Artigo IJF 2011 - Multiaxial fatigue life estimation based on a piecewise ruled S–N surface

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Multiaxial fatigue life estimation based on a piecewise ruled S–N surface

E.N. Mamiya

⇑

, F.C. Castro, R.D. Algarte, J.A. Araújo

Department of Mechanical Engineering, University of Brasilia, 70.910-900 Brasilia, DF, Brazil

a r t i c l e

i n f o

Article history: Received 1 July 2010

Received in revised form 11 October 2010 Accepted 14 October 2010

Available online 21 October 2010 Keywords:

Multiaxial fatigue Life estimation High cycle fatigue Non-proportional loading

a b s t r a c t

A multiaxial model for fatigue life estimation, formulated in terms of a piecewise ruled S–N surface, is proposed. The ﬁrst ruled surface considers the sum of a deviatoric stress amplitude and the maximum hydrostatic stress as an exponential function of the number of cycles to failure. For small magnitudes of the hydrostatic stress, another surface is deﬁned by considering only the effect of the deviatoric stress amplitude upon the expected fatigue life. The deviatoric stress amplitude was computed by the maxi-mum prismatic hull concept. The model proved successful when compared with other multiaxial criteria for available data.

1. Introduction

In real service conditions, material points of mechanical compo-nents are subjected to multiaxial stress histories which can induce fatigue failure. During the last years, the industry has increasingly demanded fast and accurate multiaxial life estimation methods so as to face the challenges in computer-aided design associated with complex geometries and loading scenarios.

Although there are several approaches for life estimation of

metallic materials reported in the literature (see[1,2] and

refer-ences therein), those associated with the concept of critical

plane—as for instance the ones proposed by Findley[3], Fatemi

and Socie[4]and McDiarmid[5]—have gained widespread usage.

The main difference among them relies upon the fatigue damage measure which is considered in the search for the critical plane. In recent years, however, other models have also been proposed as an alternative approach to classical critical plane models: Morel

[6]presented a critical plane model associated with the

accumu-lated plastic strain at the grain level. Papadopoulos[7]considered

an approach based on a generalized shear stress amplitude deﬁned, on each material plane, as the average of the resolved shear stress amplitude over all gliding directions. In the critical plane life

esti-mation model formulated by Lazzarin and Susmel[8], the shear

stress amplitude on each material plane is deﬁned as the radius of the minimum circumference enclosing the shear stresses.

Jabbado and Maitournam [9] proposed a mesoscopic plastic

shakedown framework for life estimation in which lifetime is

related to the accumulated plastic meso-strain per stabilized cycle. Life estimation models based on measures of fatigue damage deﬁned in terms of the stress history projected onto the deviatoric

stress space have been developed by Freitas, Li and Santos[10]—

where a deviatoric stress amplitude is provided by the minimum

circumscribed ellipse—and by Cristofori et al.[11]—where

projec-tions of the multiaxial loading path are considered in the evalua-tion of the fatigue damage.

In this paper, a stress-based model for multiaxial fatigue life estimation is proposed. Its main feature is the definition of a piece-wise ruled S–N surface: the first ruled surface is built by consider-ing the sum of a deviatoric stress amplitude and the maximum hydrostatic stress as an exponential function of the number of cy-cles to failure. For sufficiently small magnitudes of the hydrostatic stress, a second ruled S–N surface is defined by considering only the effect of the deviatoric stress amplitude upon the expected fa-tigue life. As a measure of deviatoric stress amplitude, we consider

the one proposed by Mamiya and co-workers[12–14], associated

with the maximum prismatic hull enclosing the stress history, which successfully differentiates proportional and nonproportional stress histories. Although the concept of prismatic hull is well established and tested within the setting of multiaxial nonpropor-tional fatigue endurance, this measure has not been extended and validated yet in the medium-high cycle fatigue regime.

Estimations of the model were well correlated with experimen-tal results obtained from combined axial load or bending and tor-sion applied on four metals, with reported lives ranging from

5 103_{to 1.3 10}6_{cycles. The proposed model is very efﬁcient}

in terms of computation time, which can be crucial, for instance, in ﬁnite element-based shape optimization procedures, if the objective function includes the life to failure. Another appealing

⇑ Corresponding author.

E-mail addresses: mamiya@unb.br (E.N. Mamiya), fabiocastro@unb.br (F.C. Castro),alex07@unb.br(J.A. Araújo).

Contents lists available atScienceDirect

International Journal of Fatigue

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feature of the model is the fact that only four material parameters have to be identiﬁed. Such parameters can be easily determined from a set of experimental data—tensile and torsion fatigue tests, for instance—in the life range of interest.

The paper is organized as follows: in Section2, the life

estima-tion model is formulated. Secestima-tion3is devoted to the presentation

of the stress amplitude based on the concept of the maximum pris-matic hull enclosing the deviatoric stresses. The life estimation

model is assessed in Section4, considering available experimental

data obtained under complex combined axial (or bending) and torsional loading conditions. Discussion and conclusions are

presented in Sections5and6, respectively.

2. The fatigue life estimation model

Under macroscopically elastic uniaxial loading conditions, fati-gue life can be described in terms of a Wöhler (or S–N) curve, which associates the stress amplitude with life (described in terms of the number of cycles to failure) of a specimen or engineering component. The most widely used S–N curve, known as Basquin’s rule[15], is given as

S ¼ aNbf; ð1Þ

where S is the uniaxial stress amplitude, Nfis the number of cycles

to failure, while a and b are material parameters.

