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A new energy-critical plane parameter for fatigue life assessment of various metallic materials subjected to in-phase and out-of-phase multiaxial fatigue loading conditions
International Journal of Fatigue 22 (2000) 295–305 www.elsevier.com/locate/ijfatigue
A new energy-critical plane parameter for fatigue life assessment of various metallic materials subjected to in-phase and out-of-phase multiaxial fatigue loading conditions A. Varvani-Farahani
*
Department of Mechanical Engineering, Ryerson Polytechnic University, Toronto, Ontario, Canada M5B 2K3 Received 2 December 1999; accepted 12 December 1999
Abstract A new multiaxial fatigue parameter for in-phase and out-of-phase straining is proposed. The parameter proposed is the sum of the normal energy range and the shear energy range calculated for the critical plane on which the stress and strain Mohr’s circles are the largest during the loading and unloading parts of a cycle. The normal and shear energies used in this parameter are divided by the tensile and shear fatigue properties, respectively. The proposed parameter, unlike many other parameters, does not use an empirical fitting factor. The proposed parameter successfully correlates multiaxial fatigue lives for: (a) various in-phase and out-of phase multiaxial fatigue straining conditions, (b) tests in which a mean stress was applied normal to the maximum shear plane, and (c) out-of-phase tests in which there was additional hardening. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Multiaxial fatigue model; In-phase and out-of-phase strain paths; Shear and normal energies; Critical plane; Mean stress effect; Strain hardening
1. Introduction Many engineering components that undergo fatigue loading experience multiaxial stresses, in which two or three principal stresses fluctuate with time, i.e. the corresponding principal stresses are out-of-phase or the principal directions change during a cycle of loading. Extensive reviews of multiaxial fatigue life prediction methods are presented by Garud [1], Brown and Miller [2], and You and Lee [3]. Fatigue analysis using the concept of a critical plane of maximum shear strain is very effective because the critical plane concept is based on the fracture mode or the initiation mechanism of cracks. In the critical plane concept, after determining the maximum shear strain plane, many researchers define fatigue parameters as combinations of the maximum shear strain (or stress) and normal strain (or stress) on that plane to explain multiaxial fatigue behavior [4–7]. However, critical
* Tel.: +1-416-979-5000. E-mail address: [email protected] (A. Varvani-Farahani).
plane parameters have been criticized for lack of adherence to rigorous continuum mechanics fundamentals. To compensate this lack, Liu [8], Chu et al. [9], and Glinka et al. [10] used the energy criterion in conjunction with the critical plane approach. Liu [8] calculated the virtual strain energy (VSE) in the critical plane by the use of crack initiation modes and stress–strain Mohr’s circles. In the calculation of VSE, Liu included both elastic energy and plastic energy while the elastic energy was not considered in Garud’s model [11]. Chu et al. [9] formulated normal and shear energy components based on the Smith–Watson–Topper parameter. They determined the critical plane and the largest damage parameter from the transformation of strains and stresses onto planes spaced at equal increments using a generalized Mroz model. Glinka et al. [10] proposed a multiaxial life parameter based on the summation of the products of normal and shear strains and stresses on the critical shear plane. In the present study a multiaxial fatigue parameter for various in-phase and out-of-phase strain paths is proposed. The parameter is given by the sum of the normal energy range and the shear energy range calculated for the critical plane at which the stress and strain Mohr’s
0142-1123/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 2 - 1 1 2 3 ( 0 0 ) 0 0 0 0 2 - 5
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Nomenclature ⌬(gap/2), ⌬eap the shear and axial strain ranges respectively ⌬(γmax/2), ⌬en maximum shear strain range and normal strain range acting on the critical plane, respectively ⌬ea, ⌬sa, ⌬ta applied tensorial strain range, stress range, and shear stress rage, respectively ⌬eij, ⌬sij the strain and stress tensor ranges (where i and j=1, 2, 3) ⌬tmax, ⌬sn maximum shear stress range and normal stress range, respectively E the elastic modulus e1, e2, e3 principal strains (e1⬎e2⬎e3) ee, ep elastic strain and plastic strain respectively f phase delay between strains on the axial and torsional axes ne, np, neff elastic, plastic, and effective Poisson’s ratios respectively q1, q2 angles during loading and unloading parts of a cycle respectively at which the Mohr’s circles are the largest s1, s2, s3 principal stresses (s1⬎s2⬎s3) mean normal stress s nm snmax maximum normal stress snmin minimum normal stress s⬘f, ε⬘f the axial fatigue strength coefficient and axial fatigue ductility coefficient, respectively t⬘f, g⬘f the shear fatigue strength coefficient and shear fatigue ductility coefficient, respectively
circles are the largest during the loading and unloading parts of a cycle. The normal and shear energies in this parameter have been weighted by the tensile and shear fatigue properties, respectively, and the parameter requires no empirical fitting factor. This parameter takes into account the effect of the mean stress applied normal to the maximum shear plane. The proposed parameter also increases when there is additional hardening caused by out-of-phase straining, while strain-based parameters fail to take into account this effect. The proposed parameter gives a good correlation of multiaxial fatigue lives for various in-phase and out-of-phase straining conditions.
2. Materials and multiaxial fatigue data Table 1 lists the references for in-phase and out-ofphase multiaxial fatigue data used in this study and tabulates the fatigue properties of the materials used. Fatigue coefficients sf⬘ and ef⬘ are the axial fatigue strength coefficient and axial fatigue ductility coefficient, respectively and tf⬘ and gf⬘ are the shear fatigue strength coefficient and shear fatigue ductility coefficient, respectively. These coefficients are illustrated in Fig. 1 for (a) uniaxial and (b) torsional fatigue loading life–strain curves.
3. In-phase and out-of-phase strain paths In this study, for convenience of presentation, inphase and out-of-phase strain paths have been categorized into three kinds: (a) in-phase strain paths (strain his-
tories A1, A2, A3, A4, A5, and A6), (b) linear out-ofphase strain paths (strain histories B1 and B2), and finally (c) non-linear out-of-phase strain paths (C1, C2, C3, C4, and C5). 3.1. In-phase strain paths In in-phase straining, both axial and shear strain cycles are alternating with no phase difference. Strain paths in in-phase straining are linear. For the in-phase straining data used in this study, the strain histories, strain paths, and strain and stress Mohr’s circles are presented in Figs. 2a and 2b. The largest stress and strain Mohr’s circles during the loading part (at q1) and unloading part (at q2) of a cycle, for which the maximum shear stress and strain and corresponding normal stress and normal strain values are calculated, are illustrated in Fig. 2. In this figure, the light Mohr’s circles are the largest during the loading part, and the dark Mohr’s circles are the largest during unloading part of a cycle. The strain histories A1, A2, and A3 correspond to uniaxial straining, torsional straining, and in-phase combined axial and torsional straining, respectively. The linear in-phase strain paths shown in Fig. 2b have mean values. Strain history A4 is a combined axial and torsional strain path with an axial mean strain. Strain history A5 has a torsional mean strain, and finally strain history A6 has both axial and torsional mean strains. 3.2. Linear out-of-phase strain paths In out-of-phase alternating straining, there is a phase difference between the axial and shear strain cycles.
