Continuous-time, high-cycle fatigue model for nonproportional stress with validation for 7075-T6 aluminum alloy

: The continuous-time, high-cycle fatigue model of Ottosen et al. (2008) is modified by introducing a quadratic, instead of linear, endurance surface. This quadratic endurance surface induces an elliptical Haigh diagram, consistent with experimental fatigue data for ductile materials. With this modification we fit the model to uniaxial fatigue data, and predict the fatigue behavior for nonproportional, biaxial stress for AA7075-T6 in fully reversed shear with a superimposed, static tension/compression. However, the fatigue life is sometimes overestimated for a combination out-of-phase shear and tensile stress fluctuations.


Introduction
Fatigue failure is the result of a long process of mechanical deterioration due to a fluctuating stress during the life-time of the product.There is also a stochastic aspect to fatigue, as fatigue life is often associated with a large scatter.These inherent properties of fatigue makes experimental work challenging: each experiment is destructive, takes a lot of time, and needs to be repeated for statistical relevance.The parameter space is also huge, since fatigue exhibits size scaling and is sensitive to, e.g., heat treatment, surface treatment and environmental conditions.Maybe most importantly, fatigue depends on the infinitedimensional stress history of the material.This underscores the need for fatigue models that can extrapolate experimental fatigue data for a simple type of stress history, such as cyclic uniaxial stress, to predict the fatigue life for an arbitrary stress history.
A widely-used fatigue modeling approach is cycle-counting, where the stress history is transformed into one-dimensional load reversals, and the resulting fatigue damage accumulation is obtained by employing, e.g., the Palmgren-Miner rule.The critical plane method is a successful technique for the stress-history transformation step.There are, however, important applications when cycle-counting becomes cumbersome.In complete machinery simulations (CMS), where the realistic operation of an entire machine is simulated using an explicit scheme and massively parallelized computing, over-night simulations produce prohibitively large quantities of stress-history data for storage.Thus, cycle-counting methods that need access to the stress history in a post-processing step become impractical [1].Thus, there is a need for incremental methods [2][3][4][5], with which fatigue damage can be integrated in parallel with the simulation, so that only the final state of the simulation needs to be stored.A trial implementation of the continuous-time fatigue (CTF) model by Ottosen et al. [4] in LS-DYNA (Livermore Software Technology) presents a recent example [1].
In this work, we consider high-cycle fatigue (HCF) of isotropic materials.Even when a metal is subjected to fluctuating stresses well below its yield limit, fatigue damage accumulates and may lead to fatigue failure.The HCF regime is characterized by the applied stress being at all times confined to the linear elastic range, which typically give a lower limit of the number of cycles to failure between 10 3 and 10 4 .The HCF process starts with a crack-initiation phase where mi- crostructural damage accumulates, leading to subsequent nucleation, growth and coalescence of microcracks.This crack-initiation ends in damage localization at a dominant crack, which then propagates to failure [6].From a macroscopic viewpoint, the microdamage is spatially distributed during the crack-initiation phase, and has a negligible effect on the elastic material properties.Since crack propagation is relatively faster than crack initiation, the crack-initiation phase is assumed to dominate fatigue life for cyclic applied stress [6].
Ottosen et al. [4] formulated a CTF model, also known as the continuum damage approach, for isotropic materials.This model is based on an endurance surface that depends on the first stress invariant and on the von Mises effective stress reduced by a backstress.Similarly to kinematic hardening in plasticity [7], fatigue damage develops when the stress state reaches this endurance surface, which also causes the backstress to move.Although the material behavior is predominantly elastic in HCF, we can understand the presence of a backstress as inelastic deformation associated with microdamage in individual grains or grain boundary segments.With the CTF model, the evolution of the damage and backstress is controlled by ordinary differential equations (ODEs) [8,9].
In the original CTF formulation for isotropic materials, where the endurance function is a linear combination of the first stress invariant and the effective stress, it is possible to find an analytical solution for the damage per cycle under cyclic, proportional stress [4,9].Uniqueness of the solution to the governing ODEs of the CTF model has been demonstrated for the case of proportional stress [9].The model also gives a fair least-squares fit to uniaxial fatigue data at different constant mean stresses, and the predicted endurance stress is in qualitative agreement with experiments [4].
Several works in the literature that modify and enhance the applicability of the CTF model demonstrate its versatility.The CTF model has been modified to include stress gradient effects [8], and there is ongoing work to extend the CTF model to the regime of low-cycle fatigue [10].The CTF framework has also been adapted to transversely isotropic materials [11][12][13].Owing to the relative simplicity of the governing equations, the CTF model can be employed as a fatigue constraint in topology optimization [14,15].However, the model also exhibits weaknesses that needs to be addressed before its full potential can be reached.
As a result from the original CTF formulation, the Haigh diagram-that is the relation between the endurance limit and the mean stress for cyclic, uniaxial stress-becomes linearly dependent on the mean stress [4].From experimental work, however, it is known that the Haigh diagram is typically nonlinear in the HCF regime.Particularly, the Haigh diagram of a ductile material can be closely fit by an ellipse [16].Thus, to extend the validity range of the CTF model, we seek a revised formulation that induces a more realistic Haigh diagram.
It has previously been claimed that the general framework of the CTF model should, in principle, make it possible to predict biaxial and nonproportional stress.Brighenti et al. [17] suggested a more elaborate variant of the endurance surface depending on several stress invariants modified by a backstress.It was not possible, however, to use the model parameters fit for proportional stress data to predict the fatigue behavior for nonproportional stress [17].It has also been demonstrated that the original CTF formulation overestimates fatigue life for rotary stress states [9].In conclusion, there is currently little evidence that the CTF model can extrapolate fatigue data significantly outside the data set to which it is fit.A modification of the model appears to be needed to address this issue.
In this study, we revisit the CTF model for isotropic materials, as formulated by Ottosen et al. [4], and investigate a modified endurance surface which is quadratic in the first stress invariant and the effective stress.The intention is to extend the range of faithful extrapolation across the HCF regime.With this nonlinear formulation, we no longer find an analytical solution to the fatigue life for cyclic, proportional stress, as is the case for the preceding linear model [9].Instead, we present a numerical scheme that exploits parallelization to efficiently compute Wöhler curves for the new nonlinear formulation.
We fit the CTF model with quadratic endurance surface to uniaxial fatigue data for unnotched specimens and a selection of aluminum alloys, AA7075-T6, AA7050-T7 and AA2024-T3, that are frequently used in the aircraft industry and elsewhere.We also fit the model to airmelted AISI 4340 steel alloy tempered to an ultimate strength of 1380 MPa, to demonstrate the versatility of the model.The predictive capability of the new model is investigated for AA7075-T6 by comparing the predicted life to experimental results for a variety of cyclic, biaxial stress states, including nonproportional stress.Interestingly, the model, when fit only to uniaxial fatigue data of AA7075-T6, exhibits fair agreement with biaxial as well as nonproportional fatigue data.

