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On the use of the Prismatic Hull method in a critical plane-based multiaxial fatigue criterion
International Journal of Fatigue xxx (2014) xxx–xxx
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International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue
On the use of the Prismatic Hull method in a critical plane-based multiaxial fatigue criterion Andrea Carpinteri, Camilla Ronchei, Andrea Spagnoli, Sabrina Vantadori ⇑ Dept. of Civil-Environmental Engineering and Architecture, University of Parma, Parco Area delle Scienze 181/A, 43124 Parma, Italy
a r t i c l e
i n f o
Article history: Received 4 March 2014 Received in revised form 2 May 2014 Accepted 15 May 2014 Available online xxxx Keywords: Critical plane approach Multiaxial high-cycle fatigue Shear stress amplitude Stress-base criterion
a b s t r a c t In the present paper, the modified Carpinteri–Spagnoli (C–S) criterion is combined with the Prismatic Hull (PH) method proposed by Mamiya, Araújo et al. [15] to evaluate an equivalent shear stress amplitude for multiaxial fatigue loading. More precisely, the PH method, originally formulated for synchronous loading and successively extended to asynchronous loading, is here adopted to compute the shear stress amplitude acting on the critical plane, used in the modified C–S criterion. By comparing some experimental data available in the literature with the theoretical estimations, the multiaxial fatigue strength evaluations derived through the modified C–S criterion are shown to be improved if the shear stress amplitude is determined by applying the PH method instead of the Minimum Circumscribed Circle (MCC) method, especially for particular shear stress paths as is here discussed. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Many metallic structural components, such as crankshafts, axles, steel rails, wheels and fan discs in jet engines, are subjected to multiaxial high-cycle fatigue during their service life [1]. Therefore, the fatigue strength of the aforementioned structural components is complex to be evaluated through theoretical models, because it depends on many factors related to stress paths, material properties and possible stress concentrations [2–4]. Several criteria are available in the literature to assess the above structural components under multiaxial stresses, but a unique method generally recognized as completely reliable does not exist. An interesting review of the most relevant criteria is reported in Ref. [5], where they are classified into three categories, that is: stress-, strain-, and energy-based criteria. Within the framework of a total life approach, under high-cycle fatigue (i.e. in presence of more than about 105 loading cycles, for metallic materials) and constant-amplitude cyclic loading, most of criteria are stress-based and aim at reducing a given multiaxial stress state to an equivalent uniaxial stress to be compared with a single parameter experimentally determined through the uniaxial S–N curve [5]. Several fatigue experimental tests have shown that the polycrystalline metallic materials are characterised by plastic deformations that occur in heterogeneous manner only at the mesoscopic scale, while no plasticity is detected at the ⇑ Corresponding author. E-mail address: [email protected] (S. Vantadori).
macroscopic one. More precisely, at the mesoscopic level, the fatigue limit of such materials is identified by (elastic) shakedown condition, while the plastic shakedown develops in the finite lifetime domain. According to the critical review reported in Ref. [6], the above stress theories can be classified into four viewpoints: empirical formulas, application of stress invariants, integral approach, and critical plane approach. As far as the fatigue criteria related to the use of stress invariants are concerned, most of them are based on the combined effects of normal stress and maximum shear stress. The significant difference among them depends on the definition of the maximum shear stress [7–16]. Regarding the fatigue criteria based on the critical plane approach, whose use has increased during recent years, it is important to highlight that they are able not only to reduce a given multiaxial stress state to an equivalent uniaxial one, but also to describe the plane of the early crack nucleation [17–23]. The modified Carpinteri–Spagnoli (C–S) criterion [24–26] belongs to the last category mentioned above. It is a simplified version of the original C–S criterion [27], as the computation of the critical plane orientation in the modified criterion is easier than that proposed in the original version. This simplified criterion takes into account a fatigue damage parameter given by a nonlinear (quadratic) combination of the equivalent normal stress amplitude, Na,eq [24–26], and the amplitude, Ca, of the shear stress vector acting on the critical plane. More precisely, Na,eq takes into account the linear relationship (proposed by Goodman) between the normal stress amplitude and the normal stress mean value.
http://dx.doi.org/10.1016/j.ijfatigue.2014.05.007 0142-1123/Ó 2014 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Carpinteri A et al. On the use of the Prismatic Hull method in a critical plane-based multiaxial fatigue criterion. Int J Fatigue (2014), http://dx.doi.org/10.1016/j.ijfatigue.2014.05.007
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A. Carpinteri et al. / International Journal of Fatigue xxx (2014) xxx–xxx
Nomenclature ai(H) bi Bi C Ca fy I N Na,eq PXYZ P123 ^2 ^3 ^ P1 Sw t T w x X
a
i-th half-side of prismatic hull i-th component of an orthonormal basis of R5 i-th components of an orthonormal basis of dev3 shear stress vector acting on the critical plane shear stress amplitude material yield stress error index normal stress vector perpendicular to the critical plane equivalent normal stress amplitude fixed frame principal stress axes frame weighted mean principal stress frame stress vector at material point P, acting on the critical plane time observation time interval unit vector normal to the critical plane deviatoric stress vector at material point P in R5 deviatoric stress tensor at material point P in dev3 phase angle between longitudinal (axial) normal stress rx and tangential (hoop/circumferential) normal stress
ry b
phase angle between longitudinal (axial) normal stress
rx and shear stress sxy
c d
angle between the specimen’s longitudinal axis X and the normal w to the critical plane (Fig. 2) ^ of r1,max and the angle between the averaged direction 1 normal w to the critical plane (Fig. 2)
Different methods are available in the literature to define Ca [13–16,28–34]: the so-called Longest Chord method [29], the Longest Projection method [30], the Minimum Circumscribed Ellipsoid method [31,32], the Minimum Circumscribed Circle method [33,34] and the Prismatic Hull (PH) method [13–16]. The modified C–S criterion employs Ca computed by means of the Minimum Circumscribed Circle (MCC) method [33,34]. Accordingly, Ca is the radius of the minimum circle bounding the closed P path described by the shear stress vector C on the critical plane during the observation time interval T. Although the MCC method is able to uniquely define Ca, the following remarks have to be made: (i) Such a method is not able to distinguish between proportional and non-proportional loading, while experimental evidences highlight that, for a given value of the radius of the Minimum Circumscribed Circle, non-proportional loading may produce a greater damage than proportional one; (ii) The MCC method requires time-consuming algorithms in order to compute the centre and the radius of the circle P bounding . The problem of the high computational time is relevant when, for instance, asynchronous sinusoidal loadings are examined, although different strategies exist in order to optimise the procedure used to determine the Minimum Circumscribed Circle [34], and an analytical procedure can be applied in the case of synchronous loadings [33]. To improve the capabilities of the modified C–S criterion and make it a useful tool in fatigue design, the Prismatic Hull (PH) method proposed by Mamiya, Araujo et al. [13–16] to evaluate an equivalent shear stress amplitude, called seq, for multiaxial fati-
four-dimensional quantity used to identify the orthonormal basis of R5 j material parameter q material parameter kxy ratio between the signal pulsations xxy/xx ky ratio between the signal pulsations xy/xx r stress tensor at material point P raf,1 fully reversed normal stress fatigue limit rp, max maximum value of the hydrostatic stress rn, n = 1, 2, 3 principal stresses, with r1 P r2 P r3 ru material ultimate tensile stress rx(t) longitudinal (axial) normal stress ry(t) tangential (hoop/circumferential) normal stress sxy(t) shear stress saf,1 fully reversed shear stress fatigue limit seq equivalent shear stress amplitude /, h,w principal Euler angles ^ ^h; w ^ /; weighted principal Euler angles xx pulsation of the sinusoidal longitudinal normal stress rx xy pulsation of the sinusoidal tangential normal stress ry xxy pulsation of the sinusoidal shear stress sxy
H
Subscripts a amplitude m mean value max maximum value
gue criteria is herein employed. More precisely, the PH method, originally formulated for synchronous loading [13,14] and successively extended to asynchronous loading [15,16], is now adopted to compute the shear stress amplitude, Ca, used in the modified C–S criterion. Note that, when the path of the shear stress is plane, the method is also named Maximum Rectangular Hull (MRH) method [16,35], since the prismatic hull circumscribing the shear stress path degenerates in a rectangular hull. The PH method and the MRH method are able to distinguish between proportional and non-proportional loading, and no complex computation algorithms are needed. Some experimental tests available in the literature [36–43], related to in- and out-of-phase synchronous and asynchronous loading, are hereafter analysed in order to evaluate the fatigue strength estimation capabilities of the PH method implemented in the modified C–S criterion. 2. Main steps of the modified C–S criterion The two main steps of the modified C–S criterion [24–26] are: (1) Determination of the critical plane orientation, which is linked to the averaged principal stress directions; (2) Evaluation of suitable stress components, related to the verification plane (critical plane), to be used in the fatigue strength assessment. 2.1. Averaged principal stress axes and orientation of the critical plane The orientation of the critical plane is assumed to be linked to the directions of the principal stress axes. Such directions are
Please cite this article in press as: Carpinteri A et al. On the use of the Prismatic Hull method in a critical plane-based multiaxial fatigue criterion. Int J Fatigue (2014), http://dx.doi.org/10.1016/j.ijfatigue.2014.05.007
A. Carpinteri et al. / International Journal of Fatigue xxx (2014) xxx–xxx
3
According to the modified C–S criterion [24–26], the multiaxial fatigue limit condition is expressed in the following form:
Na;eq
2
þ
raf ;1
2
Ca
saf ;1
¼1
ð2Þ
where Na,eq is the equivalent normal stress amplitude, and Ca is the amplitude of the shear stress vector. The amplitude Na,eq can be computed through the following relationship: Fig. 1. Cylindrical specimen: XYZ coordinate system with its origin in P. The instantaneous principal stress directions 1, 2 and 3 are defined through the principal Euler angles /, h, w.
generally time-varying under fatigue loading and, therefore, averaged principal stress directions are taken into account by deducing them through appropriate weight functions, as is described in the following. The origin of the fixed XYZ coordinate system is at a generic point P of the structural component being examined (Fig. 1). At a given time instant t, the location of the principal stress directions 1, 2 and 3 (being r1 P r2 P r3 ) can be defined through the principal Euler angles /, h, w. Mean principal stress directions are determined by averaging the instantaneous values of the principal Euler angles through a suitable weight function so that the averaged principal stress axes ^ 2; ^ 3 ^ coincide with the instantaneous ones at the time instant 1; when r1 achieves its maximum value during the observation time interval T [24–26]. Such an assumption makes the implementation of the criterion rather simple. The normal unit vector w of the critical plane is defined through ^ direction, in the averaged principal plane 1 ^3 ^ a rotation d of the 1 ^ ^ (from 1 to 3Þ. The off angle d is expressed as follows [24–27]:
d¼
" # saf ;1 2 3p 1 8 raf ;1
Na;eq ¼ Na þ raf ;1
Nm
ru
ð3Þ
where Na and Nm are the normal stress amplitude and mean value, respectively, and ru is the material ultimate tensile strength. When ru is not available, the yield stress fy might be used instead of the ultimate tensile strength, especially in the case of mild metals. Since the N direction is fixed with respect to time, the mean value Nm and the amplitude Na of its modulus can readily be computed. On the other hand, the definition of the amplitude Ca of the C vector modulus is not unique, and different methods are available in the literature [13–16,28–34]. In particular, the Prismatic Hull (PH) method [13–16] allows us to compute the amplitude Ca with a simple algorithm. In Refs. [24–27], the Minimum Circumscribed Circle (MCC) method [33,34] is implemented in the modified C–S criterion to compute Ca. In the present paper, Ca is computed by employing the Prismatic Hull (PH) method [13–16]. 3. The Prismatic Hull (PH) method Now the main steps of the PH method and its implementation in the modified C–S criterion are discussed. 3.1. Main steps of the PH method
ð1Þ
where raf,1 and saf,1 are the fatigue limit for fully reversed normal stress and for fully reversed shear stress, respectively.
