A comparative study of Glinka and Neuber approaches for fatigue strength assessment on 42CrMoS4‐QT specimens

In fatigue strength assessment, the methods based on ideal elastic stresses according to Basquin and the less established method based on elastic‐plastic stress quantities according to Manson, Coffin and Morrow are applied. The former calculates loads using linear‐elastic stresses, the latter requires elastic‐plastic evaluation parameters, such as stresses and strains. These can be determined by finite element analysis (FEA) with a linear‐elastic constitutive law, and subsequent conversion to elastic‐plastic loads, using the macro support formula by Neuber. In this contribution, an alternative approach to approximate elastic‐plastic parameters proposed by Glinka is compared to the the strain‐life method using Neuber's formula, as well as the stress‐life method of Basquin. Several component tests on 42CrMoS4‐QT specimens are investigated. To determine the input data for the fatigue strength evaluations, the entire test setup is computed by FEA. The nodal displacements from these validated full‐model simulations are used as boundary conditions for a submodel simulation of a notch, whose results serve as input for the fatigue strength assessments. It is shown that all approaches provide a reliable assessment of components. Our key result is that the strain‐life method using the concept by Glinka for notch stress computation, yields improved results in fatigue strength assessments.


| INTRODUCTION
In all areas of technology, highly-stressed components are often made from 42CrMoS4-QT quenched and tempered steel.The aim is to use as little material as possible, especially with regard to lightweight construction.It is therefore important that machine components are not unnecessarily oversized.However, this must be balanced with the imperative of preventing any damage that might potentially pose a threat to human safety.It is therefore essential that components are designed according to meaningful and safe assessment methods.
Haibach [1] and Radaj's [2] monographs provide insights into safe assessment methods that were developed during the 19th century, when significant advancements were made in this field of research.The foundational research of August Wöhler, which is summarised by Stüssi, [3] must be acknowledged as the starting point for initial investigations.Wöhler's discovery that components subjected to alternating stresses have a lower load capacity than those under static stresses was a crucial finding.Additionally, Haigh's contributions encompass investigations into the impact of mean stresses on fatigue strength. [4]Neuber played a defining role in establishing the theory of stress concentration.His extensive findings, encompassing a comprehensive monograph, have been instrumental in shaping this scientific discipline. [5]Early in the fatigue life assessment of machine components, Gassner also merits recognition.His expansive studies on fatigue strength, [6][7][8][9][10][11][12] among others, culminated in the derivation of a lifetime curve from Wöhler curves.The lifetime curve operates as a load spectrum-specific evaluation criterion.
The aforementioned, fundamental findings on fatigue life assessment were expanded in the second half of the 19th century to include the following items [13] : mean stress effects based on Haigh, [4] statistical, geometrical and technological size effect, influence of surface roughness, temperature effects and stress concentration effects.According to the latest investigations, these influencing variables must be considered to be frequency dependent, see Thumann. [14]ecent publications underscore the sustained significance of 42CrMoS4 investigations.Starke et al. [15][16][17] outline a testing approach necessitating only three specimens to define the Wöhler curve.Subsequently, Starke and Wu [18] progress this methodology, presenting an enhanced version through a new, single specimen test strategy.This strategy serves to yield more insight into fatigue behaviour while concurrently curtailing experimental effort and expenses.In the very high cycle fatigue region, Pyttel et al [19,20] have delved into the influence of defects on fatigue strength and failure mechanisms of the corresponding quenched and tempered steel.By Homayonifar et al. [21] , the frequency dependence of 42CrMoS4 in relation to its hardness is studied.Moreover, Lang et al. [22] chronicle frequency-dependent uniaxial tension-compression-fatigue tests with loads at 50 Hz and 1 kHz on 42CrMo4 specimens tempered at various temperatures to encompass a broad ultimate strength spectrum.Weber et al. present material tests on 42CrMoS4 in conjunction with finite element simulations, encompassing diverse manufacturing processes within the analyses aimed at fatigue life assessment.
