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Fatigue crack initiation behaviour of notched 34CrNiMo6 steel bars under proportional bending-torsion loading
International Journal of Fatigue 130 (2020) 105268
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Fatigue crack initiation behaviour of notched 34CrNiMo6 steel bars under proportional bending-torsion loading
T
R. Brancoa, , J.D. Costaa, F. Bertob, A. Kotousovc, F.V. Antunesa ⁎
a
CEMMPRE, Department of Mechanical Engineering, University of Coimbra, 3030-788 Coimbra, Portugal Department of Mechanical and Industrial Engineering, NTNU, 7491 Trondheim, Norway c School of Mechanical Engineering, The University of Adelaide, Adelaide, SA 5005, Australia b
ARTICLE INFO
ABSTRACT
Keywords: Multiaxial fatigue Bending-torsion Notch effect Interacting stress concentration Multi-crack initiation Strain energy density Theory of critical distances 34CrNiMo6 high-strength steel
Fatigue crack initiation behaviour of notched 34CrNiMo6 high-strength steel bars with transverse blind holes and lateral U-shaped notches subjected to proportional bending-torsion loading is studied. Crack initiation at the geometric discontinuities is examined in situ using a high-magnification digital camera. Crack initiation mechanisms and failures modes are investigated after the fatigue testing program via scanning electron microscopy. Fatigue crack initiation lifetime is predicted using an averaged Total Strain Energy Density approach in conjunction with the Theory of Critical Distances. Crack initiation sites, crack surface angles and surface crack paths at the geometric discontinuities are successfully determined a priori from the maximum value of the first principal stress, the first principal direction, and the first principal stress field at the notch surface, respectively. Overall, fatigue lifetime predictions are very well correlated with the experimental observations.
1. Introduction Modern automotive industry, driven by recent environmental policies, faces an urgent need for the production of lighter vehicles with reduced fuel consumption and lower pollutant emissions [1,2]. Despite the constant development of new materials, high-strength steels remain key materials in this challenging scenario, mainly due to their balanced features, in particular the low cost, excellent strength-to-weight ratio, and good corrosion resistance, among others [3,4]. The above-mentioned goals, namely the weight reduction, are often met via a smarter design, which is likely to introduce sudden and abrupt geometrical changes. The loading complexity acting on such components, in many cases characterised by a multiaxial nature, combined with the stress concentration phenomena occurring at the critical geometric discontinuities, leads to non-trivial and challenging problems [5–7]. Moreover, advanced engineering components can contain two or more geometric details, such as holes and notches, placed in enough proximity to each other that may cause stress concentration interactions, making the problem much more difficult [8]. Different fatigue lifetime prediction models have been proposed to deal with notched components via local pseudo-elastic stresses. Among the most successful approaches, we can mention those based on the Strain Energy Density [9–14] and those based on the Theory of Critical Distances [15–18]. The underlying idea behind the SED-based ⁎
approaches is that the damage caused by cyclic loading is a function of the mechanical energy input into the material [19,20]. Regarding the TCD-based approaches, the key concept is the determination of a critical stress, or strain, nearby the geometric detail, considering a characteristic material length [21]. However, despite the comprehensive research efforts, there are several unresolved issues. Among them, we can mention, for instance, the interaction of stress concentrations. As far as the selected material is concerned, 34CrNiMo6 highstrength steel is a typical medium-carbon low-alloy steel and one of the prime choices for several automotive components. Although this steel has been studied over the last decades, a little attention has been paid to the multiaxial fatigue behaviour of notched components. Literature review shows that the investigated geometric discontinuities have only encompassed notched shafts and notched round bars [22–26]. Nevertheless, the study of holes, either under in-phase or out-of-phase multiaxial loading conditions, has not been addressed. Interaction of stress concentrations caused by two or more geometric discontinuities, particularly holes and notches, have also not been addressed. The present paper aims at studying the fatigue behaviour of notched bars made of 34CrNiMo6 high-strength steel undergoing proportional bending-torsion loading. The geometric discontinuities studied here encompass a lateral U-shaped notched with a transverse blind hole, and a lateral U-shaped notch without transverse blind hole. The fatigue crack initiation lifetime is predicted from the Total Strain Energy
Corresponding author. E-mail address: [email protected] (R. Branco).
https://doi.org/10.1016/j.ijfatigue.2019.105268 Received 14 June 2019; Received in revised form 13 August 2019; Accepted 9 September 2019 Available online 10 September 2019 0142-1123/ © 2019 Elsevier Ltd. All rights reserved.
International Journal of Fatigue 130 (2020) 105268
R. Branco, et al.
The loading scenarios were replicated by two pairs of forces placed at one end of the specimen, while the other end was fixed. Torsion moments were created by the FT forces applied on a plane normal to the longitudinal axis of the specimen with opposite directions. Bending moments were created by the FB forces applied parallelly to the longitudinal axis of the specimen but with opposite directions. The relationship between the normal and shear stresses was established by adjusting the proportionality constant λ (see Fig. 2) which assumes the values λ = 1/2 and λ = 1 for B/T = 2 and B/T = 1, respectively.
Table 1 Nominal chemical composition (wt%) of DIN 34CrNiMo6 high-strength steel [27]. C
Si
Mn
Cr
Mo
Ni
0.34
≤0.40
0.65
1.50
0.22
1.50
Density approach on the basis of an effective stress determined using a linear-elastic framework in conjunction with the Theory of Critical Distances. In practical engineering problems, holes can be introduced during the fabrication, assembly or repair tasks, as well as due to mistakes made by manufacturing such as mislocated holes, or extra holes, which are prone to origin interaction of stress concentrations. Thus, the proposed methodology can be very helpful in an industrial context, since it provides a tool for rapid multiaxial life assessment of notched components, simple to implement, and founded on linearelastic analyses.
3. Results and discussion 3.1. Experimental fatigue crack initiation and propagation behaviour The trajectory described by a crack when subjected to specific loading histories is a relevant matter in the context of mechanical design. Its accurate prediction is a complex issue, particularly under multiaxial loading. Fig. 3 shows representative surface crack paths observed in the experimental tests for the different geometric discontinuities and bending-to-torsion ratios studied here. It is clear that the surface crack paths are remarkably affected by both the notch configuration and the multiaxial loading history. Overall, the increase of the B/T ratio, regardless of the geometric discontinuity, leads to straighter surface trajectories. This can be explained by the smaller shear stress levels, which reduce the degree of mixed-mode propagation. The surface crack paths, under in-phase bending-torsion scenarios, can be satisfactorily anticipated for both geometric discontinuities through the first principal stress (σ1) field at the notch surface. Fig. 4 plots the first principal stress field computed from the linear-elastic numerical models described in the previous section. In fact, as represented in Fig. 3, the surface crack trajectories predicted here (dashed lines) are quite close to those observed in the experiments. This outcome suggests that the proposed approach is sufficiently sensitive to determine, a priori, the surface crack paths for different geometric discontinuities subjected to in-phase bending-torsion loading. A reliable determination of the fatigue crack initiation sites is another critical point in the design of mechanical components. Fig. 3 exhibits some examples of the crack initiation sites, represented by the small circles, observed in the experiments for the different geometric discontinuities and multiaxial loading scenarios. The analysis of results shows that the fatigue crack initiation sites are also dependent on the geometric discontinuity and the bending-to-torsion ratio. In the blind hole geometries, there is always the nucleation of two cracks at the hole border in symmetrical positions with the respect to a vertical axis passing through the centre of the hole (Fig. 3(a) and 3(b)). Furthermore, the radial angle formed by a straight line joining the initiation sites and the vertical axis raises with increasing values of B/T. Regarding the lateral notch geometries, the effect of the B/T ratio is also clearly observed, i.e. for lower levels of shear stress (i.e. higher B/T ratios), the crack tends to nucleate in the centre of the notch (Fig. 3(c)), while at higher levels of shear stress (i.e. lower B/T ratios), the crack tends to shift progressively to the curved edge of the notch (Fig. 3(d)). The crack nucleation in these geometric discontinuities when subjected to in-phase bending-torsion loading can be predicted, in a precise manner, by the coordinate at the notch surface with maximum value of the first principal stress. The numerical predictions for the blind hole and lateral notch configurations can be observed in Fig. 4. A comparison of the experimental observations (small circles) and the numerical predictions (small squares) is exhibited in Fig. 3. As can be seen, there is an excellent correlation between both types of data, particularly in the case of the blind hole configuration, which is a major asset.
