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A stress-based fatigue criterion to assess high-cycle fatigue under in-phase multiaxial loading conditions
Theoretical and Applied Fracture Mechanics 73 (2014) 3–8
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A stress-based fatigue criterion to assess high-cycle fatigue under in-phase multiaxial loading conditions Krzysztof M. Golos a,b, Daniel K. Debski a,⇑, Marek A. Debski c a
Warsaw University of Technology, Narbutta 84 Street, 02-524 Warsaw, Poland Institute of Mechanised Construction and Rock Mining, Racjonalizacji 6/8 Street, 02-673 Warsaw, Poland c Institute of Aviation, Aleja Krakowska 110/114 Street, 02-256 Warsaw, Poland b
a r t i c l e
i n f o
Article history: Available online 22 July 2014 Keywords: High-cycle fatigue Multiaxial fatigue Biaxial cyclic loading Stress-based criterion
a b s t r a c t In this paper, a new criterion for in-phase multiaxial fatigue life prediction in high-cycle fatigue is proposed. The new stress-based fatigue criterion is based on the modification of Gough’s limit curve. In the case of cyclic plane stress loading, the particular form of the criterion is discussed in detail. The values of parameters in the proposed criterion can be obtained based on fatigue limits in bending and torsion. This allows use of the proposed criterion with almost no limitations to engineering calculations for different metals. Analytical predictions are compared to available experimental data from the literature for six different materials. Additionally, the predictions based on the proposed criterion are compared with the classic Gough’s criterion. The predictions based on the proposed criterion seem to be more precise for the analysed materials. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Despite great progress in fatigue analysis in recent decades, there is a need for reliable methods to predict fatigue life under multiaxial cyclic loading. It is desirable to actually base such a proposal on general principles of solid mechanics and the evaluation of constants contained in the criterion, from the standard uniaxial tests. The multiaxial fatigue tests are complex and expensive. Therefore, several proposals have been made to correlate uniaxial fatigue test data. A review of various failure criteria for multiaxial fatigue is given by Krempl [1], Garud [2], Brown and Miller [3], Macha and Sonsino [4]. These approaches can be divided into three categories: stressbased methods, strain-based methods and energy-based approaches. In the low-cycle fatigue regime, when relatively large plastic strains occur, the strain-based criteria are recommended. In the high-cycle regime, with plastic strain decreasing in the material, the stress-based criteria are often used. To obtain a unified description of both low- and high-regimes, a strain energy strategy has been proposed. This approach is based on the assumption that the energy is proportional to the fatigue damage calculated in different ways. Feltner and Morrow [5] proposed a criterion based on hysteresis energy; Garud [6] presented the plastic strain energy ⇑ Corresponding author. Tel.: +48 22 234 82 61, mobile: +48 500 131 215. E-mail addresses: [email protected] (K.M. Golos), [email protected]. pl (D.K. Debski). http://dx.doi.org/10.1016/j.tafmec.2014.07.005 0167-8442/Ó 2014 Elsevier Ltd. All rights reserved.
model for multiaxial loading; and Golos and Ellyin [7,8] modelled uniaxial, multiaxial fatigue and cumulative damage in terms of the total strain energy density. Among different energy-based approaches, a non-dimensional parameter that holds terms for material properties was proposed by Varvani-Farhani [9]. This criterion also includes the mean stress effect and additional hardening for out-of-phase loading conditions. The energy-based approach seems to be promising because energy is a scalar quantity and includes both stress and strain components of the loading path. In the case of stress-based criteria, we can distinguish two main methodologies. The first one is based on the analysis of the cyclic stress invariants, and the second one considers the stress state at the selected plane. The invariant stress criterion can be expressed in a general form as follows [10]:
FðI1 ; I2 ; I3 ; Nf Þ ¼ 0
ð1Þ
where I1 = r1 + r2 + r3, I2 = r1r2 + r1r3 + r2r3, and I3 = r1 r2 r3; r1, r2, and r3 are principal stresses; and Nf is the number of cycles to failure. According to the second methodology, fatigue analysis is performed on one critical section in the element. This plane is called critical and is usually different for selected fatigue models. It is often assumed that the failure occurs at a given number of cycles when the function of a normal stress and a shear stress reaches the critical state.
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A brief overview of several stress-based high-cycle fatigue criteria is first discussed in this paper. Several criteria proposed in recent decades are aimed at reducing a given multiaxial stress state to an equivalent uniaxial stress loading. Although there are many proposed criteria for biaxial loading, most of them are limited to specific materials or loading conditions. To the authors’ knowledge, there is no existing biaxial fatigue criterion that is universally accepted. The main goal of this paper is to propose a general form of a criterion for in-phase multiaxial fatigue failure based on the equivalent stress concept. One of the fundamental problems of calculating fatigue strength is the determination of the stress safety space corresponding to a given complex state of stress. Unlike the previous stress-based models, equivalent stress methods for multiaxial in-phase loading, based on the distortion strain energy density, are theoretically correlated with the fatigue limit and depend on the material parameters. This allows the use of the proposed criterion for engineering calculations – with almost no limitations and for different metals. The current criterion is compared with Gough’s model using the available experimental data for six different materials and various cyclic load paths. 2. Biaxial stress-based high-cycle fatigue criteria Because the main focus of this paper is high-cycle fatigue, only the stress-based approach commonly used in high-cycle fatigue design procedures is reviewed in this section. Large numbers of criteria are present in the literature – only a few common criteria are briefly discussed here. Generally, as was mentioned above, the stress-based criteria can be divided into two categories, namely, those based on stress invariants related to the number of cycles to failure (including equivalent stress- and average stress-based models) and those based on critical plane stress (calculated in different ways). Concerning older theories, the authors feel that attention has been taken from the more widely used theories in high-cycle fatigue. Gough and Pollard [11–13] proposed two criteria for metals under combined in-phase bending and torsion. For ductile metals, the equation takes the form of an ellipse quadrant:
r
2
þ
raf ;1
2
s
saf ;1
¼1
ð2Þ
where raf,1 and saf,1 are fatigue limits in reversed bending and torsion, respectively. For brittle metals, the ellipse rewritten in the form:
r
2
raf ;1 r 1 þ raf ;1 saf ;1
raf ;1
raf ;1 2 þ
saf ;1
s saf ;1
2 ¼1 ð3Þ
has been proposed. Findley [14] proposed a criterion based on the linear combination of the axial stress and the shear stress squared (parabola) in the following form:
r
raf ;1
þ
s
saf ;1
2
¼1
ð4Þ
Considering the phase difference between loading, Lee [15] modified the ellipse quadrant of Gough as follows:
"
ra;eq ¼ r 1 þ
2 #1=2
raf ;1 s saf ;1 r
ð5Þ
In recent years, criteria based on the critical plane approach have attracted increasing attention [16–19].
