy, ASTM A723 steel, 7075-T561 aluminum alloy and 1045 HRC 55 steel, and obtained better prediction than those obtained through SWT parameter in its original form. Yu et al. (Yu et al., 2017) modified SWT damage parameter by multiplying it for the elastic modulus E and also derived an alternative version of the SWT parameter in shear terms, so that it can be for shear crack materials. They obtained better results in terms of life prediction with GH4169 alloy. Song et al. (Song et al., 2021) integrated the SWT with Itoh’s hardening coefficient α*, Itoh’s non-proportionality severity factor f_NP, and a stress concentration factor k_t to take stress concentration into account, and applied it to 10CrNi3MoV notched specimens.

Brown and miller (BM)

Brown and Miller criterion (BM criterion) (Brown and Miller, 1973), represents a milestone in the development of critical plane models for MLCF. Many critical plane models that were proposed in the following decades were developed from it, building a large legacy for such criterion in the years. The BM criterion is based on two assumptions. The first one is that the localized deformation phenomenon that generates the nucleation of fatigue cracks typically occurs along persistent slip bands of the material (Kanazawa et al., 1977; Taira, 1969). Since slip bands are typically closely aligned with the planes of maximum shear strain, Brown and Miller assumed that fatigue cracks nucleate on such planes. As a result, when using the BM criterion, the critical plane is identified as the plane featuring the maximum shear strain amplitude ( $Δ γ_{\max}$ ). The second assumption on which BM criterion relies is that crack growth occurs due to decohesion of the slip bands (Tomkins, 1968). For this reason, BM parameter identifies the strain normal to the critical plane ( $ε_{n}$ ) as a promoter of crack growth. As a result, once the critical plane is found, the BM damage parameter is:

B M = \frac{Δ γ_{\max}}{2} + K_{B M} Δ ε_{n}

(33)

where

Δ ε_{n}

is the range of the strain normal to the critical plane. K_BM is a fitting parameter (K_BM

\approx

0.3 for ductile materials when their lives are close to the fatigue limit (Ramberg and Osgood, 1943)) representing the material sensitivity to normal strain for the crack growth. The BM parameter can then be inserted into a strain-life relationship equation as []:

\frac{Δ γ_{\max}}{2} K_{B M} Δ ε_{n} = α_{1} \frac{σ_{f}' - 2 σ_{n, mean}}{E} {(2 N_{f})}^{b} + α_{2} ε_{f}' {(2 N_{f})}^{c}

(34)

where

σ_{n, mean}

is the mean stress normal to the critical plane, while α₁ and α₂ are:

α_{1} = (1 + ν_{e}) + K_{B M} (1 - ν_{e})

(35)

α_{2} = (1 + ν_{p}) + K_{B M} (1 - ν_{p})

(36)

where ν_e and ν_p are the elastic and plastic Poisson’s ratio respectively. Since it is based on the assumption that cracks nucleate on the plane of maximum principal strain range, the BM model gives accurate results when predicting the lives of materials characterized by shear fracture, and the same applies for criteria deriving from it (Dileep et al., 2015). Since BM criterion uses two strain parameters, it belongs to the family of critical plane criteria relying on strain parameters only. Unfortunately, because no stress parameters are taken into consideration, this criterion is not suitable to account for non-proportional hardening, and thus to analyze non-proportional loading (Fatemi and Socie, 1988). Despite this strong limitation, BM criterion represents a bridgehead for the development of critical plane criteria for MLCF.

Kandil-Brown-Miller (KBM)

Kandil, Brown and Miller (KBM) proposed a simplified version of BM criterion that can be employed when combined tension and torsion take place (Kandil et al., 1982):

\frac{Δ γ_{\max}}{2} + K_{KBM} Δ ε_{n} = α_{1} \frac{σ_{f}'}{E} {(2 N_{f})}^{b} + α_{2} ε'_{f}' {(2 N_{f})}^{c}

(37)

where K_KBM is a fitting parameter playing the same role of K_BM, while the coefficients α₁ and α₂ are be calculated as in equation (35) and equation (36), with K_KBM instead of K_BM:

α_{1} = (1 + ν_{e}) + K_{KBM} (1 - ν_{e})

(38)

α_{2} = (1 + ν_{p}) + K_{KBM} (1 - ν_{p})

(39)

Ma et al. (Ma et al., 2020), developed two alternative models of KBM criterion. The first one, which they named KBM-P criterion, employs the critical plane as the plane of maximum normal strain range rather than the plane of maximum shear strain range. The second one, named KBM-U model, integrates the KBM-P model with the multiaxial strain ratio (ratio between applied torsional and axial strains). Tests on pure Titanium revealed that the two models outperformed the original one. KBM-P criterion was also analyzed in (Ma et al., 2021) and integrated with the effect of mean axial and torsional strain in (Ma et al., 2023). Chen et al. (Chen et al., 2006), modified the KBM criterion by identifying the critical plane through a weight function that averages the instantaneous values of the maximum shear strain planes over a period of time. They observed an improvement of the criterion accuracy in predicting the lives of components. Moreover, they could apply the criterion also to variable amplitude loading. Li. et al (Li et al., 2011), integrated in the KBM criterion the maximum stress normal to the maximum shear strain range plane. The modified criterion was validated using fatigue data of multiple steels, such as AISI 316, C40, SAE 1045, and J55 steel, and proved to give better predictions in both proportional and non-proportional loading scenarios. A further advantage is that the model does not employ the fitting parameter K_KBM or any other fitting parameter.

Socie, Wail and Dittmer (SWD)

Socie et al. (Socie et al., 1985) proposed a criterion (SWD criterion) based on BM model, thus identifying the critical plane as the plane of maximum shear strain amplitude ( $Δ γ_{\max}$ ). In addition, it incorporates the mean stress normal to the critical plane ( $σ_{n, mean}$ ), to account for the fact that crack opening reduces the friction forces between slip planes blocking crack growth:

SWD = \frac{Δ γ_{\max}}{2} + \frac{Δ ε_{n}}{2} + \frac{σ_{n, mean}}{E}

(40)

When expressed through a strain-life relation the criterion takes the form:

\frac{Δ γ_{\max}}{2} \frac{Δ ε_{n}}{2} + \frac{σ_{n, mean}}{E} = \frac{τ'_{f}}{G} {(2 N_{f})}^{b} + γ'_{f} {(2 N_{f})}^{c}

(41)

where

τ'_{f}

is the shear fatigue strength coefficient,

γ'_{f}

is the shear fatigue ductility coefficient, and G is the shear modulus.