Theory
The theoretical frame in this work is developed starting from the CTF model of Ottosen et al. [4].The CTF model for HCF in formulated in terms of stress.The stress and its time derivative are assumed to be given functions of time t.Here, is the set of all symmetric, second-order tensors x , and a superscript ' ' denotes the transpose.For convenience, we also define the deviatoric subspace of sym , where x tr( ) denotes the trace of a tensor x.

Continuous-time fatigue model
We introduce the backstress dev and damage d d [0, ] c as state variables, where = d 0 for the pristine material, and = d d c for critical failure, where d c is a constant.The HCF damage process is primarily driven by some effective stress ¯( , ) to be defined later.In addition, HCF is modulated by the first stress invariant = I tr( ) 1 . An endurance surface, is defined using a convex endurance function 1 .For stress states inside this endurance surface, i.e. for < 0, no damage develops.Outside the surface, 0, damage develops during onloading, as defined below.
The evolution of state is described by the initial value problems (IVPs) where a superposed dot denotes the time derivative, × G: sym dev dev is a material function, and > > g ( ) 0, 0, and denotes the unit ramp function, and is the Heaviside step function.
As previously suggested [14], the damage-dependence of the righthand side of Eq. ( 2b) is eliminated by a change of damage variable where D c is the critical damage at failure.The damage evolution Eq. (2b) is thus reduced to = D g H ( ) ( ) without loss of generality.To reformulate the IVPs as ODEs, we proceed to eliminate from the right-hand side of Eq. (2a) (cf.Refs.[8,9]).By differentiating with respect to time, and inserting Eq. (2a), we obtain where is the unit tensor, and the colon operator denotes the Frobenius inner product 1 2 of two tensors x 1 and x 2 .By rearranging Eq. ( 4), while using that To ensure uniqueness of , , , the left-hand side of Eq. ( 5) must be monotonic in .This is achieved by imposing the conditions where the first inequality originates from the empirical observation that the rate of damage development increases with the effective stress.Then, we can define a rate function : , such that The IVPs of backstress and damage can now be written as ODEs, which allows for integration using standard ODE solvers.
Considering condition ( 6), a natural choice for G is for some material function G I ( ¯, ) 0 1 with the units of stress.