The PH method [13–16] has been proposed to evaluate an equivalent shear stress amplitude, called seq, used in the following multiaxial fatigue criterion based on the stress invariants:
seq þ jrp;max 6 q 2.2. Fatigue strength assessment At each time instant t, the stress vector Sw related to the critical plane orientation may be decomposed in two components (Fig. 2), that is, the normal stress component N (perpendicular to the critical plane) and the shear stress component C (lying on the critical plane). During the observation time interval T, the direction of the normal stress vector N(t) is fixed with respect to time, while the P shear stress vector C(t) describes a closed path on the critical plane (as long as Sw is periodic).
ð4Þ
where rp,max is the maximum value of the hydrostatic stress during the observation time interval T, and j and q are material parameters. Let us consider a given material point P and its deviatoric stress tensor, X, defined in a space from R3 to R3 , and such a space is called dev3 in the following [13–15]. In dev3, the stress path of X is described by the set of the deviatoric stress states, X(t), so that, by considering an orthonormal basis Bi (with i = 1, . . . , 5) of dev3, we have:
XðtÞ ¼
5 X xi ðtÞBi
ð5Þ
i¼1
Fig. 2. Cylindrical specimen: components of the stress vector Sw acting on the critical plane.
Fig. 3. Rectangular hull identified by orientation H: the half lengths, Cu,a(H) and P Cv,a(H), of the rectangle enclosing the shear stress path on the critical plane are shown.
Please cite this article in press as: Carpinteri A et al. On the use of the Prismatic Hull method in a critical plane-based multiaxial fatigue criterion. Int J Fatigue (2014), http://dx.doi.org/10.1016/j.ijfatigue.2014.05.007
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A. Carpinteri et al. / International Journal of Fatigue xxx (2014) xxx–xxx
Table 1 Loading conditions of the experimental tests being examined. Material
No
Path type
a (°)
b (°)
kxy ; ky (–)
rx,a (MPa)
rx,m (MPa)
ry,a (MPa)
ry,m (MPa)
sxy,a (MPa)
sxy,m (MPa)
c (°)
Hard steel [36]
1 2 3
(a.1)
0 0 0
0 0 0
1 1 1
131.8 245.3 299.1
0.0 0.0 0.0
0.0 0.0 0.0
0.0 0.0 0.0
167.1 122.7 62.8
0.0 0.0 0.0
75 64 53
Mild steel [36]
4 5 6
(a.1)
0 0 0
0 0 0
1 1 1
99.9 180.3 213.2
0.0 0.0 0.0
0.0 0.0 0.0
0.0 0.0 0.0
120.9 90.2 44.8
0.0 0.0 0.0
78 67 56
Cast iron [36]
7 8 9
(a.1)
0 0 0
0 0 0
1 1 1
56.3 83.4 95.2
0.0 0.0 0.0
0.0 0.0 0.0
0.0 0.0 0.0
68.0 41.6 19.7
0.0 0.0 0.0
40 29 18
30NCD16 [37]
10
(a.1)
0
0
1
485.0
0.0
0.0
0.0
280.0
0.0
66
St35 [38]
11
(a.2)
0
0
1
160.0
176.0
160.0
176.0
0.0
0.0
61
42CrMo4V [38]
12
(a.2)
0
0
1
402.0
442.0
201.0
221.0
0.0
0.0
39
Hard steel [36]
13 14 15 16 17 18 19
(b.1)
0 0 0 0 0 0 0
30 30 60 60 90 90 90
1 1 1 1 1 1 1
140.4 249.7 145.7 252.4 150.2 258.0 304.5
0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0
169.9 124.9 176.3 126.2 181.7 129.0 63.9
0.0 0.0 0.0 0.0 0.0 0.0 0.0
75 63 77 59 80 41 41
Mild steel [36]
20 21 22 23 24
(b.1)
0 0 0 0 0
60 60 90 90 90
1 1 1 1 1
103.6 191.4 108.9 201.1 230.2
0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0
125.4 95.7 131.8 100.6 48.3
0.0 0.0 0.0 0.0 0.0
80 62 84 45 45
Cast iron [36]
25 26 27 28
(b.1)
0 0 0 0
60 60 90 90
1 1 1 1
67.6 93.7 73.2 101.9
0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0
81.6 46.9 88.4 21.1
0.0 0.0 0.0 0.0
42 25 46 7
St35 [38]
29 30
(b.1)
90 180
0 0
1 1
140.0 120.0
154.0 132.0
140.0 120.0
154.0 132.0
0.0 0.0
0.0 0.0
46 46
42CrMo4V [38]
31
(b.1)
90
0
1
402.0
442.0
201.0
221.0
0.0
0.0
39
30NCD16 [37]
32 33 34 35 36 37 38 39 40
(b.1)
0 0 0 0 0 0 0 0 0
0 0 0 45 45 60 90 90 90
1 1 1 1 1 1 1 1 1
211.0 480.0 590.0 480.0 565.0 470.0 473.0 480.0 540.0
300.0 300.0 300.0 300.0 300.0 300.0 300.0 0.0 300.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
365.0 277.0 148.0 277.0 141.0 270.0 273.0 277.0 135.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
69 59 51 57 48 54 41 41 41
25CrMo4 [39]
41 42 43 44
(b.2)
0 0 0 0
60 60 90 90
1 1 1 1
155.0 220.0 159.0 233.0
340.0 340.0 340.0 340.0
0.0 0.0 0.0 0.0
170.0 170.0 170.0 170.0
155.0 110.0 159.0 117.0
0.0 0.0 0.0 0.0
44 42 41 41
25CrMo4 [40]
45 46 47 48 49 50
(b.2)
0 60 90 90 90 180
90 90 45 90 135 90
1 1 1 1 1 1
208.0 225.0 222.0 205.0 215.0 224.0
255.0 255.0 255.0 255.0 255.0 255.0
156.0 169.0 167.0 154.0 161.0 168.0
210.0 210.0 210.0 210.0 210.0 210.0
104.0 113.0 111.0 103.0 108.0 112.0
0.0 0.0 0.0 0.0 0.0 0.0
39 37 42 37 40 37
34Cr4 [41]
51
(c)
0
0
2
200.0
244.0
200.0
244.0
0.0
0.0
44
25CrMo4 [42]
52 53 54 55
(c)
0 0 0 0
0 90 0 0
1/4 2 2 8
210.0 220.0 242.0 196.0
0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0
105.0 110.0 121.0 98.0
0.0 0.0 0.0 0.0
48 60 58 60
En24T [43]
56
(c)
180
0
3
260.0
0.0
260.0
0.0
0.0
0.0
55
St35 [38]
57 58
(c)
0 90
0 0
2 2
130.0 140.0
143.0 154.0
130.0 140.0
143.0 154.0
0.0 0.0
0.0 0.0
90 46
Hence, the stress path of X can also be described, in a more simply way, by employing the set of the deviatoric stress components:
xðtÞ fx1 ðtÞ; x2 ðtÞ; x3 ðtÞ; x4 ðtÞ; x5 ðtÞgT
ð6Þ
that corresponds, if referred to an orthonormal basis bi (with i = 1, . . . , 5) of R5 , to a hyper-curve. By considering, for example, the following orthonormal basis of dev3 [13–15]:
Please cite this article in press as: Carpinteri A et al. On the use of the Prismatic Hull method in a critical plane-based multiaxial fatigue criterion. Int J Fatigue (2014), http://dx.doi.org/10.1016/j.ijfatigue.2014.05.007
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A. Carpinteri et al. / International Journal of Fatigue xxx (2014) xxx–xxx
0
pffiffiffi 2= 6
1
0
1
0 0 0 0 pffiffiffi C B C C B 0 1= 2 C B ¼ 0 0 2 A @ A pffiffiffi pffiffiffi 0 0 1= 2 0 0 1= 6 pffiffiffi pffiffiffi 1 0 1 0 0 1= 2 0 0 0 1= 2 B pffiffiffi C B C B3 ¼ B B4 ¼ B 0 0 C 0 0C @ 1= 2 A @ 0 A pffiffiffi 1= 2 0 0 0 0 0 0 1 0 0 0 pffiffiffi C B B5 ¼ B 0 1= 2 C @0 A pffiffiffi 0 1= 2 0 B B1 ¼ B @
0
0 pffiffiffi 1= 6
ð7Þ
the deviatoric stress components, xi(t), in R5 are expressed as follows:
rffiffiffi 3 1 X x ðtÞ ¼ pffiffiffi 2rx ðtÞ ry ðtÞ rz ðtÞ 2 6 1 1 x2 ðtÞ ¼ pffiffiffi ½X y ðtÞ X z ðtÞ ¼ pffiffiffi ry ðtÞ rz ðtÞ 2 2 pffiffiffi pffiffiffi x3 ðtÞ ¼ 2X xy ðtÞ ¼ 2sxy ðtÞ pffiffiffi pffiffiffi x4 ðtÞ ¼ 2X xz ðtÞ ¼ 2sxz ðtÞ pffiffiffi pffiffiffi x5 ðtÞ ¼ 2X yz ðtÞ ¼ 2syz ðtÞ
x1 ðtÞ ¼
ð8Þ
where Xx, Xy, Xz, Xxy, Xyz, Xxz are the components of the deviatoric stress tensor, X, and rx, ry, rz, sxy, syz, sxz are the components of the stress tensor, r. Note that, at each time instant t, the Euclidean norm of the tensor X is equal to that of the vector x [15]. The PH method evaluates the equivalent shear stress amplitude, seq, by taking into account all the prismatic hulls (function of the considered orthonormal basis) which both enclose the path of the vector x and are tangent to such a path. For a given hull, the half values of its sides, ai (with i = 1, . . . , 5), can be determined as follows:
ai ¼
1 max xi ðtÞ min xi ðtÞ 06t
ð9Þ
Note that the above hulls might have a dimension smaller than five, depending on the stress state. For example, when the stress state is characterised only by the stress components rx, ry or rx, sxy, the hulls degenerate into rectangles [16,35], while the hulls degenerate into parallelepipeds when the stress components different from zero are rx, ry, sxy. For the particular case of synchronous cyclic loading, it has been demonstrated [13,14] that seq in Eq. (10) is a constant with respect to H. Therefore, any orthonormal basis of R5 leads to the same value of seq. 3.2. Application of the PH method to the critical plane shear stress vector Now the PH method is adopted to compute the amplitude of the shear stress vector C, by applying Eq. (10) not to the components of the five-dimensional vector x, but to the components of the twodimensional vector C, with the path of the shear stress C belonging to the critical plane. In such a case, the enclosing prismatic hull degenerates into a rectangle, and the four-dimensional quantity H degenerates in the scalar quantity H. Let us take into account a unit vector u coincident with the intersection between the critical plane and the plane defined by the normal unit vector w and the Z-axis, and a unit vector v normal to u so that uvw represents a right-hand orthogonal system (Fig. 2). The vector C is decomposed in two components along the directions of u and v, respectively (Fig. 3). Consider: (i) the rectangular hulls which both enclose the path of the vector C and are tangent to such a path, and (ii) the R2 basis defined by the unit vectors u and v, which are identified by an orientation angle H with respect to an arbitrary time-independent direction (see the horizontal dashed line in Fig. 3). Then, the rectangular hull half-sides can be computed according to Eq. (11):
C u;a ðHÞ ¼
1 max C u ðt; HÞ min C u ðt; HÞ 06t
ð12aÞ
C v ;a ðHÞ ¼
1 max C v ðt; HÞ min C v ðt; HÞ 06t
ð12bÞ
5
All the orthonormal bases of R are examined, in order to determine the specific prismatic hull, enclosing and tangent to the path of x, which maximises the following relationship:
seq
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X5 ¼ max ½a ðHÞ2 i¼1 i
ð10Þ
H
where H is a four-dimensional quantity used to identify the orthonormal basis bi and:
1 max xi ðt; HÞ min xi ðt; HÞ ai ðHÞ ¼ 06t
Finally, according to Eq. (10), the amplitude Ca of the shear stress vector is given by:
C a ¼ max
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½C u;a ðHÞ2 þ ½C m;a ðHÞ2
ð13Þ
06H
ð11Þ
where all R2 bases are taken into account by varying 0 6 H < p=2.