2. Experimental and numerical procedures The material utilised in this study is the 34CrNiMo6 high-strength steel, supplied in a quenched and tempered condition. Its chemical composition and its main mechanical properties are compiled, respectively, in Tables 1 and 2. Multiaxial fatigue testing program has been conducted under inphase constant-amplitude pulsating loading (R = 0) using 16 mm-diameter and 14 mm-diameter notched round bars. Table 2 summarises the maximum nominal stress amplitude (σa) and the maximum nominal mean stress (σm) applied in the tests. The geometric discontinuities, as schematised in Fig. 1, encompassed: (a) a lateral U-shaped notch with a central transverse blind hole (Fig. 1(b)); and (b) a lateral U-shaped notch without central transverse blind hole (Fig. 1(c)). For the sake of simplicity, the former and the latter geometries are hereafter termed blind hole and lateral notch, respectively. The loading scenarios included ratios of the bending moment to torsion moment (B/T) equal to 1 and 2. For each geometric configuration and loading case, three local stress levels were studied. Crack initiation and crack growth were monitored in-situ via a high-resolution digital camera. Images were periodically recorded with sampling rates between 1000 and 5000 cycles. Tests were performed in air, at room temperature, with sinusoidal waves, and cyclic frequencies in the range 3–6 Hz. Stress and strain fields at the notch region were computed from three-dimensional numerical models using COSMOS/M software. Meshes were developed in a parametric framework using 8-node isoparametric brick elements and assuming a homogeneous, linear-elastic and isotropic material. Fig. 2(a) shows, as an example, a finite element mesh developed for the blind hole configuration. At the geometric discontinuity, the mesh was carefully refined (Fig. 2(b)) while, at remote regions, a coarser pattern was adopted. The assembled model had 89,584 elements and 97,704 nodes. As far as the lateral notch configuration is concerned, the mesh details at geometric discontinuity are displayed in Fig. 2(c). It had 76,608 elements and 99,823 nodes. Table 2 Mechanical properties of the DIN 34CrNiMo6 high strength steel [23]. Mechanical property
Value
Yield strength, σYS (MPa) Tensile strength, σUTS (MPa) Young’s modulus, E (GPa) Poisson’s ratio, ν Cyclic hardening coefficient, K’ (MPa) Cyclic hardening exponent, n’ Fatigue limit stress range, Δσ0 (MPa) Stress intensity factor range threshold, ΔKth0 (MPa∙m0.5)
967 1035 209.8 0.296 1361.6 0.1041 353 7.12
3.2. Crack initiation angles Examples of the experimental crack angles at the notch surface in 2
International Journal of Fatigue 130 (2020) 105268
R. Branco, et al.
Fig. 1. Specimen geometries: (a) front view; (b) lateral U-shaped notch with transverse blind hole; (c) lateral U-shaped notch without transverse blind hole (dimensions in millimetres).
the early stage of growth for the blind hole and lateral notch configurations subjected to different bending-to-torsion ratios are displayed in Fig. 5. These angles are controlled by the geometric discontinuity and the multiaxial loading scenario. In general, it is possible to conclude that: (i) as the B/T ratio increases, the angle with respect to the horizontal axis decreases; and (ii) the angles of the blind holes are higher than those of the lateral notches when subjected to the same loading scenario. Although the prediction of crack initiation angles under multiaxial loading conditions remains to be a difficult problem [25,27][28–30], whose governing variables are not completely understood yet, the first principal direction accounted for at the initiation site has allowed to achieve very good results. As can be distinguished in Fig. 6, which plots the experimental observations against the numerical predictions, the maximum errors are lower than 5°, while the average errors for the blind hole and lateral notch configurations are equal to 2.6° and 1.4°, respectively.
such features, fracture surfaces were examined by optical microscopy (OM) and scanning electron microscopy (SEM). Fig. 7 shows typical topologies of fracture surfaces observed in the experiments by OM for the different geometric discontinuities and multiaxial loading scenarios. Overall, it can be noted that the fatigue crack growth process is quite complex and reveals a high degree of out-of-plane propagation, which increases as the B/T ratio decreases or, in other words, when the shear stress level climbs. Despite the obvious limitations of the top views to assess three-dimensional effects, the front views clearly show higher curvatures in the fracture surfaces of the blind hole configurations (Fig. 7(a) and (b)) than those of the lateral notch configurations (Fig. 7(c) and (d)) under the same loading history. Furthermore, irrespective of the geometric discontinuity, surface topologies are tendentially flatter for increasing values of the B/T ratio, i.e. in the cases whose propagation conditions are closer to mode-I loading [31,32]. The crack front shape changes during the fatigue crack growth stage can also help to understand the mixed-mode propagation behaviour. These changes can be evaluated from the successive macroscopic progression marks recorded on the fracture surfaces via the beach-marking technique (see the top views of Fig. 7). In fact, the crack front profiles of the blind hole configurations are remarkably different from the lateral notch configurations, particularly for smaller crack lengths. On the
3.3. Fracture surface topologies Fracture surface topologies generally contain useful information on the failure modes and fatigue mechanisms. Here, in order to evaluate
Fig. 2. Example of the assembled model (a) developed to compute the stress and strain fields at the notch region; (b) mesh detail for the blind hole; (c) mesh detail of the lateral notch. 3
International Journal of Fatigue 130 (2020) 105268
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Fig. 3. Macroscopic appearance of the fracture surfaces: (a) blind hole with B/T = 2; (b) blind hole with B/T = 1; (c) lateral notch with B/T = 2; (d) lateral notch with B/T = 1.
contrary, for longer crack lengths, since the hole effect progressively disappears, the crack front profiles become relatively similar. In this stage of propagation, as the crack grows, crack front profiles tend to be straighter. Furthermore, irrespective of the geometric discontinuity, crack front profiles are more curved, for a similar crack length, when the shear stress levels are higher, i.e. when the bending-to-torsion ratios decrease. These evidences can be explained by the nature of the fatigue crack growth process. In the blind hole geometry, as above mentioned, there is the nucleation of two corner cracks at the hole border with profiles close to quarter-elliptical shapes evolving, in a second stage, to a unique surface crack with both tips along the surface of the U-shaped notch surface. In a third stage, as the crack propagates, each tip reaches the circular portion of the specimen and, then, the crack grows as a through crack until the failure has been reached. Regarding the lateral notch geometry, crack initiation takes place somewhere at the U-shaped notch (more precisely, closer to centre under higher B/T ratios, and closer to the edge under lower B/T ratios). It should be also noted that this notch configuration, at least when subjected to in-phase bending-torsion
scenarios, is very prone to multi-crack initiation phenomena [23,25], as demonstrate the fatigue steps marked on the fracture surfaces (see the arrows of Fig. 7(c) and (d)). A plausible explanation for this behaviour is the existence of an extensive area at the notch surface with high stress levels (see Fig. 4(c) and (d)) in this case, unlike the case of the blind hole configuration, whose maximum stresses occur only at two highly confined regions (see Fig. 4(a)) and (b)). These several neighbouring cracks at the U-shaped notch gradually coalesce and, in a subsequent stage, turn into a single surface crack. As in the previous case, the crack tips advance in the direction of the circular part of the specimen, resulting in a through crack which propagates until the failure is reached. The fatigue damage mechanisms associated with the failure process of the 34CrNiMo6 high-strength steel under proportional bending-torsion loading are summarised in Fig. 8. In the blind hole configuration, as can be distinguished in Fig. 8(a), there is the nucleation of two cracks at the hole border, as a consequence of the highest stress concentration effects registered in such places (see Fig. 4(a) and (b)). The central fatigue step, identified in Fig. 8(a) by the arrow, results from the coalescence of both cracks. A close look at this region, magnified in
Fig. 4. First principal stress field at the geometric discontinuities computed through the numerical models: (a) blind hole with B/T = 2; (b) blind hole with B/T = 1; (c) lateral notch with B/T = 2; (d) lateral notch with B/T = 1. 4
International Journal of Fatigue 130 (2020) 105268
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Fig. 5. Experimental surface crack angles at the early stage of growth: (a) blind hole with B/T = 2; (b) blind hole with B/T = 1; (c) lateral notch with B/T = 2; (d) lateral notch with B/T = 1.
3.4. Fatigue crack initiation life
30 Hole
25 Measured angle (º)
Notch
20
B/T=2 B/T=1 B/T=2 B/T=1
Fatigue crack initiation lifetime was estimated from the experimental relations of the surface crack length (b) and the number of loading cycles (N). For the sake of simplicity, such relations are designated here as b-N curves. Fig. 9 displays typical b-N curves derived in the present study for the blind hole and the lateral notch configurations under different combined bending-torsion loading histories and different nominal stress levels. Note that due to the coalescence of defects, the crack length was accounted for via the sum of the individual flaws. This modus operandi provided a single b-N curve for each test, which allowed a simpler comparison of results. Briefly, the analysis of results suggests that: (i) fatigue crack initiation is faster in the blind hole geometry than that in the lateral notched geometry, as a consequence of the higher stress concentration levels introduced by the hole (see, for example, series 3 and 6); (ii) under the same nominal normal stress amplitude, regardless of the geometric discontinuity, the higher the B/T ratio, the slower the fatigue crack initiation (see, for examples, series 1 and 3, or series 5 and 6), which can be explained by the reduction of the shear stress level; (iii) fatigue crack initiation, fixing the bending-totorsion ratio and the geometric discontinuity, speeds up with increasing values of the nominal stress level (see, for example, series 2 and 3, or series 4 and 6). The hole depth plays a fundamental role in the fatigue crack initiation process of the blind hole configuration, as well as in the earlier stage of crack growth, but loses its importance in the other stages. When the hole depth is relatively small (series 2 and 3), respectively equal to 0.3 mm, crack initiation occurs much more rapidly than that in the cases without hole (series 4 and 6). However, in a second stage, the fatigue crack growth rates of the blind hole geometries slow down and seem to converge to the curves of the specimens without holes tested under identical loading histories (i.e. series 2 seems to converge to series 4; and series 3 seems to converge to series 6). It is also interesting to note that crack initiation of series 1 is faster than that of series 2,
15 10
Y
5 0
Z
0
5
10 15 20 Predicted angle (º)
25
30
Fig. 6. Experimental versus predicted crack angles at the early stage of growth.
