Evaluation of new multiaxial damage parameters on low carbon steel

Most mechanical components are subjected to the complex fatigue loading conditions, where both amplitude and direction of loading cycles change over the time. The estimation of damage caused by these complex loading scenarios are often done by simplified uniaxial fatigue theories, which ultimately leads to higher factor of safety during the final design considerations. Critical plane-based fatigue theories have been considered more accurate for computing the fatigue damage for multiaxial loading conditions in comparison to energy-based and equivalent stress-based theories. Two recently developed fatigue theories have been evaluated in this work for the available test data. Test data includes significant amount of biaxial load paths.


INTRODUCTION
ver the years, the analysis of complex multiaxial fatigue loading has mainly been done with the simplified uniaxial approach.This has led to the large amount of knowledge generated in this area, both from experimental [1,2] and numerical [3,4] points of view.Advanced experiments have allowed a number of features to be characterized, including surface and bulk behavior [5,6], crack-closure mechanisms [7,8], evolution of the process zone thru the thickness [9,10], crack branching processes [11], effect of variable amplitude loads such as crack acceleration [12] O or retardation [13,14] or mixed-mode local changes in the volume [15].However, uniaxial approach does not represent the real world scenario and computation of fatigue life thru these approach often results either in under prediction or over prediction of fatigue life for the machine component.Because of the complexities and computational cost involved with the multiaxial fatigue tests, very limited amount of fatigue data are publicly available [16].Among several other fatigue models, critical plane-based fatigue models provide more accurate prediction of fatigue lives.Critical plane-based fatigue models are based upon the determination of the location of fatigue crack initiation and microcrack -propagation [17].The damage is computed at the plane where crack initiates, and this plane is called critical plane, however; the definition of critical plane is still controversial among the fatigue researchers.Some researchers define this plane as the plane of maximum fatigue damage, whereas other group of researchers believe that considering the plane of maximum shear stress as critical plane will be more logical and computationally cheaper.Fatigue data from the multiaxial test on St52-3N steel specimen have been analyzed in this paper by using new Fatemi-Socie model [18] and Suman-Kallmeyer model [19].It is well known fact that the shear stresses at the critical plane provokes the fatigue crack initiation whereas, presence of tensile stress in combination of shear stress creates condition for the microcrack to propagate.Both of the damage parameters [18,19] discussed in this paper consider the interaction of normal and shear stress at the critical plane.Both of these damage parameters [18,19] have been used to predict the fatigue life of the test specimen, and the correlation between experimental and predicted fatigue life has been analyzed.

MATERIALS AND METHODS
aterial and test data have been taken from the previous biaxial fatigue test program at the University of Malaga [20].The material that has been considered for this paper is St52-3N.The chemical composition for this steel is: 0.17% C, 1.235% Mn, 0.225% Si, 0.010% P, 0.0006% S, 0.032% Al, 0.072% Cr, 0.058% Ni and 0.016% Mo.This is a low carbon steel, and has been widely used in the structural applications in the construction, manufacturing, ship building and offshore industries [21] that combines good fatigue resistance with low environmental impact for applications where no energy is consumed during the use phase of the component [22].Monotonic tension, compression and torsion tests were conducted to evaluate the monotonic properties of the St52-3N steel (Tab.1).Tension tests were conducted on the solid specimens, whereas, compression and torsion tests were conducted on the tubular specimens [23].Biaxial tests require both torsional and longitudinal extensometer.It also needs a technician with good skill to place four pins of these devices [24].As per the ASTM, load was also recommended to be applied at relatively low frequency (normally less than 4 Hz) because the weak connection between extensometer pins and the sample is preserved.A series of in-phase biaxial tests were conducted with the strain levels that were chosen with an intension to fail the samples between 10 4 and 10 6 .The design of these bi-axial fatigue tests was done on the basis of the previous uniaxial test results.Tab. 4 shows the applied axial strain amplitudes (ɛa), and shear strain amplitudes (ɤa) data for these tests.The corresponding axial stress amplitudes (σ a ) and shear stress amplitudes (τ a ) are also shown in Tab. 4 with corresponding fatigue life.Total of 13 samples were tested for this test program.All the load path for these bi-axial tests were kept proportional (Tab.4).

Fatemi-Socie damage parameter II (FSDP-II)
atemi-Socie has recently modified their earlier developed fatigue model [25].The previous Fatemi-Socie model was based on the concept that the range of shear strain is primary reason to initiate the fatigue crack and the tensile normal stress creates secondary effects of opening the crack face.Fatemi-Socie also introduced the normal stress component to account for the rotation of principle plane axis at the critical plane.Fatemi-Socie have recently modified their damage parameter and introduced a shear stress range term to account for the interaction of normal and shear stress at the critical plane.Fatemi & Socie model resorts on the ratio of maximum normal stress and the range of shear stress to model the interaction.A material dependent parameter k has also been introduced in the parameter.Modified Fatemi-Socie fatigue damage parameter has been shown in Eq. 1 [18].