When multiaxial loading is observed, the concept of stress amplitude has to be extended so as to take into account the evolu-tion of each component of the stress tensor, which most often is out-of-phase and/or asynchronous. When considering ductile met-als, the fatigue damage can be related to the development of cyclic plastic deformations at the grain level, leading to the formation of persistent slip bands and later to the initiation of microcracks, even if the material shows an essentially elastic behavior at macroscopic

level[16–18]. When plasticity at mesoscopic level plays a major

role in fatigue degradation, the stress amplitude is frequently de-ﬁned in terms of quantities computed in the deviatoric stress space. Within this setting, several measures of multiaxial deviatoric

stress amplitude

s

aare available in the literature, including the ones

proposed by Crossland[19], Sines[20], Deperrois[21], Freitas and

co-workers[22], amongst others. In this paper, we shall consider

the deviatoric stress amplitude

s

a:¼ 1 ffiffiffi 2 p max H ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X5 i¼1a 2 ið

H

Þ r ; ð2Þ

deﬁned in terms of the maximum Euclidian norm among all pris-matic hulls (with orientationHand radii ai(H)) enclosing the stress

history in the deviatoric space, as proposed by Mamiya et al.[12].

Notice that this deviatoric stress amplitude is a measure of shear stresses but it should not be understood as a conventional shear stress in a material plane.

Tensile normal stresses are assumed to contribute to fatigue degradation by acting (essentially in mode I) upon eventually

existing embryocracks in the material (see [16] and references

therein). In this context, we shall quantify the contribution of the tractive normal stresses upon fatigue failure by considering the maximum hydrostatic stress

r

H max:¼ max t

1

3trð

r

ðtÞÞ; ð3Þ

acting upon the material point along the stress cycle, where the

operator tr denotes the trace of the Cauchy stress tensor

r

(t) at a

time instant t.

With measures

s

aand

r

Hmaxconsidered as the driving forces of

the fatigue degradation process, a number of stress-based life

esti-mation models ([1,16,23]and references therein) considers a linear

combination of such quantities, as follows:

Nb f:

s

a

Nb f þ

r

H max

a

Nb f ¼ 1; ð5Þ

which can be represented by a straight line connecting points (0, 1) and (1, 0) in space

r

H max=

a

Nbf

s

a=

a

Nbf

. Eq. (5)is

repre-sented in Fig. 1, together with data produced by Lee [24] for

SM45C steel and by Dubar[25]for 30CND16 steel. Here it should

be remarked that, in order to produce the normalized graphics in

Fig. 1, the material parameters

a

and b have to be previously iden-tiﬁed (e.g. from fully reversed bending or push–pull tests). It can be observed that, for sufﬁciently large values of the maximum

hydro-static stress

r

Hmax, a good correlation between data and the model

Fig. 1. Data for SM45C steel[24]and 30NCD16 steel[25]represented in normalized spacesa=ðaNbfÞ rH max= aNbf

. The blue line represents the normalized S–N curve, Eq.

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described in Eq.(5)is veriﬁed. Further examination ofFig. 1allows us to conclude that low levels of positive maximum hydrostatic stresses have no or negligible inﬂuence on fatigue life. Therefore, in such cases, we claim that a new rule given by

s

a¼

c

Ndf ð6Þ

should be considered. Here

c

and d are additional material

parame-ters, now identiﬁed from a set of data associated with loading his-tories without hydrostatic stresses (fully reversed torsion tests, for instance). It is worth of notice that, in another scenario involving

rolling contact stress histories, Bernasconi and co-workers[26]

ob-served that the multiaxial fatigue limit estimated in terms of the

Dang Van criterion [27] does not depend on the instantaneous

hydrostatic stress for small positive values.

Based on the aforementioned phenomenological observations, the multiaxial fatigue life estimation model proposed in this paper will then be constructed considering two distinct domains: one associated with stress histories characterized by the presence of

signiﬁcant values of maximum hydrostatic stresses—described by

expression(5)—and another for situations involving smaller

hydro-static stresses—described by expression(6). Thus:

s

aþ

r

H max¼

a

Nbf for \significant levels" of

r

H max;

s

a¼

c

Ndf for \lower levels" of

r

H max;

(

ð7Þ

or, equivalently, in terms of estimated lives, we have:

Nf ¼ 1

að

s

aþ

r

H maxÞ

1=b

for \significant levels" of

r

H max; 1

c

s

a

1=d

for \lower levels" of

r

H max:

8 < :

ð8Þ

In Eqs.(7) and (8)above, the terms ‘‘signiﬁcant’’ and ‘‘lower’’ levels

of hydrostatic stress have to be made more precise.Fig. 2illustrates

schematically the S–N surfaces corresponding to the proposed life estimation model. The intersection of both surfaces can be obtained

by equating the two life expressions(8):

Fig. 2. Fatigue life ruled surfaces in spacerHmaxsa N and the common boundary (bolded) line deﬁned byrH max¼a sca

b=d

sa.

(a)

(b)

Fig. 3. (a) An arbitrary h-oriented rectangle enclosing the stress path in the deviatoric stress space and a h-oriented prismatic hull (the smallest rectangle with orientation h enclosing the stress path). (b) A prismatic hull with arbitrary orientation h and the one corresponding to the maximum value of the amplitudesadeﬁned in Eq.(19).

