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УДК 620.178.3
Влияние ориентации критической плоскости на оценку многоцикловой усталости при многоосном нагружении
A. Carpinteri, C. Ronchei, D. Scorza, S. Vantadori
Пармский университет, Парма, 43124, Италия
В статье выполнена оценка многоцикловой усталости при многоосном нагружении элементов конструкций вплоть до их разрушения с использованием модифицированного критерия Carpinteri-Spagnoli для описания усталости при многоосном нагружении на основе подхода критической плоскости. Согласно данному критерию положение критической плоскости связано с направлениями усредненных главных напряжений через угол разориентации 8. Следовательно, используемый параметр усталостного разрушения определяется нелинейной комбинацией амплитуд эквивалентных нормальных и сдвиговых напряжений, действующих на критическую плоскость. Представлены варианты исходного выражения для угла разориентации 8 в рамках модифицированного критерия Carpinteri-Spagnoli. Показано, что модифицированные выражения tagoda согласуются с предложенным ранее авторами подходом, в котором угол разориентации является функцией отношения между пределом упругости при симметричном сдвиговом напряжении и пределом упругости при симметричном нормальном напряжении. Такие выражения могут применяться в расчетах для металлов с разной степенью сопротивления усталостному разрушению. Проведено сравнение имеющихся экспериментальных данных с теоретическими оценками. Для материалов с высокой и очень высокой степенью сопротивления усталостному разрушению показано, что модифицированные соотношения для 8 дают более точные результаты по сравнению с результатами, полученными с помощью исходного выражения для 8.
Ключевые слова: нагрузка с постоянной амплитудой, ориентация критической плоскости, оценка усталостной прочности, модифицированный критерий Carpinteri-Spagnoli, многоцикловая усталость при многоосном нагружении
Critical plane orientation influence on multiaxial high-cycle fatigue assessment
A. Carpinteri, C. Ronchei, D. Scorza, and S. Vantadori
University of Parma, Parma, 43124, Italy
In the present paper, the multiaxial fatigue lifetime of structural components failing in the high-cycle fatigue regime is evaluated by employing the modified Carpinteri-Spagnoli (C-S) multiaxial fatigue criterion based on the critical plane approach. In the above criterion, the critical plane position is linked to averaged principal stress directions through an off-angle 8. Then, the fatigue damage parameter used is determined by a nonlinear combination of an equivalent normal stress amplitude and the shear stress amplitude acting on the critical plane. In the present paper, some modifications of the original expression for the off-angle 8 are implemented in the modified Carpinteri-Spagnoli criterion. In particular, modified expressions recently proposed by tagoda et al. are in accordance with the assumption originally developed by Carpinteri and co-workers, that is, the off-angle is a function of the ratio between the fatigue limit under fully reversed shear stress and that under fully reversed normal stress. Such expressions can be employed for metals ranging from mild to very hard fatigue behaviour. Some experimental data available in the literature are compared with the theoretical estimations and, only for materials with hard and very hard fatigue behaviour, the modified 8 relationships are shown to yield fatigue lifetime results slightly better than those determined through the original 8 expression.
Keywords: constant amplitude loading, critical plane orientation, fatigue lifetime estimation, modified Carpinteri-Spagnoli criterion, multiaxial high-cycle fatigue
Nomenclature
C—shear stress vector acting on the critical plane; C(t)—modulus of the shear stress vector C; Ca —shear stress amplitude;
m—slope of the S—N curve for fully reversed normal stress (R = -1);
m* —slope of the S-N curve for fully reversed shear stress
(R = -1);
N—normal stress vector perpendicular to the critical plane; N(t)—modulus of the normal stress vector N; N'aeq —equivalent normal stress amplitude; t—time;
© Carpinteri A., Ronchei C., Scorza D., Vantadori S., 2015
Sw —stress vector at material point P and related to the critical plane;
w—normal unit vector perpendicular to the critical plane; 8—angle between the averaged direction 1 of the maximum principal stress and the normal w to the critical plane
(Fig. 1);
aaf, _1 —fully reversed normal stress fatigue limit;
au —material ultimate tensile strength;
taf _1 —fully reversed shear stress fatigue limit.
