High-cycle fatigue model calibration with a deterministic optimization approach

A parameter identification approach is proposed to calibrate the Ottosen high cycle fatigue model using numerical optimization with regularization. The damage evolution was predicted by a continuum approach based on a moving endurance surface in the stress space, so the stress states outside the endurance surface may lead to damage evolution. The calibration of the model relied on uniaxial and multiaxial experimental data. The predictions of the calibrated models were in fair agreement with the experimental data for the 7075-T7451 and 7050-T6 aluminum alloys subjected to cyclic uniaxial and multiaxial loadings.


Introduction
Fatigue is widely known as one of the most common failure phenomena in metallic components, and it leads to a severe concern in mechanical design.Consequently, there is significant amount of work focused on enhancing the predictive capabilities of such a process, which shows three main stages [1]: An initial stage of nucleation and growth of micro-cracks and voids leading to formation of a macrocrack.Then two sequential stages of stable and further unstable crack growth happen until failure occurs.Modeling such a process turns out to be a challenging task due to the rich phenomena, and the wide ranges of length and time scales involved.Moreover, the experimental evidence shows that when the stress level is low enough (high cycle fatigue conditions) the duration of the first stage of fatigue is considerably larger than the duration of final two stages [1].Thus, the fatigue modeling objectives, have been mainly directed to predict the crack initiation process, in which a small portion of the volume of the material accumulates damage, while the overall mechanical response is purely elastic [2].
Significant effort has been directed to pose fatigue limit criteria, which is commonly defined with a stress based approach that describes a mechanical threshold for fatigue crack creation.Papadopoulos et al. [3] presented a detailed summary of the most common stress based criteria, which are: Critical plane, stress invariants and stress averages.The critical plane criteria are based on the values of normal and shear stresses ( and , respectively) to define the orientation of E-mail addresses: arturo.rubioruiz@tuni.fi(A.Rubio Ruiz), timo.saksala@tuni.fi(T.Saksala), mikko.hokka@tuni.fi(M.Hokka), reijo.kouhia@tuni.fi(R. Kouhia). 1 The authors would like to express their deepest sorrow for the loss of a fried and colleague Dr. Djebar Baroudi who untimely passed away on March 31, 2023.critical planes and the mechanical requirements for crack formation.Findley [4] presented a criteria in which the orientation of the critical plane is obtained by maximization of a linear combination of  and .Later on, Matake [5] presented a criterion in which the orientation of the critical plane depends only on the value of the , yielding close agreement between model predictions and experimental observations [3].The stress invariant approaches are based on inequality conditions containing the hydrostatic component of the stress (  ) and the second invariant of the deviatoric stress ( 2 ).Crossland [6] proposed a criteria in which √  2 and the maximum value of   are linearly combined.Moreover, the most widely used stress invariant criteria was presented by Sines [7] which consists of a linear combination between √  2 and the mean value of   .This type of approaches do not account for the direction of crack propagation.The stress average criteria are based on averaging the values of  and  over all cutting planes, the work of Liu and Zenner [8] is an example of these approaches.Additionally, theories based on mesoscopic phenomena have been developed to describe fatigue limit criteria.Such formulations were introduced by Dang Van [9] and Papadopoulos provided several contributions [3].
Considerable amount of work can be found on the description of damage evolution in the per cycle basis, as in the case of the well known Palmgren-Miner rule [10].However, these approaches require a well defined load cycle, which might be difficult to define under complex https://doi.org/10.1016/j.ijfatigue.2023.107747Received 10 March 2023; Received in revised form 17 May 2023; Accepted 24 May 2023 loading conditions.Thus, the cycle counting techniques were developed to define equivalent cycles from complex loading histories [11], and the most popular cycle counting approach is the Rainflow method [12][13][14].
Alternatively, the integration of a damage parameter over time has been studied by different authors.The work of Morel [2] presented an approach based on the work of Dang Van [9] and Papadopoulos [3] in which the accumulated plastic strain occurring at mesoscale is considered as the damage variable.Ottosen et al. [15] presented a macroscale continuum based high cycle fatigue (HCF) model in which a moving endurance surface is constructed and the movement is controlled by evolution equation for a back stress like tensor describing the center of the endurance surface in the deviatoric plane.In addition, an evolution equation for a scalar damage variable is defined.This model is inherently multiaxial and does not need any heuristic cycle-counting methods and cycle based damage accumulation rules.This model was later enhanced to account for stress gradient effects that might be significant in complex geometries [16].It can be naturally extended to anisotropy [17], to stochastic loadings [18] and to combine with low cycle fatigue modeling [19].Ottosen et al. [15] presented an approach to calibrate the five material parameters of their model using a theoretically derived system of nonlinear equations to calculate the damage evolution per cycle under tension-compression loading regimes.Lindström et al. [20] studied the properties of the Ottosen model and provided an analytical expression to estimate the damage evolution per cycle.Such expression was utilized by these authors in a parameter identification process for the 7075-T7451 aluminum alloy and the AISI 4340 steel alloy using experimental evidence of samples subjected to cyclic tension-compression, yielding a fair correlation between the model predictions and experimental data.However, Lindström et al. [20] concluded that the Ottosen model has limitations to correctly predict the damage evolution under multiaxial stress states, specially for stress histories which are tangential to the endurance surface.
In the present work a stable parameter estimation procedure for the Ottosen HCF model based on the Morozov discrepancy principle is proposed.The information fed to the model include tension-compression, pure shear and multiaxial In-Phase (IP) and Out-of-Phase(OP) stress histories.Additionally, the modification of the Ottosen HCF model presented in [16] is used to account for stress gradients.Thus, the calibration process is fed with multiaxial cases of notched samples, whose stress histories were computed by finite element (FE) simulations performed in ABAQUS.Parameter identification inverse problems are often ill-posed, and they are not expected to have unique solutions.The crucial issue in handling ill-posed inverse problems is to find a solution which is stable with respect to measurement noise.The Morozov discrepancy principle [21] ensure such stability of the identified parameters.The developed procedure is used to identify the model parameters based on data from uniaxial and multiaxial loading cases for the 7075-T7451 and 7050-T6 aluminum alloys.A weighted least-square residual is used with the inverse of the measurement variances thus emphasizing data with high accuracy.The attained model predictions for both aluminum alloys in uniaxial tension-compression are in good agreement with the experimental data, yielding average relative residuals that are comparable to the experimental data dispersion.Moreover, the predicted multiaxial fatigue lives of both alloys are fairly close to the experimental readings.