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Nf ¼ 1

a

ð

s

aþ

r

H maxÞ 1=b ¼ 1

c

s

a 1=d ð9Þ

which gives the common boundary line of the two ruled surfaces (as illustrated inFig. 2):

r

H max¼

a

s

a

c

b=d

s

a: ð10Þ

It means that ‘‘signiﬁcant’’ levels of hydrostatic stress has to be

a 1=d otherwise: 8 > < > : ð12Þ

It should be emphasized that the material parameters

a

and b in

Eqs. (11) or (12) have to be identiﬁed from loading programs

involving signiﬁcant levels of maximum hydrostatic stresses

r

Hmax

(a)

(b)

(c)

Fig. 4. The deviatoric stress amplitude can be associated with more than one prismatic hull, as illustrated for (a) quadrilateral, (b) triangular and (c) elliptic paths.

Table 1

Tensile properties of the materials. SM45C steel

30NCD16 steel

6082-T6 Al 7075-T651 Al Yield stress (MPa) 418 1045ab

301b

501b

Tensile strength (MPa) 731 1156a ₃₄₃ ₅₆₁

Young’s modulus (GPa) 208.6 – 69.4 71.7

a

Average between batches 4 and 5.

b

0.2% offset.

Table 2

SM45C steel – loading parameters and observed lives Nexp

fj from fatigue tests performed by Lee[24]. Estimated lives N FD

f from Findley[3], N CST

f from Cristofori et al.[11]and N PR f

from proposed piecewise-ruled model.

j rxa(MPa) rxm(MPa) rxya(MPa) rxym(MPa) u(°) Nexp_{f |} (cycles) NFD

f | (cycles) NCSTfj (cycles) NPRfj (cycles)

(a) Fully reversed bending

1 411 0 0 0 0 15,000 10,282 16,465 16,465 2 388 0 0 0 0 26,100 21,418 29,773 29,773 3 372 0 0 0 0 53,000 36,630 45,912 45,912 4 364 0 0 0 0 74,000 48,322 57,417 57,417 5 353 0 0 0 0 93,700 71,444 78,726 78,726 6 336 0 0 0 0 103,000 134,002 130,796 130,796 7 323 0 0 0 0 166,000 221,556 196,272 196,272 8 314 0 0 0 0 213,000 317,595 262,479 262,479 9 313 0 0 0 0 327,000 330,770 271,233 271,233 10 294 0 0 0 0 445,000 734,667 516,515 516,515 11 291 0 0 0 0 723,000 837,243 573,983 573,983

(b) Fully reversed torsion

12 0 0 278 0 0 10,400 17,487 8141 8141 13 0 0 266 0 0 23,300 30,684 18,412 18,412 14 0 0 254 0 0 19,500 55,256 43,240 43,240 15 0 0 253 0 0 30,000 58,105 46,513 46,513 16 0 0 246 0 0 109,000 83,079 78,152 78,152 17 0 0 244 0 0 166,000 92,186 90,888 90,888 18 0 0 230 0 0 332,000 195,736 271,093 271,093 19 0 0 229 0 0 142,000 206,912 293,844 293,844 20 0 0 224 0 0 403,000 274,130 442,013 442,013 21 0 0 218 0 0 1,130,000 387,447 730,325 730,325 (c) Bending–torsion 22 390 0 151 0 0 8500 2239 3814 8080 23 349 0 148 0 0 24,000 6508 9960 20,533 24 325 0 153 0 0 32,000 10,263 15,017 32,408 25 372 0 93 0 0 38,000 13,154 19,707 25,989 26 309 0 134 0 0 100,000 28,020 38,866 67,916 27 265 0 225 0 90 12,000 3715 4050 12,113 28 392 0 118 0 90 12,700 9483 7624 12,025 29 417 0 78 0 90 13,000 6596 8383 10,214 30 346 0 173 0 90 16,000 6696 5225 14,247 31 245 0 216 0 90 20,000 7366 8553 31,940 32 245 0 211 0 90 25,000 8968 10,595 43,158 33 304 0 186 0 90 26,000 8434 8261 26,728 34 304 0 152 0 90 57,000 34,834 25,630 53,925 35 314 0 127 0 90 100,000 81,110 41,441 68,147 36 286 0 143 0 90 120,000 75,821 54,263 101,028 37 167 0 211 0 90 290,000 39,176 53,581 231,246 38 265 0 132 0 90 350,000 205,511 140,861 223,940

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(completely reversed traction–compression tests, for instance),

while

c

and d should be calibrated by considering loading programs

without hydrostatic stresses (completely reversed torsion tests, for instance). The choice of tests for material parameter identiﬁcation are of utmost importance as it must maintain coherence with the phenomenological observation: Indeed, it is reasonable to assume that one should not calibrate a model that correlates deviatoric (shear) and hydrostatic stresses using data that do not contain the inﬂuence of both variables. The same can be reasoned when consid-ering data with low levels of hydrostatic stresses to calibrate the

second rule(12)which is devised for situations with no or

negligi-ble maximum normal stresses. Thus, although torsion S–N curves may not always be available in the current literature, and temporar-ily it may represent an additional difﬁculty in the application of the model, it is reasonable to expect that the torsion tests are the best suited to identify parameters when the goal is to estimate fatigue lives under multiaxial loading with low levels of hydrostatic stresses.