Fig. 8(b), shows traces of plastic deformation, namely fatigue striations with radial orientation, which denotes a ductile failure mode. With regard to the lateral notch configuration, cracks initiate from surface irregularities (see the arrow of Fig. 8(c)) or from inclusions and pores located in the vicinity or at the notch surface (see the arrow of Fig. 8(d)), which act as local stress raisers, easing the crack nucleation. It is also possible to observe, in Fig. 8(c), at least three fatigue steps, as the result of the coalescence of neighbouring defects; a magnified example of a fatigue step is exhibited in Fig. 8(d). Additionally, there is a population of non-metallic particles, composed by globular shapes, disperse throughout the fracture surfaces (see Fig. 8(a) and (c)). 5
International Journal of Fatigue 130 (2020) 105268
R. Branco, et al.
B/T=2
B/T=1
B/T=2
B/T=1
(c)
(d)
3 mm
(a)
(b)
Fig. 7. Top views and front views of the fracture surfaces: (a) blind hole with B/T = 2; (b) blind hole with B/T = 1; (c) lateral notch with B/T = 2; (d) lateral notch with B/T = 1.
however the curves cross each other after some propagation causing an inversion of the fatigue crack growth rates. The higher nominal stress in case 2, despite the smaller hole depth (0.30 mm against 0.60 mm in case 1) and the greater B/T ratio, explains the faster initiation. Nevertheless, as the crack grows, the hole effect of case 2 disappears first, causing a gradual reduction of the fatigue crack growth rates. On the contrary, in case 1, the hole effect acts more time, which combined with a stronger
shear stress level, results in a higher fatigue crack growth rate. The number of cycles to crack initiation (Ni) was determined from the b-N curves, which were fitted to the experimental data by using power functions and polynomial functions for the lateral notch and blind hole configurations, respectively (see Fig. 9). The crack initiation length (b0) inserted into the b-N functions was defined on the basis of the El-Haddad [33] parameter (a0), i.e.
Fig. 8. SEM micrographs: (a and b) blind hole with B/T = 2; (c and d) lateral notch with B/T = 2. 6
International Journal of Fatigue 130 (2020) 105268
R. Branco, et al.
Fig. 9. Surface crack length versus number of loading cycles for different geometric discontinuities, bending-to-torsion ratios, and nominal stress levels.
a0 =
1
Kth
2
0
Table 3 Summary of the multiaxial fatigue test program.
(1)
B/T
where ΔKth is the range of the threshold value of the stress intensity factor, and Δσ0 is the fatigue limit of the unnotched specimen. Such constants are evaluated at the same stress ratio of the notched component to be assessed. For the 34CrNiMo6 high-strength steel, under pulsating loading conditions, based on the values of the fatigue limit stress range (Δσ0) and the stress intensity factor range threshold (ΔΚth0) presented in Table 2, a0 is equal to 129 μm [23]. Since the experimental observations have been conducted at the notch surface, the relationship between the crack length in the in-depth direction (a0) and the corresponding crack length at the notch surface (b0) was estimated from the smallest crack front profile recorded on the fracture surfaces in the beachmarking tests (see Fig. 7). In short, as mentioned above, the crack front profiles of the blind hole configuration in the early stage of growth were close to quarter-elliptical shapes, while in the lateral notch configuration the crack front profiles were close to semi-elliptical shapes [23]. Therefore, the crack length measured at the hole surface was designated as b0 (semi-axis length) and the surface crack length at the notch was designated as 2b0 (axis length). The dimensionless crack front shapes accounted for as the semi-axes ratio (a0/ b0) were approximately equal to 0.89 for the blind hole configuration and 0.80 for the lateral notch configuration. Table 3 compiles the number of cycles to crack initiation computed here using the abovedescribed methodology.
a
D (mm)
h (mm)
a
(MPa)
m
(MPa)
Ni (cycles)
Np (cycles)
Blind hole (σa/τa = 4, σm/τm = 4) 2 16 0.3 224 2 14 0.6 179 2 14 0.3 179
239 194 194
10,557 17,111 59,878
10,313 10,466 39,417
Blind hole (σa/τa = 2, σm/τm = 2) 1 16 1.3 224 1 14 0.5 179 1 14 1.4 298
239 194 313
2406 15,320 1250
1733 8230 953
Lateral 2 2 2
notch (σa/τa = 4, σm/τm = 4) 16 – 179 16 – 224 16 – 298
194 239 313
102,386 49,103 24,207
143,925 63,794 15,787
Lateral 1 1 1
notch (σa/τa = 2, σm/τm = 2) 16 – 179 16 – 224 16 – 298
194 239 313
83,277 26,420 8314
95,799 37,653 8775
=
32B D3
( ) 1
R
2
m
=
32B D3
( ) 1+R 2
a
=
16T D3
( ) 1
R
2
m
=
16T D3
( ). 1+R 2
the initiation sites against the number of cycles to crack initiation (Ni) for the geometric discontinuities and the loading scenarios studied here. It is clearly seen that these local stress states can be very good correlated with the fatigue crack initiation lifetime. This objectively demonstrates the capabilities of the local von Mises stress to account for the fatigue damage in these notch configurations when subjected to inphase combined bending-torsion [23,34]. Fig. 12 also displays a fatigue design curve for 95% failure probability with upper and lower bounds defined from the mean curve with two-sided confidence levels equal to 95%. Although both geometrical discontinuities are treated as a single statistical group, despite their specific features, the scatter band index (Tσ) is relatively small and is in line with the values found for other notched components made of different materials [17,35]. The second step encompasses the computation of an effective stress at the fatigue process zone. Here, effective stresses have been computed from the local von Mises stress range (ΔσvM) by applying the Line Method (LM) of the Theory of Critical Distances. This approach has already been successfully used in notched samples of 34CrNiMo6 highstrength steel undergoing proportional bending-torsion loads [12,34]. Fig. 12 exhibits typical variations of the local von Mises stress range over a straight line emanating from the crack initiation site, in a direction normal to the geometric discontinuity, for both the blind hole
3.5. Multiaxial fatigue lifetime prediction Fatigue crack initiation lifetime for the notched samples, as schematised in Fig. 10, was predicted using the approach proposed by Branco et al. [12] based on an effective value of the Total Strain Energy Density (TSED). The modus operandi consists of four main tasks: (1) reduction of the multiaxial stress state to an equivalent uniaxial stress state at the notch region (Fig. 10(a)); (2) computation of an effective stress at the fatigue process zone (Fig. 10(b)); (3) generation of a representative stress–strain circuit on the basis of the effective stress and calculation of the total strain energy density (ΔWT) defined by the sum of both the elastic positive and the plastic components (Fig. 10(c)); and (4) prediction of crack initiation life from a fatigue curve defined using smooth specimens subjected to uniaxial strain-controlled conditions (Fig. 10(d)). Regarding the first step, the multiaxial stress states have been reduced to equivalent uniaxial stress states through the von Mises stress. Fig. 11 plots the numerical von Mises stress amplitude (ΔσvM,max/2) at 7
International Journal of Fatigue 130 (2020) 105268
R. Branco, et al.
Fig. 10. Schematic representation of the model employed to estimate the fatigue crack initiation lifetime.
DLM ≤ 0.5, since the curves are perfectly overlapped. On the contrary, as the distance from the crack initiation site increases, the dimensionless stress profiles behave differently, moving away from one another. The next step deals with the generation of a stable hysteresis loop from the effective stress range computed in the previous step, whose main aim is the calculation of thetotal strain energy density at the fatigue process zone. In this study, hysteresis loops have been generated through the Equivalent Strain Energy Density [36] concept, as in previous studies [12,34], using the mechanical properties listed in Table 2. Briefly, the procedure calculates the maximum stress and the maximum strain, i.e. the stress–strain coordinates of Point A; and, then, supported by an auxiliary coordinate system centred on Point A, it calculates the stress range and the strain range, allowing the definition of Point B. More details about these calculations can be found in references [12,23,34,37]. After that, the generated stress–strain loops are used to compute the averaged total strain energy density defined as the sum of the elastic positive ( We+) and the plastic ( Wp) strain density components. Fig. 13 shows several examples of the stress–strain loops computed for the different geometric discontinuities and multiaxial loading scenarios studied here. The amount of plastic deformation depends not only on the geometric discontinuity, but also on the bending-to-torsion ratio and the nominal stress level. This is, therefore, consistent with the conclusions drawn in the previous sections: (i) the introduction of a hole induces higher local stresses, leading to more plastic deformation at the fatigue process zone and, consequently, lower fatigue crack initiation lives; (ii) higher bending-to-torsion ratios result in smaller shear stress levels, reducing the plastic strain energy density, and rising the number of cycles to fatigue crack initiation; and (iii) decreasing values of the nominal stress give rise to longer fatigue crack initiation lives and reduced values of plastic deformation. The final step is devoted to the estimation of fatigue crack initiation
3162
vM,max/2
(MPa)
vM/2 = 10
Hole
4.230 N -0.278 i
Notch 1000
= 1.573 vM/2
316
B/T=2 B/T=1 B/T=2 B/T=1
= 10 4.033 Ni -0.278
95% survival probability (two-sided 95% confidence level) 33 10
1055
1044
1066
Ni (cycles) Fig. 11. Relationship between the maximum von Mises stress amplitude and the number of cycles to fatigue crack initiation.
1.2 Hole Notch
0.8
B/T=2 B/T=1 B/T=2 B/T=1
0.6
vM/
vM,max (-)
1.0
0.4 0.2 0.0
0
0.25
0.5
0.75
1
1.25
1.5
1100
d/DLM (-)
(MPa)
Fig. 12. Dimensionless variation of the local von Mises stress with the distance from the crack initiation site for both the blind hole and the lateral notch configurations under bending-torsion loading.
and the lateral notch configurations under different B/T ratios. For comparison purposes, stresses are divided by the maximum value of the local von Mises stress range (ΔσvM/ΔσvM,max) which occurs at the initiation site; and the distance from the crack initiation site (d) is divided by the critical distance of the LM (d/DLM). Not surprisingly, these dimensionless functions are notch-sensitive; the stress reduction near the crack initiation site is notoriously faster in the lateral notch configurations than that in the blind hole configurations, which explains the higher crack initiation lifetimes found in the former geometries. Moreover, it is interesting to note that the dimensionless stress profiles, near the initiation sites, are not significantly affected by the bending-to-torsion ratios, particularly when d/
700
B/T=2
300
m
A
B/T=1
-100 -500 -900
B/T=2
0
0.5
B
Hole Notch
B/T=1
1
1.5
2
2.5
3
(%) Fig. 13. Stress–strain hysteresis loops generated using the Equivalent Strain Energy Density concept from an effective stress computed via the Theory of Critical Distances. 8
International Journal of Fatigue 130 (2020) 105268
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106 B/T=2 B/T=1
Notch
B/T=2 B/T=1
Hole Notch
5.0 Probability density function
Predicted life (cycles)
105
Hole
6.0
Np = 2Ni
104
103
4.0 3.0 2.0 1.0
Ni = 2Np
102 102
103
104
105
0.0
106
Unsafe
-0.8
Experimental life (cycles)
Safe
-0.4
0.0 Error, EN
0.4
0.8
Fig. 14. Experimental fatigue crack initiation life versus numerical predictions computed using the TSED approach.