Carpinteri et al. [18–21] used the maximum normal stress and the shear stress amplitude on the critical plane as parameters to modify Gough’s criterion. The calculation of the critical plane is performed in two steps. First, the weighted mean direction of the maximum principal stress under multiaxial random loading is estimated [20–25]. According to this concept, the fatigue failure assessment is presented by considering a quadratic combination of the maximum normal stress (rmax) and the shear stress amplitude (sa) in the following form:
rmax raf ;1
2
þ
sa
2
saf ;1
¼1
ð6Þ
Several different multiaxial fatigue criteria based on stress invariants are discussed in [26]. The criterion formulated by Sines [27] is probably the most popular invariant-based approach. According to this method, the stress state is in its fatigue limit condition when the following state is achieved:
pffiffiffiffiffiffiffi I2;a þ krH k
ð7Þ
where I2a is the second invariant of the stress tensor, rH is the hydrostatic stress amplitude, and k and k are material constants that can be calculated by considering two fatigue limits. Papadopoulos et al. [26] proposed the fatigue model based on the average stress approach, which can be expressed as:
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffi r2a 2 hT a i2 þ k½I1a þ I1m ¼ þ sa þ k½I1a þ I1m ¼ f 3
ð8Þ
where hTai2 is the average quantity within the volume, I1a is the first invariant of the stress tensor, ra is the bending stress amplitude, sa is the torsion stress amplitude, and k and f are material parameters depending on the fatigue limit. Brighenti and Carpinteri [34] proposed a fatigue damage model to evaluate the life assessment of notched structural elements under multiaxial stress histories. Ottosen et al. [35] presented high-cycle fatigue modelling using multiaxial load histories (inphase and out-of-phase). Stephanov [36] showed a method for multiaxial fatigue life estimation under non-proportional stresses. 3. Equivalent stress based criterion for multiaxial cyclic loading The stress at the point in the structure subjected to multiaxial cyclic loading can be expressed through the stress tensor as:
2
rx ðtÞ 6 s ðtÞ 6 yx ½Drij ðtÞ=2 ¼ 6 4 szx ðtÞ
sxy ðtÞ sxz ðtÞ 3 ry ðtÞ syz ðtÞ 7 7 7 szy ðtÞ rz ðtÞ 5
ð9Þ
This stress state can be expressed in the principal axes of the coordinate system as:
2
3 r1 ðtÞ 0 0 r2 ðtÞ 0 7 7 7 0 r3 ðtÞ 5
60 6 ½Drij ðtÞ=2 ¼ 6 40
ð10Þ
In this paper, the form strain energy density approach presented by Huber has been used to analyse multiaxial fatigue in high-cycle loading. This approach leads to similar conclusions as obtained by von Mises and Hencky. Therefore, here it will be described as Huber–Mises–Hencky’s (HMH) hypothesis [28]. According to this approach, failure under cyclic multiaxial fatigue in high-cycle loading is proposed to occur when the maximum distortion strain energy density exceeds the distortion strain energy density at the yield stress obtained from a uniaxial tensile test, i.e., /f 6 /f,yield.
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The distortion strain energy density can be expressed as:
/f ¼
1 þ mh 6E
ðrx ry Þ2 þ ðrx rz Þ2 þ ðrz ry Þ2 þ 6ðs2xy þ s2yz þ s2zx Þ
i
ð11Þ In the uniaxial state of stress ry – 0, rx = rz = 0, and
sxy = syz = szx = 0: /f ¼
1þm 2 r 3E y
ð12Þ
Accepting that strains in both states are equal and that ry = req, then the expression for equivalent stress can be obtained:
req
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðrx ry Þ2 þ ðrx rz Þ2 þ ðrz ry Þ2 þ 6ðs2xy þ s2yz þ s2zx Þ
1 ¼ pffiffiffi 2
ð13Þ
element. In a two-dimensional state, Fig. 1 implies that one of the coordinates does not play a role in the description of the problem. The tensor component associated with the z direction is equal to zero. Under plain stress loading, rz = 0 and syz = szx = 0 or r3 = 0, and the expression for the equivalent stress takes one of the following forms:
req ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2x þ r2y rx ry þ 3s2xy
ð20Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r21 þ r22 r1 r2
ð21Þ
or
req ¼
where the principal stresses are given by:
r1 ; r2 ¼
rx þ ry 2
In the principal stress system, Eq. (10) becomes:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
req ¼ pffiffiffi ðr1 r2 Þ2 þ ðr1 r3 Þ2 þ ðr3 r2 Þ2
ð14Þ
2
However, in bulk materials of the engineering structures subjected to cyclic loading, the distribution of defects such as impurities, microholes, and microcracks creates a local stress concentration. As a result, the bulk fatigue strength of a material is reduced. In the proposed approach, the effects of defects have been averaged. It is assumed, from a macroscopic point of view, that the material is homogeneous. It is postulated that in highcycle multiaxial fatigue, the equivalent stress cannot exceed the value of the fatigue limit in tension–compression. Mathematically, the failure criterion is req 6 raf,1. Then:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 raf ;1 ¼ pffiffiffi ðrx ry Þ2 þ ðrx rz Þ2 þ ðrz ry Þ2 þ 6ðs2xy þ s2yz þ s2zx Þ 2 ð15Þ Dividing both sides by raf,1: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ! 1 u rx ry 2 rx rz 2 rz ry 2 sxy 2 syz 2 szx 2 1 ¼ pffiffiffi t þ þ þ6 þ þ raf ;1 raf ;1 raf ;1 raf ;1 raf ;1 raf ;1 2
ð16Þ Then, the general form of the proposed criterion for multiaxial high-cycle loading can be expressed as:
rx ry 2 rx rz 2 rz ry 2 þ þ raf ;1 raf ;1 raf ;1 2 2 ! sxy syz szx 2 ¼2 þ6 þ þ raf ;1 raf ;1 raf ;1
ð17Þ
or by using principal stresses:
r1 r2 raf ;1
2
r1 r3 þ raf ;1
2
r3 r2 þ raf ;1
2
1 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðrx ry Þ2 þ 4s2xy
¼2
2
rx
raf ;1
rx ry raf ;1
2
ry raf ;1
þ
2
þ3
sxy raf ;1
pffiffiffi
rx ry 2 rx rz 2 rz ry 2 þ þ raf ;1 raf ;1 raf ;1 2 2 ! sxy syz szx 2 þ2 þ þ ¼2 saf ;1 saf ;1 saf ;1
2
rx
raf ;1
3.1. Criterion for plane cyclic stress loading The plane state of stress is considered because the initiation of fatigue damage usually takes place in the surface of a structural
¼1
ð23Þ
rx ry raf ;1
2
ry raf ;1
þ
2
sxy saf ;1
þ
2
¼1
ð24Þ
or using the principal stresses:
r1 raf ;1
2
r1 r2 raf ;1
2
þ
r2 raf ;1
2
¼1
ð25Þ
3.2. Criterion for bending and torsion loading Assuming pagreement ffiffiffi
with
the
HMH
hypothesis
that
raf ;1 =saf ;1 ¼ 3 and req = raf,1, the equation of the Gough ellipse curve is also expressed as:
1¼
2
r
raf ;1
þ
2
s
ð26Þ
saf ;1
Investigations so far have revealed that Eq. (26) does not fit well to the fatigue experimental results. By assuming that raf,1/saf,1 = 2 in Eq. (3), the equation of the limit curve is obtained in accordance with Eq. (2) or (26). The same result can bepffiffiffiachieved using the HMH hypothesis for the raf ;1 =saf ;1 ¼ 3 parameter. Another equivalent stress in the plane stress state for cyclic loading can be expressed as suggested by Debski [29] in the form:
ð27Þ
3
3
That yields the result raf,1/saf,1 = 4/3. Similarly, as derived above, the following equation can be obtained:
1¼ ð19Þ
2
or
ð18Þ
raf ;1 =saf ;1 ¼ 3, Eq.