Table 2 Static, fatigue properties and extreme values of the error index, DI, obtained by applying both the MCC and the PH method to calculate Ca for each examined material. The DI values obtained by applying the Crossland [44], Papadopoulos et al. [45] and Gonçalves et al. [14] criteria are also reported. Authors
Ref.
Nishihara and Kawamoto
[36]
Froustey and Lasserre Bhongbhibhat
[37]
Zenner et al. Troost et al. Heidenreich et al. Kaniut McDiarmid
[39] [40] [41] [42] [43]
[38]
saf ;1 raf ;1
DI (MCC) (%)
DI (PH) (%)
DI (Crossland) (%)
DI (Papadopoulos) (%)
DI (Gonçalves et al.) (%)
196.2 137.3 91.2 410.0
0.63 0.58 0.95 0.62
10/1 5/1 3/14 18/3
10/1 5/1 3/14 18/3
18/2 15/1 3/27 27/4
3/7 4/13 3/37 8/4
3/4 4/12 3/15 4/8
130.0 315.0 228.0 228.0 204.0 228.0 270.0
0.57 0.65 0.63 0.67 0.59 0.67 0.67
36/15 1 27/19 24/9 27 46/23 28
36/15 1 20/11 16/1 27 46/23 15
35/10 7/2 23/13 19/8 9 27/21 13
35/10 7/8 9/4 0/21 2 14/0 18
34/11 1/14 4/0 4/24 10 26/13 4
ru/fy
raf,1
saf,1
(MPa)
(MPa)
(MPa)
Hard steel Mild steel Cast iron 30NCD16
681 374 181 1880
313.9 235.4 96.1 660.0
St35 42CrMo4V 25CrMo4 25CrMo4 34Cr4 25CrMo4 En24T
340 1003 780 660 550 – –
230.0 485.00 361.0 340.0 343.0 340.0 405.0
Material
Please cite this article in press as: Carpinteri A et al. On the use of the Prismatic Hull method in a critical plane-based multiaxial fatigue criterion. Int J Fatigue (2014), http://dx.doi.org/10.1016/j.ijfatigue.2014.05.007
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A. Carpinteri et al. / International Journal of Fatigue xxx (2014) xxx–xxx
St35 - Bhongbhibhat (1986) Specimens No. 11,29 0.6 0.2
29
-0.6
(a) 0
100
150
ry ¼ ry;a sinðxy t aÞ þ ry;m
ð14bÞ
sxy ¼ sxy;a sinðxxy t bÞ þ sxy;m
ð14cÞ
200
P (a) The shear stress path is represented by a line segment; P (b) The shear stress path is represented by an ellipse; P (c) The shear stress path is represented by a closed curve with a generic shape.
25CrMo4 - Zenner et al. (1985) Specimens No.41 - 44
100
44 43
0
In case (a), we can have the following sub-cases: 42
(a.1) For synchronous proportional loading (in-phase input signals with zero mean values), the line segment passes through the origin of the Puvw frame; (a.2) For synchronous affine loading (in-phase input signals with non-zero mean values), the line segment does not pass through the origin of the Puvw frame.
41
-100
(b) -200 300 200
0
50 100 150 200 250 300 En24T - McDiarmid (1985) Specimen No.56
In case (b), produced by synchronous non-proportional loading (out-of-phase signals), we can have the following sub-cases:
56 100
(b.1) One semi-axis is much greater than the other one; (b.2) The two semi-axes have the same order of magnitude.
0 -100
Finally, case (c) is produced by asynchronous loading.
-200
(c) -300 -150 -100 -50
0
50
4. Comparison with experimental data
100 150
SHEAR STRESS COMPONENT, Cu [MPa]
Now some experimental data available in the literature, related to smooth cylindrical metallic specimens (round bars and thinwalled tubes) subjected to synchronous and asynchronous, sinusoidal, in- and out-of-phase loading, are examined. The multiaxial stress states corresponding to fatigue limit conditions are reported in Table 1 (where kxy ¼ xxy =xx or ky ¼ xy =xx ). For each experimental test, the critical plane orientation, i.e. the angle c (Fig. 2) formed by the normal unit vector w with respect to the specimen’s longitudinal axis X, is also reported. A total number of 58 specimens are analysed. The static and fatigue properties of the materials being examined are shown in Table 2. More precisely, in order to evaluate the capabilities of
Fig. 4. Shear stress paths for the experimental tests in Refs. [38,39,43]: (a) specimens No. 11 and 29; (b) No. 41–44; and (c) No. 56.
3.3. Some types of shear stress paths on the critical plane
500
MCC method PH method 400 Path type: 300
(a1)
( b 1)
(a2)
(b2)
(c )
200
57 58
5556
51 52 53 54
50
47 48 49
4546
42 43 44
4041
37 38 39
3536
32 33 34
2526
22 23 24
2021
17 18 19
1516
12 13 14
1011
5
0
6 7 8 9
100
1 2 3 4
SHEAR STRESS AMPLITUDE, C a [MPa]
The plane stress condition of biaxial normal and shear stresses at the generic point P of a cylindrical body (Fig. 1) subjected to an asynchronous out-of-phase sinusoidal loading can be expressed as follows:
3031
SHEAR STRESS COMPONENT, Cv [MPa]
200
50
ð14aÞ
where rx is the longitudinal (axial) normal stress, ry is the tangential (hoop/circumferential) normal stress, and sxy is the shear stress. By varying the mean value, the pulsations and the phases of the above signals, the shear stress vector C describes closed paths of different shapes on the critical plane and, according to the above definition of the critical plane, three cases can occur:
-0.2
-1.0
SHEAR STRESS COMPONENT, C v [MPa]
11
0.0
rx ¼ rx;a sinðxx tÞ þ rx;m
27 28 29
SHEAR STRESS COMPONENT, C v [MPa]
1.0
SPECIMEN No. Fig. 5. The MCC method [33,34] and the PH method [13–16]: Ca values, by highlighting (for each specimen) the stress path type of the corresponding shear stress vector C.
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A. Carpinteri et al. / International Journal of Fatigue xxx (2014) xxx–xxx
0.5
RELATIVE FREQUENCY
specimens No. 11 and 12 [38], characterised by shear stress P paths represented by line segments of type (a.2); specimens No. 13–28 [36], No. 29–31 [38], and No. 32–40 [37], P characterised by shear stress path represented by ellipses of type (b.1); specimens No. 41–44 [39] and No. 45–50 [40], characterised by P shear stress paths represented by ellipses of type (b.2); specimens No. 51 [41], No. 52–55 [42], No. 56 [43] and No. 57 P and 58 [38], characterised by shear stress paths represented by generic closed curves of type (c).
MCC method
(a) 0.4
0.3
0.2
0.1
Some of the stress paths being analysed are shown in Fig. 4, by plotting the components Cu and Cv of the shear vector C, evaluated with respect to the two orthogonal unit vectors, u and v, on the critical plane. In Fig. 5, the values of the amplitude Ca are plotted for each of the 58 specimens examined. Such values are determined by applying either the MCC method [33,34] or the PH method [13–16] to the modified C–S criterion. Note that the PH method introduces a quite significant increase (up to 30%) in the value of the amplitude Ca with respect to the value obtained from the MCC method, especially for those tests with the following load features: the sides of the rectangle bounding the shear stress path, according to Eq. (13), are of the same order of magnitude (see No. 41–50, and No. 56). Such an increase makes the results less non-conservative, as can be remarked by examining the error index I values plotted in Fig. 6, where I (positive values of I correspond to conservative estimations) is computed as follows:
0 0.5
PH method
RELATIVE FREQUENCY
(b) 0.4
0.3
0.2
0.1
0 -30
-25
-20
-15
-10
-5
0
ERROR INDEX, I [%]
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi r
Fig. 6. Error index for the specimens No. 41–50, and No. 56 according to the modified C–S criterion, by applying: (a) the MCC method [33,34] and (b) the PH method [13–16].
I¼
specimens No. 1–9 [36], and No. 10 [37], characterised by shear P stress paths represented by line segments of type (a.1);
ERROR INDEX, I [%]
raf ;1 100%
raf ;1
ð15Þ
(a)
MCC method PH method
20
2
More precisely, Fig. 6 shows the relative frequency of the error index according to the modified C–S criterion by applying both the MCC method and the PH method for specimens No. 41–50, and No. 56. Further, the minimum and the maximum values (extreme values) of the error index I for each examined material are reported in Table 2. The values of the error index, made by implementing both the MCC method and the PH method in the modified C–S criterion,
the PH method implementation in the modified C–S criterion, the aforementioned experimental tests are sorted in Table 1 on the basis of the shear stress path types listed in the previous Section, that is:
40
N2a;eq þ C 2a s af ;1 af ;1
I>0 CONSERVATIVE
0 -20 Proportional loadings
-40 1
2
3
4
5
6
7
8
9
10
SPECIMEN No.
ERROR INDEX, I [%]
50 30
(b)
MCC method PH method
I>0 CONSERVATIVE
10 -10 -30
56 57 58
55
51 52 53 54
50
46 47 48 49
45
41 42 43 44
40
36 37 38 39
35
31 32 33 34
30
26 27 28 29
25
21 22 23 24
20
16 17 18 19
15
11 12 13 14
Non-proportional loadings
-50
SPECIMEN No. Fig. 7. Error index values calculated by applying both the MCC method and the PH method to calculate Ca: (a) proportional loading; (b) non-proportional loading.
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A. Carpinteri et al. / International Journal of Fatigue xxx (2014) xxx–xxx
are plotted in Fig. 7(a) and (b) for proportional and non-proportional loading, respectively. Positive values of the error index indicate conservative results. The averaged absolute value of the error index related to the MCC method implementation is equal to about 3% for proportional loading and to 15% for non-proportional one. By implementing the PH method, the averaged absolute value of the error index is equal to about 3% for proportional loading and to 13% for nonproportional one. From such a comparison made by taking into account all the data examined, the PH method and the MRC method have the same accuracy for both proportional and non-proportional loading,
RELATIVE FREQUENCY
0.5
τaf,-1 / σaf,-1 ≤ 0.6
(a)
0.4
0.3
For mild metals (saf,1/raf,1 < 0.6, see Fig. 8(a)), the absolute error index value falls in the range of ±10% for 57% of the examined experimental tests; For hard metals (0.6 < saf,1/raf,1 < 0.9, see Fig. 8(b)), the corresponding absolute error index value falls in the range of ±10% for 59% of the examined experimental tests; For extremely hard metals (saf ;1 =raf ;1 P 0:9, see Fig. 8(c)), I falls in the range of ±10% for 86% of the examined experimental tests.
