Fracture Surface Behavior of 34CrNiMo6 High-Strength Steel Bars with Blind Holes under Bending-Torsion Fatigue

The present study evaluates the fracture surface response of fatigued 34CrNiMo6 steel bars with transverse blind holes subjected to bending with torsion loading. The analysis of the geometric product specification was performed by means of height parameters Sx, functional volume parameters Vx, and fractal dimension Df. Surface topography measurements were carried out using an optical profilometer with focus variation technology. The experimental results show that the doubling the bending to torsion moment ratio B/T from B/T = 1 to B/T = 2, maintaining the same normal stress amplitude, greatly reduces both Sa, Vv as well as the fractal dimension Df of the analyzed specimen fractures by 32.1%, 29.8%, and 16.0%, respectively. However, as expected, a two-fold increase in the B/T ratio, maintaining the same normal stress amplitude, resulted in a larger number of cycles to fatigue crack initiation, Ni, which can be explained by the lower shear stress level. These experiments prove that parameters Sx, Vx, Df are smaller for larger Ni values, which is an important finding. In addition, it was found a high consistency of surface topography measurements for the two sides of the broken specimens. The proposed methodology is both reliable and applicable for other engineering applications involving different geometries and loading conditions.


Introduction
Efficient design and safe service operation of mechanical components subjected to cyclic loading is a constant goal of modern industry. Therefore, engineering materials with different geometries and shapes are increasingly studied under complex fatigue conditions and synergistic effects [1][2][3][4]. Viespoli et al. [4] studied the failure mechanisms of severe geometric discontinuities in terms of creep and fatigue interaction. They showed that the plastic behavior of the cable resulted in a minimal notch sensitivity for cracks starting from different positions associated with lower geometrical stress concentration factors. Martínez et al. [5] also have attempted to estimate the fatigue life in wires with blind micro holes. Their assessment of the geometric discontinuities in AA 6201-T81 wires showed good agreement with calculations based on Theory of Critical Distances. Loading generation also needs to be as close as possible to the real service conditions, which requires the development of specific fatigue machines either for uniaxial or multiaxial loading [6][7][8][9]. Multiaxial loading, and especially bending-torsion, is an important and interesting case, albeit quite complicated. Giannakis and Savaidis [10] took up stressed automotive antiroll bars and implemented an innovative procedure to calculate the fatigue life. Parallel to the tests, new calculation models have been developed for the determination of fatigue life, especially for loading combinations, such as bending-torsion [11][12][13]. Furthermore, a huge effort has been put on the development of advanced numerical models, particularly those based on the finite element method, to address multiaxial fatigue problems [14][15][16][17][18]. For practical applications such as cables and wire ropes [19], implantable medical leads [20], or containership structures [21]. All these methods are, in general, supplemented by material tests at various scales [22][23][24][25][26]. The morphology of the material can be examined in the depth of the material [27][28][29][30][31] and on its surface [32][33][34][35]. In some works, scientists have tried to combine all fractographic methods and link them to fracture mechanisms or fatigue life, especially using surface roughness parameters evaluated from the fracture surfaces [36][37][38]. Examples of quantitative fractography cases are presented in the works by Goldsmith et al. [39] and by Kobayashi et al. [40]. However, as far as the authors know, there are no studies in the literature connecting multiaxial loadings and surface fracture parameters in round bars with blind holes subjected to multiaxial loading.
Thus, this paper attempts to link the fracture surface topography parameters, including fractal dimension Df, with the multiaxial fatigue loading, particularly bending-torsion loading, in circular cross-section geometries containing transversal blind holes. Surface analysis was taken out on the whole fracture surface area, without partition into the three fatigue stages, i.e., initiation, propagation, and final rupture. More specifically, the paper aims to investigate the effect of the bending moment to torsion moment ratio on fatigue crack initiation of round bars with transversal blind holes; and on fracture surface topography. It is also focused on the identification and optimization of surface topography parameters in the context of fatigue crack mechanisms, where fracture topography is evaluated via the entire fracture area method [41][42][43].
Following the Introduction, the paper is organized as follows: Section 2 describes the materials and methods used in this research. Section 3 gathers information on the experimental fatigue behavior and the fracture surface fractography results. Section 4 presents the main outcomes about the fracture surface analysis conducted using different fractographic parameters for both sides of the specimens. The paper ends with a summary of the most relevant findings. Finally, for the sake of clarity, two appendixes were added: Appendix A shows the 3D views for the investigated fracture surfaces, both before and after extracting the region of interest (ROI), while Appendix B plots four variants of calculating the fractal dimension with its parameters.