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Table 1 Fatigue properties of materials used in this study Materials and fatigue data
E (GPa)
ef⬘
sf⬘ (MPa)
G (GPa)
gf⬘
tf⬘ (MPa)
Ni–Cr–Mo–V steela [12]b 1045 steel [13–15]b Inconel 718 [16]b Haynes 188 [17]b Waspaloy [18,19]c Mild steel [20]c Stainless steel [21]b
200
1.14
680
77
1.69
444
206 208.5 170.2 362 210 185
0.26 2.67 0.489 0.381 0.1516 0.171
948 1640 823 2610 1009 1000
79.2 80.2 65.5 139.2 80.8 71
0.413 3.62 1.78 0.516 0.322 0.413
505 1030 635 1640 431 709
a b c
Ni–Cr–Mo–V steel is known as Rotor steel. Fatigue properties are given by referenced papers. Fatigue properties are calculated from uniaxial and torsional fatigue life–strain data.
Fig. 1.
Schematic presentation of fatigue life–strain curves for (a) uniaxial loading and (b) torsional loading.
Strain history B1 (box) and strain history B2 (two-box), shown in Fig. 2c, are linear out-of-phase strain histories. 3.3. Non-linear out-of-phase strain paths In the non-linear out-of-phase strain histories examined there is a phase delay between the axial strain and torsional strain. Strain paths are elliptical and as the phase difference increases the elliptical path becomes larger in its minor diameter and finally at a 90° phase difference the strain path becomes circular. Strain histories C1, C2, C3, and C4 present out-of-phase axial and torsional straining with phase delays of 30°, 45°, 60°, and 90°, respectively. Strain history C5 corresponds to a 90° out-of-phase strain path containing an axial mean strain value. Fig. 2d presents non-linear out-of-phase strain paths, strain histories, and strain and stress Mohr’s circles. The maximum shear strains for in-phase and out-ofphase paths were numerically calculated at 10° increments through a cycle and are presented in Figs. 3a and 3b respectively.
4. Proposed parameter and analysis Fig. 4 illustrates a thin-walled tubular specimen subjected to combined axial and torsional fatigue. The strain and stress tensors for a thin-walled tubular specimen subjected to axial and torsional fatigue are given by Eqs. (1) and (2), respectively:
冢
冉冊 gap 2
−veff⌬eap ⌬
⌬eij ⫽
冉冊
⌬
gap 2
0
冢
0
冣
⌬eap
0
0
−veff⌬eap
冣
(1)
⌬ta 0
⌬sij ⫽ ⌬ta ⌬sa 0 0
0
0
0
(2)
298
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Fig. 2. (a) Strain history, strain path, and Mohr’s circle presentation for in-phase strain paths. (b) In-phase strain histories, paths, and Mohr’s circle presentations for in-phase paths containing mean strain values. (c) Linear out-of-phase strain history, path, and strain–stress Mohr’s circles. (d) Strain histories, paths, and Mohr’s circle presentations for non-linear out-of-phase strain paths without and with mean strain values.
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299
Fig. 2. (continued)
where axial and shear strain ranges ⌬eap and ⌬(gap/2) respectively are given by Eqs. (3) and (4) as: ⌬eap⫽⌬ea sin q
冉 冊 冉冊
gap ga ⌬ ⫽⌬ sin(q⫺f) 2 2
(3) (4)
where ea and ga/2 are the applied axial and shear amplitude strains, respectively. The angle q is the angle during a cycle of straining at which the Mohr’s circle is the largest and has the maximum value of shear strain. Angle f corresponds to the phase delay between strains on the axial and torsional axes. In Eq. (2) ⌬sa and ⌬ta are the ranges of axial and shear stresses, respectively. In Eq. (1) neff is the effective Poisson’s ratio which is given by: neff⫽
neee+npep ee+ep
(5)
where ne=0.3 is the elastic Poisson’s ratio and np=0.5 is the plastic Poisson’s ratio. The axial elastic and plastic strains are given by Eqs. (6a) and (6b), respectively: sa ee⫽ E ep⫽eap⫺
(6a) sa E
(6b)
The range of maximum shear strain and the corresponding normal strain range on the critical plane at which both strain and stress Mohr’s circles are the larg-
est during loading (at the angle q1) and unloading (at the angle q2) of a cycle (see Fig. 2) are calculated as: ⌬
冉 冊冉 冊 冉 冊 冉 冊 冉 冊 gmax e1−e3 ⫽ 2 2 e1+e3 2
⌬en⫽
⫺
q1
e1−e3 2
e1+e3 2
⫺
q1
(7a)
q2
(7b)
q2
where e1, e2, and e3 are the principal strain values (e1⬎e2⬎e3) which are calculated from the strain Mohr’s circle (see Fig. 5a) as: e1⫽(1⫺neff)
冋
e2⫽⫺neffeap e3⫽(1⫺neff)
冉 冊册
eap 1 2 gap ⫹ e (1⫹neff)2⫹ 2 2 ap 2
冋
2 1/2
冉 冊册
eap 1 2 gap ⫺2 eap(1⫹neff)2⫹ 2 2
(8a) (8b)
2 1/2
(8c)
Similarly, the range of maximum shear stress and the corresponding normal stress range are calculated from the largest stress Mohr’s circle during loading (at the angle q1) and unloading (at the angle q2) of a cycle as:
冉 冊 冉 冊 冉 冊 冉 冊
⌬tmax⫽ ⌬sn⫽
s1−s3 2
s1+s3 2
⫺
q1
⫺
q1
s1−s3 2
s1+s3 2
(9a)
q2
q2
(9b)
300
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Fig. 2. (continued)
Fig. 3.
Maximum shear strain through loading part of a cycle for various (a) in-phase loading and (b) out-of-phase loading conditions.
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Fig. 4. (a) Thin-walled tubular specimen subjected to combined axial and torsional fatigue, (b) 3D presentation of strain state and (c) stress state.