1. Introduction
One of the main causes of failure in engineering metallic structural components is fatigue failure. Structural components are frequently subjected to complex time-varying loading, which produces multiaxial stress/strain states. In the literature, several multiaxial fatigue life estimation criteria have been proposed [1-4], but nowadays a universally accepted method does not exist yet.
Amongst the different criteria proposed by researchers, those based on the critical plane approach are characterised by high effectiveness and wide application range [5, 6]. The fatigue failure analysis performed according to such an approach is physics-based, since the critical plane concept is related to the observation that fatigue cracks nucleate and early grow on a particular material plane (the so-called critical plane). Moreover, such criteria prove to be very successful in a broad range of applications, including notched components [7], non-proportional loadings [8, 9], and variable amplitude loadings [10-12].
The above criteria are characterized by different strategies to determine the orientation of the critical plane: for instance, several researchers define the critical plane as the material plane where the amplitude or the value of some stress/strain components or a combination of them exhibits its maximum value [13-17]. Alternatively, the position of the critical plane can be deduced by means of the weight function method [18-20], which consists in searching the averaged directions of the principal stress axes by employing suitable weight functions which take into account the main factors influencing the fatigue behaviour of the material.
Note that the above methodologies represent just some of those reported in the literature [21] (see also Ref. [22] for a comparison of different methods to determine the critical plane position).
In spite of that, the common strategy of all critical plane-based criteria is to employ a combination of stresses/strains acting on the critical plane itself to perform the fatigue life assessment [1]. Within the framework of such criteria, Carpinteri and co-workers proposed the modified Carpinteri-Spagnoli (C-S) criterion [23], which is a simplified version of the original one [24]. More precisely, the critical plane orientation is linked through an off-angle, 8, to the averaged directions of the principal stress axes, 1, 2, and 3. Such directions are computed by taking into account a weight function, which is deemed to be dependent on both the maximum principal stress g1 (t) and two material pa-
rameters, a.
af, -1
and t
af,-1
Fig. 1. Framework of the modified Carpinteri-Spagnoli criterion
deduced from the S-N curves for fully reversed normal and shear stress, respectively. Further, a nonlinear combination of an equivalent normal stress amplitude and the shear stress amplitude acting on the critical is employed as fatigue damage parameter (see the review on such a criterion reported in Ref. [25]).
A capability of the modified Carpinteri-Spagnoli criterion is that it can be applied for different (i) multiaxial loading conditions, i.e. under constant [23, 24] and variable amplitude loading [26, 27], and (ii) structural component configurations, i.e. for smooth [23, 24] and notched structures [28], in the latter case the fatigue strength being strongly affected by the gradients of stress/strain components [29-34].
By taking as starting point the original assumption developed by Carpinteri and co-workers related to the critical plane orientation [23, 24], Lagoda et al. [35] have recently proposed some modifications to the original expression adopted to determine the off-angle 8.
In the present paper, the orientation of the critical plane is computed by taking into account the above expressions by Lagoda et al., implementing them in the modified Carpinteri-Spagnoli criterion in order to verify whether they are able to improve this criterion in terms of fatigue lifetime estimation. The validation of such an implementation is performed by employing experimental data available in the literature [36-41] related to stress-controlled fatigue tests performed on smooth specimens under biaxial loadings.
2. Formulation of the modified Carpinteri-Spagnoli criterion
The high-cycle multiaxial fatigue criterion, known as the modified Carpinteri-Spagnoli criterion [23], is a simplified version of the original one proposed in Ref. [24]. In particular, the modifications are related to a simplified weighting procedure to determine the averaged principal stress axes, and to the effect of the non-zero normal mean stress on fatigue limit. Such modifications introduced in
the simplified version make the implementation of the criterion rather simple.
The main steps of the modified Carpinteri-Spagnoli criterion (Fig. 1), applied to multiaxial constant amplitude cyclic loading, are discussed in the following sections.