Material and Fatigue Test Procedure
The material tested in this research was the DIN 34CrNiMo6 high strength steel, a martensitic steel, whose elemental composition and mechanical properties are summarized in Tables 1 and 2, respectively. This steel exhibits a martensitic matrix containing small amounts of ferrite and bainite. The microstructure is the result of a rapid quenching that transforms most of the austenite into martensite. The strengthening mechanisms are associated with the precipitation of a fine dispersion of alloy carbides during tempering. The bending-torsion fatigue testing program, originally performed in a previous study, has been conducted in a 100 kN DARTEC servo-hydraulic machine (Dartec Ltd., Bournemouth, UK) connected to a custom-made gripping system [44]. The specimens, whose geometry is presented in Figure 1, were prepared in a high-precision computer numerical control turning center from extruded 20 mm-diameter round bars and were tested under in-phase constant-amplitude for pulsating loading conditions (R = 0). This circular cross-section geometry encompasses a lateral U-shaped notch along with a 1.25 mmdiameter central hole whose depth (h) varies between 0.3 and 1.4 mm (see Table 3).  The tests were performed in air, at room temperature, with sinusoidal waves, and cyclic frequencies in the range 3-6 Hz, using a conventional servo-hydraulic machine connected to a custom-made gripping system. More details about the experimental apparatus can be seen in the paper by Branco et al. [45] Table 3 precises, inter alia, the nominal normal stress amplitude (σ a ) and the nominal normal mean stress (σ m ) applied in each individual specimen during the tests. The bending moment to torsion moment ratios, B/T, also presented in Table 3, were equal to 1 and 2. The number of cycles to fatigue crack initiation for each case was calculated using the El-Haddad parameter (a 0 ) which can be defined by the following equation: where ∆K th is the range of the threshold value of the stress intensity factor and ∆σ 0 is the fatigue limit stress range of the unnotched specimen. The two constants are evaluated under the same stress ratio (R = 0, in this case) of the multiaxially loaded geometry. The values obtained in this research, originally calculated in a previous study [44], are listed in Table 3.

Fracture Surface Measurement
The fracture surface parameters were determined using a profilometer (Alicona Imaging GmbH, Graz, Austria) using the Focus Variation Method. This non-contact measuring system uses a white light source to project light beams onto the specimen's surface. More information about the methodology used to carry out the surface texture evaluations can be found in the paper by Macek et al. [46]. Reflected light rays appear from the measured surface and are processed via a precise sensor. In this study, the total area of the fracture surface was investigated using an objective magnification of 10×. The main measurement parameters are summarized in Table 4. To perform the scanning of the total area, the imagefield function was used. The fracture surface was divided into 19 rows and 13 columns, and then the individual images were stitched together to map the entire fracture region. Surface fractography studies were carried out on the entire fracture area using height parameters Sx and functional parameters (volume) Vx defined according to ISO 25178 [47], as well as the fractal dimension Df. This standard, whose main title is "Geometrical product specifications (GPS)-Surface texture: Areal", introduces the terminology and the main definitions associated with surface texture, and describes the main parameter used in the evaluation of surface texture. Regarding the Sx parameters, as defined in Table 5, Sq and Sa are the root-mean square height and the arithmetical mean height of the surface, respectively; Sz is the maximum height of the surface, that is, the height between the highest peak and the deepest valley; the maximum peak height, Sp, is calculated as the difference for height between the highest peak and the deepest valley, Sz, and the maximum pit height, Sv. Skewness Ssk is a measure of the symmetry of the height distribution and can thus be used to point at superiority of peaks on the surface for Ssk > 0 whether valley for Ssk < 0. In pursuance of Krolczyk et al. [48], kurtosis Sku indicates appearance on the surface of excessively high peaks or deep valleys for Sku > 3, or their absence on the surface for Sku < 3. The functional parameters included: Vm, Vv, Vmp, Vmc, Vvc, and Vvv [47,49]. Table 5 defines the selected parameters according to the ISO 25178 standard.
The fractal dimension Df was calculated using the enclosing boxes method (EBM) from the extracted final fracture surface areas (see Figure 2) with four variants of calculations, i.e., EBM and EBM in real units for two resolutions: coarse and fine. The EBM divides the profile into smaller sections with a width ε and calculates the field Aε of all fields covering the entire profile [50,51]. This is an iterative procedure in which the width of the field is changed to plot, ln(Aε)/ln(ε). The EBM in real units considers real Z-spacing values to calculate the enclosed area. The resolution of the graph determines the number of iterations and, therefore, the calculation time. For the fine resolution, 59 points in the plot were considered, while for the course resolution, the analysis used 16 points.  Where A is the definition area; z is the surface height in position x, y; x, y are the lengths in perpendicular directions.

