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A multiaxial criterion for notch high-cycle fatigue using a critical-point method
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Engineering Fracture Mechanics 75 (2008) 1864–1874 www.elsevier.com/locate/engfracmech
A multiaxial criterion for notch high-cycle fatigue using a critical-point method q Andrea Carpinteri, Andrea Spagnoli *, Sabrina Vantadori, Danilo Viappiani Department of Civil and Environmental Engineering and Architecture, University of Parma, Viale Usberti 181/A, 43100 Parma, Italy Received 2 October 2006; received in revised form 9 November 2006; accepted 10 November 2006 Available online 26 December 2006
Abstract In the present paper, a high-cycle critical plane-based multaxial fatigue criterion, recently proposed by the first two authors to determine the fatigue strength of smooth components, is extended to notched ones by using the so-called point method. Accordingly, once the location of the ‘hot spot’ (crack initiation point) on the notch surface is determined, the orientation of the critical plane (where to perform fatigue strength assessment) is assumed to be correlated with some averaged principal stress directions in the hot spot itself. Some experimental results related to round bars with a surface circular notch (an artificially drilled surface hole) submitted to three types of cyclic loading (bending, torsion and combined in-phase bending and torsion) are compared with the theoretical predictions of the criterion herein proposed. The comparison, which is instrumental in highlighting the notch size-effect (as the hole diameter varies) under uniaxial and biaxial far-field stress conditions, appears to be quite satisfactory. 2006 Elsevier Ltd. All rights reserved. Keywords: Critical distance; Critical plane; Metals; Multiaxial HCF criterion; Stress/strain gradient
1. Introduction Since the fundamental works of Neuber [1] and Peterson [2], notch fatigue has attracted a great deal of attention in the research community. Among others and in line with such works, some recent approaches to notch fatigue are based on the idea of considering a certain volume in the vicinity of the notch surface to assess the fatigue strength of the component. In this way, the influence of stress/strain gradient due to the notch on the fatigue strength can be taken into account. It is worth mentioning: the critical layer approach of Flavenot and Skalli [3]; the method of Pluvinage and co-workers [4] which is based on an elastic–plastic analysis in the notch vicinity; the general non-local criterion of Seweryn and Mroz [5]; the method of Sheppard [6] which is based on the average value of the circumferential stress in the notch vicinity; the q
This article appeared in its original form in Fracture of Nano and Engineering Structures: Proceedings of the 16th European Conference of Fracture, Alexandroupolis, Greece, July 3–7, 2006 (Edited by E.E. Gdoutos, 2006). Springer, Dordrecht, The Netherlands. ISBN 1-4020-4971-4. * Corresponding author. E-mail address: [email protected] (A. Spagnoli). 0013-7944/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2006.11.002
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Nomenclature shear stress vector acting on the critical plane (Fig. 3) ‘intrinsic’ crack length Eq. (1) slope of the S–N curve for fully reversed normal stress normal stress vector acting on the critical plane fixed coordinate system coordinate system of the principal stress axes coordinate system of the averaged principal stress axes stress vector acting on the critical plane angle between the averaged direction ^ 1 of the maximum principal stress and the normal to the critical plane DKI,th threshold range of the stress intensity factor for long cracks under Mode I k biaxiality ratio (k = sxy,a/rx,a) r stress tensor raf fatigue limit for fully reversed normal stress req,a equivalent stress amplitude r1 ; r2 ; r3 principal stresses, with r1 P r2 P r3 rx applied normal stress sxy applied shear stress saf fatigue limit for fully reversed shear stress saf/raf fatigue limit ratio /, h, w principal Euler angles ^ ^h; w ^ averaged principal Euler angles /;
C L m N PXYZ P123 P^ 1^2^ 3 Sw d
Subscripts a amplitude (of a given stress component) m mean value (of a given stress component) max maximum value (of a given stress component)
finite-volume-energy based approaches of Lazzarin and Zambardi [7] and Yosibash et al. [8]. Moreover, it is worth recalling the criterion of Papadopoulos [9] which, conversely to the previous proposals, explicitly accounts for stress gradient instead of considering a finite volume around the notch tip. Another paper to be mentioned is Ref. [10] of Tanaka. In this line, a recent proposal to estimate the fatigue limit of notched components is that by Taylor [11]. Accordingly, the fatigue limit condition in a notched structural element occurs either when the amplitude of a relevant stress component (e.g. the normal stress perpendicular to the notch bisector), calculated from a linear elastic analysis at a given distance from the notch tip (point method), is equal to raf (raf = normal stress fatigue limit measured from a smooth specimen), or when the averaged value of such a stress amplitude over a line (line method) or a semi-circular area (area method) ahead of the notch tip is equal to raf. Such so-called critical-distance methods assume that the distance from the notch tip, the length of the line and the radius of the area ahead of the notch tip are equal to L/2, 2L and L, respectively, where L is a material constant (characteristic length of the material) equal to the ElHaddad ‘intrinsic’ crack length [12], that is 2 1 DK I;th L¼ p raf
ð1Þ
DKI,th being the threshold range of the stress intensity factor for long cracks under Mode I. Taylor has shown that these methods can successfully be applied to predict the fatigue limit of structural components containing either sharp, blunt or short notches [11,13] subjected to uniaxial cyclic loading. The
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predicting capability of the line method seems to depend on a theoretical argument in the case of sharp notches, which are characterised by non-propagating cracks at the fatigue limit [14]. As a matter of fact, the stress intensity factor range of a non-propagating crack of length 2L originating from the tip of the notch has been shown to attain the threshold value DKth of the material under a stress field which, in the uncracked body, presents an averaged value of the normal stress amplitude along the crack line equal to raf [15]. When a notched component is subjected to a far-field multiaxial cyclic loading, appropriate stress components ahead of the notch tip must be selected in order to exploit the above critical-distance methods. As has been shown in the works by Susmel and Taylor [16], Susmel [17] and Naik et al. [18], critical plane-based multiaxial criteria can be used in conjunction with one of the critical-distance methods. In the present paper, an extension of the Carpinteri–Spagnoli criterion presented in Refs [19–24] to multiaxial notch fatigue by using the point method is proposed. 2. The Carpinteri–Spagnoli criterion for multiaxial high-cycle fatigue According to this criterion, the fatigue limit estimation under multiaxial high-cycle fatigue is performed by considering the time-varying stress tensor (determined from a linear elastic analysis) at a given point of the material. In the case of stress components assumed to be periodic functions of time, the main steps of the criterion are as follows [19–23]: • Averaged directions of the principal stresses are determined on the basis of their instantaneous directions (two material parameters are required at this step: the fatigue limit raf under fully reversed normal stress, and the corresponding slope m of the S–N curve in the high-cycle regime). • The orientation of the verification material plane (hereafter termed ‘critical plane’, where to perform fatigue strength assessment, see below) and that of the final fracture plane are linked to the averaged directions of the principal stress axes (one further material parameter is required at this step: the fatigue limit saf under fully reversed shear stress). • The maximum value of normal stress and the amplitude of shear stress, both acting on the critical plane, are determined in a loading cycle. • The fatigue strength estimation is performed through a quadratic combination of normal and shear stress components acting on the critical plane. At a given material point P, the direction cosines of the instantaneous principal stress directions 1, 2 and 3 (being r1(t) P r2(t) P r3(t)) with respect to a fixed PXYZ frame can be worked out from the time-varying stress tensor r(t). The orthogonal coordinate system P123 with origin at point P and axes coincident with the principal directions (Fig. 1) can be defined through the ‘principal Euler angles’, /, h, w, which represent three counter-clockwise sequential rotations around the Z-axis, Y 0 -axis and three-axis, respectively (0 6 / < 2p; 0 6 h 6 p; 0 6 w < 2p). The procedure to obtain the principal Euler angles from the direction cosines of the principal stress directions consists of two stages described in Ref. [16], and is rather lengthy, although very simple. The ranges of such Euler angles at the end of this two-stage procedure are reduced as follows: 0 6 /, h 6 p/2 and p/2 < w 6 p/2. As is well known, the instantaneous directions of the principal stresses do not vary with time in the case of a proportional loading (i.e. the loading case characterised by a time-varying stress tensor which can formally be written as r(t) = f(t)r, where f(t) is a scalar time function and r is a time-constant stress tensor), whereas the opposite occurs for non-proportional loading. However, it is worth noticing that, also in the case of a proportional loading, principal stress directions might vary with respect to time as a result of principal stress ordering r1 P r2 P r3 [19]. This occurs, for instance, even in the simple case of uniaxial tension-compression [21]. The averaged directions of the principal stress axes ^1; ^2 and ^3 are obtained from the averaged values ^ ^ of the principal Euler angles. Such values are computed by independently averaging the instantaneous /; ^ h; w values /(t), h(t), w(t), as follows [19]: Z T Z T Z T 1 ^¼ 1 ^¼ 1 / /ðtÞW ðtÞ dt ^ h¼ hðtÞW ðtÞ dt w wðtÞW ðtÞ dt ð2Þ W 0 W 0 W 0
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Z 1
X
θ
θ
P
ψ
3
φ
Y
Y' 2
Fig. 1. Principal stress directions 1, 2, 3 described through the Euler angles /, h, w.