2.1. Orientation of the critical plane
According to the original idea developed by Brown and Miller in 1973 [42], the fatigue crack evolution process can be separated into two stages: (i) stage 1, where a crack nucleates (usually on the external surface of the structural component being examined) along a shear slip plane (mode II, fatigue crack initiation plane); (ii) stage 2, where crack propagates in the plane normal to the direction of the maximum principal stress (mode I, final fatigue fracture plane).
Since it has been experimentally observed that some metallic materials exhibit fatigue crack propagation mechanisms for which stage 2 is predominant over stage 1 (e.g. metals characterized by hard and very hard fatigue behaviour) and vice versa, a suitable procedure to define the critical plane orientation should be able to take into account both mode II and mode I mechanisms.
According to the modified Carpinteri-Spagnoli criterion, the procedure here reported considers each of the above mechanisms. In more detail, the position of the final fracture plane is linked to that of the principal stress directions, being fatigue life strongly influenced by the principal stresses [18]. Since the above directions are generally time-varying under fatigue loading, averaged principal stress directions can be computed by using suitable weight functions. Then, the critical plane orientation is correlated to such averaged directions by means of an off-angle, as is specified below.
Let us consider the stress state at a material point P on the body surface: the averaged directions of principal stress axes can be determined on the basis of their instantaneous directions by means of the averaged values of the principal Euler angles. By assuming the weight function W(t) reported in Ref. [23], the averaged principal stress axes 1, 2, 3 coincide with the instantaneous ones at the time instant when the maximum principal stress g1 attains its peak value over the loading cycle. The normal to the final fatigue fracture plane, which is the one experimentally observed post mortem at the macrolevel, is assumed to be coincident with the averaged direction 1 of the maximum principal stress gp
The orientation of the critical plane, which is the verification plane for fatigue life evaluation, is linked to the above averaged maximum principal stress direction 1. In more detail, the off-angle 8 proposed by Carpinteri et al. [23-28] (named 81 in the following), between the normal w to the critical plane (where w belongs to the principal plane 23, as is shown in Fig. 1) and the averaged direction 2, is given by the empirical expression here reported:
8! = 3/2 [1 - (t afH/g'f,_ ,)2] • 45°, (1)
where gaf -1 is the fully reversed normal stress fatigue limit,
and taf,-1 is the fully reversed shear stress fatigue limit. According to Eq. (1), the off-angle is equal to 0o for taf _1j gaf -1 = 1 (very hard metals), whereas it is equal to 45o for taf _xj gaf _1 = 1/V3 (value at the border between hard and mild metals). It is important to highlight that the above equation is in line with the fact that, when stage 2 is predominant, the critical plane is coincident with the final fatigue fracture plane, while when stage 1 is predominant, the critical plane is coincident with the fatigue crack initiation plane.
Recently, Lagoda et al. [35] have proposed some modifications to the original expression of the off-angle 81, that is, the critical plane orientation is determined through a procedure which follows the same logic used in the modified Carpinteri-Spagnoli criterion, where the off-angle is a function of the fatigue limit ratio taf, _1j gaf, _1 and can be employed for metals ranging from mild to very hard fatigue behaviour. The relationships reported in Ref. [35] are as follows:
82 = 9/8[1 _ (taf^/gaf^)4] • 45°, (2)
83 = ITTl [1 _ (taf,_1 /gaf,_1 )3 ] • 45°, (3)
84 = [1 _ (taf'"^gaf,_1 )] • 45°, (4)
85 = [1 _ (taf^/gaf^)]2 • 45(5)
The diagram shown in Fig. 2 represents a graphical interpretation of Eqs. (1)-(5), against the fatigue limit ratio. It can be noted that, for a borderline mild/hard metal (t af,g af, _1 ^ 0.58), the value of the off-angle computed by employing the above expressions is almost the same. On the other hand, in the case of a hard metal characterized by taf, _xf gaf, _1 = 0.7, the difference between the off-angle results can be at most equal to 170, which could affect the fatigue lifetime estimation if the modified Carpinteri-Spagnoli criterion were applied.
