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Fatigue & Fracture of Engineering Materials & Structures Volume 25 Issue 1 2002 [Doi 10.1046%2Fj.1460-2695.2002.00462.x] L. Susmel; P. Lazzarin -- A Bi-parametric Wöhler Curve for

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    A bi-parametric Wohler curve for high cycle multiaxial fatigueassessment

    L . S U S M E L 1 a n d P. L A Z Z A R I N 2

    1Department of Mechanical Engineering, University of Padova, Padova, Italy,2Department of Management and Engineering, University of Padova,

    Vicenza, ItalyReceived in final form 27 July 2001

    A B S T R A C T This paper presents a method for estimating high-cycle fatigue strength under multiaxialloading conditions. The physical interpretation of the fatigue damage is based on thetheory of cyclic deformation in single crystals. Such a theory is also used to single outthose stress components which can be considered significant for crack nucleation andgrowth in the so-called Stage I regime. Fatigue life estimates are carried out by meansof a modified Wohler curve which can be applied to both smooth and blunt notchedcomponents, subjected to either in-phase or out-of-phase loads. The modified Wohler

    curve plots the fatigue strength in terms of the maximum macroscopic shear stressamplitudes, the reference planewhere such amplitudes have to be evaluatedbeingthought of as coincident with the fatigue microcrack initiation plane. The position ofthe fatigue strength curve also depends on the stress component normal to such a planeand the phase angle as well. About 450 experimental data taken from the literature areused to check the accuracy of the method under multiaxial fatigue conditions.

    Keywords cyclic deformation; high cycle fatigue; multiaxial loads; notch; singlecrystal.

    N O M E N C L A T U R E E(%) =fatigue strength error index in percentagekt=inverse slope of the fatigue curves in the modified Wohler diagram

    Kf,ax=axial fatigue strength reduction factor

    Kf,tors =torsional fatigue strength reduction factorm=unit vector of the easy glide directionm*=direction of maximum resolved shear macro-stressM=generic direction on the D plane

    Ms

    ,Ts

    =Papadopoulos integralsn=unit vector nornal to the generic Dplane

    N=number of cyclesNf=number of cycles to failure

    NRef=number of cycles assumed as a reference valueOxyz=reference frame

    Oab=reference frame on the generic plane DX,Y=parameters depending on the applied loadingsa,b=parameters depending on the material fatigue strength

    d=out-of-phase anglew,h=angles which define the position of a generic plane D

    w*,h*=angles which define the initiation planew,h=angles which define the plane on which the T

    svalue is maximum

    Correspondence: P. Lazzarin, Department of Management and

    Engineering, University of Padova, Stradella S.Nicola 3-36100

    Vicenza, Italy.

    E-mail: [email protected]

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    64 L . S U S M E L a n d P. L A Z Z A R I N

    D=generic planeC=accumulated plastic strain

    mcpl=microscopic plastic shear strainmt=microscopic shear stressmt0y=microscopic initial yield limit of a crystalr=stress ratio related to the initiation plane

    sA2

    =fully reversed axial fatigue limit

    sH,max=maximum hydrostatic stresssn,max(w*,h*)=maximum stress normal to the initiation planesT =tensile strengtht=shear stress

    ta(w*,h*)=shear stress amplitude on the plane of the maximum shear stressamplitude

    tA,Ref=fatigue strength corresponding to NRefcyclestA2

    =fully reversed torsional fatigue limittm (w*,h*)=mean shear stress on the initiation plane

    tr=resolved shear stresstr,a=resolved shear stress amplitudeV=easy glide plane

    fatigue problems are based on a stress approach andI N T R O D U C T I O N

    their influence on fatigue assessment techniques has beencrucial up to now. This fact results in two main conse-Mechanical components frequently work under multiax-

    ial fatigue loadings and, for this reason, the problem of quences: a large amount of experimental data obtainedunder load control is reported in the technical literature,the multiaxial fatigue assessment has long been investi-

    gated and continues to be investigated by many research- and theSNWohler curves continue to be used regularly,at least in some European countries, by engineersers. The state of the art shows the approaches vary

    mainly as a function of the fatigue life and are different engaged in fatigue strength problems.In this paper a multiaxial life estimation method basedfor low-cycle fatigue and high-cycle fatigue. The most

    popular low-cycle fatigue life estimation techniques are on a modified, non-conventional, Wohler curve is pre-sented. The method takes advantage of some recentbased on a strain approach (see, for example, the critical

    plane based criteria proposed by Socie and coworkers,14 findings of multiaxial fatigue studies and, in particular,uses the theory of cyclic deformation in single crystalsBrown and Miller5 and Wang and Brown6 as well as the

    energy criterion introduced by Ellyin,79). These criteria and the initiation plane concept21,22 to give a physicalinterpretation of fatigue damage and to single out thoseare sometimes extended to high-cycle fatigue, where the

    plastic strain contribution becomes negligible.1012 On stress components that are considered to be really sig-nificant to crack initiation and growth during thethe contrary, all the multiaxial criteria devoted solely to

    high-cycle fatigue are based exclusively on stress. so-called Stage I regime, as defined in Millerspapers.23,24This holds true, for example, for the microscopic-

    approach-based criteria proposed by Dang Van13 orPapadopoulos14,15 and for the critical-plane-based cri-