The model expressed by Eqs.(11) or (12)should be applied to

materials for which fatigue degradation is mainly driven by shear stresses. In order to establish a precise bound upon the

applicabil-ity of the model, let us consider a completely reversed traction–

a

ffiffiffi 3

p : ð15Þ

The application of this relation in the ﬁrst expression(12)gives:

Nf ¼ 1

a

s

aþ

s

a ffiffiffi 3 p 1=b ; ð16Þ or, equivalently: Table 3

30NCD16 steel – loading parameters and observed lives Nexp

fj from fatigue tests performed by Dubar[25]. Estimated lives N FD

f from Cristofori et al.[11]and

NPR

f from proposed piecewise-ruled model.

j rxa(MPa) rxm(MPa) rxya(MPa) rxym(MPa) u(°) Nexp_fj (cycles) NFD

fj (cycles) NCSTfj (cycles) NPRfj (cycles)

1 765 0 0 0 0 120,000 116,909 117,002 117,002 2 790 0 0 0 0 90,000 87,631 84,511 84,511 3 795 0 0 0 0 80,000 82,812 79,286 79,286 4 780 0 0 0 0 100,000 98,232 96,135 96,135 5 725 0 0 0 0 200,000 189,202 201,434 201,434 6 708 0 0 0 0 250,000 234,045 256,080 256,080 7 720 0 0 0 0 210,000 201,311 216,041 216,041 8 752 0 0 0 0 140,000 136,325 139,153 139,153 9 820 0 0 0 0 51,000 62,742 57,965 57,965 10 785 0 0 0 0 95,000 92,763 90,117 90,117 11 715 0 0 0 0 230,000 214,288 231,821 231,821

(b) Fully reversed torsion

12 0 0 482 0 0 120,000 117,381 116,963 116,963 13 0 0 500 0 0 90,000 84,501 86,547 86,547 14 0 0 505 0 0 80,000 77,290 79,755 79,755 15 0 0 495 0 0 100,000 92,467 93,995 93,995 16 0 0 450 0 0 200,000 217,291 205,644 205,644 17 0 0 440 0 0 250,000 265,786 247,335 247,335 18 0 0 446 0 0 210,000 235,398 221,293 221,293 19 0 0 470 0 0 140,000 147,146 143,876 143,876 20 0 0 527 0 0 51,000 52,737 56,187 56,187 21 0 0 497 0 0 95,000 89,185 90,933 90,933 22 0 0 445 0 0 230,000 240,182 225,411 225,411

(c) Bending or torsion with mean normal stress

23 660 290 0 0 0 250,000 159,944 417,876 115,241 24 695 290 0 0 0 120,000 105,837 240,565 73,342 25 620 450 0 0 0 140,000 148,151 735,542 90,453 26 640 450 0 0 0 51,000 117,094 516,309 70,122 27 0 290 460 0 0 120,000 46,502 129,893 171,675 28 0 450 430 0 0 250,000 38,680 201,875 298,743 29 0 450 460 0 0 120,000 23,938 110,806 171,675 (d) Bending–torsion 30 600 0 335 0 0 80,000 51,918 62,667 117,168 31 600 0 335 0 90 100,000 122,121 62,667 117,168 32 548 0 306 0 0 200,000 116,961 149,300 246,615 33 562 0 315 0 90 200,000 214,460 115,333 197,483

(e) Bending–torsion with mean normal stress

34 500 290 290 0 0 120,000 59,167 235,964 167,190

35 500 290 290 0 90 210,000 109,082 235,964 167,190

36 490 450 285 0 0 95,000 34,774 248,416 90,096

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s

a¼ ffiffiffi 3 p 1 þpffiffiffi3

a

N b f: ð17Þ

On the other hand, the deviatoric stress amplitude

s

afrom

expres-sion (17) (describing a loading program with positive maximum

hydrostatic stress) should not be larger than the one associated

with the second of expressions(11)(where the inﬂuence of

hydro-static stresses is not taken into account). This restriction is moti-vated by the experimental observation that the number of cycles to failure decreases in the presence hydrostatic stresses. As a conse-quence, the following inequality must be veriﬁed:

ffiffiffi 3 p 1 þpffiffiffi3

a

N b f 6

c

N d f: ð18Þ

Thus, for a given set of parameters

a

, b,

c

and d, the model described

by S–N rules(11) or (12)can be applied for multiaxial fatigue life

estimation if these parameters satisfy inequality(18).

3. The prismatic hull as a measure of deviatoric stress amplitude

Within the setting of multiaxial loading, the deﬁnition of devi-atoric stress amplitude is far from trivial. A number of different

ap-proaches have been proposed in the literature ([21,27,23,10,13,11],

amongst others). The authors proposed an alternative measure, based on the concept of the maximum prismatic hull, which pro-vided very good estimation of fatigue limits for a wide range of

complex multiaxial loadings[12]. In this paper, we consider such

measure of the deviatoric stress amplitude for the fatigue life

esti-mation model(12).