Fatemi and Socie (FS)

Fatemi and Socie (FS) criterion was proposed in 1988 as a modification of BM parameter (Fatemi and Socie, 1988). Among multiaxial fatigue criteria, it is one of the most employed, if not the most employed, for MLCF. FS criterion relies on the same hypothesis made by Brown and Miller that the initiation of cracks occurs on the planes of maximum shear strain amplitude, which are identified as the critical planes. However, differently from BM criterion, FS criterion relies on the stress component perpendicular to the plane as secondary damage parameter promoting crack growth. As a result, FS criterion falls in the category of critical plane criteria relying on a combination of stresses and strains. The FS damage parameter is calculated as:

F S = \frac{Δ γ_{\max}}{2} (1 + K_{F S} \frac{σ_{n, \max}}{σ_{y}})

(42)

where K_FS is a data fitting coefficient,

σ_{n, \max}

is the maximum normal stress acting on the critical plane, and σ_y is the material cyclic yield strength. Just as the coefficient K_BM of BM criterion represents the material sensitivity to normal strain for the crack growth, the coefficient K_FS, represents the material sensitivity to normal stress for the crack growth. K_FS varies with the failure life (Lopez-Crespo et al., 2015):

K_{F S} = [\frac{\frac{τ_{f}'}{G} {(2 N_{f})}^{b_{γ}} + γ_{f}' {(2 N_{f})}^{c_{γ}}}{(1 + ν_{e}) \frac{σ_{f}'}{E} {(2 N_{f})}^{b} + (1 + v_{p}) ε_{f}' {(2 N_{f})}^{c}}] \frac{2 σ_{y}}{σ_{f}' {(2 N_{f})}^{b}}

(43)

where b_γ is the shear fatigue strength exponent and c_γ is the shear fatigue ductility exponent. As for previously listed criteria, it can be implemented into a LCF strain-life relation:

\frac{Δ γ_{\max}}{2} (1 + K_{F S} \frac{σ_{n, \max}}{σ_{y}}) = \frac{τ_{f}'}{G} {(2 N_{f})}^{b_{γ}} + γ_{f}' {(2 N_{f})}^{c_{γ}}

(44)

The fact that also $σ_{n, \max}$ is included makes the FS parameter criterion to analyze MLCF also under non-proportional loading. Indeed, $σ_{n, \max}$ allows to take into account for cyclic hardening, since it changes every cycle as the material hardens (Cruces et al., 2018). FS approach shows a wide range of adaptability and gives good results both in proportional and non-proportional loading, justifying its wide employment.

Huang et al (Huang et al., 2020) proposed an alternative version of FS criterion by substituting K_FS with a non-proportionality coefficient that can take into account both the effect of the strain path and additional material hardening. They used to successfully evaluate the fatigue behavior of beam-to-column welded joints made out of Q235 low carbon steel, and compared it with SW criterion, obtaining better and more conservative results.

Wang and Brown (WB)

Wang and Brown criterion (WB criterion) was proposed in 1993 (Wang and Brown, 1993) as a modification of BM criterion, so that it could take both proportional and non-proportional loading into account. WB damage parameter looks very similar to the KBM one, but rather than the overall range of the normal strain $Δ ε_{n}$ , it employs a different range of the normal strain, called normal strain excursion, $Δ ε_{n}^{*}$ :

W B = \frac{Δ γ_{\max}}{2} + K_{W B} Δ ε_{n} *

(45)

where K_WB is a fitting parameter playing the same role of K_BM.

Δ ε_{n}^{*}

is calculated considering two adjacent major turning points of the maximum shear strain and measuring the excursion of the normal strain between those two points. To have a better understanding about it, Figure 11 should be observed. It portrays the time histories of the normal strain

ε_{n}^{*}

and of the maximum shear strain γ_max on the critical plane. γ_max has been ordered through cycle counting. Three major turning points can be identified in points A, E and I defining two reversals of γ_max. Points A and E are two adjacent turning points.

Δ ε_{n}^{*}

is the excursion of the normal strain between the points A and E. If the loading were in phase and proportional,

ε_{n}^{*}

and γ_max reach the minimum and maximum values at the tame time instants, and

Δ ε_{n}^{*}

would be equal to the overall range of the normal strain. In this case, the WB criterion is equal to the KBM one. However, when the load is out of phase (as in Figure 11), it can be observed that this is no longer true, and

Δ ε_{n}^{*}

is different from the overall range of the normal strain. Employing

Δ ε_{n}^{*}

instead of the overall range of the normal strain enables the WB criterion to evaluate MLCF lives also under non-proportional loading conditions. Inserting the WB parameter in a strain-life equation, what is obtained is:

\frac{Δ γ_{\max}}{2} + K_{W B} Δ ε_{n}^{*} = α_{1} \frac{σ_{f}'}{E} {(2 N_{f})}^{b} + α_{2} ε_{f}' {(2 N_{f})}^{c}

(46)

where the coefficients α₁ and α₂ are calculated as in equation (35) and equation (36), with K_WB instead of K_BM:

α_{1} = (1 + ν_{e}) + K_{W B} (1 - ν_{e})

(47)

α_{2} = (1 + ν_{p}) + K_{W B} (1 - ν_{p})

(48)

The coefficient K_WB is calculated as (Cruces et al., 2018):

K_{W B} = \frac{\frac{τ_{f}'}{G} {(2 N_{f})}^{b_{γ}} + γ_{f}' {(2 N_{f})}^{c_{γ}} - (1 + ν_{e}) \frac{σ_{f}'}{E} {(2 N_{f})}^{b} - (1 + ν_{p}) ε_{f}' {(2 N_{f})}^{c}}{(1 - ν_{e}) \frac{σ_{f}'}{E} {(2 N_{f})}^{b} + (1 - ν_{p}) ε_{f}' {(2 N_{f})}^{c}}

(49)

Brown et al. (Brown et al., 1996) later modified the criterion by integrating $σ_{n, mean}$ , so that also the effect of mean stress could be accounted for:

\frac{Δ γ_{\max}}{2} K_{W B} Δ ε_{n}^{*} = α_{1} \frac{σ_{f}' - σ_{n, mean}}{E} {(2 N_{f})}^{b} + α_{2} ε_{f}' {(2 N_{f})}^{c}

(50)

They applied the criterion to EN15R steel, and (being it a ductile material), they observed no effect of mean stress for many of the loading scenarios they tested, especially the proportional ones. However, the effect of mean stress could often be observed in the case of variable amplitude and non-proportional loading cases.

