New research findings on non-proportional low cycle fatigue

One of the challenges regarding multiaxial fatigue damage predictions is non-proportional loading. Relevant studies have shown that these multiaxial loadings cause significant additional hardening and reduction in durability due to non-proportionality. Fatigue life predictions due to non-proportional loadings are based on an equivalent nonproportional strain range that considers a material constant related to additional hardening and a non-proportionality factor. In this paper an analysis of the non-proportional factor for three multiaxial loadings forming a square in γ/√3 – ε coordinates is carried out. One of the observations revealed by this analysis is the sensitivity of the non-proportional factor to variable shear strain rate.


Introduction
The stress or strain state in a material point is defined as a multiaxial state. This multiaxiality is influenced by the external load, geometry, residual stress, etc. The problem of multiaxiality becomes more complicated when the stress/strain components vary over time and even more when they vary in an independent manner or at different frequencies. The mode of time variation of the stress/strain state components introduces effects that complicate the prediction of fatigue damage. Multiaxial loadings are non-proportional if the principal directions and the corresponding principal stresses/strains change over a loading cycle. Such loadings are very common in the operation of mechanical components, can be complex as a variation over time and it is difficult to quantify their effect on the fatigue life. However, considerable efforts have been made in this direction for development of methods, algorithms and/or criteria for analysing complex stress/strains histories and, respectively, the assessment of non-proportional effects on fatigue damage. Based on an experimental study of low-cycle fatigue tests under combined axial and torsional strains with different phase angles, Kanazawa, Brown and Miller, [1][2][3], highlighted a number of significant issues. These can be summarized as: a) during out-of-phase biaxial tests, fatigue cracks initiate on the plane of maximum shear strain; b) the fatigue life is governed by the shear strain range and the amplitude of normal strain on the plane of maximum shear strain; c) the out-of-phase loading condition with phase angle of 90° and strain ratio λ = γa/εa = 1.5 (where γa is the amplitude of shear strain and εa is the amplitude of normal strain) gave the lowest lifetime. Another important observation of their studies is the additional hardening phenomenon due to the rotation of the principal directions during a cycle. Considering this phenomenon quantified through a parameter, it was possible to correlate out-of-phase data in a single stress-strain curve. Very important contributions were made by Socie and his collaborators who started by experimentally highlighting two cracking modes of materials (Mode II -shear cracking and Mode I -tensile-cracking) that need to be considered in lifetime predictions of components subjected to multiaxial fatigue, [4,5]. Itoh et al, [6,7], explain the rotation of principal directions by the degree of non-proportionality that is found to be maximum for a load which in γ/√3 -ε coordinates forms a square. Also, an equivalent non-proportional strain range is defined which correlates the non-proportional experimental data. This strain range includes a material constant that describes the additional cyclic hardening under 90° out-ofphase and a factor that expresses the severity of non-proportional loading obtained directly from the strain history. Non-proportional multiaxial loadings result in the initiation of large number of microcracks compared to proportional loadings. These microcracks propagate in different directions due to the rotation of the principal directions and the activation of multiple slip systems. At the same time, the large number of microcracks initiated is responsible for reducing the fatigue life, [8]. Shamsaei et al., [9,10], found that the fatigue life of 1050 QT steel without additional non-proportional hardening is more sensitive to multiaxial non-proportional loadings than 304L stainless steel which shows significant nonproportional cyclic hardening. This observation supports the idea that additional nonproportional hardening is not the only effect responsible for reducing the fatigue life. An alternative method for evaluation the non-proportional loading effects was proposed by Freitas et al., [11]. They evaluate the effective shear stress/strain amplitude throughout a complex loading based on the geometric characteristics of a minimum ellipse embodying the multiaxial load path. Recent studies suggest a link between the microstructure of the material and the non-proportional loading effects. The non-proportional hardening may be related to stacking faults and dislocation structures, which have different forms produced by different loading paths, [12]. A new non-proportionality damage factor based on accumulative path length traversed within half cycle is proposed in [13] and [14]. This factor can vary from zero for proportional load to 1 for a semi-circular load. These studies are focused on the overall correlation of non-proportional experimental data through an equivalent parameter that includes a factor corresponding to additional cyclic hardening and respectively a factor corresponding to the loading path.
Based on analysis of low cycle multiaxial fatigue data, a slight inconsistency was observed between experimental data and correlations used for non-proportional loadings which describes a square in γ/√3 -ε coordinates. Also, in most cases these loadings determine the lowest fatigue life, even lower than non-proportional 90° out-of-phase loadings.
An analysis of the non-proportionality factor for square loadings is carried out in this paper. Thus, three non-proportional loadings are proposed in the paper which in γ/√3 -ε coordinates describe a square. For these loadings the non-proportional load factor and implicitly their effect on fatigue damage has been analysed.