301
strain, the ranges of shear strains and stresses are calculated by multiplying the second term of Eqs. (7a) and (9a) by sin a and in calculating the ranges of normal strains and stresses the second terms of Eqs. (7b) and (9b) are multiplied by (1+cos a). For strain history A5, containing both axial and shear mean strains, the second terms of equations 7 and 9 become zero. The range of maximum shear stress ⌬tmax and shear strain ⌬(gmax/2) obtained from the largest stress and strain Mohr’s circles at angles q1 and q2 during the loading and unloading parts of a cycle and the corresponding normal stress range ⌬sn and the normal strain range ⌬en on that plane are the components of the proposed parameter. Both the normal and shear strain energies are weighted by the axial and shear fatigue properties, respectively:
冉
冉 冊冊
1 1 γmax (⌬sn⌬en)⫹ ⌬tmax⌬ ⫽f(Nf) (sf⬘ef⬘) (tf⬘gf⬘) 2
Fig. 5. (a) Strain Mohr’s circle, and (b) stress Mohr’s circle.
where s1, s2, and s3 are the principal stress values (s1⬎s2⬎s3) and they are calculated from the stress Mohr’s circle (see Fig. 5b) as:
(11)
where sf⬘ and ef⬘ are the axial fatigue strength coefficient and axial fatigue ductility coefficient, respectively, and tf⬘ and gf⬘ are the shear fatigue strength coefficient and shear fatigue ductility coefficient, respectively. Multiaxial fatigue energy-based models have been long discussed in terms of normal and shear energy weights. In Garud’s approach [11] he found that an empirical weighting factor of C=0.5 in the shear energy part of his model (Eq. (12)) gave a good correlation of multiaxial fatigue results for 1% Cr–Mo–V steel for both in-phase and out-of-phase loading conditions.
sa s1⫽ ⫹12[s2a⫹4t2a]1/2 2
(10a)
s2⫽0
(10b)
⌬e⌬s⫹C⌬g⌬t⫽f(Nf)
sa s3⫽ ⫺12[s2a⫹4t2a]1/2 2
(10c)
Tipton [22] found that a good multiaxial fatigue life correlation was obtained for 1045 steel with a scaling factor C of 0.90. Andrews [23] found that a C factor of 0.30 yielded the best correlation of multiaxial life data for AISI 316 stainless steel. Chu et al. [9] weighted the shear energy part of their formulation by a factor of C=2 to obtain a good correlation of fatigue results. The formulations of Liu [8] and Glinka et al. [10] provided an equal weight of normal and shear energies. The empirical factor (C) suggested by each of the above authors gave a good fatigue life correlation for a specific material which suggests that C is material dependent. In the present study, the proposed model correlates multiaxial fatigue lives by normalizing the normal and shear energies using the axial and shear material fatigue properties, respectively, and hence the parameter uses no empirical weighting factor.
In strain paths with no mean strain, the largest strain and stress Mohr’s circles, obtained during loading (at q1) and unloading (at q2) in a cycle, have equal diameters. In these strain paths, to achieve the plane of maximum shear strain, the plane P (obtained at q1) and the plane Q (obtained at q2) should rotate anti-clockwise with the angle of a=tan−1[(⌬ea/2)/(⌬ga/2)]q1 on the Mohr’s circles (see Fig. 2a, history A3). For strain paths having a mean strain, the largest Mohr’s circles obtained at q1 and q2 do not have equal diameters (see Fig. 2b). To achieve the same critical plane, both planes P and Q on Mohr’s circles have to rotate through an angle a (see Fig. 2b, histories A4 and A5). The ranges of shear strain and normal strain for strain histories containing axial and shear mean strains are shown in Fig. 2b. For the strain history A4 which has an axial mean strain, the ranges of shear strains and stresses are calculated by multiplying the second terms of Eqs. (7a) and (9a) by cos a and the ranges of normal strains and stresses are calculated by multiplying the second terms of Eqs. (7b) and (9b) by sin a. For strain history A5 which has a mean shear
(12)
4.1. Out-of-phase strain hardening Under out-of-phase loading, the principal stress and strain axes rotate during fatigue loading often causing additional cyclic hardening of materials. A change of
302
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loading direction allows more grains to undergo their most favorable orientation for slip, and leads to more active slip systems in producing dislocation interactions and dislocation tangles to form dislocation cells. Interactions strongly affect the hardening behavior and as the degree of out-of-phase increases, the number of active slip systems increases. Socie [21] performed in-phase and 90° out-of-phase fatigue tests with the same shear strain range on 304 stainless steel. Even though both loading histories had the same shear strain range, cyclic stabilized stress–strain hysteresis loops in the 90° outof-phase tests had stress ranges twice as large as those of the in-phase tests. Socie and Marquis [24] concluded that the higher magnitude of strain and stress ranges in the out-of-phase tests was due to the effect of an additional strain hardening in the material. During out-of-phase straining the magnitude of the normal strain and stress ranges is larger than that for inphase straining with the same applied shear strain ranges per cycle. The proposed parameter via its stress range term increases with the additional hardening caused by out-of-phase tests whereas critical plane models that include only strain terms do not change when there is strain path-dependent hardening. To calculate the additional hardening for out-of-phase fatigue tests, these approaches may be modified by a proportionality factor like the one proposed by Kanazawa et al. [25]. 4.2. Mean stress correction Under multiaxial fatigue loading mean tensile and compressive stresses have a substantial effect on fatigue life. Sines [26] showed compressive mean stresses are beneficial to the fatigue life while tensile mean stresses are detrimental. He also showed that a mean axial tensile stress superimposed on torsional loading has a significant effect on the fatigue life. In 1942 Smith [27] reported experimental results for 27 different materials from which it was concluded that mean shear stresses have very little effect on fatigue life and endurance limit. Sines [26] reported his findings and Smith’s results by plotting mean stress normalized by monotonic yield stress versus the amplitude of alternating stress normalized by fatigue limit (R=⫺1) values (see Fig. 6). Figures 6a and 6b show the effect of a static tensile and compressive stress for various materials on axial and torsional fatigue, respectively. The relation is linear as long as the maximum stress during a cycle does not exceed the yield stress of the material [24]. Concerning the effect of mean strain on fatigue life, Bergmann et al. [28] found almost no effect in the low-cycle fatigue region and very little effect in the high-cycle fatigue region. Mean stress effects are included in fatigue parameters in different ways [24]. One approach was applied earlier by Fatemi and Socie [29] to incorporate mean stress
Fig. 6. The effect of axial mean stress on (a) pull–push fatigue loading and (b) torsional fatigue loading [26].
using the maximum value of normal stress during a cycle to modify the damage parameter. Considering the effect of axial mean stress, a similar mean stress correction factor (1+sm n /sf⬘) in Eq. (11) showed a good correlation of multiaxial fatigue data containing mean stress values for both in-phase and out-of-phase straining conditions. This correction is based on the mean normal stress applied to the critical plane. To take into account the effect of mean axial stress on the proposed parameter, Eq. (11) is rewritten as:
冉
冉 冊冊
1 (1+sm gmax n /sf⬘) ⌬tmax⌬ (⌬sn⌬en)⫹ (sf⬘ef⬘) (tf⬘gf⬘) 2
(13)
⫽f(Nf) where the normal mean stress sm n acting on the critical plane is given by: max min 1 sm n ⫽2(sn ⫹sn )
(14)
and smin are the maximum and miniIn Eq. (14), smax n n mum normal stresses which are calculated from the stress Mohr’s circles.