Fig. 15. Probability density functions of fatigue life prediction error for the blind hole and the lateral notch configurations.
lifetime. According to the proposed approach (Fig. 10(d)), the number of cycles to fatigue crack initiation is predicted by inserting the values of ΔWT calculated in the previous step into a fatigue master curve developed for the 34CrNiMo6 high-strength steel on the basis of a series of uniaxial low-cycle fatigue tests performed using smooth samples. The fitted function of the TSED approach, calculated in a previous research [27], is given by Eq. (2).
discontinuities causing interacting stress concentrations introduced in the design, fabrication, assembly, or repair stages. Last but not least, it is based on linear elastic simulations and requires a limited number of material properties, therefore is faster and cheaper, which make it very attractive to the industry.
WT
(MJ/m3)
= 2165.37(2Nf )
0.6854+
0.7049
4. Conclusions
(2)
In this paper, fatigue crack initiation behaviour of notched bars made of 34CrNiMo6 high-strength steel with blind transverse holes and lateral U-shaped notches subjected to proportional bending-torsion loading has been studied. Crack initiation lifetime has been predicted using an averaged strain energy density approach. The following conclusions can be drawn:
The fatigue crack initiation lives accounted for through the proposed methodology (Np) are summarised in Table 3. Fig. 14 plots the predicted values (Np) against the experimental observations (Ni) for the various geometric discontinuities and multiaxial loading scenarios studied here. In order to simplify the analysis, conservative and unconservative scatter bands with factors of two (i.e. Nia2Np and Npa2Ni) are drawn. Fatigue life predictions, as can be seen in the Fig. 14, are within the delimited area, which is an important outcome. A close analysis shows a mixed trend with safe and unsafe predictions. Safe predictions have mainly been found for the blind hole, while the unsafe ones have mainly occurred for the lateral notch. The capabilities of the predictive approach can be evaluated, in a more objective basis, via a statistical study of the fatigue life prediction errors (EN). Here, EN is defined by the following equation [17]
EN = log10
Ni Np
• Crack initiation sites, crack initiation angles, and surface crack paths
•
(3)
•
where Ni is the experimental crack initiation life and Np represents the predicted crack initiation life. Briefly, the closer to zero is the mean error, the more accurate is the model; and the smaller is the standard deviation, the more concentrated are the fatigue life prediction errors in relation to the mean value. These features can be graphically inferred from Fig. 15, which plots the probability density functions for both geometric discontinuities. In fact, mean values are close to zero in both cases, but in opposite sides; the mean error of the hole configuration is in the safe region, while the mean error of the lateral notch is in unsafe region. On the other hand, in the former geometric discontinuity, there is a clear concentration of values in relation to the mean error, in contrast to what happens in the latter method whose dispersion is relatively greater. Overall, these outcomes strongly validate the viability of the proposed approaches to predict the crack initiation lifetime in notched bars made of 34CrNiMo6 high-strength steel subjected to bending-torsion loading and to handle sudden changes in the geometric
• • • 9
at the notch surface have been successfully predicted in both notch configurations from linear-elastic finite-element models via the maximum value of the first principal stress at the notch surface, the first principal direction at the initiation site, and the first principal stress field at the notch surface, respectively; Lateral notch configurations under bending-torsion loading are very prone to multi-crack initiation phenomena, which can be explained by the high stress levels acting at an extensive area of the notch. The existence of geometric defects and pores in such regions are likely to make the multi-crack initiation much easier; By contrast, in the blind hole configuration, maximum stress levels are confined to two small sites at the hole border. Thus, such sites become natural places to crack initiation, avoiding the possibility of crack nucleation in other locations; There is a remarkable difference between the crack front shape developments of the blind hole and the lateral notch configurations at the early stage of crack growth due to the hole effect. Nevertheless, as the crack extends, this effect gradually disappears, leading to crack shapes much closer; Fatigue crack initiation life has been successfully predicted through the TSED approach. Overall, experimental and predicted fatigue lives have been very well correlated in the entire range with all points within scatter bands of two. The proposed method has shown good predictive capabilities to deal with different geometric discontinuities, since the mean errors were statistically similar but in different opposite sides, i.e.
International Journal of Fatigue 130 (2020) 105268
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•
predominantly in the safe region for the blind hole and predominantly in opposite region for the lateral notch. The reduced cost associated with the determination of the material constants and the ability to incorporate a linear-elastic framework makes it simpler and cheaper, which is a major asset in the context of mechanical design.
[15] Susmel L, Taylor D. The theory of critical distances to estimate lifetime of notched components subjected to variable amplitude uniaxial fatigue loading. Int J Fatigue 2011;33:900–11. https://doi.org/10.1016/j.ijfatigue.2011.01.012. [16] Faruq N, Susmel L. Proportional/nonproportional constant/variable amplitude multiaxial notch fatigue: cyclic plasticity, non‐zero mean stresses, and critical distance/plane. Fatigue Frac Eng Mat Struct (in press). https://doi.org/10.1111/ffe. 13036. [17] Hu Z, Berto F, Hong Y, Susmel L. Comparison of TCD and SED methods in fatigue lifetime assessment. Int J Fatigue 2019;123:105–34. https://doi.org/10.1016/j. ijfatigue.2019.02.009. [18] Liu B, Yan X. An extension research on the theory of critical distances for multiaxial notch fatigue finite life prediction. Int J Fatigue 2019;123:105–34. https://doi.org/ 10.1016/j.ijfatigue.2018.08.017. [19] Golos K, Ellyin F. Generalization of cumulative damage criterion to multilevel cyclic loading. Theo Appl Fract Mech 1987;7:169–76. https://doi.org/10.1016/01678442(87)90032-2. [20] Correia J, Apetre N, Attilio A, De Jesus A, Muñiz-Calvente M, Calçada R, et al. Generalized probabilistic model allowing for various fatigue damage variables. Int J Fatigue 2017;100:187–94. https://doi.org/10.1016/j.ijfatigue.2017.03.031. [21] Taylor D. Geometrical effects in fatigue: a unifying theoretical model. Int J Fatigue 1999;21:413–20. https://doi.org/10.1016/S0142-1123(99)00007-9. [22] Fröschl J. Fatigue behaviour of forged components: Technological effects and multiaxial fatigue PhD Thesis Austria: Montanuniversität Leoben; 2006. [23] Branco R, Costa JD, Antunes FV. Fatigue behaviour and life prediction of lateral notched round bars under bending–torsion loading. Eng Fract Mech 2014;119:66–84. https://doi.org/10.1016/j.engfracmech.2014.02.009. [24] Hannemann R, Köster P, Sander M. Investigations on crack propagation in wheelset axles under rotating bending and mixed mode loading. Procedia Struct Integrity 2017;5:861–8. https://doi.org/10.1016/j.prostr.2017.07.104. [25] Branco R, Costa JD, Berto F, Antunes FV. Effect of loading orientation on fatigue behaviour in severely notched round bars under non-zero mean stress bendingtorsion. Theor Appl Fract Mech 2017;92:185–97. https://doi.org/10.1016/j.tafmec. 2017.07.015. [26] Leitner M, Grün F. Multiaxial fatigue assessment of crankshafts by local stress and critical plane approach. Frattura ed Integrità Strutturale 2016;38:47–53. https:// doi.org/10.3221/IGF-ESIS.38.06. [27] Branco R, Costa JDM, Antunes FV, Perdigão S. Monotonic and cyclic behavior of DIN 34CrNiMo6 tempered alloy steel. Metals 2016;6:98. https://doi.org/10.3390/ met6050098. [28] Reis L, Li B, Freitas M. Crack initiation and growth path under multiaxial fatigue loading in structural steels. Int J Fatigue 2009;31:1660–8. https://doi.org/10. 1016/j.ijfatigue.2009.01.013. [29] Lopez-Crespo P, Moreno B, Lopez-Moreno A, Zapatero J. Study of crack orientation and fatigue life prediction in biaxial fatigue with critical plane models. Eng Fract Mech 2015;136:115–30. https://doi.org/10.1016/j.engfracmech.2015.01.020. [30] Freitas M. Multiaxial fatigue: From materials testing to life prediction. Theor Appl Fract Mech 2017;92:360–72. https://doi.org/10.1016/j.tafmec.2017.05.008. [31] Branco RF, Antunes FV, Martins R. Modelling fatigue crack propagation in CT specimens. Fatigue Fract Eng Mater Struct 2008;31:452–65. https://doi.org/10. 1111/j.1460-2695.2008.01241.x. [32] Barata R, Martins RF, Albarran T, Santos T, Mourão A. Failure analysis of a pull rod actuator of an ATOX raw mill used in the cement production process. Eng Fail Anal 2017;76:99–114. https://doi.org/10.1016/j.engfailanal.2016.12.023. [33] El Haddad MH, Topper TH, Smith KN. Prediction of non propagating cracks. Eng Fract Mech 1979;11:573–84. https://doi.org/10.1016/0013-7944(79)90081-X. [34] Branco R, Prates P, Costa JD, Berto F, Kotousov A. New methodology of fatigue life evaluation for multiaxially loaded notched components based on two uniaxial strain-controlled tests. Int J Fat 2018;111:308–20. https://doi.org/10.1016/j. ijfatigue.2018.02.027. [35] Abreu LM, Costa JD, Ferreira JAM. Fatigue behaviour of AlMgSi tubular specimens subjected to bending–torsion loading. Int J Fatigue 2009;31:1327–36. https://doi. org/10.1016/j.ijfatigue.2009.03.004. [36] Glinka G. Calculation of inelastic notch-tip strain–stress histories under cyclic loading. Eng Fract Mech 1985;22:839–54. https://doi.org/10.1016/0013-7944(85) 90112-2. [37] Branco R, Prates PA, Costa JD, Borrego LP, Antunes FV. Rapid assessment of multiaxial fatigue lifetime in notched components using an averaged strain energy density approach. Int J Fat 2019;124:89–98. https://doi.org/10.1016/j.ijfatigue. 2019.02.005.