0.6 0.7 0.8 0.9 1.0 Fatigue limit ratio Taf _i/aaf _i
Fig. 2. Different expressions of the off-angle (see Eqs. (1)-(5)) against the fatigue limit ratio taf _1j gaf _ 1
2.2. Evaluation of multiaxialfatigue lifetime
At each time instant t, the stress vector Sw related to the critical plane orientation may be decomposed in two components: the normal stress component N (perpendicular to the critical plane) and the shear stress component C (lying on the critical plane). During the observation time interval T, the direction of the normal stress vector N(t) is fixed with respect to time, and consequently its mean value Nm and amplitude Na can readily be computed. On the other hand, the amplitude Ca of the shear stress vector C(t) is not uniquely defined, because of the time-varying direction of vector C which describes a closed path on the critical plane during a loading cycle. Different methods to define the value of Ca are available in the literature: for instance, the prismatic hull method [43] allows us to compute the amplitude Ca with a simple algorithm. Note that the prismatic hull method has recently been implemented in the modified Carpinteri-Spagnoli criterion [44, 45]. In the present paper, the min-max procedure proposed by Papadopoulos [46] to define Ca by using the minimum circumscribed circle method is adopted.
The multiaxial fatigue limit condition can be written by equating an equivalent uniaxial stress amplitude tfeqa to the fully reversed normal stress fatigue limit aaf _i [23, 25]:
CTeq,a = -
: N'eq)2 + (fl/ Taf,-l)2(Ca)2 = ^f ,-1, (6)
where
(<eq) = Na + °af,-1
' N ^
N
Nm
(7)
au being the ultimate tensile strength [47]. It is worth noticing that the above equivalent normal stress amplitude Na>eq is introduced in the modified Carpinteri-Spagnoli criterion in order to take into account the following experimental findings: a tensile mean stress, superimposed upon an alternating normal stress, strongly decreases the fatigue strength of metals.
Using the Basquin-like relationships for both fully reversed normal stress and fully reversed shear stress [25], the number Nf of loading cycles to failure is determined by solving Eq. (8) through an iterative procedure:
(N' )2 +
/ \2 / \2m / \2m*
T WW. M \
af,-1
af,-1
Nf
No
No Nf
(Ca)2 =
= a.
af,-1
( Nf ^
No
(8)
where N0 is the reference number of loading cycles (for example N0 = 2 -106), and m and m* are the slopes of S-N curves for fully reversed normal and shear stress, respectively.
3. Experimental validation and discussion
In the present paper, the different relationships of the off-angle 8, described in the previous section, are implemented in the modified Carpinteri-Spagnoli criterion in order to verify whether they are able to improve the lifetime estimation for some experimental tests available in the literature [36-41].
The examined experimental data are related to specimens made of SM45C steel [36], 30CrNiMo8 steel [37, 38], 6082-T6 aluminium alloy [39, 40] and S355J0 alloy steel [41] subjected to synchronous, sinusoidal, in- and out-of-phase loading (with zero and non-zero mean value). More precisely, in order to evaluate the capability of the above implementations, the tested materials have been selected on the basis of their values of fatigue limit ratio, that is: (i) for SM45C steel and 30CrNiMo8 steel, the fatigue limit ratio is typical of very hard metals; (ii) for 6082-T6 aluminium alloy, the fatigue limit ratio is typical of borderline mild/hard metals; (iii) for S355J0 alloy steel, the fatigue limit ratio is typical of hard metals.
The relevant static and fatigue properties of the examined materials are listed in Table 1.
