Influence of Pre-Strain and Notching on the Fatigue Life of DD11 Low-Carbon Steel

Abstract

Structural applications commonly adopt low-carbon steels, with the fatigue concept being one of the primary causes of failure. In this research, the aim was to study the fatigue behaviour of DD11 low-carbon steel, considering also specific conditions, like the effect of pre-deformation and influence of stress intensity factor. After determining the geometry and performing static tests to extrapolate the mechanical properties of the material, the fatigue behaviour of the base material was analysed, following the actual standards. Then, two conditions, a pre-strain equal to 11% and a notch, simulated with a hole and without pre-deformation, were studied. The results showed an absence of influence on the fatigue limit for the material with a pre-strain effect, and regarding the notching tests conducted, there was a low sensitivity to fatigue of the material.

1. Introduction

The fatigue life of a component has always been difficult to predict due to the multiplicity of variables that are involved in the process. This phenomenon affects several industrial fields, such as automotive [1,2], lifting equipment [3,4], but also pressure vessels [5] or railway vehicles [6]. In all these fields, steel is one of the most adopted families of materials due to multiple factors, for example, its mechanical properties, its reliability or the vast knowledge about its behaviour. Low-carbon steel, a good compromise between mechanical properties and processing aptitude, is a widely used material in this family.

Regarding the fatigue behaviour of these materials, there are several studies concerning both low-cycle fatigue (LCF) [7] and high-cycle fatigue (HCF) [8], but also concerning the particular situation of the material: Kim et al. [9] analysed the fatigue behaviour of a low-carbon steel and the fatigue behaviour of welded joints obtained through the plasma arc welding process; the authors found that the cyclic stress response in the HCF mode showed an initial cyclic softening, followed by cyclic stabilisation, while for the welded joints, the extremely low cycle fatigue (ELCF) mode at large strains showed no cyclic stabilisation after initial cyclic hardening. Tsutsumi et al. [10] analysed the very high cycle fatigue (VHCF) through ultrasonic fatigue tests (20 kHz) of annealed and 10% pre-strained specimens for 0.13% carbon steel, discovering that fatigue lives and fatigue limits under these conditions were higher than those under conventional tension–compression fatigue tests. Mayer [11] studies concerned the differences between the fatigue limits with constant amplitude and two-step variable amplitude, finding that two-step variable amplitude endurance and fatigue crack growth experiments could show detrimental or beneficial influences of low-load cycles on fatigue damage, and the maximum load amplitude of the variable amplitude repeat sequence determined which effect prevailed. Regarding the effect of pre-strain on the tensile and fatigue strength of low-carbon steel, several studies have been reported, but always regarding low-carbon steels with high strength due to their particular production processes or chemical compositions [12,13,14], or observing the effect of heat treatments [15]. Regarding the effect of notching, the main studies [16] analyse the stress concentration factor when the stress ratio R = −1, but not many studies consider other values of R. Another research study conducted by Liao et al. [17] highlights the possible factors which influence the fatigue notch factor $K_f$, pointing out that different stress ratios also produce different values of $K_f$. Zhang et al. [18,19] investigated the effect of multiple corrosion pits on high-strength steel wires and the influence of their geometric parameters on the stress concentration factor of those materials, also developing innovative models to predict the fatigue life of steel bridge decks’ welds based on machine learning approaches [20].

However, in the literature, limited works have been reported pertaining to the influence of pre-strain or notching on the behaviour of low-carbon, hot-rolled mild steel subjected to cyclic loads with $R\ne -1$, especially if the loads bring the material over the yield strength.

With this in mind, this research article analyses the influence of pre-strain and notching on the fatigue life of low-carbon steel (DD11) through several experimental tests, which can give useful additional information about the fatigue material properties in real industrial applications, for example, agricultural or industrial wheels [21], where DD11 steel is widely use for the rim manufacturing [22]. In fact, after the production process, this component has areas either plastically deformed or with notches, like the valve hole, which are subjected to cyclic load during service. An example of this application can be seen in Figure 1.