Shang and Wang (SW)

Shang and Wang (SW) criterion is a critical plane criterion based on the same parameters of WB model, i.e., the maximum shear strain amplitude to identify the critical plane and the normal strain excursion $Δ ε_{n}^{*}$ (De-Guang and De-Jun, 1998). However, it employs them in a different way, combining them as an

S W = \frac{Δ ε_{e q}}{2} = \sqrt{{(ε_{n}^{*})}^{2} + \frac{1}{3} {(\frac{Δ γ_{\max}}{2})}^{2}}

(51)

A strain-life equation can be obtained relating the damage parameter to the Basquin-Coffin-Manson equation:

\sqrt{{(ε_{n}^{*})}^{2} + \frac{1}{3} {(\frac{Δ γ_{\max}}{2})}^{2}} = \frac{σ_{f}'}{E} {(2 N_{f})}^{b} + ε_{f}' {(2 N_{f})}^{c}

(52)

Sun et al. (Sun et al., 2010) modified the SW parameter to improve its accuracy, by using an effective Poisson ratio ν_eff:

\frac{Δ ε_{e q}}{2} = \sqrt{{(ε_{n}^{*})}^{2} + \frac{ν_{eff} (2 - ν_{eff})}{{(1 + ν_{eff})}^{2}} {(\frac{Δ γ_{\max}}{2})}^{2}}

(53)

where

ν_{eff} = (ν_{e} ε_{e} + ν_{p} ε_{p}) / (ε_{e} + ε_{p})

. They tested the modified criterion and obtained good correlation for GH4169 alloy, Haynes 188, and mild steel, yet with little difference between the results obtained using the modified criterion and those obtained using the original one.

Lohr and Ellison (LE)

Lohr and Ellison (Lohr and Ellison, 1980) proposed a critical plane criterion (LE criterion) whose damage parameter is calculated through an equivalent strain. The criterion is similar to BM model, with the difference that the critical plane is not identified as the plane featuring the highest shear strain amplitude, but as the plane inclined of 45° with respect to the free surface of the material (Karolczuk and Macha, 2005; Ma et al., 2020).The damage parameter is:

L E = \frac{Δ γ_{45}}{2} + K_{L E} ε_{n, 45}

(54)

where

Δ γ_{45}

and

ε_{n, 45}

are the shear and normal strain range on the critical plane. K_LE is a meterial coefficient. Unfortunately, it was observed that LE criterion is suitable only for proportional in-phase loading (RBong-Ryul and Soon-Bok, 1996).

Chen, Xu, huang (CXH I, CXH II)

Chen et al. (Chen et al., 1999) proposed two critical plane criteria, one for tensile fracture materials and one for shear fracture materials (CXH I and CXH II criterion, respectively). CXH I criterion consists of a modification of the SWT model (as expressed by Socie for multiaxial fatigue), and therefore the critical plane is identified as the plane featuring the maximum range of normal strain ( $Δ ε_{n, \max}$ ). The criterion integrates the SWT damage parameter with the contribute of the shear stress and the shear strain on the critical plane, without which Chen et al. state that SWT criterion would be unconservative for nonproportional loading:

CXH I = Δ ε_{n, \max} Δ σ_{n} + Δ τ Δ γ

(55)

where

Δ σ_{n}

Δ τ

, and

Δ γ

are the normal stress, shear stress and shear strain ranges calculated on the critical plane. CXH I damage parameter can then be inserted into a strain-life relationship of the Basquin-Coffin-Manson type as:

Δ ε_{n, \max} Δ σ_{n} + Δ τ Δ γ = 4 \frac{σ_{f}'^{2}}{E} {(2 N_{f})}^{2 b} + 4 σ_{f}' ε_{f}' {(2 N_{f})}^{b + c}

(56)

CXH II criterion works for shear failure materials and therefore it identifies the critical plane as the one having the largest shear strain range ( $Δ γ_{\max}$ ). In this case it integrates the contribute of the normal stress and the normal strain on the critical plane:

CXH I I = Δ γ_{\max} Δ τ + Δ ε_{n} Δ σ_{n}

(57)

Also in this case $Δ σ_{n}$ , $Δ τ$ , and $Δ ε_{n}$ are calculated on the critical plane. The strain-life form of the criterion is:

Δ γ_{\max} Δ τ + ε_{n} Δ σ_{n} = 4 \frac{τ_{f}'^{2}}{G} {(2 N_{f})}^{2 b_{γ}} + 4 τ_{f} γ_{f}' {(2 N_{f})}^{b_{γ} + c_{γ}}

(58)

Liu et al. (Liu et al., 2020) stated that both CXH models, being unable to distinguish the contribution to the damage of the normal and the shear strain, tend to be conservative. Thus, they integrated them with weight factors that could do such distinction. Experimental verifications on 16MnR, AISI 304, pure titanium, and titanium alloy BT9, proved that the modified criteria outperformed the original ones, as well as the SWT and the FS ones. Zheng et al. (Zheng et al., 2022) integrated CXH I criterion with a weight factor for the damage contribution of shear fatigue, obtaining an improvement in the accuracy of the life predictions for 316LN stainless steel. Jiang et al. (Jiang, 2000; Jiang et al., 2007) applied weight factors to both the tangential and the normal components of CXH I criterion, and applied it to multiple steels (1070, 1045 and 304 SS) with satisfactory results. The model was also successfully employed by Benedetti et al (Benedetti et al., 2014) on shot-peened and non-peened Al7075-T6 for HCF.