Instantiation of the CTF model
For isotropic materials, the following instantiation of the fatigue model is commonly used [4,9] Since ¯is non-differentiable at = s , becomes ill-defined at that point.This is resolved by requiring that < 0 for = ¯0, and setting to an arbitrary constant at = ¯0.The evolution of state is driven by the effective stress and the first stress invariant, where E is the Young's modulus, which merely functions as a reference stress.In this work, the endurance function is chosen as a bivariate, quadratic polynomial where A is a positive definite, symmetric material parameter matrix, and a is a material parameter vector.This endurance function can be specialized by constraining A. Borrowing terminology from the field of plasticity, we obtain a Bresler-Pister (BP) type endurance surface by setting = = A A 0 11 12 [18], or a Drucker-Prager (DP) type surface for = A 0 [19].The original CTF model by Ottosen et al. uses a DP type endurance function.A geometric interpretation of the endurance surface and the evolution of the backstress is given in Fig. 1.This particular choice for with Eqs.(10a)-(10c) yields where, using that = A A , we have In summary, the isotropic CTF model with quadratic endurance surface consists of Eq. (10a) and Eqs.(11a), (11b), ( 12)-( 14), (15a), (15b), where the IVPs, (11a) and (11b), are integrated for a given stress history using standard methods.

Periodic, proportional stress
The possibility to efficiently integrate the CTF model for cyclic, proportional stress is crucial for constructing a feasible parameter-fitting procedure for uniaxial fatigue data.For proportional applied stress, the stress, the deviatoric stress and the backstress can be expressed as [9] = e S t ( ) , (16a) We insert Eqs.(16a)-(16c), (17) into the equations of the fatigue model.For cyclic, proportional stress between a minimum stress S and a maximum stress S ^, after a transient, the backstress fluctuates between a minimum value and a maximum value ^(Fig.2).The endurance limit is obtained when the amplitude of this fluctuation vanishes, that is when for some unknown constant .Thus, to find this endurance limit we seek a solution to = = S S ( , ) ( ^, ) 0, i.e. the nonlinear system of equations Another limit of the HCF regime is found from the von Mises yield criterion = s 3/2 y , with y the yield stress, which becomes for proportional stress.We introduce the mean stress and the effective stress , which is the equivalent amplitude in a von Mises sense [9].This gives For a constant mean stress, we substitute Eq. ( 20) into Eqs.(18a) and (18b).With this substitution, the difference between Eqs. (18a) and (18b) is linear in .Thus, it is straight-forward to eliminate by substitution, resulting in a univariate, fourth-order polynomial equation in a for which one real root is the fatigue limit ( ) a m (A).The sought root is such that S S ] , ^[ and 0 a .The existence and uniqueness of this root is tested each time the polynomial equation is solved.The Haigh diagram induced by the parameters for AA7075-T6 (Table 1) in tension/compression takes an elliptical shape in the general case (Fig. 3), which is consistent with experimental observations for ductile materials [16].
In the case of uniaxial tension/compression, = = e tr( ) 1, at a constant stress ratio = R S S / ^, we have a , so that Eq. ( 20) gives We substitute Eq. ( 21) into Eqs.(18a) and (18b), and then, solving for S S ] , ^[ and 0 a , we obtain the fatigue limit R ( ) a (A).
Moreover, the yield limit from Eq. ( 19) is

L (24b)
We only need to solve these reduced IVPs for cases of proportional stress.