In this study, after a general presentation of the material characteristics, the focus is put on the geometry of the specimens (Section 2). Then, the test machines used and the procedures followed for the tensile and fatigue testing are presented in detail (Section 3). In Section 4, a finite element (FE) model developed for a numerical simulation of the tests is introduced. It is used as a further verification of the experimental results from the lab testing, analysing the stress distribution in the gauge area of the specimens and around the notch. It is further highlighted in Section 5, where the overall results of tensile and fatigue testing are presented and discussed in detail, also in reference to the FEA results. A low sensitivity of the material to high pre-strain is found. Regarding the notch effect, outcomes also show a lower sensitivity of the material than theoretical calculations, due to a local plasticisation of the material around the notch itself. Reasons for that are discussed in detail and the overall results are then summarized in Section 6, where future developments of this research are also suggested.

2. Material and Specimens

In the current study, the specimen material was DD11 low-carbon steel (1.0332 to the UNI EN 10027-2:2015 standard [23]). According to the UNI EN 10111:2008 standard [24],

Table 1. Chemical composition of DD11 steel (wt%).

C	Mn	P	S	Si	Al	Cu	Ni	Cr	N
0.07	0.42	0.010	0.008	0.05	0.016	0.030	0.012	0.020	0.004

shows the chemical composition of the material, while the mechanical properties are shown and discussed in Section 5.1. The specimens were obtained with the longest dimension perpendicular to the rolling direction. This means a different behaviour compared to the longitudinal direction due to the manufacturing process of the material. The location of the material samples from the original strip was chosen in accordance with ISO 377:2017 [25].

All of the specimens used in this work were characterised by a flat geometry and featured a rectangular cross section; in fact, they were directly machined from a hot-rolled DD11 steel sheet, as can be seen in Figure 2.

Overall, two types of specimens were produced: smooth and notched; Figure 3 illustrates their geometries. This geometry was chosen to satisfy the following:

• International standards for both tensile tests and fatigue tests (respectively, ISO 6892-1 [27] and ISO 12107:2012 [28]);
• Specifications of the test machines;
• Base material strip, which had a nominal thickness of 7 mm (in Figure 2, it can be seen in section A-A).

Figure 3. Photos of the specimens adopted: here, it can be noticed that the specimens were obtained directly from the coil strip through milling. (a) Smooth specimen. (b) Notched specimen, with a zoom to highlight the presence of the hole.

Figure 3. Photos of the specimens adopted: here, it can be noticed that the specimens were obtained directly from the coil strip through milling. (a) Smooth specimen. (b) Notched specimen, with a zoom to highlight the presence of the hole.

3. Experimental Analysis

3.1. Test Machines for Tensile Tests and for Uniaxial Fatigue Tests

Regarding the test machines used, the servo-hydraulic universal testing machine Instron 8501 with a maximum load capacity of 100 kN was adopted for the tensile tests on the material, while for the fatigue tests, the Rumul Mikrotron resonant machine (Russenberger Prüfmaschinen AG, Neuhausen am Rheinfall, Switzerland) with a maximum load capacity of 20 kN was utilized, as shown in Figure 4.

3.2. Tensile Tests

Before analysing the fatigue behaviour of the material, uniaxial tensile tests must be conducted with the aim of a better comprehension of the behaviour of the material, as they allow for a physical interpretation of the stress values. Furthermore, the fatigue strength of the material could be estimated as a first approximation from the tensile strength [29] for an appropriate choice of load levels.

The tensile tests were performed on the tensile testing machine in displacement control, as the transverse speed was set to 2 mm/min, falling within the range specified by ISO 6892-1 [27], which states that the transverse speed must be between 1 mm/min and 5 mm/min. An extensometer was installed on the gauge area of the specimens (see Figure 5a) with the purpose of recording accurate deformation data for the linear–elastic region where the estimation of the Young modulus had to be made. However, it was decided to remove the extensometer after reaching a displacement of 0.15 mm; the choice of this parameter was guided by an earlier estimation of the elastic region.