Model description
In this section a brief description of the Ottosen high cycle fatigue model is provided, for more details about it refer to Ref. [15].

Ottosen high cycle fatigue model
The model is based on the concept of a moving endurance surface () in the stress space, where the endurance surface serves as an indicator to stress states leading to fatigue damage.Thereby, the stress states outside the endurance surface may induce fatigue damage growth, while those inside it do not.
The original form of the endurance surface is defined as [15] where  oe is the endurance limit, which turns out to be the fatigue strength of fully reversed loading, and  is a dimensionless material parameter in control of the mean stress effect on .The stress tensor is denoted as  and σeff is the von Mises type effective stress computed from the reduced deviatoric stress  −  as More complex form of the endurance surface with six material parameters has been proposed by Brighenti et al. [22].Notice that  is the deviatoric stress tensor,  is the identity tensor, and  is a deviatoric back stress tensor defining the center point of the endurance surface in the deviatoric plane.
Then the endurance surface is given by  = 0, and every stress state inside of the endurance surface ( < 0) do not cause fatigue damage evolution.Stress states outside of the endurance surface ( ≥ 0) lead to damage growth if β ≥ 0, where the superimposed dot indicates time derivative.Such condition is required to ensure that the damage () in the material do not decrease, since it evolves according to the following expression where  and  are dimensionless material parameters.Similarly,  evolves proportionally to β and parallel to the direction of  −  when  ≥ 0 and β ≥ 0. Then  evolves by where  is a dimensionless parameter describing the capability of the material to adapt to stress states outside of  by moving of the endurance surface.
The expression for β can be obtained by direct time differentiation of (1), resulting in equation where ∶ denotes the double dot product of two second order tensors  ∶  = tr( T ).
As pointed out by Morel [2], the damage evolution occurs by mesoscale plasticity phenomena in which only a small portion of the material volume behaves inelastically while the macroscopic behavior is purely elastic.Then the elastic problem can be assumed to be independent of the damage evolution.Thus, the damage can be computed after the stress history is known by integration of the system of Eqs.(1) to (5) with the following initial conditions where  stands for time.Notice that the initial conditions hold only for initially unloaded and undamaged materials.The fatigue life of the material is computed by integration of the ODE system until the damage parameter meets the condition for complete failure  = 1.