In what follows, we discuss the concept of the maximum prismatic hull and the procedures necessary for the calculation of the corresponding deviatoric stress amplitude. For the sake of simplicity, we shall consider biaxial stress histories described by

components s1(t) and s2(t) in the deviatoric space. We deﬁne the

Table 4

6082-T6 Al alloy – loading parameters and observed lives Nexp

f | from fatigue tests performed by Susmel and Petrone[28]. Estimated lives N FD

f from Cristofori

et al.[11]and NPR

j rxa(MPa) rxm(MPa) rxya(MPa) rxym(MPa) u(°) Nexp_fj (cycles) NFD

1 224 1 4 0 0 52,990 45,438 52,562 30,967 2 190 0 5 7 0 159,000 148,407 165,187 115,190 3 188 1 4 0 0 197,275 172,034 177,961 125,737 4 180 4 4 1 0 244,403 241,249 239,639 178,018 5 162 0 3 1 0 421,560 527,083 503,479 414,608 6 165 2 4 1 0 437,636 461,544 440,749 356,906 7 145 1 4 1 0 1,060,730 1,223,486 1,084,503 1,002,259 8 145 1 4 0 0 1,235,690 1,237,180 1,084,503 1,002,259

(a) Fully reversed torsion

9 14 1 138 0 0 14,695 20,539 18,931 18,233 10 18 3 139 0 0 23,052 18,854 17,988 17,062 11 15 1 111 0 0 67,690 105,172 106,381 103,132 12 16 1 111 0 0 113,455 104,449 106,235 102,789 13 13 3 99 0 0 196,555 247,024 264,316 258,130 14 24 0 98 0 0 449,997 250,242 274,141 264,661 15 15 2 86 1 0 497,990 701,949 792,916 783,051 16 15 1 87 0 0 1,100,000 650,855 723,756 714,475 (d) Bending–torsion, set 1 17 70 3 118 0 0 71,255 40,555 48,390 41,536 18 71 1 117 1 1 78,730 42,272 51,097 43,608 19 59 1 100 1 7 230,750 143,778 173,765 157,027 20 61 0 98 0 18 516,985 160,771 195,279 176,164 21 53 1 83 1 2 1,018,775 549,699 680,894 650,462 22 52 2 82 0 2 1,289,550 609,414 751,006 721,503 23 79 1 129 1 129 20,730 21,345 24,028 19,794 24 79 4 116 0 125 41,490 45,810 49,543 41,762 25 69 1 110 0 126 188,882 70,081 80,356 69,364 26 68 2 99 0 128 234,725 143,856 166,691 147,180 27 68 2 99 0 125 368,080 145,195 166,691 147,180 28 60 3 94 0 126 1,016,280 224,540 264,413 240,218 (d) Bending–torsion, set 2 29 147 2 106 1 4 31,000 22,201 30,104 21,036 30 151 4 104 0 3 64,090 22,600 30,398 21,139 31 163 2 81 0 5 124,460 47,960 62,079 43,067 32 147 1 90 1 8 132,215 46,497 62,004 44,370 33 146 3 76 1 6 232,370 94,994 117,856 88,434 34 118 3 82 1 5 315,795 141,240 179,243 145,991 35 119 1 72 1 0 694,062 240,992 303,078 253,260 36 188 0 106 0 89 5590 39,482 14,253 8656 37 189 5 106 0 94 27,420 39,287 13,808 8470 38 189 1 106 0 88 34,015 38,650 14,030 8470 39 171 4 99 1 91 44,750 70,889 25,391 16,650 40 190 4 105 0 91 47,020 42,477 14,080 8595 41 149 0 68 0 93 114,845 628,348 154,405 116,499 42 151 0 67 0 94 273,325 578,300 151,903 114,012 43 155 1 72 1 92 445,560 458,039 110,626 80,378 44 152 1 47 2 91 456,725 734,321 316,842 252,647

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h-oriented prismatic hull as the smallest rectangle with orientation

hcontaining the stress path, as illustrated inFig. 3a. The amplitude

of the h-oriented prismatic hull is given by

s

aðhÞ :¼ 1 ffiffiffi 2 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 1hþ a22h q ; ð19Þ

where a1hand a2hare the distances of the centre of the h-oriented

rectangle to its faces:

a1h¼ 1 2 maxt s1hðtÞ mint s1hðtÞ ; ð20Þ a2h¼ 1 2 maxt s2hðtÞ mint s2hðtÞ ; ð21Þ Table 5

7075-T651 Al alloy – loading parameters and observed lives Nexp

fj from fatigue tests performed by Zhao and Jiang[29]. Estimated lives N FD

f from Cristofori

et al.[11]and NPR

j rxa(MPa) rxm(MPa) rxya(MPa) rxym(MPa) u(°) k Nexp_fj (cycles) NFD

(a) Fully reversed axial loading (dog-bone specimens)

1 349.7 0. 0. 0. 0. 1. 7144 12,306 8469 8469

2 361.6 0. 0. 0. 0. 1. 9112 10,056 6528 6528

3 299.6 0. 0. 0. 0. 1. 27,011 31,279 28,209 28,209

4 240.0 0. 0. 0. 0. 1. 99,287 119,234 158,469 158,469

5 200.2 0. 0. 0. 0. 1. 919,687 356,019 649,659 649,659

(b) Fully reversed torsional loading

6 0. 0. 203.3 0. 0. 1. 18,842 53,184 67,085 67,085 7 0. 0. 219.0 0. 0. 1. 20,271 33,953 46,143 46,143 8 0. 0. 190.3 0. 0. 1. 178,065 79,236 93,540 93,540 9 0. 0. 167.8 0. 0. 1. 308,144 169,277 176,157 176,157 10 0. 0. 192.0 0. 0. 1. 328,816 75,096 89,447 89,447 11 0. 0. 136.5 0. 0. 1. 403,731 588,166 497,665 497,665 12 0. 0. 146.9 0. 0. 1. 428,510 377,674 343,967 343,967 13 0. 0. 110.8 0. 0. 1. 805,783 2,070,305 1,421,250 1,421,250 14 0. 0. 124.5 0. 0. 1. 913,545 1,024,692 790,633 790,633