ð22Þ
Therefore, the general criterion for plane stress cyclic loading can be expressed as either:
r 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi req ¼ þ r 2 þ 4s2
In agreement with the hypothesis that (17) can be rewritten as:
1 3
r
raf ;1
2 þ
2 3
r
raf ;1
þ
s
saf ;1
2 ð28Þ
It can be seen that for raf,1/saf,1 = 4/3, Eqs. (3) and (28) become identical, and for raf,1/saf,1 = 2, Eq. (3) becomes compatible with Eq. (26). Based on the data found in the literature, one can conclude that most of the experimental results are divided between the results obtained by Eq. (26) and Eq. (28). pIfffiffiffifor raf,1/saf,1 = 2 (not the HMH hypothesis that raf ;1 =saf ;1 ¼ 3), the Gough curve (3) assumes the shape of curve (26) and
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K.M. Golos et al. / Theoretical and Applied Fracture Mechanics 73 (2014) 3–8
Fig. 1. Plane stress state, where Fn(t) = plane cyclic forces of loading.
1
42CrMo4 steel
Mild steel
1
0,8
τ / τaf,-1
τ / τaf,-1
0,8
0,6
0,6
0,4
0,4 eq. (32)
eq. (32)
experimental data [24]
experimental data [22]
Gough criterion (3) 0,2
Gough criterion (3)
0,2
eq. (2)
eq.(2)
σ / σaf,-1
σ / σaf,-1
0
0 0
0,2
0,4
0,6
0,8
1
Fig. 2. Experimental [26] vs. calculated life diagrams obtained by Gough and proposed criterion for 42CrMo4 steel.
1
Hard steel
0,8
0
0,2
0,4
0,6
0,8
1
Fig. 4. Experimental [30] vs. calculated life diagrams obtained by Gough and proposed criterion for mild steel.
1
Alloy steel
τ / τaf,-1
τ / τaf,-1
0,8
0,6
0,6
0,4
0,4
eq. (32) experimental data [22]
eq. (32) Gough criterion (3) experimental data [27] eq. (2)
Gough criterion (3) 0,2
0,2
eq. (2)
σ / σaf,-1
σ / σaf,-1 0
0
0
0,2
0,4
0,6
0,8
1
Fig. 3. Experimental [30] vs. calculated life diagrams obtained by Gough and proposed criterion for hard steel.
0
0,2
0,4
0,6
0,8
1
Fig. 5. Experimental [31] vs. calculated life diagrams obtained by Gough and proposed criterion for alloy steel.
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K.M. Golos et al. / Theoretical and Applied Fracture Mechanics 73 (2014) 3–8
1
Gough’s criterion for ductile steel is a special case of the proposed extension. In the case when A = 1, B = 0 and C = 1, Eq. (29) reduces to Eq. (26). Additionally, for raf,1/saf,1 = 2, the coefficients are A = 1, B = 0 and C = 1, and for raf,1/saf,1 = 4/3, they become A = 1/3, B = 2/3 and C = 1. Consistently, in order to obtain the p values of the ffiffiffi coefficients A = 1, B = 0 and C = 1 for raf ;1 =saf ;1 ¼ 3, the results are derived from the HMH hypothesis. Based on experimental results, the following equations allow calculation of the values of A, B, and C:
30NiCrMo16 steel
τ / τaf,-1
0,8
0,6
pffiffiffi raf ;1 2 34 þ pffiffiffi A ¼ pffiffiffi ; 3 3 4 saf ;1 3 34 pffiffiffi raf ;1 2 2 3 B ¼ pffiffiffi þ pffiffiffi ; C¼1 3 3 4 saf ;1 3 34
0,4 eq. (32) experimental data [28] Gough criterion (3) 0,2
With the accuracy sufficient for engineering calculations, the approximated values of these parameters are:
eq. (2)
σ / σaf,-1
A ¼ 1:67
0 0
0,2
ð30Þ
0,4
0,6
0,8
raf ;1 raf ;1 1:90; B ¼ 1:67 þ 2:90; C ¼ 1 ð31Þ saf ;1 saf ;1
1
Therefore, the new criterion takes the form: Fig. 6. Experimental [32] vs. calculated life diagrams obtained by Gough and proposed criterion for 30NiCrMo4 steel.
τ / τaf,-1
0,4
eq. (32) Gough criterion (3) experimental data [33] eq. (2)
0,2
σ / σ af,-1 0 0
0,2
0,4
0,6
0,8
1
Fig. 7. Experimental [33] vs. calculated life diagrams obtained by Gough and proposed criterion for 0.34% C steel.
if for raf,1/saf,1 = 4/3, it assumes the shape of curve (28), then the coefficients of the Gough curve should be corrected. The following general form is proposed:
2
raf ;1
Aþ
ð32Þ
4. Comparison with experimental data
0,6
r
2 raf ;1 1:90 1:67
This allows determination pof ffiffiffi the criterion according to the HMH hypothesis if raf ;1 =saf ;1 ¼ 3 and determination of the criterion corresponding to hypothesis (27) when raf,1/saf,1 = 4/3 .
