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УДК 539.42
Исследование усталостной прочности образцов с надрезом из стали 40ХМФ13.9 при комнатной и повышенной температуре
Ф. Берто
Падуанский университет, Виченца, 36100, Италия
В работе представлен большой объем экспериментальных данных для образцов стали 40ХМФ13.9. Первая часть статьи посвящена рассмотрению усталостной прочности круглых прутков с надрезом при многоосном нагружении в условиях совместного синхронного и несинхронного растяжения и кручения. Обсуждаются результаты многоосных испытаний, а также результаты, полученные при нагружении образцов с надрезом такой же геометрической формы в условиях чистого растяжения и чистого кручения. Во второй части статьи представлены результаты усталостных испытаний при одноосном растяжении образцов, выполненных из той же марки стали, в условиях регулирования напряжений. Испытания проводились при повышении температуры от комнатной до 650 °C. Статья содержит более 180 новых результатов, полученных при проведении усталостных испытаний в течение более двух лет. Данные испытаний на усталость приведены в значениях амплитуд номинальных напряжений относительно рабочей площади. Кроме того, проведен анализ данных с использованием среднего значения плотности энергии деформации, рассчитанного для некоторого объема прутка серповидной формы, включающего область концентрации напряжений. Обнаружено, что для рассматриваемой марки стали радиус контрольного объема не зависит от режима нагружения.
Ключевые слова: многоосная усталость, высокотемпературная усталость, плотность энергии деформации, детали с надрезом, непропорциональное нагружение
Some recent results on the fatigue strength of notched specimens made of 40CrMoV13.9 at room and high temperature
F. Berto
University of Padova, Vicenza, 36100, Italy
The work summarizes a large bulk of experimental data from specimens made of 40CrMoV13.9 steel. The first part of the paper deals with multiaxial fatigue strength of notched round bars tested under combined tension and torsion loading, both in-phase and out-of-phase. The results from multi-axial tests are discussed together with those obtained under pure tension and pure torsion loading from notched specimens with the same geometry. The second part of the paper summarizes data from uniaxial-tension stress-controlled fatigue tests on specimens made of the same steel. Tests are performed varying temperature, from room temperature up to 650 °C. Altogether more than 180 new fatigue data are summarised in the present work, corresponding to more than two-years of testing programme. All fatigue data are presented first in terms of nominal stress amplitudes referred to the net area and then re-analysed in terms of the mean value of the strain energy density evaluated over a given, crescent shape volume embracing the stress concentration region. For the specific steel, the radius of the control volume is found to be independent of the loading mode.
Keywords: multiaxial fatigue, high temperature fatigue, strain energy density, notched components, nonproportional loading
Nomenclature
cw — parameter taking into account the nominal load ratio in the strain energy density evaluation; E — Young's modulus;
F(2a) — function used in the evaluation of the averaged strain energy density for blunt notches;
H(2a, Rc/p) — function used in the evaluation of the averaged strain energy density for blunt notches;
Q(T) — averaging temperature depending function for blunt notches; k — inverse slope of the fatigue curves; Kf — fatigue strength reduction factor;
Ktbn — theoretical stress concentration factor (on the net area); N — number of cycles to failure; Na — reference number of cycles to failure; R — nominal load ratio;
Rc — radius of the control volume under pure tension;
T<, TT — stress-based scatter index (for 10-90 % probabilities of survival);
Tw — strain energy-based scatter index (for 10-90 % probabilities of survival);
W — averaged value of the strain energy density over the control volume (circular sector); 2a — V-notch opening angle; ® — phase angle; X — biaxiality ratio Ta / <5a; p — notch tip radius;
<a — nominal stress amplitude due to axial loading (referred to the net transverse sectional area specimens);
Ta — nominal shear stress amplitude due to torsion loading (referred to the net transverse sectional area specimens);
<5A — fatigue strength at Na cycles to failure (both amplitudes referred to the net area); v — Poisson's ratio.
© Berto F., 2014
1. Introduction
The work summarizes a large bulk of experimental data from specimens made of 40CrMoV13.9 steel. The paper is twofold. The first part deals with multiaxial fatigue strength of notched round bars tested under combined tension and torsion loading, both in-phase and out-of-phase. The results from multiaxial tests are discussed together with those obtained under pure tension and pure torsion loading from notched specimens with the same geometry. The second part of the paper is focused on uniaxial-tension stress-controlled fatigue tests on specimens made of the same steel. Tests are performed varying temperature, from room temperature up to 650 °C.
With reference to cracked and notched components a number of multiaxial predictive models were proposed in the last years [1-6], often as variants of the critical plane approach formulated at the end of 1980s by Fatemi, Socie and Kurath [7, 8]. An overview on multiaxial fatigue is due to Fatemi and Shamsaei [9] where some approximated models for fatigue life assessments are applied also in the presence of blunt notches. A comparison among different failure criteria, the volume-based strain energy density (SED) approach among the others, was carried out also by Nieslony [5] considering a large bulk of experimental data from notched specimens.
A thermodynamic analysis of cyclic plastic deformation was carried out by Ye et al. [4] to establish an energy transition relation for describing the elastic-plastic stress and strain behavior of the notch-tip material element in bodies subjected to uniaxial and multiaxial cyclic loads. Neuber's rule [10] and Glinka's ESED method [11, 12] represent two particular cases of Ye's formulation.
Recently different multiaxial fatigue criteria based on stress, strain and energy have been proposed [13, 14].
Dealing more specifically with energy-based approaches, worth mentionning is the pioneering work dated 1923 due to Jasper, who used an energy-based parameter to analyze fatigue strength under tension-compression loadings [15]. In the last years well known are the contributions by Ellyin [16, 17] who developed an energy based formulation where a combination of both the plastic and elastic strain work could be efficiently used as a damage parameter in multiaxial fatigue. In a review by Macha and Sonsino on energy based multiaxial fatigue life criteria, it is shown that energy based formulations are good candidates when dealing with loadings presenting complex temporal histories [18].
In a very recent contribution by Saintier et al. [19], a fatigue life assessment method is proposed for proportional and non-proportional multiaxial variable amplitude loadings in the range 104-107 cycles. This method, which is based on the non-local energy, takes its origins from a previous contribution by Palin-Luc and Lasserre [20], later extended to multiaxial loading under constant amplitude by Banvillet
[21]. The new proposal consists of a complete reformulation and extension of the previous energy-based fatigue strength criterion. It includes two major extensions with respect to the past: the former is represented by a fatigue criterion for multiaxial variable amplitude loadings, the latter by an incremental fatigue life assessment method for proportional and non-proportional multiaxial variable amplitude loadings. The criterion is applicable to unnotched components as well as to components with notches with low stress concentration effects. The new method has been successfully applied to a variety of materials [19].
Worth mentioning is also a recent papers by Tanaka [22]. Circumferentially notched bars of austenitic stainless steel SUS316L and carbon steel SGV410 with three different notch-tip radii were fatigued under cyclic torsion without and with static tension. The main result was that in pure cyclic torsion, the fatigue life of circumferentially notched bars of austenitic stainless steel SUS316L was found to be longer than that of smooth bars. This effect was found to be longer with increasing stress concentration under the same amplitude of the nominal shear stress on the net transverse area. The anomalous behavior of the notch-strengthening effect was ascribed to the larger retardation of fatigue crack propagation by crack surface contact for the sharper notches. The superposition of static tension on cyclic torsion of SUS316L reduced the retardation due to the smaller amount of crack surface contact, which gave rise to the well-known notch-weakening of the fatigue strength. A "notch-strengthening" effect was found also in SGV410 [22] under cyclic torsion with and without static tension, as well as in a recent work by the present authors dealing with sharply V-notched specimens made of 39NiCrMo3 steel [23]. Here all data from multiaxial loads were reanalysed in terms of strain energy density averaged over a control volume.