    S O M E P R E L I M I N A R Y D E F I N I T I O N Steria due to McDiarmid16,17 Matake18 and Findley.19

    Recently, the critical plane approach has been reviewed Consider a body subjected to a cyclic loading and pointO located on its surface (Fig. 1a). This point is thoughtand modified by Carpinteri and Spagnoli.20 Their

    new criterion correlates the critical plane orientation of as the critical point for fatigue strength, so that it isconsidered as coincident with the centre of the absolutewith the weighted mean principal stress directions.

    Accordingly, the fatigue failure assessment is performed reference frame Oxyz (Fig. 1a). The position of a mate-rial plane D, having n as normal unit vector, can beby considering a non-linear combination of the maxi-

    mum normal stress and the shear stress amplitude acting determined by means of the spherical co-ordinateswandh: the former is the angle between the projection of theon the critical plane.

    The older and universally known, studies on uniaxial unit vector non the xy plane and the x-axis, the latter

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    A B I - P A R A M E T R I C W O H LE R C UR V E 65

    Fig. 1 Reference frame and definition of

    the spherical co-ordinateshandw.

    the angle between the normal n and the z-axis22 (seeT H E M U L T I A X I A L F A T I G U E D A M A G E

    Fig. 1b).A C C O R D I N G T O T H E T H E O R Y O F T H E C Y C L I C

    Considering the generic plane D, in each instant t itD E F O R M A T I O N I N S I N G L E C R Y S T A L S

    is possible to subdivide the stress state into two compo-nents: the normal stress sn(t) and the shear stress t(t). Consider the generic body shown in Fig. 3(a). A volume

    V containing point O is considered and its dimensionsThe amplitude, the mean value and the maximum valueof the stress component normal to D can be expressed are defined in such a way that the stress state can be

    assumed to be a constant and equal to the value relatedby the following relationships:22

    to point O. In addition, the material is thought of ashomogeneous and isotropic.s

    n,a(w,h )=

    1

    2

    GmaxtT

    sn

    (w,h, t) mintT

    sn

    (w,h, t)

    H In a crystal the microscopic cyclic stress state creates

    persistent slip bands25 and such bands are parallel to ansn,m (w,h )=

    1

    2GmaxtT sn(w,h, t) +mintT sn(w,h, t)H (1) easy glide plane V[Fig. 3(b)]. After a certain number ofcycles, a microcrack initiates in the crystal because of amicrostress concentration effect due to the presence ofsn,max(w,h )=sn,a (w,h )+sn,m(w,h )a deep intrusion21,26 [Fig. 3(b)].

    The definition of the amplitude and the mean value ofThe damage phenomenon is mainly influenced by the

    the shear stress tangential to a generic plane is muchvalue of a microplastic shear strain acting on the easy

    more complex, mainly because the vector t(t) changesglide plane V and calculated with respect to the easy

    its magnitude and direction during the cyclic load. Inglide directionm.21,27 The theory of cyclic deformation

    order to define ta and tm, we intend here to use the in single crystals makes use of the cumulated plasticconcept of the minimum circumscribed circle according

    strain C to weigh the fatigue damage of the single crystal.to Papadopoulos.22 More precisely, the shear stress

    In particular, C is defined as follows:21

    amplitude is equal to the radius of the minimum circlethat circumscribes the curve Y (the curve Y being

    C= N

    i=1

    |mcpl,im| (2)described by the tip of the shear stress vector t(t) on theDplane during the cyclic load), whereas the mean shear

    value is defined as the magnitude of the vector that joins wheremcpl,i is the microscopic plastic shear strain ampli-tude in thei-th cycle andNis the total number of cycles.point O with the centre of the minimum circumscribed

    circle (Fig. 2). Under the hypothesis of a purely elastic macroscopicstrain, the macroscopic shear stress versus the micro-Finally, it is important to highlight that in the present

    paper the stresses will always be referred to the net area. scopic plastic shear stress relationship can be expressed

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    66 L . S U S M E L a n d P. L A Z Z A R I N