0.2
0.1
0.0
RELATIVE FREQUENCY
0.5
0.6 ≤ τaf,-1 / σaf,-1 ≤ 0.9
(b) Finally, in order to compare the obtained evaluations with those determined through other multiaxial criteria available in the literature, the Crossland [44], Papadopoulos et al. [45] and Gonçalves et al. [14] criteria are applied to each experimental test in Table 1, and the results are reported in Table 2.
0.4
0.3
5. Conclusions
0.2
In the present paper, the shear stress amplitude Ca acting on the critical plane is computed by implementing both the MCC method and the PH method in the modified C–S criterion. For the experimental data here examined, the use of Ca implemented through the PH method provides quite satisfactory multiaxial fatigue strength estimations for experimental tests characterised by shear stress paths not too flatten. For example, the relative decrease of the absolute error value by applying the PH method is equal to 88% for specimen No. 47.
0.1
0.0
RELATIVE FREQUENCY
0.5
but the PH method is computationally more efficient than the MRC one. Note that this result (same accuracy) is due to the fact that we have considered the averaged absolute value of the error index computed taking into account all the tests. As a matter of fact, as highlighted in Fig. 5, the PH method increases the value of Ca only for the tests characterised by shear stress paths not too flatten, that is, when the corresponding sides of the bounding rectangle, chosen according to Eq. (13), are of the same order of magnitude. If only the tests that have such a loading feature were considered (No. 41–50 and No. 56), the averaged absolute value of I would be equal to 21% by implementing the MCC method, and equal to 13% by implementing the PH method, that is, the multiaxial fatigue strength evaluations derived through the modified C–S criterion are improved if the shear stress amplitude is determined by applying the PH method instead of the MCC method. The relative frequency of the error index by applying the PH method is plotted in Fig. 8, where the experimental data are collected on the basis of the ratio between fully reversed shear stress fatigue limit, saf,1, and the fully reversed normal stress fatigue limit, raf,1. The following remarks can be made:
τaf,-1 / σaf,-1 ≥ 0.9
(c)
0.4
Acknowledgements
0.3
The authors gratefully acknowledge the financial support of the Italian Ministry of Education, University and Research (MIUR) under the project PRIN 2009 No. 2009Z55NWC_003.
0.2
References
0.1
0.0 -50 -40 -30 -20 -10
0
10
20
30
40
50
ERROR INDEX, I [%] Fig. 8. Error index according to the modified C–S criterion by applying the PH method [13–16] and collecting the experimental data on the basis of saf,1/raf,1 ratio: (a) saf,1/raf,1 6 0.6; (b) 0.6 < saf,1/raf,1 < 0.9; and (c) saf,1/raf,1 P 0.9.
[1] Carpinteri A. Handbook of fatigue crack propagation in metallic structures. Amsterdam, The Netherlands: Elsevier Science Publisher B.V.; 1994. [2] Berto F, Lazzarin P, Kotousov A, Pook LP. Induced out-of-plane mode at the tip of blunt lateral notches and holes under in-plane shear loading. Fatigue Fract Eng Mater Struct 2012;35:538–55. [3] Pook LP. A 50-year retrospective review of three-dimensional effects at cracks and sharp notches. Fatigue Fract Eng Mater Struct 2013;36:699–723. [4] Berto F, Lazzarin P, Marangon C. Fatigue strength of notched specimens made of 40CrMoV13.9 under multiaxial loading. Mater Des 2014;54:57–66.
Please cite this article in press as: Carpinteri A et al. On the use of the Prismatic Hull method in a critical plane-based multiaxial fatigue criterion. Int J Fatigue (2014), http://dx.doi.org/10.1016/j.ijfatigue.2014.05.007
A. Carpinteri et al. / International Journal of Fatigue xxx (2014) xxx–xxx [5] Wang Y-Y, Yao W-X. Evaluation and comparison of several multiaxial fatigue criteria. Int J Fatigue 2004;26:17–25. [6] You B-R, Lee S-B. A critical review on multiaxial fatigue assessments of metals. Int J Fatigue 1996;18:235–44. [7] Crossland B., Effect of large hydrostatic pressures on the torsional fatigue strength of an alloy steel. In: Proceedings of the international conference on fatigue of metals. ImechE London (UK); 1956. p. 138–49. [8] Sines G, Ohgi G. Fatigue criteria under combined stresses or strains. J Eng Mater Technol 1981;103:82–90. [9] Hashin Z. Fatigue failure criteria for combined cyclic stress. Int J Fract 1981;17:101–9. [10] Dietmann H, Bhongbhibhat T, Schmid A. Multiaxial fatigue behaviour of steels under in-phase and out-of-phase loading, including different wave forms and frequencies. In: Kussmaul K, McDiarmid DL, Socie D, editors. Fatigue under biaxial and multiaxial loadings, ESIS 10. London (UK): Mechanical Eng Publications; 1991. p. 449–64. [11] Ballard P, Dang Van K, Deperrois A, Papadopoulos YV. High cycle fatigue and a finite element analysis. Fatigue Fract Eng Mater Struct 1995;18:397–411. [12] Li B, Santos JLT, Freitas M. A unified numerical approach for multiaxial fatigue limit evaluation. Mech Struct Mach 2000;28:85–103. [13] Mamiya EN, Araújo JA. Fatigue limit under multiaxial loadings: on the definition of the equivalent shear stress. Mech Res Commun 2002;29:141–51. [14] Gonçalves CA, Araújo JA, Mamiya EN. Multiaxial fatigue: a simple stress based criterion for hard metals. Int J Fatigue 2005;27:177–87. [15] Mamiya EN, Araújo JA, Castro FC. Prismatic hull: a new measure of shear stress amplitude in multiaxial high cycle fatigue. Int J Fatigue 2009;31:1144–53. [16] Araújo JA, Dantas AP, Castro FC, Mamiya EN, Ferreira JLA. On the characterization of the critical plane with a simple and fast alternative measure of the shear stress amplitude in multiaxial fatigue. Int J Fatigue 2011;33:1092–100. [17] Findley WN. A theory for the effect of mean stress on fatigue of metals under combined torsion and axial load or bending. J Eng Ind Trans ASME 1959;81:301–6. [18] Dang Van K. Sur la résistance a la fatigue des métaux. These de Doctorat es Sciences. Sci Techniq l’Armement 1973;47:647–722. [19] Matake T. An explanation on fatigue limit under combined stress. Bull JSME 1977;20:257–63. [20] McDiarmid DL. A general criterion for high cycle multiaxial fatigue failure. Fatigue Fract Eng Mater Struct 1991;14:429–53. [21] Robert J-L. Contribution à l’étude de la fatigue multiaxiale sous sollicitations périodiques ou aléatoires. Thèse Laboratoire de Mécanique des solides (INSA) de Lyon; 1992. [22] McDiarmid DL. A shear stress based critical-plane in the case of components weakened by quite blunt criterion of multiaxial fatigue failure for design and life prediction. Fatigue Fract Eng Mater Struct 1994;17:1475–84. [23] Susmel L, Lazzarin P. A bi-parametric Wöhler curve for high cycle multiaxial fatigue assessment. Fatigue Fract Eng Mater Struct 2002;25:63–78. [24] Carpinteri A, Spagnoli A, Vantadori S. Multiaxial fatigue life estimation in welded joints using the critical plane approach. Int J Fatigue 2009;31:188–96. [25] Carpinteri A, Spagnoli A, Vantadori S. Multiaxial fatigue assessment using a simplified critical plane-based criterion. Int J Fatigue 2011;33:969–76.
9
[26] Carpinteri A, Spagnoli A, Vantadori S, Bagni C. Structural integrity assessment of metallic components under multiaxial fatigue: the C–S criterion and its evolution. Fatigue Fract Eng Mater Struct 2013;36:870–83. [27] Carpinteri A, Spagnoli A. Multiaxial high-cycle fatigue criterion for hard metals. Int J Fatigue 2001;23:135–45. [28] Susmel L. Multiaxial notch fatigue. Cambridge (UK): Woodhead Publishing Limited; 2009. [29] Lemaitre J, Chaboche JL. Mechanics of solid materials. Cambridge (UK): Cambridge University Press; 1990. [30] Grubisic V, Simbürger A. Fatigue under combined out of phase multiaxial stresses. In: Proceedings of international conference on fatigue testing and design. London (UK): Society of Environmental Engineers; 1976. p. 27.1–8. [31] Zouain N, Mamiya EN, Comes F. Using enclosing ellipsoids in multiaxial fatigue strength criteria. Eur J Mech – A/Solids 2006;25:51–71. [32] Li B, Reis L, de Freitas M. Comparative study of multiaxial fatigue damage models for ductile structural steel and brittle materials. Int J Fatigue 2009;31:1895–906. [33] 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. Fatigue Fract Eng Mater Struct 1998;21:269–85. [34] Bernasconi A. Efficient algorithms for calculation of shear stress amplitude and amplitude of the second invariant of the shear deviator in fatigue criteria applications. Int J Fatigue 2002;24:649–57. [35] Araujo JA, Carpinteri A, Ronchei C, Spagnoli A, Vantadori S. An alternative definition of the shear stress amplitude based on the Maximum Rectangular Hull method and application to the C–S (Carpinteri–Spagnoli) criterion. Fatigue Fract Eng Mater Struct 2014;37. http://dx.doi.org/10.1111/ffe.12180. [36] Nishihara T., Kawamoto M., The strength of metals under combined alternating bending and torsion with face difference. Kyoto Imperial University, Memoirs of the College of Engineering, vol. 11; 1945. p. 85–112. [37] Froustey C, Lasserre S. Multiaxial fatigue endurance of 30NCD16 steel. Int J Fatigue 1989;11:169–75. [38] Bhongbhibhat T. Festigkeitsverhalten von Stählen unter mehrachsiger phasenverschobener Schwingbeanspruchung mit unterschiedlichen Schwingungsformen und Frequenzen. Diss Uni Stuttgart 1986. [39] Zenner H, Heidenreich R, Richter I. Dauerschwingfestigkeit bei nich synchroner mehrachsiger Beanspruchung. Z Werkstofftech 1985;16:101–12. [40] Troost A, Akin O, Klubberg F. Dauerfestigkeitsverhalten metallischer Werkstoffe bei zweiachsiger Beanspruchung durch drei phasenverschobene schwingende Lastspannungen. Konstruktion 1987;39:479–88. [41] Heidenreich R, Zenner H, Richter I. Dauerschwingfestigkeit bei mehrachsiger Beanspruchung. Forschungshefte FKM, Heft 105; 1983. [42] Kaniut C. Zur Betriebsfestigkeit metallischer Werkstoffe bei mehrachsiger Beanspruchung. Dizertation RWTH, Aachen, Germany; 1983. [43] McDiarmid DL. Fatigue under out-of-phase biaxial stresses of different frequencies. In: Miller KJ, Brown MW, editors. Multiaxial fatigue. ASTM STP 853; 1985. p. 606–21. [44] Crossland B. Effect of large hydrostatic pressures on the torsional fatigue strength of an alloy steel. Proc Int Conf Fatigue Met 1956:138–49. [45] Papadopoulos IV, Davoli P, Gorla C, Filippini M, Bernasconi A. A comparative study of multiaxial high-cycle fatigue criteria for metals. Int J Fatigue 1997;19:219–35.