Dealing with multiaxial fatigue strength data from specimens with mild notches, the deviatoric strain energy density at the notch tip has already been used by Park and Nelson
[24] to predict the fatigue behavior under multiaxial stresses. That point-wise approach cannot be applied to sharply notched components due to the high stress gradients surrounding the notch tips. With the aim to overcome this problem, a volume-based strain energy density approach originally proposed for plane problems in the presence of stress singularity due to sharp, zero radius, V-notches and cracks
[25] has been extended to multiaxial fatigue conditions [23, 26-30] as well as to the case of quasi-brittle failure of a variety of brittle and quasi-brittle materials (ceramics [31], graphite [32], polymethylmethacrylate [33], duraluminium [34]) under static loads. In parallel, the link between averaged strain energy density and the notch rounding concept was discussed in detail by Radaj et al. [35, 36] with reference to welded joints under fatigue loading conditions. The fact that under torsion loading plasticity might play an important role also when tensile loading induces effects fully contained within the linear elastic field is deeply discussed
in [37]. The different inverse slope of the fatigue curves under tension and torsion loadings has been motivated by means of some elastoplastic finite element analysis aimed to show the difference effect of plasticity under the above mentioned loading conditions. An extensive review of the volume-based strain energy density approach was presented in [38, 39].
2. Recent works on Cr steels at room temperature: motivation of the present work
The present work first summarizes multiaxial fatigue strength of 40CrMoV13.9 [40]. The tested material is characterized by four different heat treatments able to assure a high strength at room and elevate temperatures. The most important applications are cold or hot rolling of magnesium and aluminium alloys. The service conditions are usually characterized by a complex stress state combined with aggressive in-service conditions. The 40CrMoV13.9 steel combines very good static and fatigue properties with an excellent wear resistance, also at high temperature and in corrosive environments. At the best of author's knowledge, a complete set of data from notch specimens under torsion and combined tension and torsion loadings are not available in the literature for 40CrMoV13.9 whereas other Cr steels have been widely investigated (see, among the others, [41-46]). The phenomenon of fatigue is critical in many sectors, not only dealing with structural steels [47]. It is a common task for engineers engaged in fatigue problems to search suitable solutions for improving the life of the components both acting on the material (see, for example, [48]) and on the geometry of the components.
The absence of data from 40CrMoV13.9 steel under multiaxial fatigue loading has motivated this work which is a revised extension of a previous work [40]. Dealing with multiaxial fatigue loading at room temperature, circumferential V-notched and semicircular notched specimens have been tested under combined tension and torsion loading, both in-phase and out-of-phase. The approach based on the strain energy density averaged on a control volume will be employed to summarise all the data in a single scatterband. A single value of the control volume, evaluated accordingly to the threshold stress intensity range reported in [41, 43] for 40Cr steels characterized by similar heat treatments, will be used in the synthesis, independent of the loading mode.
3. High temperature fatigue behaviour of Cr steels
The second part of the work summarizes high temperature fatigue data from 40CrMoV13.9 steel [49] which is usually employed for hot-rolling processes and other enginnering applications at elevated temperature. Hot-rolling process is in fact increasingly required for higher mechanical performances, fatigue strength and quality of laminated products. Different steels have been employed in a large variety of applications. Dealing with unnotched speci-
mens, the fatigue behavior of different materials at high temperature has been investigated by some researchers [5052].
In [50] an experimental investigation was conducted on 22Cr-20Ni-18Co-Fe alloy at elevated temperature using plain specimens. Fatigue tests were carried out at a constant temperature (871 °C) while the strain ranged from 0.265 to 1.500 %. The fatigue properties and crack growth mechanism of a 2.25Cr-1Mo steel were investigated in [51]. The tests were conducted on unnotched specimens in a temperature range varying between 20 and 500 °C. Dealing with 1.25Cr0.5Mo steel, high-temperature stress controlled tests were carried out at different loading conditions to investigate the fatigue-creep interaction behavior at high temperatures [52]. In [53] ferritic stainless steels, such as 409L, 436L and 429EM STS were tested in a temperature range from 600 to 800 °C, which is typical for exhaust system during the common operations of a vehicle. Fully reversed axial fatigue tests were performed in [54] on smooth specimens of 18Cr-2Mo ferritic stainless steel (type 444) at room temperature, 673 and 773 K in laboratory air, with the aim to investigate the effect of temperature on high cycle fatigue behavior. In [55] the experimental investigation involved three different steels. A commercial martensitic P92 steel was tested at elevated temperature as well as two mar-tensitic steels, reinforced with either 0.007 % of boron (VY2 steel) or 0.2 % of titanium (Ti1 steel) to improve their long-term creep strength. Creep-fatigue tests were carried out with tensile holding periods at 550 °C. A physically-based model was proposed to assess the creep-fatigue lifetime of the considered steels.
In [56] the high temperature low cycle fatigue behavior of ACI HB20-type, a cast ferritic stainless steel, was investigated. Isothermal strain controlled fatigue tests were carried out at 600 and 800 °C.
While results from unnotched materials are not so rare in the past and recent literature as just discussed, only few systematic investigations have been performed on notched specimens under fatigue loading at high temperature at medium-high cycle fatigue [57-60].
In [57, 58], the notched fatigue strength of the nickelbase superalloy Inconel 718 was investigated under rotating bending loading at room temperature and 500 °C in air. The linear notch mechanics was employed to assess the fatigue strength at elevated temperature being for that material the small-scale yielding conditions satisfied also at elevated temperature.
The effect of notch types and stress concentration factors on low cycle fatigue life and cracking of the DZ125 directionally solidified superalloy was experimentally investigated in [59]. Single-edge notched specimens with V-and U-type geometries were tested at 850 °C with a stress ratio R = 0.1. The notch radius varied from 0.2 to 3.6 mm while the notch opening angle varied from 0° to 120°. The results revealed that the fatigue strength decreased with the
Table 1
Mechanical properties of 40CrMoV13.9
Ultimate tensile strength, MPa 1355
Yield stress, MPa 1127
Elongation to fracture, % 15.2
Brinell hardness 393-415
elastic stress concentration factor Kt, increasing from 1.76 to 4.35. The main conclusion of the paper was that Kt can be considered as a key parameter controlling the notch fatigue at least when the absolute dimensions of the tested notched specimens are similar.
In a recent contribution the present authors have summarized the results from uniaxial tension stress-controlled fatigue tests performed at 650 °C on Cu-Co-Be specimens [60]. Two geometries have been considered: hourglass-shaped specimens and plates weakened by a central hole. All fatigue data from unnotched and notched specimens have been reanalyzed there in terms of the mean value of the strain energy density evaluated, for the notched specimens, over a finite size control volume surrounding the highly stressed zone at the hole edge. This has permitted to summarize all fatigue data in a quite narrow scatter band.
As for multiaxial fatigue loading at the best of author's knowledge, a complete set of data from unnotched and notched specimens at high temperature is not available in the literature for 40CrMoV13.9. With the aim to fill this lack, in the second part of the present paper this topic is investigated reviewing a recent contribution [49] which experimentally investigated the behavior of this steel at different temperatures ranging from room temperature up to 650 °C. The tests were performed under uniaxial tension and stress-controlled conditions. A final synthesis of the high temperature results together with previous data from multiaxial tests (at room temperature) on the same material [40] is carried out by means of the strain energy density approach, as recently made for Cu-Co-Be alloys tested at elevated temperature [60].
4. Strain energy density approach
Dealing with strain energy density approach a large amount of papers has been published in the last years by Lazzarin and co-authors. Some papers are devoted to the application of strain energy density under static loadings [61-84] and different combinations of the applied load. The fatigue assessment of notched components and welded struc-
tures has been carried out in [85-93] while some interesting reviews related to these topics can be found in [94-96]. The parallelism between strain energy density approach and the fictitious notch rounding approach has been discussed in [97-104].
The extension of strain energy density to 3D models and the capability of the approach to automatically take into account the effects due to a finite thickness are discussed in [105-122]. Some recent applications deal also with higher order terms [123-125] and periodic notches [126-128].