Material and methods
The analysis performed in this paper has as a starting point three sets of low cycle multiaxial fatigue data, where the square non-proportional loadings have been considered among others. These experimental data include tests conducted by Socie [5], Itoh et al. [6] and Nobah et al. [15]. The experimental analysis by Itoh et al. reveals a maximum degree of non-proportionality in the case of square loading that determined the lowest fatigue life. Continuous loading path similar to the square loading are more damaging compared to discontinuous loading path.
For Nobah et al. tests, the same axial and shear strain amplitudes used for 90° out-of-phase and square loadings gave lower fatigue lives in the case of square loading. The same trend is maintained for Socie's tests. These experimental evidences indicates the need for a detailed analysis of non-proportional square loadings.
On the other hand, the current approach uses an equivalent non-proportional strain range to correlate the non-proportional fatigue data with the proportional ones. This parameter is defined by the following relationship: where ΔεI is the principal strain range, α is a parameter related to the additional hardening under nonproportional loading and is defined as the ratio of stress amplitudę in 90° out-ofphase loading to that from in-phase loading. The nonproportionality factor, fNP, express the severity of nonproportional loading and is obtained directly from the strain history. Practically, the correlation of non-proportional and proportional fatigue data depends on the product of α·fNP. Considering that α is a material constant experimentally determined under the same conditions for each tested materials, attention should be directed to the non-proportional load factor. Also, taking into account the fact that there are materials that do not exhibit additional non-proportional hardening but they are sensitive to non-proportional fatigue loadings, the analysis in this paper focuses on the non-proportional load factor for square loadings. Therefore, three non-proportional loadings are proposed in the paper which in γ/√3 -ε coordinates describe a square. The three loadings are shown in Figs. 1 -3. The three loadings, although having different variations of axial and shear strains, in γ/√3 -ε coordinates they describe a square with the same limits, Fig. 4.   Fig. 4. The square loading described by the three non-proportional loadings Two techniques have been used in this paper to analyse the non-proportionality factor for the three loadings. On the one hand, the method developed by Itoh et all. [6], based on the rotation of the principal strain directions, was used to analyse the severity of each nonproportional loading. The second method, developed by Mei and Dong, [13], based on the accumulative Moment of Load Path (MLP) concept was applied for analysing the dimensionless non-proportionality induced damage factor, gNP.
According to Itoh's approach, the severity of non-proportional low cycle fatigue is calculated by measuring the rotation of the maximum principal strain direction and the strain amplitude after rotation, fig. 5. This severity is expressed by a non-proportional factor, fNP, defined as integral form: where εI(t) is the maximum absolute value of principal strain at time t and given by equation (3), εI,max is the maximum value that principal strain records during the loading path, T is the period of a cycle and ξ(t) is the angle between εI,max and εI(t). The constant 1.57 is chosen to make fNP unity for 90° out-of-phase loading. The Mei and Dong approach is based on a normalized integral form of the load path nonproportionality induced fatigue damage, Fig. 6, given by the following relationship : where DNP is the total load path non-proportionality induced fatigue damage, Dmax is the maximum possible non-proportional fatigue damage induced by a semi-circular load path with radius R.
The term √β ε is a fatigue equivalency parameter between pure cyclic tensile strain and pure shear strain. For this analysis β ε = 1/3. Two evidences are clear based on the definitions of the two non-proportional factors. The dimensionless non-proportional factor gNP is calculated with respect to ε -γ√β ε strain plane while Itoh et al.'s non-proportional factor fNP is defined with respect to a polar coordinate based maximum principal strain plane as a function of its direction (angle). Also, the nonproportional factor gNP is defined with respect to the reference semi-circular load path wich corresponds to conditions yielding the maximum possible damage among all paths between any two positions forming one half cycle in ε -γ√β ε plane, while Itoh's et al.'s factor is calculated with respect to the maximum principal strain within a loading event.