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Contents lists available at ScienceDirect
International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue
On the use of the Prismatic Hull method in a critical plane-based multiaxial fatigue criterion Andrea Carpinteri, Camilla Ronchei, Andrea Spagnoli, Sabrina Vantadori ⇑ Dept. of Civil-Environmental Engineering and Architecture, University of Parma, Parco Area delle Scienze 181/A, 43124 Parma, Italy
a r t i c l e
i n f o
Article history: Received 4 March 2014 Received in revised form 2 May 2014 Accepted 15 May 2014 Available online xxxx Keywords: Critical plane approach Multiaxial high-cycle fatigue Shear stress amplitude Stress-base criterion
a b s t r a c t In the present paper, the modified Carpinteri–Spagnoli (C–S) criterion is combined with the Prismatic Hull (PH) method proposed by Mamiya, Araújo et al. [15] to evaluate an equivalent shear stress amplitude for multiaxial fatigue loading. More precisely, the PH method, originally formulated for synchronous loading and successively extended to asynchronous loading, is here adopted to compute the shear stress amplitude acting on the critical plane, used in the modified C–S criterion. By comparing some experimental data available in the literature with the theoretical estimations, the multiaxial fatigue strength evaluations derived through the modified C–S criterion are shown to be improved if the shear stress amplitude is determined by applying the PH method instead of the Minimum Circumscribed Circle (MCC) method, especially for particular shear stress paths as is here discussed. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Many metallic structural components, such as crankshafts, axles, steel rails, wheels and fan discs in jet engines, are subjected to multiaxial high-cycle fatigue during their service life [1]. Therefore, the fatigue strength of the aforementioned structural components is complex to be evaluated through theoretical models, because it depends on many factors related to stress paths, material properties and possible stress concentrations [2–4]. Several criteria are available in the literature to assess the above structural components under multiaxial stresses, but a unique method generally recognized as completely reliable does not exist. An interesting review of the most relevant criteria is reported in Ref. [5], where they are classified into three categories, that is: stress-, strain-, and energy-based criteria. Within the framework of a total life approach, under high-cycle fatigue (i.e. in presence of more than about 105 loading cycles, for metallic materials) and constant-amplitude cyclic loading, most of criteria are stress-based and aim at reducing a given multiaxial stress state to an equivalent uniaxial stress to be compared with a single parameter experimentally determined through the uniaxial S–N curve [5]. Several fatigue experimental tests have shown that the polycrystalline metallic materials are characterised by plastic deformations that occur in heterogeneous manner only at the mesoscopic scale, while no plasticity is detected at the ⇑ Corresponding author. E-mail address: [email protected] (S. Vantadori).
macroscopic one. More precisely, at the mesoscopic level, the fatigue limit of such materials is identified by (elastic) shakedown condition, while the plastic shakedown develops in the finite lifetime domain. According to the critical review reported in Ref. [6], the above stress theories can be classified into four viewpoints: empirical formulas, application of stress invariants, integral approach, and critical plane approach. As far as the fatigue criteria related to the use of stress invariants are concerned, most of them are based on the combined effects of normal stress and maximum shear stress. The significant difference among them depends on the definition of the maximum shear stress [7–16]. Regarding the fatigue criteria based on the critical plane approach, whose use has increased during recent years, it is important to highlight that they are able not only to reduce a given multiaxial stress state to an equivalent uniaxial one, but also to describe the plane of the early crack nucleation [17–23]. The modified Carpinteri–Spagnoli (C–S) criterion [24–26] belongs to the last category mentioned above. It is a simplified version of the original C–S criterion [27], as the computation of the critical plane orientation in the modified criterion is easier than that proposed in the original version. This simplified criterion takes into account a fatigue damage parameter given by a nonlinear (quadratic) combination of the equivalent normal stress amplitude, Na,eq [24–26], and the amplitude, Ca, of the shear stress vector acting on the critical plane. More precisely, Na,eq takes into account the linear relationship (proposed by Goodman) between the normal stress amplitude and the normal stress mean value.
http://dx.doi.org/10.1016/j.ijfatigue.2014.05.007 0142-1123/Ó 2014 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Carpinteri A et al. On the use of the Prismatic Hull method in a critical plane-based multiaxial fatigue criterion. Int J Fatigue (2014), http://dx.doi.org/10.1016/j.ijfatigue.2014.05.007
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A. Carpinteri et al. / International Journal of Fatigue xxx (2014) xxx–xxx
Nomenclature ai(H) bi Bi C Ca fy I N Na,eq PXYZ P123 ^2 ^3 ^ P1 Sw t T w x X
a
i-th half-side of prismatic hull i-th component of an orthonormal basis of R5 i-th components of an orthonormal basis of dev3 shear stress vector acting on the critical plane shear stress amplitude material yield stress error index normal stress vector perpendicular to the critical plane equivalent normal stress amplitude fixed frame principal stress axes frame weighted mean principal stress frame stress vector at material point P, acting on the critical plane time observation time interval unit vector normal to the critical plane deviatoric stress vector at material point P in R5 deviatoric stress tensor at material point P in dev3 phase angle between longitudinal (axial) normal stress rx and tangential (hoop/circumferential) normal stress
ry b
phase angle between longitudinal (axial) normal stress
rx and shear stress sxy
c d
angle between the specimen’s longitudinal axis X and the normal w to the critical plane (Fig. 2) ^ of r1,max and the angle between the averaged direction 1 normal w to the critical plane (Fig. 2)
Different methods are available in the literature to define Ca [13–16,28–34]: the so-called Longest Chord method [29], the Longest Projection method [30], the Minimum Circumscribed Ellipsoid method [31,32], the Minimum Circumscribed Circle method [33,34] and the Prismatic Hull (PH) method [13–16]. The modified C–S criterion employs Ca computed by means of the Minimum Circumscribed Circle (MCC) method [33,34]. Accordingly, Ca is the radius of the minimum circle bounding the closed P path described by the shear stress vector C on the critical plane during the observation time interval T. Although the MCC method is able to uniquely define Ca, the following remarks have to be made: (i) Such a method is not able to distinguish between proportional and non-proportional loading, while experimental evidences highlight that, for a given value of the radius of the Minimum Circumscribed Circle, non-proportional loading may produce a greater damage than proportional one; (ii) The MCC method requires time-consuming algorithms in order to compute the centre and the radius of the circle P bounding . The problem of the high computational time is relevant when, for instance, asynchronous sinusoidal loadings are examined, although different strategies exist in order to optimise the procedure used to determine the Minimum Circumscribed Circle [34], and an analytical procedure can be applied in the case of synchronous loadings [33]. To improve the capabilities of the modified C–S criterion and make it a useful tool in fatigue design, the Prismatic Hull (PH) method proposed by Mamiya, Araujo et al. [13–16] to evaluate an equivalent shear stress amplitude, called seq, for multiaxial fati-
four-dimensional quantity used to identify the orthonormal basis of R5 j material parameter q material parameter kxy ratio between the signal pulsations xxy/xx ky ratio between the signal pulsations xy/xx r stress tensor at material point P raf,1 fully reversed normal stress fatigue limit rp, max maximum value of the hydrostatic stress rn, n = 1, 2, 3 principal stresses, with r1 P r2 P r3 ru material ultimate tensile stress rx(t) longitudinal (axial) normal stress ry(t) tangential (hoop/circumferential) normal stress sxy(t) shear stress saf,1 fully reversed shear stress fatigue limit seq equivalent shear stress amplitude /, h,w principal Euler angles ^ ^h; w ^ /; weighted principal Euler angles xx pulsation of the sinusoidal longitudinal normal stress rx xy pulsation of the sinusoidal tangential normal stress ry xxy pulsation of the sinusoidal shear stress sxy
H
Subscripts a amplitude m mean value max maximum value
gue criteria is herein employed. More precisely, the PH method, originally formulated for synchronous loading [13,14] and successively extended to asynchronous loading [15,16], is now adopted to compute the shear stress amplitude, Ca, used in the modified C–S criterion. Note that, when the path of the shear stress is plane, the method is also named Maximum Rectangular Hull (MRH) method [16,35], since the prismatic hull circumscribing the shear stress path degenerates in a rectangular hull. The PH method and the MRH method are able to distinguish between proportional and non-proportional loading, and no complex computation algorithms are needed. Some experimental tests available in the literature [36–43], related to in- and out-of-phase synchronous and asynchronous loading, are hereafter analysed in order to evaluate the fatigue strength estimation capabilities of the PH method implemented in the modified C–S criterion. 2. Main steps of the modified C–S criterion The two main steps of the modified C–S criterion [24–26] are: (1) Determination of the critical plane orientation, which is linked to the averaged principal stress directions; (2) Evaluation of suitable stress components, related to the verification plane (critical plane), to be used in the fatigue strength assessment. 2.1. Averaged principal stress axes and orientation of the critical plane The orientation of the critical plane is assumed to be linked to the directions of the principal stress axes. Such directions are
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3
According to the modified C–S criterion [24–26], the multiaxial fatigue limit condition is expressed in the following form:
Na;eq
2
þ
raf ;1
2
Ca
saf ;1
¼1
ð2Þ
where Na,eq is the equivalent normal stress amplitude, and Ca is the amplitude of the shear stress vector. The amplitude Na,eq can be computed through the following relationship: Fig. 1. Cylindrical specimen: XYZ coordinate system with its origin in P. The instantaneous principal stress directions 1, 2 and 3 are defined through the principal Euler angles /, h, w.
generally time-varying under fatigue loading and, therefore, averaged principal stress directions are taken into account by deducing them through appropriate weight functions, as is described in the following. The origin of the fixed XYZ coordinate system is at a generic point P of the structural component being examined (Fig. 1). At a given time instant t, the location of the principal stress directions 1, 2 and 3 (being r1 P r2 P r3 ) can be defined through the principal Euler angles /, h, w. Mean principal stress directions are determined by averaging the instantaneous values of the principal Euler angles through a suitable weight function so that the averaged principal stress axes ^ 2; ^ 3 ^ coincide with the instantaneous ones at the time instant 1; when r1 achieves its maximum value during the observation time interval T [24–26]. Such an assumption makes the implementation of the criterion rather simple. The normal unit vector w of the critical plane is defined through ^ direction, in the averaged principal plane 1 ^3 ^ a rotation d of the 1 ^ ^ (from 1 to 3Þ. The off angle d is expressed as follows [24–27]:
d¼
" # saf ;1 2 3p 1 8 raf ;1
Na;eq ¼ Na þ raf ;1
Nm
ru
ð3Þ
where Na and Nm are the normal stress amplitude and mean value, respectively, and ru is the material ultimate tensile strength. When ru is not available, the yield stress fy might be used instead of the ultimate tensile strength, especially in the case of mild metals. Since the N direction is fixed with respect to time, the mean value Nm and the amplitude Na of its modulus can readily be computed. On the other hand, the definition of the amplitude Ca of the C vector modulus is not unique, and different methods are available in the literature [13–16,28–34]. In particular, the Prismatic Hull (PH) method [13–16] allows us to compute the amplitude Ca with a simple algorithm. In Refs. [24–27], the Minimum Circumscribed Circle (MCC) method [33,34] is implemented in the modified C–S criterion to compute Ca. In the present paper, Ca is computed by employing the Prismatic Hull (PH) method [13–16]. 3. The Prismatic Hull (PH) method Now the main steps of the PH method and its implementation in the modified C–S criterion are discussed. 3.1. Main steps of the PH method
ð1Þ
where raf,1 and saf,1 are the fatigue limit for fully reversed normal stress and for fully reversed shear stress, respectively.