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Fig. 7. Multiaxial fatigue life correlation for various in-phase and out-of-phase strain histories and seven different materials: (a) Ni–Cr–Mo–V steel [12], (b) 1045 steel [13–15], (c) Inconel 718 [16], (d) Haynes 188 [17], (e) Waspaloy [18,19], (f) stainless steel [21], and (g) mild steel [20].
304
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5. Correlations of fatigue data using the proposed parameter In order to assess the capability of the proposed parameter to correlate multiaxial fatigue lives for both inphase and out-of-phase loading conditions, fatigue data for different materials and various in-phase and out-ofphase strain paths available in the literature were used. Figures 7a–7g present multiaxial fatigue life correlations based on the proposed parameter (Eq. (13)) for seven different materials subjected to the various inphase and out-of-phase strain paths and strain histories which are shown in Fig. 2. A very good correlation of multiaxial fatigue lives is obtained for Ni–Cr–Mo–V steel (Fig. 7a), Inconel 718 (Fig. 7c), Haynes 188 (Fig. 7d), and Waspaloy (Fig. 7e) for the various in-phase and out-of-phase conditions within a factor of 1.5 for both low-cycle and high-cycle fatigue lives. Fatigue life correlation for 1045 steel (Fig. 7b), stainless steel (Fig. 7f), and mild steel (Fig. 7g) fell within factors of 2, 2.5, and 2 respectively.
6. Discussion Energy-critical plane parameters [8–10] including the parameter proposed in this paper are defined on specific planes and account for states of stress through combinations of the normal and shear strain and stress ranges. These parameters depend upon the choice of the critical plane and the stress and strain ranges acting on that plane. For the proposed parameter, the critical plane is defined by the largest shear strain and stress Mohr’s circles during the loading and unloading parts of a cycle and the parameter consists of tensorial stress and strain range components acting on this critical plane. The critical plane in Liu’s parameter [8], on the other hand, is associated with two different physical modes of failure and the parameter consists of Mode I and Mode II energy components. Liu’s parameter does not account for the effect of mean stress. Chu et al. [9] formulated normal and shear energy components based on the Smith–Watson–Topper parameter. They determined the critical plane and the largest damage parameter from the transformation of strains and stresses onto planes spaced at equal increments using a generalized Mroz model. This parameter is based on the maximum value of the damage parameter rather than being defined on planes of maximum stress or strain. Glinka et al. [10] proposed a multiaxial fatigue life parameter based on the summation of the products of normal and shear strains and stresses on the critical plane which is assumed to be the plane of maximum shear strain. In their papers Liu, Chu et al., and Glinka et al. reported that their parameters
were capable of correlating multiaxial fatigue life results for both in-phase and out-of-phase loading paths. The proposed parameter successfully correlated multiaxial fatigue lives within a factor that varied with materials from 1.5 to 2.5 for both low- and high-cycle fatigue regimes for various in-phase and out-of-phase multiaxial fatigue straining conditions. The poorest correlation— a factor of 2.5 in fatigue life in stainless steel—may be due to crack growth mechanism in this material. Observations of crack formation and early crack growth for this material reported by Socie [21] showed that in tensile loading Mode I failures were observed at all strain amplitudes. In torsion, Mode II shear failures were observed at high strain amplitude and Mode I failures at low strain amplitudes. However, for other materials studied in the present paper, in the early stage of crack growth a Mode II crack was dominant independent of stress state. 7. Conclusions A multiaxial fatigue parameter is proposed by the sum of the normal energy range and the shear energy range calculated for the critical plane on which the stress and strain Mohr’s circles are the largest during the loading and unloading parts of a cycle. The normal and shear energies in this parameter have been weighted by the tensile and shear fatigue properties, respectively. The proposed parameter successfully correlated multiaxial fatigue lives and crack growth rates by taking into account: (a) various in-phase and out-of-phase multiaxial fatigue straining conditions, (b) the effect of a mean stress applied normal to the maximum shear plane, and (c) the proposed parameter via its stress range term increases when there is an additional hardening caused by out-of-phase tests, whereas critical plane approaches that include only strain terms do not change when there is strain path-dependent hardening. The proposed parameter has shown a very good correlation of multiaxial low-cycle and high-cycle fatigue lives for various in-phase and out-of-phase straining conditions for different material fatigue data reported in the literature. Acknowledgements The author wishes to thank Prof. T.H. Topper and Prof. G. Glinka of the University of Waterloo, Canada for helpful discussions in this research work. References [1] Garud YS. Multiaxial fatigue: a survey of the state of the art. J Test Eval 1981;9(3):165–78.