    Fig. 2 taandtmdefinitions based on the

    minimum inscribed circle concept due to

    Papadopoulos.

    by the following equation:27,28 in a transcrystalline mode in the persistent slip bands(PSBs).25 Second, the material can be thought of as

    tm=h(mcp m) (3) homogeneous and isotropic. Such hypotheses make itpossible to affirm that, from a statistical point of view,where h is a monotonic function. Equations (2) and (3)

    show that the cumulated plastic strain value depends each plane locates an equal number of grains thathave an easy glide plane coincident with the consideredboth on the macro-shear stress (calculated along the easy

    glide direction m) and the number of cycles to failure plane [Fig. 4(a),(b)]; moreover, each direction on a planelocates an equal number of grains that have an easyN. Papadopoulos was able to demonstrate27 according

    to the elastic shakedown state concept, that the scalar glide direction coincident with the analysed direction[Fig. 4(c)]. These simple remarks, together with Eq. (3)entity C can be expressed for an infinite number of

    loading cycles as: suggest that the microscopic fatigue crack occurs on theplane of maximum macroscopic shear stress amplitude

    and the maximum fatigue damage is produced along theC2=

    1

    g(ta mmt0

    y) (4)direction of maximum macroscopic resolved shear stressamplitude m*. Since m* direction is implicitly andwheregis a dimensional constant positive and mt0yis theunambiguously determined by using the ta definitioninitial yield limit of the crystal.proposed by Papadopoulos (Fig. 2), the shear stressEquation (4) shows that, when N2, the accumu-amplitude calculated by means of the minimum circum-lated plastic strain is proportional only to the amplitudescribed circle is always proportional to the plastic strainof the resolved macro-shear stress13,21,22 and no longercumulated by the unfavourably positioned crystals, i.e.to N.tais proportional to the fatigue damage of the unfavour-Note that the proposed considerations are hereably orientated crystals. This last statement shows thatexpressed under the hypothesis of purely elastic macro-it is not necessary to calculate the fatigue damage inscopic deformation and they are related to an easy glideeach direction, as alone by Papadopoulos27 since the taplane and an easy glide direction of a single crystal. Thedefinition intrinsically represents the shear stress fatiguesituation in a real material is much more complex, mainlydamage calculated along each direction.because of the presence of grain boundaries, non-metallic

    Papadopouloss theory was proposed according to theinclusions, precipitates, defects, and so on. As a conse-elastic shakedown state concept. A fatigue limit is reachedquence, it is not possible to knowa priorithe easy glideif, at the mesoscopic scale, some plastically deformingplane and the direction of the grains which are likelygrains tend to recover a purely elastic response. In thatto break.case initiation is avoided. In other words, PapadopoulosNevertheless, the concepts of easy glide plane/direc-uses asymptotic quantities and then deals with infinitetion can be applied to a real material if two furtherfatigue life by comparing the maximum C

    2 with thesimplifying hypotheses are accepted. First, in a polycrys-

    corresponding shakedown value. In order to extendtal at room temperature the fatigue cracks occur mainly

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    A B I - P A R A M E T R I C W O H LE R C UR V E 67

    Papadopouloss theory to finite fatigue life, Morel28

    proposed a fatigue life prediction method built accordingto a microscopic description of the damage accumulation.

    The initiation process of a crack is treated as a phenom-enon taking place on a scale of the order of a grain or afew grains. A crack is supposed to initiate as a conse-quence of the failure of some grains which are less

    resistant to plastic deformation. Such grains are assumedto behave as crystals following three successive phases:hardening, saturation and softening. In order to accu-rately describe all these phases, the damage variablechosen is always the accumulated plastic strain at itsmesoscopic scale. Its estimation requires the location ofthe plane subjected to maximum damage. This operationis achieved by maximizing the measure of the accumu-lated plastic mesostrain on every material plane of anelementary volumeV.

    Morels method, which gives very satisfactory resultsin high cycle fatigue28 demonstrated that the fatiguedamage in each phase is always proportional to the

    macroscopic resolved shear stress vector acting in aneasy glide direction. It is interesting to note that, byproceeding on parallel tracks, the combined use of thesingle crystal cyclic deformation theory and the mini-mum circumscribed circle concept makes it possible tosingle out the same entity as a fundamental stress compo-nent for the multiaxial fatigue damage.

    Finally, on the basis of the previous considerations, itis also possible to state that the plane of maximummacroscopic shear stress amplitude can be thought of ascoincident with the fatigue microcrack initiation plane.