(c) Torsional loading with mean normal stress

15 0. 200.0 188.5 0. 0. 1. 12,739 7697 31,069 47,501 16 0. 200.0 192.0 0. 0. 1. 52,986 87,557 265,929 89,447 17 0. 293.1 196.7 0. 0. 1. 84,946 79,517 390,354 79,201 18 0. 289.8 134.2 0. 0. 1. 30,192 10,022 154,767 103,721 19 0. 288.7 135.4 0. 0. 1. 25,167 9871 146,168 100,852 20 0. 391.6 131.6 0. 0. 1. 14,489 3905 107,494 38,519

(d) Proportional axial-torsional loading

21 127.2 0. 170.5 0. 0. 1. 59,194 40,556 51,694 105,954

22 166.1 0. 110.4 0. 0. 1. 136,646 117,809 161,980 297,131

23 201.3 0. 130.0 0. 0. 1. 45,500 40,230 47,753 72,954

24 153.5 0. 100.3 0. 0. 1. 662,627 199,455 294,768 549,375

25 137.4 0. 79.7 0. 0. 1. 1,018,000 542,425 950,098 1,319,396

(e) Out-of-phase axial-torsional loading

26 280.4 0. 181.8 0. 90. 1. 10,191 5691 4855 5466

27 200.9 0. 131.6 0. 90. 1. 29,439 41,130 46,262 70,850

28 200.6 0. 115.8 0. 90. 1. 41,747 62,072 68,910 104,267

(f) Asynchronous axial-torsional loading

29 205.8 0. 137.5 0. 0. 2. 35,804 – – 15,097

30 147.5 0. 86.9 0. 0. 2. 225,000 – – 294,108

31 203.8 0. 136.3 0. 0. 4. 12,708 – – 9610

Table 6

Material parameters of the fatigue models. nid= number of data considered in each identiﬁcation procedure.

Prismatic Hull nid a(MPa) b c(MPa) d

SM45C steel 21 962.0 0.0972 452.4 0.0541 30NCD16 steel 22 2208.2 0.0989 1995.4 0.122 6082-T6 Al 16 972.1 0.144 470.7 0.125 7075-T651 Al 8 1018.3 0.129 1851.8 0.199 CST nid ffiffiffiffiffiffiJ2A p q¼1ðMPaÞ jq=1 ffiffiffiffiffiffi J2A p q¼0ðMPaÞ jq=0 SM45C steel 21 148.8 10.29 206.4 18.50 30NCD16 steel 22 333.6 10.12 341.2 8.21 6082-T6 Al 16 76.8 6.97 76.9 8.01 7075-T651 Al 8 100.0 7.78 103.5 5.03 Findley nid j a(MPa) b SM45C steel 21 0.45 656.2 0.0785 30NCD16 steel 22 0.27 1835.8 0.112 6082-T6 Al 16 0.12 516.9 0.132 7075-T651 Al 8 0.55 1408.7 0.166

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In expressions(20) and (21), s1h(t) and s2h(t) account for the

descrip-tion of the stress history, in the deviatoric space, written in terms of

a basis with orientation h. An important feature of the quantity

s

a(h)

is its ability to distinguish between proportional and

nonproportion-al stress paths. Motivated by the good correlations reported in[12]

for multiaxial fatigue strength analysis, we propose the deviatoric

Fig. 6. SM45C steel – comparison of estimated and observed fatigue lives[24], considering models proposed by Cristofori et al. and in this paper. Parameter identiﬁcation obtained from fully reversed bending and torsion data.

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1. Project the discretized stress path {

r

(tk),k = 1:N} onto the devi-atoric space: s1ðtkÞ ¼ 1 ffiffiffi 6 p 2

r

xðtkÞ

r

yðtkÞ ; ð23Þ s2ðtkÞ ¼ 1_ﬃﬃ 2

pryðtkÞ for push—pull loadings in orthogonal directions;

ffiffiffi 2 p

rxyðtkÞ for push—pull=torsion loadings;

(

ð24Þ 2. Set

s

a= 0;

3. For h = 0:Dh:90°

3.1. Describe the stress path in a h-oriented basis (s1h, s2h)

s1hðtkÞ s2hðtkÞ ¼ cos h sin h sin h cos h _s 1ðtkÞ s2ðtkÞ ; ð25Þ

3.2. Compute the amplitudes a1hand a2hof the h-oriented

pris-matic hull along directions s1hand s2h:

a1h¼ 1 2 maxtk s1hðtkÞ min tk s1hðtkÞ ; ð26Þ a2h¼ 1 2 maxtk s2hðtkÞ min tk s2hðtkÞ ; ð27Þ

algorithm for its calculation, are provided in[12].

Fig. 7. 30NCD16 steel – comparison of estimated and observed fatigue lives[25], considering models proposed by Cristofori et al. and in this paper. Parameter identiﬁcation obtained from fully reversed bending and torsion data.