5. Multiaxial tests at room temperature
5.1. Material properties at room temperature and specimens geometry for multiaxial tests
Static tensile tests were carried out to evaluate the elastic and the strength properties of 40CrMoV13.9 steel, see Table 1. The chemical composition of the material is reported in Table 2. The heat treatment schedules that characterize the tested material are shown in Table 3. The material was first quenched at 920 °C and subsequently tempered at about 580 °C twice. A final stress relieving treatment at 570 °C was carried out. The final microstructure was characterized by a high strength bainitic martensitic structure.
The geometries of the specimens tested in the present investigation are shown in Fig. 1. The axis-symmetric V-notched specimens were characterized by a depth d = = 4 mm, and an opening angle equal to 90°. Constant was also the notch root radius, 1 mm. The semicircular specimens presented instead a notch root radius of 4 mm
(Fig. 1, b).
All tests were performed under load control on a MTS 809 servohydraulic biaxial testing device (±100 kN, ±1100 Nm, ±75mm/±55°). The load was measured by a MTS cell with ±0.5 % error at full scale. The specimens were tested under pure tension, pure torsion and multiaxial tension-torsion loading with different biaxiality ratios (A = Ta/ = 0.6 and 1.0). In particular, 10 series of fatigue test were conducted, according to the following subdivision:
- two series of tests on notched specimens (V and semicircular notches) under pure torsion fatigue loading (nominal load ratio R = -1);
- one series of tests on V-notched specimens under pure tension fatigue loading (R = -1);
- two series of tests on V-notched specimens under combined tension-torsion fatigue loading, with biaxiality ratio A = 1, load ratio R = -1, load phase angle O = 0° and 90°;
Table 2
Chemical composition wt. %, balance Fe
C Mn Si S P Cr Ni Mo V Al W
0.380 0.500 0.270 0.006 0.003 3.050 0.240 1.040 0.240 0.013 0.005
Heat treatment schedules
Table 3
Heat treatment Heating ratio, °C/h Temperature, °C Holding, h Cooling
1 Quenching 100 920° ± 10° 3 Water
2 Tempering 1 100 580° ± 10° 5 Air
3 Tempering 2 100 590° ± 10° 5 Air
4 Stress relieving 100 570° ± 10° 3 Air
- two series of tests on V-notched specimens under combined tension-torsion fatigue loading, under constant biaxiality ratio X = 0.6, load ratio R = -1, load phase angle O = 0° and 90°;
- two series of tests on semicircular specimens under combined tension-torsion fatigue loading, with biaxiality ratio X = 1, load ratio R = -1, load phase angle O = 0° and 90°;
- one series of tests on semicircular specimens under combined tension-torsion fatigue loading, with biaxiality ratio X = 0.6, load ratio R = -1, load phase angle O = 0°.
5.2. Results from the fatigue tests
Before being tested, all specimens have been polished in order to both eliminate surface scratches or machining marks and to make the observation of the fatigue crack path easier. As stated above, fatigue tests have been carried out on a MTS 809 servohydraulic biaxial machine with a 100 kN axial load cell and a torsion load cell of 1100 Nm. All tests have been performed under load control, with a frequency
Fig. 1. Geometry of V-notches and semicircular notches (a) and details of the notch tip (b)
ranging from 1 to 10 Hz, as a function of the geometry and load level. At the end of the fatigue tests, the notch root and the fracture surfaces were examined using optical and electronic microscopy.
The results of statistical analyses carried out by assuming a log-normal distribution are summarised in Table 4. In particular, it is summarised the mean value of the nominal stress amplitudes at different number of cycles, the inverse slope k of the Wohler curves and the scatter index T, which gives the width of the scatterband related to 10-90 % probabilities of survival. All failures from 10 to 5 -106 have been processed in the statistical analysis whereas run-outs were excluded.
Figure 2 depicts fatigue data from V-notched specimens tested under tension, torsion and combined tension and torsion with different load phases (0° and 90°). In this case the biaxiality ratio was equal to 1. It is evident from the figure that the out of phase loading causes a slight but visible detrimental effect on the fatigue life of the V-notched specimens, almost independent of the number of cycles.
Figure 3 summarizes fatigue data from V-notched specimens tested under multiaxial fatigue at a biaxiality ratio X = 0.6, both in-phase and out-of-phase. The two mean curves are compared with the curve from pure tension. It is possible to observe that multiaxial loading decreases the fatigue life with respect to the tensile loading. Also for X = 0.6 the out-of-phase loading is more damaging.
Figure 4 reports the fatigue curves related to semicircular notches. In particular it summarises the curves from pure torsion, multiaxial loading with X = 1 (in phase and out-of-phase) and multiaxial loading with X = 0.6 (only in phase). In this case it is interesting to observe that for this particular notch radius and specimen geometry the out-of-phase loading is slightly beneficial with respect to in-phase loading at high cycle regime while the fatigue strength is the same at low cycle regime. This means that the non-proportional loading effect may be different not only for different materials but also for the same material by moving from a sharply notched specimen to a specimen with a large notch root radius.
A distinction between sharp and blunt notches has been drawn [129] in Atzori-Lazzarin's diagram mainly on the basis of the product of the squared stress concentration factor and a characteristic length of the material, eventually modified by a shape factor. Notch sensitivity and defect
Table 4
Results from fatigue tests. Mean values, Ps = 50 %. Stresses referred to the net area
Series Load Number of k Ta or Tt aa or Ta, MPa
specimens n 106 2-106 5 • 106
1 Torsion, R = -1 16 T 13.33 1.144 281.16 266.91 249.18
2 Multiaxial, R = -1, O = 0o, X = 1.0 14 a 9.39 1.176 182.6 169.91 153.84
3 Multiaxial, R = -1, O = 90o, X = 1.0 13 a 7.67 1.428 145.83 133.23 118.23
4 Tension, R = -1 12 a 8.42 1.335 253.69 233.64 209.55
5 Multiaxial, R = -1, O = 0o, X = 0.6 12 a T 9.9 1.202 214.61 128.77 200.10 120.06 182.41 109.45
6 Multiaxial, R = -1, O = 90o, X = 0.6 13 a T 7.63 1.485 187.45 112.47 171.17 102.7 151.8 91.08
7 Torsion, R = -1, semicircular 11 T 14.27 1.160 367.17 349.76 328.01
8 Multiaxial, R = -1, O = 0o, X = 1.0, semicircular 15 a 7.68 1.268 239.57 218.9 194.28
9 Multiaxial, R = -1, O = 90o, X = 1.0, semicircular 17 a 10.79 1.260 263.82 247.4 227.26
10 Multiaxial, R = -1, O = 0o, X = 0.6, semicircular 12 a T 11.45 1.307 332.64 199.58 313.10 187.86 289.02 173.41
sensitivity were seen as two sides of the same medal. The local strain energy density allows a natural transition between the fatigue strength of the parent material and that of the notched components [130], independent on the notch acuity, keeping constant Rc.
5.3. Fracture surfaces of V-notches
The failure surface of a V-notched specimen broken after 970000 cycles under torsion is shown in Fig. 5, a. The outer diameter of the fracture surface is largely flat, suggesting a mode III fracture mechanism. The classical mode I "factory roof" morphology is also evident in large areas of the centre of the fracture surface together with evidence of abrasion mainly on the bottom hand side of the image.
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Fig. 3. Data from V-notches under multiaxial loading, comparison with pure tension, 2a = 90°, p = 1 mm
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Fig. 2. Data from V-notches under pure tension, torsion and multiaxial loading, 2a = 90°, p = 1 mm
Fig. 4. Data from semicircular notches under pure torsion and multiaxial loading, p = 4 mm
Fig. 5. Fracture surfaces from V-notched specimens under pure torsion (a) and pure tension (b). N = 970000 (a), 84000 (b); Ta= 280 (a), 340 MPa (b)
Figure 5, b shows the image of a specimen broken at 83000 cycles under tension loading. The surface is flat, some small signs of abrasion are present and it is well distinguishable the zone corresponding to the final static fai-
lure which generally tends to decrease in size at high cycle regime.