    This statement fully agrees both with Millers Stage I

    concept23,24

    and the experimental data obtained by Socieand Bannantine.30 when testing AISI 304, Inconel 718and SAE 1045 specimens. Such data demonstrated, in

    Fig. 3 The elementary volumeV, the micro-stress state affecting particular, that shear crack nucleation was followed bythe single crystal (a) and the persistent slip bands in a single crystal crack growth on planes of maximum principal strain,(b). even if the final failure was controlled by Mode I. 30

    Fatigue crack initiation as well as fatigue crack growth

    Fig. 4 Unfavourably positioned crystals

    located by the D1(a) and D2(b) planes and

    by them1andm2directions on the D plane.

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    68 L . S U S M E L a n d P. L A Z Z A R I N

    are conditioned by the stress normal to the initiation In Eq. (5) the maximum value of normal stress is usedto take into account the influence of mean stress onplane. The influence of this component during the

    growth can be explained by transferring Socies fatigue multiaxial fatigue strength.1,30

    It is worth noting that the parameter r makes itdamage model1,2 to the microscopic scale and byobserving that the normal macro-stresses are transmitted possible to take into account directly the phase angle

    influence on multiaxial fatigue strength, as shown inon the microscopic scale without being altered:27 anormal traction stress component opens the microcrack Fig. 6. In particular, Fig. 6(b), (c) show that the r ratio

    varies as a function of the phase angle d, even when theand, consequently, favours growth; on the other hand, acompression stress component slows down the growth applied load amplitudes are kept constant. This holdstrue both for axial/torsional loads [Fig. 6(b)] and for abecause of the friction between the faced microcrack

    surfaces [Fig. 5(b)]. In the same way, the normal stress combined application of an axial load and an internalpressure [Fig. 6(c)]. Obviously, the presence of out-of-component also influences the persistent slip band (PSB)

    formation process, i.e. the microcrack initiation process, phase loadings results, in general, in variations of r,shear stress amplitude and initiation plane.because a compression component inhibits the PSB

    laminar flow, whereas a traction component favours their Consider now a modified loglog Wohler diagramwherethe abscissa is the number of cycles to failure andflow [Fig. 5(a)].

    In conclusion, the theory of cyclic deformation in the ordinate the shear stress amplitudeta(w*,h*) calcu-lated on the initiation plane [Fig. 7(a)]. Two fatiguesingle crystals suggests that the fatigue damage in a

    polycrystal depends on the maximum shear stress ampli- curves are usually available for materials: the fullyreversed axial and the torsional fatigue curves. Whentude (determined by the minimum circumscribed circle

    concept) and on the stress component normal to the plotted in the modified Wohler diagram, they are rep-resented by two different straight lines: the axial and thecrack initiation plane. The model presented here should

    be used to describe the fatigue damage only during torsional fatigue curves correspond tor=1 and r=0,respectively [Fig. 7(a)]. In general, such curves haveStage I (shear crack), according to Millers definition.23,24

    Then, from a rigorous point of view, the predicted different values of the inverse slope kt

    [being kt(r)=

    tana(r)] and are identified by the values tA,Ref (r=1)fatigue life should be thought of as coincident with thenumber of cycles required to exhaust Stage I. and tA,Ref(r=0) at a number of cycles to failure NRef,

    to be assumed as reference value (for example, 2 106

    cycles in some design codes). If one would use the VonT H E N E W M E T H O D F R A M EMises hypothesis simply to estimate the trend, the ratio

    Consider a body subjected to a multiaxial cyclic loadbetween the torsion and tension reference values is equal

    [Fig. 1(a)] and define the plane of maximum shear stressto:

    amplitude by using the minimum circumscribed circleconcept (Fig. 2). tA,

    2(r=0)

    tA,2

    (r=1)=

    2

    3$1.155 (6)With reference to this plane, it is possible to define

    for the crack initiation plane the stress ratio r as follows:Then it is natural to think that, in the modified Wohlerdiagram, the more r increases, the more the fatigue

    r=sn.,max

    ta(w*,h*) (5)

    curve moves downwards. If experimental data were so

    Fig. 5 Application of the Socie fatigue

    damage model to interpret both the

    persistent slip band laminar flow (a) and the

    micro-crack growth (b).