where T is the period of the cyclic load and W ¼ 1 1 r1 ðtÞ m W ðtÞ ¼ h r1 ðtÞ raf 2 raf
RT 0
W ðtÞ dt. The weight function W(t) is given by ð3Þ
where h[Æ] is the Heaviside function (h[x] = 1 for x > 0, h[x] = 0 for x 6 0). The adopted weight function is such that it includes into the averaging procedure those positions of the principal directions for which the maximum principal stress r1 is greater than half of the normal stress fatigue limit raf under fully reversed normal stress (a discussion on such a weight function is presented elsewhere [19]). As has been pointed out by Brown and Miller [25], fatigue crack propagation can be distinguished into two stages: a first one (Stage 1) in which a crack nucleates (near the external surface of a structural component) along a shear slip plane (Mode II, fatigue crack initiation plane), and a second one (Stage 2) in which crack propagation occurs in a plane normal to the direction of the maximum principal stress (Mode I, final fatigue fracture plane). By comparing theoretical and experimental results [21], the normal to the estimated final fatigue fracture plane of Stage 2, which is the one observed post mortem at the macro level, has been assessed to be coincident with the averaged direction ^ 1 of the maximum principal stress r1. This implies that, according to the present criterion, the orientation of the final fatigue fracture plane depends on the time-varying stress state and the material parameters raf and m. The orientation of the critical plane, where to perform fatigue strength assessment, has been proposed to be correlated with the averaged directions of the principal stress axes through the off angle d (d = angle between the normal w to the critical plane and the averaged direction ^1, of the maximum principal stress r1, where w belongs to the principal plane ^ 1^ 3 as is shown in Fig. 2) [21]. The following empirical expression for d is proposed: " 2 # 3p saf 1 d¼ ð4Þ 8 raf Accordingly, the off angle d should be regarded as a material parameter (depending on the fatigue limit ratio) which influences the orientation of the critical pffiffiffi plane. Eq. (4) is valid for hard metals which are characterised by values of the ratio saf/raf ranging from 1= 3 to 1 (note that the lower limit of saf/raf corresponds to the Von Mises strength criterion of mild metals for static loading). In the light of the above, the orientation of the critical plane depends on the time-varying pffiffiffi stress state as well as the material parameters raf, saf and m. When the ratio saf/raf tends to 1= 3 (threshold between mild and hard metals), the off angle d tends to p/4. In this case, the critical plane tends to coincide with the actual fatigue crack initiation plane (Stage 1). This is justified by the fact that, in the high-cycle fatigue regime, the microcrack initiation stage is the critical one for such materials. On the other hand, when the ratio saf/raf tends to 1 (threshold between hard and extremely hard metals), the off angle d tends to zero. This is in line with the fact that such materials have predominantly
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Z
w ˆ1
δ
θˆ
X
P
ˆ3
ψˆ
φˆ
Y
Y' ˆ2
Fig. 2. Correlation between averaged principal stress directions ^1^2^3 and normal w to the critical plane.
Stage 2 crack growth and, hence, the critical plane is assumed to be the final fatigue fracture plane. In conclusion, the introduction of the off angle d allows us to consider both shear stress (Mode II) and normal stress (Mode I) mechanisms (the governing mechanism depends on the fatigue behaviour of the material and the stress state) of crack initiation pointed out by Socie [26]. The stress vector Sw acting on the critical plane at point P can be expressed as follows (Fig. 3): Sw ¼ rw
ð5Þ
where w is the vector normal to the critical plane. Then, the normal stress vector N and the shear stress vector C lying on the critical plane are: N ¼ ðw Sw Þw
ð6Þ
C ¼ Sw N
ð7Þ
For multiaxial constant amplitude cyclic loading, the vectors N and C are periodic functions of time. The direction of the normal stress N(t) is fixed with respect to time: consequently, the mean value Nm and the amplitude Na of the normal stress can readily be calculated. On the other hand, the definitions of the shear stress mean value Cm and amplitude Ca are not unique, owing to the generally time-varying direction of C. The procedure proposed by Papadopoulos [27] has been adopted in Ref. [22,23] to determine the mean value Cm and the amplitude Ca of the shear stress vector C:
Z
critical plane
N(t)
Σ
X
Qm
w
Sw(t)
P C(t) critical plane
Ca Cm
C(t)
P
C′
Q'
Y Fig. 3. Stress components acting on the critical plane.
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0
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C m ¼ min max kCðtÞ C k 0 C
06t6T
ð8Þ
C a ¼ max kCðtÞ C m k 06t6T
where the symbol k k indicates the Euclidean norm of a vector, C(t) is the shear vector at the time instant t, C0 is a vector chosen according to the procedure presented in Ref. [27] (see Fig. 3, where R is the closed path described by the shear stress vector C during a loading cycle). Note that, for proportional loading, we have r(t) = f(t)r and, hence, Sw(t) = r(t) w = f(t)rw (see Eq. (5)). Therefore, the direction of Sw and that of C (see Eq. (7)) are fixed with respect to time, since f(t) is a scalar time function. As a multiaxial fatigue limit condition, the following non-linear combination of the maximum normal stress (Nmax = Nm + Na) and the shear stress amplitude (Ca) acting on the critical plane has been proposed [22,23]: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ffi raf 2 req;a ¼ N max þ C 2a ¼ raf ð9Þ saf Such a criterion is to some extent reminiscent of the well-known Gough ellipse [28]. However, differently from the Gough’s criterion, the proposed criterion considers stress components acting on the critical plane and, hence, appears to be sensitive to loading non-proportionality (e.g. for sinusoidal loading, the critical plane orientation depends on the phase angles between the stress components [23]). Note that some well-established experimental findings are included in Eq. (9), namely: the mean shear stress Cm does not affect the high-cycle fatigue strength of hard metals [28], whereas the effect of the mean normal stress Nm is taken into account. 3. Extension of the Carpinteri–Spagnoli criterion to multiaxial notch fatigue using the critical-point method An extension of the multiaxial Carpinteri–Spagnoli criterion [19–24] using the critical-point method of Taylor [11] is hereafter proposed to estimate the fatigue limit of notched components. The application of the critical-distance method (originally conceived for uniaxial loading) to multiaxial fatigue requires to tackle some issues, including: position of the crack initiation point (the so-called hot spot) on the notch surface; variation of the characteristic length L (the intrinsic crack length) as a function of the notch loading modes (Mode I/Mode II/Mode III); direction along which the characteristic length should be considered to determine the position of the critical point (such a direction is the so-called ‘assumed crack path’); stress components (at the critical point) to be considered for multiaxial fatigue limit estimation. The extension proposed is based on some simple assumptions which deal with the above issues. Consider a traction-free notch surface contained in a body submitted to a periodic fatigue loading. At any point on the notch surface, a principal stress is always null and its direction is normal to the notch surface. The point H of crack initiation on the notch surface (see Fig. 4 where, for the sake of simplicity, a plane stress/ strain situation is considered) is assumed as that point experiencing the maximum value of req,a Eq. (9). Hence, with respect to the coordinate systemxy, we have
ðxH ; y H Þ ¼ max req;a ðx; yÞ ð10Þ x;y2S n
where Sn indicates the notch surface. Note that, under any applied multiaxial loading, the components of the stress tensor r(t) at any point on the notch surface are proportional (r(t) = f(t)r). Hence, the principal stress directions might vary with respect to time only as a result of principal stress ordering (r1 P r2 P r3) [19]. Since according to the weight function in Eq. (3) only the time instants related to r1 > raf/2 are considered in the averaging procedure, the averaged principal stress directions at any point on the notch surface are those depicted in Fig. 4 (^ 1 is tangent to the notch surface and ^3 is normal to such a surface). Note that the maximisation procedure of Eq. (10) indicates that the hot spot corresponds to the point on the notch surface experiencing the maximum amplitude of the maximum principal stress r1. The orientation of the critical plane at point H is computed by considering the off angle d (Fig. 4). Then, it is thought that such an orientation does not change up to point P which is at a distance L/2 from the notch surface (Fig. 4). As an engineering assumption, the length L is taken to be that related to Mode I loading (see Eq. (1))
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w( P) 3ˆ(H)
w(H)
y
O
1ˆ(H)
1ˆ(P)
P
δ
L/2 H
x
Fig. 4. Critical plane orientation (w(H) w(P) is the normal to the critical plane) in the vicinity of a general traction-free notch surface (plane stress/strain conditions are assumed).