In a previous investigation [23] on subcomponents of forming and cutting dies for sheet metal manufacturing, the authors carried out component tests on specimens made of steel 42CrMoS4-QT for several load-cases, showing a ratesensitivity of this material.In order to extend the findings in the field of fatigue strength evaluation of 42CrMoS4, the present work uses this experimental data for the validation of numerical fatigue strength assessment.Specifically, two concepts for fatigue strength assessment are compared.[27] The latter requires elastic-plastic evaluation parameters, such as notch-stresses and notch-strains.These can be determined using FEA with a linear-elastic constitutive law, which are converted into elastic-plastic notchstresses and notch-strains using either the Neuber [5] or the Glinka [28] approach.This leads to three different approaches for the numerical fatigue strength assessment which were performed and evaluated in this study: 1. the evaluation concept according to Basquin, 2. the evaluation concept according to Manson, Coffin and Morrow, (a) using either the Neuber approach or (b) using the Glinka approach for calculating the notch-stresses.
In this contribution, the component tests are computed using FEA with a submodel approach (see, e.g., Wagner [29] ).A comparison of the numerical results with the experimental data reveals, that applying the Glinka approach yields the most accurate life-time assessment for 42CrMoS4-QT specimens.However, the Glinka method is not commonly available in commercial software, for which no obvious reason can be found in the literature.Hence, the key statement of this contribution is that the method of Glinka to compute notch-stresses is at least comparable to other methods and thus should be made available for fatigue strength assessment in commercial software.
In Section 2, the FE model for an explicit simulation of the experimental setup and a static-implicit submodel of a notch region of interest are explained.Section 3 briefly describes the stress-and strain-life methods and the fatigue strength assessment.A discussion in Section 4 focuses on comparing these two concepts for fatigue strength assessment of 42CrMoS4, especially regarding the computation of the notch stress and the implications for fatigue strength assessment.
This section describes the methodical procedure for modelling and performing the transient numerical simulations of the component tests as well as the submodel technology for the static-mechanical investigation of the notch in the specimen.The FE software LS-DYNA [30] is used for the simulations since it offers highly accurate and efficient capabilities for transient as well as static simulations, see Thumann. [14]1 | Outset Increasingly, press systems with elevated stroke rates are being employed to enhance efficiency in the realm of sheet metal manufacturing.The escalation in stroke rates leads to heightened structural-dynamic loads on components, e.g. from the processes of forming and cutting.To assess the endurance capability of critical subcomponents within a complex entity like a press die, force-controlled fatigue tests on these components were executed in Thumann et al. [23] These tests are conducted on simplified specimens intentionally designed to replicate the critical part geometries found in a die, such as a slide stopper in this instance, see Figure 1 and Figure 2. The relevant root radius of the specimen is chosen to be r ¼ 0:5mm, see Section 2.3.1.The primary role of a slide stopper is to accurately position slides at specific F I G U R E 1 Schematic of the clamping device for fatigue tests with an inserted 42CrMoS4 specimen, adapted from Thumann et al. [23] F I G U R E 2 FE discretised experimental setup.
terminal points.Owing to substantial restoring forces, slide stoppers endure substantial strain, and there have been instances of failure in the press shop previously.The load functions in the present report are chosen based on measurements of such a system.For an in-depth description of the specimen and the test setup, see Thumann et al. [23] The fatigue testing was conducted using an Instron 8800 Servohydraulic Test System.For securing the specimens within the testing machine, a specialised clamping arrangement was developed (depicted in Figure 1).This arrangement incorporates a strengthened clamping mechanism that firmly holds the specimens, while a retention pin ensures a form-locking fixation.This precautionary measure effectively prevents any slippage of the specimen during the course of cyclic loading.In essence, the assembly for the clamping system is divided into an upper and a lower sub-assembly.These two sub-assemblies are linked together by the specimen.The upper sub-assembly is connected to the fixed load cell.The load application is controlled via the load cell.The lower sub-assembly is linked to the moveable cylinder piston.The piston movement is set by the defined load functions, see Figure 3.

| Transient FE simulation of the test setup
The transient FE simulation of the test setup described in Section 2.2.1 is generally used to resolve the entire deformation behaviour of the system.Since the tests were carried out on a hydropulser, there was no possibility with this equipment to measure displacements directly on the specimen.Hence, we have to take into account the deformation of the machine also in the simulation, since applying measured forces in the load cells of the hydropulser on the probe directly would not represent reality correctly.But, the deformation in the area of the notch in particular should be simulated static-implicit with more detail, as shown in Section 2.3.