Enhancement of the fatigue criterion to consider stress gradient effects
Ottosen et al. [16] presented a modification of their original fatigue criterion to include a simple correction of the parameter  oe as where  corr oe is the correction of  oe and  is the Neuber parameter with unit of length.The scalar  is the scaled effective stress gradient, and  eff is the von Mises stress.For more details about this modification refer to Ottosen et al. [16]

Model evaluation and numerical methods
The model described in the previous section was evaluated under four different types of periodic loading scenarios: uniaxial tensioncompression, pure torsion, multiaxial tension-compression with torsion both IP and OP, and multiaxial loading in notched samples were stress gradients play a role.This section describes the explicit forms of the fatigue model to evaluate the fatigue life for each of the these loading scenarios.

The periodic uniaxial tension-compression loading case
In the case of cyclic tension-compression, the only non-zero component of the stress tensor is  11 , which varies in time according to where  m and  a are the mean stress and amplitude, respectively, and  is the period of a cycle.In the current scenario, the only non-zero components of the -tensor are  11 = ,  22 =  33 = − 1 2  and the nonzero deviatoric stress tensor components are  11 = 2  3 ,  22 =  33 = − 1  3 .With this uniaxial normal stress state, the system ( 1)-( 6) reduces to two ODEs describing the evolution of  and .Such equations can be integrated numerically using the forward Euler method.The computation of the state at time   is given as Then, if both conditions   ≥ 0 and β ≥ 0 are met the evolution of  and  are obtained as where the time step is  =  +1 −   .If either   ≥ 0 or β ≥ 0 are not met then

The periodic pure torsion loading case
In the case of cyclic pure shear, the only non-zero components of the stress tensor are  12 and  21 , which evolve in time as where  m and  a are the mean stress and amplitude, respectively.In this case the only non-zero components of the -tensor are  12 =  21 = , and the non zero deviatoric stress tensor components are  12 =  21 = .Then the system (1)-( 6) reduces to two ODEs describing the evolution of  and  in the form If both conditions   ≥ 0 and β ≥ 0 are met the evolution of  and  are computed according to Eq. (11).If either   ≥ 0 or β ≥ 0 are not met then the state at  +1 is given by Eq. (12).

The periodic multiaxial loading case
The considered multiaxial loading cases include different combinations of tension-compression and torsion, which are shown in Fig. 1 in terms of normal load (  ) and torsional moment (  ).
No simplifications are used during the integration of the multiaxial cases.Thus, the fatigue life prediction in such scenarios required the integration of the system (1)-( 6), which has seven ODEs, one for the damage variable and six for the independent entries of the tensor .The stress tensor components for the instant  in the case of homogeneous fully reversed cyclic torsion with a constant tension (loading path in Fig. 1a) are computed as where  is the constant normal stress and  a is the amplitude of the shear stress, and the rest of the tensor components are constant and equal to zero.The stress tensor components for the instant  in the case of homogeneous cyclic tension-compression with torsion (loading paths in Figs.1b, 1c, 1d) are computed as where  is the phase angle between axial and torsional loads.It is noteworthy that some of the considered multiaxial cases have geometries leading to heterogeneous stress states with large stress gradients.In such cases, the correction of the parameter  oe in Eq. ( 8) was used.In this work, the considered heterogeneous loading cases are: The stress history and stress gradients in the samples are calculated using FE models implemented in ABAQUS [25].Detailed descriptions of the simulation set-ups are presented hereafter.

FE model of thin waled pipes with a passing hole
The modeled geometry is a thin walled pipe of 1.5 mm thick, with a passing hole of diameter of 1 mm in the middle section of the specimen, as shown in Fig. 2. The elements used in the FE discretization were 10node quadratic solid tetrahedrons (C3D10).The element size far from the hole is 1.5 mm and a local mesh refinement near the hole resulted in an average element size of 0.02 mm in the edge of the hole, as shown in Fig. 2c.The local refinement is attained using a radial geometry around the hole, as shown in Fig. 2c.
The loading path applied to the numerical samples is the one shown in Fig. 1b.The mechanical loads were applied as time dependent point loads at reference placed at the edges of the sample as shown in Fig. 2c.Then rigid body constraints of the sample edges are used to attain an homogeneous distribution of the loads at the ends of the sample.The complete description of the samples and experimental procedure is shown in the work of Chaves et al. [23].