Generally for V-notches, in presence of torsion loading, the crack nucleated at 30 % of the fatigue life and all the
Fig. 6. Fracture surfaces from V-notched specimens under in-phase multiaxial loading. Multiaxial fatigue, N = 456750 (a) and 151000 (b), oa= 220 MPa, X= 1, O = 0°
Fig. 7. Fracture surfaces from V-notched specimens under multiaxial fatigue, X = 0.6, O = 0°
rest of the life was spent in crack propagation. Under pure tension approximately 80 % of the fatigue life was spent in crack nucleation and only the final part of the life was spent in crack propagation. By avoiding to interrupt the tests, the presence of the crack was detected by means of an optical microscope used to monitor the notch tip state during the fatigue life of the specimen. Generally a limited but distinguishable quantity of debris and powder was emanated from the notch tip when a visible crack was detected and started to propagate significantly.
Fatigue damage is generally described as the nucleation and growth of cracks to final failure, although the differentiation of two stages is "qualitatively distinguishable but quantitatively ambiguous", as underlined by Jiang and Feng [131]. No demarcation line being drawn between fatigue crack initiation and early microcrack propagation inside the
control volume, both phases are thought of as strictly dependent on the averaged strain energy density initially present on the uncracked specimens.
Fracture surfaces of some specimens tested under multiaxial conditions are shown in Fig. 6. All the specimens were from tests at the same phase angle, O = 0, and at the same biaxiality ratios: X = 1. A general comment on the two figures is the presence of the "factory roof" morphology extending from the initial notch root. This changes to a single, inclined fracture surface as one of the mode I crack becomes dominant. The detail differences between the inphase and out-phase loading cases are discussed below.
By considering in-phase loading cases shown in Fig. 6, some observations can be made:
- At medium-high cycle regime, few cracks nucleate, and the crack propagation phase takes a significant fraction of the fatigue life of the specimen. Under low-cycle fatigue, the number of initiated cracks generally increases, and the propagation phase is much reduced (see Fig. 7).
- The inclination of the planes of crack propagation is less than the 45° that occurs under pure torsional loading. This is due to the change in the angle of the opening displacements from 45° to 0° as the level of axial load increases.
- The irregular surface corresponding to the final static failure is not perpendicular with the specimen axis, indicating a nonuniform crack propagation front from the outside toward the centre of the specimen.
Fig. 8. Fracture surfaces from V-notched specimens under multiaxial fatigue, X = 0.6 (a) and 1.0 (b), O = 90°, N = 94698 (a) and 27 500 (b), oa= 280 (a) and 260 MPa (b)
Fig. 9. Typical crack propagating from a semicircular notch under multiaxial loading under in-phase (a) and out-of-phase loading (b)
The fracture surface morphology of specimens tested was strongly influenced by the phase angle. Signs of microabrasions may be observed on all fracture surfaces and the extent to which the rubbing occurred appears to depend on phase angle. Most visible abrasion appears to occur at X = = 0.6 or X = 1 and O = 90° resulting in a smooth featureless fracture topography. Small spherical oxide debris, evidence of wear, can be observed in the scanning electron micrographs (Fig. 8).
The main features were as follows:
- The fracture surfaces are generally flatter than those from in-phase tests. This may be due to the increased abrasion observed in these tests.
- In the zone of crack initiation microabrasions due to the interference between the mating surfaces are well visible. During the tests, in fact, the two surfaces of the cracks slightly scrape each other.
5.4. Fracture surfaces of semicircular notches
Semicircular notches were characterized by different fracture surfaces with respect to V-notches. For this kind of notches the main part of the fatigue life was spent in the crack initiation and only a small part, approximately equal to 10 %, was spent in the crack propagation with visible debris emanated by the notch tip. Some differences were observed under in-phase and out-of-phase loading. Figure 9 shows the final crack propagating at notch tip under inphase and out-of-phase loading: under in-phase loading the crack branching is visible (Fig. 9, a) while it is absent under out-of-phase loading (Fig. 9, b). This effect does not influence substantially the fracture surfaces which generally do not present clear planes of crack propagation (Fig. 10). They appear only slightly more flat under out-of-phase loading (Fig. 10, b).
5.5. A synthesis in terms of linear elastic strain energy density averaged over a control volume
The averaged strain energy density criterion states that brittle failure occurs when the mean value of the strain en-
ergy density over a given control volume is equal to a critical value Wcr.
This critical value varies from material to material but does not depend on the notch geometry and sharpness and the criterion can be applied to sharp notches (also with p = 0) as well as to blunt notches. In Beltrami's original formulation Wcr does not depend on the loading mode.
In the particular case of 40CrMoV13.9 steel the radius of the control volume Rc has been estimated here by using the fatigue properties of a quenched and tempered 40Cr steel reported in [41, 43] which presented a chemical composition, a static strength analogous to those of the material considered herein. Also the heat treatments were almost coincident. In particular the threshold value of stress intensity factor range (AKth = 6 MPa • m0-5 at R = 0) from pure
Fig. 10. Typical fracture surface from a semicircular notch under multiaxial loading in-phase (a) and out-of-phase (b)
Fig. 11. Critical volume for V-shaped notches (a) and semicircular notches (b)
tension fatigue and the fatigue limit range of plain material, Aga at R = 0 (430 MPa), taken from [41] have been used. Moreover in [51] the trend of the threshold stress intensity factor and of the ultimate tensile strength is plotted as a function of the tempering treatment temperature. For tempering temperature higher than 500 °C, AKth reaches almost a plateau value equal to 6 MPa • m05.
A convenient expression for the critical radius is as follows [25]:
V^eT AK th'2
R =
AG,
(1)
where eT is equal 0.1345 [25]. By using Eq. (1) in combination with the material properties reported above, the estimated control radius is equal to 0.05 mm. A characteristic length equal to about 0.05 was found by Li et al. [41] for the 12CrNi3 steel, characterized by a tempering temperature equal to 600 °C and by a threshold stress intensity factor range equal to 6.4 MPa • m05.
Dealing with V-shaped notches the control volume is as shown in Fig. 11, a; the volume assumes a crescent shape and it is centred at the distance r0 from the notch tip. The distance r0 depends on the notch radius p and the notch opening angle 2 a according to the simple Eqs. (2), (3) derived from Neuber's conformal mapping: 2(n-a)
n
q -1
(2)
(3)
For semicircular notches under mode I and III loadings, the volume assumes the crescent shape shown in Fig. 11, b, where Rc is the depth measured along the notch bisector line. The outer radius of the crescent shape is equal to Rc +p/2, being p/2 the distance between the notch tip and the origin of the local coordinate system.
The value of the strain energy density averaged over the control volume has been calculated numerically by using the FE code ANSYS 12.0® both for V-notches and semicircular notches.
Being in this case the nominal load ratio constant and equal to R = -1 the synthesis based on the strain energy
density has been carried out directly with the values obtained by the numerical simulations. Without entering in the details, it is worth mentioning that the effect of different nominal load ratios can be quantified by introducing the parameter cw, as made in previous works [26]. As applied here the averaged strain energy density approach is not sensitive to the variation of the phase angle. To fully take into account the effect of the out-of-phase loading, which varies from material to material, the radius of the control volume should be updated by using a specific information from cracked or sharply V-notched specimens tested under out-of-plane loading conditions. Two parameters are not sufficient, a third parameter is necessary.
Figure 12 shows the final synthesis based on the averaged strain energy density. Since all data come from specimens tested under the same nominal load ratio, R = -1, the coefficient cw = 0.5 is not introduced herein, cw should be taken into account when the comparison would involve different steels tested under different nominal load ratios.