3.3. Fatigue Tests

In this research, axial fatigue tests were performed at a frequency of ca. 80 Hz, with load ratio R = 0.1 (except for two series of tests where the load ratio was R = 0.01 and R = 0.2) and a sinusoidal load function. The reason for these particular load ratio values was driven by the need to avoid compression and by a limitation on maximum stress to maintain the same failure mode and stay away from the LCF field; moreover, considering one of the applications of this material shown previously, the load ratios chosen also matched the classic stress behaviour of agricultural wheels.

As the failure criterion, the complete fracture of the specimen was assumed. Nevertheless, if specimens reached 2,000,000 cycles, the tests were stopped and considered runouts. The reason for this specific number was its extensive use in literature (examples are [30,31,32]). In this study, different conditions were analysed:

• Fatigue behaviour of the material through different stress ratios, defining a Wohler’s curve.
• Fatigue limit of the material with an 11% pre-strain level, through the staircase method. A similar value of pre-strain was already investigated in the literature with a similar material [33]. To understand this percentage, Figure 5b shows this pre-deformation on the stress–strain graph of the material.
• The fatigue limit of the material, considering the notch effect, was determined using the staircase method.

To understand the staircase method, the example of the base material with R = 0.1 is reported in

Table 2. Staircase test procedure of the base-material C specimens.

${\mathit{\sigma }}_{\mathit{a}}$	Specimen ID
[MPa]	R1.1	R1.2	R1.3	R1.4	R1.5	R1.6	R1.7	R1.8	R1.9	R1.10	R1.11
145			X		X				X		X
142.5		O		O		X		O		O
140	O						O

and

Table 3. Evaluation of the fatigue limit of the notched PS specimens.

Stress [MPa]	Level	Events		Values
${\mathit{S}}_{\mathit{i}}$	$\mathit{i}$	X	O	${\mathit{f}}_{\mathit{i}}$	${\mathit{if}}_{\mathit{i}}$	${\mathit{i}}^{\mathbf{2}}{\mathit{f}}_{\mathbf{i}}$
145	2	4	0	4	8	16
142.5	1	1	4	1	1	1
140	0	0	2	0	0	0
Sum	-	5	6	$C=5$	$A=9$	$B=17$

. A staircase method is usually defined by a minimum of 15 specimens, but having described the whole curve for the base material, it was possible to adopt only 6 specimens according to the UNI ISO 3800 standard [34], which is based on the homoscedasticity hypothesis, i.e., that the standard deviations of both fatigue limit and finite life region are the same. Here, 11 tests are shown to validate the hypothesis previously made.

Table 2 shows the tests for the staircase method, their amplitude stress, and whether the test was a runout (marked with “O”) or whether the specimen reached failure (marked with “X”). In this method, each test is dependent on the previous, so if the previous specimen fails, the new test should be taken to a lower level of stress amplitude and vice versa; if a previous specimen is a runout, the new test should be taken to an upper level of stress amplitude. Table 3 elaborates the results: the level starts as zero, the lowest level of stress amplitude (in this scenario, 140 MPa), and the values to evaluate coefficients C, A and B are taken based on the least number of events, so here, the failure tests (“X”) are chosen as being less than the runouts. Coefficient C is the sum of the events chosen, coefficient A is the sum of each event multiplied by its level of stress amplitude, while coefficient B is the sum of each event multiplied by the square of its level of stress amplitude. These coefficients are needed to evaluate the fatigue limit ${\sigma }_{a50}$ and its standard deviation ($Std.Dev.$) through Equation (1) [34]:

$${\sigma }_{a50}={\sigma }_0+\Delta S\left({\displaystyle \frac{A}{C}}\pm 0.5\right);Std.Dev.=1.62·\Delta S\left({\displaystyle \frac{C·E-A^2}{C^2}}+0.029\right)$$

(1)

where ${\sigma }_0$ is the lowest level of stress amplitude of the staircase method, $\Delta S$ is the step level of the stress amplitude (in this case chosen equal to 2.5 MPa), and the sign before the value 0.5 in Equation (1) depends on the analysed event: a minus sign if the analysed events are tests reaching failure or plus sign if they are runouts.