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    A B I - P A R A M E T R I C W O H LE R C UR V E 69

    Fig. 6 Influence of phase angle don stress components for a cylindrical specimen (a) subjected to a stress state generated by axial force and

    internal pressure (b) and by axial force and torsional moment (c, d).

    numerous as to make it possible to evaluate the functions life assessment the nominal net section stresses[Fig. 7(b)].tA,ref(r) andkt(r), a fatigue life prediction for a multiaxial

    stress state could be carried out by using the followingexpression:

    F O R M A L I Z A T I O N O F T H E M E T H O D F O R H I G H -

    C Y C L E F A T I G U E P R O B L E M SNf=

    CtA,Ref(r)ta(w*,h*)D

    k(r)

    NRef (7)

    The accuracy of the method proposed for multiaxialfatigue life estimations largely depends on the numberwhere Nf is the number of cycles to failure. Equation

    (7) represents a bi-parametric non conventional Wohler of experimental data used to determine the functionstA,Ref(r) and kt(r). The precision is obviously expectedcurve given in terms ofrand the shear stress component

    amplitude. This procedure can also be used to estimate to increase with the number of modified Wohler curvesavailable to perform the calibration of the model.the fatigue life of notched components, by altering the

    fatigue curves by means of the fatigue strength reduction In most cases, the experimental data of fatigue strengthrefer to fully reversed axial and torsional loading con-factor Kf

    31 and then involving in the multiaxial fatigue

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    70 L . S U S M E L a n d P. L A Z Z A R I N

    Fig. 7 Modified Wohler curves (a) and

    fatigue curves valid for smooth and notched

    components (b).

    ditions, so that only two different fatigue limit valuesta(w*,h*)+[tA,Ref(r=0)tA,Ref(r=1)]

    sn,max

    ta(w*,h*)and two different inverse slope values are available to

    determine the functions tA,Ref(r)andkt(r). This directlytA,Ref(r=0) (10)implies giving tA,Ref(r) and kt(r) only in terms of two

    simple linear functions. When reference shear stress values correspond to fatigueBy assuming the linearity for tA,Ref(r) and using the limits, Eq. (10) becomes:

    uniaxial and torsional reference shear stresses to calibratethe model, tA,Ref(r) can be written as follows:

    ta(w*,h*)+

    CtA2

    sA2

    2

    D

    sn,max

    ta(w*,h*)tA

    2 (11)

    tA,Ref(r)=tA,Ref(r=0) (8)It is worth noting that, while the qualitative interpret-

    +r[tA,Ref(r=1)tA,Ref(r=0)] ation of the multiaxial fatigue damage is based on thetheory of cyclic deformation in single crystals, the quanti-The condition to be assured under multiaxial high cycletative evaluation uses a modified Wohler curve in con-fatigue is:

    junction with the crack initiation plane concept.In order to be applied, Eq. (11) requires the values ofta(w*,h*)tA,Ref(r) (9)

    the fully reversed fatigue limits sA2

    and tA2

    . Suchvalues are summarized in Table 1 for all the materialsBy introducing Eq. (8) into Eq. (9) one obtains:

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    A B I - P A R A M E T R I C W O H LE R C UR V E 71

    Table 1 Data related to smooth specimens

    Materials Refs sA2

    (MPa) tA2

    (MPa) sT(MPa) Applied loads* No. data

    Carbon steel 0.35% C [42] 215.8 127.2 570 PPT 6

    Hardened steel 0.51% C [43] 313.9 196.2 694 BT 24

    Soft steel 0.1% C [43] 235.4 137.3 382 BT 15

    Cast iron 3.87% C [43] 96.1 91.2 185 BT 12

    Duraluminium 3.81% Cu [43] 156 100 443 BT 14Grey cast iron 3.32% Cu [44] 143 110 279 BT 15