Wöhler diagram for cyclic, proportional stress
Wöhler curves for cyclic, proportional tension/compression can be constructed for a constant mean stress m or for a constant stress ratio R. Herein, we describe how to numerically integrate a Wöhler curve for a constant stress ratio; integration for a constant mean stress is analogous.
The Wöhler curve for cyclic, proportional tension/compression and a constant R is the graph of the effective stress amplitude N R ( ; ) .Herein, the integration is carried out using an applied stress k t a (25) with = T 2 / , and with = T 1 s the period, which is an arbitrary constant since the governing IVPs are rate-independent.Here, S and S âre computed using Eq. ( 21).The exponential ramp with coefficient = k 1 3 is included to suppress any excessive transient response from the IVPs.The IVPs (24a) and (24b) are integrated using the explicit sixth order Adams-Bashforth method (lsode in Octave [20]) to give the  of incremental damage.For cyclic, proportional stress, this sequence converges to a constant value [9].The fatigue life is computed as where we use = D 1 c and = n 100 in this work, and the superscript 1 denotes the inverse of a function.Using this method, a Wöhler curve can be computed in a few seconds on a laptop (Intel® Core TM i5-5200U CPU).

Results and discussion
Using the framework above, we fit the CTF model to fatigue data from uniaxial experiments of AA7075-T6, and then investigate the predictive capability of the model for this material subjected to nonproportional stress.We also fit the model to a range of materials, AA7050-T7, AA2024-T3 and AISI 4340 steel, frequently used in the aircraft industry.

Eight model parameters,
and L, are introduced above.For the fitting procedure, we consider cyclic tension/ compression of unnotched cylindrical specimens.This type of experiment is prolific in the literature.Each data point is a triplet s , where N i f is the measured life at a given effective stress amplitude i a and stress ratio R i .We define the squared error of the model w.r.t. the tensile test data as , , , ) log ( , ; , , , , ) , where is the model prediction for the stress amplitude.A least-squares fit is obtained by minimizing J .We assume that the prediction errors in the base 10 logarithm of the stress are normally distributed with zero mean.Then, the estimator of the variance of this error is where p is the dimensionality of the parameter space, and s is the estimator of the standard deviation.Moreover, the 95% confidence intervals for the fitted parameters are obtained using the method in Section 2.3.1 of Ref. [21] (nlparci in Octave [20]), where the interference region is calculated using a linearization of the nonlinear model at the optimal set of parameters.Throughout this work, minimization problems are solved using the Nelder-Mead simplex algorithm (fminsearch in Octave [20]) with an indicator function to enforce inequality constraints.
In the following, we fit the model to uniaxial fatigue data for AA7075-T6, AA7050-T7 and AA2024-T3 aluminum alloys, and for AISI 4340 steel alloy, respectively, in the range R 1 1/2 .The ex- perimental data are obtained from plots in Ref. [22], which we digitize using the Engauge Digitizer Software [23].In addition, we use fatigue data for AA7050-T7 from Ref. [24].

Parameter fit for a DP type endurance function
We begin with an investigation of the quality of the parameter fit for the original model of Ottosen with a DP type endurance function.Thus we investigate the minimization problem ( , , , , ).
We find that the minimization of problem (29) converges for AA7075-T6, AA7050-T7 and AISI 4340, whereas the parameters C K , and L diverge to large values for AA2024-T3.Thus, J is nonconvex for AA2024-T3 and potentially for many other materials.When examining the 95% confidence intervals of the parameters of AA7075-T6, AA7050-T7 and AISI 4340, we find that the intervals of a 1 and a 2 are relatively tight (tighter than ± 7% and ± 20%, respectively), while the intervals of