| FE model setup
Figure 2 shows the FE discretised simulation model with the boundary conditions according to the component tests carried out in Thumann et al. [23] Table 1 lists the individual components considered in the model.The parts columns (1), cylinder piston (13) and all metric bolts (8) are discretised with beam elements.The base plates (2, 14) and the ball joint (12) consist of linear, reduced-integrated shell elements.To resemble the behaviour of a ball joint as realistically as possible, it is modelled with shell and beam elements.The angular motion is realised by nodal rotational degrees of freedom of connected beam elements.
All other components are almost exclusively meshed with linear, reduced-integrated hexahedral elements; for details, see, for example, Wagner. [29]The lateral plates (7) and the clamping claw (9) are hidden in the figure to make the relevant parts underneath visible.All parts are modelled as deformable bodies with elastic material behaviour to resemble the real experimental behaviour of the test setup as closely as possible.The additional numerical effort is negligible as compared to modelling parts of the system as rigid bodies.
The boundary conditions are chosen as follows and depicted in Figure 2: The degrees of freedom of the corner nodes of the lower base plate are locked as a kinematic boundary condition for the support.This support condition has been chosen for simplicity reasons, even though it does not model exactly the real experimental setup.However, since the lower base plate is very massive, it is assumed that the results are not influenced by this.A force curve acts as a kinetic force boundary condition on the cylinder piston.The three applied load functions, represented as force-time curves in Figure 3, were used as input signals for the component test in Thumann et al [23] and are therefore used as load boundary conditions in the FE simulations.For the sake of completeness, gravity is introduced as another force boundary condition, even though its influence should be negligible in this application.Bolt preloads [31] act on the bolts as displacement boundary conditions.
The contacts in the model are summarised in Table 2.These are mainly frictional penalty contacts with the same uniform coefficient of friction μ ¼ 0:2.This value is chosen from experience since adhesion is assumed in this setup, but small local relative motions should be possible.The tied contact specimen-fixation represents the positive-locking connection of the two components contained in the real structure by means of a cylindrical pin.
The tied contact notch-specimen is included in the model because the cutting planes for subsequent submodel simulations have to be defined using the LS-DYNA solver.For this reason, the specimen is divided into two components and firmly connected by a tied contact, see Figure 2. The calculated nodal displacements at the cutting planes (see Figure 5) are used as kinematic displacement boundary conditions in the following submodel simulation.
The material data of all components to define the linear-elastic constitutive behaviour are listed below: Based on previous studies, the following solver settings are defined (for details, see Wagner [29] ): • An explicit time integration method is used for the simulation of the overall component model, where the submodel in Section 2.3 is calculated statically by use of an implicit solver.The simulation of the overall movement is a shorttime dynamic process that is strongly characterised by contact modelling.For this, the explicit FEM is a very efficient calculation method but implicit time integration could be used as well.• A dynamic relaxation is carried out at the beginning of the calculation.This ensures that an approximate equilibrium position is established before the actual transient analysis, taking into account the dead weight and the screw preload forces.• Hourglass stabilisation is activated to avoid zero-energy modes of the reduced-integrated elements used.
• In order to reduce the CPU time, the loading speed from the component tests of Load 1 and 2 is reduced from T ¼ 1s to T ¼ 0:2s.This causes oscillations that are reduced by an artificially introduced global system damping.

| Results and validation
The deformation behaviour of the test setup is examined to validate the transient model simulation.For this purpose, the path of the cylinder piston resulting from the introduction of force by the hydraulic cylinder is observed and the measured values from the simulation and the tests are compared, see Figure 4.In all simulations, the deviations e ¼ s m À s c between the measured and calculated displacements s m and s c , respectively, are e < 0:05mm, which is considered to be reasonably small.For a detailed discussion, see Thumann. [14]ence, the explicit full model is considered to yield accurate results that can be used as boundary condition in the submodel simulation.

| Static-mechanical submodel simulation
The static-mechanical submodel simulations are used to generate FE results for the subsequent fatigue strength analyses.