although notch loading conditions are in general multimodal (i.e. involving Modes I, II and/or III). Clearly, such assumption is a strong simplification of the physical reality of the problem, but it might have the advantage of avoiding the knowledge of fatigue threshold parameters under mixed mode loading which are often difficult to be determined experimentally (e.g. see Ref. [29] ). In line with the point method of Taylor [11], the equivalent normal stress req,a Eq. (9) is calculated from the stress tensor (obtained from linear-elastic analysis) at point P, bearing in mind that the orientation of the critical plane (where to calculate the normal Nmax and shear stress Ca components) depends on the averaged principal stress directions at point H and the off angle d. In other words, the critical plane orientation does not varies moving from point H to point P; the underlying physical assumption is that, at fatigue limit (non-propagating) cracks tend to nucleate on the critical plane defined by the surface conditions, and continue to propagate in a self-similar manner along that direction. The limiting cases for the critical planeporientation are as follows: if saf/raf = 1 the crack path HP is normal to the notch surface, while if ffiffiffi saf =raf ¼ 1= 3 the crack path HP is p/4 off the normal to the notch surface. Note that if two principal stress components (e.g. r2 = r3 = 0) are null at point H, two principal directions are indeterminate (e.g. in the plane normal to the principal direction of r1) and, hence, also the critical plane orientation is indeterminate. In such a case (e.g. under plane stress conditions), it is assumed that the principal direction of r3 belongs to the stress plane (xy plane in Fig. 4), so that the critical plane (which is coincident with the crack path as is shown above) is the plane of maximum stress gradient (this is in line with the principle of the ‘focus path’ introduced by Taylor and co-workers [30] to deal with the application of the criticaldistance theory to three-dimensional notches). From the stress tensor at point P, the averaged direction ^1 of the maximum principal stress can be calculated. It can be conjectured, although this is beyond the scope of the present paper, that the crack plane during Stage 2 propagation (final fatigue fracture plane) is normal to the direction ^1 at point P (Fig. 4). 4. Comparison with experimental results for circular notches Some experimental tests by Endo and Murakami [31] and Endo [32,33] related to proportional loading are considered for comparison. Further comparisons for non-proportional loading will be considered in the near future. The experiments concern round bars with artificially drilled surface holes (the hole diameter D ranges from 40 to 500 lm, Fig. 5). The bars are subjected to fully reversed bending or torsion [31], or fully reversed bending, torsion and combined in-phase bending and torsion [32,33]. The materials being analysed are: 0.46%C pannealed steel (with Vickers hardness equal to 170, raf = 245 MPa, saf = 145 MPa, DKI,th = 5.3 MPa m, L = 150 lm), and 0.37%C annealed steel (with Vickers hardness equal to 160, raf = 233 MPa, p saf = 145 MPa, DKI,th = 7.6 MPa m, L = 340 lm). The threshold range of the stress intensity factor for Mode I is estimated from the values reported in Ref. [34]. The experimental tests by Endo and Murakami allowed the determination of fatigue limit conditions in terms of applied normal and shear stress amplitudes. These fatigue limit conditions are in general characterised by non-propagating cracks at the hole edge [31–33].
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τxy,a τxy,a
y ˆ 1(H)
w(H)
δ H
σx,a O
3ˆ(H)
P
β
σx,a
α x
τxy,a
D
τxy,a Fig. 5. Critical plane orientation (w(H) is the normal to the critical plane) in the vicinity of a circular hole under in-phase axial and shear stresses (plane stress conditions are assumed).
From Eq. (4), the values of the off angle d are equal to 44 and 41 for the tests by Endo and Murakami [31] and Endo [32,33], respectively. The experimental tests by Endo and Murakami allows us to investigate the notch size effect (notch fatigue limit against D/L, with D = hole diameter), while the influence of notch acuity (e.g. D/q, with q = notch radius of curvature at hot spot) on notch fatigue limit clearly cannot be investigated since the D/q ratio is constant for circular holes. Note that the experimental results of Endo and Murakami in Figs. 6 and 7 below are presented against D (instead of D/L), since, for each series of tests, L assumes a constant value. The applied far-field stress amplitudes rx,a (due to bending) and sxy,a (due to torsion) are shown in Fig. 5, where the xy plane is tangent to the bar surface in the hole center. Stress field on the bar surface in the vicinity of the hole is here determined according to the Kirsch solution for plane stress state [35]. The different values of the biaxiality ratio k = sxy,a/rx,a being considered in the experimental tests are: 0 (pure bending), 0.5, 1, 2, 1 (pure torsion). In Table 1, the hot spot position (see angle a in Fig. 5) as a function of the different values of the biaxiality ratio is reported along with the critical plane orientation (see angle b in Fig. 5). Fig. 6 shows experimental results by Endo and Murakami [31] for pure bending or torsion, and theoretical results determined through the Carpinteri–Spagnoli criterion. The comparison is quite satisfactory with an error within about 15%. Note that a smaller notch size effect in torsion with respect to that in bending is shown by experiments and well predicted by the proposed criterion. As is stated in Ref. [31], non-propagating cracks develop at the specimen surface (instead of at the hole edge) in the fatigued specimens under torsion with D < 100 lm. Fig. 7 illustrates some experimental results by Endo [32,33] for pure bending, torsion and combined bending and torsion. Again, although to a less degree, the comparison with the theoretical results is quite satisfactory with an error within about 25%. It is worth noticing that, in a recent paper by Taylor [36], the same results by Endo and Murakami [31] were compared with the results deduced by combining the line method of Taylor [11] with the Susmel–Lazzarin criterion (e.g. see Ref. [37]). In particular, different paths emanating from the hole were considered to average stresses according to the line method. It is interesting to note that Taylor [36] concluded that crack paths forming a 45 angle with the normal to the hole at the hot spot gave the best comparisons, which is in line with the results of the present investigation, since the crack path HP (Fig. 5) forms an angle with the normal to the hole at point H equal to 44 and 41 for the tests by Endo and Murakami [31] and Endo [32,33], respectively. Finally, it is interesting to note that the curves related to torsion or combined bending and torsion (k 5 0) in Figs. 6 and 7 exhibit a non-monotonic trend. Such a trend is due to the variation of the values of the stress
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1000 Exp. - bending Exp. - torsion Theory - bending
Fatigue limit (MPa)
Theory - torsion
100
0
100
1000
Hole diameter (μm) Fig. 6. Experimental results [31] and theoretical predictions of fatigue limit as a function of the hole diameter for bending (in terms of applied normal stress amplitude) and torsion (in terms of applied shear stress amplitude).
1000
Exp. - λ = 0.0 (bending) Exp. - λ = 0.5 Exp. - λ = 1.0
Fatigue limit (MPa)
Exp. - λ = 2.0 Exp. - λ = ∞ (torsion) Theory λ = 0.0 (bending)
λ = 0.5 λ =1
100
λ = ∞ (torsion)
λ =2
0
100
1000
Hole diameter (μm) Fig. 7. Experimental results [32,33] and theoretical predictions of fatigue limit as a function of the hole diameter for bending, torsion and combined bending and torsion (in terms of applied maximum principal stress amplitude).