C K
, and L all exceed ± 100%.This indicates that C K , and L co-vary in the optimization to produce essentially the same quality of fit for a large range of values.
We propose that the problem of nonconvexity and that of interdependence between C K , and L are related.The CTF model parameters control both the transient and the steady-state evolution of the state variables.The transient of the damage has very limited influence on fatigue life.Thus, when fitting the model parameters to fatigue life measurements, we anticipate that there could be a large set of equally good solutions, each one corresponding to a different damage transient.
Herein, in contrast to previous work, we regard C as a numerical stability parameter rather than a material parameter.As C , considering Eq. (11a), the stress state comes to lie almost on the endurance surface during onloading.Thus, by choosing a sufficiently large value for C, the backstress evolution of the CTF model becomes essentially the same as that of kinematic hardening in plasticity.By solving minimization problem (29) subject to an additional constraint = C C 0 with {1, 10, 100} 0 and fatigue data for AA7075-T6, and comparing the different evolutions of the backstress and damage due to cyclic stress (example in Fig. 4), we find that C indeed controls transient, and that the same asymptotic damage per cycle is predicted for different values of C 0 (Fig. 4b).Moreover, the difference in the evolution of state is negligible between = C 10 0 and = C 100 0 (Fig. 4ab).The observation yields approximately the same prediction explains the nonconvexity sometimes observed for J , and also shows that a constraint = C 10 is sufficient to obtain the C-independent limiting behavior, which we take to be the desired behavior.

Parameter fit for different endurance functions
In the following, we fit the CTF model to uniaxial fatigue data by solving the minimization problem We fit the CTF model to uniaxial fatigue data for unnotched cylindrical specimens of AA7075-T6 aluminum alloy.The parameters obtained for DP, BP and EL type endurance functions are given with a 95% confidence interval in Table 1, together with the estimator of the standard deviation.As judged by the eye, a fair fit is obtained for all three types of endurance functions; only the fit to the BP type function is shown in Fig. 5.
By adding the constraint = C C 0 , as rationalized in Section 3.1.1,the four remaining parameters of the DP type model come to have relatively tight confidence intervals (Table 1).The parameter K, which controls the horizontal shift of the Wöhler curves, has the widest confidence interval.This is likely related to data points near the endurance limit, where the sensitivity to K vanishes.
For the BP type model, the confidence interval of the added parameter A 22 becomes greater than ± 100%, and that of a 2 also becomes rather large at ± 49%, while the other parameters are determined with the same accuracy as in the DP case (Table 1).Thus, the introduction of 22 cannot be defended from a purely statistical standpoint.However, it will be demonstrated later (Section 3.2) that the BP model enhances the predictive capability for fatigue due to nonproportional stress.
With an EL type endurance surface, the confidence intervals of parameters A A a , , 11 22 1 and a 2 all become larger than ± 100%.Thus, the dimensionality of the parameter space is too large to render a physically meaningful model.However, the fit and convergence properties remain acceptable.

Extrapolation to nonproportional stress
To investigate whether the proposed CTF formulation extrapolates the fatigue behavior across the HCF regime, it is necessary to compare the model predictions to experimental data that spans as much as possible of this regime.The material AA7075-T6 is particularly wellcharacterized experimentally owing to the work of Zhao and Jiang [25].In their experiments, Zhao and Jiang consider tubular specimens subjected to axial strain and torsion, with many data points in the HCF regime.The tubular geometry maintains an approximately plane state of stress [26], which is crucial to suppress stress-gradient effects.
In experiments of Zhao and Jiang, the material is subjected to strain-controlled, fully reversed shear in the HCF regime.Moreover, they perform several nonproportional fatigue tests, including fully reversed shear with a superimposed static tension or compression, as well as a number of more unconventional strain paths.When comparing the CTF model predictions to these experiments, we remove all data points outside the linear elastic regime.

Fully reversed shear with static tensile stress
For fully reversed simple shear, = e tr( ) 0 and = 3/2 , with shear stress amplitude a and zero tensile stress = 0 t , it is possible to integrate the stress-strain relation using the method described in Section 2.4.We compare the prediction of the DP, BP and EL type model to experimental data for AA7075-T6 [25] in Fig. 6a.Compared to the DP type model, the BP and EL type models give a slightly better agreement with experimental data.No conclusion can, however, be drawn regarding the prediction of the simple shear endurance limit, since it is not well-resolved by the experiments.
For fully reversed shear with shear stress amplitude and a static tensile stress 0 t , we integrate the damage to obtain the predicted cycles to failure A a N C K L ( , ; , , , , ) f a t and compare to corresponding experimental data [25] at the stresses compiled in the inset table of Fig. 6b.Each data point is a triplet frs , where N i f is the measured life at a given effective shear stress amplitude a and tensile stress i t .The superposed static stress ranges from tensile to compressive, spanning a large region of the HCF regime.
We introduce the squared error of the fatigue life prediction, 0.57 for the DP, BP and EL type model, respectively, indicating an en- hanced predictive capability for models with a quadratic endurance function.This is also confirmed when plotting the predicted fatigue life for the DP, BP and EL type model against measured fatigue life of AA7075-T6 with this type of nonproportional stress (Fig. 6b).As expected, the prediction becomes more conservative for a compressive stress < 0 t with a quadratic endurance function, but there is also an improvement in the prediction for > 0 t .This observation, together with the finding in Section 3.1.2,suggests that a BP type model provides a good balance between the predictive capability and the dimensionality of the parameter space.