FE model of notched shafts
The modeled geometry is a solid shaft with a notch in the middle section, as shown in Fig. 3.The core of the shaft is meshed using 6node linear triangular prisms (C3D6), while the rest of the specimen is meshed using 8-node trilinear brick elements (C3D8R).The largest element size far from the notch is 2 mm, and a local mesh refinement at the notch resulted in an average element size of 0.1 mm, as shown in Fig. 2c.
The loading paths applied to the numerical samples are shown in Figs.1b, 1c, 1d.The mechanical loads have been applied as time dependent point loads at reference points placed at the edges of the sample as shown in Fig. 2c.Then a rigid body constraints of the sample edges are used to attain a homogeneous distribution of the loads at the ends of the sample.The complete description of the samples and experimental procedure is shown in the work of Tao et al. [24].

Periodic state and damage extrapolation
Ottosen et al. [15] pointed out that under the application of a periodic proportional loading the evolution of the ODE system reaches a periodic state after a small number of loading cycles ().Such periodic state leads to an exponential convergence to linear growth of  with , as shown by Lindström et al. [20].The evolution of  =  11 and  for fully reversed tension-compression is shown Fig. 4. Notice that the behavior of  11 reaches a periodic behavior immediately, while  shows a stepped evolution that can be accurately described with a linear fit.Thus, integrating the ODE system up to failure ( = 1) turns out to be unnecessary, since the fatigue life can be computed from a linear extrapolation of the simulated damage growth.The same convergence to a linear growth of  with  is observed for multiaxial cases as shown in Fig. 5a.Fig. 5b shows that the extrapolated fatigue life (  ) for pure torsion and multiaxial loading cases do not change when more than 50 cycles are computed before the extrapolation.Thus, a minimum of 500 cycles was set to evaluated the fatigue models before doing the linear extrapolation of the damage to compute the fatigue life.

Model calibration approach
The Ottosen high cycle fatigue model has a set of five material parameters p = [, , , ,  oe ] that can be calibrated using experimental S-N curves.Such curves represent the number of loading cycles before failure ( f ) of samples subjected to cyclic stresses with a given stress amplitude ( a ) and stress ratio ( =  min ∕ max ).The calibration procedure consists of the solution of an inverse problem using numerical optimization algorithms to minimize a cost function ( ).In this work, the inverse problem is defined using the Morozov regularization principle [21] to avoid nonphysical fitting of the experimental data dispersion.Besides, additional function penalizations are included in the cost function as described below: subjected to    18), per S-N curve included in the calibration.Thus, when a simulated S-N curve did not meet the condition in Eq. ( 18) the following penalization was applied to the cost function The weights   are used to enhance the fit to the data with lower experimental uncertainty and these are computed as follows where  , stands for the estimated mean  a at every  exp , which are computed by local regression smoothing of every experimental curve.Notice that the weighs   are the square inverse of the estimated experimental standard deviation   .
The experimental S-N curves have runout data points, which correspond to those experiments in which the tested sample did not fail at a given amount of loading cycles with an applied  a and -value.The contribution of runouts to the cost function  ro is calculated as follows where  is a constant set to 0.1 in this work.The limit  sn is the number of S-N curves included in the calibration.Then, the final cost function is given by  +  ro .In addition to uniaxial fatigue tests, the presented calibration admits multiaxial loading cases as those described in Section 3.3.The contribution of multiaxial cases to the cost function is calculated as follows where the limit  ma exp equals the number of experimental multiaxial tests included in the calibration and the weight  was fixed to a value of 10.The values  exp  are the experimentally measured fatigue lives and the values  num  are the numerically predicted fatigue lives.Moreover, the penalization in Eq. ( 22) has the same form for multiaxial tests of notched samples.The calibration approach has been entirely implemented in MATLAB [26] using the ''patternsearch'' optimization algorithm.