Results
Tables 6-8 summarize the main variables of the fracture surface measurements, namely Sx and Vx, and fractal dimension Df, respectively, for the different specimen geometries subjected to in-phase bending-torsion loading. For the B/T = 2 ratio, both sides of the same specimen (BT1-3a and BT1-3b) were evaluated to study the consistency of surface topography measurements based on the two fracture surfaces.   The fracture surfaces of 5 selected specimens, previously subjected to fatigue loading, were measured. All fractographic parameters were calculated on the whole fracture surface. The entire surface was reduced to eliminate the final break, discontinuities and "nonsampling" areas. Original pseudo-color views of the fracture surfaces, on the left-hand side, and photo simulations of extracted fractures areas, on the right-hand side, are presented in Figure 2. All analyzed samples in their original state as well as the corresponding extracted areas are presented in Appendix A, in Figures A1 and A2, respectively.   Figure 3 presents a summary of results of the fracture surface measurements, as a scatter plot, for the different tested specimens. It is clearly seen that all values of both Sx and Vx are higher for B/T = 1 (σ/τ = 2) than for B/T = 2 (σ/τ = 4). A similar tendency can be noticed for the fractal dimension, Df, determined by the EBM described above. However, its values show greater differentiation and dispersion at B/T = 1.

Cracking Mechanisms
The typical locations of fatigue crack initiation as well as the fatigue crack paths at the early stage of growth at the notch surface for the two B/T ratios are exhibited in Figure 4. In this geometry, as can be seen, there is a multi-crack initiation phenomenon. Two cracks appear at the hole surface in diametrically opposite coordinates. These locations are affected by the loading scenario. For higher B/T ratios, the angle formed by the line that connects the two initiation sites (black and white dots) is lower, which is explained by the fact that the crack is closer to Mode-I (see Figure 4a). In the absence of shear stresses, this angle should be 0 • . On the other hand, as the B/T ratio decreases, i.e., the shear stress level increases, this angle rises, which is associated with the higher degree of mixed-mode loading of these cases (see Figure 4b). Another important difference is concerned with the crack paths at the early stage of growth. As can be seen in Figure 4, these angles are similar for both sides of the hole, respectively equal to 13 • for B/T = 2 and 28 • for B/T = 1. It is also clear that this angle is affected by the loading scenario. The increase of the B/T ratio leads to smaller angles which can be justified by the different shear stress levels of the two loading cases. As referred to above, the crack front subjected to higher B/T ratios is closer to Mode-I, while the other is subjected to a higher degree of mixed-mode loading [53]. Figure 5 shows the typical aspect of fracture surfaces observed by scanning electron microscopy (Zeiss, Jena, Germany) of the multiaxial fatigue tests for B/T = 2 (Figure 5a-c) and B/T = 1 (Figure 5d-f). Overall, as can be expected, we can see the main micromechanisms associated with cyclic loading, namely traces of plastic deformation and secondary cracks. The multi-crack initiation phenomenon can be also inferred from the images. In general, when two cracks coalesce, it is visible a fatigue step is caused by the junction of two different planes of propagation. Representative examples of fatigue steps are exhibited in Figure 5a,d (see the red arrows). Particularly in the former case, coalescence of both cracks occurred at the middle point of the hole. In the other case, this junction of both propagation planes is slightly deviated to one side of the hole. The traces of plastic deformation caused by cyclic loading are clearly visible in Figure 5b,c which show a magnification of the surface region near the hole boundary in the area where both cracks coalesced. Figure 5e,f shows the fracture surface close to the initiation site (i.e., at the hole boundary). It is possible to see the radial convergence of the fatigue marks to the vertex (see the green arrow) caused by the cyclic loading. The region where the crack coalesced for the case BT2-1 (see Figure 5a-c) is analyzed, in more detail, in Figure 6 using profilometer scans. Zone I is obtained on the fatigue step which resulted from the junction of both cracks initiated at the hole surface in diametrically positions of the circular boundary (see Figure 4); while Zone II was taken in a region close to the fatigue step but in a region not affected by the junction of both cracks. As can be seen, the differences in the z-axis coordinates represented by the pseudo-color views are significant. In the former case, the maximum values are more than three times greater than the latter case.