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Table 1 Position of the hot spot (crack initiation point, see angle a in Fig. 5) on the hole surface and direction of the critical plane (see angle b in Fig. 5) k
jaj (deg)
jbj (deg)
0.0 0.5 1.0 2.0 1
90 70 60 50 45
90 d 70 d 60 d 50 d 45 d
equal to L/2 in Fig. 5) from the hole surface as the hole components at a constant distance (see the distance HP diameter D is made to vary. When uniaxial loading is considered (pure bending, k = 0) the normal stress and the shear stress acting on the critical plane at point P monotonically increase with increasing D, while under biaxial loading (torsion or combined bending and torsion, k 5 0) a non-monotonic trend characterized by a minimum is observed for the shear stress. Consequently, when k 5 0 the amplitude of the equivalent normal stress req,a (function of the normal stress and shear stress acting on the critical plane, see Eq. (9)) at point P, which is used for fatigue limit assessment, presents a minimum as the hole diameter varies. 5. Concluding remarks The present study represents a first attempt to predict fatigue limit conditions for multiaxially loaded notched components by combining a critical plane-based criterion recently proposed by the first two authors (Carpinteri–Spagnoli criterion) with a critical-distance method. In particular, following the philosophy of the point method, fatigue limit conditions occur in a notched component submitted to far-field multiaxial loading when the amplitude of the equivalent normal stress, defined according to the Carpinteri–Spagnoli criterion, attains the normal stress fatigue limit for smooth components in a point at a distance from the hot spot on the notch surface which is half the ElHaddad intrinsic crack length. Such a distance is measured along a direction which depends on the material fatigue limit ratio. The comparison with some experimental results concerning circular notches casts a promising light on the present proposal. Acknowledgement The authors gratefully acknowledge the research support for this work provided by the Italian Ministry for University and Technological and Scientific Research (MIUR). References [1] Neuber H. Theory of notch stresses: principles for exact stress calculation. Edwards; 1946. [2] Peterson RE. Notch sensitivity. In: Sines G, Waisman JL, editors. Metal fatigue. McGraw-Hill; 1959. p. 293–306. [3] Flavenot J-F, Skalli NA. A critical depth criterion for the evaluation of long-life fatigue strength under multiaxial loading and a stress gradient. In: Brown MW, Miller KJ, editors. Biaxial and multiaxial fatigue ECF 3. Mechanical Engineering Publications; 1989. p. 355–65. [4] Qylafku G, Azari Z, Kadi N, Gjonaj M, Pluvinage G. Application of a new model proposal for fatigue life prediction on notches and key-seats. Int J Fatigue 1999;21:753–60. [5] Seweryn A, Mroz Z. On the criterion of damage evolution for variable multiaxial stress states. Int J Solids Struct 1998;35:1589–616. [6] Sheppard SD. Field effects in fatigue crack initiation: long life fatigue strength. Trans ASME J Mech Des 1991;113:188–94. [7] Lazzarin P, Zambardi R. A finite-volume-energy based approach to predict the static and fatigue behaviour of components with sharp V-shaped notches. Int J Fract 2001;112:275–98. [8] Yosibash Z, Bussiba A, Gilad I. Failure criteria for brittle elastic materials. Int J Fract 2004;125:307–33. [9] Papadopoluos IV, Panoskaltsis VP. Invariant formulation of a gradient dependent multiaxial high-cycle fatigue criterion. Engng Fract Mech 1996;55:513–28. [10] Tanaka K. Engineering formulae for fatigue strength reduction due to crack-like notches. Int J Fract 1983;22:R39–45. [11] Taylor D. Geometrical effects in fatigue: a unifying theoretical model. Int J Fatigue 1999;21:413–20.
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[12] ElHaddad MH, Dowling NF, Topper TH, Smith KN. J integral applications for short fatigue cracks at notches. Int J Fract 1980;16:15–24. [13] Taylor D, Wang G. The validation of some methods of notch fatigue analysis. Fatigue Fract Engng Mater Struct 2000;23:387–94. [14] Smith RA, Miller KJ. Prediction of fatigue regimes in notched components. Int J Mech Sci 1978;20:201–6. [15] Taylor D. A mechanistic approach to critical-distance methods in notch fatigue. Fatigue Fract Engng Mater Struct 2001;24:215–24. [16] Susmel L, Taylor D. Two methods for predicting the multiaxial fatigue limits of sharp notches. Fatigue Fract Engng Mater Struct 2003;26:821–33. [17] Susmel L. A unifying approach to estimate the high-cycle fatigue strength of notched components subjected to both uniaxial and multiaxial cyclic loadings. Fract Engng Mater Struct 2004;27:391–411. [18] Naik RA, Lanning DB, Nicholas T, Kallmeyer AR. A critical plane gradient approach for the prediction of notched HCF life. Int J Fatigue 2005;27:481–92. [19] Carpinteri A, Brighenti R, Macha E, Spagnoli A. Expected principal stress directions for multiaxial random loading-part I: theoretical aspects of the weight function method. Int J Fatigue 1999;21:83–8. [20] Carpinteri A, Brighenti R, Macha E, Spagnoli A. Expected principal stress directions for multiaxial random loading – part II: numerical simulation and experimental assessment through the weight function method. Int J Fatigue 1999;21:89–96. [21] Carpinteri A, Brighenti R, Spagnoli A. A fracture plane approach in multiaxial high-cycle fatigue of metals. Fatigue Fract Engng Mater Struct 2000;23:355–64. [22] Carpinteri A, Spagnoli A. Multiaxial high-cycle fatigue criterion for hard metals. Int J Fatigue 2001;23:135–45. [23] Spagnoli A, A new high-cycle fatigue criterion applied to out-of-phase biaxial stress state. Int. J Mech Sci 2001;43:2581–95. [24] Carpinteri A, Spagnoli A, Vantadori S. A multiaxial fatigue criterion for random loading. Fatigue Fract Engng Mater Struct 2003;26:515–22. [25] Brown MW, Miller KJ. A theory for fatigue failure under multiaxial stress–strain condition. Proc Inst Mech Engng 1973;187:745–55. [26] Socie DF. Multiaxial fatigue damage models. J Engng Mater Technol 1987;109:293–8. [27] Papadopoulos IV. Critical plane approaches in high-cycle fatigue: on the definition of the amplitude and mean value of the shear stress acting on the critical plane. Fract Engng Mater Struct 1998;21:269–85. [28] Gough HJ, Pollard HV, Clenshaw WJ. Some experiments on the resistance of metals to fatigue under combined stresses. Aeronautical Research Council Reports, R and M 2522, HMSO, 1951. [29] Tong J, Yates JR, Brown MW. Some aspects of fatigue thresholds under mode III and mixed mode III and I loadings. Int J Fatigue 1996;18:279–85. [30] Bellett D, Taylor D, Marco S, Mazzeo E, Guillois J, Pircher T. The fatigue behaviour of three-dimensional stress concentrations. Int J Fatigue 2005;27:207–21. [31] Endo M, Murakami Y. Effect of an artificial small defect on torsional fatigue strength of steels. J Engng. Mater Technol (ASME Trans) 1987;109:124–9. [32] Endo M. Effects of small defects on the fatigue strength of steel and ductile iron under combined axial/torsional loading. In: Ravichandran KS, Ritchie RO, Murakami Y, editors. Small fatigue cracks: mechanics. Mechanisms and applications. Elsevier; 1999. p. 375–87. [33] Endo M. The multiaxial fatigue strength of specimens containing small defects. In: Carpinteri A, de Freitas M, Spagnoli A, editors. Biaxial/multiaxial fatigue and fracture ESIS publication, vol. 31. Elsevier; 2003. p. 243–64. [34] Murakami Y. Metal fatigue: effects of small defects and nonmetallic inclusions. Elsevier; 2002. [35] Kirsch G. Die teorie der elastizita¨t und die bedu¨rfnisse der festigkeitslehre. VDI 1898;42:797–807. [36] Taylor D. Notch size effects in multiaxial fatigue. In: Susmel L, Tovo R, editors. Progettazione a fatica in presenza di multiassialita` tensionali. SG Editoriali; 2005. p. 3–13. [37] Susmel L, Lazzarin P. A bi-parametric Wohler curve for high cycle multiaxial fatigue assessment. Fatigue Fract Engng Mater Struct 2002;25:63–78.