The PH method [13–16] has been proposed to evaluate an equivalent shear stress amplitude, called seq, used in the following multiaxial fatigue criterion based on the stress invariants:
seq þ jrp;max 6 q 2.2. Fatigue strength assessment At each time instant t, the stress vector Sw related to the critical plane orientation may be decomposed in two components (Fig. 2), that is, the normal stress component N (perpendicular to the critical plane) and the shear stress component C (lying on the critical plane). During the observation time interval T, the direction of the normal stress vector N(t) is fixed with respect to time, while the P shear stress vector C(t) describes a closed path on the critical plane (as long as Sw is periodic).
ð4Þ
where rp,max is the maximum value of the hydrostatic stress during the observation time interval T, and j and q are material parameters. Let us consider a given material point P and its deviatoric stress tensor, X, defined in a space from R3 to R3 , and such a space is called dev3 in the following [13–15]. In dev3, the stress path of X is described by the set of the deviatoric stress states, X(t), so that, by considering an orthonormal basis Bi (with i = 1, . . . , 5) of dev3, we have:
XðtÞ ¼
5 X xi ðtÞBi
ð5Þ
i¼1
Fig. 2. Cylindrical specimen: components of the stress vector Sw acting on the critical plane.
Fig. 3. Rectangular hull identified by orientation H: the half lengths, Cu,a(H) and P Cv,a(H), of the rectangle enclosing the shear stress path on the critical plane are shown.
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A. Carpinteri et al. / International Journal of Fatigue xxx (2014) xxx–xxx
Table 1 Loading conditions of the experimental tests being examined. Material
No
Path type
a (°)
b (°)
kxy ; ky (–)
rx,a (MPa)
rx,m (MPa)
ry,a (MPa)
ry,m (MPa)
sxy,a (MPa)
sxy,m (MPa)
c (°)
Hard steel [36]
1 2 3
(a.1)
0 0 0
0 0 0
1 1 1
131.8 245.3 299.1
0.0 0.0 0.0
0.0 0.0 0.0
0.0 0.0 0.0
167.1 122.7 62.8
0.0 0.0 0.0
75 64 53
Mild steel [36]
4 5 6
(a.1)
0 0 0
0 0 0
1 1 1
99.9 180.3 213.2
0.0 0.0 0.0
0.0 0.0 0.0
0.0 0.0 0.0
120.9 90.2 44.8
0.0 0.0 0.0
78 67 56
Cast iron [36]
7 8 9
(a.1)
0 0 0
0 0 0
1 1 1
56.3 83.4 95.2
0.0 0.0 0.0
0.0 0.0 0.0
0.0 0.0 0.0
68.0 41.6 19.7
0.0 0.0 0.0
40 29 18
30NCD16 [37]
10
(a.1)
0
0
1
485.0
0.0
0.0
0.0
280.0
0.0
66
St35 [38]
11
(a.2)
0
0
1
160.0
176.0
160.0
176.0
0.0
0.0
61
42CrMo4V [38]
12
(a.2)
0
0
1
402.0
442.0
201.0
221.0
0.0
0.0
39
Hard steel [36]
13 14 15 16 17 18 19
(b.1)
0 0 0 0 0 0 0
30 30 60 60 90 90 90
1 1 1 1 1 1 1
140.4 249.7 145.7 252.4 150.2 258.0 304.5
0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0
169.9 124.9 176.3 126.2 181.7 129.0 63.9
0.0 0.0 0.0 0.0 0.0 0.0 0.0
75 63 77 59 80 41 41
Mild steel [36]
20 21 22 23 24
(b.1)
0 0 0 0 0
60 60 90 90 90
1 1 1 1 1
103.6 191.4 108.9 201.1 230.2
0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0
125.4 95.7 131.8 100.6 48.3
0.0 0.0 0.0 0.0 0.0
80 62 84 45 45
Cast iron [36]
25 26 27 28
(b.1)
0 0 0 0
60 60 90 90
1 1 1 1
67.6 93.7 73.2 101.9
0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0
81.6 46.9 88.4 21.1
0.0 0.0 0.0 0.0
42 25 46 7
St35 [38]
29 30
(b.1)
90 180
0 0
1 1
140.0 120.0
154.0 132.0
140.0 120.0
154.0 132.0
0.0 0.0
0.0 0.0
46 46
42CrMo4V [38]
31
(b.1)
90
0
1
402.0
442.0
201.0
221.0
0.0
0.0
39
30NCD16 [37]
32 33 34 35 36 37 38 39 40
(b.1)
0 0 0 0 0 0 0 0 0
0 0 0 45 45 60 90 90 90
1 1 1 1 1 1 1 1 1
211.0 480.0 590.0 480.0 565.0 470.0 473.0 480.0 540.0
300.0 300.0 300.0 300.0 300.0 300.0 300.0 0.0 300.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
365.0 277.0 148.0 277.0 141.0 270.0 273.0 277.0 135.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
69 59 51 57 48 54 41 41 41
25CrMo4 [39]
41 42 43 44
(b.2)
0 0 0 0
60 60 90 90
1 1 1 1
155.0 220.0 159.0 233.0
340.0 340.0 340.0 340.0
0.0 0.0 0.0 0.0
170.0 170.0 170.0 170.0
155.0 110.0 159.0 117.0
0.0 0.0 0.0 0.0
44 42 41 41
25CrMo4 [40]
45 46 47 48 49 50
(b.2)
0 60 90 90 90 180
90 90 45 90 135 90
1 1 1 1 1 1
208.0 225.0 222.0 205.0 215.0 224.0
255.0 255.0 255.0 255.0 255.0 255.0
156.0 169.0 167.0 154.0 161.0 168.0
210.0 210.0 210.0 210.0 210.0 210.0
104.0 113.0 111.0 103.0 108.0 112.0
0.0 0.0 0.0 0.0 0.0 0.0
39 37 42 37 40 37
34Cr4 [41]
51
(c)
0
0
2
200.0
244.0
200.0
244.0
0.0
0.0
44
25CrMo4 [42]
52 53 54 55
(c)
0 0 0 0
0 90 0 0
1/4 2 2 8
210.0 220.0 242.0 196.0
0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0
105.0 110.0 121.0 98.0
0.0 0.0 0.0 0.0
48 60 58 60
En24T [43]
56
(c)
180
0
3
260.0
0.0
260.0
0.0
0.0
0.0
55
St35 [38]
57 58
(c)
0 90
0 0
2 2
130.0 140.0
143.0 154.0
130.0 140.0
143.0 154.0
0.0 0.0
0.0 0.0
90 46
Hence, the stress path of X can also be described, in a more simply way, by employing the set of the deviatoric stress components:
xðtÞ fx1 ðtÞ; x2 ðtÞ; x3 ðtÞ; x4 ðtÞ; x5 ðtÞgT
ð6Þ
that corresponds, if referred to an orthonormal basis bi (with i = 1, . . . , 5) of R5 , to a hyper-curve. By considering, for example, the following orthonormal basis of dev3 [13–15]:
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A. Carpinteri et al. / International Journal of Fatigue xxx (2014) xxx–xxx
0
pffiffiffi 2= 6
1
0
1
0 0 0 0 pffiffiffi C B C C B 0 1= 2 C B ¼ 0 0 2 A @ A pffiffiffi pffiffiffi 0 0 1= 2 0 0 1= 6 pffiffiffi pffiffiffi 1 0 1 0 0 1= 2 0 0 0 1= 2 B pffiffiffi C B C B3 ¼ B B4 ¼ B 0 0 C 0 0C @ 1= 2 A @ 0 A pffiffiffi 1= 2 0 0 0 0 0 0 1 0 0 0 pffiffiffi C B B5 ¼ B 0 1= 2 C @0 A pffiffiffi 0 1= 2 0 B B1 ¼ B @
0
0 pffiffiffi 1= 6
ð7Þ
the deviatoric stress components, xi(t), in R5 are expressed as follows:
rffiffiffi 3 1 X x ðtÞ ¼ pffiffiffi 2rx ðtÞ ry ðtÞ rz ðtÞ 2 6 1 1 x2 ðtÞ ¼ pffiffiffi ½X y ðtÞ X z ðtÞ ¼ pffiffiffi ry ðtÞ rz ðtÞ 2 2 pffiffiffi pffiffiffi x3 ðtÞ ¼ 2X xy ðtÞ ¼ 2sxy ðtÞ pffiffiffi pffiffiffi x4 ðtÞ ¼ 2X xz ðtÞ ¼ 2sxz ðtÞ pffiffiffi pffiffiffi x5 ðtÞ ¼ 2X yz ðtÞ ¼ 2syz ðtÞ
x1 ðtÞ ¼
ð8Þ
where Xx, Xy, Xz, Xxy, Xyz, Xxz are the components of the deviatoric stress tensor, X, and rx, ry, rz, sxy, syz, sxz are the components of the stress tensor, r. Note that, at each time instant t, the Euclidean norm of the tensor X is equal to that of the vector x [15]. The PH method evaluates the equivalent shear stress amplitude, seq, by taking into account all the prismatic hulls (function of the considered orthonormal basis) which both enclose the path of the vector x and are tangent to such a path. For a given hull, the half values of its sides, ai (with i = 1, . . . , 5), can be determined as follows:
ai ¼
1 max xi ðtÞ min xi ðtÞ 06t
ð9Þ
Note that the above hulls might have a dimension smaller than five, depending on the stress state. For example, when the stress state is characterised only by the stress components rx, ry or rx, sxy, the hulls degenerate into rectangles [16,35], while the hulls degenerate into parallelepipeds when the stress components different from zero are rx, ry, sxy. For the particular case of synchronous cyclic loading, it has been demonstrated [13,14] that seq in Eq. (10) is a constant with respect to H. Therefore, any orthonormal basis of R5 leads to the same value of seq. 3.2. Application of the PH method to the critical plane shear stress vector Now the PH method is adopted to compute the amplitude of the shear stress vector C, by applying Eq. (10) not to the components of the five-dimensional vector x, but to the components of the twodimensional vector C, with the path of the shear stress C belonging to the critical plane. In such a case, the enclosing prismatic hull degenerates into a rectangle, and the four-dimensional quantity H degenerates in the scalar quantity H. Let us take into account a unit vector u coincident with the intersection between the critical plane and the plane defined by the normal unit vector w and the Z-axis, and a unit vector v normal to u so that uvw represents a right-hand orthogonal system (Fig. 2). The vector C is decomposed in two components along the directions of u and v, respectively (Fig. 3). Consider: (i) the rectangular hulls which both enclose the path of the vector C and are tangent to such a path, and (ii) the R2 basis defined by the unit vectors u and v, which are identified by an orientation angle H with respect to an arbitrary time-independent direction (see the horizontal dashed line in Fig. 3). Then, the rectangular hull half-sides can be computed according to Eq. (11):
C u;a ðHÞ ¼
1 max C u ðt; HÞ min C u ðt; HÞ 06t
ð12aÞ
C v ;a ðHÞ ¼
1 max C v ðt; HÞ min C v ðt; HÞ 06t
ð12bÞ
5
All the orthonormal bases of R are examined, in order to determine the specific prismatic hull, enclosing and tangent to the path of x, which maximises the following relationship:
seq
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X5 ¼ max ½a ðHÞ2 i¼1 i
ð10Þ
H
where H is a four-dimensional quantity used to identify the orthonormal basis bi and:
1 max xi ðt; HÞ min xi ðt; HÞ ai ðHÞ ¼ 06t
Finally, according to Eq. (10), the amplitude Ca of the shear stress vector is given by:
C a ¼ max
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½C u;a ðHÞ2 þ ½C m;a ðHÞ2
ð13Þ
06H
ð11Þ
where all R2 bases are taken into account by varying 0 6 H < p=2.