Acknowledgment This research is sponsored by FEDER funds through the program COMPETE (under project T449508144-00019113) and by national funds through FCT – Portuguese Foundation for Science and Technology, under the project PTDC/EMS-PRO/1356/2014. The authors are also thankful to Prof. Amílcar Ramalho and Dr. Pedro Antunes from the University of Coimbra for the help provided in the SEM analysis. References [1] Ghosh M, Ghosh A, Roy A. Renewable and sustainable materials in automotive industry. reference module in materials science and materials engineering. Elsevier 2019. https://doi.org/10.1016/b978-0-12-803581-8.11461-4. [2] Mayyas A, Qattawi A, Omar M, Shana D. Design for sustainability in automotive industry: a comprehensive review. Renew Sust Energ Rev 2012;16:1845–62. https://doi.org/10.1016/j.rser.2012.01.012. [3] Schmitt JH, Iung T. New developments of advanced high-strength steels for automotive applications. C R Physique 2018;19:641–56. https://doi.org/10.1016/j. crhy.2018.11.004. [4] Tisza M, Czinege I. Comparative study of the application of steels and aluminium in lightweight production of automotive parts. Int J Lightweight Mater Manuf 2018;1:229–38. https://doi.org/10.1016/j.ijlmm.2018.09.001. [5] Carpinteri A, Spagnoli A, Vantadori S, Viappiani D. A multiaxial criterion for notch high-cycle fatigue using a critical-point method. Eng Fract Mech 2008;75:1864–74. https://doi.org/10.1016/j.engfracmech.2006.11.002. [6] Liao D, Zhu S, Correia J, Jesus A, Calçada R. Computational framework for multiaxial fatigue life prediction of compressor discs considering notch effects. Eng Fract Mech 2018;202:423–35. https://doi.org/10.1016/j.engfracmech.2018.08.009. [7] Rozumek D, Macha E. Elastic-plastic fatigue crack growth in 18G2A steel under proportional bending with torsion loading. Fatigue Fract Eng Mater Struct 2006;29:135–45. https://doi.org/10.1111/j.1460-2695.2006.00972.x. [8] Whitley D. Interacting stress concentration factors and their effect on fatigue of metallic aerostructures. Missouri University of Science and Technology, Department of Mechanical and Aerospace Engineering, Doctoral Dissertations 1830, 2013. http://scholarsmine.mst.edu/doctoral_dissertations/1830. [9] Berto F, Campagnolo A, Lazzarin P. Fatigue strength of severely notched specimens made of Ti–6Al–4V under multiaxial loading. Fatigue Fract Eng Mater Struct 2015;38:503–17. https://doi.org/10.1111/ffe.12272. [10] Meneghetti G, Campagnolo A, Berto F, Tanaka K. Notched Ti-6Al-4V titanium bars under multiaxial fatigue: Synthesis of crack initiation life based on the averaged strain energy density. Theo App Frac Mech 2018;96:509–33. https://doi.org/10. 1016/j.tafmec.2018.06.010. [11] Zhu SP, Liu Y, Liu Q, Yua ZY. Strain energy gradient-based LCF life prediction of turbine discs using critical distance concept. Int J Fat 2018;113:33–42. https://doi. org/10.1016/j.ijfatigue.2018.04.006. [12] Branco R, Costa JD, Berto F, Antunes FV. Fatigue life assessment of notched round bars under multiaxial loading based on the total strain energy density approach. Theo Appl Fract Mech 2018;97:340–8. https://doi.org/10.1016/j.tafmec.2017.06. 003. [13] Gasiak G, Pawliczek R. Application of an energy model for fatigue life prediction of construction steels under bending, torsion and synchronous bending and torsion. Int J Fatigue 2003;25:1339–46. https://doi.org/10.1016/S0142-1123(03)00055-0. [14] Lesiuk G, Szata M, Rozumek D, Marciniak Z, Correia J, de Jesus A. Energy response of S355 and 41Cr4 steel during fatigue crack growth process. J Strain Anal Eng Design 2018;53:663–75. https://doi.org/10.1177/0309324718798234.
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Fatigue crack initiation behaviour of notched 34CrNiMo6 steel bars under proportional bending-torsion loading
T
R. Brancoa, , J.D. Costaa, F. Bertob, A. Kotousovc, F.V. Antunesa ⁎
a
CEMMPRE, Department of Mechanical Engineering, University of Coimbra, 3030-788 Coimbra, Portugal Department of Mechanical and Industrial Engineering, NTNU, 7491 Trondheim, Norway c School of Mechanical Engineering, The University of Adelaide, Adelaide, SA 5005, Australia b
ARTICLE INFO
ABSTRACT
Keywords: Multiaxial fatigue Bending-torsion Notch effect Interacting stress concentration Multi-crack initiation Strain energy density Theory of critical distances 34CrNiMo6 high-strength steel
Fatigue crack initiation behaviour of notched 34CrNiMo6 high-strength steel bars with transverse blind holes and lateral U-shaped notches subjected to proportional bending-torsion loading is studied. Crack initiation at the geometric discontinuities is examined in situ using a high-magnification digital camera. Crack initiation mechanisms and failures modes are investigated after the fatigue testing program via scanning electron microscopy. Fatigue crack initiation lifetime is predicted using an averaged Total Strain Energy Density approach in conjunction with the Theory of Critical Distances. Crack initiation sites, crack surface angles and surface crack paths at the geometric discontinuities are successfully determined a priori from the maximum value of the first principal stress, the first principal direction, and the first principal stress field at the notch surface, respectively. Overall, fatigue lifetime predictions are very well correlated with the experimental observations.
1. Introduction Modern automotive industry, driven by recent environmental policies, faces an urgent need for the production of lighter vehicles with reduced fuel consumption and lower pollutant emissions [1,2]. Despite the constant development of new materials, high-strength steels remain key materials in this challenging scenario, mainly due to their balanced features, in particular the low cost, excellent strength-to-weight ratio, and good corrosion resistance, among others [3,4]. The above-mentioned goals, namely the weight reduction, are often met via a smarter design, which is likely to introduce sudden and abrupt geometrical changes. The loading complexity acting on such components, in many cases characterised by a multiaxial nature, combined with the stress concentration phenomena occurring at the critical geometric discontinuities, leads to non-trivial and challenging problems [5–7]. Moreover, advanced engineering components can contain two or more geometric details, such as holes and notches, placed in enough proximity to each other that may cause stress concentration interactions, making the problem much more difficult [8]. Different fatigue lifetime prediction models have been proposed to deal with notched components via local pseudo-elastic stresses. Among the most successful approaches, we can mention those based on the Strain Energy Density [9–14] and those based on the Theory of Critical Distances [15–18]. The underlying idea behind the SED-based ⁎
approaches is that the damage caused by cyclic loading is a function of the mechanical energy input into the material [19,20]. Regarding the TCD-based approaches, the key concept is the determination of a critical stress, or strain, nearby the geometric detail, considering a characteristic material length [21]. However, despite the comprehensive research efforts, there are several unresolved issues. Among them, we can mention, for instance, the interaction of stress concentrations. As far as the selected material is concerned, 34CrNiMo6 highstrength steel is a typical medium-carbon low-alloy steel and one of the prime choices for several automotive components. Although this steel has been studied over the last decades, a little attention has been paid to the multiaxial fatigue behaviour of notched components. Literature review shows that the investigated geometric discontinuities have only encompassed notched shafts and notched round bars [22–26]. Nevertheless, the study of holes, either under in-phase or out-of-phase multiaxial loading conditions, has not been addressed. Interaction of stress concentrations caused by two or more geometric discontinuities, particularly holes and notches, have also not been addressed. The present paper aims at studying the fatigue behaviour of notched bars made of 34CrNiMo6 high-strength steel undergoing proportional bending-torsion loading. The geometric discontinuities studied here encompass a lateral U-shaped notched with a transverse blind hole, and a lateral U-shaped notch without transverse blind hole. The fatigue crack initiation lifetime is predicted from the Total Strain Energy
Corresponding author. E-mail address: [email protected] (R. Branco).
https://doi.org/10.1016/j.ijfatigue.2019.105268 Received 14 June 2019; Received in revised form 13 August 2019; Accepted 9 September 2019 Available online 10 September 2019 0142-1123/ © 2019 Elsevier Ltd. All rights reserved.
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The loading scenarios were replicated by two pairs of forces placed at one end of the specimen, while the other end was fixed. Torsion moments were created by the FT forces applied on a plane normal to the longitudinal axis of the specimen with opposite directions. Bending moments were created by the FB forces applied parallelly to the longitudinal axis of the specimen but with opposite directions. The relationship between the normal and shear stresses was established by adjusting the proportionality constant λ (see Fig. 2) which assumes the values λ = 1/2 and λ = 1 for B/T = 2 and B/T = 1, respectively.