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[16]
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and periodic compressive overstrains on fatigue crack growth mode and crack growth rate, ASTM STP 1360. In: Jerina KL, Paris PC, editors. Fatigue and Fracture Mechanics. West Conshohocken, PA: American Society for Testing and Materials, 2000:299–312. Koch JL. Proportional and non-proportional biaxial fatigue of Inconel 718. Report no. 121, Department of Mechanical and Industrial Engineering, University of Illinois at Urbana-Champaign, 1985. Kalluri S, Bonacuse PJ. In-phase and out-of-phase axial-torsional fatigue behaviour of Haynes 188 superalloy at 760°C. In: McDowell DL, Ellis R, editors. Advances in multiaxial fatigue, ASTM STP 1191. Philadelphia: American Society for Testing and Materials, 1993:133–50. Lerch BA, Jayaraman N. A study of fatigue damage mechanisms in Waspaloy from 25 to 800°C. Mater Sci Eng 1984;66:151–66. Jayaraman N, Ditmars MM. Torsional and biaxial (tension– torsion) fatigue damage mechanisms in Waspaloy at room temperature. Int J Fatigue 1989;11:309–18. Doquet V, Pineau A. Multiaxial low-cycle fatigue behaviour of a mild steel. In: Kussmaul K, McDiarmid D, Socie D, editors. Fatigue under biaxial and multiaxial loading, ESIS10. London: Mechanical Engineering Publication, 1991:81–101. Socie D. Multiaxial fatigue damage models. J Eng Mater Technol 1987;109:293–8. Tipton SM. Fatigue behaviour under multiaxial loading in the presence of a notch: methodologies for the prediction of life to crack initiation and life spent in crack propagation. PhD thesis, Mechanical Engineering Department, Stanford University, Stanford, CA, 1984. Andrews RM. High temperature fatigue of AISI 316 stainless steel under complex biaxial loading. PhD thesis, University of Sheffield, UK, 1986. Socie D, Marquis, G, editors. Multiaxial fatigue. Society of Automotive Engineers, 2000, in press. Kanazawa K, Miller KJ, Brown MW. Cyclic deformation of 1% Cr–Mo–V steel under out-of-phase loads. Fatigue Eng Mater Struct 1979;2:217–28. Sines G. The prediction of fatigue fracture under combined stresses at stress concentrations. Bull Jpn Soc Mech Eng 1961;4(15):443–53. Smith JO. Effect of range of stress on fatigue strength of metals. University of Illinois, Engineering Experiment Station, Bull no. 334, 1942;39(26). Bergmann J, Klee S, Seeger T. Effect of mean strain and mean stress on the cyclic stress–strain and fracture behaviour of steel StE70. Materialpruefung 1977;19(1):10–7. Fatemi A, Socie DF. A critical plane approach to multiaxial fatigue damage including out of phase loading. Fatigue Fract Eng Mater Struct 1988;11:149–65.
A new energy-critical plane parameter for fatigue life assessment of various metallic materials subjected to in-phase and out-of-phase multiaxial fatigue loading conditions A. Varvani-Farahani
*
Department of Mechanical Engineering, Ryerson Polytechnic University, Toronto, Ontario, Canada M5B 2K3 Received 2 December 1999; accepted 12 December 1999
Abstract A new multiaxial fatigue parameter for in-phase and out-of-phase straining is proposed. The parameter proposed is the sum of the normal energy range and the shear energy range calculated for the critical plane on which the stress and strain Mohr’s circles are the largest during the loading and unloading parts of a cycle. The normal and shear energies used in this parameter are divided by the tensile and shear fatigue properties, respectively. The proposed parameter, unlike many other parameters, does not use an empirical fitting factor. The proposed parameter successfully correlates multiaxial fatigue lives for: (a) various in-phase and out-of phase multiaxial fatigue straining conditions, (b) tests in which a mean stress was applied normal to the maximum shear plane, and (c) out-of-phase tests in which there was additional hardening. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Multiaxial fatigue model; In-phase and out-of-phase strain paths; Shear and normal energies; Critical plane; Mean stress effect; Strain hardening
1. Introduction Many engineering components that undergo fatigue loading experience multiaxial stresses, in which two or three principal stresses fluctuate with time, i.e. the corresponding principal stresses are out-of-phase or the principal directions change during a cycle of loading. Extensive reviews of multiaxial fatigue life prediction methods are presented by Garud [1], Brown and Miller [2], and You and Lee [3]. Fatigue analysis using the concept of a critical plane of maximum shear strain is very effective because the critical plane concept is based on the fracture mode or the initiation mechanism of cracks. In the critical plane concept, after determining the maximum shear strain plane, many researchers define fatigue parameters as combinations of the maximum shear strain (or stress) and normal strain (or stress) on that plane to explain multiaxial fatigue behavior [4–7]. However, critical
* Tel.: +1-416-979-5000. E-mail address: [email protected] (A. Varvani-Farahani).
plane parameters have been criticized for lack of adherence to rigorous continuum mechanics fundamentals. To compensate this lack, Liu [8], Chu et al. [9], and Glinka et al. [10] used the energy criterion in conjunction with the critical plane approach. Liu [8] calculated the virtual strain energy (VSE) in the critical plane by the use of crack initiation modes and stress–strain Mohr’s circles. In the calculation of VSE, Liu included both elastic energy and plastic energy while the elastic energy was not considered in Garud’s model [11]. Chu et al. [9] formulated normal and shear energy components based on the Smith–Watson–Topper parameter. They determined the critical plane and the largest damage parameter from the transformation of strains and stresses onto planes spaced at equal increments using a generalized Mroz model. Glinka et al. [10] proposed a multiaxial life parameter based on the summation of the products of normal and shear strains and stresses on the critical shear plane. In the present study a multiaxial fatigue parameter for various in-phase and out-of-phase strain paths is proposed. The parameter is given by the sum of the normal energy range and the shear energy range calculated for the critical plane at which the stress and strain Mohr’s
0142-1123/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 2 - 1 1 2 3 ( 0 0 ) 0 0 0 0 2 - 5
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Nomenclature ⌬(gap/2), ⌬eap the shear and axial strain ranges respectively ⌬(γmax/2), ⌬en maximum shear strain range and normal strain range acting on the critical plane, respectively ⌬ea, ⌬sa, ⌬ta applied tensorial strain range, stress range, and shear stress rage, respectively ⌬eij, ⌬sij the strain and stress tensor ranges (where i and j=1, 2, 3) ⌬tmax, ⌬sn maximum shear stress range and normal stress range, respectively E the elastic modulus e1, e2, e3 principal strains (e1⬎e2⬎e3) ee, ep elastic strain and plastic strain respectively f phase delay between strains on the axial and torsional axes ne, np, neff elastic, plastic, and effective Poisson’s ratios respectively q1, q2 angles during loading and unloading parts of a cycle respectively at which the Mohr’s circles are the largest s1, s2, s3 principal stresses (s1⬎s2⬎s3) mean normal stress s nm snmax maximum normal stress snmin minimum normal stress s⬘f, ε⬘f the axial fatigue strength coefficient and axial fatigue ductility coefficient, respectively t⬘f, g⬘f the shear fatigue strength coefficient and shear fatigue ductility coefficient, respectively
circles are the largest during the loading and unloading parts of a cycle. The normal and shear energies in this parameter have been weighted by the tensile and shear fatigue properties, respectively, and the parameter requires no empirical fitting factor. This parameter takes into account the effect of the mean stress applied normal to the maximum shear plane. The proposed parameter also increases when there is additional hardening caused by out-of-phase straining, while strain-based parameters fail to take into account this effect. The proposed parameter gives a good correlation of multiaxial fatigue lives for various in-phase and out-of-phase straining conditions.