The scatter index Tw, related to the two curves with probabilities of survival Ps = 10 and 90 %, is 1.96, to be compared with the variation of the strain energy density range, from about 2.5 to about 0.75 MJ/m3, Tw = 1.96 becomes equal to 1.40 when reconverted to an equivalent local stress range with probabilities of survival Ps = 10 and 90 %. The inverse slope of the scatterband is equal to 5.0.
At the light of the present results and application of the strain energy density criterion it is clear that a single value of the control volume, independent of the loading modes, is sufficient to characterize the multiaxial fatigue behavior of 40CrMoV13.9 steel. This result was not known a priori and usually depends on the tested material. By comparing Fig. 12 with Figs. 2-4, based on nominal stresses at the net
Fig. 12. Synthesis by means of local strain energy density (SED) of multiaxial series data. Steel 40CrMoV13.9, Rc = 0.05 mm, 120 new data, ^ — tension, R = -1; o — torsion, R = -1; o — multiaxial, R = -1, X = 1.0, O = 0°; a — multiaxial, R = -1, X = 1.0, O = 90°; □ — multiaxial, R = -1, X = 0.6, O = 0°; + — multiaxial, R = -1, X = 0.6, O = 90°; ■ — semicircular, torsion, R = = -1; ♦ — semicircular, multiaxial, R = -1, X = 1.0, O = 0°; * — semicircular, multiaxial, R = -1, X = 1.0, O = 90°; ♦ — semicircular, multiaxial, R = -1, X = 0.6, O = 0°
Table 5
Mechanical properties of 40CrMoV13.9 at high temperature
Room temperature Temperature 650 0C
Young's modulus E, GPa 206 135
Ultimate tensile strength ar, MPa 1355 610
Yield stress ay, MPa 1127 520
Elongation to fracture A, % 15.2 23.5
HRC 52 35
area, it is evident the unifying capacity of the strain energy density which is able to summarize all the data in a single scatterband independent of the loading mode and notch geometry.
Fig. 13. Fatigue test equipment for high temperature tests
6. Tensile tests at high temperature
6.1. Material properties at high temperature
As stated before, the material investigated in the present study is usually employed for hot-rolling of metals where the material is usually subjected to a combination of mechanical and thermal loading conditions. Some static tensile tests on a standard specimen were carried out to evaluate the elastic and strength properties of 40CrMoV 13.9 steel at 650 °C: Young's modulus E is equal to 135 GPa, Gy is equal to 520 MPa and Gr to 610 MPa. The data-sheet reports the following mechanical properties at room temperature (25 °C): elastic modulus E is equal to 206 GPa, tensile strength of about 1300 MPa and a yield strength of 1100 MPa with a percent elongation of 15 %. The properties (at room temperature and 650 °C) are also summarized in Table 5.
6.2. Testing equipment for high temperature fatigue
The fatigue tests are conducted on a servohydraulic MTS 810 test system with a load cell capacity of 250 kN. The system is provided with a MTS Model 653 High Temperature Furnace, as shown in Fig. 13. It is ideal for a wide variety of high-temperature tests, including tension, compression, bending and fatigue testing of different materials, metallic and not. It has a center-split design that enables easy access to both grippers and specimens. The furnace includes the MTS digital PID Temperature Control System and is configured for two heating zones which can be independently temperature-controlled through high precision thermocouples. The zone at constant temperature is 80 mm length. The heating elements are made of silicon carbide. An insulation plate situated between the upper and lower elements helps to ensure reliable zone separation and pre-cut insulation reduces heat loss. This furnace is particularly well-suited for applications that require a lower thermal gradient on a fatigue (or tensile) specimen. The nominal temperature for this furnace ranges from 100 to 1400 °C
and the control point stability is about ±1 °C. Since the wedge grips are affected by the heat of the furnace, they are equipped with a cooling system that keeps the temperature low enough in order to not provoke any damage to the testinstruments.
6.3. Procedure for fatigue testing at elevate temperature
The concerned stress-controlled fatigue tests were carried out at temperature values ranging from 20 to 650 °C: the specimen was heated to reach the desired temperature and after a short waiting period (20 min) necessary to assure a uniform temperature, the test was started. The temperature was maintained constant until specimen failures thank to the PID temperature control system. The uniaxial tensile fatigue tests were carried out over a range of cyclic stresses at the constant frequency of 5 Hz; the nominal load ratio R was kept constant and equal to 0.
Two specimen geometries were considered:
- hourglass shaped (smooth) specimens with a theoretical stress concentration factor close to 1.0 (Fig. 14, a),
1. 250 .1
15
Thickness = 5 [p = 1045 5
300 b
90°
Fig. 14. Hour-glass shaped specimen geometry, dimensions in mm (a), notched specimen geometry, dimensions in mm (b)
Fig. 15. High temperature tests on plain (a) and notched specimens (b)
- plates weakened by lateral symmetric V-notches, with a net cross section of 20 mm x 5 mm and a total length of 300 mm (Fig. 14, b). The notches were characterized by a depth a equal to 5 mm, an opening angle 2a equal to 90o and a notch tip radius p = 1 mm. This geometry results in a theoretical stress concentration factor Kth,n = 3.84 (on the net transverse sectional area).
The specimens were designed to avoid an increase of temperature near the grippers and the length of 300 mm, which is higher than the length usually adopted at room temperature, was chosen for this reason. Figure 15 shows an image of the tested specimens.
As stated above the stress-controlled fatigue tests were carried out at different temperatures. More precisely, the hour-glass shaped specimens were tested at room temperature, 360 and 650 oC; the V-notched specimens were tested at room temperature, 360, 500 and 650 oc. Overall, eight fatigue curves were obtained by testing more than 60 specimens.
6.4. Results from high temperature fatigue
The fatigue data were statistically reanalysed by using a log-normal distribution and are plotted in term of nominal stress ranges (referred to the net area) in Figs. 16 and 17. More specifically, Figure 16 shows the fatigue data of the hourglass specimens, the W ohler curve (mean curve, Ps = 50 %), the Haibach scatter band referred to 10 and
Fig. 16. Data from hourglass shaped specimens at different temperatures. R = 0
90 % probabilities of survival (for a confidence level equal to 95 %) and the inverse slope k of the curves. Data from specimens tested at room temperature and at T = 360 °C are found to belong to the same scatterband, with a value of the scatter index quite low, TG = 1.29. The scatter of the specimens tested at T = 650 °C, instead, is higher being TG = 2.00, which show also a strong decrease of the fatigue strength combined with a strong variation of the slope. A vertical line is drawn in correspondence of one million cycles where the mean values of the stress range are given to make the comparison easier. At 106 cycles the stress range is equal to 675.14 MPa when T< 360 °C, while it is equal to 95.23 MPa at 650 °C.
Fatigue data of the specimens weakened by lateral V-notches are shown in Fig. 17 at different temperatures. The run-out specimens (marked by tilted arrow) were excluded from the statistical analysis. It is evident that up to 500 °C there are no differences with respect to the room temperature, whereas a substantial decrease of fatigue strength can be observed at 650 °C. The scatter-band related to the specimens tested at T = 650 °C is compared with that summarising data obtained for T< 500 °C. At one million cycle, the value of the stress referred to a probability of survival of 50 % decreases from 213.12 to 74.32 MPa. The variation of the slope is also strong, it decreases from k = 5.14 to k = 2.91. Conversely, the values of the scatter index are comparable.
Fig. 17. Data from V-notched specimens at different temperatures. R = 0
Table 6
Results from fatigue tests at different temperatures. Stresses referred to the net area
Specimen, T k T °max> MPa (106 cycles)
10 % 50 % 90 %
HG, amb. to 360 °C 7.28 1.29 766.27 675.14 595.54
HG, 650 °C 2.48 2.00 134.76 95.23 67.29
V, amb. to 500 °C 5.14 1.48 259.66 213.12 174.92
V, 650 °C 2.91 1.63 94.83 74.32 58.25
HG stands for hour-glass shaped specimens while V stands for V-notched ones
For the sake of completeness, the results are also summarized in details in Table 6.