1
r
raf ;1 saf ;1 raf ;1 r s 2 þ 2:90 þ 1:67 ¼1 þ raf ;1 saf ;1 saf ;1
0.34% C steel
0,8
r raf ;1
Bþ
2
s
saf ;1
C¼1
ð29Þ
Data on six different types of steel are used to validate the proposed criterion. The experimental data related to in-phase biaxial loading are retrieved from the publications [26,30–33]. The summary of reported material properties is given in Table 1. It is important to emphasize that all material constants listed in the above table are available in the cited papers. The proposed criterion was subsequently applied to in-phase reversed bending tests and reversed torsion or tension–compression tests. The s/saf,1 vs. r/raf,1 diagrams reported in Figs. 2–7 clearly indicate that the use of the proposed criterion correctly predicts the fatigue life of tested materials under the assumed loading paths. For comparison, the predictions based on Gough’s model are also depicted in these figures. It is shown that the results obtained by the proposed criterion are generally more accurate in comparison to Gough’s model. 5. Scatter index analysis The scatter index is given by:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Xj 2 r0 ¼ r i¼1 i j
ð33Þ
Table 1 Material properties. Material
raf,1 (MPa)
saf,1 (MPa)
raf,1/saf,1
Rm (MPa)
Re (MPa)
A (%)
42CrMo4 steel [26] Hard steel [30] Mild steel [30] Alloy steel [31] 30NiCrMo16 steel [32] 0.34%C steel [33]
398 313.9 235.4 590 658 –
260 196.2 137.3 360 428 –
1.53 1.60 1.71 1.64 1.54 1.75
1025 704.1 518.8 900–1100 1200 700
750 500 340 700 1050 430
10 9 20 12 9 17
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K.M. Golos et al. / Theoretical and Applied Fracture Mechanics 73 (2014) 3–8
Fig. 8. Experimental data (1) vs. calculated life diagrams (2) obtained using Eq. (32), Eqs. (3) or (2).
Table 2 Scatter index analysis. r0
42CrMo4 steel Fig. 2
Hard steel Fig. 3
Mild steel Fig. 4
Alloy steel Fig. 5
30NiCrMo16 steel Fig. 6
0.34% C steel Fig. 7
Eq. (32) Eq. (3) Eq. (2)
0.091 0.095 0.091
0.035 0.042 0.034
0.041 0.052 0.039
0.026 0.030 0.031
0.059 0.061 0.068
0.018 0.010 0.016
where j is the number of experimental data, and ri is defined in Fig. 8. The scatter index analysis for Eq. (32), Eqs. (3) and (2) is presented in Table 2. 6. Conclusions The modification of Gough’s limit curve has been presented in the form of the stress-based criterion for multiaxial fatigue life prediction Eq. (32). The criterion for cyclic plane stress loading is proposed as a function of axial and shear stress for combined loading, and the A and B factors are expressed in terms ofprffiffiffiaf,1/saf,1. The criterion described by Eq. (32) for raf ;1 =saf ;1 ¼ 3 is a criterion according to the HMH hypothesis and Eq. (2) or (26). When raf,1/saf,1 = 4/3, Eq. (32) becomes Eq. (28). The proposed criterion could be applied to engineering calculations for different metals. Experimental data for six different materials was used to validate the criterion. The comparison analysis using the scatter index given by Eq. (33) is presented above. The most suitable equation is the one with the smallest scatter index value (Table 2, bolded values). It seems that the proposed criterion could produce more precise analyses of materials. References [1] E. Krempl, The influence of state of stress on low-cycle fatigue of structural materials: a literature survey and interpretive report, ASTM STP 549, 1974, pp. 46. [2] Y.S. Garud, Multiaxial fatigue: a survey of the state-of-the-art, J. Test. Eval. 9 (3) (1981) 165–178. [3] M.W. Brown, K.J. Miller, Two decades of progress in the assessment of multiaxial low-cycle fatigue life, in: C.B. Leis, P. Rabbe (Eds.), Low-cycle fatigue and life prediction, ASTM STP 770, Philadelphia, PA, ASTM, 1982, pp. 482–499. [4] E. Macha, C.M. Sonsino, Energy criteria of multiaxial fatigue failure, Fatigue Fract. Eng. Mater. Struct. 22 (1999) 1053–1070.
[5] C.E. Feltner, J.D. Morrow, Microplastic strain hysteresis energy as a criterion for fatigue fracture, J. Basic Eng. 83 (1961) 15–22. Series D. [6] Y.S. Garud, A new approach to the evaluation of fatigue under multiaxial loadings, J. Eng. Mater. Technol. 103 (1981) 118–125. [7] K. Golos, F. Ellyin, A total strain energy density theory for cumulative damage, J. Press. Vessel Technol. Trans. ASME 110 (1988) 38–41. [8] F. Ellyin, K. Golos, Z. Xia, In-phase and out-of-phase multiaxial fatigue, transactions ASME, J. Eng. Mater. Technol. 113 (1991) 112–118. [9] A. Varvani-Farhani, A new energy based plane parameter for fatigue life assessment of various metallic materials subjected to in-phase and outof-phase multiaxial fatigue loading condition, Int. J. Fatigue 22 (2000) 295–305. [10] Z. Hashin, Fatigue failure criteria for combined cyclic stress, Int. J. Fracture 17 (2) (1981) 101–109. [11] H.J. Gough, H.V. Pollard, The strength of metals under combined alternating stress, Proc. Inst. Mech. Eng. 131 (1935) 3–18. [12] H.J. Gough, H.V. Pollard, Properties of some materials for cast crankshafts, with special reference to combined alternating stresses, Proc. Inst. Automobile Eng. 31 (1937) 821–893. [13] H.J. Gough, Engineering steels under combined cyclic and static stresses, J. Appl. Mech. (June 1950) 113–125. [14] W.N. Findley, J.J. Coleman, B.C. Hanley, Theory for combined bending and torsion fatigue with data for SAE 4340 steel, Proc. Int. Conf. on Fatigue of Metals, London, 1956, pp. 150–157. [15] S.B. Lee, A criterion for fully reversed out-of-phase torsion and bending, in: K.J. Miller, M.W. Brown (Eds.), Multiaxial Fatigue, ASTM STP 853, ASTM, Philadelphia, 1985, pp. 553–568. [16] A. Fatemi, D.E. Socie, A critical plane approach to multiaxial fatigue damage including out of phase loading, Fatigue Fract. Eng. Mater. Struct. 11 (1988) 149–165. [17] D.E. Socie, P. Kurath, J. Koch, A multiaxial fatigue damage parameter, biaxial and multiaxial fatigue, EGF 3, M.W. Brown, K.J. Miller (Eds.), Mechanical Engineering, London, 1989, pp. 535–550. [18] A. Carpinteri, A. Spagnoli, S. Vantadori, Multiaxial assessment using a simplified critical plane-based criterion, Int. J. Fatigue 33 (2011) 969–976. [19] A. Carpinteri, A. Spagnoli, S. Vantadori, C. Bagni, Structural integrity assessment of metallic components under multiaxial fatigue: the C–S criterion and its evolution, Fatigue Fract. Eng. Mater. Struct. 36 (2013) 870– 883. [20] A. Carpinteri, A. Spagnoli, S. Vantadori, Reformulation in the frequency domain of a critical plane-based multiaxial fatigue criterion, Int. J. Fatigue (2014), http://dx.doi.org/10.1016/j.ijfatigue.2014.01.0080. [21] J.A. Araújo, A. Carpinteri, C. Ronchei, A. Spagnoli, Vantadori S., An alternative definition of the shear stress amplitude based on the Maximum Rectangular Hull method and application to the C-S (Carpinteri–Spagnoli) criterion, Fatigue Fract. Eng. Mater. Struct. 