Suman & Kallmeyer damage parameter (SKDP)
The importance of interaction of normal and shear stress on the critical plane has recently been investigated by Suman & Kallmeyer [19].They have used the product of normal and shear stress at the critical plane to model this interaction.The product term in Suman & Kallmeyer model represents the maximum value of the product of normal and shear stress at the critical plane.By considering this product term, Suman & Kallmeyer were able to overcome the ambiguity caused by the non-proportional loading where both normal and shear stress peaks do not occur at the same time point.This product term can model the interaction effects for wide range of in-phase and out of phase fatigue data.Apart from that, this formulation (Eq 3) also has significantly less number of material dependent parameter in comparison to the model previously developed by the same group of researchers, and provides excellent correlation between experimental and predicted fatigue lives of the steel and titanium specimen.Suman & Kallmeyer fatigue model also captures the effect of strain hardening due to LCF loading and the mean shear stress at the critical plane.

RESULTS
n this study, most of the load path used are proportional and sinusoidal.Due to the proportionality, both shear and normal stress peaks in these tests happened at the same time point.The critical plane stresses for the test IP1 (Tab.

Suman-Kallmeyer Damage parameter
In contrast to Fatemi-Socie damage parameter, the difference in the location of maximum shear stress range plane and the plane where the value of Suman & Kallmeyer damage parameter value is maximum is much smaller.While computing the damage parameter for the proportional load path, computation of maximum shear plane was comparatively easy.Life-Life plots were also created for the test data by using both Fatemi-Socie II [18]

CONCLUSION
wo recently developed fatigue theories have been studied on experimental data.The experimental data were acquired on biaxial tests combining tension-compression and torsion under proportional load.The two fatigue theories were based on the critical plane approach.One of them is an enhanced formulation of the Fatemi-Socie damage parameter.The other theory uses the product of the normal and the shear stress at the critical plane.Both theories provided an overall good prediction for the loading cases analyzed.While the new Fatemi-Socie damage parameter tended to overestimate the fatigue life preditions, the Suman-Kallmeyer tended to underestimate the life preditions.

Fig 1
shows the geometry and dimensions of the tubular specimen.

Figure 1 :
Figure 1: Geometry of the tubular hollow specimen used in the experiments.All dimensions are in mm.

4 )
are presented in Fig 2 and Fig 4. The peaks of normal and shear stresses are presented by solid and dashed lines respectively, and the damage parameter values are presented by the green solid line (damage parameter is obtained with the maximum load).

IFatemi-
Socie damage parameter IIIt can be observed from the plots in Fig2that the maximum value of Fatemi-Socie damage parameter and the maximum value for shear stress range is not at the same time point, which implies that the plane of maximum shear and plane of maximum damage parameter is far apart.
FSDP-II was computed for all the test data shown in Tab. 4, and the damage parameter value was presented against the cycles to failure.The plot is shown in Fig3, and it can be noticed that the FSDP-II models most of the LCF fatigue behavior data but it produces significant scatter in the HCF data range.

Figure 3 :
Figure 3: Fatigue life correlation for St52-3N steel based on Fatemi II model

Figure 4 :
Figure 4: IP 1 -σ, τ and SKDP at max peak cycle (A) and min peak cycle (B) SKDP was computed for the test data shown in Tab. 4, and the material dependent parameters were optimized with a dual power law fit line.After several iterations of computations to optimize these damage parameter, it was observed that fit was good for larger w & k values for this data set.The direction of shear stress in this parameter is not taken into the account because both positive and negative shear stress create same crack initiation phenomenon.The damage parameter vs cycles to failure for the Suman-Kallmeyer damage parameter is shown in fig 5, and it can be observed from the curve that both LCF and HCF data have collapsed very well along the fitting line.Comparison of fig 3 and fig 5 appears to indicate that the Suman-Kallmeyer damage parameter provided better correlation for the test data.
damage parameter and the damage parameter proposed by Suman-Kallmeyer [19].Red square in fig 6 represent the life predicted by FSDP II, and green triangles represent the life predicted by SKDM model.By observing this plot (fig 6) and the data, the FSDP II tends to overestimate the fatigue life and the SKDP tends to underestimate the fatigue life.Similar level of correlation is observed on both damage parameters.The two new theories give overall good predictions for the test data under study.

Figure 6 :
Figure 6: Predicted vs. experimental life for St52-3N using Fatemi II model and Sandip model.

Table 4 :
Strain and stress conditions for in-phase strain conditions and the obtained fatigue life in number of cycles.