Table 2 Static, fatigue properties and extreme values of the error index, DI, obtained by applying both the MCC and the PH method to calculate Ca for each examined material. The DI values obtained by applying the Crossland [44], Papadopoulos et al. [45] and Gonçalves et al. [14] criteria are also reported. Authors
Ref.
Nishihara and Kawamoto
[36]
Froustey and Lasserre Bhongbhibhat
[37]
Zenner et al. Troost et al. Heidenreich et al. Kaniut McDiarmid
[39] [40] [41] [42] [43]
[38]
saf ;1 raf ;1
DI (MCC) (%)
DI (PH) (%)
DI (Crossland) (%)
DI (Papadopoulos) (%)
DI (Gonçalves et al.) (%)
196.2 137.3 91.2 410.0
0.63 0.58 0.95 0.62
10/1 5/1 3/14 18/3
10/1 5/1 3/14 18/3
18/2 15/1 3/27 27/4
3/7 4/13 3/37 8/4
3/4 4/12 3/15 4/8
130.0 315.0 228.0 228.0 204.0 228.0 270.0
0.57 0.65 0.63 0.67 0.59 0.67 0.67
36/15 1 27/19 24/9 27 46/23 28
36/15 1 20/11 16/1 27 46/23 15
35/10 7/2 23/13 19/8 9 27/21 13
35/10 7/8 9/4 0/21 2 14/0 18
34/11 1/14 4/0 4/24 10 26/13 4
ru/fy
raf,1
saf,1
(MPa)
(MPa)
(MPa)
Hard steel Mild steel Cast iron 30NCD16
681 374 181 1880
313.9 235.4 96.1 660.0
St35 42CrMo4V 25CrMo4 25CrMo4 34Cr4 25CrMo4 En24T
340 1003 780 660 550 – –
230.0 485.00 361.0 340.0 343.0 340.0 405.0
Material
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St35 - Bhongbhibhat (1986) Specimens No. 11,29 0.6 0.2
29
-0.6
(a) 0
100
150
ry ¼ ry;a sinðxy t aÞ þ ry;m
ð14bÞ
sxy ¼ sxy;a sinðxxy t bÞ þ sxy;m
ð14cÞ
200
P (a) The shear stress path is represented by a line segment; P (b) The shear stress path is represented by an ellipse; P (c) The shear stress path is represented by a closed curve with a generic shape.
25CrMo4 - Zenner et al. (1985) Specimens No.41 - 44
100
44 43
0
In case (a), we can have the following sub-cases: 42
(a.1) For synchronous proportional loading (in-phase input signals with zero mean values), the line segment passes through the origin of the Puvw frame; (a.2) For synchronous affine loading (in-phase input signals with non-zero mean values), the line segment does not pass through the origin of the Puvw frame.
41
-100
(b) -200 300 200
0
50 100 150 200 250 300 En24T - McDiarmid (1985) Specimen No.56
In case (b), produced by synchronous non-proportional loading (out-of-phase signals), we can have the following sub-cases:
56 100
(b.1) One semi-axis is much greater than the other one; (b.2) The two semi-axes have the same order of magnitude.
0 -100
Finally, case (c) is produced by asynchronous loading.
-200
(c) -300 -150 -100 -50
0
50
4. Comparison with experimental data
100 150
SHEAR STRESS COMPONENT, Cu [MPa]
Now some experimental data available in the literature, related to smooth cylindrical metallic specimens (round bars and thinwalled tubes) subjected to synchronous and asynchronous, sinusoidal, in- and out-of-phase loading, are examined. The multiaxial stress states corresponding to fatigue limit conditions are reported in Table 1 (where kxy ¼ xxy =xx or ky ¼ xy =xx ). For each experimental test, the critical plane orientation, i.e. the angle c (Fig. 2) formed by the normal unit vector w with respect to the specimen’s longitudinal axis X, is also reported. A total number of 58 specimens are analysed. The static and fatigue properties of the materials being examined are shown in Table 2. More precisely, in order to evaluate the capabilities of
Fig. 4. Shear stress paths for the experimental tests in Refs. [38,39,43]: (a) specimens No. 11 and 29; (b) No. 41–44; and (c) No. 56.
3.3. Some types of shear stress paths on the critical plane
500
MCC method PH method 400 Path type: 300
(a1)
( b 1)
(a2)
(b2)
(c )
200
57 58
5556
51 52 53 54
50
47 48 49
4546
42 43 44
4041
37 38 39
3536
32 33 34
2526
22 23 24
2021
17 18 19
1516
12 13 14
1011
5
0
6 7 8 9
100
1 2 3 4
SHEAR STRESS AMPLITUDE, C a [MPa]
The plane stress condition of biaxial normal and shear stresses at the generic point P of a cylindrical body (Fig. 1) subjected to an asynchronous out-of-phase sinusoidal loading can be expressed as follows:
3031
SHEAR STRESS COMPONENT, Cv [MPa]
200
50
ð14aÞ
where rx is the longitudinal (axial) normal stress, ry is the tangential (hoop/circumferential) normal stress, and sxy is the shear stress. By varying the mean value, the pulsations and the phases of the above signals, the shear stress vector C describes closed paths of different shapes on the critical plane and, according to the above definition of the critical plane, three cases can occur:
-0.2
-1.0
SHEAR STRESS COMPONENT, C v [MPa]
11
0.0
rx ¼ rx;a sinðxx tÞ þ rx;m
27 28 29
SHEAR STRESS COMPONENT, C v [MPa]
1.0
SPECIMEN No. Fig. 5. The MCC method [33,34] and the PH method [13–16]: Ca values, by highlighting (for each specimen) the stress path type of the corresponding shear stress vector C.
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A. Carpinteri et al. / International Journal of Fatigue xxx (2014) xxx–xxx
0.5
RELATIVE FREQUENCY
specimens No. 11 and 12 [38], characterised by shear stress P paths represented by line segments of type (a.2); specimens No. 13–28 [36], No. 29–31 [38], and No. 32–40 [37], P characterised by shear stress path represented by ellipses of type (b.1); specimens No. 41–44 [39] and No. 45–50 [40], characterised by P shear stress paths represented by ellipses of type (b.2); specimens No. 51 [41], No. 52–55 [42], No. 56 [43] and No. 57 P and 58 [38], characterised by shear stress paths represented by generic closed curves of type (c).
MCC method
(a) 0.4
0.3
0.2
0.1
Some of the stress paths being analysed are shown in Fig. 4, by plotting the components Cu and Cv of the shear vector C, evaluated with respect to the two orthogonal unit vectors, u and v, on the critical plane. In Fig. 5, the values of the amplitude Ca are plotted for each of the 58 specimens examined. Such values are determined by applying either the MCC method [33,34] or the PH method [13–16] to the modified C–S criterion. Note that the PH method introduces a quite significant increase (up to 30%) in the value of the amplitude Ca with respect to the value obtained from the MCC method, especially for those tests with the following load features: the sides of the rectangle bounding the shear stress path, according to Eq. (13), are of the same order of magnitude (see No. 41–50, and No. 56). Such an increase makes the results less non-conservative, as can be remarked by examining the error index I values plotted in Fig. 6, where I (positive values of I correspond to conservative estimations) is computed as follows:
0 0.5
PH method
RELATIVE FREQUENCY
(b) 0.4
0.3
0.2
0.1
0 -30
-25
-20
-15
-10
-5
0
ERROR INDEX, I [%]
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi r
Fig. 6. Error index for the specimens No. 41–50, and No. 56 according to the modified C–S criterion, by applying: (a) the MCC method [33,34] and (b) the PH method [13–16].
I¼
specimens No. 1–9 [36], and No. 10 [37], characterised by shear P stress paths represented by line segments of type (a.1);
ERROR INDEX, I [%]
raf ;1 100%
raf ;1
ð15Þ
(a)
MCC method PH method
20
2
More precisely, Fig. 6 shows the relative frequency of the error index according to the modified C–S criterion by applying both the MCC method and the PH method for specimens No. 41–50, and No. 56. Further, the minimum and the maximum values (extreme values) of the error index I for each examined material are reported in Table 2. The values of the error index, made by implementing both the MCC method and the PH method in the modified C–S criterion,
the PH method implementation in the modified C–S criterion, the aforementioned experimental tests are sorted in Table 1 on the basis of the shear stress path types listed in the previous Section, that is:
40
N2a;eq þ C 2a s af ;1 af ;1
I>0 CONSERVATIVE
0 -20 Proportional loadings
-40 1
2
3
4
5
6
7
8
9
10
SPECIMEN No.
ERROR INDEX, I [%]
50 30
(b)
MCC method PH method
I>0 CONSERVATIVE
10 -10 -30
56 57 58
55
51 52 53 54
50
46 47 48 49
45
41 42 43 44
40
36 37 38 39
35
31 32 33 34
30
26 27 28 29
25
21 22 23 24
20
16 17 18 19
15
11 12 13 14
Non-proportional loadings
-50
SPECIMEN No. Fig. 7. Error index values calculated by applying both the MCC method and the PH method to calculate Ca: (a) proportional loading; (b) non-proportional loading.
Please cite this article in press as: Carpinteri A et al. On the use of the Prismatic Hull method in a critical plane-based multiaxial fatigue criterion. Int J Fatigue (2014), http://dx.doi.org/10.1016/j.ijfatigue.2014.05.007
8
A. Carpinteri et al. / International Journal of Fatigue xxx (2014) xxx–xxx
are plotted in Fig. 7(a) and (b) for proportional and non-proportional loading, respectively. Positive values of the error index indicate conservative results. The averaged absolute value of the error index related to the MCC method implementation is equal to about 3% for proportional loading and to 15% for non-proportional one. By implementing the PH method, the averaged absolute value of the error index is equal to about 3% for proportional loading and to 13% for nonproportional one. From such a comparison made by taking into account all the data examined, the PH method and the MRC method have the same accuracy for both proportional and non-proportional loading,
RELATIVE FREQUENCY
0.5
τaf,-1 / σaf,-1 ≤ 0.6
(a)
0.4
0.3
For mild metals (saf,1/raf,1 < 0.6, see Fig. 8(a)), the absolute error index value falls in the range of ±10% for 57% of the examined experimental tests; For hard metals (0.6 < saf,1/raf,1 < 0.9, see Fig. 8(b)), the corresponding absolute error index value falls in the range of ±10% for 59% of the examined experimental tests; For extremely hard metals (saf ;1 =raf ;1 P 0:9, see Fig. 8(c)), I falls in the range of ±10% for 86% of the examined experimental tests.