Table 4. Legend of the letters used for specimen identification.

ID	#	Purpose
T	3	Tensile tests on the base material
R1	26	Fatigue tests on the base material at $R=0.1$
R2	15	Fatigue tests on the base material at $R=0.2$
R01	15	Fatigue tests on the base material at $R=0.01$
PS	15	Fatigue tests on axially deformed specimens (${ϵ}_p=11\%$)
N	15	Fatigue tests on notched specimens

shows the alphanumeric coding used to label the specimens implemented in this research work. For the tensile tests, specimens labelled with the letter T were implemented. To define the fatigue behaviour of the material, R1, R2, R01 specimens were used, while for investigating the fatigue limit of the material with pre-deformation, specimens labelled with PS were adopted. To investigate the notch effect, specimens with the letter N were used.

It must be highlighted that positive values of load ratios were chosen to avoid a possible buckling effect, which can happen with a negative load ratio due to the contribution of the compression. Another reason of such load ratios lay in one of its mechanical applications: considering the agricultural wheel, there is a mean stress due to the pressure of the tyre, and an alternate stress due to the rotation of the wheel itself. The reason for the different load ratios for R01 specimens is that studying the fatigue with different load ratios makes it possible to define the entire Wohler’s curve of the material, i.e., R = −1. This result is achievable through the so-called Walker’s method [35]. We also implemented a third different load ratio for R2 specimens to validate the results of the numerical simulations.

Furthermore, the fractography analysis is shown, with the goal of fully describing the behaviour of the material under fatigue loads.

4. Numerical Simulations

To generalise the experimental results, numerical analysis based on the finite element method (FEM) was carried out. The simulations were performed through the software Solidworks^® (2024), and for the smooth specimen, a solid mesh of 86,315 quadratic tetrahedral elements (TET10) and 127,813 nodes was implemented, as shown in Figure 6a.

For the notched specimen, a solid mesh consisting of 163,583 quadratic tetrahedral elements (TET10) and 242,584 nodes was implemented, as shown in Figure 6b. Both final meshes resulted from an h-refinement process with a destination precision of 98%. This strategy implies progressively decreasing the characteristic size of the finite elements (denoted as h) in areas necessitating higher accuracy, typically guided by error estimators supplied by the solver. The target accuracy of 98% relates to the convergence criterion established in SolidWorks^®, wherein the solver evaluates the outcomes of successive mesh refinements and assesses the percentage of convergence towards an asymptotic solution. A local mesh control was also applied around the hole, featuring an average element size of 0.1 mm, with an element size/diameter hole ratio equal to 1/20. This guaranteed that the stress gradients surrounding the geometric discontinuity were accurately captured with adequate resolution, enhancing the overall h-refinement strategy.

About the boundary conditions, to represent the actual behaviour of the specimen stressed on the experimental machine, Figure 6c shows where the fixed constraints (defined by green arrows) and the forces/displacements controlled (indicated by purple arrows) were applied. Due to this configuration, the mesh’s refinement at the edge of the fixed constraint was justified; however, that area of the specimen was not the focus of this research study, as the gauge area was the one analysed.

5. Results and Discussion

5.1. Results of Tensile Tests

Regarding the tensile tests, as previously explained, three specimens were tested. From the experimental results, the stress–strain curve of the material was derived and then implemented in the numerical simulations to analyse the notch behaviour.

5.1.1. Experimental Results of Tensile Tests

Table 5. Summary of the tensile tests results. The supplier values are in accordance with the UNI EN 10025-2 standard [36].

Specimen ID	E [MPa]	${\mathit{\sigma }}_{\mathit{y}}$ [MPa]	${\mathit{\sigma }}_{\mathit{r}}$ [MPa]
Supplier’s values	207,000	278	391
T1	228,580	290.06	400.38
T2	212,578	294.28	396.71
T3	208,229	278.32	389.28
Mean	216,462	287.56	395.46
Std. Dev.	10,717	8.27	5.66

enhances the data obtained from the tensile specimens and compares them with the supplier’s data, while Figure 7 shows the tensile tests conducted. It is noticeable that the average of the values is higher than the declared values for every mechanical property analysed.