The drastic decrease of the fatigue properties observed at 650 °C is linked to the specific heat treatments made on the material. The maximum temperature in the tempering treatment was equal to 590 °C and the last stress relieving treatment was carried out at 570 °C. Experimental data clearly document that under long time exposure at temperatures higher than 570-590 °C all beneficial effects due to the heat treatments are lost.
6.5. Fracture surfaces by scanning electron microscopy
The final geometrical configuration of some broken specimens was analysed together to their fracture surfaces. Figure 18 compares plain specimens tested at room and elevated temperature (650 °C), with comparable fatigue lives. It is evident that the specimen tested at high temperature shows, as expected, a more ductile fracture if compared with the specimen tested at room temperature. It is also evident the visible necking on the net sectional area before the final failure. The fracture surfaces were also found to be different when analyzed by means of the scanning electron microscope (SEM) at the same magnification value (see Fig. 19). At high temperature some signs of transgra-nular fracture can be detected, as well as the formation of
Fig. 19. Comparison between the fracture surface of plain specimen tested at 650 °C (a) and room temperature (b), considering the same magnification value, Ag = 140 (a) and 900 MPa (b), N = 284049 (a) and 155000 (b)
numerous microvoids (Fig. 19, a). These microvoids are large if compared to that of room temperature fracture surfaces (Fig. 19, b). At last, the fracture surface of plain specimens tested at high temperature is relatively clean, without any oxidation of the surface occurred during the tests.
Dealing with the fracture surfaces of V-notches, no significant plastic deformation was noted at failure in the vicinity of the notch tip at room temperature and at elevated temperature. Figure 20 shows two specimens failed approximately at the same number of cycles. The fracture surfaces of the V-notched specimens were found very similar at different temperatures, almost flat in the crack initiation zone.
Fig. 18. Fracture surface of plain specimen tested at 650 °C (a) and room temperature (b), Ag = 140 (a) and 900 MPa (b), N= 284049 (a) and 155000 (b)
Fig. 20. Fracture surfaces of the V-notched specimens tested at room temperature (a) and high temperature (b), Ag = 250 (a) and 100 MPa (b), N = 295000 (a) and 304000 (b)
Fig. 21. Details of the fracture surface of a notched specimen tested at room temperature, Ao = 250 MPa, N = 295000
This is most due to the presence of the V-notch, that leads to a brittle behavior of the material also at high temperatures.
This observation was also supported by the scanning electron microscope micrographs (Figs. 21 and 22). At room temperature and at T = 650 oc, the fracture surfaces were found to be analogous near the notch tip where the crack initiation took place. No evident sign of microvoids formation can be seen, nor evident signs of oxidation. At 650 oc, two clear distinguishable zones of the fracture surfaces can be identified (see Fig. 22): the bottom zone corresponds to the crack initiation and early propagation zone, the upper one corresponds to the final static failure. In this second zone, far away from the notch tip, small microvoids similar
(but smaller in diameter) to those detected on plain specimens (tested at high temperature) were observed.
6.6. A synthesis in terms of linear elastic strain energy density averaged over a control volume: a close form expression
Dealing with blunt notches under fatigue loading the following expression can be used to evaluate the strain energy density range [38, 39]:
AW = cw F (2a)H
( R ') KL Aon
2a,:
(4)
Fig. 22. Fracture surface of a notched specimen tested at 650 oc at different magnification values, Ao = 100 MPa, N = 304000
Here Aon is the stress range, Kth n is the theoretical stress concentration factor (both referred to the net sectional area), E is the Young's modulus, F(2a) depends on the notch opening angle and is equal to 0.705 for 2a = 90o. Finally H depends both on the notch angle and the critical radius-notch tip radius ratio.
In order to unify in a common diagram the fatigue results from both R = -1 and R = 0, the weighting parameter cw has to be applied, according to the simple rule reported in [26]. This rule states, in particular, cw = 1.0 for R = 0 and cw = 0.5 for R = -1. As a result of the reduction of the effective strain energy density, the fatigue strength range for R = -1 should theoretically increase according to a factor V2 with respect to the R = 0 case.
6.7. Synthesis based on strain energy density of the new high temperature data from 40CrMoV13.9
The new high temperature data from unnotched and notched specimens made of 40CrMoV13.9 are summarized in this section by using the strain energy density approach. On the basis of the experimental evidences of the present work, the synthesis in terms of strain energy density has been carried out up to 500 oc considering the same critical radius used in [40] for multiaxial fatigue data. In fact, as visible from Figs. 16 and 17, no reduction in the fatigue strength has been detected until 500 oc both for unnotched and notched specimens.
In the medium and high cycle fatigue regime the critical strain energy density range for unnotched specimens can be simply evaluated by using the following expression:
Fig. 23. Synthesis by means of local strain energy density of new fatigue data from plain and V-notched specimens between room temperature and 500 oc. R = 0; ♦ — plain, Troom; ° — plain, T = = 360 oc; ■ — V-notched, Troom; a — V-notched, T = 360 oc; ♦ — V-notched, T = 500 oc
AW = AG 2. 2 E n
(5)
In Eq. (5) Aon is the nominal stress range referred to the net sectional area. As already said, the weighting parameter cw has to be applied to take into account different values of the nominal load ratio. Being the actual tests referred to R = 0, cw is equal to 1.0. Since Eq. (5) is applied at different temperatures, the Young's modulus has to be updated as a function of the temperature.
The Young's modulus E is equal to 206 GPa at room temperature and 135 GPa at 650 oc. In the intermediate cases a linear trend has been assumed. For a temperature of 360 oc the Young's modulus E results to be 165 GPa and at 500 oc it is equal to 150 GPa.
For the notched specimens Eq. (4) can be directly applied. For the specific case of 2a = 90o and Rc/ p = 0.05, parameters F and H are equal to 0.7049 and 0.5627, respectively [38, 39]. The stress concentration factor referred to the net area is equal to 3.84.
By using Eqs. (4), (5) the new data from the tests carried out at room temperature up to 500 oc can be summarized in Fig. 23 in a single narrow scatterband, characterized by an inverse slope k equal to 5.31 and a scatter index Tw equal to 2.25. Involving now the room temperature data from multiaxial tests on the same material [40] (Fig. 12), it is possible to obtain a single scatterband, as shown in Fig. 24. It must be underlined that the multiaxial tests has been carried out at R = -1, and for this reason the weighting parameter cw = 0.5 introduced previously in the Sect. 4.1, has to be employed in order to convert the data in terms of effective strain energy density range. Owing to the strain energy density approach it has been possible to summarise in a single scatterband all the fatigue data, independently of the specimen geometry, of the loading condition, and of the temperature, up to 500 oc.
Dealing with data carried out at 650 oc, the fatigue strength of unnotched and notched specimens has been found strongly lower than the corresponding data from tests
carried out at T < 500 oc. For this specific temperature (T= = 650 oc), which is important in practical industrial applications, in particular for hot rolling of aluminum alloys, an empirical formula has been proposed for notched specimens by modifying Eq. (4). This allows us to take into account the notch sensitivity of this material at temperatures higher than 500 oc [49]:
AW = ow Q (T) L
f_ fo
F (2a) H
2a R P
Y th,n AG 2
(6)
where Q(T) is the notch sensitivity function at a specific temperature T (this function has to be set (as a function of the temperature T) by equating at high cycle fatigue (106 cycles) the strain energy density value from plain specimens and those from notched specimens), f is the test frequency of notched specimens at high temperature, and f0 is the test frequency of unnotched specimens at the same temperature, L is a function related to the sensitivity of the material to the load frequency and depends on the ratio f / f0. Function L is required to be equal to 1.0 if f = f0, a condition respected in all tests of the present analysis. The critical radius Rc is kept constant and equal to that obtained at room temperature (Rc = 0.05 mm). Dealing with our specific case Q(T= 650 oc) = 0.18, Eq. (6) can be rewritten by substituting the numerical value of each function [49]:
AW = 1.0 • 0.18 • 1.0 • 0.7049 • 0.5627-
KIn AG2
= 0.07139
K t2h,n AG 2
(7)
where Kt2hn = 3.84.