Table 1 Nominal chemical composition (wt%) of DIN 34CrNiMo6 high-strength steel [27]. C
Si
Mn
Cr
Mo
Ni
0.34
≤0.40
0.65
1.50
0.22
1.50
Density approach on the basis of an effective stress determined using a linear-elastic framework in conjunction with the Theory of Critical Distances. In practical engineering problems, holes can be introduced during the fabrication, assembly or repair tasks, as well as due to mistakes made by manufacturing such as mislocated holes, or extra holes, which are prone to origin interaction of stress concentrations. Thus, the proposed methodology can be very helpful in an industrial context, since it provides a tool for rapid multiaxial life assessment of notched components, simple to implement, and founded on linearelastic analyses.
3. Results and discussion 3.1. Experimental fatigue crack initiation and propagation behaviour The trajectory described by a crack when subjected to specific loading histories is a relevant matter in the context of mechanical design. Its accurate prediction is a complex issue, particularly under multiaxial loading. Fig. 3 shows representative surface crack paths observed in the experimental tests for the different geometric discontinuities and bending-to-torsion ratios studied here. It is clear that the surface crack paths are remarkably affected by both the notch configuration and the multiaxial loading history. Overall, the increase of the B/T ratio, regardless of the geometric discontinuity, leads to straighter surface trajectories. This can be explained by the smaller shear stress levels, which reduce the degree of mixed-mode propagation. The surface crack paths, under in-phase bending-torsion scenarios, can be satisfactorily anticipated for both geometric discontinuities through the first principal stress (σ1) field at the notch surface. Fig. 4 plots the first principal stress field computed from the linear-elastic numerical models described in the previous section. In fact, as represented in Fig. 3, the surface crack trajectories predicted here (dashed lines) are quite close to those observed in the experiments. This outcome suggests that the proposed approach is sufficiently sensitive to determine, a priori, the surface crack paths for different geometric discontinuities subjected to in-phase bending-torsion loading. A reliable determination of the fatigue crack initiation sites is another critical point in the design of mechanical components. Fig. 3 exhibits some examples of the crack initiation sites, represented by the small circles, observed in the experiments for the different geometric discontinuities and multiaxial loading scenarios. The analysis of results shows that the fatigue crack initiation sites are also dependent on the geometric discontinuity and the bending-to-torsion ratio. In the blind hole geometries, there is always the nucleation of two cracks at the hole border in symmetrical positions with the respect to a vertical axis passing through the centre of the hole (Fig. 3(a) and 3(b)). Furthermore, the radial angle formed by a straight line joining the initiation sites and the vertical axis raises with increasing values of B/T. Regarding the lateral notch geometries, the effect of the B/T ratio is also clearly observed, i.e. for lower levels of shear stress (i.e. higher B/T ratios), the crack tends to nucleate in the centre of the notch (Fig. 3(c)), while at higher levels of shear stress (i.e. lower B/T ratios), the crack tends to shift progressively to the curved edge of the notch (Fig. 3(d)). The crack nucleation in these geometric discontinuities when subjected to in-phase bending-torsion loading can be predicted, in a precise manner, by the coordinate at the notch surface with maximum value of the first principal stress. The numerical predictions for the blind hole and lateral notch configurations can be observed in Fig. 4. A comparison of the experimental observations (small circles) and the numerical predictions (small squares) is exhibited in Fig. 3. As can be seen, there is an excellent correlation between both types of data, particularly in the case of the blind hole configuration, which is a major asset.
2. Experimental and numerical procedures The material utilised in this study is the 34CrNiMo6 high-strength steel, supplied in a quenched and tempered condition. Its chemical composition and its main mechanical properties are compiled, respectively, in Tables 1 and 2. Multiaxial fatigue testing program has been conducted under inphase constant-amplitude pulsating loading (R = 0) using 16 mm-diameter and 14 mm-diameter notched round bars. Table 2 summarises the maximum nominal stress amplitude (σa) and the maximum nominal mean stress (σm) applied in the tests. The geometric discontinuities, as schematised in Fig. 1, encompassed: (a) a lateral U-shaped notch with a central transverse blind hole (Fig. 1(b)); and (b) a lateral U-shaped notch without central transverse blind hole (Fig. 1(c)). For the sake of simplicity, the former and the latter geometries are hereafter termed blind hole and lateral notch, respectively. The loading scenarios included ratios of the bending moment to torsion moment (B/T) equal to 1 and 2. For each geometric configuration and loading case, three local stress levels were studied. Crack initiation and crack growth were monitored in-situ via a high-resolution digital camera. Images were periodically recorded with sampling rates between 1000 and 5000 cycles. Tests were performed in air, at room temperature, with sinusoidal waves, and cyclic frequencies in the range 3–6 Hz. Stress and strain fields at the notch region were computed from three-dimensional numerical models using COSMOS/M software. Meshes were developed in a parametric framework using 8-node isoparametric brick elements and assuming a homogeneous, linear-elastic and isotropic material. Fig. 2(a) shows, as an example, a finite element mesh developed for the blind hole configuration. At the geometric discontinuity, the mesh was carefully refined (Fig. 2(b)) while, at remote regions, a coarser pattern was adopted. The assembled model had 89,584 elements and 97,704 nodes. As far as the lateral notch configuration is concerned, the mesh details at geometric discontinuity are displayed in Fig. 2(c). It had 76,608 elements and 99,823 nodes. Table 2 Mechanical properties of the DIN 34CrNiMo6 high strength steel [23]. Mechanical property
Value
Yield strength, σYS (MPa) Tensile strength, σUTS (MPa) Young’s modulus, E (GPa) Poisson’s ratio, ν Cyclic hardening coefficient, K’ (MPa) Cyclic hardening exponent, n’ Fatigue limit stress range, Δσ0 (MPa) Stress intensity factor range threshold, ΔKth0 (MPa∙m0.5)
967 1035 209.8 0.296 1361.6 0.1041 353 7.12
3.2. Crack initiation angles Examples of the experimental crack angles at the notch surface in 2
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Fig. 1. Specimen geometries: (a) front view; (b) lateral U-shaped notch with transverse blind hole; (c) lateral U-shaped notch without transverse blind hole (dimensions in millimetres).
the early stage of growth for the blind hole and lateral notch configurations subjected to different bending-to-torsion ratios are displayed in Fig. 5. These angles are controlled by the geometric discontinuity and the multiaxial loading scenario. In general, it is possible to conclude that: (i) as the B/T ratio increases, the angle with respect to the horizontal axis decreases; and (ii) the angles of the blind holes are higher than those of the lateral notches when subjected to the same loading scenario. Although the prediction of crack initiation angles under multiaxial loading conditions remains to be a difficult problem [25,27][28–30], whose governing variables are not completely understood yet, the first principal direction accounted for at the initiation site has allowed to achieve very good results. As can be distinguished in Fig. 6, which plots the experimental observations against the numerical predictions, the maximum errors are lower than 5°, while the average errors for the blind hole and lateral notch configurations are equal to 2.6° and 1.4°, respectively.
such features, fracture surfaces were examined by optical microscopy (OM) and scanning electron microscopy (SEM). Fig. 7 shows typical topologies of fracture surfaces observed in the experiments by OM for the different geometric discontinuities and multiaxial loading scenarios. Overall, it can be noted that the fatigue crack growth process is quite complex and reveals a high degree of out-of-plane propagation, which increases as the B/T ratio decreases or, in other words, when the shear stress level climbs. Despite the obvious limitations of the top views to assess three-dimensional effects, the front views clearly show higher curvatures in the fracture surfaces of the blind hole configurations (Fig. 7(a) and (b)) than those of the lateral notch configurations (Fig. 7(c) and (d)) under the same loading history. Furthermore, irrespective of the geometric discontinuity, surface topologies are tendentially flatter for increasing values of the B/T ratio, i.e. in the cases whose propagation conditions are closer to mode-I loading [31,32]. The crack front shape changes during the fatigue crack growth stage can also help to understand the mixed-mode propagation behaviour. These changes can be evaluated from the successive macroscopic progression marks recorded on the fracture surfaces via the beach-marking technique (see the top views of Fig. 7). In fact, the crack front profiles of the blind hole configurations are remarkably different from the lateral notch configurations, particularly for smaller crack lengths. On the
3.3. Fracture surface topologies Fracture surface topologies generally contain useful information on the failure modes and fatigue mechanisms. Here, in order to evaluate
Fig. 2. Example of the assembled model (a) developed to compute the stress and strain fields at the notch region; (b) mesh detail for the blind hole; (c) mesh detail of the lateral notch. 3
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Fig. 3. Macroscopic appearance of the fracture surfaces: (a) blind hole with B/T = 2; (b) blind hole with B/T = 1; (c) lateral notch with B/T = 2; (d) lateral notch with B/T = 1.