37 (2014) 764–771. [22] A. Carpinteri, A. Karolczuk, E. Macha, S. Vantadori, Expected position of the fatigue fracture plane by using the weighted mean principal Euler angles, Int. J. Fracture 185 (2002) 87–99. [23] A. Carpinteri, A. Spagnoli, S. Vantadori, Multiaxial fatigue life estimation in welded joints using the critical plane approach, Int. J. Fatigue 31 (2009) 188– 196. [24] A. Carpinteri, E. Macha, R. Brighenti, A. Spagnoli, Expected principal stress directions under multiaxial random loading, Part I, theoretical aspects of the weight function method, Int. J. Fatigue 21 (1999) 83–88. [25] A. Carpinteri, R. Briehenti, E. Macha, A. Spagnoli, Expected principal stress directions under multiaxial random loading, Part II, numerical simulation and experimental assessment through the weight function method, Int. J. Fatigue 21 (1999) 89–96. [26] I.V. Papadopoulos, P. Davoli, C. Gorla, M. Filippini, A. Bernasconi, A comparative study of multiaxial high-cycle fatigue criteria for metals, Int. J. Fatigue 19 (3) (1997) 219–235. [27] G. Sines, Behavior of metals under complex stresses, in: G. Sines, J.L. Waisman (Eds.), Metal Fatigue, McGraw-Hill, New York, 1959, pp. 145–169. [28] R.M. Jones, Deformation Theory of Plasticity, Bull Ridge Corporation, 2008. [29] M.A. Debski, Hypothesis of maximum stress at the edge of the circular hole notch, Aeronaut. Astronaut. Eng. 6 (1987) 18. [30] A. Carpinteri, A. Spagnoli, Multiaxial high-cycle fatigue criterion for hard metals, Int. J. Fatigue 23 (2001) 135–145. [31] V.P. Kogajev, Calculations on the strength with variables in time stresses, Mashinostroenie, Moscow, 1977. [32] A. Banvillet, T. Pain-Luc, S. Lessere, A volumetric energy based high cycle multiaxial fatigue criterion, Int. J. Fatigue 25 (2003) 755–769. [33] T.J. Dolan, Physical Properties, Metals Engineering Design, ASME Handbook, 1953, pp. 93–99. [34] R. Brighenti, A. Carpinteri, A notch multiaxial-fatigue approach based on damage mechanics, Int. J. Fatigue 39 (2012) 122–133. [35] N.S. Ottosen, R. Stenström, M. Ristinmaa, Continuum approach to high-cycle fatigue modeling, Int. J. Fatigue 30 (2008) 996–1006. [36] S.H. Stephanov, A curvilinear integral method for multiaxial fatigue life computing under non-proportional, arbitrary or random stressing, Int. J. Fatigue 15 (1993) 467–472.
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Theoretical and Applied Fracture Mechanics journal homepage: www.elsevier.com/locate/tafmec
A stress-based fatigue criterion to assess high-cycle fatigue under in-phase multiaxial loading conditions Krzysztof M. Golos a,b, Daniel K. Debski a,⇑, Marek A. Debski c a
Warsaw University of Technology, Narbutta 84 Street, 02-524 Warsaw, Poland Institute of Mechanised Construction and Rock Mining, Racjonalizacji 6/8 Street, 02-673 Warsaw, Poland c Institute of Aviation, Aleja Krakowska 110/114 Street, 02-256 Warsaw, Poland b
a r t i c l e
i n f o
Article history: Available online 22 July 2014 Keywords: High-cycle fatigue Multiaxial fatigue Biaxial cyclic loading Stress-based criterion
a b s t r a c t In this paper, a new criterion for in-phase multiaxial fatigue life prediction in high-cycle fatigue is proposed. The new stress-based fatigue criterion is based on the modification of Gough’s limit curve. In the case of cyclic plane stress loading, the particular form of the criterion is discussed in detail. The values of parameters in the proposed criterion can be obtained based on fatigue limits in bending and torsion. This allows use of the proposed criterion with almost no limitations to engineering calculations for different metals. Analytical predictions are compared to available experimental data from the literature for six different materials. Additionally, the predictions based on the proposed criterion are compared with the classic Gough’s criterion. The predictions based on the proposed criterion seem to be more precise for the analysed materials. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Despite great progress in fatigue analysis in recent decades, there is a need for reliable methods to predict fatigue life under multiaxial cyclic loading. It is desirable to actually base such a proposal on general principles of solid mechanics and the evaluation of constants contained in the criterion, from the standard uniaxial tests. The multiaxial fatigue tests are complex and expensive. Therefore, several proposals have been made to correlate uniaxial fatigue test data. A review of various failure criteria for multiaxial fatigue is given by Krempl [1], Garud [2], Brown and Miller [3], Macha and Sonsino [4]. These approaches can be divided into three categories: stressbased methods, strain-based methods and energy-based approaches. In the low-cycle fatigue regime, when relatively large plastic strains occur, the strain-based criteria are recommended. In the high-cycle regime, with plastic strain decreasing in the material, the stress-based criteria are often used. To obtain a unified description of both low- and high-regimes, a strain energy strategy has been proposed. This approach is based on the assumption that the energy is proportional to the fatigue damage calculated in different ways. Feltner and Morrow [5] proposed a criterion based on hysteresis energy; Garud [6] presented the plastic strain energy ⇑ Corresponding author. Tel.: +48 22 234 82 61, mobile: +48 500 131 215. E-mail addresses: [email protected] (K.M. Golos), [email protected]. pl (D.K. Debski). http://dx.doi.org/10.1016/j.tafmec.2014.07.005 0167-8442/Ó 2014 Elsevier Ltd. All rights reserved.
model for multiaxial loading; and Golos and Ellyin [7,8] modelled uniaxial, multiaxial fatigue and cumulative damage in terms of the total strain energy density. Among different energy-based approaches, a non-dimensional parameter that holds terms for material properties was proposed by Varvani-Farhani [9]. This criterion also includes the mean stress effect and additional hardening for out-of-phase loading conditions. The energy-based approach seems to be promising because energy is a scalar quantity and includes both stress and strain components of the loading path. In the case of stress-based criteria, we can distinguish two main methodologies. The first one is based on the analysis of the cyclic stress invariants, and the second one considers the stress state at the selected plane. The invariant stress criterion can be expressed in a general form as follows [10]:
FðI1 ; I2 ; I3 ; Nf Þ ¼ 0
ð1Þ
where I1 = r1 + r2 + r3, I2 = r1r2 + r1r3 + r2r3, and I3 = r1 r2 r3; r1, r2, and r3 are principal stresses; and Nf is the number of cycles to failure. According to the second methodology, fatigue analysis is performed on one critical section in the element. This plane is called critical and is usually different for selected fatigue models. It is often assumed that the failure occurs at a given number of cycles when the function of a normal stress and a shear stress reaches the critical state.