Liu (LIU I, LIU II)

Liu (1993) proposed two critical plane criteria, one for tensile failure (LIU I) and one for shear failure (LIU II) modes. Both models are based on a parameter called virtual strain energy ( $Δ$ W_V SE), but they employ it differently. Given a material plane, $Δ$ W_V SE is defined as the sum of the energy related only to normal strain ( $Δ$ $W_{VSE, n}$ ) and the energy related only to shear strain ( $Δ$ $W_{VSE, s}$ ) on that plane (Macha and Sonsino, 1999):

Δ W_{VSE} = Δ W_{VSE, n} + Δ W_{VSE, s} = Δ σ_{n} Δ ε_{n} + \frac{Δ τ Δ γ}{2}

(59)

where

Δ σ_{n}

Δ ε_{n}

Δ τ

and

Δ γ

are the normal stress, normal strain, shear stress and shear strain ranges on the material plane under analysis. LIU I model identifies the critical plane as the plane where the virtual normal strain energy

Δ

W_{VSE, s}

is maximum. The corresponding damage parameter (

Δ

W_I) is represented by the virtual strain energy on the critical plane. Therefore, first the plane with maximum virtual normal strain energy is found, and then the respective shear strain energy on the same plane is summed:

Δ W_{I} = {(Δ W_{VSE, n})}_{\max} + Δ W_{VSE, s} = {(Δ σ_{n} Δ ε_{n})}_{\max} + \frac{Δ τ Δ γ}{2}

(60)

Conversely, LIU II model identifies the critical plane as the plane where the virtual shear strain energy $Δ$ $W_{VSE, n}$ is maximum. The corresponding damage parameter ( $Δ W_{I I}$ ) is calculated finding the plane with maximum virtual shear strain energy and summing the respective normal strain energy on the same plane:

Δ W_{I I} = Δ W_{VSE, n} + {(Δ W_{VSE, s})}_{\max} = Δ σ_{n} Δ ε_{n} + {(\frac{Δ τ Δ γ}{2})}_{\max}

(61)

The damage parameters can be integrated into strain-life equations as (Reis et al., 2009):

Δ W_{I} = 4 \frac{σ_{f}'^{2}}{E} {(2 N_{f})}^{b} + 4 σ_{f}' ε_{f}' {(2 N_{f})}^{b + c}

(62)

Δ W_{I I} = 4 \frac{τ_{f}'^{2}}{G} {(2 N_{f})}^{b_{γ}} + 4 τ_{f}' γ_{f}' {(2 N_{f})}^{b_{γ} + c_{γ}}

(63)

Calvo et al. (Calvo et al., 2011) modified both LIU I and LIU II criteria so that they could incorpo-rate the effect of mean stress.

Chu, Conle, and Bonnen (CCB)

Chu et al. (Chu et al., 1993) proposed a criterion (CCB criterion) that like LIU I and LIU II criteria combines normal and shear and strain energies. However, differently from them, it employs the maximum stresses rather than the stress ranges, in an attempt to include the effect of mean stress. The criterion relies on a damage parameter that is based on strain energy. Moreover, this critical plane criterion focuses on finding the maximum value of the damage parameter rather than looking for planes of maximum stress or strain amplitude. The CCB damage parameter is calculated as:

CCB = {(τ_{\max} \frac{Δ γ}{2} + σ_{n, \max} \frac{Δ ε}{2})}_{\max}

(64)

where

Δ ε

and

Δ γ

are the normal stress and normal strain calculated on the critical plane, while τ_max is the maximum shear stress and

σ_{n, \max}

is the maximum stress normal to it.

Glinka, Shen and Plumtree (GSP)

Glinka et al. (Glinka et al., 1995) proposed a criterion (GSP criterion) that identifies the plane featuring the maximum shear strain amplitude as the critical plane, just like the BM criterion. What is employed as damage parameter is the strain energy density on such plane:

GSP = \frac{Δ τ Δ γ}{2} + \frac{Δ σ_{n} Δ ε_{n}}{2}

(65)

where

Δ σ_{n}

Δ ε_{n}

Δ τ

and

Δ γ

are the normal stress, normal strain, shear stress and shear strain ranges on the material plane under analysis (the critical plane). A disadvantage of this model is that it cannot account for the effect of mean stress. Indeed, for given stress and strain ranges, the GSP parameter gives the same value regardless of the mean values of the ranges. Consequently, lives under stress and strain ranges with a low mean value result to be the same of lives under the same stress and strain ranges with a high mean value, which is contradictory to reality. As a result, the same authors (Glinka et al., 1995) modified the criterion so that it could account for the mean stress:

GSP = \frac{Δ τ Δ γ}{2} (\frac{σ'_{f}}{σ'_{f} - σ_{n, \max}} + \frac{τ'_{f}}{τ'_{f} - τ_{\max}})

(66)

where τ_max is the maximum absolute value of the shear stress calculated on the critical plane and

σ_{n, \max}

is the maximum absolute value of the stress perpendicular to it. Although both versions of the GSP parameter give satisfactory results for proportional loading, they are not suitable for non-proportional loading (Pan et al., 1999). Pan et al. (Pan et al., 1999) modified equation (66) to improve the GSP criterion suitability to non-proportional loading analysis. They considered the different influence on fatigue life of strain energy in the shear direction and of strain energy in the normal direction:

G S P_{PAN} = \frac{Δ τ Δ γ}{2} + \frac{σ'_{f} ε'_{f}}{τ'_{f} γ'_{f}} \frac{Δ σ_{n} Δ ε_{n}}{2}

(67)

Later, the criterion was further elaborated (Han et al., 2002), so that the damage parameter could be linked to the fatigue life N_f, in a BCM-type relationship and so that it could include the effect of mean stress:

(1 + \frac{σ_{n, mean}}{σ'_{f}}) \frac{Δ τ Δ γ}{2} + \frac{σ'_{f} ε'_{f}}{τ'_{f} γ'_{f}} \frac{Δ σ_{n} Δ ε_{n}}{2} = \frac{τ_{f}'^{2}}{G} {(2 N_{f})}^{2 b_{γ}} + τ'_{f} γ'_{f} {(2 N_{f})}^{b_{γ} + c_{γ}}

(68)

where the factor (1 +

σ_{n, mean}

σ'_{f}

) is employed to take the mean stress effect into account.