Engineering Fracture Mechanics 75 (2008) 1864–1874 www.elsevier.com/locate/engfracmech
A multiaxial criterion for notch high-cycle fatigue using a critical-point method q Andrea Carpinteri, Andrea Spagnoli *, Sabrina Vantadori, Danilo Viappiani Department of Civil and Environmental Engineering and Architecture, University of Parma, Viale Usberti 181/A, 43100 Parma, Italy Received 2 October 2006; received in revised form 9 November 2006; accepted 10 November 2006 Available online 26 December 2006
Abstract In the present paper, a high-cycle critical plane-based multaxial fatigue criterion, recently proposed by the first two authors to determine the fatigue strength of smooth components, is extended to notched ones by using the so-called point method. Accordingly, once the location of the ‘hot spot’ (crack initiation point) on the notch surface is determined, the orientation of the critical plane (where to perform fatigue strength assessment) is assumed to be correlated with some averaged principal stress directions in the hot spot itself. Some experimental results related to round bars with a surface circular notch (an artificially drilled surface hole) submitted to three types of cyclic loading (bending, torsion and combined in-phase bending and torsion) are compared with the theoretical predictions of the criterion herein proposed. The comparison, which is instrumental in highlighting the notch size-effect (as the hole diameter varies) under uniaxial and biaxial far-field stress conditions, appears to be quite satisfactory. 2006 Elsevier Ltd. All rights reserved. Keywords: Critical distance; Critical plane; Metals; Multiaxial HCF criterion; Stress/strain gradient
1. Introduction Since the fundamental works of Neuber [1] and Peterson [2], notch fatigue has attracted a great deal of attention in the research community. Among others and in line with such works, some recent approaches to notch fatigue are based on the idea of considering a certain volume in the vicinity of the notch surface to assess the fatigue strength of the component. In this way, the influence of stress/strain gradient due to the notch on the fatigue strength can be taken into account. It is worth mentioning: the critical layer approach of Flavenot and Skalli [3]; the method of Pluvinage and co-workers [4] which is based on an elastic–plastic analysis in the notch vicinity; the general non-local criterion of Seweryn and Mroz [5]; the method of Sheppard [6] which is based on the average value of the circumferential stress in the notch vicinity; the q
This article appeared in its original form in Fracture of Nano and Engineering Structures: Proceedings of the 16th European Conference of Fracture, Alexandroupolis, Greece, July 3–7, 2006 (Edited by E.E. Gdoutos, 2006). Springer, Dordrecht, The Netherlands. ISBN 1-4020-4971-4. * Corresponding author. E-mail address: [email protected] (A. Spagnoli). 0013-7944/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2006.11.002
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Nomenclature shear stress vector acting on the critical plane (Fig. 3) ‘intrinsic’ crack length Eq. (1) slope of the S–N curve for fully reversed normal stress normal stress vector acting on the critical plane fixed coordinate system coordinate system of the principal stress axes coordinate system of the averaged principal stress axes stress vector acting on the critical plane angle between the averaged direction ^ 1 of the maximum principal stress and the normal to the critical plane DKI,th threshold range of the stress intensity factor for long cracks under Mode I k biaxiality ratio (k = sxy,a/rx,a) r stress tensor raf fatigue limit for fully reversed normal stress req,a equivalent stress amplitude r1 ; r2 ; r3 principal stresses, with r1 P r2 P r3 rx applied normal stress sxy applied shear stress saf fatigue limit for fully reversed shear stress saf/raf fatigue limit ratio /, h, w principal Euler angles ^ ^h; w ^ averaged principal Euler angles /;
C L m N PXYZ P123 P^ 1^2^ 3 Sw d
Subscripts a amplitude (of a given stress component) m mean value (of a given stress component) max maximum value (of a given stress component)
finite-volume-energy based approaches of Lazzarin and Zambardi [7] and Yosibash et al. [8]. Moreover, it is worth recalling the criterion of Papadopoulos [9] which, conversely to the previous proposals, explicitly accounts for stress gradient instead of considering a finite volume around the notch tip. Another paper to be mentioned is Ref. [10] of Tanaka. In this line, a recent proposal to estimate the fatigue limit of notched components is that by Taylor [11]. Accordingly, the fatigue limit condition in a notched structural element occurs either when the amplitude of a relevant stress component (e.g. the normal stress perpendicular to the notch bisector), calculated from a linear elastic analysis at a given distance from the notch tip (point method), is equal to raf (raf = normal stress fatigue limit measured from a smooth specimen), or when the averaged value of such a stress amplitude over a line (line method) or a semi-circular area (area method) ahead of the notch tip is equal to raf. Such so-called critical-distance methods assume that the distance from the notch tip, the length of the line and the radius of the area ahead of the notch tip are equal to L/2, 2L and L, respectively, where L is a material constant (characteristic length of the material) equal to the ElHaddad ‘intrinsic’ crack length [12], that is 2 1 DK I;th L¼ p raf
ð1Þ
DKI,th being the threshold range of the stress intensity factor for long cracks under Mode I. Taylor has shown that these methods can successfully be applied to predict the fatigue limit of structural components containing either sharp, blunt or short notches [11,13] subjected to uniaxial cyclic loading. The
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predicting capability of the line method seems to depend on a theoretical argument in the case of sharp notches, which are characterised by non-propagating cracks at the fatigue limit [14]. As a matter of fact, the stress intensity factor range of a non-propagating crack of length 2L originating from the tip of the notch has been shown to attain the threshold value DKth of the material under a stress field which, in the uncracked body, presents an averaged value of the normal stress amplitude along the crack line equal to raf [15]. When a notched component is subjected to a far-field multiaxial cyclic loading, appropriate stress components ahead of the notch tip must be selected in order to exploit the above critical-distance methods. As has been shown in the works by Susmel and Taylor [16], Susmel [17] and Naik et al. [18], critical plane-based multiaxial criteria can be used in conjunction with one of the critical-distance methods. In the present paper, an extension of the Carpinteri–Spagnoli criterion presented in Refs [19–24] to multiaxial notch fatigue by using the point method is proposed. 2. The Carpinteri–Spagnoli criterion for multiaxial high-cycle fatigue According to this criterion, the fatigue limit estimation under multiaxial high-cycle fatigue is performed by considering the time-varying stress tensor (determined from a linear elastic analysis) at a given point of the material. In the case of stress components assumed to be periodic functions of time, the main steps of the criterion are as follows [19–23]: • Averaged directions of the principal stresses are determined on the basis of their instantaneous directions (two material parameters are required at this step: the fatigue limit raf under fully reversed normal stress, and the corresponding slope m of the S–N curve in the high-cycle regime). • The orientation of the verification material plane (hereafter termed ‘critical plane’, where to perform fatigue strength assessment, see below) and that of the final fracture plane are linked to the averaged directions of the principal stress axes (one further material parameter is required at this step: the fatigue limit saf under fully reversed shear stress). • The maximum value of normal stress and the amplitude of shear stress, both acting on the critical plane, are determined in a loading cycle. • The fatigue strength estimation is performed through a quadratic combination of normal and shear stress components acting on the critical plane. At a given material point P, the direction cosines of the instantaneous principal stress directions 1, 2 and 3 (being r1(t) P r2(t) P r3(t)) with respect to a fixed PXYZ frame can be worked out from the time-varying stress tensor r(t). The orthogonal coordinate system P123 with origin at point P and axes coincident with the principal directions (Fig. 1) can be defined through the ‘principal Euler angles’, /, h, w, which represent three counter-clockwise sequential rotations around the Z-axis, Y 0 -axis and three-axis, respectively (0 6 / < 2p; 0 6 h 6 p; 0 6 w < 2p). The procedure to obtain the principal Euler angles from the direction cosines of the principal stress directions consists of two stages described in Ref. [16], and is rather lengthy, although very simple. The ranges of such Euler angles at the end of this two-stage procedure are reduced as follows: 0 6 /, h 6 p/2 and p/2 < w 6 p/2. As is well known, the instantaneous directions of the principal stresses do not vary with time in the case of a proportional loading (i.e. the loading case characterised by a time-varying stress tensor which can formally be written as r(t) = f(t)r, where f(t) is a scalar time function and r is a time-constant stress tensor), whereas the opposite occurs for non-proportional loading. However, it is worth noticing that, also in the case of a proportional loading, principal stress directions might vary with respect to time as a result of principal stress ordering r1 P r2 P r3 [19]. This occurs, for instance, even in the simple case of uniaxial tension-compression [21]. The averaged directions of the principal stress axes ^1; ^2 and ^3 are obtained from the averaged values ^ ^ of the principal Euler angles. Such values are computed by independently averaging the instantaneous /; ^ h; w values /(t), h(t), w(t), as follows [19]: Z T Z T Z T 1 ^¼ 1 ^¼ 1 / /ðtÞW ðtÞ dt ^ h¼ hðtÞW ðtÞ dt w wðtÞW ðtÞ dt ð2Þ W 0 W 0 W 0
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Z 1
X
θ
θ
P
ψ
3
φ
Y
Y' 2
Fig. 1. Principal stress directions 1, 2, 3 described through the Euler angles /, h, w.