Fig. 8. 6082-T6 Al alloy – comparison of estimated and observed fatigue lives[28], considering models proposed by Cristofori et al. and in this paper. Parameter identiﬁcation obtained from fully reversed bending and torsion data.

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4. Model assessment

In order to assess the proposed fatigue life estimation model, combined axial load or bending and torsion experimental data

ta-ken from the literature [24,25,28,29]were considered. The tests

were conducted on SM45C structural steel, 30NCD16 steel (batches 4 and 5), 6082-T6 and 7075-T651 aluminum alloys. The tensile

properties of the materials are listed inTable 1. The stress histories

where subscripts m and a stand respectively for the mean value and

the amplitude of the stress components,

x

is the loading frequency,

kis the frequency ratio and

u

is the phase angle.

Tables 2–5report the loading programs for each material and

the number of cycles to failure. Data from 5 103_{to 1.3 10}6

cy-cles were selected for the present study.

For the sake of comparison, we present the lives estimated from

the fatigue models proposed by Findley[3] and Cristofori et al.

[11]. Findley’s model can be expressed as

max

h;/ f

s

aþ

jr

n maxg ¼

a

N b

f; ð30Þ

where

s

ais the radius of the minimum circumference enclosing the

shear stress vectors[23]and

r

nmaxis the maximum normal stress,

both computed at material planes oriented by the angles h and /,

while

a

(11)

Table 6 lists the material parameters of the fatigue models which, in all cases, were obtained from ﬁtting of fully reversed ax-ial load or bending and torsion data. The number of data consid-ered during parameter identiﬁcation, for each material and model, is denoted as nid.

Tables 2–5list the lives obtained from the proposed model, to-gether with the estimations provided by the CST and Findley mod-els.Fig. 5illustrates the proposed piecewise ruled S–N surface for

data produced with SM45C steel by Lee[24].Figs. 6–9compare

ob-served versus estimated lives from the CST and the proposed mod-el for the four materials. Data smod-elected for parameter identification are also showed. In all figures, the central diagonal line accounts for a perfect agreement between estimated and recorded lives, while the dashed lines represent factor two and tree error band-widths. For steels, the estimations of the proposed model lie essen-tially within the factor two bandwidth. This is a very good result in the finite life regime, especially if we take into account the usually observed scattering of fatigue data. For aluminum alloys, the majority of the results fell within the factor three bandwidth.

The histogram inFig. 10provides a comparison between

esti-mated Nest

f and observed N

exp

f lives considering all fatigue models

and experimental data, except the ones used for material parame-ter identiﬁcation. The error index is deﬁned as

E :¼N est f Nexp f : ð32Þ

A perfect agreement between estimated and observed lives is at-tained when E = 1. Estimations are non-conservative if E > 1 and conservative if E < 1. Most of the estimated lives—88.3% of the experimental data—falls within a factor 3 bandwidth, while 66.2% of the estimated lives lies within a factor 2 bandwidth. From this

histogram, and alsoFigs. 6–9, it is observed that the estimations

provided by the proposed model and the CST model are quite sim-ilar, the difference lying essentially for loading programs with mean stresses, which represent a small fraction of the total number of experimental data.

5. Discussion

The proposed multiaxial fatigue model aims to estimate life within the framework of macroscopically elastic time varying stresses. The piecewise ruled S–N surface was based on the assumption that: (i) the fatigue life is insensitive to low levels of hydrostatic stresses and (ii) for ‘‘signiﬁcant levels’’ of maximum hydrostatic stress the sum of the deviatoric stress amplitude and the maximum hydrostatic stress is an exponential function of the number of cycles to failure. This linear relation provided a very good representation of the experimental data for normalized max-imum hydrostatic stresses

r

H max=

a

Nbf

up to 0.52. For higher lev-els of hydrostatic stress, data should be produced in order to

evaluate whether the proposed relation between

s

aand

r

Hmaxstill

holds. In other linear models available in the literature, the

hydro-static stress is usually multiplied by a material parameter

j

, which

varies between 0 and 1. Clearly, in this work, two extreme

condi-tions were considered:

j

= 0 for low levels of hydrostatic stress

and

j

= 1 for higher levels. Speciﬁcally for the material and loading

programs reported here, such assumption proved simple and at the same time satisfactory. For different materials and test conditions,

it may be required to seek a value for

j

which can be identiﬁed by

basic tests under uniaxial conditions.

As a measure of deviatoric stress amplitude, the maximum pris-matic hull enclosing the stress history is proposed. It successfully distinguishes between proportional and nonproportional stress his-tories. For biaxial stress histories, the search for the deviatoric stress amplitude depends on a simple two-dimensional rotation

algo-rithm. However, it should be emphasized here that the orientations of the hull corresponding to the deviatoric stress amplitude has no physical meaning connected to crack orientations, for instance.

The assessment was carried out with available data produced under combined axial load or bending and torsion, and considered the effects of out-of-phase, asynchronous and mean stresses on fa-tigue live. For most data, the results were slightly better when compared with estimates provided by two other models available in the literature. However, the life estimates were signiﬁcantly im-proved by the proposed model for loading situations involving mean normal stresses.

6. Conclusion

A model for multiaxial fatigue life estimation based on a piece-wise S–N surface was proposed in this paper. It produced estimates within a factor two when considering multiaxial data within the medium-high cycle regime for steel alloys and within a factor three for most alluminum alloys data. Life estimates were essentially the same as those obtained from the model proposed by Cristofori and co-workers, except for data involving mean normal stresses, where the criterion here proposed performed better. Findley’s model per-formed worse for steel data.