Varvani and Farahani (VF)

Varvani and Farahani (Varvani-Farahani, 2000) proposed a critical plane criterion (VF criterion) based on strain energy. The critical plane is identified as the plane of maximum shear strain range ( $Δ γ_{\max}$ ). Moreover, the time instant when the Mohr circles are the largest is to be considered. At that instant, the damage parameter of the VF criterion is calculated as the sum of the normal and shear strain energies:

V F = \frac{1}{(σ'_{f} ε'_{f})} (Δ σ_{n} Δ ε_{n}) + \frac{1 + σ_{n, mean} / σ'_{f}}{τ'_{f} γ'_{f}} (Δ τ_{\max} \frac{Δ γ_{\max}}{2})

(69)

where

Δ σ_{n}

Δ ε_{n}

, and

Δ τ_{\max}

are the normal stress range, normal strain range, and shear stress range calculated on the critical plane. The factor (1 +

σ_{n, mean}

/σ′_f) is employed to take the mean stress effect into account. The shear stress amplitude

Δ τ_{\max}

is characterized by the “max” label, because it is calculated at the time instant where the Mohr’s circles are the largest, hence at the time instant when it is the largest. This is not to be confused with the reason why the shear strain

Δ γ_{\max}

has the “max” label, which is because it is calculated on the plane where it is the largest. The original work of Varvani and Farahani does not explicit the BCM-type function that binds the damage parameter to the fatigue life N_f. However, other authors (Han et al., 2002) reformulated the criterion and expressed its relationship with N_f:

(\frac{τ'_{f} γ'_{f}}{σ'_{f} ε'_{f}}) Δ σ_{n} Δ ε_{n} + (1 + \frac{σ_{n, m}}{σ'_{f}}) (Δ τ_{\max} \frac{Δ γ_{\max}}{2}) = 2 \frac{τ_{f}'^{2}}{G} {(2 N_{f})}^{2 b_{γ}} + 2 τ'_{f} γ'_{f} {(2 N_{f})}^{b_{γ} + c_{γ}}

(70)

Conclusions

A review of currently available multiaxial fatigue criteria for low-cycle fatigue life prediction is presented. The review is carried out through a systematic approach, which makes it unbiased, objective and replicable. Moreover, it is enriched with the basic theoretical notions that are necessary to understand how MLCF criteria work and can be employed. A complete classification of MLCF criteria is also provided, specifying the differences with other classifications. These features make the review particularly useful for readers approaching the subject for the first time.

The systematic review analyzes more than thirty different models between criteria and their modifications. What can be observed is that there are multiple available criteria that can be employed to analyze MLCF. Most of the models have their own strengths and shortcomings, and in many cases they cannot be properly compared as their validity has been proven on a restricted number of metals or only under certain loading conditions. As a result, no consensus can be found about a definitive criterion proving its higher accuracy for all materials or loading conditions (RBong-Ryul and Soon-Bok, 1996). This is due to the uncontrolled boom of newly developed criteria that took place starting from the 1990s (Papuga et al., 2022), which is well described by Stefanov (Stefanov, 1996): “It is fairly traditional that each author develops his own criterion for fatigue life prediction and verifies it by his own experimental data. Then some other author’s data are not satisfied by that criterion, and a new one is suggested. Thus too many proposed criteria have been accumulated […]. Many criteria have remained isolated each from other, without comparison or competition.” Consequently, further analysis of the available MLCF criteria is needed, in order to effectively compare criteria and obtain a better understanding of their accuracy and capabilities. In conclusion, the continuous development of new materials requires the criteria to be continuously tested to asses their reliability. Future steps intended by the authors of this work are the quantitative comparison between different criteria to assess their reliability when applied to materials yet to be investigated, such as additively manufactured materials, or when applied to complex loading conditions, such as variable amplitude loading.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Lorenzo Pagliari

References

A. Code (1988), Case n-47-23 of Asme Boiler and Pressure Vessel Code. New York: American Society of Mechanical Engineers

Ahn

S-H

Nam

K-W

Yoo

Y-S

, et al. (2002) Fracture behavior of straight pipe and elbow with local wall thinning. Nuclear Engineering and Design 211(2–3): 91–103.

Basquin

(1910) The exponential law of endurance tests. American Society for Testing and Materials Proceedings 10: 625–630.

Beaver

(1985) Biaxial fatigue and fracture of metals: a review. Metals Forum 8(1): 14–29.

Benedetti

Fontanari

Bandini

, et al. (2014) Multiaxial fatigue resistance of shot peened high-strength aluminum alloys. International Journal of Fatigue 61: 271–282.

Bolotin

(1999) Mechanics of Fatigue. 1st ed. Boca Raton: CRC Press.

Borodii

Strizhalo

(2000) Analysis of the experimental data on a low cycle fatigue under nonproportional straining. International Journal of Fatigue 22(4): 275–282.

Borodii

(1996) An Engineering method for Determining Nonproportionality Parameter under Biaxial Fatigue Loading. Tech. rep. New York, NY: American Society of Mechanical Engineers.

Brighenti

Carpinteri

(2012) A notch multiaxial-fatigue approach based on damage mechanics. International Journal of Fatigue 39: 122–133.

10.

Brighenti

Carpinteri

Corbari

(2013) Damage mechanics and Paris regime in fatigue life assessment of metals. International Journal of Pressure Vessels and Piping 104: 57–68.

11.

Brown

Miller

(1973) A theory for fatigue failure under multiaxial stress-strain conditions. Proceedings of the Institution of Mechanical Engineers 187(1): 745–755.

12.

Brown

Surer

Wang

(1996) An analysis of mean stress in multiaxial random fatigue. Fatigue & Fracture of Engineering Materials & Structures 19(2–3): 323–333.

13.

Calvo

Canales

Gómez

, et al. (2011) Probabilistic formulation of the multiaxial fatigue damage of Liu. International Journal of Fatigue 33(3): 460–465.

14.

Campbell

F C

(2012) Fatigue and Failure, Understanding the Basics. 1st ed. Materials Park: ASM International.

15.

Carpinteri

Spagnoli

Vantadori

(2017) A review of multiaxial fatigue criteria for random variable amplitude loads. Fatigue & Fracture of Engineering Materials & Structures 40(7): 1007–1036.

16.

Chen

Jin

Kim

(2006) A weight function-critical plane approach for low-cycle fatigue under variable amplitude multiaxial loading. Fatigue & Fracture of Engineering Materials & Structures 29(4): 331–339.