where T is the period of the cyclic load and W ¼ 1 1 r1 ðtÞ m W ðtÞ ¼ h r1 ðtÞ raf 2 raf
RT 0
W ðtÞ dt. The weight function W(t) is given by ð3Þ
where h[Æ] is the Heaviside function (h[x] = 1 for x > 0, h[x] = 0 for x 6 0). The adopted weight function is such that it includes into the averaging procedure those positions of the principal directions for which the maximum principal stress r1 is greater than half of the normal stress fatigue limit raf under fully reversed normal stress (a discussion on such a weight function is presented elsewhere [19]). As has been pointed out by Brown and Miller [25], fatigue crack propagation can be distinguished into two stages: a first one (Stage 1) in which a crack nucleates (near the external surface of a structural component) along a shear slip plane (Mode II, fatigue crack initiation plane), and a second one (Stage 2) in which crack propagation occurs in a plane normal to the direction of the maximum principal stress (Mode I, final fatigue fracture plane). By comparing theoretical and experimental results [21], the normal to the estimated final fatigue fracture plane of Stage 2, which is the one observed post mortem at the macro level, has been assessed to be coincident with the averaged direction ^ 1 of the maximum principal stress r1. This implies that, according to the present criterion, the orientation of the final fatigue fracture plane depends on the time-varying stress state and the material parameters raf and m. The orientation of the critical plane, where to perform fatigue strength assessment, has been proposed to be correlated with the averaged directions of the principal stress axes through the off angle d (d = angle between the normal w to the critical plane and the averaged direction ^1, of the maximum principal stress r1, where w belongs to the principal plane ^ 1^ 3 as is shown in Fig. 2) [21]. The following empirical expression for d is proposed: " 2 # 3p saf 1 d¼ ð4Þ 8 raf Accordingly, the off angle d should be regarded as a material parameter (depending on the fatigue limit ratio) which influences the orientation of the critical pffiffiffi plane. Eq. (4) is valid for hard metals which are characterised by values of the ratio saf/raf ranging from 1= 3 to 1 (note that the lower limit of saf/raf corresponds to the Von Mises strength criterion of mild metals for static loading). In the light of the above, the orientation of the critical plane depends on the time-varying pffiffiffi stress state as well as the material parameters raf, saf and m. When the ratio saf/raf tends to 1= 3 (threshold between mild and hard metals), the off angle d tends to p/4. In this case, the critical plane tends to coincide with the actual fatigue crack initiation plane (Stage 1). This is justified by the fact that, in the high-cycle fatigue regime, the microcrack initiation stage is the critical one for such materials. On the other hand, when the ratio saf/raf tends to 1 (threshold between hard and extremely hard metals), the off angle d tends to zero. This is in line with the fact that such materials have predominantly
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Z
w ˆ1
δ
θˆ
X
P
ˆ3
ψˆ
φˆ
Y
Y' ˆ2
Fig. 2. Correlation between averaged principal stress directions ^1^2^3 and normal w to the critical plane.
Stage 2 crack growth and, hence, the critical plane is assumed to be the final fatigue fracture plane. In conclusion, the introduction of the off angle d allows us to consider both shear stress (Mode II) and normal stress (Mode I) mechanisms (the governing mechanism depends on the fatigue behaviour of the material and the stress state) of crack initiation pointed out by Socie [26]. The stress vector Sw acting on the critical plane at point P can be expressed as follows (Fig. 3): Sw ¼ rw
ð5Þ
where w is the vector normal to the critical plane. Then, the normal stress vector N and the shear stress vector C lying on the critical plane are: N ¼ ðw Sw Þw
ð6Þ
C ¼ Sw N
ð7Þ
For multiaxial constant amplitude cyclic loading, the vectors N and C are periodic functions of time. The direction of the normal stress N(t) is fixed with respect to time: consequently, the mean value Nm and the amplitude Na of the normal stress can readily be calculated. On the other hand, the definitions of the shear stress mean value Cm and amplitude Ca are not unique, owing to the generally time-varying direction of C. The procedure proposed by Papadopoulos [27] has been adopted in Ref. [22,23] to determine the mean value Cm and the amplitude Ca of the shear stress vector C:
Z
critical plane
N(t)
Σ
X
Qm
w
Sw(t)
P C(t) critical plane
Ca Cm
C(t)
P
C′
Q'
Y Fig. 3. Stress components acting on the critical plane.
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0
1869
C m ¼ min max kCðtÞ C k 0 C
06t6T
ð8Þ
C a ¼ max kCðtÞ C m k 06t6T
where the symbol k k indicates the Euclidean norm of a vector, C(t) is the shear vector at the time instant t, C0 is a vector chosen according to the procedure presented in Ref. [27] (see Fig. 3, where R is the closed path described by the shear stress vector C during a loading cycle). Note that, for proportional loading, we have r(t) = f(t)r and, hence, Sw(t) = r(t) w = f(t)rw (see Eq. (5)). Therefore, the direction of Sw and that of C (see Eq. (7)) are fixed with respect to time, since f(t) is a scalar time function. As a multiaxial fatigue limit condition, the following non-linear combination of the maximum normal stress (Nmax = Nm + Na) and the shear stress amplitude (Ca) acting on the critical plane has been proposed [22,23]: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ffi raf 2 req;a ¼ N max þ C 2a ¼ raf ð9Þ saf Such a criterion is to some extent reminiscent of the well-known Gough ellipse [28]. However, differently from the Gough’s criterion, the proposed criterion considers stress components acting on the critical plane and, hence, appears to be sensitive to loading non-proportionality (e.g. for sinusoidal loading, the critical plane orientation depends on the phase angles between the stress components [23]). Note that some well-established experimental findings are included in Eq. (9), namely: the mean shear stress Cm does not affect the high-cycle fatigue strength of hard metals [28], whereas the effect of the mean normal stress Nm is taken into account. 3. Extension of the Carpinteri–Spagnoli criterion to multiaxial notch fatigue using the critical-point method An extension of the multiaxial Carpinteri–Spagnoli criterion [19–24] using the critical-point method of Taylor [11] is hereafter proposed to estimate the fatigue limit of notched components. The application of the critical-distance method (originally conceived for uniaxial loading) to multiaxial fatigue requires to tackle some issues, including: position of the crack initiation point (the so-called hot spot) on the notch surface; variation of the characteristic length L (the intrinsic crack length) as a function of the notch loading modes (Mode I/Mode II/Mode III); direction along which the characteristic length should be considered to determine the position of the critical point (such a direction is the so-called ‘assumed crack path’); stress components (at the critical point) to be considered for multiaxial fatigue limit estimation. The extension proposed is based on some simple assumptions which deal with the above issues. Consider a traction-free notch surface contained in a body submitted to a periodic fatigue loading. At any point on the notch surface, a principal stress is always null and its direction is normal to the notch surface. The point H of crack initiation on the notch surface (see Fig. 4 where, for the sake of simplicity, a plane stress/ strain situation is considered) is assumed as that point experiencing the maximum value of req,a Eq. (9). Hence, with respect to the coordinate systemxy, we have
ðxH ; y H Þ ¼ max req;a ðx; yÞ ð10Þ x;y2S n
where Sn indicates the notch surface. Note that, under any applied multiaxial loading, the components of the stress tensor r(t) at any point on the notch surface are proportional (r(t) = f(t)r). Hence, the principal stress directions might vary with respect to time only as a result of principal stress ordering (r1 P r2 P r3) [19]. Since according to the weight function in Eq. (3) only the time instants related to r1 > raf/2 are considered in the averaging procedure, the averaged principal stress directions at any point on the notch surface are those depicted in Fig. 4 (^ 1 is tangent to the notch surface and ^3 is normal to such a surface). Note that the maximisation procedure of Eq. (10) indicates that the hot spot corresponds to the point on the notch surface experiencing the maximum amplitude of the maximum principal stress r1. The orientation of the critical plane at point H is computed by considering the off angle d (Fig. 4). Then, it is thought that such an orientation does not change up to point P which is at a distance L/2 from the notch surface (Fig. 4). As an engineering assumption, the length L is taken to be that related to Mode I loading (see Eq. (1))
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w( P) 3ˆ(H)
w(H)
y
O
1ˆ(H)
1ˆ(P)
P
δ
L/2 H
x
Fig. 4. Critical plane orientation (w(H) w(P) is the normal to the critical plane) in the vicinity of a general traction-free notch surface (plane stress/strain conditions are assumed).