5.1.2. Numerical Results of Tensile Tests

For a better comprehension of the experimental tests, some numerical simulations were performed. One goal of this study was to extend the knowledge about the strength concentration factor; therefore, Figure 8 shows the numerical result for a linear static simulation of the notched specimen.

The stress applied was through a tensional force equal to 3000 N in the elastic field of the material. On the smooth specimen, a stress value of 107 MPa was analytically evaluated (having a gauge area equal to 28 ${\mathrm{mm}}^2$), while for the notched specimen, the value was 253 MPa, as it can be seen from the FEM results; the stress concentration factor $K_t$ is defined in Equation (2) [38]:

$$K_t={\displaystyle \frac{{\sigma }_{max,n}}{{\sigma }_n}}$$

(2)

where ${\sigma }_{max,n}$ is the value of the stress at the edge of the hole, and ${\sigma }_n$ is the nominal stress calculated at a location without a hole. This value was equal to 2.36, which is in line with the value found in the literature [38,39] for the example of a finite-width plate with a circular hole.

5.2. Results of Fatigue Tests

5.2.1. Results for the Fatigue Behaviour of the Material

Regarding the fatigue tests, the first study concerned the behaviour of the material; Figure 9 hence displays HCF and staircase data along with the bilinear model that was inferred from it. The two thinner lines represent the S–N curves for 10% and 90% probabilities of failure with a confidence level of 95% [28].

The stress–life curve inferred above was based on fatigue tests carried out at a stress ratio of 0.1, which means that every single test was characterised by a positive (hence, tensile) mean stress in addition to the pulsating component typical of fatigue. It is well known that a tensile mean stress has a detrimental effect on fatigue life, and a number of different methods exist to obtain the equivalent fully reversed fatigue strength when the stress ratio differs from −1 (i.e., the stress ratio of a traditional Wöhler curve).

In this work, as already mentioned in Section 3, the Walker method was adopted since it offers the advantage of a better fit of the data thanks to the experimental parameter $\gamma$, which can be interpreted as an indicator of the material’s sensitivity to mean stresses in fatigue. The method is based on a bilinear regression of high-cycle fatigue data acquired at different stress ratios. Therefore, HCF tests at the stress ratios of 0.01 and 0.2 were carried out. As previously defined, the choice of these particular values was driven by the need to avoid compression and by a limitation on maximum stress to maintain the same failure mode and stay away from the LCF field, respectively.

A comprehensive stress–life plot of all the tests performed is available in Figure 10. It can be noted that the linear regression curves at 0.1 and 0.01 stress ratios feature similar slopes, which are quite different from the one at a ratio R = 0.2. This variation can be explained by the high maximum stresses and the main presence of the mean stress.

The next step was to apply the Walker equation: this equation correlates the equivalent fully reversed stress amplitude ${\sigma }_{ar}$ with the actual parameters of a cycle. It can be expressed as Equation (3):

$${\sigma }_{ar}={\sigma }_{\max }^{1-\gamma }{\sigma }_a^{\gamma }={\sigma }_a{\left({\displaystyle \frac{2}{1-R}}\right)}^{1-\gamma }$$

(3)

and can be used to estimate the equivalent Wöhler curve based on fatigue tests carried out at load ratios other than $-1$. For fully reversed loading, the material is assumed to still follow the Basquin equation in HCF, as defined in Equation (4) [40]:

$${\sigma }_{ar}=A_p{N}_f^b$$

(4)

Overall, the unknown parameters A, b and $\gamma$ were found. The method involved a multiple linear regression, modelled as Equation (5):

$$y=m_1{x}_1+m_2{x}_2+dwherey=log{N}_f$$

(5)

Here, y is the dependent variable, while Equation (6) defines the independent variables:

$$x_1=log{\sigma }_{\max };x_2=log\left({\displaystyle \frac{1-R}{2}}\right)$$

(6)