By considering Eq. (7) applied to notched specimens and Eq. (5) applied to plain specimens, the strain energy density master curve for 40CrMoV13.9 at 650 °C has been obtained. The fatigue data from tests at 650 °C are plotted in terms of averaged strain energy density range over a control volume in Fig. 25, considering the critical radius previ-
Fig. 24. Synthesis by means of local strain energy density of new fatigue data up to 500 °C and room temperature multiaxial fatigue data. Rc = 0.05 mm; o — multiaxial data, Troom, R = -1; a — plain, Troom, R = 0; a — plain, T = 360 °C, R = 0; ■ — V-notched, Troom, R = 0; o — V-notched, T = 360 °C, R = 0; ♦ — V-notched, T = 500 °C, R = 0
Fig. 25. Synthesis by means of local strain energy density of the new fatigue data at 650 °C; plain and notched specimens are summarised in the same scatterband. R = 0, Rc = 0.05 mm, 0(0 = 0.18
ously derived at room temperature. It is possible to observe that the scatterband is quite narrow, with the scatter index being Tw = 2.56 that results in Tg = 1.60 in terms of equivalent local stress range. The inverse slope of the scatterband is equal to 1.43. Owing to the strain energy density approach it is possible to summarise in a single scatterband all the fatigue data at the same temperature, independently of the specimen geometry.
7. Conclusions
Fatigue tests and metallographic analysis were carried out on 40CrMoV 13.9 steel under multiaxial loading at room temperature and under uniaxial loading at high temperature. The work has been motivated by the fact that, at the best of author's knowledge, only a limited number of works dealing with this steel are available in the literature and no results seem to be available from notched components made of this steel tested under multiaxial loading or at elevated temperatures.
The main results are summarised as follows:
1. A large bulk of results from multiaxial tests from V-notched and semicircular notched specimens made of 40CrMoV13.9 steel are discussed together with those obtained under pure tension and pure torsion loading from notched specimens with the same geometry.
2. All fatigue data are presented first in terms of nominal stress amplitudes and then reanalysed in terms of the mean value of the strain energy density evaluated over a finite size semicircular sector surrounding the tip of the notch. The synthesis permits to obtain a very narrow band characterised by a scatter index equal to 1.96. The synthesis has been carried out with a constant value of the control radius Rc, independent of the loading conditions.
3. The tested alloy exhibits a good high temperature fatigue behaviour up to 500 °C. Until that temperature no reduction in the fatigue strength with respect to the room temperature has been detected.
4. Above 500 °C, a significant reduction in fatigue strength is shown both for plain and V-notched specimens.
5. At 650 °C the notch sensitivity of the present steel seems to be quite low. The inverse slope k is also very similar. It is equal to 2.48 for hourglass shaped specimens and 2.91 for plates weakened by lateral notches.
6. The high temperature mitigates the stress concentration effect: the experimental value of Kf has been found to be equal to 1.48, which is very low if compared with the theoretical stress concentration factor (Kthn = 3.84).
7. The strong reduction of the fatigue properties of the present material at 650 °C can be attributed to the specific thermal treatments performed on it. In fact the material was first quenched at 920 °C and subsequently tempered at about 580 °C twice. A final stress relieving treatment at 570 °C was carried out. The tempered temperature and final stress relieving temperatures were lower than 650 °C. It seems that for a long time exposure at temperatures higher than 570-590 °C the beneficial effects due to these specific treatments were lost.
8. All data from tests carried at room temperature up to 500 °C are summarized in terms of mean strain energy density over a control volume with Rc = 0.05 mm. A sound agreement in terms of strain energy density has been found between the present results and those recently obtained from a large bulk of multiaxial tests performed at room temperature on the same material.
9. A quite narrow scatter band characterized by a limited value of the scatter index has been obtained by summarizing together new data and old multiaxial data up to 500 °C.
10. A specific master curve based on strain energy density has been proposed for the considered steel tested at T= = 650 °C. The scatterband makes possible to summarize together data from plain and notched specimens. Dealing with notched specimens an empirical expression has been also proposed for the strain energy density calculation. The equation can be directly employed for practical applications of 40CrMoV13.9 steel at 650 °C.
Acknowledgements
This work is dedicated to Prof. Paolo Lazzarin. He was a top researcher in the field of fatigue and fracture mechanics. He was a wonderful teacher and my mentor for 13 years. I am strongly indebted with him.
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75. Berto F., Ayatollahi M.R. Fracture assessment of Brazilian disc specimens weakened by blunt V-notches under mixed mode loading by means of local energy // Mater. Des. - 2011. - V. 32. - P. 2858-2869.
76. Berto F., Elices M., Lazzarin P., Zappalorto M. Fracture behaviour of notched round bars made of PMMA subjected to torsion at room temperature // Eng. Fract. Mech. - 2012. - V. 90. - P. 143-160.
77. Berto F., Lazzarin P., Ayatollahi M.R. Brittle fracture of sharp and blunt V-notches in isostatic graphite under pure compression loading // Carbon. - 2013. - V. 63. - P. 101-116.
78. Pegorin F., Kotousov A., Berto F., Swain M.V., Sornsuwan T. Strain energy density approach for failure evaluation of occlusal loaded ceramic tooth crowns // Theor. Appl. Fract. Mech. - 2012. - V. 58. -P. 44-50.
79. Berto F., Lazzarin P., Ayatollahi M.R. Brittle fracture of sharp and blunt V-notches in isostatic graphite under pure compression loading // Carbon. - 2013. - V. 63. - P. 101-116.
80. Berto F., Cendon D.A.A., Lazzarin P., Elices M. Fracture behaviour of notched round bars made of PMMA subjected to torsion at -60°C // Eng. Fract. Mech. - 2013. - V. 102. - P. 271-287.
81. Torabi A.R., Berto F. Fracture assessment of blunt V-notched graphite specimens by means of the strain energy density // Strength Mater. - 2013. - V. 45. - P. 635-647.
82. Lazzarin P., Berto F., Ayatollahi M.R. Brittle failure of inclined keyhole notches in isostatic graphite under in-plane mixed mode loading // Fatigue Fract. Eng. Mater. Struct. - 2013. - V. 36. - P. 942-955.
83. Berto F. A brief review of some local approaches for the failure assessment of brittle and quasi-brittle materials // Adv. Mater. Sci. Eng. -2014. - V. 2014. - P. 1-10.
84. Salavati H., Alizadeh Y., Berto F. Effect of notch depth and radius on the critical fracture load of bainitic functionally graded steels under mixed mode I + II loading // Phys. Mesomech. - 2014. - V. 17. -No. 3. - P. 178-189.
85. Lazzarin P., Berto F., Gomez F., Zappalorto M. Some advantages derived from the use of the strain energy density over a control volume in fatigue strength assessments of welded joints // Int. J. Fatigue. - 2008. - V. 30. - P. 1345-1357.
86. Livieri P., Berto F. Local strain energy approach applied to fatigue analysis of welded rectangular hollow section joints // Int. J. Mater. Prod. Technol. - 2007. - V. 30. - P. 124-140.
87. Lazzarin P., Berto F., Radaj D. Fatigue-relevant stress field parameters of welded lap joints: pointed slit tip compared with keyhole notch // Fatigue Fract. Eng. Mater. Struct. - 2009. - V. 32. - P. 713-735.
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89. Ferro P., Lazzarin P., Berto F. Fatigue properties of ductile cast iron containing chunky graphite // Mater. Sci. Eng. A. - 2012. - V. 554. -P. 122-128.
90. Taghizadeh K., Berto F., Barati E. Local strain energy density applied to martensitic steel plates weakened by U-notches under mixed mode loading // Theor. Appl. Fract. Mech. - 2012. - V. 59. - P. 21-28.