2. Materials and multiaxial fatigue data Table 1 lists the references for in-phase and out-ofphase multiaxial fatigue data used in this study and tabulates the fatigue properties of the materials used. Fatigue coefficients sf⬘ and ef⬘ are the axial fatigue strength coefficient and axial fatigue ductility coefficient, respectively and tf⬘ and gf⬘ are the shear fatigue strength coefficient and shear fatigue ductility coefficient, respectively. These coefficients are illustrated in Fig. 1 for (a) uniaxial and (b) torsional fatigue loading life–strain curves.
3. In-phase and out-of-phase strain paths In this study, for convenience of presentation, inphase and out-of-phase strain paths have been categorized into three kinds: (a) in-phase strain paths (strain his-
tories A1, A2, A3, A4, A5, and A6), (b) linear out-ofphase strain paths (strain histories B1 and B2), and finally (c) non-linear out-of-phase strain paths (C1, C2, C3, C4, and C5). 3.1. In-phase strain paths In in-phase straining, both axial and shear strain cycles are alternating with no phase difference. Strain paths in in-phase straining are linear. For the in-phase straining data used in this study, the strain histories, strain paths, and strain and stress Mohr’s circles are presented in Figs. 2a and 2b. The largest stress and strain Mohr’s circles during the loading part (at q1) and unloading part (at q2) of a cycle, for which the maximum shear stress and strain and corresponding normal stress and normal strain values are calculated, are illustrated in Fig. 2. In this figure, the light Mohr’s circles are the largest during the loading part, and the dark Mohr’s circles are the largest during unloading part of a cycle. The strain histories A1, A2, and A3 correspond to uniaxial straining, torsional straining, and in-phase combined axial and torsional straining, respectively. The linear in-phase strain paths shown in Fig. 2b have mean values. Strain history A4 is a combined axial and torsional strain path with an axial mean strain. Strain history A5 has a torsional mean strain, and finally strain history A6 has both axial and torsional mean strains. 3.2. Linear out-of-phase strain paths In out-of-phase alternating straining, there is a phase difference between the axial and shear strain cycles.
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Table 1 Fatigue properties of materials used in this study Materials and fatigue data
E (GPa)
ef⬘
sf⬘ (MPa)
G (GPa)
gf⬘
tf⬘ (MPa)
Ni–Cr–Mo–V steela [12]b 1045 steel [13–15]b Inconel 718 [16]b Haynes 188 [17]b Waspaloy [18,19]c Mild steel [20]c Stainless steel [21]b
200
1.14
680
77
1.69
444
206 208.5 170.2 362 210 185
0.26 2.67 0.489 0.381 0.1516 0.171
948 1640 823 2610 1009 1000
79.2 80.2 65.5 139.2 80.8 71
0.413 3.62 1.78 0.516 0.322 0.413
505 1030 635 1640 431 709
a b c
Ni–Cr–Mo–V steel is known as Rotor steel. Fatigue properties are given by referenced papers. Fatigue properties are calculated from uniaxial and torsional fatigue life–strain data.
Fig. 1.
Schematic presentation of fatigue life–strain curves for (a) uniaxial loading and (b) torsional loading.
Strain history B1 (box) and strain history B2 (two-box), shown in Fig. 2c, are linear out-of-phase strain histories. 3.3. Non-linear out-of-phase strain paths In the non-linear out-of-phase strain histories examined there is a phase delay between the axial strain and torsional strain. Strain paths are elliptical and as the phase difference increases the elliptical path becomes larger in its minor diameter and finally at a 90° phase difference the strain path becomes circular. Strain histories C1, C2, C3, and C4 present out-of-phase axial and torsional straining with phase delays of 30°, 45°, 60°, and 90°, respectively. Strain history C5 corresponds to a 90° out-of-phase strain path containing an axial mean strain value. Fig. 2d presents non-linear out-of-phase strain paths, strain histories, and strain and stress Mohr’s circles. The maximum shear strains for in-phase and out-ofphase paths were numerically calculated at 10° increments through a cycle and are presented in Figs. 3a and 3b respectively.
4. Proposed parameter and analysis Fig. 4 illustrates a thin-walled tubular specimen subjected to combined axial and torsional fatigue. The strain and stress tensors for a thin-walled tubular specimen subjected to axial and torsional fatigue are given by Eqs. (1) and (2), respectively:
冢
冉冊 gap 2
−veff⌬eap ⌬
⌬eij ⫽
冉冊
⌬
gap 2
0
冢
0
冣
⌬eap
0
0
−veff⌬eap
冣
(1)
⌬ta 0
⌬sij ⫽ ⌬ta ⌬sa 0 0
0
0
0
(2)
298
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Fig. 2. (a) Strain history, strain path, and Mohr’s circle presentation for in-phase strain paths. (b) In-phase strain histories, paths, and Mohr’s circle presentations for in-phase paths containing mean strain values. (c) Linear out-of-phase strain history, path, and strain–stress Mohr’s circles. (d) Strain histories, paths, and Mohr’s circle presentations for non-linear out-of-phase strain paths without and with mean strain values.
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299
Fig. 2. (continued)
where axial and shear strain ranges ⌬eap and ⌬(gap/2) respectively are given by Eqs. (3) and (4) as: ⌬eap⫽⌬ea sin q
冉 冊 冉冊
gap ga ⌬ ⫽⌬ sin(q⫺f) 2 2
(3) (4)
where ea and ga/2 are the applied axial and shear amplitude strains, respectively. The angle q is the angle during a cycle of straining at which the Mohr’s circle is the largest and has the maximum value of shear strain. Angle f corresponds to the phase delay between strains on the axial and torsional axes. In Eq. (2) ⌬sa and ⌬ta are the ranges of axial and shear stresses, respectively. In Eq. (1) neff is the effective Poisson’s ratio which is given by: neff⫽
neee+npep ee+ep
(5)
where ne=0.3 is the elastic Poisson’s ratio and np=0.5 is the plastic Poisson’s ratio. The axial elastic and plastic strains are given by Eqs. (6a) and (6b), respectively: sa ee⫽ E ep⫽eap⫺
(6a) sa E
(6b)
The range of maximum shear strain and the corresponding normal strain range on the critical plane at which both strain and stress Mohr’s circles are the larg-
est during loading (at the angle q1) and unloading (at the angle q2) of a cycle (see Fig. 2) are calculated as: ⌬
冉 冊冉 冊 冉 冊 冉 冊 冉 冊 gmax e1−e3 ⫽ 2 2 e1+e3 2
⌬en⫽
⫺
q1
e1−e3 2
e1+e3 2
⫺
q1
(7a)
q2
(7b)
q2
where e1, e2, and e3 are the principal strain values (e1⬎e2⬎e3) which are calculated from the strain Mohr’s circle (see Fig. 5a) as: e1⫽(1⫺neff)
冋
e2⫽⫺neffeap e3⫽(1⫺neff)
冉 冊册
eap 1 2 gap ⫹ e (1⫹neff)2⫹ 2 2 ap 2
冋
2 1/2
冉 冊册
eap 1 2 gap ⫺2 eap(1⫹neff)2⫹ 2 2
(8a) (8b)
2 1/2
(8c)
Similarly, the range of maximum shear stress and the corresponding normal stress range are calculated from the largest stress Mohr’s circle during loading (at the angle q1) and unloading (at the angle q2) of a cycle as:
冉 冊 冉 冊 冉 冊 冉 冊
⌬tmax⫽ ⌬sn⫽
s1−s3 2
s1+s3 2
⫺
q1
⫺
q1
s1−s3 2
s1+s3 2
(9a)
q2
q2
(9b)
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Fig. 2. (continued)
Fig. 3.