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K.M. Golos et al. / Theoretical and Applied Fracture Mechanics 73 (2014) 3–8
A brief overview of several stress-based high-cycle fatigue criteria is first discussed in this paper. Several criteria proposed in recent decades are aimed at reducing a given multiaxial stress state to an equivalent uniaxial stress loading. Although there are many proposed criteria for biaxial loading, most of them are limited to specific materials or loading conditions. To the authors’ knowledge, there is no existing biaxial fatigue criterion that is universally accepted. The main goal of this paper is to propose a general form of a criterion for in-phase multiaxial fatigue failure based on the equivalent stress concept. One of the fundamental problems of calculating fatigue strength is the determination of the stress safety space corresponding to a given complex state of stress. Unlike the previous stress-based models, equivalent stress methods for multiaxial in-phase loading, based on the distortion strain energy density, are theoretically correlated with the fatigue limit and depend on the material parameters. This allows the use of the proposed criterion for engineering calculations – with almost no limitations and for different metals. The current criterion is compared with Gough’s model using the available experimental data for six different materials and various cyclic load paths. 2. Biaxial stress-based high-cycle fatigue criteria Because the main focus of this paper is high-cycle fatigue, only the stress-based approach commonly used in high-cycle fatigue design procedures is reviewed in this section. Large numbers of criteria are present in the literature – only a few common criteria are briefly discussed here. Generally, as was mentioned above, the stress-based criteria can be divided into two categories, namely, those based on stress invariants related to the number of cycles to failure (including equivalent stress- and average stress-based models) and those based on critical plane stress (calculated in different ways). Concerning older theories, the authors feel that attention has been taken from the more widely used theories in high-cycle fatigue. Gough and Pollard [11–13] proposed two criteria for metals under combined in-phase bending and torsion. For ductile metals, the equation takes the form of an ellipse quadrant:
r
2
þ
raf ;1
2
s
saf ;1
¼1
ð2Þ
where raf,1 and saf,1 are fatigue limits in reversed bending and torsion, respectively. For brittle metals, the ellipse rewritten in the form:
r
2
raf ;1 r 1 þ raf ;1 saf ;1
raf ;1
raf ;1 2 þ
saf ;1
s saf ;1
2 ¼1 ð3Þ
has been proposed. Findley [14] proposed a criterion based on the linear combination of the axial stress and the shear stress squared (parabola) in the following form:
r
raf ;1
þ
s
saf ;1
2
¼1
ð4Þ
Considering the phase difference between loading, Lee [15] modified the ellipse quadrant of Gough as follows:
"
ra;eq ¼ r 1 þ
2 #1=2
raf ;1 s saf ;1 r
ð5Þ
In recent years, criteria based on the critical plane approach have attracted increasing attention [16–19].
Carpinteri et al. [18–21] used the maximum normal stress and the shear stress amplitude on the critical plane as parameters to modify Gough’s criterion. The calculation of the critical plane is performed in two steps. First, the weighted mean direction of the maximum principal stress under multiaxial random loading is estimated [20–25]. According to this concept, the fatigue failure assessment is presented by considering a quadratic combination of the maximum normal stress (rmax) and the shear stress amplitude (sa) in the following form:
rmax raf ;1
2
þ
sa
2
saf ;1
¼1
ð6Þ
Several different multiaxial fatigue criteria based on stress invariants are discussed in [26]. The criterion formulated by Sines [27] is probably the most popular invariant-based approach. According to this method, the stress state is in its fatigue limit condition when the following state is achieved:
pffiffiffiffiffiffiffi I2;a þ krH k
ð7Þ
where I2a is the second invariant of the stress tensor, rH is the hydrostatic stress amplitude, and k and k are material constants that can be calculated by considering two fatigue limits. Papadopoulos et al. [26] proposed the fatigue model based on the average stress approach, which can be expressed as:
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffi r2a 2 hT a i2 þ k½I1a þ I1m ¼ þ sa þ k½I1a þ I1m ¼ f 3
ð8Þ
where hTai2 is the average quantity within the volume, I1a is the first invariant of the stress tensor, ra is the bending stress amplitude, sa is the torsion stress amplitude, and k and f are material parameters depending on the fatigue limit. Brighenti and Carpinteri [34] proposed a fatigue damage model to evaluate the life assessment of notched structural elements under multiaxial stress histories. Ottosen et al. [35] presented high-cycle fatigue modelling using multiaxial load histories (inphase and out-of-phase). Stephanov [36] showed a method for multiaxial fatigue life estimation under non-proportional stresses. 3. Equivalent stress based criterion for multiaxial cyclic loading The stress at the point in the structure subjected to multiaxial cyclic loading can be expressed through the stress tensor as:
2
rx ðtÞ 6 s ðtÞ 6 yx ½Drij ðtÞ=2 ¼ 6 4 szx ðtÞ
sxy ðtÞ sxz ðtÞ 3 ry ðtÞ syz ðtÞ 7 7 7 szy ðtÞ rz ðtÞ 5
ð9Þ
This stress state can be expressed in the principal axes of the coordinate system as:
2
3 r1 ðtÞ 0 0 r2 ðtÞ 0 7 7 7 0 r3 ðtÞ 5
60 6 ½Drij ðtÞ=2 ¼ 6 40
ð10Þ
In this paper, the form strain energy density approach presented by Huber has been used to analyse multiaxial fatigue in high-cycle loading. This approach leads to similar conclusions as obtained by von Mises and Hencky. Therefore, here it will be described as Huber–Mises–Hencky’s (HMH) hypothesis [28]. According to this approach, failure under cyclic multiaxial fatigue in high-cycle loading is proposed to occur when the maximum distortion strain energy density exceeds the distortion strain energy density at the yield stress obtained from a uniaxial tensile test, i.e., /f 6 /f,yield.