91. Berto F., Lazzarin P. Fatigue strength of Al7075 notched plates based on the local SED averaged over a control volume // Sci. China Physics. Mech. Astron. - 2013. - V. 57. - P. 30-38.
92. Lazzarin P., Berto F., Atzori B. A synthesis of data from steel spot welded joints of reduced thickness by means of local SED // Theor. Appl. Fract. Mech. - 2013. - V. 63-64. - P. 32-39.
93. Ferro P. The local strain energy density approach applied to pre-stressed components subjected to cyclic load // Fatigue Fract. Eng. Mater. Struct. - 2014. - V 37. - P. 1268-1280.
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95. Radaj D. State-of-the-art review on the local strain energy density concept and its relation to the J-integral and peak stress method // Fatigue Fract. Eng. Mater. Struct. - 2014. - doi: 10.1111/ffe.12231.
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97. Berto F., Lazzarin P., Radaj D. Fictitious notch rounding concept applied to sharp V-notches: Evaluation of the microstructural support factor for different failure hypotheses. Part I: Basic stress equations // Eng. Fract. Mech. - 2008. - V. 75. - P. 3060-3072.
98. Berto F., Lazzarin P., Radaj D. Fictitious notch rounding concept applied to sharp V-notches: Evaluation of the microstructural support factor for different failure hypotheses // Eng. Fract. Mech. - 2009. -V. 76. - P. 1151-1175.
99. Berto F., Lazzarin P. Fictitious notch rounding approach of pointed V-notch under in-plane shear // Theor. Appl. Fract. Mech. - 2010. -V. 53. - P. 127-135.
100. Berto F., Zappalorto M. Fictitious notch rounding concept applied to V-notches with end holes under mode I loading // Int. J. Fract. -2011.- V. 171. - P. 91-98.
101. Berto F., Zappalorto M. The fictitious notch rounding approach applied to V-notches with root holes subjected to mode I loading // J. Strain Anal. Eng. Des. - 2012. - V. 47. - P. 176-186.
102. Berto F., Lazzarin P., Radaj D. Fictitious notch rounding concept applied to V-notches with root holes subjected to in-plane shear loading // Eng. Fract. Mech. - 2012. - V. 79. - P. 281-294.
103. Berto F. Fictitious notch rounding concept applied to V-notches with end holes under mode 3 loading // Int. J. Fatigue. - 2012. - V 38. -P. 188-193.
104. Radaj D., Lazzarin P., Berto F. Generalised Neuber concept of fictitious notch rounding // Int. J. Fatigue. - 2013. - V. 51. - P. 105-115.
105. Berto F., Lazzarin P., Wang C.H. Three-dimensional linear elastic distributions of stress and strain energy density ahead of V-shaped notches in plates of arbitrary thickness // Int. J. Fract. - 2004. -V. 127. - P. 265-282.
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107. Kotousov A., Lazzarin P., Berto F., Harding S. Effect of the thickness on elastic deformation and quasi-brittle fracture of plate components // Eng. Fract. Mech. - 2010. - V. 77. - P. 1665-1681.
108. Berto F., Lazzarin P., Kotousov A., Harding S. Out-of-plane singular stress fields in V-notched plates and welded lap joints induced by in-plane shear load conditions // Fatigue Fract. Eng. Mater. Struct. -2011. - V. 34. - P. 291-304.
109. Berto F., Lazzarin P., Kotousov A. On the presence of the out-of-plane singular mode induced by plane loading with Kn = K = 0 // Int. J. Fract. - 2010. - V. 167. - P. 119-126.
110. Berto F., Lazzarin P., Kotousov A. On higher order terms and out-of-plane singular mode // Mech. Mater. - 2011. - V. 43. - P. 332-341.
111. Berto F., Lazzarin P., Kotousov A., PookL.P. Induced out-of-plane mode at the tip of blunt lateral notches and holes under in-plane shear loading // Fatigue Fract. Eng. Mater. Struct. - 2012. - V 35. - P. 538555.
112. Kotousov A., Berto F., Lazzarin P., Pegorin F. Three dimensional finite element mixed fracture mode under anti-plane loading of a crack // Theor. Appl. Fract. Mech. - 2012. - V. 62. - P. 26-33.
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Berto Filippo, Prof., University of Padova, Italy, berto@gest.unipd.it
113. Berto F., Lazzarin P., Marangon C. The effect of the boundary conditions on in-plane and out-of-plane stress field in three dimensional plates weakened by free-clamped V-notches // Phys. Mesomech. -
2012. - V. 15. - No. 1. - P. 26-36.
114. Berto F., Kotousov A., Lazzarin P., Pegorin F. On a coupled mode at sharp notches subjected to anti-plane loading // Eur. J. Mech. A. Solids. - 2013. - V. 38. - P. 70-78.
115. Berto F., Kotousov A., Lazzarin P., Pook L.P. On scale effect in plates weakened by rounded V-notches and subjected to in-plane shear loading // Int. J. Fract. - 2012. - V 180. - P. 111-118.
116. Berto F., Marangon C. Three-dimensional effects in finite thickness plates weakened by rounded notches and holes under in-plane shear // Fatigue Fract. Eng. Mater. Struct. - 2013. - V. 36. - P. 11391152.
117. Berto F. A review on coupled modes in V-notched plates of finite thickness: A generalized approach to the problem // Phys. Mesomech.
2013. - V. 16. - No. 4. - P. 378-390.
118. Kotousov A., Lazzarin P., Berto F., Pook L.P. Three-dimensional stress states at crack tip induced by shear and anti-plane loading // Eng. Fract. Mech. - 2013. - V. 108. - P. 65-74.
119. Afshar R., Berto F., Lazzarin P., Pook L.P. Analytical expressions for the notch stress intensity factors of periodic V-notches under tension by using the strain energy density approach // J. Strain Anal. Eng. Des. - 2013. - V. 48. - P. 291-305.
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121. Zappalorto M., Lazzarin P. Three-dimensional elastic stress fields ahead of notches in thick plates under various loading conditions // Eng. Fract. Mech. - 2013. - V. 108. - P. 75-88.
122. Pook L.P, Berto F., Campagnolo A., Lazzarin P. Coupled fracture mode of a cracked disc under anti-plane loading // Eng. Fract. Mech. -
2014. - V. 128. - P. 22-36.
123. Berto F., Lazzarin P. On higher order terms in the crack tip stress field // Int. J. Fract. - 2010. - V. 161. - P. 221-226.
124. Berto F., Lazzarin P. Multiparametric full-field representations of the in-plane stress fields ahead of cracked components under mixed mode loading // Int. J. Fatigue. - 2013. - V. 46. - P. 16-26.
125. Berto F., Lazzarin P., Livieri P. On the second non-singular stress term of the V-notch solution: a new engineering solution // Int. J. Fract. - 2013. - V. 181. - P. 83-98.
126. Afshar R., Berto F. Stress concentration factors of periodic notches determined from the strain energy density // Theor. Appl. Fract. Mech. -2011. - V. 56. - P. 127-139.
127. Berto F., Lazzarin P., Afshar R. Simple new expressions for the notch stress intensity factors in an array of narrow V-notches under tension // Int. J. Fract. - 2012. - V. 176. - P. 237-244.
128. Lazzarin P., Afshar R., Berto F. Notch stress intensity factors of flat plates with periodic sharp notches by using the strain energy density // Theor. Appl. Fract. Mech. - 2012. - V. 60. - P. 38-50.
129. Atzori B., Lazzarin P., Meneghetti G. Fracture mechanics and notch sensitivity // Fatigue Fract. Eng. Mater. Struct. - 2003. - V. 26. -P. 257-267.
130. Lazzarin P., Berto F. From Neuber's elementary volume to Kitagawa and Atzori's diagrams: An interpretation based on local energy // Int. J. Fract. - 2005. - V. 135. - P. L33-L38.
131. Jiang Y., Feng M. Modeling of fatigue crack propagation // J. Eng. Mater. Technol. - 2004. - V 126. - P. 77-86.
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