17.

Chen

Huang

(1999) A critical plane-strain energy density criterion for multiaxial low-cycle fatigue life under non-proportional loading. Fatigue & Fracture of Engineering Materials & Structures 22(8): 679–686.

18.

Chu

C-C

Conle

Bonnen

(1993) Multiaxial stress-strain modeling and fatigue life prediction of SAE axle shafts. ASTM Special Technical Publication 1191: 37–37.

19.

Concli

(2022) Numerical study of the impact of shot peening on the tooth root fatigue performances of gears using critical plane fatigue criteria. Applied Sciences 12(16): 8245.

20.

Concli

Fraccaroli

Maccioni

(2021a) Gear root bending strength: A new multiaxial approach to translate the results of single tooth bending fatigue tests to meshing gears. Metals 11(6): 863.

21.

Concli

Fraccaroli

Nalli

, et al. (2022) High and low-cycle-fatigue properties of 17–4 ph manufactured via selective laser melting in as-built, machined and hipped conditions. Progress in Additive Manufacturing 7(1): 99–109.

22.

Concli

Gerosa

Panzeri

, et al. (2023) High and low cycle fatigue properties of selective laser melted aisi 316l and alsi10mg. International Journal of Fatigue 177: 107931.

23.

Concli

Maccioni

Fraccaroli

, et al. (2021b) Early crack propagation in single tooth bending fatigue: Combination of finite element analysis and critical-planes fatigue criteria. Metals 11(11): 1871.

24.

Concli

Maccioni

Fraccaroli

, et al. (2022) Effect of gear design parameters on stress histories induced by different tooth bending fatigue tests: a numerical-statistical investigation. Applied Sciences 12(8): 3950.

25.

Cruces

Lopez-Crespo

Moreno

, et al. (2018) Multiaxial fatigue life prediction on s355 structural and offshore steel using the sks critical plane model. Metals 8(12): 1060.

26.

de Castro e Sousa

Nussbaumer

(2019) Multiaxial ultra low cycle fatigue in welded high strength steel structural components. Journal of Constructional Steel Research 153: 473–482.

27.

De-Guang

De-Jun

(1998) A new multiaxial fatigue damage model based on the critical plane approach. International Journal of Fatigue 20(3): 241–245.

28.

Deng

Xiong

, et al. (2022) Analysis of biaxial proportional low-cycle fatigue and biaxial accumulative plasticity of hull inclined-crack plate. International Journal of Naval Architecture and Ocean Engineering 14: 100423.

29.

Dileep

Esakki Muthu

Udayanan

, et al. (2015) Multiaxial fatigue damage prediction and life estimation of a centrifugal impeller for a turboshaft engine. Journal of Failure Analysis and Prevention 15(6): 883–891.

30.

Dufailly

Lemaitre

(1995) Modelling very low cycle fatigue. International Journal of Damage Mechanics 4(2): 153–170.

31.

Ellyin

(1974) A criterion for fatigue under multiaxial states of stress. Mechanics Research Communications 1(4): 219–224.

32.

Ellyin

Kujawski

(1993) A multiaxial fatigue criterion including mean-stress effect. ASTM Special Technical Publication 1191: 55–55.

33.

Farhat

(2021) Operation, Maintenance, and Repair of Land-Based Gas Turbines. Amsterdam: Elsevier.

34.

Fatemi

Shamsaei

(2011) Multiaxial fatigue: an overview and some approximation models for life estimation. International Journal of Fatigue 33(8): 948–958.

35.

Fatemi

Socie

(1988) A critical plane approach to multiaxial fatigue damage including out-of-phase loading. Fatigue & Fracture of Engineering Materials & Structures 11(3): 149–165.

36.

Filippini

Foletti

Papadopoulos

, et al. (2003) A multiaxial fatigue life criterion for non-symmetrical and non-proportional elasto-plastic deformation. European Structural Integrity Society 31: 383–400.

37.

Fontanesi

Giacopini

(2013) Multiphase cfd–cht optimization of the cooling jacket and fem analysis of the engine head of a v6 diesel engine. Applied Thermal Engineering 52(2): 293–303.

38.

Foti

Razavi

Fatemi

, et al. (2023) Multiaxial fatigue of additively manufactured metallic components: a review of the failure mechanisms and fatigue life prediction methodologies. Progress in Materials Science 137: 101126.

39.

Garud

(1981) A new approach to the evaluation of fatigue under multiaxial loadings. Journal of Engineering Materials and Technology 103(2): 118–125.

40.

Garud

(1981) Multiaxial fatigue: A survey of the state of the art. A Journal of Testing and Evaluation 9(3): 165–178.

41.

Glinka

Shen

Plumtree

(1995) A multiaxial fatigue strain energy density parameter related to the critical fracture plane. Fatigue & Fracture of Engineering Materials & Structures 18(1): 37–46.

42.

Glinka

Wang

Plumtree

(1995) Mean stress effects in multiaxial fatigue. Fatigue & Fracture of Engineering Materials & Structures 18(7–8): 755–764.

43.

Han

Chen

Kim

(2002) Evaluation of multiaxial fatigue criteria under irregular loading. International Journal of Fatigue 24(9): 913–922.

44.

Wang

(2013) Combined fatigue experiments on full scale turbine components. Aircraft Engineering and Aerospace Technology 85(1): 4–9.

45.

Huang

Zhao

(2020) Extremely-low-cycle fatigue damage for beam-to-column welded joints using structural details. Materials 13(7): 1768.

46.

Ince

Glinka

(2014) A generalized fatigue damage parameter for multiaxial fatigue life prediction under proportional and non-proportional loadings. International Journal of Fatigue 62: 34–41.

47.

Itoh

Sakane

Ohnami

, et al. (1995) Nonproportional low cycle fatigue criterion for type 304 stainless steel. Journal of Engineering Materials and Technology 117(3): 285–292.

48.

J, Coffin

(1954) A study of the effects of cyclic thermal stresses on a ductile metal,. Journal of Fluids Engineering 76(6): 931–949.

49.

Jahed

Varvani-Farahani

(2006) Upper and lower fatigue life limits model using energy-based fatigue properties. International Journal of Fatigue 28(5-6): 467–473.

50.