The fitting parameters are described in Equation (7). The regression was performed in Matlab^®.

$$m_1={\displaystyle \frac{1}{b}}{m}_2={\displaystyle \frac{\gamma }{b}}d=-{\displaystyle \frac{1}{b}}log{A}_p$$

(7)

Figure 11 displays the experimental points in the three-dimensional coordinate system $(x_1,x_2,y)$ along with the planar surface that interpolates them. By rearranging the surface’s parameters according to the above formulas, the values in Equation (8) were obtained:

$$\gamma =0.5588;{\sigma }_{ar}=376N_f^{-0.0432}$$

(8)

Using Equation (8), it is possible to estimate the fatigue behaviour of the material with a stress ratio equal to −1 and thus the equivalent fully reversed fatigue limit, by evaluating that Basquin equation at the ultimate number of cycles of $2·{10}^6$; the value is shown in Equation (9).

$${\sigma }_{lim,R=-1}=201\mathrm{MPa}$$

(9)

To check these results, some numerical simulations were performed. Since the curve R = 0.2 was very different from the others, the fatigue analysis was based on using that stress ratio value. Figure 12 shows the fatigue behaviour for a finite life, with a linear regression defined by the Basquin equation and a confidence interval. It must be highlighted that this equation is valid only in HCF; for LCF values, the results can differ.

A fatigue analysis was executed with ${\sigma }_a=140$ MPa and another with ${\sigma }_a=145$ MPa. The mean stress correction theory adopted in this research was Walker’s method, as previously described. However, for the numerical simulations, the Wöhler curve was partially built using the Basquin equation while remaining in the HCF area of the stress–life cycle plot, and the authors also tested whether other mean stress theories could be adopted.

Usually the literature indicates that experimental tests for ductile steel fall between Goodman’s and Gerber’s theory [41]. Here, the analysis showed that the Walker method was preferred for a number of cycles lower than ${10}^6$ cycles, while the Goodman theory was a good choice for this material between ${10}^6$ and $2·{10}^6$ cycles (due to the experimental tests, the analysis stopped at that last value), indicating the minimum number of cycles required to reach failure inside the failure probability band of Figure 12. Specifically, in Figure 13, it is possible to see the results of this analysis, adopting the Goodman theory for the analysis performed at ${\sigma }_a=140$ MPa and Walker’s method for the analysis performed at ${\sigma }_a=145$ MPa. In particular, it is possible to see the areas where the specimen reaches failure at the edge of the gauge length, as confirmed by the experimental results.

5.2.2. Results of Fatigue Limit for Pre-Stretched or Notched Specimens

Regarding the study of pre-strained specimens, a pre-strain level of tensile plastic deformation equal to 11% was applied to a group of smooth specimens so as to investigate the effect of pre-strain on the fatigue limit (as previously cited, a similar value of pre-strain was already investigated in the literature with a similar material [33]). In terms of stress, observing Figure 5b, it is possible to define a parameter $\chi$ in Equation (10):

$$\chi ={\displaystyle \frac{{\sigma }_{ps,11\%}}{{\sigma }_y}}={\displaystyle \frac{390\mathrm{MPa}}{287\mathrm{MPa}}}=1.36$$

(10)

where ${\sigma }_{ps,11\%}$ is the stress to reach the level of plastic deformation equal to 11%, and ${\sigma }_y$ is the yield stress of the material. The specimen identification for this group of specimens is referred to as PS. This level of pre-strain being located near the plateau of the $\sigma$–$\epsilon$ curve, the stretching process was conducted using displacement control.

For the notched tests, the fatigue limit was analysed.

Table 6. Summary of the fatigue limits’ results.