contrary, for longer crack lengths, since the hole effect progressively disappears, the crack front profiles become relatively similar. In this stage of propagation, as the crack grows, crack front profiles tend to be straighter. Furthermore, irrespective of the geometric discontinuity, crack front profiles are more curved, for a similar crack length, when the shear stress levels are higher, i.e. when the bending-to-torsion ratios decrease. These evidences can be explained by the nature of the fatigue crack growth process. In the blind hole geometry, as above mentioned, there is the nucleation of two corner cracks at the hole border with profiles close to quarter-elliptical shapes evolving, in a second stage, to a unique surface crack with both tips along the surface of the U-shaped notch surface. In a third stage, as the crack propagates, each tip reaches the circular portion of the specimen and, then, the crack grows as a through crack until the failure has been reached. Regarding the lateral notch geometry, crack initiation takes place somewhere at the U-shaped notch (more precisely, closer to centre under higher B/T ratios, and closer to the edge under lower B/T ratios). It should be also noted that this notch configuration, at least when subjected to in-phase bending-torsion
scenarios, is very prone to multi-crack initiation phenomena [23,25], as demonstrate the fatigue steps marked on the fracture surfaces (see the arrows of Fig. 7(c) and (d)). A plausible explanation for this behaviour is the existence of an extensive area at the notch surface with high stress levels (see Fig. 4(c) and (d)) in this case, unlike the case of the blind hole configuration, whose maximum stresses occur only at two highly confined regions (see Fig. 4(a)) and (b)). These several neighbouring cracks at the U-shaped notch gradually coalesce and, in a subsequent stage, turn into a single surface crack. As in the previous case, the crack tips advance in the direction of the circular part of the specimen, resulting in a through crack which propagates until the failure is reached. The fatigue damage mechanisms associated with the failure process of the 34CrNiMo6 high-strength steel under proportional bending-torsion loading are summarised in Fig. 8. In the blind hole configuration, as can be distinguished in Fig. 8(a), there is the nucleation of two cracks at the hole border, as a consequence of the highest stress concentration effects registered in such places (see Fig. 4(a) and (b)). The central fatigue step, identified in Fig. 8(a) by the arrow, results from the coalescence of both cracks. A close look at this region, magnified in
Fig. 4. First principal stress field at the geometric discontinuities computed through the numerical models: (a) blind hole with B/T = 2; (b) blind hole with B/T = 1; (c) lateral notch with B/T = 2; (d) lateral notch with B/T = 1. 4
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Fig. 5. Experimental surface crack angles at the early stage of growth: (a) blind hole with B/T = 2; (b) blind hole with B/T = 1; (c) lateral notch with B/T = 2; (d) lateral notch with B/T = 1.
3.4. Fatigue crack initiation life
30 Hole
25 Measured angle (º)
Notch
20
B/T=2 B/T=1 B/T=2 B/T=1
Fatigue crack initiation lifetime was estimated from the experimental relations of the surface crack length (b) and the number of loading cycles (N). For the sake of simplicity, such relations are designated here as b-N curves. Fig. 9 displays typical b-N curves derived in the present study for the blind hole and the lateral notch configurations under different combined bending-torsion loading histories and different nominal stress levels. Note that due to the coalescence of defects, the crack length was accounted for via the sum of the individual flaws. This modus operandi provided a single b-N curve for each test, which allowed a simpler comparison of results. Briefly, the analysis of results suggests that: (i) fatigue crack initiation is faster in the blind hole geometry than that in the lateral notched geometry, as a consequence of the higher stress concentration levels introduced by the hole (see, for example, series 3 and 6); (ii) under the same nominal normal stress amplitude, regardless of the geometric discontinuity, the higher the B/T ratio, the slower the fatigue crack initiation (see, for examples, series 1 and 3, or series 5 and 6), which can be explained by the reduction of the shear stress level; (iii) fatigue crack initiation, fixing the bending-totorsion ratio and the geometric discontinuity, speeds up with increasing values of the nominal stress level (see, for example, series 2 and 3, or series 4 and 6). The hole depth plays a fundamental role in the fatigue crack initiation process of the blind hole configuration, as well as in the earlier stage of crack growth, but loses its importance in the other stages. When the hole depth is relatively small (series 2 and 3), respectively equal to 0.3 mm, crack initiation occurs much more rapidly than that in the cases without hole (series 4 and 6). However, in a second stage, the fatigue crack growth rates of the blind hole geometries slow down and seem to converge to the curves of the specimens without holes tested under identical loading histories (i.e. series 2 seems to converge to series 4; and series 3 seems to converge to series 6). It is also interesting to note that crack initiation of series 1 is faster than that of series 2,
15 10
Y
5 0
Z
0
5
10 15 20 Predicted angle (º)
25
30
Fig. 6. Experimental versus predicted crack angles at the early stage of growth.
Fig. 8(b), shows traces of plastic deformation, namely fatigue striations with radial orientation, which denotes a ductile failure mode. With regard to the lateral notch configuration, cracks initiate from surface irregularities (see the arrow of Fig. 8(c)) or from inclusions and pores located in the vicinity or at the notch surface (see the arrow of Fig. 8(d)), which act as local stress raisers, easing the crack nucleation. It is also possible to observe, in Fig. 8(c), at least three fatigue steps, as the result of the coalescence of neighbouring defects; a magnified example of a fatigue step is exhibited in Fig. 8(d). Additionally, there is a population of non-metallic particles, composed by globular shapes, disperse throughout the fracture surfaces (see Fig. 8(a) and (c)). 5
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B/T=2
B/T=1
B/T=2
B/T=1
(c)
(d)
3 mm
(a)
(b)
Fig. 7. Top views and front views of the fracture surfaces: (a) blind hole with B/T = 2; (b) blind hole with B/T = 1; (c) lateral notch with B/T = 2; (d) lateral notch with B/T = 1.
however the curves cross each other after some propagation causing an inversion of the fatigue crack growth rates. The higher nominal stress in case 2, despite the smaller hole depth (0.30 mm against 0.60 mm in case 1) and the greater B/T ratio, explains the faster initiation. Nevertheless, as the crack grows, the hole effect of case 2 disappears first, causing a gradual reduction of the fatigue crack growth rates. On the contrary, in case 1, the hole effect acts more time, which combined with a stronger
shear stress level, results in a higher fatigue crack growth rate. The number of cycles to crack initiation (Ni) was determined from the b-N curves, which were fitted to the experimental data by using power functions and polynomial functions for the lateral notch and blind hole configurations, respectively (see Fig. 9). The crack initiation length (b0) inserted into the b-N functions was defined on the basis of the El-Haddad [33] parameter (a0), i.e.
Fig. 8. SEM micrographs: (a and b) blind hole with B/T = 2; (c and d) lateral notch with B/T = 2. 6
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Fig. 9. Surface crack length versus number of loading cycles for different geometric discontinuities, bending-to-torsion ratios, and nominal stress levels.
a0 =
1
Kth
2
0
Table 3 Summary of the multiaxial fatigue test program.
(1)
B/T
where ΔKth is the range of the threshold value of the stress intensity factor, and Δσ0 is the fatigue limit of the unnotched specimen. Such constants are evaluated at the same stress ratio of the notched component to be assessed. For the 34CrNiMo6 high-strength steel, under pulsating loading conditions, based on the values of the fatigue limit stress range (Δσ0) and the stress intensity factor range threshold (ΔΚth0) presented in Table 2, a0 is equal to 129 μm [23]. Since the experimental observations have been conducted at the notch surface, the relationship between the crack length in the in-depth direction (a0) and the corresponding crack length at the notch surface (b0) was estimated from the smallest crack front profile recorded on the fracture surfaces in the beachmarking tests (see Fig. 7). In short, as mentioned above, the crack front profiles of the blind hole configuration in the early stage of growth were close to quarter-elliptical shapes, while in the lateral notch configuration the crack front profiles were close to semi-elliptical shapes [23]. Therefore, the crack length measured at the hole surface was designated as b0 (semi-axis length) and the surface crack length at the notch was designated as 2b0 (axis length). The dimensionless crack front shapes accounted for as the semi-axes ratio (a0/ b0) were approximately equal to 0.89 for the blind hole configuration and 0.80 for the lateral notch configuration. Table 3 compiles the number of cycles to crack initiation computed here using the abovedescribed methodology.
a
D (mm)
h (mm)
a
(MPa)
m
(MPa)
Ni (cycles)
Np (cycles)
Blind hole (σa/τa = 4, σm/τm = 4) 2 16 0.3 224 2 14 0.6 179 2 14 0.3 179
239 194 194
10,557 17,111 59,878
10,313 10,466 39,417
Blind hole (σa/τa = 2, σm/τm = 2) 1 16 1.3 224 1 14 0.5 179 1 14 1.4 298
239 194 313
2406 15,320 1250
1733 8230 953
Lateral 2 2 2
notch (σa/τa = 4, σm/τm = 4) 16 – 179 16 – 224 16 – 298
194 239 313
102,386 49,103 24,207
143,925 63,794 15,787
Lateral 1 1 1
notch (σa/τa = 2, σm/τm = 2) 16 – 179 16 – 224 16 – 298
194 239 313
83,277 26,420 8314
95,799 37,653 8775
=
32B D3
( ) 1
R
2
m
=
32B D3
( ) 1+R 2
a
=
16T D3
( ) 1
R
2
m
=
16T D3
( ). 1+R 2
the initiation sites against the number of cycles to crack initiation (Ni) for the geometric discontinuities and the loading scenarios studied here. It is clearly seen that these local stress states can be very good correlated with the fatigue crack initiation lifetime. This objectively demonstrates the capabilities of the local von Mises stress to account for the fatigue damage in these notch configurations when subjected to inphase combined bending-torsion [23,34]. Fig. 12 also displays a fatigue design curve for 95% failure probability with upper and lower bounds defined from the mean curve with two-sided confidence levels equal to 95%. Although both geometrical discontinuities are treated as a single statistical group, despite their specific features, the scatter band index (Tσ) is relatively small and is in line with the values found for other notched components made of different materials [17,35]. The second step encompasses the computation of an effective stress at the fatigue process zone. Here, effective stresses have been computed from the local von Mises stress range (ΔσvM) by applying the Line Method (LM) of the Theory of Critical Distances. This approach has already been successfully used in notched samples of 34CrNiMo6 highstrength steel undergoing proportional bending-torsion loads [12,34]. Fig. 12 exhibits typical variations of the local von Mises stress range over a straight line emanating from the crack initiation site, in a direction normal to the geometric discontinuity, for both the blind hole
3.5. Multiaxial fatigue lifetime prediction Fatigue crack initiation lifetime for the notched samples, as schematised in Fig. 10, was predicted using the approach proposed by Branco et al. [12] based on an effective value of the Total Strain Energy Density (TSED). The modus operandi consists of four main tasks: (1) reduction of the multiaxial stress state to an equivalent uniaxial stress state at the notch region (Fig. 10(a)); (2) computation of an effective stress at the fatigue process zone (Fig. 10(b)); (3) generation of a representative stress–strain circuit on the basis of the effective stress and calculation of the total strain energy density (ΔWT) defined by the sum of both the elastic positive and the plastic components (Fig. 10(c)); and (4) prediction of crack initiation life from a fatigue curve defined using smooth specimens subjected to uniaxial strain-controlled conditions (Fig. 10(d)). Regarding the first step, the multiaxial stress states have been reduced to equivalent uniaxial stress states through the von Mises stress. Fig. 11 plots the numerical von Mises stress amplitude (ΔσvM,max/2) at 7
International Journal of Fatigue 130 (2020) 105268
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Fig. 10. Schematic representation of the model employed to estimate the fatigue crack initiation lifetime.