although notch loading conditions are in general multimodal (i.e. involving Modes I, II and/or III). Clearly, such assumption is a strong simplification of the physical reality of the problem, but it might have the advantage of avoiding the knowledge of fatigue threshold parameters under mixed mode loading which are often difficult to be determined experimentally (e.g. see Ref. [29] ). In line with the point method of Taylor [11], the equivalent normal stress req,a Eq. (9) is calculated from the stress tensor (obtained from linear-elastic analysis) at point P, bearing in mind that the orientation of the critical plane (where to calculate the normal Nmax and shear stress Ca components) depends on the averaged principal stress directions at point H and the off angle d. In other words, the critical plane orientation does not varies moving from point H to point P; the underlying physical assumption is that, at fatigue limit (non-propagating) cracks tend to nucleate on the critical plane defined by the surface conditions, and continue to propagate in a self-similar manner along that direction. The limiting cases for the critical planeporientation are as follows: if saf/raf = 1 the crack path HP is normal to the notch surface, while if ffiffiffi saf =raf ¼ 1= 3 the crack path HP is p/4 off the normal to the notch surface. Note that if two principal stress components (e.g. r2 = r3 = 0) are null at point H, two principal directions are indeterminate (e.g. in the plane normal to the principal direction of r1) and, hence, also the critical plane orientation is indeterminate. In such a case (e.g. under plane stress conditions), it is assumed that the principal direction of r3 belongs to the stress plane (xy plane in Fig. 4), so that the critical plane (which is coincident with the crack path as is shown above) is the plane of maximum stress gradient (this is in line with the principle of the ‘focus path’ introduced by Taylor and co-workers [30] to deal with the application of the criticaldistance theory to three-dimensional notches). From the stress tensor at point P, the averaged direction ^1 of the maximum principal stress can be calculated. It can be conjectured, although this is beyond the scope of the present paper, that the crack plane during Stage 2 propagation (final fatigue fracture plane) is normal to the direction ^1 at point P (Fig. 4). 4. Comparison with experimental results for circular notches Some experimental tests by Endo and Murakami [31] and Endo [32,33] related to proportional loading are considered for comparison. Further comparisons for non-proportional loading will be considered in the near future. The experiments concern round bars with artificially drilled surface holes (the hole diameter D ranges from 40 to 500 lm, Fig. 5). The bars are subjected to fully reversed bending or torsion [31], or fully reversed bending, torsion and combined in-phase bending and torsion [32,33]. The materials being analysed are: 0.46%C pannealed steel (with Vickers hardness equal to 170, raf = 245 MPa, saf = 145 MPa, DKI,th = 5.3 MPa m, L = 150 lm), and 0.37%C annealed steel (with Vickers hardness equal to 160, raf = 233 MPa, p saf = 145 MPa, DKI,th = 7.6 MPa m, L = 340 lm). The threshold range of the stress intensity factor for Mode I is estimated from the values reported in Ref. [34]. The experimental tests by Endo and Murakami allowed the determination of fatigue limit conditions in terms of applied normal and shear stress amplitudes. These fatigue limit conditions are in general characterised by non-propagating cracks at the hole edge [31–33].
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τxy,a τxy,a
y ˆ 1(H)
w(H)
δ H
σx,a O
3ˆ(H)
P
β
σx,a
α x
τxy,a
D
τxy,a Fig. 5. Critical plane orientation (w(H) is the normal to the critical plane) in the vicinity of a circular hole under in-phase axial and shear stresses (plane stress conditions are assumed).
From Eq. (4), the values of the off angle d are equal to 44 and 41 for the tests by Endo and Murakami [31] and Endo [32,33], respectively. The experimental tests by Endo and Murakami allows us to investigate the notch size effect (notch fatigue limit against D/L, with D = hole diameter), while the influence of notch acuity (e.g. D/q, with q = notch radius of curvature at hot spot) on notch fatigue limit clearly cannot be investigated since the D/q ratio is constant for circular holes. Note that the experimental results of Endo and Murakami in Figs. 6 and 7 below are presented against D (instead of D/L), since, for each series of tests, L assumes a constant value. The applied far-field stress amplitudes rx,a (due to bending) and sxy,a (due to torsion) are shown in Fig. 5, where the xy plane is tangent to the bar surface in the hole center. Stress field on the bar surface in the vicinity of the hole is here determined according to the Kirsch solution for plane stress state [35]. The different values of the biaxiality ratio k = sxy,a/rx,a being considered in the experimental tests are: 0 (pure bending), 0.5, 1, 2, 1 (pure torsion). In Table 1, the hot spot position (see angle a in Fig. 5) as a function of the different values of the biaxiality ratio is reported along with the critical plane orientation (see angle b in Fig. 5). Fig. 6 shows experimental results by Endo and Murakami [31] for pure bending or torsion, and theoretical results determined through the Carpinteri–Spagnoli criterion. The comparison is quite satisfactory with an error within about 15%. Note that a smaller notch size effect in torsion with respect to that in bending is shown by experiments and well predicted by the proposed criterion. As is stated in Ref. [31], non-propagating cracks develop at the specimen surface (instead of at the hole edge) in the fatigued specimens under torsion with D < 100 lm. Fig. 7 illustrates some experimental results by Endo [32,33] for pure bending, torsion and combined bending and torsion. Again, although to a less degree, the comparison with the theoretical results is quite satisfactory with an error within about 25%. It is worth noticing that, in a recent paper by Taylor [36], the same results by Endo and Murakami [31] were compared with the results deduced by combining the line method of Taylor [11] with the Susmel–Lazzarin criterion (e.g. see Ref. [37]). In particular, different paths emanating from the hole were considered to average stresses according to the line method. It is interesting to note that Taylor [36] concluded that crack paths forming a 45 angle with the normal to the hole at the hot spot gave the best comparisons, which is in line with the results of the present investigation, since the crack path HP (Fig. 5) forms an angle with the normal to the hole at point H equal to 44 and 41 for the tests by Endo and Murakami [31] and Endo [32,33], respectively. Finally, it is interesting to note that the curves related to torsion or combined bending and torsion (k 5 0) in Figs. 6 and 7 exhibit a non-monotonic trend. Such a trend is due to the variation of the values of the stress
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1000 Exp. - bending Exp. - torsion Theory - bending
Fatigue limit (MPa)
Theory - torsion
100
0
100
1000
Hole diameter (μm) Fig. 6. Experimental results [31] and theoretical predictions of fatigue limit as a function of the hole diameter for bending (in terms of applied normal stress amplitude) and torsion (in terms of applied shear stress amplitude).
1000
Exp. - λ = 0.0 (bending) Exp. - λ = 0.5 Exp. - λ = 1.0
Fatigue limit (MPa)
Exp. - λ = 2.0 Exp. - λ = ∞ (torsion) Theory λ = 0.0 (bending)
λ = 0.5 λ =1
100
λ = ∞ (torsion)
λ =2
0
100
1000
Hole diameter (μm) Fig. 7. Experimental results [32,33] and theoretical predictions of fatigue limit as a function of the hole diameter for bending, torsion and combined bending and torsion (in terms of applied maximum principal stress amplitude).