Maximum shear strain through loading part of a cycle for various (a) in-phase loading and (b) out-of-phase loading conditions.
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Fig. 4. (a) Thin-walled tubular specimen subjected to combined axial and torsional fatigue, (b) 3D presentation of strain state and (c) stress state.
301
strain, the ranges of shear strains and stresses are calculated by multiplying the second term of Eqs. (7a) and (9a) by sin a and in calculating the ranges of normal strains and stresses the second terms of Eqs. (7b) and (9b) are multiplied by (1+cos a). For strain history A5, containing both axial and shear mean strains, the second terms of equations 7 and 9 become zero. The range of maximum shear stress ⌬tmax and shear strain ⌬(gmax/2) obtained from the largest stress and strain Mohr’s circles at angles q1 and q2 during the loading and unloading parts of a cycle and the corresponding normal stress range ⌬sn and the normal strain range ⌬en on that plane are the components of the proposed parameter. Both the normal and shear strain energies are weighted by the axial and shear fatigue properties, respectively:
冉
冉 冊冊
1 1 γmax (⌬sn⌬en)⫹ ⌬tmax⌬ ⫽f(Nf) (sf⬘ef⬘) (tf⬘gf⬘) 2
Fig. 5. (a) Strain Mohr’s circle, and (b) stress Mohr’s circle.
where s1, s2, and s3 are the principal stress values (s1⬎s2⬎s3) and they are calculated from the stress Mohr’s circle (see Fig. 5b) as:
(11)
where sf⬘ and ef⬘ are the axial fatigue strength coefficient and axial fatigue ductility coefficient, respectively, and tf⬘ and gf⬘ are the shear fatigue strength coefficient and shear fatigue ductility coefficient, respectively. Multiaxial fatigue energy-based models have been long discussed in terms of normal and shear energy weights. In Garud’s approach [11] he found that an empirical weighting factor of C=0.5 in the shear energy part of his model (Eq. (12)) gave a good correlation of multiaxial fatigue results for 1% Cr–Mo–V steel for both in-phase and out-of-phase loading conditions.
sa s1⫽ ⫹12[s2a⫹4t2a]1/2 2
(10a)
s2⫽0
(10b)
⌬e⌬s⫹C⌬g⌬t⫽f(Nf)
sa s3⫽ ⫺12[s2a⫹4t2a]1/2 2
(10c)
Tipton [22] found that a good multiaxial fatigue life correlation was obtained for 1045 steel with a scaling factor C of 0.90. Andrews [23] found that a C factor of 0.30 yielded the best correlation of multiaxial life data for AISI 316 stainless steel. Chu et al. [9] weighted the shear energy part of their formulation by a factor of C=2 to obtain a good correlation of fatigue results. The formulations of Liu [8] and Glinka et al. [10] provided an equal weight of normal and shear energies. The empirical factor (C) suggested by each of the above authors gave a good fatigue life correlation for a specific material which suggests that C is material dependent. In the present study, the proposed model correlates multiaxial fatigue lives by normalizing the normal and shear energies using the axial and shear material fatigue properties, respectively, and hence the parameter uses no empirical weighting factor.
In strain paths with no mean strain, the largest strain and stress Mohr’s circles, obtained during loading (at q1) and unloading (at q2) in a cycle, have equal diameters. In these strain paths, to achieve the plane of maximum shear strain, the plane P (obtained at q1) and the plane Q (obtained at q2) should rotate anti-clockwise with the angle of a=tan−1[(⌬ea/2)/(⌬ga/2)]q1 on the Mohr’s circles (see Fig. 2a, history A3). For strain paths having a mean strain, the largest Mohr’s circles obtained at q1 and q2 do not have equal diameters (see Fig. 2b). To achieve the same critical plane, both planes P and Q on Mohr’s circles have to rotate through an angle a (see Fig. 2b, histories A4 and A5). The ranges of shear strain and normal strain for strain histories containing axial and shear mean strains are shown in Fig. 2b. For the strain history A4 which has an axial mean strain, the ranges of shear strains and stresses are calculated by multiplying the second terms of Eqs. (7a) and (9a) by cos a and the ranges of normal strains and stresses are calculated by multiplying the second terms of Eqs. (7b) and (9b) by sin a. For strain history A5 which has a mean shear
(12)
4.1. Out-of-phase strain hardening Under out-of-phase loading, the principal stress and strain axes rotate during fatigue loading often causing additional cyclic hardening of materials. A change of
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loading direction allows more grains to undergo their most favorable orientation for slip, and leads to more active slip systems in producing dislocation interactions and dislocation tangles to form dislocation cells. Interactions strongly affect the hardening behavior and as the degree of out-of-phase increases, the number of active slip systems increases. Socie [21] performed in-phase and 90° out-of-phase fatigue tests with the same shear strain range on 304 stainless steel. Even though both loading histories had the same shear strain range, cyclic stabilized stress–strain hysteresis loops in the 90° outof-phase tests had stress ranges twice as large as those of the in-phase tests. Socie and Marquis [24] concluded that the higher magnitude of strain and stress ranges in the out-of-phase tests was due to the effect of an additional strain hardening in the material. During out-of-phase straining the magnitude of the normal strain and stress ranges is larger than that for inphase straining with the same applied shear strain ranges per cycle. The proposed parameter via its stress range term increases with the additional hardening caused by out-of-phase tests whereas critical plane models that include only strain terms do not change when there is strain path-dependent hardening. To calculate the additional hardening for out-of-phase fatigue tests, these approaches may be modified by a proportionality factor like the one proposed by Kanazawa et al. [25]. 4.2. Mean stress correction Under multiaxial fatigue loading mean tensile and compressive stresses have a substantial effect on fatigue life. Sines [26] showed compressive mean stresses are beneficial to the fatigue life while tensile mean stresses are detrimental. He also showed that a mean axial tensile stress superimposed on torsional loading has a significant effect on the fatigue life. In 1942 Smith [27] reported experimental results for 27 different materials from which it was concluded that mean shear stresses have very little effect on fatigue life and endurance limit. Sines [26] reported his findings and Smith’s results by plotting mean stress normalized by monotonic yield stress versus the amplitude of alternating stress normalized by fatigue limit (R=⫺1) values (see Fig. 6). Figures 6a and 6b show the effect of a static tensile and compressive stress for various materials on axial and torsional fatigue, respectively. The relation is linear as long as the maximum stress during a cycle does not exceed the yield stress of the material [24]. Concerning the effect of mean strain on fatigue life, Bergmann et al. [28] found almost no effect in the low-cycle fatigue region and very little effect in the high-cycle fatigue region. Mean stress effects are included in fatigue parameters in different ways [24]. One approach was applied earlier by Fatemi and Socie [29] to incorporate mean stress
Fig. 6. The effect of axial mean stress on (a) pull–push fatigue loading and (b) torsional fatigue loading [26].