    42CrMo4 [45] 398 260 1025 BT 9

    34Cr4 [45] 410 256 795 BT 14

    30NCD16 [46] 660 410 1880 BT 10

    CK45 [47] 423 287 PPT 8

    SAE4340 [48] 462 286 PPT 3

    SAE1045 [49] 211.5 125.5 621 PPT 5

    XC18 [50] 332 186 1530 BT 3

    FGS 800-2 [51] 294 220 815 BT 3

    EN24T [52] 405 270 850 PPPr 11

    25CrMo4 [15] 361 228 PPTPr 8

    EN25T [53, 54] 476 273 PPPr 5

    St35 [55] 189 122 PPPr 8

    0.1% C steel (normalized) [41] 268.6 151.3 430.7 BT 10

    0.4% C steel (normalized) [41] 331.9 206.9 648.8 BT 5

    0.4% C steel (spheroidized) [41] 274.8 155.9 477.0 BT 5

    0.9% C steel (pearlitic) [41] 352.0 240.8 847.5 BT 5

    3% Ni steel [41] 342.7 205.3 526.4 BT 5

    3/3.5% Ni steel [41] 352.0 267.1 722.5 BT 5

    Cr-Va steel [41] 429.1 257.8 751.8 BT 5

    3.5% NiCr steel (normal impact) [41] 540.3 352.0 895.3 BT 10

    3.5% NiCr steel (low impact) [41] 509.4 324.2 896.9 BT 5

    NiCrMo steel (6070 tons) [41] 725.6 484.7 1000.3 BT 5

    NiCrMo steel (7580 tons) [41] 660.7 342.7 1242.7 BT 5

    NiCr steel [41] 810.4 452.3 1667.2 BT 5

    SILAL cast iron [41] 240.8 219.2 230.0 BT 5

    NICROSILAL cast iron [41] 253.2 211.5 219.2 BT 5

    Brass [17] 83.0 74.0 BT 2

    Hard steel [17] 460.0 275.0 BT 2Soft steel [17] 196.0 186.0 BT 2

    Carbon steel [56] 261.0 160.0 BT 3

    Cast iron [57] 151.0 92.0 PPPr 5

    CrMo steel [58] 713.2 425.3 946.5 BT 6

    CrMo steel [58] 688.9 412.8 946.5 BT 6

    CrMo steel [58] 509.0 306.9 954.0 BT 6

    CrMo steel [58] 589.6 367.6 944.8 BT 3

    CrMo steel [58] 593.3 350.7 944.8 BT 6

    CrMo steel [58] 628.3 366.6 954.0 BT 6

    NiCrMo steel S81 [58] 589.7 331.9 1103.8 BT 6

    NiCrMoVa steel [58] 660.7 342.7 1242.7 BT 6

    CrMoVa steel DTD551 [58] 667.8 398.3 1397.1 BT 6

    CrMoVa steel DTD551 [58] 659.9 386.5 1397.1 BT 6

    CrMoVa steel DTD551 [58] 706.1 412.5 1368.7 BT 3CrMoVa steel DTD551 [58] 737.7 447.4 1368.7 BT 6

    NiCr steel [58] 666.7 369.7 1398.4 BT 6

    NiCr steel [58] 653.2 339.6 1398.4 BT 6

    NiCr steel [58]] 771.9 452.3 1667.2 BT 5

    *B, bending; PP, pushpull; T, torsion; Pr, internal/external pressure.

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    72 L . S U S M E L a n d P. L A Z Z A R I N

    we will consider in the re-analysis of multiaxial fatigue factor ranging only between 1.17 and 2.48. It is theauthors opinion that in the presence of severe or sharpdata. Equation (11) will be applied also to fatigue data

    characterized by values ofr greater than 1.0. V-shaped notches the method should be re-formulatedon the basis of the notch stress intensity factors (N-SIFs)of the relevant geometries, by upgrading some pro-

    V A L I D A T I O N O F T H E M E T H O Dcedures already illustrated for plane fatigue prob-lems.3336 Such procedures initially proposed for aEquation 11 of the proposed criterion can be rewritten

    in the form: material obeying a linear elastic law, have recently beenextended37 also to cases of small scale plasticity whereX+aYb (12)

    plastic N-SIFs substitute elastic N-SIFs. The moresevere the notch is, the easier the identification of thewhere parameters Xand Ydepend on the applied loads

    and a and b are, on the contrary, two constants which crack initiation plane should be. It is reasonable to thinkthat, in the presence of sharp V-shaped notches thedepend on the material fatigue properties. The accuracy

    of the multiaxial fatigue predictions can be quantified by geometry, and no longer the stress ratior, fully controlsthe crack initiation and propagation phenomena. Datausing the fatigue strength error index E%:12

    already reported by Ritchie et al.38 related to sharpV-shaped notches subjected to combined tension andE % =

    X

    baY1 (13)

    torsion loads, seem to strongly support this hypothesis.Figure 8 plots tA

    2as a function ofsA

    2, by using theifE=0 the fatigue life estimation is exact; ifE0 the pre-estimating the fully reversed torsion fatigue limit bydiction is conservative.using the Von Mises hypothesis is generally a goodThe degree of accuracy of Eq. (11) has been checkedapproximation which, in most cases, is also in the safeby using hundreds of fatigue strength data taken fromdirection; this judgement holds true for both smooththe literature and already re-organized into a database.32

    and notched specimens. Note that each point of theIn particular, the validation of the method involves adiagram represents a series of experimental data. Twototal number of 447 experimental data subdivided intoaspects complicated the re-analysis shown in Fig. 8:72 different series. The data refer to cylindrical speci-(a) sometimes only the mean values were reported in themens, both smooth and notched, made of differentoriginal papers and the information about the statisticalmaterials. Information about the different series of datadispersion of the experimental data was often not avail-is given in Table 1 and in Table 2 for smooth and notchedable; (b) some of the published fatigue limits werespecimens, respectively. It is worth noting that theobtained by extrapolating run-out data. These consider-notches are quite blunt, the fatigue strength reduction