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K.M. Golos et al. / Theoretical and Applied Fracture Mechanics 73 (2014) 3–8
The distortion strain energy density can be expressed as:
/f ¼
1 þ mh 6E
ðrx ry Þ2 þ ðrx rz Þ2 þ ðrz ry Þ2 þ 6ðs2xy þ s2yz þ s2zx Þ
i
ð11Þ In the uniaxial state of stress ry – 0, rx = rz = 0, and
sxy = syz = szx = 0: /f ¼
1þm 2 r 3E y
ð12Þ
Accepting that strains in both states are equal and that ry = req, then the expression for equivalent stress can be obtained:
req
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðrx ry Þ2 þ ðrx rz Þ2 þ ðrz ry Þ2 þ 6ðs2xy þ s2yz þ s2zx Þ
1 ¼ pffiffiffi 2
ð13Þ
element. In a two-dimensional state, Fig. 1 implies that one of the coordinates does not play a role in the description of the problem. The tensor component associated with the z direction is equal to zero. Under plain stress loading, rz = 0 and syz = szx = 0 or r3 = 0, and the expression for the equivalent stress takes one of the following forms:
req ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2x þ r2y rx ry þ 3s2xy
ð20Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r21 þ r22 r1 r2
ð21Þ
or
req ¼
where the principal stresses are given by:
r1 ; r2 ¼
rx þ ry 2
In the principal stress system, Eq. (10) becomes:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
req ¼ pffiffiffi ðr1 r2 Þ2 þ ðr1 r3 Þ2 þ ðr3 r2 Þ2
ð14Þ
2
However, in bulk materials of the engineering structures subjected to cyclic loading, the distribution of defects such as impurities, microholes, and microcracks creates a local stress concentration. As a result, the bulk fatigue strength of a material is reduced. In the proposed approach, the effects of defects have been averaged. It is assumed, from a macroscopic point of view, that the material is homogeneous. It is postulated that in highcycle multiaxial fatigue, the equivalent stress cannot exceed the value of the fatigue limit in tension–compression. Mathematically, the failure criterion is req 6 raf,1. Then:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 raf ;1 ¼ pffiffiffi ðrx ry Þ2 þ ðrx rz Þ2 þ ðrz ry Þ2 þ 6ðs2xy þ s2yz þ s2zx Þ 2 ð15Þ Dividing both sides by raf,1: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ! 1 u rx ry 2 rx rz 2 rz ry 2 sxy 2 syz 2 szx 2 1 ¼ pffiffiffi t þ þ þ6 þ þ raf ;1 raf ;1 raf ;1 raf ;1 raf ;1 raf ;1 2
ð16Þ Then, the general form of the proposed criterion for multiaxial high-cycle loading can be expressed as:
rx ry 2 rx rz 2 rz ry 2 þ þ raf ;1 raf ;1 raf ;1 2 2 ! sxy syz szx 2 ¼2 þ6 þ þ raf ;1 raf ;1 raf ;1
ð17Þ
or by using principal stresses:
r1 r2 raf ;1
2
r1 r3 þ raf ;1
2
r3 r2 þ raf ;1
2
1 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðrx ry Þ2 þ 4s2xy
¼2
2
rx
raf ;1
rx ry raf ;1
2
ry raf ;1
þ
2
þ3
sxy raf ;1
pffiffiffi
rx ry 2 rx rz 2 rz ry 2 þ þ raf ;1 raf ;1 raf ;1 2 2 ! sxy syz szx 2 þ2 þ þ ¼2 saf ;1 saf ;1 saf ;1
2
rx
raf ;1
3.1. Criterion for plane cyclic stress loading The plane state of stress is considered because the initiation of fatigue damage usually takes place in the surface of a structural
¼1
ð23Þ
rx ry raf ;1
2
ry raf ;1
þ
2
sxy saf ;1
þ
2
¼1
ð24Þ
or using the principal stresses:
r1 raf ;1
2
r1 r2 raf ;1
2
þ
r2 raf ;1
2
¼1
ð25Þ
3.2. Criterion for bending and torsion loading Assuming pagreement ffiffiffi
with
the
HMH
hypothesis
that
raf ;1 =saf ;1 ¼ 3 and req = raf,1, the equation of the Gough ellipse curve is also expressed as:
1¼
2
r
raf ;1
þ
2
s
ð26Þ
saf ;1
Investigations so far have revealed that Eq. (26) does not fit well to the fatigue experimental results. By assuming that raf,1/saf,1 = 2 in Eq. (3), the equation of the limit curve is obtained in accordance with Eq. (2) or (26). The same result can bepffiffiffiachieved using the HMH hypothesis for the raf ;1 =saf ;1 ¼ 3 parameter. Another equivalent stress in the plane stress state for cyclic loading can be expressed as suggested by Debski [29] in the form:
ð27Þ
3
3
That yields the result raf,1/saf,1 = 4/3. Similarly, as derived above, the following equation can be obtained:
1¼ ð19Þ
2
or
ð18Þ
raf ;1 =saf ;1 ¼ 3, Eq.
ð22Þ
Therefore, the general criterion for plane stress cyclic loading can be expressed as either:
r 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi req ¼ þ r 2 þ 4s2
In agreement with the hypothesis that (17) can be rewritten as:
1 3
r
raf ;1
2 þ
2 3
r
raf ;1
þ
s
saf ;1
2 ð28Þ
It can be seen that for raf,1/saf,1 = 4/3, Eqs. (3) and (28) become identical, and for raf,1/saf,1 = 2, Eq. (3) becomes compatible with Eq. (26). Based on the data found in the literature, one can conclude that most of the experimental results are divided between the results obtained by Eq. (26) and Eq. (28). pIfffiffiffifor raf,1/saf,1 = 2 (not the HMH hypothesis that raf ;1 =saf ;1 ¼ 3), the Gough curve (3) assumes the shape of curve (26) and
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K.M. Golos et al. / Theoretical and Applied Fracture Mechanics 73 (2014) 3–8
Fig. 1. Plane stress state, where Fn(t) = plane cyclic forces of loading.
1
42CrMo4 steel
Mild steel
1
0,8
τ / τaf,-1
τ / τaf,-1
0,8
0,6
0,6
0,4
0,4 eq. (32)
eq. (32)
experimental data [24]
experimental data [22]
Gough criterion (3) 0,2
Gough criterion (3)
0,2
eq. (2)
eq.(2)
σ / σaf,-1
σ / σaf,-1
0
0 0
0,2
0,4
0,6
0,8
1
Fig. 2. Experimental [26] vs. calculated life diagrams obtained by Gough and proposed criterion for 42CrMo4 steel.
1
Hard steel
0,8
0
0,2
0,4
0,6
0,8
1
Fig. 4. Experimental [30] vs. calculated life diagrams obtained by Gough and proposed criterion for mild steel.
1
Alloy steel
τ / τaf,-1
τ / τaf,-1
0,8
0,6
0,6
0,4
0,4
eq. (32) experimental data [22]
eq. (32) Gough criterion (3) experimental data [27] eq. (2)
Gough criterion (3) 0,2
0,2
eq. (2)
σ / σaf,-1
σ / σaf,-1 0
0
0
0,2
0,4
0,6
0,8
1
Fig. 3. Experimental [30] vs. calculated life diagrams obtained by Gough and proposed criterion for hard steel.
0
0,2
0,4
0,6
0,8
1
Fig. 5. Experimental [31] vs. calculated life diagrams obtained by Gough and proposed criterion for alloy steel.