The accuracy of the proposed fatigue lifetime estimation can be evaluated by means of the root mean square error method [35]. In particular, the value of the root mean square logarithmic error is computed as follows:
^RMS
1
I log2 (Nf,exp /Nf cal )
(9)
where n is the total number of data, Nf,exp is the experimental multiaxial fatigue life, and Nfcal is the theoretical multiaxial fatigue life determined by considering the different 8 expressions. Finally, the mean square error TRMS is given by:
trms = 10£rms. (10)
Figure 3 shows the mean square error determined for the different materials, in accordance to the five off-angle expressions discussed in Sect. 2. The analysis of the results clearly proves that the modified Carpinteri-Spagnoli criterion produces evaluations within the scatter band 3, being
Static and fatigue properties for examined materials
Table 1
Material au, MPa °af,-1> MPa m Taf,-1, MPa * m
SM45C steel [36] 731 254.25 -0.10 209.41 -0.05
30CrNiMo8 steel [37, 38] 1014 427.37 -0.13 371.52 -0.04
6082-T6 aluminium alloy [39, 40] 290 152.83 -0.11 87.90 -0.15
S355J0 alloy steel [41] 611 276.58 -0.15 183.70 -0.09
Fig. 3. Mean square error related to SM45C steel (a), 30CrNiMo8 steel (b), 6082-T6 aluminium alloy (c), S355J0 alloy steel (d). The values of the off-angle computed according to the Eqs. (1)-(5) are reported in the legend
the value of the TRMS lower than 3 (note that, if all the computed results fell within the scatter band 2, the value of TRMS would be equal to 2).
Moreover, the analysis of the results in terms of the mean square error for the examined materials indicates that:
(a) SM45C steel: the most accurate result is obtained for 8 = 85. In particular, 85 significantly decreases the TRMS value (up to 25.5%) with respect to that deduced by using the original 81 expression;
(b) 30CrNiMo8 steel: higher accuracy is gained for 8 = 84. Note that, by implementing Eq. (4) in the modified Carpinteri-Spagnoli criterion, the value of TRMS decreases to 6% in comparison with that determined for 81 expression;
(c) 6082-T6 aluminum alloy: similar accuracy is obtained by using the different 8 expressions above, since the value of the five off-angles is almost the same;
(d) S355J0 alloy steel: the most accurate result is determined by using 8 = 85, with a decrease of the TRMS value up to 20.4% with respect to that deduced by employing 81.
In conclusion, we can highlight that the effectiveness of the implementation of the 8 relationships proposed by Lagoda in the modified Carpinteri-Spagnoli criterion can only be observed for materials characterized by fatigue limit ratio typical of hard and very hard metals. More precisely, for SM45C steel, 30CrNiMo8 steel and S355J0 alloy steel,
higher accuracy level in terms of fatigue life evaluation is obtained when estimating the critical plane orientation by employing 84 and 85 instead of 81.
4. Conclusions
In the present paper, the orientation of the critical plane connected with averaged principal stress directions is computed by taking into account both the original off-angle expression and the modified ones. In particular, the modified expressions of 8, recently proposed by Lagoda et al. in accordance to the original idea developed by Carpinteri and co-workers, depend on the fatigue limit ratio, and can be employed for metals ranging from mild to very hard fatigue behaviour. The different 8 relationships are implemented in the modified Carpinteri-Spagnoli criterion in order to estimate the fatigue lifetime. The comparison with some experimental data related to stress-controlled fatigue tests appears to be satisfactory, with a value of the mean square error lower that 3 for each examined material. In particular, for hard and very hard metals, such an implementation provides fatigue lifetime results better than those determined through the original 8 expression. Further analysis is needed to check whether the same trend may be observed for other materials characterised by hard and very hard fatigue behaviour.
Acknowledgments
The authors gratefully acknowledge the research support for this work provided by the Italian Ministry for University and Technological and Scientific Research (MIUR).
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Поступила в редакцию 06.10.2015 г.
Сведения об авторах
Andrea Carpinteri, University of Parma, Italy, andrea.carpinteri@unipr.it
Camilla Ronchei, University of Parma, Italy, camilla.ronchei@nemo.unipr.it
Daniela Scorza, University of Parma, Italy, daniela.scorza@nemo.unipr.it
Sabrina Vantadori, Dr., Assoc. Prof., University of Parma, Italy, sabrina.vantadori@unipr.it