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Table 1 Position of the hot spot (crack initiation point, see angle a in Fig. 5) on the hole surface and direction of the critical plane (see angle b in Fig. 5) k
jaj (deg)
jbj (deg)
0.0 0.5 1.0 2.0 1
90 70 60 50 45
90 d 70 d 60 d 50 d 45 d
equal to L/2 in Fig. 5) from the hole surface as the hole components at a constant distance (see the distance HP diameter D is made to vary. When uniaxial loading is considered (pure bending, k = 0) the normal stress and the shear stress acting on the critical plane at point P monotonically increase with increasing D, while under biaxial loading (torsion or combined bending and torsion, k 5 0) a non-monotonic trend characterized by a minimum is observed for the shear stress. Consequently, when k 5 0 the amplitude of the equivalent normal stress req,a (function of the normal stress and shear stress acting on the critical plane, see Eq. (9)) at point P, which is used for fatigue limit assessment, presents a minimum as the hole diameter varies. 5. Concluding remarks The present study represents a first attempt to predict fatigue limit conditions for multiaxially loaded notched components by combining a critical plane-based criterion recently proposed by the first two authors (Carpinteri–Spagnoli criterion) with a critical-distance method. In particular, following the philosophy of the point method, fatigue limit conditions occur in a notched component submitted to far-field multiaxial loading when the amplitude of the equivalent normal stress, defined according to the Carpinteri–Spagnoli criterion, attains the normal stress fatigue limit for smooth components in a point at a distance from the hot spot on the notch surface which is half the ElHaddad intrinsic crack length. Such a distance is measured along a direction which depends on the material fatigue limit ratio. The comparison with some experimental results concerning circular notches casts a promising light on the present proposal. Acknowledgement The authors gratefully acknowledge the research support for this work provided by the Italian Ministry for University and Technological and Scientific Research (MIUR). References [1] Neuber H. Theory of notch stresses: principles for exact stress calculation. Edwards; 1946. [2] Peterson RE. Notch sensitivity. In: Sines G, Waisman JL, editors. Metal fatigue. McGraw-Hill; 1959. p. 293–306. [3] Flavenot J-F, Skalli NA. A critical depth criterion for the evaluation of long-life fatigue strength under multiaxial loading and a stress gradient. In: Brown MW, Miller KJ, editors. Biaxial and multiaxial fatigue ECF 3. Mechanical Engineering Publications; 1989. p. 355–65. [4] Qylafku G, Azari Z, Kadi N, Gjonaj M, Pluvinage G. Application of a new model proposal for fatigue life prediction on notches and key-seats. Int J Fatigue 1999;21:753–60. [5] Seweryn A, Mroz Z. On the criterion of damage evolution for variable multiaxial stress states. Int J Solids Struct 1998;35:1589–616. [6] Sheppard SD. Field effects in fatigue crack initiation: long life fatigue strength. Trans ASME J Mech Des 1991;113:188–94. [7] Lazzarin P, Zambardi R. A finite-volume-energy based approach to predict the static and fatigue behaviour of components with sharp V-shaped notches. Int J Fract 2001;112:275–98. [8] Yosibash Z, Bussiba A, Gilad I. Failure criteria for brittle elastic materials. Int J Fract 2004;125:307–33. [9] Papadopoluos IV, Panoskaltsis VP. Invariant formulation of a gradient dependent multiaxial high-cycle fatigue criterion. Engng Fract Mech 1996;55:513–28. [10] Tanaka K. Engineering formulae for fatigue strength reduction due to crack-like notches. Int J Fract 1983;22:R39–45. [11] Taylor D. Geometrical effects in fatigue: a unifying theoretical model. Int J Fatigue 1999;21:413–20.
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A. Carpinteri et al. / Engineering Fracture Mechanics 75 (2008) 1864–1874
[12] ElHaddad MH, Dowling NF, Topper TH, Smith KN. J integral applications for short fatigue cracks at notches. Int J Fract 1980;16:15–24. [13] Taylor D, Wang G. The validation of some methods of notch fatigue analysis. Fatigue Fract Engng Mater Struct 2000;23:387–94. [14] Smith RA, Miller KJ. Prediction of fatigue regimes in notched components. Int J Mech Sci 1978;20:201–6. [15] Taylor D. A mechanistic approach to critical-distance methods in notch fatigue. Fatigue Fract Engng Mater Struct 2001;24:215–24. [16] Susmel L, Taylor D. Two methods for predicting the multiaxial fatigue limits of sharp notches. Fatigue Fract Engng Mater Struct 2003;26:821–33. [17] Susmel L. A unifying approach to estimate the high-cycle fatigue strength of notched components subjected to both uniaxial and multiaxial cyclic loadings. Fract Engng Mater Struct 2004;27:391–411. [18] Naik RA, Lanning DB, Nicholas T, Kallmeyer AR. A critical plane gradient approach for the prediction of notched HCF life. Int J Fatigue 2005;27:481–92. [19] Carpinteri A, Brighenti R, Macha E, Spagnoli A. Expected principal stress directions for multiaxial random loading-part I: theoretical aspects of the weight function method. Int J Fatigue 1999;21:83–8. [20] Carpinteri A, Brighenti R, Macha E, Spagnoli A. Expected principal stress directions for multiaxial random loading – part II: numerical simulation and experimental assessment through the weight function method. Int J Fatigue 1999;21:89–96. [21] Carpinteri A, Brighenti R, Spagnoli A. A fracture plane approach in multiaxial high-cycle fatigue of metals. Fatigue Fract Engng Mater Struct 2000;23:355–64. [22] Carpinteri A, Spagnoli A. Multiaxial high-cycle fatigue criterion for hard metals. Int J Fatigue 2001;23:135–45. [23] Spagnoli A, A new high-cycle fatigue criterion applied to out-of-phase biaxial stress state. Int. J Mech Sci 2001;43:2581–95. [24] Carpinteri A, Spagnoli A, Vantadori S. A multiaxial fatigue criterion for random loading. Fatigue Fract Engng Mater Struct 2003;26:515–22. [25] Brown MW, Miller KJ. A theory for fatigue failure under multiaxial stress–strain condition. Proc Inst Mech Engng 1973;187:745–55. [26] Socie DF. Multiaxial fatigue damage models. J Engng Mater Technol 1987;109:293–8. [27] Papadopoulos IV. Critical plane approaches in high-cycle fatigue: on the definition of the amplitude and mean value of the shear stress acting on the critical plane. Fract Engng Mater Struct 1998;21:269–85. [28] Gough HJ, Pollard HV, Clenshaw WJ. Some experiments on the resistance of metals to fatigue under combined stresses. Aeronautical Research Council Reports, R and M 2522, HMSO, 1951. [29] Tong J, Yates JR, Brown MW. Some aspects of fatigue thresholds under mode III and mixed mode III and I loadings. Int J Fatigue 1996;18:279–85. [30] Bellett D, Taylor D, Marco S, Mazzeo E, Guillois J, Pircher T. The fatigue behaviour of three-dimensional stress concentrations. Int J Fatigue 2005;27:207–21. [31] Endo M, Murakami Y. Effect of an artificial small defect on torsional fatigue strength of steels. J Engng. Mater Technol (ASME Trans) 1987;109:124–9. [32] Endo M. Effects of small defects on the fatigue strength of steel and ductile iron under combined axial/torsional loading. In: Ravichandran KS, Ritchie RO, Murakami Y, editors. Small fatigue cracks: mechanics. Mechanisms and applications. Elsevier; 1999. p. 375–87. [33] Endo M. The multiaxial fatigue strength of specimens containing small defects. In: Carpinteri A, de Freitas M, Spagnoli A, editors. Biaxial/multiaxial fatigue and fracture ESIS publication, vol. 31. Elsevier; 2003. p. 243–64. [34] Murakami Y. Metal fatigue: effects of small defects and nonmetallic inclusions. Elsevier; 2002. [35] Kirsch G. Die teorie der elastizita¨t und die bedu¨rfnisse der festigkeitslehre. VDI 1898;42:797–807. [36] Taylor D. Notch size effects in multiaxial fatigue. In: Susmel L, Tovo R, editors. Progettazione a fatica in presenza di multiassialita` tensionali. SG Editoriali; 2005. p. 3–13. [37] Susmel L, Lazzarin P. A bi-parametric Wohler curve for high cycle multiaxial fatigue assessment. Fatigue Fract Engng Mater Struct 2002;25:63–78.