Validation of the model calibration
The optimization approach described in Section 3.5 is used to calibrate the Ottosen model for the 7050-T7451 and the 7075-T6 aluminum alloys.The uniaxial tension-compression S-N curves used for the model calibration of both alloys are those presented in [27].The multiaxial fatigue data with homogeneous stress fields for the 7075-T6 aluminum alloy is the one presented by Zhao and Jiang [28], while the data with stress gradients is from the work of Chaves et al. [23].The multiaxial cases with stress gradients used for the 7050-T7451 aluminum alloy are those presented by Tao et al. [24].The calibrated parameters for the two aluminum alloys are presented in Table 1.
The model predictions for the 7050-T7451 alloy are compared with experimental fatigue data obtained under uniaxial and multiaxial loadings in Fig. 6.Uniaxial tension-compression Wöhler curves with different stress ratios are computed using the numerical scheme in Section 3.1, and the obtained results are compared with the experimental data from [27] in Fig. 6a.The numerically predicted Wöhler curves are in good agreement with the experimental data, showing mean relative residuals under 40%, which is comparable to the experimental dispersion.The fatigue lives for multiaxial cases have been calculated using the stress histories computed by the FE model discussed in Section 3.3.2.The predicted fatigue lives are compared with the experimental data presented by Tao et al. [24] in Fig. 7b.The attained mean relative residuals for the multiaxial cases is around 15%, which indicates a very close match between the model predictions and the experimental data.
The model predictions for the 7075-T6 alloy are compared with experimental data obtained under uniaxial and multiaxial loadings in Fig. 7. Two different types of S-N curves were computed: four uniaxial tension-compression S-N curves with different stress ratios were computed using the scheme in Section 3.1, and one S-N curve for fully reverse torsion were solved using the scheme in Section 3.2.A comparison of the experimental data in [27,28] with the simulated S-N curves is presented in Fig. 7a.The simulated S-N curves are in good agreement with the experimental data, showing a mean relative deviation from the model to the experimental data under 50%, which is comparable to the experimental dispersion.The fatigue lives for multiaxial cases were computed using the stress histories generated with Equations ((15),( 16)), and the predicted fatigue lives are compared with the experimental data reported by Zhao and Jiang [28] in Fig. 7b.The multiaxial fatigue lives of thin pipes with a passing hole in the middle were computed using the stress histories generated by the FE model discussed in Section 3.3.1.Then, the predicted fatigue lives are compared with the experimental data presented by Chaves et al. [23] in Fig. 7b.A quantitative comparison of the numerical and experimental results is presented in table in Table 2.The model predictions in multiaxial cases are in fair agreement with the experiments, deviating in average by a factor of two, which is smaller than the experimental dispersion of the experimental data reported by Chaves et al. [23].
The simulated fatigue lives for both alloys are in good match with the experimental data obtained under different loading conditions.The constraints in the Morozov regularization approach lead to residuals for the uniaxial cases that are in the order of the experimental data dispersion, while providing solutions that also match the rest of the loading cases.In the 7050-T7451 alloy the multiaxial fatigue predictions accurately describe the experimental evidence for notched shafts despite of the phase angle between axial and torsional loads.Moreover, there is not enough experimental evidence to enable a comparison between the model residuals and the statistical dispersion of the experimental data.The experimental data set used to calibrate the model for the 7075-T6 alloy contains a wide variety of loading scenarios which were simulated with a fairly close match in most cases.More importantly, the results for the multiaxial cases are conservative in 50% of the cases included in the validation.

Comparison of the proposed calibration approach with a classical calibration method
One very important advantage of the calibration procedure proposed in this paper over other typical calibration procedures is that it addresses the ill-posed nature of the parameter identification inverse problems.This allows to identify sets of parameters that are stable with respect to the experimental scatter.Moreover, the proposed calibration procedure includes multiaxial fatigue data that enhances the model  predictive capacity under multiaxial and non homogeneous loading conditions.A comparison between the fit quality obtained with the proposed calibration procedure and a reference calibration approach is presented in this section.The reference calibration approach is based on the minimization of the mean square deviation between the experimental and simulated S-N curves.Thus, the cost function is defined as, A set of parameters is identified using Eq. ( 23) and the fatigue experimental data of the 7075-T6 alloy tested under uniaxial tensioncompression [27] and pure torsion [23,28].A quasi-newton algorithm with BFGS Hessian approximation is used to identify the model parameters, which are shown in Table 3 .
A comparison between the fit quality obtained with the proposed approach and the fit quality obtained using the calibration procedure is presented In Fig. 8.The fit quality obtained with both methods is comparable in the uniaxial tension-compression, pure torsion and multiaxial homogeneous cases.Moreover, the identification of parameters with the classical approach is sensitive to experimental scatter, whereas the method proposed in this paper provides robustness against this feature.The model calibrated with the proposed approach provides fatigue lives estimations in multiaxial loading of notched samples that are within the experimental scatter.The predictions of the model calibrated with the reference approach yielded significantly larger deviations, which is expected since no heterogeneous stress data was fed into it, then the Neuber parameter was equal to zero.
It is noteworthy that the calibration approach presented in this work requires an extensive amount of experimental data since it uses the standard deviation as limit for the calibration procedure.The standard deviation of poor experimental data sets is not well defined, and in such cases the constraints in Eq. ( 18) cannot be defined.Additionally, the multiaxial fatigue tests are not so common in the literature, thus the use of this approach is limited by the availability of experimental evidence.