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K.M. Golos et al. / Theoretical and Applied Fracture Mechanics 73 (2014) 3–8
1
Gough’s criterion for ductile steel is a special case of the proposed extension. In the case when A = 1, B = 0 and C = 1, Eq. (29) reduces to Eq. (26). Additionally, for raf,1/saf,1 = 2, the coefficients are A = 1, B = 0 and C = 1, and for raf,1/saf,1 = 4/3, they become A = 1/3, B = 2/3 and C = 1. Consistently, in order to obtain the p values of the ffiffiffi coefficients A = 1, B = 0 and C = 1 for raf ;1 =saf ;1 ¼ 3, the results are derived from the HMH hypothesis. Based on experimental results, the following equations allow calculation of the values of A, B, and C:
30NiCrMo16 steel
τ / τaf,-1
0,8
0,6
pffiffiffi raf ;1 2 34 þ pffiffiffi A ¼ pffiffiffi ; 3 3 4 saf ;1 3 34 pffiffiffi raf ;1 2 2 3 B ¼ pffiffiffi þ pffiffiffi ; C¼1 3 3 4 saf ;1 3 34
0,4 eq. (32) experimental data [28] Gough criterion (3) 0,2
With the accuracy sufficient for engineering calculations, the approximated values of these parameters are:
eq. (2)
σ / σaf,-1
A ¼ 1:67
0 0
0,2
ð30Þ
0,4
0,6
0,8
raf ;1 raf ;1 1:90; B ¼ 1:67 þ 2:90; C ¼ 1 ð31Þ saf ;1 saf ;1
1
Therefore, the new criterion takes the form: Fig. 6. Experimental [32] vs. calculated life diagrams obtained by Gough and proposed criterion for 30NiCrMo4 steel.
τ / τaf,-1
0,4
eq. (32) Gough criterion (3) experimental data [33] eq. (2)
0,2
σ / σ af,-1 0 0
0,2
0,4
0,6
0,8
1
Fig. 7. Experimental [33] vs. calculated life diagrams obtained by Gough and proposed criterion for 0.34% C steel.
if for raf,1/saf,1 = 4/3, it assumes the shape of curve (28), then the coefficients of the Gough curve should be corrected. The following general form is proposed:
2
raf ;1
Aþ
ð32Þ
4. Comparison with experimental data
0,6
r
2 raf ;1 1:90 1:67
This allows determination pof ffiffiffi the criterion according to the HMH hypothesis if raf ;1 =saf ;1 ¼ 3 and determination of the criterion corresponding to hypothesis (27) when raf,1/saf,1 = 4/3 .
1
r
raf ;1 saf ;1 raf ;1 r s 2 þ 2:90 þ 1:67 ¼1 þ raf ;1 saf ;1 saf ;1
0.34% C steel
0,8
r raf ;1
Bþ
2
s
saf ;1
C¼1
ð29Þ
Data on six different types of steel are used to validate the proposed criterion. The experimental data related to in-phase biaxial loading are retrieved from the publications [26,30–33]. The summary of reported material properties is given in Table 1. It is important to emphasize that all material constants listed in the above table are available in the cited papers. The proposed criterion was subsequently applied to in-phase reversed bending tests and reversed torsion or tension–compression tests. The s/saf,1 vs. r/raf,1 diagrams reported in Figs. 2–7 clearly indicate that the use of the proposed criterion correctly predicts the fatigue life of tested materials under the assumed loading paths. For comparison, the predictions based on Gough’s model are also depicted in these figures. It is shown that the results obtained by the proposed criterion are generally more accurate in comparison to Gough’s model. 5. Scatter index analysis The scatter index is given by:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Xj 2 r0 ¼ r i¼1 i j
ð33Þ
Table 1 Material properties. Material
raf,1 (MPa)
saf,1 (MPa)
raf,1/saf,1
Rm (MPa)
Re (MPa)
A (%)
42CrMo4 steel [26] Hard steel [30] Mild steel [30] Alloy steel [31] 30NiCrMo16 steel [32] 0.34%C steel [33]
398 313.9 235.4 590 658 –
260 196.2 137.3 360 428 –
1.53 1.60 1.71 1.64 1.54 1.75
1025 704.1 518.8 900–1100 1200 700
750 500 340 700 1050 430
10 9 20 12 9 17
8
K.M. Golos et al. / Theoretical and Applied Fracture Mechanics 73 (2014) 3–8
Fig. 8. Experimental data (1) vs. calculated life diagrams (2) obtained using Eq. (32), Eqs. (3) or (2).
Table 2 Scatter index analysis. r0
42CrMo4 steel Fig. 2
Hard steel Fig. 3
Mild steel Fig. 4
Alloy steel Fig. 5
30NiCrMo16 steel Fig. 6
0.34% C steel Fig. 7
Eq. (32) Eq. (3) Eq. (2)
0.091 0.095 0.091
0.035 0.042 0.034
0.041 0.052 0.039
0.026 0.030 0.031
0.059 0.061 0.068
0.018 0.010 0.016
where j is the number of experimental data, and ri is defined in Fig. 8. The scatter index analysis for Eq. (32), Eqs. (3) and (2) is presented in Table 2. 6. Conclusions The modification of Gough’s limit curve has been presented in the form of the stress-based criterion for multiaxial fatigue life prediction Eq. (32). The criterion for cyclic plane stress loading is proposed as a function of axial and shear stress for combined loading, and the A and B factors are expressed in terms ofprffiffiffiaf,1/saf,1. The criterion described by Eq. (32) for raf ;1 =saf ;1 ¼ 3 is a criterion according to the HMH hypothesis and Eq. (2) or (26). When raf,1/saf,1 = 4/3, Eq. (32) becomes Eq. (28). The proposed criterion could be applied to engineering calculations for different metals. Experimental data for six different materials was used to validate the criterion. The comparison analysis using the scatter index given by Eq. (33) is presented above. The most suitable equation is the one with the smallest scatter index value (Table 2, bolded values). It seems that the proposed criterion could produce more precise analyses of materials. References [1] E. Krempl, The influence of state of stress on low-cycle fatigue of structural materials: a literature survey and interpretive report, ASTM STP 549, 1974, pp. 46. [2] Y.S. Garud, Multiaxial fatigue: a survey of the state-of-the-art, J. Test. Eval. 9 (3) (1981) 165–178. [3] M.W. Brown, K.J. Miller, Two decades of progress in the assessment of multiaxial low-cycle fatigue life, in: C.B. Leis, P. Rabbe (Eds.), Low-cycle fatigue and life prediction, ASTM STP 770, Philadelphia, PA, ASTM, 1982, pp. 482–499. [4] E. Macha, C.M. Sonsino, Energy criteria of multiaxial fatigue failure, Fatigue Fract. Eng. Mater. Struct. 22 (1999) 1053–1070.
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