## 1. Introduction

The phenomenon of material fatigue is a known effect which is of fundamental importance when designing structures working under variable load conditions. Despite the large number of research works, it was not possible to develop a single, universal calculation algorithm that can be used in a wide range of construction materials as well as load conditions of the structure. The formation and accumulation of fatigue damage is a complex phenomenon and depends on many factors, e.g., the type and state of material, geometry of the element, type of loading or stress state [

1,

2,

3].

The fatigue of materials is usually referred to simple stress states, e.g., uniaxial tension-compression, where material characteristics are limited to determining the relationship between the number of cycles to failure and the level of the applied load. In the case of operational loads, the most common is a complex stress state, and in this case, we refer to the results of calculations to uniaxial characteristics using the appropriate hypotheses (calculation models) [

1,

4,

5,

6,

7].

Among the large number of fatigue hypotheses and computational models, it is difficult to find one universal that would contain a wide range of factors describing the fatigue cracks creation. Moreover, neither of them can offer a comprehensive tool that makes them adequate to use in any kind of the material, geometry and load conditions. The literature in the area contains a variety of fatigue criteria. They are based on various assumptions and parameters describing the process of fatigue. One can distinguish criteria based on stresses, strains and the so-called energy, taking into account both, the state of stress and strain.

In the case of a complex stress state, such parameters as the stress amplitude and the mean stress value cause a change in the orientation of the principal stresses, so their impact on the fatigue life is difficult to predict. The mean value of stress, different from zero, often comes as a result of the own weight of the operating element or of the entire structure, and can arise from preloading of carriers (e.g., V-belts in the gear units). Technological aspects of manufacture of constructions, technologies of joining elements (e.g., welding, explosive welding) are a source of residual stresses, which introduce significant mean loads to the structure. In the situation when the object does not transfer loads with the mean value of the load, omitting the residual stress can introduce a significant error in estimating fatigue life [

8,

9,

10].

The aim of this paper is comparison of the efficiency for selected and most frequently used fatigue life estimation criteria taking into account the effect of the mean stress in multiaxial loading conditions: Gerber [

11], Findley [

12], Dang Van [

13], Carpinteri-Spagnoli [

3], Smith-Watson-Topper (SWT) parameter [

14], and the stress models by: Kluger-Łagoda [

15,

16], which were modified with the use of two-parameter fatigue characteristics describing the relation between the values of amplitude and mean stress of Pawliczek-Gasiak model [

17]. The paper presents experimental results with calculations for the analyzed models.

## 3. Experimental Study

A fatigue test was performed in specimens made of 2017A-T4 [

15] and 6082-T6 [

15,

30] aluminum alloys and also of S355J0 [

31] steel alloy. Strength properties of the analyzed materials are provided in

Table 1. For the 2017A-T4 aluminum alloy and S355J0 steel alloy, the tests included bending, torsion conditions and two combinations of constant-amplitude of proportional bending with torsion, for which τ(t) = σ(t) and τ(t) = 0.5σ(t) with zero and non-zero mean value. For the 6082-T6 aluminum alloy, additional combinations of constant-amplitude of bending with torsion, for which τ(t) = 0.25σ(t) were analyzed.

The tests projected in this study were performed in room temperature by using the MZGS100 fatigue testing machine that can apply the control of the resultant moment of loading the specimen, on specimens presented in

Figure 1. The loads were of a sinusoidal nature with a frequency of about 25–29 Hz. The amplitudes and the mean value of the load were changed according to the test requirements. For each combination of load at least two or three specimens were used. The nominal stress amplitude and nominal mean stress value were used in calculations.

For S355J0 steel, functions for calculations of material sensitivity factor on the asymmetry of cycles for normal and shear stresses were defined experimentally [

31]:

—For bending: ψ_{σ}(N) = 3.124⋅N^{−0.162},

—For torsion ψ_{σ} (N) = 2.890⋅N^{−0.148}.

## 4. Analysis of Results

A wide range of empirical research tests allows for estimation of the effectiveness of fatigue life prediction using the analyzed models regarding the impact of mean loading. The estimation involved the computational models for each material in cases of various loading. The application of various materials for the purpose of this study allowed for performing estimation of fatigue life prediction in selected models in relation to the material grade.

Figure 2 illustrates an exemplary comparison of the calculated fatigue life (N

_{cal}) and the one derived experimentally (N

_{exp}) for S355J0 steel under various loading conditions using the Gerber’s model (

Figure 2a) and a comparison of the applied computational models according to one type of loading-bending (

Figure 2b). The equivalent stress σ

_{eq} was determined using the Gerber model (Equation (3)). This stress was used to determine the calculated fatigue life N

_{cal} using a standard S-N curve for bending (see

Table 1, slope factor and intercept).