DLM ≤ 0.5, since the curves are perfectly overlapped. On the contrary, as the distance from the crack initiation site increases, the dimensionless stress profiles behave differently, moving away from one another. The next step deals with the generation of a stable hysteresis loop from the effective stress range computed in the previous step, whose main aim is the calculation of thetotal strain energy density at the fatigue process zone. In this study, hysteresis loops have been generated through the Equivalent Strain Energy Density [36] concept, as in previous studies [12,34], using the mechanical properties listed in Table 2. Briefly, the procedure calculates the maximum stress and the maximum strain, i.e. the stress–strain coordinates of Point A; and, then, supported by an auxiliary coordinate system centred on Point A, it calculates the stress range and the strain range, allowing the definition of Point B. More details about these calculations can be found in references [12,23,34,37]. After that, the generated stress–strain loops are used to compute the averaged total strain energy density defined as the sum of the elastic positive ( We+) and the plastic ( Wp) strain density components. Fig. 13 shows several examples of the stress–strain loops computed for the different geometric discontinuities and multiaxial loading scenarios studied here. The amount of plastic deformation depends not only on the geometric discontinuity, but also on the bending-to-torsion ratio and the nominal stress level. This is, therefore, consistent with the conclusions drawn in the previous sections: (i) the introduction of a hole induces higher local stresses, leading to more plastic deformation at the fatigue process zone and, consequently, lower fatigue crack initiation lives; (ii) higher bending-to-torsion ratios result in smaller shear stress levels, reducing the plastic strain energy density, and rising the number of cycles to fatigue crack initiation; and (iii) decreasing values of the nominal stress give rise to longer fatigue crack initiation lives and reduced values of plastic deformation. The final step is devoted to the estimation of fatigue crack initiation
3162
vM,max/2
(MPa)
vM/2 = 10
Hole
4.230 N -0.278 i
Notch 1000
= 1.573 vM/2
316
B/T=2 B/T=1 B/T=2 B/T=1
= 10 4.033 Ni -0.278
95% survival probability (two-sided 95% confidence level) 33 10
1055
1044
1066
Ni (cycles) Fig. 11. Relationship between the maximum von Mises stress amplitude and the number of cycles to fatigue crack initiation.
1.2 Hole Notch
0.8
B/T=2 B/T=1 B/T=2 B/T=1
0.6
vM/
vM,max (-)
1.0
0.4 0.2 0.0
0
0.25
0.5
0.75
1
1.25
1.5
1100
d/DLM (-)
(MPa)
Fig. 12. Dimensionless variation of the local von Mises stress with the distance from the crack initiation site for both the blind hole and the lateral notch configurations under bending-torsion loading.
and the lateral notch configurations under different B/T ratios. For comparison purposes, stresses are divided by the maximum value of the local von Mises stress range (ΔσvM/ΔσvM,max) which occurs at the initiation site; and the distance from the crack initiation site (d) is divided by the critical distance of the LM (d/DLM). Not surprisingly, these dimensionless functions are notch-sensitive; the stress reduction near the crack initiation site is notoriously faster in the lateral notch configurations than that in the blind hole configurations, which explains the higher crack initiation lifetimes found in the former geometries. Moreover, it is interesting to note that the dimensionless stress profiles, near the initiation sites, are not significantly affected by the bending-to-torsion ratios, particularly when d/
700
B/T=2
300
m
A
B/T=1
-100 -500 -900
B/T=2
0
0.5
B
Hole Notch
B/T=1
1
1.5
2
2.5
3
(%) Fig. 13. Stress–strain hysteresis loops generated using the Equivalent Strain Energy Density concept from an effective stress computed via the Theory of Critical Distances. 8
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106 B/T=2 B/T=1
Notch
B/T=2 B/T=1
Hole Notch
5.0 Probability density function
Predicted life (cycles)
105
Hole
6.0
Np = 2Ni
104
103
4.0 3.0 2.0 1.0
Ni = 2Np
102 102
103
104
105
0.0
106
Unsafe
-0.8
Experimental life (cycles)
Safe
-0.4
0.0 Error, EN
0.4
0.8
Fig. 14. Experimental fatigue crack initiation life versus numerical predictions computed using the TSED approach.
Fig. 15. Probability density functions of fatigue life prediction error for the blind hole and the lateral notch configurations.
lifetime. According to the proposed approach (Fig. 10(d)), the number of cycles to fatigue crack initiation is predicted by inserting the values of ΔWT calculated in the previous step into a fatigue master curve developed for the 34CrNiMo6 high-strength steel on the basis of a series of uniaxial low-cycle fatigue tests performed using smooth samples. The fitted function of the TSED approach, calculated in a previous research [27], is given by Eq. (2).
discontinuities causing interacting stress concentrations introduced in the design, fabrication, assembly, or repair stages. Last but not least, it is based on linear elastic simulations and requires a limited number of material properties, therefore is faster and cheaper, which make it very attractive to the industry.
WT
(MJ/m3)
= 2165.37(2Nf )
0.6854+
0.7049
4. Conclusions
(2)
In this paper, fatigue crack initiation behaviour of notched bars made of 34CrNiMo6 high-strength steel with blind transverse holes and lateral U-shaped notches subjected to proportional bending-torsion loading has been studied. Crack initiation lifetime has been predicted using an averaged strain energy density approach. The following conclusions can be drawn:
The fatigue crack initiation lives accounted for through the proposed methodology (Np) are summarised in Table 3. Fig. 14 plots the predicted values (Np) against the experimental observations (Ni) for the various geometric discontinuities and multiaxial loading scenarios studied here. In order to simplify the analysis, conservative and unconservative scatter bands with factors of two (i.e. Nia2Np and Npa2Ni) are drawn. Fatigue life predictions, as can be seen in the Fig. 14, are within the delimited area, which is an important outcome. A close analysis shows a mixed trend with safe and unsafe predictions. Safe predictions have mainly been found for the blind hole, while the unsafe ones have mainly occurred for the lateral notch. The capabilities of the predictive approach can be evaluated, in a more objective basis, via a statistical study of the fatigue life prediction errors (EN). Here, EN is defined by the following equation [17]
EN = log10
Ni Np
• Crack initiation sites, crack initiation angles, and surface crack paths
•
(3)
•
where Ni is the experimental crack initiation life and Np represents the predicted crack initiation life. Briefly, the closer to zero is the mean error, the more accurate is the model; and the smaller is the standard deviation, the more concentrated are the fatigue life prediction errors in relation to the mean value. These features can be graphically inferred from Fig. 15, which plots the probability density functions for both geometric discontinuities. In fact, mean values are close to zero in both cases, but in opposite sides; the mean error of the hole configuration is in the safe region, while the mean error of the lateral notch is in unsafe region. On the other hand, in the former geometric discontinuity, there is a clear concentration of values in relation to the mean error, in contrast to what happens in the latter method whose dispersion is relatively greater. Overall, these outcomes strongly validate the viability of the proposed approaches to predict the crack initiation lifetime in notched bars made of 34CrNiMo6 high-strength steel subjected to bending-torsion loading and to handle sudden changes in the geometric
• • • 9
at the notch surface have been successfully predicted in both notch configurations from linear-elastic finite-element models via the maximum value of the first principal stress at the notch surface, the first principal direction at the initiation site, and the first principal stress field at the notch surface, respectively; Lateral notch configurations under bending-torsion loading are very prone to multi-crack initiation phenomena, which can be explained by the high stress levels acting at an extensive area of the notch. The existence of geometric defects and pores in such regions are likely to make the multi-crack initiation much easier; By contrast, in the blind hole configuration, maximum stress levels are confined to two small sites at the hole border. Thus, such sites become natural places to crack initiation, avoiding the possibility of crack nucleation in other locations; There is a remarkable difference between the crack front shape developments of the blind hole and the lateral notch configurations at the early stage of crack growth due to the hole effect. Nevertheless, as the crack extends, this effect gradually disappears, leading to crack shapes much closer; Fatigue crack initiation life has been successfully predicted through the TSED approach. Overall, experimental and predicted fatigue lives have been very well correlated in the entire range with all points within scatter bands of two. The proposed method has shown good predictive capabilities to deal with different geometric discontinuities, since the mean errors were statistically similar but in different opposite sides, i.e.
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•
predominantly in the safe region for the blind hole and predominantly in opposite region for the lateral notch. The reduced cost associated with the determination of the material constants and the ability to incorporate a linear-elastic framework makes it simpler and cheaper, which is a major asset in the context of mechanical design.
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10