| FE-model setup
Figure 5 shows the discretised submodel.The interface areas between the overall model and the submodel are indicated.There, the displacements from the overall model are calculated and transferred with a mapping to the boundaries of the submodel on the same surfaces as displacement boundary conditions.The figure illustrates the advantage of the submodel technique: in contrast to the transient model of the test setup, the notch now contains the root radius of r ¼ 0:5mm, which can be discretised using a fine FE-mesh.This value has been chosen because it is a typical but critical value in such components in deep drawing tools.A smaller notch radius can not be easily produced by milling in large-scale toolmaking.
Another advantage is the possibility of a structured meshing of the problem domain using hexahedral elements with a quadratic basis function.This generates the best possible input data in the form of mechanical stresses for subsequent fatigue strength analyses.In addition, the stress distribution over a problem domain can be well approximated with relatively few elements.
The FE discretisation shown in Figure 5 consists of six elements over the radius.The FE discretisation is varied to determine mesh convergence with regard to the maximum mechanical stresses occurring in the notch area.Meshes with two to eight elements over the radius are used in this study, see Table 3.The nodal displacements from the transient simulation are applied as kinematic displacement boundary conditions at the interface boundaries in the submodel simulations.
The constitutive behaviour corresponds to that of the transient simulation.It follows that the stress load determined from linear-elastic FE calculations is used for the fatigue strength analyses.In the following evaluations according to Basquin, linear stresses are directly included in the calculation.For the evaluation according to Manson, Coffin and Morrow, elastic-plastic load values as stated by Neuber [5] or Glinka [28] must be determined.
The submodel simulations are static analyses.The inertia effects are not taken into account as these are already included in the transient model simulation.The convergence criteria are left at the program-internal standard settings.

| Results
Figure 6 shows the distributions of the von-Mises equivalent stresses σ v on the submodel for the three load cases at maximum load.As expected, the maximum values occur in the middle of the notch radius, the maximum stress value is given under the picture, respectively.Correspondingly, the diagram in Figure 7 displays the maximum equivalent stresses over the number of elements in the notch region for a representative convergence study carried out for Load 1. Due to the linearity of the submodel calculation the other load-cases yield shifted but similar results.
The change in the maximum stresses with mesh refinement is used as a criterion in order to determine a suitable discretisation.A comparative calculation is carried out with the ABAQUS [32] solver to confirm the validity of the criterion for the mesh refinement.In both calculations, there are only minor changes between Mesh number 3 (six elements F I G U R E 6 Distribution of von-Mises equivalent stresses over the notch for the three load cases at maximum load for Mesh number 3 in Table 3. over notch) and Mesh number 4 (eight elements over notch).Fatigue strength evaluations are therefore based on the FE results obtained from Mesh 3. The slight difference in the converged stress result of about 5.9 % can be explained by the fact, that the used element types-H20 in LS-DYNA and C3D20 in ABAQUS-are both twenty-node serendipity hexahedral elements but use a different quadrature rule for the computation of stresses.

| FATIGUE STRENGTH CALCULATIONS
The evaluation concept based on linear-elastic stresses according to Basquin [24] (stress-life method) and the evaluation concept based on elastic-plastic strains according to Manson, Coffin and Morrow [25][26][27] (strain-life method) are compared with regard to the fatigue strength analyses.Moreover, the methods of Neuber and Glinka for computing the notch stresses in the strain-life method are investigated.
The stress values used for the two concepts, the determination of the component Wöhler curves and the evaluation results are described in the following.

| Load inputs
The calculation results of the FE analyses according to Section 2.3 with linear-elastic material behaviour are used for both evaluation concepts.
Due to the simple load-time curve, the load collective for the stress-life analysis is created manually and not according to a common counting method such as rainflow counting.The strain-life method, on the other hand, requires a sequence-dependent count to include the mean stress redistributions. [1,2]

| Stress-life method after Basquin
When verifying the fatigue strength according to the stress-life method, the FKM guideline [13] can be consulted.If this guideline is used, either the surface stresses σ x , σ y and τ xy or the principal stresses σ I , σ II , σ III can be used.The principal stresses are employed in the present work.Figure 8 shows the resulting stress curve taking the first principal stress σ I for Load 1 as an example.All load cycles that form a complete oscillation are counted.In the present case, there are four load cycles whose location and magnitude are defined using the mean stresses σ m and amplitude stresses σ a .The corresponding stress components of all load cycles are summarised in Table 4.It should be noted that the third principal stress must be zero, due to the free surface condition.Since a discretisation method is used for the computation, this is not completely fulfilled.However, the values are several orders of magnitude smaller than the in-plane principal stresses and therefore have no influence on the result.For this reason, the values are not given in Table 4.The same applies to Table 9 further on.