Fractured Specimen Both Sides Comparison
An important issue in fractographic analysis of fracture surfaces caused by fatigue loading is the coherency degree of the measurements carried from both sides of the same specimen. In order to check whether the fracture surfaces obtained for the tested cases can be analyzed using any of the two sides of the same specimen, a comparison for a specific case is presented in Figure 7, which visually compares the fracture surfaces of the two sides of the BT1-3 specimen. The validation was made by mirroring the BT1-3a side with respect to the x-axis by inverting the x-coordinates and the heights of the Z-axis. Overall, both figures are similar, either in the original figures or in the pseudo-color representation, which suggests that the fractographic parameters are likely to be quite similar. The surfaces prepared in this way, together with the measurement results, could also be used for further comparative analyses.  Figure 8a,b present a comparison carried out using the Sx and Vx parameters for both sides of specimen, respectively, excluding Sku and Ssk, whose map is shown in Figure 8d. Both the Sx and Vx parameters showed very high compliance for both sides of the specimen, with coefficients of determination close to 1, which confirms the independence of these parameters relatively of the mounting of the specimen on the gripping system. The same is true for the fractal dimension Df (see Figure 8c), whose coefficient of determination R 2 is equal to 0.9633. The Ssk parameter provides information about the asymmetry of the surface. The Ssk parameter value indicates the predominance of peaks on the surface for Ssk > 0. The Sku parameter demonstrates absence of inordinately high peaks or deep valleys for Sku < 3. As can be seen in Figure 8d, there are no dependencies on the specimen side for the Ssk and Sku parameters.

Effect of B/T Ratio on the Fracture Surface Parameters
In order to better understand the effect of B/T ratio on fracture surface parameters, a detailed analysis based on the Sa and Vv parameters was performed. These parameters turned out to be the most fitted, which was also confirmed in the papers [41,43]. Moreover, for the Df parameters, the one determined using the EBM in real units with fine resolution was selected as the most accurate. Selected cases of the EBM estimations for the BT1-2 specimen are shown in Figure 9 for the sake of clarity. In addition, all fractal dimension Df plots and parameters for the four calculation conditions considered in this study are presented in Appendix B. To estimate the fractal dimension Df a line is fitted using the least-squares method. The absolute value of the slope of the fitted line is the estimation of the fractal dimension Df. The densification of the measurement points increases the slope of the curve angle, which is reflected in a slightly higher value of the fractal dimension Df.   Figure 10a) and for both functional parameters Vx or Vv (see Figure 10b). Regarding the fractal dimension Df, which is represented in Figure 10c, the values for the B/T = 2 show a similar trend than those presented in the Figure 10a,b. On the contrary, for the cases B/T = 1, the values are more distant and show an opposite trend, i.e., the values of the BT1-3a case is higher than the value of the BT1-2 case.     This methodology, connecting the applied nominal loading with the fracture mechanics based on topographic parameters, provides important clues to improve the materials performance as well as to mitigate the fatigue damage mechanisms [32,[41][42][43]46]. It can be used, for instance, in the field of forensic engineering to trace back to the origin of structural failures and correlate them with the applied loads, establishing the dependence between the loading scenario and the characteristic features of their surfaces.

Conclusions
The effect of the bending moment to torsion moment ratio (B/T) on fracture surface parameters in notched round bars made of high-strength steel was studied. Two different values of B/T (2 and 1) were considered in the multiaxial fatigue testing program. After the fatigue tests, a quantitative analysis of the entire fracture surface of broken specimens was performed using height parameters Sx, functional volume parameters Vx, and fractal dimension Df. The following conclusions can be drawn as follows: − The approach of analyzing the entire surface of the fracture is a valid concept when trying to estimate the causes of the destruction of high-strength steels subjected to bending-torsion loading; − Height (Sx), functional (Vx) and fractal dimension Df fracture surface texture parameters determined in the entire area of the fracture surface showed dependence on bending moment to torsion moment ratio; − The bending moment to torsion moment ratio has a strong influence on the crack initiation sites, crack paths in the early stage of growth, and the number of cycles to fatigue crack initiation lives. − The number of cycles to fatigue crack initiation, which is closed related to the loading scenario and stress level, significantly affects the height (Sx), functional (Vx) and fractal dimension Df fracture surface texture parameters; − The comparison of the surface topography measurements obtained for the two fracture surfaces of the same specimen demonstrated an independence of these parameters relatively to the specimen side selected in the analysis; − Resolution used in the EBM has a significant impact on the results of the calculated fractal dimension Df. The most accurate values in this study were those based determined in real units with fine resolution; − Regardless of the surface texture parameters used, it was found that the smallest average values of Sa, Vv and Df occurred for the higher bending moment to torsion moment ratio.