    Table 2 Bending/torsion experimental data obtained using notched specimens

    Materials Refs Specimen shape sNA2

    (MPa) tNA2

    (MPa) Kf,ax Kf,tors No. data

    0.4% C steel (normalized) [41] Sharp V 179.1 176.0 1.85 1.18 5

    3% Ni steel [41] Sharp V 209.9 151.3 1.63 1.36 5

    3/3.5% Ni steel [41] Sharp V 302.6 183.7 1.47 1.45 5

    Cr-Va steel [41] Sharp V 216.1 160.6 1.99 1.61 5

    3.5% NiCr steel (normal impact) [41] Sharp V 268.6 236.2 2.01 1.49 5

    3.5% NiCr steel (low impact) [41] Sharp V 247.0 182.2 2.06 1.78 5

    NiCrMo steel (75-80 tons) [41] Sharp V 271.7 240.8 2.43 1.42 5

    CrMo steel [58] Oil hole 450.9 295.5 1.53 1.40 6

    CrMo steel [58] Oil hole 223.4 162.1 2.28 1.89 6

    CrMo steel [58] Oil hole 424.7 305.5 1.39 1.20 3CrMo steel [58] Oil hole 423.4 300.3 1.40 1.17 6

    CrMo steel [58] Oil hole 423.4 288.4 1.48 1.27 6

    NiCrMoVa steel [58] Notched 271.7 240.8 2.43 1.42 5

    NiCrMoVa steel [58] Oil hole 288.6 211.5 2.29 1.62 3

    CrMoVa steel DTD551 [58] Oil hole 300.7 235.4 2.19 1.64 6

    CrMoVa steel DTD551 [58] Oil hole 297.8 220.9 2.48 2.03 6

    CrMoVa steel DTD551 [58] Oil hole 471.3 354.6 1.57 1.26 3

    CrMoVa steel DTD551 [58] Oil hole 448.1 354.9 1.65 1.26 6

    NiCr steel [58] Oil hole 312.4 225.5 2.09 1.51 6

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    the other hand the severity of the notches was not sostrong as to overwhelm the role of the shear stressamplitudes in describing multiaxial fatigue phenomenon.

    Finally, it is useful to point out that in the combinedbending/torsion tests (312 in total), parameter r wasgreater than 1.0 in 34 cases, its maximum value being4.0. In the other tests (pushpull, torsion and pressure,

    36 data in total), 24 data exhibited a r-value rangingfrom 1.1 to 3.2. Equation (11) was seen to work well inthese cases.

    A C O M P A R I S O N W I T H P A P A D O P O U L O S

    C R I T E R I O NsA(MPa)

    tA

    (MPa)

    Fig. 8 Fully reversed torsional fatigue limit versus fully reversed Some comparative analyses reported in the literatureuniaxial fatigue limit (fatigue limits of the notched specimens being demonstrated12,39 that in multiaxial high-cycle fatiguegiven in terms of nominal stresses on the net area). Papadopoulos criterion gives more accurate predictions

    than the criteria due to Crossland, Sines, McDiarmid,Matake and Dang Van.ations also hold true for the multiaxial fatigue data

    reported in Figs 9 and 10. Moreover, a rigorous vali- Papadopoulos criterion is based on the calculation of

    theTsand Ms integrals defined as follows14 (see Fig. 2):dation of the proposed method should be carried out byusing only fatigue data related either to the fatigue crackinitiation phase or to Stage I fatigue. Unfortunately, the T

    s(w,h )=SP

    2p

    j=0

    t2r,a(w,h,j)dj (14)numbers of cycles of almost all the data reported in theliterature referred to the complete rupture of the speci-mens and the fatigue limit was extrapolated at a certain M

    s=SP

    2p

    w=0Pp

    h=0

    T2s

    (w,h )sin hdhdw (15)number of cycles. In any case, it is well-known that forsmooth specimens the fatigue crack initiation life is a

    where the integrals represent the contribution of manylarge proportion of the total fatigue life and this also

    grains to the damage mechanism. Such integrals wereholds substantially true for blunt notched specimens.

    introduced because the use of the amplitude of shearHowever, it is evident that. using total life fatigue data

    stress alone (Dang Vans criterion) did not lead to goodincreases the scatter between predictions and experimen-

    predictions when dealing with out-of-phase loading.tal data. Nevertheless, the agreement is very satisfactory,confirming the validity of the method.