| Strain-life method after Manson, Coffin and Morrow
In the evaluation concept according to Manson, Coffin and Morrow, the elastic-plastic notch stress is approximately determined using the linear-elastic stress values from the FE analyses.The macro support formula from Neuber [5] is widely used for this.There exists a second method to evaluate notch stresses which is attributed to Glinka [28] .Both methods are still the subject of current research.Ball [33] provides an overview of the current state of research and comes to the conclusion that none of the methods are suitable to every use case.Ostash and Chepil [34] measure notch strains for different notch geometries and compare their experimental results with the Neuber and the Glinka approach, leading to the conclusion that the Glinka method delivers the most accurate results.However, the Glinka approximation is hardly used in commercial software applications.It is shown in the following, that this procedure is preferable and should be considered to be used in fatigue strength assessment.
Neuber relates the elastic-plastic stress and strain concentration factors α σ and α ε to the purely elastic concentration factor α k By inserting the ratios of the local values to the nominal values α σ ¼ σ pl =σ n and α ε ¼ ε pl =ε n in Equation ( 1), we get If the nominal stress σ n does not exceed the yield stress R p0:2 , the nominal strain ε n can be calculated using Hooke's law ε n ¼ σ n =E with Young's modulus E and Equation (1) changes to T A B L E 4 Load inputs for the fatigue strength assessment according to the stress-life method (FKM guideline).

Stress component in
Substituting the product of the elastic concentration factor and nominal stress α k σ n by the linear-elastic notch stress σ el from a FE analysis results in Figure 9 depicts the relationship between the elastic Hooke's notch stresses σ el and the elastic-plastic notch stresses σ pl from Equation (4).Hyperbolic curves are employed.The relation according to Ramberg and Osgood [35] serves as a constitutive equation for the description of the elastic-plastic material behaviour with the cyclic hardening coefficient K 0 , the cyclic hardening exponent n 0 and Young's modulus E.
Inserting Equation ( 5) in Equation ( 4), we arrive at an equation to iteratively compute the elastic-plastic notch stress σ pl ðε pl Þ from the linear-elastic notch stress: In contrast, the approach according to Glinka to determine the elastic-plastic notch stress is based on the deformation energy density, see Figure 9. Thus, the deformation energy density in the notch region with local plastification is assumed to be the same as with linear-elastic material behaviour [36,37] With F I G E 9 Graphic representation of Neuber's rule and the energy-based method according to Glinka. and follows (see Glinka [28] ) Analogous to Equation ( 5), the elastic-plastic notch stress σ pl can be determined iteratively from the linear-elastic stress σ el by transformation of Equation (10): As already mentioned, the Glinka approach provides a more accurate computation of notch stresses, as has been pointed out by Radaj [2] (p.176), as well as by Zeng and Fatemi, [38] than the Neuber macro support formula. [28]It can be seen from Figure 9 that the notch strain according to Glinka ε pl,G is significantly lower than the notch strain according to Neuberε pl,N for the same linear-elastic strain ε el .Both methods are considered in the evaluation of the results in Section 3.3.
In the strain-life method, stress data from the resulting hysteresis loops in the stress-strain diagram are used to estimate the fatigue strength.Figure 10 shows the hysteresis calculated according to Neuber using Load 1.These are generally determined on the basis of the load curves (Figure 3), the maximum von-Mises stresses (Figure 6) and the constitutive equation (Equation 6or Equation 11).On the left of Figure 10 the stress-strain relation is given and on the right the time sequence.The initial loading from ⓪ up to ⑧ does not contribute to the life time assessment, as the cyclic loading only starts at point⑧ onwards.
From the beginning of a load sequence at point ⑧, the stress-strain path follows the constitutive equation according to Equation ( 5) up to the maximum value of the load sequence ①.This constitutive equation describes the cyclically stabilised stress-strain curve, which differs from the stress-strain curve in a tensile test in that hardening or softening processes are taken into account during cyclical loading.The two curves do not differ if a material has neither a strengthening nor softening behaviour. [1]The cyclic hardening coefficient K 0 and the cyclic hardening exponent n 0 are determined experimentally using the incremental step test. [39]If material tests cannot be carried out, cyclically stabilised stress-strain curves can be generated using estimation formulas.These are based on material tests that are compiled in Boller and Seeger. [40]In the present investigations, the Uniform Material Law (UML) according to Bäumel jr. and Seeger [41] is used for unalloyed and weakly alloyed steels.The result for the material 42CrMoS4-QT is shown in Table 5.
After the first load, at the turning point ① and the maximum value of the load sequence, the specimen is unloaded.This extends to the minimum value of the load sequence at point ②.This sequence as well as all other loading and F I U R E 1 0 Stress-strain hysteresis with the associated load-time sequence.
unloading progressions, are represented in good approximation by doubling the stress and strain values according to the Masing model. [42]It therefore follows from Equation ( 5) that The further sequences of load changes from ② to ⑧ follow Equation (12).All subsequent load cycles begin and end at the minimum value at ⑧. Therefore, the entire stress event lies within an envelope hysteresis⑧ -①-②, which is defined by the maximum and minimum values.
The resulting load data from the hysteresis loops are used in the fatigue strength assessment in Section 3.3 and are summarised in Table 6.Note, that the hysteresis indicated in the table with -⑤-does not form a closed hysteresis in the stress-strain diagram and incorporates also③ which has been left out for the sake of the naming convention.The light green triangles without a border in Figure 10 are intended to indicate this.Since residuals should be avoided, especially in the case of load sequences with few reversal points, the stress data for this hysteresis are interpreted as closed hysteresis and used with the starting point in and a reversal point in⑤.