Unconventional, cyclic stress paths
With integral structures, free-formed designs and general boundary conditions, essentially any stress path may arise in a material.Thus, it is prudent to validate the CTF model against some unconventional stress paths.Consider a state of plane stress with respect to the x x with 0 the shear stress amplitude.Experimental fatigue data for these combinations of normal stress and shear stress are found in Ref. [25].
It was previously demonstrated that the original CTF model with a DP type endurance function can overpredict the fatigue life for 90 phase-shifted normal stress and shear stress fluctuation, Eq. (33a) [9].Indeed, this shortcoming of the CTF model persists for the fatigue model with a BP type endurance surface when compared to experiments (Fig. 7); in at least one instance, infinite life is predicted when experiments point toward a finite life of ~10 6 cycles.However, for normal and shear stress fluctuations of different frequencies, Eqs.(33b) and (33c), the BP type CTF model gives a fair or conservative prediction of the fatigue life when compared to experimental data from Ref. [25] (Fig. 7).

Parameter fit for additional materials
Parameter fits of the BP type model to uniaxial fatigue data of cylindrical specimens are also provided for AA7050-T7 and AA2024-T3 aluminum alloys, and for AISI 4340 steel alloy, respectively (Table 1).A fair model fit is observed for all investigated materials (Fig. 8).
Notably, the A 22 parameter of AA7050-T7 is zero with a narrow interval (Table 1).In this respect, AA7050-T7 is qualitatively different from the other investigated aluminum alloys, so that a DP type model is sufficient to capture its HCF behavior.Also, the relative uncertainty of the parameters for AISI 4340 is much larger than for the aluminum alloys (Table 1).This could be due to the smaller number of samples for the steel alloy.
It should be emphasized that CTF model predictions for nonproportional stress and materials AA7050-T7, AA2024-T3 or AISI 4340 should be treated with caution until additional, corroborating experimental data for nonproportional stress become available for those materials.

Conclusions
By introducing a quadratic endurance function, the accuracy and extrapolation capability of the CTF model is enhanced.This comes at the cost of not having any analytical solution for the fatigue life available for cyclic proportional stress.As a remedy, we propose numerical integration of the stress amplitude-fatigue life relation.The interpolation technique for the construction of Wöhler curves is pivotal to the implementation of fitting procedures for the CTF model with a nonlinear endurance function.For AA7075-T6, fitting the CTF model to uniaxial data of different stress ratios is sufficient to predict fatigue life for fully reversed shear with a superimposed constant tensile or compressive stress (Fig. 6ab).Also, the inclusion of a quadratic term in the expression for the endurance surface gives enhanced predictive capabilities to the CTF model for this material (Fig. 6ab).A Bresler-Pister type endurance function, which is only quadratic in the first stress invariant, yields the best results in this respect.The results are encouraging, but at this time, due to scarcity of suitable experimental data, it is not possible to investigate whether the same conclusion applies to other materials than AA7075-T6.
Rotary stress states, or any stress path persistently tangent to the endurance surface, remain a weakness of the CTF model, as previously observed [9].In many instances, the resulting misprediction of fatigue life is conservative (Fig. 7).However, the fatigue life of AA7075-T6 is incorrectly predicted to be infinite for at least one instance of out-ofphase shear and tensile stress fluctuations (Fig. 7).Rotary stress states, although unusual, presents a threat that needs to be treated separately, or perhaps by a future, gentle reformulation of the CTF model.
The CTF framework of Ottosen and co-workers shows great promise owing to its relative simplicity and the modest computational effort it requires, which is crucial to its application in CMS.Future work includes the development of a method to safely handle rotational stress states, and a proof of concept regarding the application of the CTF model to integral metal structures with realistic, dynamic boundary conditions.