0.2
0.1
0.0
RELATIVE FREQUENCY
0.5
0.6 ≤ τaf,-1 / σaf,-1 ≤ 0.9
(b) Finally, in order to compare the obtained evaluations with those determined through other multiaxial criteria available in the literature, the Crossland [44], Papadopoulos et al. [45] and Gonçalves et al. [14] criteria are applied to each experimental test in Table 1, and the results are reported in Table 2.
0.4
0.3
5. Conclusions
0.2
In the present paper, the shear stress amplitude Ca acting on the critical plane is computed by implementing both the MCC method and the PH method in the modified C–S criterion. For the experimental data here examined, the use of Ca implemented through the PH method provides quite satisfactory multiaxial fatigue strength estimations for experimental tests characterised by shear stress paths not too flatten. For example, the relative decrease of the absolute error value by applying the PH method is equal to 88% for specimen No. 47.
0.1
0.0
RELATIVE FREQUENCY
0.5
but the PH method is computationally more efficient than the MRC one. Note that this result (same accuracy) is due to the fact that we have considered the averaged absolute value of the error index computed taking into account all the tests. As a matter of fact, as highlighted in Fig. 5, the PH method increases the value of Ca only for the tests characterised by shear stress paths not too flatten, that is, when the corresponding sides of the bounding rectangle, chosen according to Eq. (13), are of the same order of magnitude. If only the tests that have such a loading feature were considered (No. 41–50 and No. 56), the averaged absolute value of I would be equal to 21% by implementing the MCC method, and equal to 13% by implementing the PH method, that is, the multiaxial fatigue strength evaluations derived through the modified C–S criterion are improved if the shear stress amplitude is determined by applying the PH method instead of the MCC method. The relative frequency of the error index by applying the PH method is plotted in Fig. 8, where the experimental data are collected on the basis of the ratio between fully reversed shear stress fatigue limit, saf,1, and the fully reversed normal stress fatigue limit, raf,1. The following remarks can be made:
τaf,-1 / σaf,-1 ≥ 0.9
(c)
0.4
Acknowledgements
0.3
The authors gratefully acknowledge the financial support of the Italian Ministry of Education, University and Research (MIUR) under the project PRIN 2009 No. 2009Z55NWC_003.
0.2
References
0.1
0.0 -50 -40 -30 -20 -10
0
10
20
30
40
50
ERROR INDEX, I [%] Fig. 8. Error index according to the modified C–S criterion by applying the PH method [13–16] and collecting the experimental data on the basis of saf,1/raf,1 ratio: (a) saf,1/raf,1 6 0.6; (b) 0.6 < saf,1/raf,1 < 0.9; and (c) saf,1/raf,1 P 0.9.
[1] Carpinteri A. Handbook of fatigue crack propagation in metallic structures. Amsterdam, The Netherlands: Elsevier Science Publisher B.V.; 1994. [2] Berto F, Lazzarin P, Kotousov A, Pook LP. Induced out-of-plane mode at the tip of blunt lateral notches and holes under in-plane shear loading. Fatigue Fract Eng Mater Struct 2012;35:538–55. [3] Pook LP. A 50-year retrospective review of three-dimensional effects at cracks and sharp notches. Fatigue Fract Eng Mater Struct 2013;36:699–723. [4] Berto F, Lazzarin P, Marangon C. Fatigue strength of notched specimens made of 40CrMoV13.9 under multiaxial loading. Mater Des 2014;54:57–66.
Please cite this article in press as: Carpinteri A et al. On the use of the Prismatic Hull method in a critical plane-based multiaxial fatigue criterion. Int J Fatigue (2014), http://dx.doi.org/10.1016/j.ijfatigue.2014.05.007
A. Carpinteri et al. / International Journal of Fatigue xxx (2014) xxx–xxx [5] Wang Y-Y, Yao W-X. Evaluation and comparison of several multiaxial fatigue criteria. Int J Fatigue 2004;26:17–25. [6] You B-R, Lee S-B. A critical review on multiaxial fatigue assessments of metals. Int J Fatigue 1996;18:235–44. [7] Crossland B., Effect of large hydrostatic pressures on the torsional fatigue strength of an alloy steel. In: Proceedings of the international conference on fatigue of metals. ImechE London (UK); 1956. p. 138–49. [8] Sines G, Ohgi G. Fatigue criteria under combined stresses or strains. J Eng Mater Technol 1981;103:82–90. [9] Hashin Z. Fatigue failure criteria for combined cyclic stress. Int J Fract 1981;17:101–9. [10] Dietmann H, Bhongbhibhat T, Schmid A. Multiaxial fatigue behaviour of steels under in-phase and out-of-phase loading, including different wave forms and frequencies. In: Kussmaul K, McDiarmid DL, Socie D, editors. Fatigue under biaxial and multiaxial loadings, ESIS 10. London (UK): Mechanical Eng Publications; 1991. p. 449–64. [11] Ballard P, Dang Van K, Deperrois A, Papadopoulos YV. High cycle fatigue and a finite element analysis. Fatigue Fract Eng Mater Struct 1995;18:397–411. [12] Li B, Santos JLT, Freitas M. A unified numerical approach for multiaxial fatigue limit evaluation. Mech Struct Mach 2000;28:85–103. [13] Mamiya EN, Araújo JA. Fatigue limit under multiaxial loadings: on the definition of the equivalent shear stress. Mech Res Commun 2002;29:141–51. [14] Gonçalves CA, Araújo JA, Mamiya EN. Multiaxial fatigue: a simple stress based criterion for hard metals. Int J Fatigue 2005;27:177–87. [15] Mamiya EN, Araújo JA, Castro FC. Prismatic hull: a new measure of shear stress amplitude in multiaxial high cycle fatigue. Int J Fatigue 2009;31:1144–53. [16] Araújo JA, Dantas AP, Castro FC, Mamiya EN, Ferreira JLA. On the characterization of the critical plane with a simple and fast alternative measure of the shear stress amplitude in multiaxial fatigue. Int J Fatigue 2011;33:1092–100. [17] Findley WN. A theory for the effect of mean stress on fatigue of metals under combined torsion and axial load or bending. J Eng Ind Trans ASME 1959;81:301–6. [18] Dang Van K. Sur la résistance a la fatigue des métaux. These de Doctorat es Sciences. Sci Techniq l’Armement 1973;47:647–722. [19] Matake T. An explanation on fatigue limit under combined stress. Bull JSME 1977;20:257–63. [20] McDiarmid DL. A general criterion for high cycle multiaxial fatigue failure. Fatigue Fract Eng Mater Struct 1991;14:429–53. [21] Robert J-L. Contribution à l’étude de la fatigue multiaxiale sous sollicitations périodiques ou aléatoires. Thèse Laboratoire de Mécanique des solides (INSA) de Lyon; 1992. [22] McDiarmid DL. A shear stress based critical-plane in the case of components weakened by quite blunt criterion of multiaxial fatigue failure for design and life prediction. Fatigue Fract Eng Mater Struct 1994;17:1475–84. [23] Susmel L, Lazzarin P. A bi-parametric Wöhler curve for high cycle multiaxial fatigue assessment. Fatigue Fract Eng Mater Struct 2002;25:63–78. [24] Carpinteri A, Spagnoli A, Vantadori S. Multiaxial fatigue life estimation in welded joints using the critical plane approach. Int J Fatigue 2009;31:188–96. [25] Carpinteri A, Spagnoli A, Vantadori S. Multiaxial fatigue assessment using a simplified critical plane-based criterion. Int J Fatigue 2011;33:969–76.
9
[26] Carpinteri A, Spagnoli A, Vantadori S, Bagni C. Structural integrity assessment of metallic components under multiaxial fatigue: the C–S criterion and its evolution. Fatigue Fract Eng Mater Struct 2013;36:870–83. [27] Carpinteri A, Spagnoli A. Multiaxial high-cycle fatigue criterion for hard metals. Int J Fatigue 2001;23:135–45. [28] Susmel L. Multiaxial notch fatigue. Cambridge (UK): Woodhead Publishing Limited; 2009. [29] Lemaitre J, Chaboche JL. Mechanics of solid materials. Cambridge (UK): Cambridge University Press; 1990. [30] Grubisic V, Simbürger A. Fatigue under combined out of phase multiaxial stresses. In: Proceedings of international conference on fatigue testing and design. London (UK): Society of Environmental Engineers; 1976. p. 27.1–8. [31] Zouain N, Mamiya EN, Comes F. Using enclosing ellipsoids in multiaxial fatigue strength criteria. Eur J Mech – A/Solids 2006;25:51–71. [32] Li B, Reis L, de Freitas M. Comparative study of multiaxial fatigue damage models for ductile structural steel and brittle materials. Int J Fatigue 2009;31:1895–906. [33] 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. Fatigue Fract Eng Mater Struct 1998;21:269–85. [34] Bernasconi A. Efficient algorithms for calculation of shear stress amplitude and amplitude of the second invariant of the shear deviator in fatigue criteria applications. Int J Fatigue 2002;24:649–57. [35] Araujo JA, Carpinteri A, Ronchei C, Spagnoli A, Vantadori S. An alternative definition of the shear stress amplitude based on the Maximum Rectangular Hull method and application to the C–S (Carpinteri–Spagnoli) criterion. Fatigue Fract Eng Mater Struct 2014;37. http://dx.doi.org/10.1111/ffe.12180. [36] Nishihara T., Kawamoto M., The strength of metals under combined alternating bending and torsion with face difference. Kyoto Imperial University, Memoirs of the College of Engineering, vol. 11; 1945. p. 85–112. [37] Froustey C, Lasserre S. Multiaxial fatigue endurance of 30NCD16 steel. Int J Fatigue 1989;11:169–75. [38] Bhongbhibhat T. Festigkeitsverhalten von Stählen unter mehrachsiger phasenverschobener Schwingbeanspruchung mit unterschiedlichen Schwingungsformen und Frequenzen. Diss Uni Stuttgart 1986. [39] Zenner H, Heidenreich R, Richter I. Dauerschwingfestigkeit bei nich synchroner mehrachsiger Beanspruchung. Z Werkstofftech 1985;16:101–12. [40] Troost A, Akin O, Klubberg F. Dauerfestigkeitsverhalten metallischer Werkstoffe bei zweiachsiger Beanspruchung durch drei phasenverschobene schwingende Lastspannungen. Konstruktion 1987;39:479–88. [41] Heidenreich R, Zenner H, Richter I. Dauerschwingfestigkeit bei mehrachsiger Beanspruchung. Forschungshefte FKM, Heft 105; 1983. [42] Kaniut C. Zur Betriebsfestigkeit metallischer Werkstoffe bei mehrachsiger Beanspruchung. Dizertation RWTH, Aachen, Germany; 1983. [43] McDiarmid DL. Fatigue under out-of-phase biaxial stresses of different frequencies. In: Miller KJ, Brown MW, editors. Multiaxial fatigue. ASTM STP 853; 1985. p. 606–21. [44] Crossland B. Effect of large hydrostatic pressures on the torsional fatigue strength of an alloy steel. Proc Int Conf Fatigue Met 1956:138–49. [45] Papadopoulos IV, Davoli P, Gorla C, Filippini M, Bernasconi A. A comparative study of multiaxial high-cycle fatigue criteria for metals. Int J Fatigue 1997;19:219–35.
Please cite this article in press as: Carpinteri A et al. On the use of the Prismatic Hull method in a critical plane-based multiaxial fatigue criterion. Int J Fatigue (2014), http://dx.doi.org/10.1016/j.ijfatigue.2014.05.007