| Component Wöhler curves
In both the stress-life and the strain-life methods, specific component Wöhler curves are generated from general material Wöhler curves.

| Stress-life method
In the FKM guideline, which is based on the concept of Basquin, [24] a Wöhler curve is shown as a straight line in the double-logarithmic diagram according to the equation T A B L E 5 Material parameters for 42CrMoS4-QT according to the UML for the constitutive equation.
T A B L E 6 Load inputs for the fatigue strength assessment according to the strain-life method.

Hysteresis
Neuber Glinka see Figure 11.This equation describes the relationship between the maximum stress amplitude S and the load repetitions N f that can be borne until failure occurs.The coefficient C and the exponent n are determined by regression analyses from Wöhler tests and are included in the equation as material parameters.
In general, Wöhler tests are carried out on unnotched specimens to determine the Wöhler curve at a constant stress ratio R with the lower stress σ low and the upper stress σ up of a load cycle.It follows that a component Wöhler curve for a stress ratio R corresponding to the load must be derived from a general Wöhler curve.According to the FKM guideline, the component Wöhler curve is calculated using the equation see Figure 11.The fatigue duration strength σ D , the limit of cycle number N D and the Wöhler curve exponent k are incorporated into the equation as material parameters.The fatigue duration strength σ D is determined by the tensile strength R m .In Table 7, the values for the tempered steel 42CrMoS4-QT are summarised according to the FKM guideline.
Taking into account shape and surface influences due to the technological size-influencing factor K d , the notch sensitivity n χ , the stress concentration factor K t and the roughness factor K r , the fatigue duration strength σ D is adapted to the actual component, see Section 3.3.
The bearable stress amplitudes σ A vary for different mean stresses σ m .This connection is described in the Haigh diagram, [4,43] which is the graphic representation of all values of fatigue duration strength σ D obtained from a series of Wöhler curves at different mean stresses σ m .In general, the bearable stress amplitude σ A decreases with an increasing tensile mean stress σ m .This correlation is taken into account in the FKM guideline by means of the mean stress factor K AK .

| Strain-life method
In their trendsetting works, [25][26][27] Manson, Coffin and Morrow related the number of cycles until crack initiation to the amplitude of the plastic deformation.Accordingly, the graphs of the elastic and plastic strain amplitudes can be represented as straight lines on a double-logarithmic scale in a Wöhler diagram (see Figure 12) The fatigue strength coefficient σ 0 f , the cyclic ductility coefficient ε 0 f , the fatigue strength exponent b, the cyclic ductility exponent c and the modulus of elasticity E are included as material parameters.These values are determined either by complex Wöhler tests [44] or by the less complex incremental step test. [39]f experimental investigations are not possible, strain Wöhler curves can be created using estimation formulas.These formulas are based on strain-controlled Wöhler tests, which are compiled in Boller and Seeger. [40]In the present investigations, the Modified Universal Slopes Equation (MUSE) according to Muralidharan and Manson [45] is used for unalloyed and weakly alloyed steels.The values for the material 42CrMoS4-QT are summarised in Table 8.
Figure 12 shows schematically the composition of the strain Wöhler curve according to Equation ( 16).The values σ 0 f =E and ε 0 f , which are based on the intersection of elastic and plastic strain Wöhler curves with the ordinate, can be read directly from the diagram.The exponents b and c in Equation ( 16) correspond to the slope of the elastic and plastic strain Wöhler curves, respectively, as depicted in Figure 12.
According to the FKM guideline, a component Wöhler curve is developed from a general material Wöhler curve.In the case of the strain-life method, this is referred to as a damage parameter Wöhler curve, which is created with the help of a defined damage parameter.Mean stress and sequence influences are taken into account with this damage parameter. [1]