Jahed

Varvani-Farahani

Noban

, et al. (2007) An energy-based fatigue life assessment model for various metallic materials under proportional and non-proportional loading conditions. International Journal of Fatigue 29(4): 647–655.

51.

Jiang (2000) A fatigue criterion for general multiaxial loading. Fatigue & Fracture of Engineering Materials & Structures 23(1): 19–32.

52.

Jiang

Hertel

Vormwald

(2007) An experimental evaluation of three critical plane multiaxial fatigue criteria. International Journal of Fatigue 29(8): 1490–1502.

53.

Kanazawa

Miller

Brown

(1977) Low-cycle fatigue under out-of-phase loading conditions. Journal of Engineering Materials and Technology 99(3): 222–228.

54.

Kanazawa

Miller

Brown

(1979) Cyclic deformation of 1% cr-mo-v steel under out-of-phase loads. Fatigue & Fracture of Engineering Materials and Structures 2(2): 217–228.

55.

Kandil

Brown

Miller

(1982) Biaxial low-cycle fatigue failure of 316 stainless steel at elevated temperatures, in. Mechanical Behaviour and Nuclear Applications of Stainless Steel at Elevated Temperatures

56.

Karolczuk

Macha

(2005) A review of critical plane orientations in multiaxial fatigue failure criteria of metallic materials. International Journal of Fracture 134(3-4): 267–304.

57.

Karunananda

Ohga

Dissanayake

, et al. (2011) Effect of high amplitude loading on fatigue life prediction of steel bridges. Procedia Engineering 14: 521–528.

58.

Lefebvre

Neale

Ellyin

(1981) A criterion for low-cycle fatigue failure under biaxial states of stress. Journal of Engineering Materials and Technology 103(1): 1–6.

59.

Reis

De Freitas

(2006a) Simulation of cyclic stress/strain evolutions for multiaxial fatigue life prediction. International Journal of Fatigue 28(5-6): 451–458.

60.

Reis

de Freitas

(2006b) Simulations of cyclic plasticity and fatigue behavior of structural steels under multiaxial loading. Materials Science Forum 514–516: 1414–1418.

61.

Zhang

Sun

, et al. (2011a) Low-cycle fatigue life prediction of various metallic materials under multiaxial loading. Fatigue & Fracture of Engineering Materials & Structures 34(4): 280–290.

62.

Zhang

Z-p

Sun

, et al. (2011b) Multiaxial fatigue life prediction for various metallic materials based on the critical plane approach. International Journal of Fatigue 33(2): 90–101.

63.

Liu

Yan

(2018) A new method for studying the effect of multiaxial strain states on low cycle non-proportional fatigue prediction. International Journal of Fatigue 117: 420–431.

64.

Liu

Wei

, et al. (2020) A novel model for low-cycle multiaxial fatigue life prediction based on the critical plane-damage parameter. Science Progress 103(3): 36850420936220.

65.

Liu

K (1993) A method based on virtual strain-energy parameters for multiaxial fatigue life prediction. ASTM Special Technical Publication 1191: 67–67.

66.

Lohr

Ellison

(1980) A simple theory for low cycle multiaxial fatigue. Fatigue & Fracture of Engineering Materials and Structures 3(1): 1–17.

67.

Lopez-Crespo

Moreno

Lopez-Moreno

, et al. (2015) Study of crack orientation and fatigue life prediction in biaxial fatigue with critical plane models. Engineering Fracture Mechanics 136: 115–130.

68.

Zhong

(2019) A modified energy-based model for low-cycle fatigue life prediction under multiaxial irregular loading. International Journal of Fatigue 128: 105187.

69.

Luo

Yao

Susmel

, et al. (2017) A survey on multiaxial fatigue damage parameters under non-proportional loadings. Fatigue & Fracture of Engineering Materials & Structures 40(9): 1323–1342.

70.

T-H

Chang

Guo

, et al. (2020) Comparison of multiaxial low cycle fatigue behavior of cp-ti under strain-controlled mode at different multiaxial strain ratios. International Journal of Fatigue 140: 105818.

71.

T-H

Gao

Chang

, et al. (2021) Low-cycle fatigue behavior and life prediction of cp-ti under non-proportional and multiaxial loading. Engineering Fracture Mechanics 254: 107930.

72.

T-H

Zhou

C-Y

Chang

, et al. (2023) Multiaxial low cycle fatigue behavior and life prediction of cp-ti under non-proportional stress-controlled mode. Fatigue & Fracture of Engineering Materials & Structures 46(1): 341–356.

73.

Macha Sonsino (1999) Energy criteria of multiaxial fatigue failure. Fatigue & Fracture of Engineering Materials & Structures 22(12): 1053–1070.

74.

Masing

(1926) Eigenspannungen und verfestigung beim messing (self stretching and hard-ening for brass. In: Proceedings of the second international congress for applied mechanics. Zurich, Swizerland, pp. 332–335.

75.

Miner

(1945) Cumulative damage in fatigue. Journal of Applied Mechanics 12(3): A159–A164.

76.

Morishita

Itoh

(2016) Evaluation of multiaxial low cycle fatigue life for type 316l stainless steel notched specimen under non-proportional loading. Theoretical and Applied Fracture Mechanics 84: 98–105.

77.

Mroz

(1967) On the description of anisotropic workhardening. Journal of the Mechanics and Physics of Solids 15(3): 163–175.

78.

Nagel

Knödel

Ummenhofer

(2017) Testing of ultra-low cycle fatigue at complex loading scenarios. Procedia Structural Integrity 5: 1377–1384.

79.

Pagliari

Fraccaroli

Concli

(2023) Numerical analysis of the impact of shot peening on the tooth root strength of alsi10mg gears using critical plane multiaxial fatigue criteria. In: International symposium on industrial engineering and automation. Berlin: Springer, pp. 667–679.

80.

Pan

W-F

Hung

C-Y

Chen

L-L

(1999) Fatigue life estimation under multiaxial loadings. International Journal of Fatigue 21(1): 3–10.

81.

Papuga

(2011) A survey on evaluating the fatigue limit under multiaxial loading. International Journal of Fatigue 33(2): 153–165.

82.

Papuga

Nesládek

Hasse

, et al. (2022) Benchmarking newer multiaxial fatigue strength criteria on data sets of various sizes. Metals 12(2): 289.