Declaration of Competing Interest
The author declares that he has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.for brevity in the expressions for the coefficient q 0 through q 4 .The coefficient of the polynomial in Eq. (A.1) are calculated by solving Eqs.(18a) and.(18b) using the SymPy symbolic toolbox [27] is the Frobenius norm, and = s dev( ) is the deviatoric stress tensor with = the deviator of a tensor x.Using Eqs.(10a)-(10c) while noting that =

Fig. 1 .
Fig. 1.Schematic drawing of the endurance surface ( ) and the deviatoric subspace dev sym .Three dimensions of dev are collapsed for illustration purposes.

Fig. 2 .
Fig.2.Sample evolution of the backstress (solid line) for the initial cycles of an applied periodic uniaxial stress S (dashed line).

y
of the effective stress amplitude for HCF and finite fatigue life is useful when numerically constructing Wöhler curves (Section 2.4).Since the stress and backstress can be represented by scalars for proportional stress, integrating the evolution of backstress and damage can be made more efficient.By using Eqs.(16a)-(16c) with Eq. (14), we obtain / ¯are given by Eqs.(15a) and (15b), respectively.The IVPs (11a) and (11b) for the backstress and damage become life N f .To obtain this relation, we integrate the damage to find the fatigue life at discrete values of ] parallelized process.Cubic interpolation in the log-log N f -

Fig. 3 .
Fig. 3. Haigh diagram for AA7075-T6 and uniaxial tension/compression for a quadratic endurance function (solid line), and a DP type endurance function (dashed line).The model is valid in the HCF regime, whose boundary-the yield limit-is indicated by dotted lines.

Fig. 4 .
Fig. 4. a: Example of an applied cyclic stress S (solid line) and evolution of the backstress for = C 1 (dashed line), = C 10 (dash-dotted line) and = C 100 (dotted line), with remaining parameters of a DP type CTF model fit to AA7075-T6 fatigue data.b: The corresponding damage evolution.The evolutions of = C 10 and = C 100 are indistinguishable by the eye.
subject to different constraints on A depending on the type of en- durance function:= A 0 for a DP type function, Fig. 5. a: Model fit to uniaxial fatigue data for AA7075-T6 corresponding to the parameters of the BP type endurance function in Table 1.The maximum stress reaches the yield stress at the indicated endpoints of the model curves.Arrows indicate runout.Experimental data points are from Ref. [22].b: Model prediction of stress amplitude against measured stress amplitude for the BP type endurance function.The dashed lines indicate one standard deviation = s 0.0448.

Fig. 6 .
Fig.6.a: Fatigue of AA7075-T6 for fully reversed shear.Experimental data (circles) from Ref.[25], where arrows indicate runout, are plotted with the model prediction for EL (solid line), BP (dashed line) and DP (dash-dotted line) type models.b: Fatigue life of AA7075-T6 for fully reversed shear of amplitude a [MPa] with a superimposed, static tensile stress t [MPa] according to inset table.The model predictions for different types of endurance functions are plotted against the experimental fatigue life.The solid line represents perfect agreement.

Fig. 7 .
Fig. 7. Comparison between fatigue experiments with AA7075-T6 from Ref. [25] and integrated fatigue life for a BP type model for unconventional stress paths that combine fluctuating normal stress with fluctuating shear stress of (a) the same frequency, (b) double frequency, and (c) quadruple frequency.The inset table gives the tensile stress amplitude 0 [MPa] and shear stress amplitude [MPa].The solid line represents perfect agreement, and the arrow indicates infinite life.

Table 1
[22]gue model parameters with 95% confidence intervals for AA7075-T6, AA7050-T7 and AA2024-T3, and for AISI 4340 steel alloy for different types endurance functions.The constraints = A 0 12 and = C 10 are applied in all cases.The Young's modulus E and yield stress y are taken from Ref.[22].