Specimen ID	${\mathit{\sigma }}_{\mathit{a},\mathit{lim}}$ [MPa]	Std. Dev. [MPa]	$\mathbf{\Delta }\mathit{S}$ [MPa]
R1	143.25	0.77	2.5
PS	143.93	4.21	5
N	121.25	2.43	2.5

shows a comparison between all the fatigue limits analysed and the fatigue limit of the material. All the tests were conducted with a stress ratio of $R=0.1$, but for the staircase method, different stress steps ($\Delta S$) were adopted. The standards [28] highlight that if there is no information available about the standard deviation, a step of about 5% of the estimated mean fatigue strength may be used as the stress step. This means that the stress step for this material should be between 5 and 10 MPa. Experimentally, it was observed that having stress steps with these values was appropriate for the pre-stretched specimens with a high pre-deformation, but for the other tests, a value equal to 2.5 MPa could achieve a lower uncertainty of the results.

The values indicate that the pre-deformation of the specimen had an almost negligible effect on the fatigue limit, although there may be a slight increase in it. To strengthen this result, a non-linear numerical simulation for the pre-strain process was performed: Figure 14 shows the findings from the numerical simulation performed.

The numerical results clearly show that the specimen presented some area on the gauge length with residual compression stress; this could explain the slight increment in the fatigue limit for the pre-strained specimens. Moreover, there was a residual tensile stress at the centre of the transition area, but due to the larger section, it did not affect the failure location of the specimen.

Comparing the fatigue limit of the material with the fatigue limit of the notched specimens, the fatigue strength concentration factor was analysed. Through Equation (11), it is possible to evaluate the fatigue notch factor $K_f$ [42]:

$$K_f={\displaystyle \frac{{\sigma }_{a,lim,smooth}}{{\sigma }_{a,lim,notched}}}$$

(11)

where ${\sigma }_{a,lim,smooth}$ is the unnotched fatigue limit of the material, and ${\sigma }_{a,lim,notched}$ is the fatigue limit of the notched material. We obtained $K_f$ = 1.18.

To understand this result, the value of the fatigue notch sensitivity factor q should be evaluated through a material parameter defined as $\alpha$ in Equation (12):

$$log\alpha =-1.079·{10}^{-9}{\sigma }_r^3+2.740·{10}^{-6}{\sigma }_r^2-3.740·{10}^{-3}{\sigma }_r+0.6404$$

(12)

where $R_m$ is the tensile strength of the material; this results in $\alpha$ equal to 0.339. With this value, it was then possible to evaluate the fatigue notch sensitivity factor q through Equation (13):

$$q={\displaystyle \frac{1}{1+\sqrt{\frac{\alpha }{\rho }}}}$$

(13)

where parameter $\alpha$ was computed previously, while $\rho$ is the notch radius, that is, 1 mm in the geometry considered. This means a notch sensitivity factor q equal to 0.632. Considering the value of $K_t$, as described in Section 5.1, is equal to 2.36, an expected value of the elastic fatigue notch factor $K_{f,el}$ (the fatigue notch factor of the studied scenario if the stress cycle is under the yield strength of the material) can be obtained through Equation (14):

$$K_{f,el}=1+q·(K_t-1)=1.85$$

(14)

This result is much higher than the $K_f$ value found experimentally. The reasons behind the decrease in the fatigue notch factor may be the stress conditions of the tests performed and the material used. As already seen in the literature [43,44,45], the decrease in $K_f$ can be due to different factors:

• The stress ratio adopted ($R=0.1$) and the loads applied bring the material in the elasto-plastic field;
• The material itself is very ductile, and combined with the geometry implemented, notch blunting phenomena can easily happen;
• Working in the high-cycle fatigue and very high cycle fatigue, repeated plastic strain accumulation can occur, avoiding the propagation of cracks.

In fact, considering that ${\sigma }_{a,lim}$ was evaluated to be 121.25 MPa for the notched specimen and the stress ratio R was equal to 0.1, through Equation (15), it was possible to evaluate the maximum stress during the fatigue limit cycle ${\sigma }_{max,lim}$:

$${\sigma }_{max,lim}={\displaystyle \frac{2·{\sigma }_{a,lim}}{1-R}}=269.44\mathrm{MPa}$$

(15)

where R is the stress ratio. Thus, the maximum local stress could be evaluated through Equation (16) thanks to the stress concentration factor $K_t$:

$${\sigma }_{max,loc}=K_t·{\sigma }_{max,lim}=633.19\mathrm{MPa}>{\sigma }_y=287.56\mathrm{MPa}$$