Application in the study of different materials also offered the assessment of the efficiency of fatigue life prediction in selected models with respect to the type of the material.

Figure 3 presents an exemplary graph comparing the stability of computational results with those obtained experimentally by using Gerber’s model.

The solid line in

Figure 2 and

Figure 3 represents perfect compatibility between the results derived in calculations and experimental ones. The dotted lines represent the range of scatter of results, for which the ratio of the number of cycles to failure derived by calculations N

_{cal} and those derived experimentally N

_{exp} corresponds to results the value of 3 and ⅓.

The feasibility of the analyzed computational models and results obtained by an empirical test was estimated by the analysis of computational scatter-band as [

1,

32]:

where E

_{m} is the mean scatter, while E

_{std} is the standard deviation to mean scatter, respectively derived from the relation:

where n is the number of specimens. The scatter band for the particular results was determined as

The scatter values derived on the basis of Equation (38) for analyzed models according to the materials used in the test are presented in

Figure 4 and

Figure 5.

In most cases, it can be seen that efficiency in estimating fatigue life depends on the manner of loading and type of material. We can note that the use of Kluger-Lagoda model (

Figure 4a) does not lead to scatter bands not exceeding the value of 0.6 and the maximum values are received for torsion in 6082-T6 aluminum alloy and S355J0 steel under bending and torsional loading with mean value of loading. By using the model by Matake (

Figure 4b) greater scatters are received for non-zero mean value of loading under bending and torsion in S355J0 steel, wherein, the case of bending relates also to aluminum alloys. For Findley’s model (

Figure 4c) the largest scatter bands are derived for S355J0 steel, regardless of the manner of loading, whereas in the case of complex loading with non-zero mean value, satisfactory results of fatigue life estimation were obtained, which were comparable for all analyzed materials. The Dang Van fatigue model (

Figure 4d) yields very similar results in terms of the nature and values as they are derived from Findley’s model. Additionally, the fatigue life prediction for S355J0 steel demonstrates large scatters. Models proposed by Papadopoulos (

Figure 4e) are characterized by wide-ranging scatter plots of results for non-zero mean value loads in all of the analyzed materials, while loading with zero mean value indicates two to three times lower scatters. The Smith-Watson-Topper model (

Figure 4f) demonstrates extended scatters for both tested aluminum alloys under torsion loading with—both for zero as well as the non-zero mean value, where the scatter band receive the values: E

_{eq} = 2 and 3 for 6082-T6 and 2017A-T4, respectively. For S355J0 steel the scatter bands are ranging from 0.5 to 1.75.

Figure 5 shows the continuation of the analysis for the remaining models.

Scatter bands of fatigue life estimation results calculated according to the model by McDiarmid is presented in

Figure 5a, wherein for loading with mean value, the poor results were obtained for S355J0 steel and 2017A-T4 alloy under torsion and bending with torsion with mean value of loading. Moreover, in the Carpinteri-Spadnoli model (

Figure 5b) the poorest estimation results are derived under loading with mean value for bending, torsion and torsion with bending.

Fatigue life estimations based on models by Goodman (

Figure 5c) and Gerber (

Figure 5d) are characterized by similarity of scatter bands, wherein, the dependence by Gerber seems to be more sensitive to the type of test material—scatter levels for all of the analyzed cases of loading are similar for three of the analyzed materials. For 2017A-T4 alloy in complex stress state with mean value, both of the analyzed dependencies indicate approximately two times wide-ranging scatter bands than the other cases of loading. As a result of the application of the model proposed by Soderberg (

Figure 5e), three to four times larger scatters with regard to the estimation results for S355J0 steel, being tested under loading with mean stress values. For S355J0 steel the fatigue life prediction was additionally performed using the model proposed by Pawliczek and Gasiak (

Figure 5f). This model is characterized by a scatter plot on the results that ranged from 0.25 to 0.4 for all the analyzed cases of loading with both, the zero and non-zero mean values of stress.

The above observations indicate that most of these models demonstrate high sensitivity to loading conditions. In certain cases, there can be seen larger discrepancies between computational results and experimental results under loading with non-zero mean values.

In order to assess the flexibility of the analyzed calculation models considering the effect of mean load and the type of loading (i.e., simple or complex), mean value of scatter bands E

_{eq} were subsequently derived for each of the materials (

Figure 6). For S355J0 steel the most adequate averaged results were obtained for models by Kluger-Lagoda, Carpinteri-Spagnoli, Goodman, Gerber and Pawliczek-Gasiak, where scatter band of the results was E

_{eq} < 0.4. In the case of aluminum alloys, the discrepancy was not equally evident in terms of the analyzed calculation models, however, the most efficient results of computations were obtained for the Kluger-Lagoda dependency, and then for Goodman and Soderberg’s models.