Results and discussion
For each loading path, the variation of the principal strain according to Itoh's method was determined, Fig. 7, as follows: (4) where ε is the applied axial strain, γ is the applied shear strain and ν is the Poisson's ratio.  Fig. 8. Applying the Mei and Dong method, mainly involves the computation of the r'·sin θ term and integrating over the loading path. For the three analysed load paths, the variations of r'·sin θ term during a loading cycle are shown in Fig. 9. Table 1 gives the values of the non-proportional factor for the analysed loadings, based on the two methods used. The model proposed by Kanazawa et al. [2], was used in this paper to estimate the equivalent stress amplitude for a given equivalent strain amplitude, for the three analysed loading paths. This model incorporates the additional non-proportional hardening in the cyclic behaviour and is defined as: where ∆ ̅/2 is the equivalent stress amplitude, ∆ ̅ /2 is the equivalent plastic strain amplitude, ′ is the proportional (uniaxial) cyclic strength coefficient, ′ is the proportional (uniaxial) cyclic strength exponent, α is the additional non-proportional hardening and F is the non-proportional load factor. The model was applied on two materials analysed by Shamsaei et al. [10], one of which does not show additional non-proportional hardening (a medium carbon 1050 steel in quenched and tempered condition), while the other exhibit additional non-proportional hardening (a 304L stainless steel). The used properties of these materials are given in Table 2.  Considering the materials loaded with the three non-proportional square loadings with an equivalent plastic strain amplitude of 0.00566, Figures 10 and 11 present the estimated equivalent stress amplitudes. A first observation that supports the Shamsaei findings on 1050 QT steel is the low sensitivity of the stress response to the loading path. In addition, the non-proportional load factor according to Mei and Dong method indicates an equivalent stress amplitude of only 1.4 % higher than the Itoh method. Instead, for 304L stainless steel, the Itoh method estimate an equivalent stress amplitude of 15 % higher than the Mei and Dong method. Also, in the case of Path I the difference is 24 % for Itoh method compared to Mei and Dong method.
Although the three loadings form the same square in γ/√3 -ε coordinates, the variation law of axial and shear strains influences the severity of the non-proportional loading. Moreover, the severity of non-proportional loading proves to be more sensitive at variable shear strain rates.
The loading Path I presents two important features, primarily, maintaining under constant shear strain for a half-cycle over which axial strain overlaps could have effects on ductility and implicitly on low cycle fatigue life. In a study conducted by Shenguyan Ji et all., [16], it is reported that the ductility decrease gradually with the increase of the shear pre-strain, instead the low cycle fatigue life is divided in three stages. Gradually increasing of the shear pre-strain determines a first stage with monotonically decreasing of fatigue life, followed by a second stage with fatigue life monotonically increasing and a third stage with decreasing the fatigue life. Depending on the amplitude of the shear pre-strain, one of the three stages in the evolution of the low cycle fatigue life can be reached, and this is determined by a combination of material ductility and microstructural transformations. A second characteristic with effect on low cycle fatigue life is the very high strain rate. The transition from the maximum to minimum and vice versa in the axial and shear strains is almost instantaneous. A recent study on structural steel, [17], has shown that yield strength and strain hardening exponent increases with increasing the strain rate. Also, increasing the strain amplitudes cause decrease of the strain rate sensitivity.
Loading Path II is easier to achieve experimentally and probably the most often used in experimental programs that generate non-proportional square loadings. It is characterized by constant strain rates of both axial and shear strains.
Loading Path III shows interest through the sinusoidal variation law of the shear strain overlapped over the axial strain maintained constantly for a half-cycle. Sinusoidal variation involves a variable strain rate of the shear strain.

Conclusion
Multiaxial fatigue loadings continuously generate non-proportional loads with significant effects on component durability. Current approaches on fatigue life predictions are cyclebased methods, but evolution of fatigue damage within a loading cycle is also important. Time dependent effects or stress relaxation are factors that may occur during a loading cycle and requires attention. In this paper, four non-proportional square loadings are analysed for the severity of non-proportionality. Two methods were used to analyse the non-proportional factor, proving that both can be sensitive to different variations of axial and shear strains. Also, the principal strain based method has found a higher non-proportional factor compared to non-proportionality induced fatigue damage method. This study requires a further analysis of stress response and respectively experimental validation.