(16)

This means a yielding around the notch area, the value of maximum local stress ${\sigma }_{max,loc}$ being greater than the yielding strength of the material ${\sigma }_y$. To be specific, since the so-called condition of reversed yielding is not satisfied, the actual condition is called initial yielding [46]. In this condition, the fatigue notch factor for mean stress $k_{fm}$ can be defined by Equation (17):

$$k_{fm}={\displaystyle \frac{{\sigma }_y-K_t·{\sigma }_{a,lim}}{|{\sigma }_{m,lim}|}}=0.018$$

(17)

where ${\sigma }_{a,lim}$ and ${\sigma }_{m,lim}$ are the alternate and the mean stress of the notch fatigue limit cycle, respectively. This result describes an almost reversed yielding condition (where the fatigue notch factor for mean stress $k_{fm}$ is equal to zero), yielding the following stresses on the notch area:

$${\sigma }_{a,loc}=K_t·{\sigma }_{a,lim}=284.94\mathrm{MPa};{\sigma }_{m,loc}=k_{fm}·{\sigma }_{m,lim}=2.66\mathrm{MPa}$$

(18)

where the ${\sigma }_{a,loc}$ and ${\sigma }_{m,loc}$ are the alternate and the mean stress of the notch fatigue limit cycle localised in the plasticised area, respectively.

5.2.3. Fractography of Fatigue Tests

Figure 15 shows the fracture’s surface of the specimens observed under each testing condition. In general, excluding the notched condition, two characteristic regions can be identified: the crack initiation and propagation area, which appears as a quarter-circle and lies on a single plane, and the final fracture region, inclined at approximately 45°, as typically observed in ductile materials. A schematic representation of the fracture surface highlighting these two regions is provided. The relative extension of these regions depends on the testing conditions. In particular, with an increasing stress ratio R, the extension of the propagation region decreases. This difference becomes significant starting from $R=0.1$. The case of $R=0.01$ shows only a slightly larger propagation zone compared to $R=0.1$, while the difference between $R=0.1$ and $R=0.2$ is much more pronounced. The pre-strain condition shows only a slightly larger propagation region, which does not define a significant change in the material behaviour, as confirmed by the experimental results. Finally, in the notched condition, the presence of the hole masks any visible propagation effect on the fracture surface. The crack initiates directly from the hole surface, and the final fracture region resembles that of the unnotched specimens.

6. Conclusions

This research study analysed the effect of pre-strain and notching on a flat steel bar made of DD11 low-carbon steel. After characterising the mechanical properties of the material along the transversal direction (with respect to the rolling direction) and defining the fatigue behaviour of this material, two different conditions were analysed. The first one was the fatigue limit of the material with pre-strained specimens, where the residual level of pre-strain was equal to 11%, while the second condition was the fatigue limit of the material with notched specimens, with the aim of analysing the notch fatigue factor of the material with a stress ratio where the loads were greater than the yield strength. To understand the results obtained, numerical simulations were carried out, and documented photographs of the fracture area were taken. The following results were obtained:

• The concentration factor obtained from the numerical results was in line with the values defined in the scientific literature.
• A high pre-strain level had a negligible influence on the material’s fatigue limit, indicating that even significant deformations (up to 11%) did not impact the material’s fatigue performance.
• The notch factor found from the tests on notched samples indicated that the material was not very sensitive to fatigue, which is understandable because the material is very flexible and the stress ratio used in these tests increased the average stress. These testing conditions caused the material to behave like it was in an almost reversed yielding condition.

As future developments, fatigue limits with other levels of pre-strain can be studied to understand the fatigue behaviour of the material in different pre-strain conditions. The idea would be to analyse higher levels of pre-strain, even higher than the ultimate tensile strength point. Furthermore, the combination of pre-strain and notching can be analysed, with the aim of a better comprehension of the notching effect for this material. Other aspects that the authors would like to observe are methods for reducing the notching effect on the fatigue behaviour of the material.