using the maximum value of normal stress during a cycle to modify the damage parameter. Considering the effect of axial mean stress, a similar mean stress correction factor (1+sm n /sf⬘) in Eq. (11) showed a good correlation of multiaxial fatigue data containing mean stress values for both in-phase and out-of-phase straining conditions. This correction is based on the mean normal stress applied to the critical plane. To take into account the effect of mean axial stress on the proposed parameter, Eq. (11) is rewritten as:
冉
冉 冊冊
1 (1+sm gmax n /sf⬘) ⌬tmax⌬ (⌬sn⌬en)⫹ (sf⬘ef⬘) (tf⬘gf⬘) 2
(13)
⫽f(Nf) where the normal mean stress sm n acting on the critical plane is given by: max min 1 sm n ⫽2(sn ⫹sn )
(14)
and smin are the maximum and miniIn Eq. (14), smax n n mum normal stresses which are calculated from the stress Mohr’s circles.
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Fig. 7. Multiaxial fatigue life correlation for various in-phase and out-of-phase strain histories and seven different materials: (a) Ni–Cr–Mo–V steel [12], (b) 1045 steel [13–15], (c) Inconel 718 [16], (d) Haynes 188 [17], (e) Waspaloy [18,19], (f) stainless steel [21], and (g) mild steel [20].
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5. Correlations of fatigue data using the proposed parameter In order to assess the capability of the proposed parameter to correlate multiaxial fatigue lives for both inphase and out-of-phase loading conditions, fatigue data for different materials and various in-phase and out-ofphase strain paths available in the literature were used. Figures 7a–7g present multiaxial fatigue life correlations based on the proposed parameter (Eq. (13)) for seven different materials subjected to the various inphase and out-of-phase strain paths and strain histories which are shown in Fig. 2. A very good correlation of multiaxial fatigue lives is obtained for Ni–Cr–Mo–V steel (Fig. 7a), Inconel 718 (Fig. 7c), Haynes 188 (Fig. 7d), and Waspaloy (Fig. 7e) for the various in-phase and out-of-phase conditions within a factor of 1.5 for both low-cycle and high-cycle fatigue lives. Fatigue life correlation for 1045 steel (Fig. 7b), stainless steel (Fig. 7f), and mild steel (Fig. 7g) fell within factors of 2, 2.5, and 2 respectively.
6. Discussion Energy-critical plane parameters [8–10] including the parameter proposed in this paper are defined on specific planes and account for states of stress through combinations of the normal and shear strain and stress ranges. These parameters depend upon the choice of the critical plane and the stress and strain ranges acting on that plane. For the proposed parameter, the critical plane is defined by the largest shear strain and stress Mohr’s circles during the loading and unloading parts of a cycle and the parameter consists of tensorial stress and strain range components acting on this critical plane. The critical plane in Liu’s parameter [8], on the other hand, is associated with two different physical modes of failure and the parameter consists of Mode I and Mode II energy components. Liu’s parameter does not account for the effect of mean stress. Chu et al. [9] formulated normal and shear energy components based on the Smith–Watson–Topper parameter. They determined the critical plane and the largest damage parameter from the transformation of strains and stresses onto planes spaced at equal increments using a generalized Mroz model. This parameter is based on the maximum value of the damage parameter rather than being defined on planes of maximum stress or strain. Glinka et al. [10] proposed a multiaxial fatigue life parameter based on the summation of the products of normal and shear strains and stresses on the critical plane which is assumed to be the plane of maximum shear strain. In their papers Liu, Chu et al., and Glinka et al. reported that their parameters
were capable of correlating multiaxial fatigue life results for both in-phase and out-of-phase loading paths. The proposed parameter successfully correlated multiaxial fatigue lives within a factor that varied with materials from 1.5 to 2.5 for both low- and high-cycle fatigue regimes for various in-phase and out-of-phase multiaxial fatigue straining conditions. The poorest correlation— a factor of 2.5 in fatigue life in stainless steel—may be due to crack growth mechanism in this material. Observations of crack formation and early crack growth for this material reported by Socie [21] showed that in tensile loading Mode I failures were observed at all strain amplitudes. In torsion, Mode II shear failures were observed at high strain amplitude and Mode I failures at low strain amplitudes. However, for other materials studied in the present paper, in the early stage of crack growth a Mode II crack was dominant independent of stress state. 7. Conclusions A multiaxial fatigue parameter is proposed by the sum of the normal energy range and the shear energy range calculated for the critical plane on which the stress and strain Mohr’s circles are the largest during the loading and unloading parts of a cycle. The normal and shear energies in this parameter have been weighted by the tensile and shear fatigue properties, respectively. The proposed parameter successfully correlated multiaxial fatigue lives and crack growth rates by taking into account: (a) various in-phase and out-of-phase multiaxial fatigue straining conditions, (b) the effect of a mean stress applied normal to the maximum shear plane, and (c) the proposed parameter via its stress range term increases when there is an additional hardening caused by out-of-phase tests, whereas critical plane approaches that include only strain terms do not change when there is strain path-dependent hardening. The proposed parameter has shown a very good correlation of multiaxial low-cycle and high-cycle fatigue lives for various in-phase and out-of-phase straining conditions for different material fatigue data reported in the literature. Acknowledgements The author wishes to thank Prof. T.H. Topper and Prof. G. Glinka of the University of Waterloo, Canada for helpful discussions in this research work. References [1] Garud YS. Multiaxial fatigue: a survey of the state of the art. J Test Eval 1981;9(3):165–78.
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