Acknowledgment

The supports provided by CNPq under Contracts 303279/2007-9 and 304773/200303279/2007-9-3 are gratefully acknowledged.

References

[1] Socie DF, Marquis GB. Multiaxial fatigue. SAE International; 2000.

[2] Ellyin F. Fatigue damage, crack growth and life estimation. Chapman & Hall; 1996.

[3] 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 1959:301–6. [4] Fatemi A, Socie DF. A critical plane approach to multiaxial fatigue damage

including out-of-phase loading. Fatigue Fract Eng Mater Struct 1988;11(3):149–65.

[5] McDiarmid DL. A shear-stress based critical-plane criterion of multiaxial fatigue failure for design and life estimation. Fatigue Fract Eng Mater Struct 1994;17(12):1475–84.

[6] Morel F. A critical plane approach for life estimation of high cycle fatigue under multiaxial variable amplitude loading. Int J Fatigue 2000;22(2):101–19. [7] Papadopoulos IV. Long life fatigue under multiaxial loading. Int J Fatigue

2001;23:839–49.

[8] Lazzarin P, Susmel L. A stress-based method to predict lifetime under multiaxial fatigue loadings. Fatigue Fract Eng Mater Struct 2003;26:1171–87. [9] Jabbado M, Maitournam MH. A high-cycle fatigue life model for variable amplitude multiaxial loading. Fatigue Fract Eng Mater Struct 2008;31(1):67–75.

[10] de Freitas M, Li B, Santos JLT. A numerical approach for high-cycle fatigue life estimation with multiaxial loading. In: Kalluri S, Bonacuse PJ, editors. Multiaxial fatigue and deformation: testing and estimation. ASTM STP 1387. West Conshohocken (PA): American Society for Testing and Materias; 2000. p. 139–56.

[11] Cristofori A, Susmel L, Tovo R. A stress invariant based criterion to estimate fatigue damage under multiaxial loading. Int J Fatigue 2008;30:1646–58. [12] 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. [13] Castro FC, Araújo J, Mamiya EN, Zouain N. Remarks on multiaxial fatigue limit

criteria based on prismatic hulls and ellipsoids. Int J Fatigue 2009;31:1875–81. [14] Mamiya EN, Araújo JA. Fatigue limit under multiaxial loadings: on the deﬁnition of the equivalent shear stress. Mech Res Commun 2002;29:141–51. [15] Basquin OH. The exponential law of endurance tests. In: Proc. annual meeting,

American society for testing materials, vol. 10; 1910. p. 625–30. [16] Suresh S. Fatigue of materials. 2nd ed. Cambridge University Press; 1998. [17] Dang Van K. Introduction to fatigue analysis in mechanical design by the

multiscale approach. In: Dang Van K, Papadopoulos I, editors. High-cycle metal fatigue from theory to applications. CISM courses and lecture, vol. 392. International Centre for Mechanical Sciences; 1999. p. 57–88.

[18] Papadopoulos IV. Multiaxial fatigue limit criterion of metals: a mesoscopic scale approach. In: Dang Van K, Papadopoulos IV, editors. High-cycle metal fatigue. CISM courses and lectures, vol. 392. New York: Springer; 1999. p. 89–143. [19] Crossland B. Effect of large hydrostatic pressures on the torsionnal fatigue

strength of an alloy steel. In: Proceedings of the international conference on fatigue of metals, Institution of Mechanical Engineers, London; 1956. p. 138–49.

(12)

[20] Sines G. Behavior of metals under complex static and alternating stresses. In: Sines G, Waisman JL, editors. Metal fatigue. New York: McGraw-Hill; 1959. p. 145–69.

[21] Deperrois A. Sur le calcul de limites d’endurance des aciers, Thèse de Doctorat, Ecole Polytechnique, Paris; 1991.

[22] Li B, Santos JLT, de Freitas M. A uniﬁed numerical approach for multiaxial fatigue limit evaluation. Int J Mech Struct Mach 2000;28:85–103.

[23] 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(3):219–35.

[24] Lee S-B. A criterion for fully reversed out-of-phase torsion and bending. In: Miller KJ, Brown MW, editors. Multiaxial fatigue. ASTM STP 853. Philadelphia: American Society for Testing and Materials; 1985. p. 553–68.

[25] Dubar L. Fatigue multiaxiale des aciers–passage de l’endurance ál’endurance limitée–prise en compte des accidents géométriques, Ph.D. Thesis, Ecole Nationale Supérieure d’Arts et Métiers; 1992.

[26] Desimone H, Bernasconi A, Beretta S. On the application of Dang Van criterion to rolling contact fatigue. Wear 2006;260:567–72.

[27] Dang Van K. Sur la résistance à la fatigue des métaux. Sciences et Techniques de L’armement 1973;47:429–53.

[28] Susmel L, Petrone N. Multiaxial fatigue life estimations for 6082-T6 cylindrical specimens under in-phase and out-of-phase biaxial loadings. In: Carpinteri A et al., editors. Biaxial/multiaxial fatigue and fracture. ESIS publication 31. Oxford: Elsevier; 2002. p. 83–104.

[29] Zhao T, Jiang Y. Fatigue of 7075-T651 aluminum alloy. Int J Fatigue 2008;30:834–49.