| Fatigue strength assessment
The linear cumulative damage model after Miner [46] is used for the fatigue strength assessment.According to Miner, the ratio of the deformation energy of a load cycle w i and the total possible deformation energy up to failure W equals the ratio of the number of load cycles n i and the bearable load cycles N f i : Using the absorbed total deformation energy up to failure W , which is made up of individual portions w i at different load levels the sum of the ratios of the deformation energies w i and the maximum possible deformation energy W yields the material failure with Substituting the terms from Equation (19) into Equation ( 17) leads to the mathematical description of the linear cumulative damage as The total damage D consists of the individual damages D i .The material failure begins with the damage sum D ¼ 1, analogous to Equation (19).
This contribution focuses on the comparison of the methods based on ideal elastic stresses according to Basquin and the less established method based on elastic-plastic stress quantities according to Manson, Coffin, and Morrow.A comparison of damage models is not the objective.Therefore, we deliberately chose the linear damage accumulation according to Miner, as this method is widely used.In accordance with Section 3.2, form and surface influences are taken into account in the fatigue strength assessment in line with the FKM guideline.Table 9 shows the development of the long-term strength σ D,i , called fatigue duration strength, for the relevant principal stresses i ¼ I; II.The material strength, which decreases with an increasing component size, is taken into account by means of the technological size-influencing factor K d . [13]In our case, the fatigue endurance strength σ D is decreased.The notch supporting effect is taken into account using the notch sensitivity n χ , thus increasing the fatigue duration strength σ D,i .This report uses the supporting number after Stieler and Siebel. [47,48]According to the FKM guideline, the influence of the roughness factor K r again reduces the strength.By considering a stress concentration factor K t > 1 for the case of notched components, the fatigue duration strength σ D increases.The fatigue duration strength σ D from Equation ( 7) is adjusted with these influencing variables depending on the stress component.
Furthermore, the effect of the mean stress influence can be seen in Table 9.According to the FKM guideline, the strength assessment is related to a specific load reversal in the total load curve.In the present work, this is the first load reversal in Figure 8, which also has the highest mean and amplitude stress values.The fatigue duration strengths σ D,i decrease due to the existing tensile mean stresses.
The third principal stress (i ¼ III) tends towards zero due to the stress-free surface.The damage due to these low third principal stresses σ III,m and σ III,a is negligibly small in the fatigue strength assessment and is thus neglected here, as already mentioned with Table 4.
As explained above, the strength assessment relates to the first cycle of the load curve in Figure 8.The existing amplitude stress σ a is compared with the bearable amplitude stress σ f .The three other load reversals are taken into account by the bearable amplitude stresses σ f ,i and are determined by the Miner elemental method.Table 10 lists the fatigue strength values σ f ,i that are decisive for the strength assessment.

| Strain-life method
As explained in Section 3.2, a damage parameter Wöhler line is derived from the general strain Wöhler line according to Equation (16) with the help of a defined damage parameter.A common damage parameter is described by Smith, Watson and Topper [44] with the equation Here, the product of the upper stress σ u and the total strain amplitude ε a,t ¼ Δε=2 is considered damaging.Young's modulus E is included as an additional quantity but is irrelevant to damage.The product σ u ε a,t can be interpreted as a strain energy density [2] that describes the upper part of the hysteresis loop in Figure 13.
The damage parameter according to Smith, Watson and Topper, however, shows a weakness with pressure medium stresses σ m < 0, since the bearable amplitude stresses σ A are predicted to be significantly too high. [49]Based on this knowledge, Bergmann [50] introduces an individual damage parameter T A B L E 1 0 Endurance strength of the component.