83.

Paul

(2016) Prediction of non-proportional cyclic hardening and multiaxial fatigue life for fcc and bcc metals under constant amplitude of strain cycling. Materials Science and Engineering: A 656: 111–119.

84.

Pereira

JCR

de Jesus

AMP

Xavier

, et al. (2014) Ultra low-cycle fatigue behaviour of a structural steel. Engineering Structures 60(3): 214–222.

85.

Radhakrishnan

(1995) Multiaxial fatigue – An overview. Sadhana 20(1): 103–122.

86.

Ramberg

Osgood

(1943) Description of stress–strain curves by three parameters. Technical Note. no. 902, Tech. rep. Washington, DC: National Advisory Committee For Aeronautics.

87.

RBong-Ryul

Soon-Bok

(1996) A critical review on multiaxial fatigue assessments of metals. International Journal of Fatigue 18(4): 235–244.

88.

Reis

De Freitas

(2009) Crack initiation and growth path under multiaxial fatigue loading in structural steels. International Journal of Fatigue 31(11–12): 1660–1668.

89.

Rolovic

Tipton

(1999) Multiaxial cyclic ratcheting in coiled tubing – Part I: Theoretical modeling. Journal of Engineering Materials and Technology 122(2): 157–161.

90.

Ryu

Choi

Huh

, et al. (2018) Effects of work hardening models on low-cycle fatigue evaluations of coiled tubing with CT-100 steel. Journal of Mechanical Science and Technology 32(11): 5055–5061.

91.

Shamsaei

Fatemi

(2009) Effect of hardness on multiaxial fatigue behaviour and some simple approximations for steels. Fatigue & Fracture of Engineering Materials & Structures 32(8): 631–646.

92.

Smith

(1970) A stress-strain function for the fatigue of metals. Journal of Materials 5: 767–778.

93.

Socie

(1987) Multiaxial fatigue damage models. Journal of Engineering Materials and Technology 109(4): 293–298.

94.

Socie

Marquis

(1999) Multiaxial Fatigue. Warrendale: SAE International.

95.

Socie

Waill

Dittmer

(1985) Biaxial fatigue of inconel 718 including mean stress effects. In: Multiaxial Fatigue. West Conshohocken, PA: ASTM International.

96.

Song

Liu

, et al. (2021) Multiaxial low cycle fatigue of notched 10crni3mov steel and its undermatched welds. International Journal of Fatigue 150: 106309.

97.

Standard (2011) Standard Practices for Cycle Counting in Fatigue Analysis. West Conshohocken, PA: ASTM International.

98.

Stefanov

(1996) The curvilinear integral method: Testing 2 (under non-proportional pulsating axial force and internal pressure). International Journal of Fatigue 18(1): 41–48.

99.

Stephens

Fatemi

Stephens

(2000) Metal Fatigue in Engineering. Hoboken: Wiley Interscience.

100.

Sun

G-Q

Shang

D-G

Bao

(2010) Multiaxial fatigue damage parameter and life prediction under low cycle loading for gh4169 alloy and other structural materials. International Journal of Fatigue 32(7): 1108–1115.

101.

Taira

Inoue

Takabashi

(1969) Low-cycle fatigue under multiaxial stresses (in the case of combined cyclic tension-compression and cyclic torsion at room temperature), Trans. The JapanSociety of Mechanical Engineers 35: 525–532.

102.

Takahashi

Ando

Matsuo

, et al. (2014) Estimation of low-cycle fatigue life of elbow pipes considering the multi-axial stress effect. Journal of Pressure Vessel Technology 136(4): 041405.

103.

Takahashi

Watanabe

Ando

, et al. (2009) Low cycle fatigue behaviors of elbow pipe with local wall thinning. Nuclear Engineering and Design 239(12): 2719–2727.

104.

Tomkins

(1968) Fatigue crack propagation – An analysis. Philosophical Magazine 18(155): 1041–1066.

105.

Varvani-Farahani

(2000) A new energy-critical plane parameter for fatigue life assessment of various metallic materials subjected to in-phase and out-of-phase multiaxial fatigue loading conditions. International Journal of Fatigue 22(4): 295–305.

106.

Wöhler

(1860) Versucheüber die festiykeit eisenbahnwagenuchsen. Fatigue & Fracture of Engineering Materials & Structures 10.

107.

Wang

Brown

(1993) A path-independent parameter for fatigue under proportional and non-proportional loading. Fatigue & Fracture of Engineering Materials & Structures 16(12): 1285–1297.

108.

Wang

Khan

Bathias

(2012) Current understanding of ultra-high cycle fatigue. Theoretical and Applied Mechanics Letters 2(3): 031002.

109.

Xin

(2013) 2 - durability and Reliability in Diesel Engine System Design. Sawston: Woodhead Publishing.

110.

Xue

McDowell

Horstemeyer

, et al. (2007) Microstructure-based multistage fatigue modeling of aluminum alloy 7075-t651. Engineering Fracture Mechanics 74(17): 2810–2823.

111.

Zhu

Liu

, et al. (2017) Multiaxial fatigue damage parameter and life prediction without any additional material constants. Materials 10(8): 923.

112.

Zarandi

Skallerud

(2020) Cyclic behavior and strain energy-based fatigue damage analysis of mooring chains high strength steel. Marine Structures 70: 102703.

113.

Zhang

(2013) Cyclic Hardening/Softening. Boston, MA: Springer US.

114.

Zhao

(2016) An evaluation of the strain-based critical plane parameters for multiaxial low-cycle fatigue evaluation, modelling. Measurement and Control B 85(1): 150–162.

115.

Zhao

J-Y

Zhou

C-Y

, et al. (2022) In-plane biaxial ratcheting effect and low-cycle fatigue behavior of cp-ti based on dic method. Fatigue & Fracture of Engineering Materials & Structures 45(5): 1464–1479.

116.

Zheng

Chen

Zhang

, et al. (2022) Multiaxial low cycle fatigue behavior and life prediction method of 316ln stainless steel at 550°C. International Journal of Fatigue 156: 106637.

117.

Zhong

(2020) Review: Energy methods for multiaxial fatigue life prediction. Journal of Harbin Institute of Technology (New Series) 27(2): 1–9.