Effect of phase angle in biaxial, non proportional stress
The calibrated models show a fairly good match with the experimental data obtained under various loading conditions including biaxial OP loadings.The effect of the phase angle in the fatigue life of 7050-T7451 aluminum notched shafts is predicted in close agreement with the experiments.However, some miss-predictions were obtained in the OP multiaxial cases of the 7075-T6 alloy.The experiments reported by Zhao and Jiang [28] show that the multiaxial fatigue life of the 7075-T6 alloy is shorter under OP loadings than IP cases.Lindström et al. [20] reported an analysis of the Ottosen HCF model showing that the predicted damage evolution under OP loads leads to a fatigue life only a quarter of that under IP stress history.Their work showed that one possible reason for such miss-prediction is that the stress moves tangentially to the endurance surface during OP loading, leading to a severe underestimation of the damage growth rate.Moreover, the comparison of the experimental data in the literature and the numerical predictions attained in this work do not show a consistent overestimation of the fatigue life in biaxial cases.Consequently, an extension of the analysis of the effect of the phase angle in biaxial fatigue is presented hereafter to see if the model predicts at least qualitatively the effect of the phase angle.
Here a combination of tension-compression and shear is considered for different values of  and different stress levels.The approach presented by Lindström [20] to create biaxial stress histories with different phase angles and equivalent amplitudes of the von Mises stress is adopted.In this approach the non-zero components of the stress tensor are computed as ) sin ) sin where  0 is a constant controlling the stress level,  is the loading frequency and  is the phase angle between tension and shear.The analysis was performed using the calibrated model for the 7050-T7451 alloy since it shows a good agreement with the experiments in both IP and OP loadings.The simulated fatigue lives as function of the  0 and  are presented in Fig. 9.This shows that the predicted fatigue life is reduced when  increases.Notice that the reduction of the predicted fatigue life with the increment of  is small for  < ∕4, and for larger phase angles the fatigue life decreases abruptly.The relative reduction of the fatigue life due to the increment of  is studied in terms of the dimensionless parameter .This parameter The simulated fatigue lives in Fig. 9a are used to calculate the contour plot of  in Fig. 10.For stress levels under 200 MPa and  < ∕4, the OP fatigue life is reduced by less than 20% while  grows quickly with the phase angle for  > ∕4 reaching a fatigue life reduction of 70% when  approaches ∕2.For stress levels above 200 MPa,  changes faster with  than at lower stress intensities, reaching a reduction of the fatigue life of 80% when  approaches ∕2.Thus, the numerical results attained in this work are qualitatively consistent with the reduction of the fatigue life with the OP fluctuations observed in the experiments in [28].

Conclusions
In the finite life regime, the Ottosen high cycle fatigue model [15] shows a periodic behavior for constant cyclic loading.Such behavior leads to a linear growth of the damage with the number of cycles.Then, the number of cycles to fatigue can be calculated from a linear extrapolation of the damage parameter evolution over a reduced number of cycles.This fact facilitates Wöhler curve simulations with low computational cost, enabling to perform parameter calibration with optimization algorithms.The damage extrapolation is applicable to various loading scenarios including uniaxial tension-compression, pure torsion, and multiaxial cases with different phase angles between loading types.Thus, the calibration presented in this work contains multiaxial data from various sources.Additionally, a modification of the fatigue criterion has been used to include multiaxial cases with strong stress gradients in the calibration.A Morozov regularization scheme was used in the calibration of the Ottosen fatigue model for the 7075-T6 and the 7050-T7451 aluminum alloys.
The present work shows that adding multiaxial data in the calibration of the Ottosen fatigue model enhances significantly its predictive capacity without adding complexity to the model formulation.The attained model residuals for both aluminum alloys in uniaxial tensioncompression are comparable to the data dispersion of the experiments used for the calibration.The results for the 7050-T7451 alloy in multiaxial cases closely match the experimental data despite of the phase angle between torsion and normal loads, yielding a mean relative residual of 15%.The multiaxial fatigue predictions for the 7050-T6 alloy are fairly close to most of the loading scenarios used in the calibration, which included: pure torsion, torsion with constant axial load and multiaxial tension-compression with different phase angles.Moreover, the model predictions under cyclic tension-compression and shear revealed a gradual reduction of the fatigue life with the phase angle, which shows at least qualitative consistency with experiments in the literature.