Load case
Endurance strength in MPa where the upper stress σ u in Equation ( 21) is described by the equivalent expression from the amplitude stress σ a and the mean stress σ m .The parameter k for weighting the mean stress σ m is defined for steel materials as follows [49] : Using the information in Table 6, Equation ( 22) yields the damage parameters P I listed in Table 11.These values are used with the damage parameter Wöhler curve and the linear cumulative damage to determine the bearable load repetitions N f .The upper stress σ u in Equation ( 21) is obtained using the elastic strain component from Equation ( 16) to determine the damage parameter Wöhler curve By substituting the total strain amplitude ε a,t in Equation ( 21) with the strain Wöhler line from Equation ( 16), and by using Equation (23), it follows that the damage parameter Wöhler line valid for any mean stress and mean strain is F I G U R E 1 3 Specific values of stress-strain-hysteresis for Smith-Watson-Topper damage parameter P SW T T A B L E 1 1 Damage parameters P I for the fatigue strength assessment according to the strain-life method.

Load Hysteresis
Damage parameter P I Table 12 shows the calculated bearable cycles N f ,i for the corresponding load level i ¼ 1,2, 3,4.Taking into account the fatigue duration strength at N D ¼ 10 6 , the bearable load repetitions N f for load 1, 2 and 3 are obtained according to the linear cumulative damage results.

| DISCUSSION OF THE RESULTS
Table 13 compares the calculation results from the stress-life and the strain-life method with the component tests according to Thumann et al [23] with a failure probability of 10 %.
The percentage deviations e are calculated from the differences between the actually endured and calculated load repetitions.
As can be seen in Table 13, the strain-life method using elastic-plastic notch stresses according to Neuber exhibits the highest deviations and thus the most pessimistic results as compared to the stress-life method used in the FKM guideline.It must be pointed out, that for Load 2 the stress-life according to the FKM guideline is overestimated, which is a critical result.On the other hand, our investigations show that the strain-life method using the elastic-plastic notch stresses according to Glinka provides similar deviations than the stress-life method and does not overestimate any load T A B L E 1 2 Bearable load repetitions according to the strain-life method.

Neuber
Glinka cases.This yields a promising argument that the method of Glinka to compute notch stresses in combination with the strain-life method should get more attention in fatigue assessments, especially in commercially available software as an additional analysis option.

| CONCLUSIONS AND OUTLOOK
The stress-life method according to Basquin (FKM guideline), as well as the strain-life methods with elastic-plastic notch stresses according to Manson, Coffin and Morrow are investigated.For the strain-life method the macro support formula of Neuber and the lesser known method by Glinka are compared.The strain-life method offers certain advantages: It is valid in the short-term strength range.On the other hand, the range of validity of the stress-life method, only begins with a cycle number of N > 10 4 . [13]In addition, the influence of the order of load sequence is taken into account in the strain-life method.The disadvantage compared to the stress-life method is the comparatively low level of commercial establishment, even though, a detailed guideline has been available since 2019, see Fiedler et al. [51] Our investigations show that all concepts generate proper results, underestimating the strength of the test objects and are therefore suitable for a reliable design of components made from 42CrMoS4-QT.One exception is a slight overestimation of one load case with the stress-life concept.
According to our experiments the method of Glinka for notch-stress calculation is at least comparable and in specific cases superior to the other here evaluated methods for the application in lifetime predictions.Nevertheless, the method remains largely unused for fatigue strength assessment.It is therefore suggested to consider that this approach should be added in commercial software as an additional analysis option for lifetime assessment.
In order to further verify the strain-life method using the computation of the notch stress approach of Glinka, the authors suggest carrying out more calculation examples.To evaluate the calculation results, either test data or a defective component with a load history should be available.

T A B L E 1
Components included in the FE model.

FF
I G U R E 4 Comparison between test and simulation results of piston displacement.Upper row: displacements, lower row: deviation.T A B L E 3 FE discretisation for mesh studies in the submodel simulations.I G U R E 5 Submodel with radius r ¼ 0:5mm in the notch.

F I G U R E 8
Time response of the first principal stress σ I for Load 1.

F G U R E 1 1
Stress-life Wöhler curves.T A B L E 7 Material parameters according to the FKM guideline.Parameter R m in MPa σ D in MPa N D k

T A B L E 9 1 |
Change in fatigue strength due to the influencing factors.Stress-life method Contacts in the simulation model.
T A B L E 2 Comparison of the calculated and the actually endured load repetitions.