Declaration of competing interest
The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Reijo Kouhia reports financial support was provided by European Union.

Fig. 2 .
Fig. 2. Geometry and EF mesh of the thin walled pipe model with a passing hole for multiaxial fatigue tests: (a) front view, (b) top view, (c) isometric view with a zoom at hole.All distance units in mm. 1. Multiaxial fatigue tests of thin walled pipes with a passing hole made of the 7075-T6 aluminum alloy with the loading path described in Fig. 1b, as described in the work of Chaves et al. [23].2. Multiaxial fatigue tests of notched shafts made of the 7050-T7451 aluminum alloy with the loading paths described in Figs.1b, 1c and 1d as described in the work of Tao et al. [24].

Fig. 3 .
Fig. 3. Geometry and EF mesh of the notched shaft model for multiaxial fatigue tests: (a) front view, (b) View of a section cut at the notch, (c) isometric view with a zoom at the notch.All distance units in mm.

Fig. 4 .
Fig. 4. Evolution of uniaxial periodic state variables during the first 10 loading cycles: (a) Evolution of stress components  11 and  11 with the number of cycles.(b) Integrated evolution of the dimensionless damage parameter  with the number of cycles and linear fit.

Fig. 5 .
Fig. 5. Integrated evolution of the dimensionless damage parameter  with the number of cycles in pure torsion and multiaxial loading scenarios (a).Predicted fatigue lives  f as function of the number of evaluated cycles  eval before extrapolation of damage.

Fig. 6 .
Fig. 6.Comparison between the calibrated model predictions and experimental data for the 7050-T7451 aluminum alloy: (a) S-N curves obtained under uniaxial tension-compression.The solid triangles pointing to the right represent the runout data.(b) Validation of multiaxial fatigue life predictions for a notched shaft.The dashed line indicate a dispersion of 40%.

Fig. 7 .
Fig. 7. Comparison between the calibrated model predictions and experimental data for the 7075-T6 aluminum alloy: (a) S-N curves obtained under uniaxial tension-compression.The solid triangles pointing to the right represent the runout data.(b) Validation of multiaxial fatigue life predictions including cases for thin pipes with a hole in he middle section.The dashed line indicate a deviation by a factor of 3.

Fig. 8 .
Fig. 8.Comparison between fit quality obtained with the proposed calibration approach (red symbols) and a classical procedure (blue symbols): (a) Uniaxial tension-compression and pure torsion.(b) Multiaxial loading in smooth and notched samples.

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Fig. 9 .
Fig. 9. Fatigue life as function of the phase angle, , and the stress parameter  0 : (a) Isometric view (b) Contour plot.

Fig. 10 .
Fig. 10.Fatigue life as function of the phase angle, , and the stress parameter  0 : (a) Isometric view (b) Contour plot. is here defined as a measurement of the reduction of the fatigue life in OP loading with respect to the IP fatigue life at a given value of  0  = 1 −   (;  0 )   ( = 0;  0 )(25) : Conceptualization, Methodology, Software implementation, Writing -original draft, Analysis of results, Writing and editing the first submitted version of the manuscript.Timo Saksala: Conceptualization, Methodology, Software implementation, Analysis of results, Writing, and editing the first submitted version of the manuscript.Djebar Baroudi: Conceptualization, Methodology, Analysis of results, Writing and editing the first submitted version of the manuscript.Mikko Hokka: Analysis of results, Writing, and editing the first submitted version of the manuscript.Reijo Kouhia: Conceptualization, Methodology, Analysis of results, Writing and editing the first submitted version of the manuscript.

Table 1
Calibrated model parameters for the 7050-T7451 and the 7075-T6 aluminum alloys.

Table 2
Comparison of simulated multiaxial fatigue lives with experimental data for the 7075-T6 aluminum alloys.

Table 3
Identified model parameters for the 7075-T6 aluminum alloys with the mean square algorithm.