    Papadopoulos subdivided the metals into hard orIn Fig. 9 the shear stress amplitudes calculated on themild on the basis of the following ruleinitiation plane ta(w*,h*) are reported versus the esti-

    mated multiaxial fatigue limitbar. Figs 9(a), (b) referMild metals:

    to data obtained with smooth specimens, either in-phaseor out-of-phase. It is evident that there is a good

    0.5tA2

    sA2

    0.6agreement between the new method and the experimen-tal results: in particular, in the case of in-phase load,

    Hard metals:predictions are given within a fatigue strength errorindex E of about 20%; in the case of out-of-phase

    0.6ta2

    sA2

    0.8loads, Eranges between 30%.Finally, an even more favourable trend can be noticed

    In the case of hard materials, by using theMsintegral,in Fig. 9(c), where notched specimen data are taken intothe criterion turns out to be:27account. The method makes it possible to summarize all

    the experimental points in a very restricted error bandM

    s+

    tA2

    sA2/3

    sA2

    /3 sH,maxtA

    2 (16)(15%

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    74 L . S U S M E L a n d P. L A Z Z A R I N

    Fig. 9 Experimental shear stress amplitude

    ta(w*,h*) versus estimated multiaxial fatigue

    limitbarfor all in-phase and out-of-phase tests carried out on smooth (a, b) and

    In-phase data (notched specimens)

    b-ar(MPa)

    ta

    (w*,h*)(MPa)

    In-phase data (smooth specimens)

    b-ar(MPa)

    ta

    (w*,h*)(MPa)

    Out-of-phase data (smooth specimens)

    ta

    (w*,h*)(MPa)

    b-ar(MPa)

    notched (c) specimens.

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    Fig. 10 Error frequency distribution

    histograms for smooth specimens (a) and

    notched specimens (b).

    Papadopoulos criterion can be written as follows:14 has been checked by using the error frequency histogramplotted in Fig. 10(a). As a small number of experimentaldata collected from the literature exhibited a ratioT

    s(w,h )+

    3p

    sA2AtA2

    sA2

    2B sH,maxtA2p (17)tA2

    sA2

    0.6The degree of accuracy in the fatigue life predictions ofthe method proposed here and Papadopouloss criterion

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    76 L . S U S M E L a n d P. L A Z Z A R I N

    such data were not considered when applyingPapadopouloss criterion.

    E%=Atxa,a

    tA2B

    2

    1Asx,a

    sA2B

    2

    AsA2

    tA2

    1BAsx,a

    sA2B A2

    sA2

    tA2B

    1In Fig. 10(a) the fatigue strength error index E isreported in the abscissa and its values have been separatedinto 5% amplitude intervals. The ordinate is the error

    (19)frequency (%) and this quantity is thought of as thenumber of data (expressed in percentage) which have an The McDiarmid criterion16,17 uses a critical plane

    error index within each considered D

    E interval. approach, the plane being that on which the shear stressFigure 10(a) shows that the performances of the two amplitude achieves its maximum value. McDiarmidscriteria are comparable: in fact, about 70% of the

    fundamental equation is:experimental data belong to the interval 10%

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    A B I - P A R A M E T R I C W O H LE R C UR V E 77

    8 Ellyin, F., Golos, K. and Xia, Z. (1991) In-phase and out-of-the uniaxial fatigue field to the multiaxial fatigue field,phase multiaxial fatigue. Trans. ASME, J. Engng Mater. Technol.and among these, the statistical re-analysis on the basis113, 112118.of the log-normal distribution as well as or the use of

    9 Ellyin, F. and Xia, Z. (1993) A general fatigue theory and itsthe fatigue strength reduction factors Kf for notched application to out-of-phase cyclic loading. Trans. ASME,components. Even if in this paper the criterion has been J. Engng Mater. Technol. 115, 411416.applied only to the high-cycle fatigue data and fatigue 10 Klesnil, M. and Lukas, P. (1992) Fatigue of Metallic Materials,

    Second revised edition. Elsevier, Amsterdam.limits, it can easily be used in the medium fatigue life,11 Lemaitre, J. and Chaboche, J.-L. (1988)Mecanique Des Materiax

    where elasticity continues to play a strongly prevailing Solides. Bordas, Paris.role with respect to plasticity.12 Petrone, N. and Susmel, L. (1999) A comparison among

    To validate the method about 450 data taken from the different criteria valid for high cycle multiaxial fatigue. In:literature have been systematically analysed. All data Proceedings of the of XXIX AIAS Conference, Vicenza (Italy),referred to smooth and notched cylindrical specimens (Edited by Atzori B. et al.) SGE, Padova, pp. 239250 (in

    Italian).made with different materials and under different surface13 Dang Van, K. (1993) Macro-micro approach in high-cyclefinishing.

    multiaxial fatigue. In: Advance in Multiaxial Fatigue, ASTMSome comparisons have demonstrated that the newSTP 1191, (Edited by D. L. McDowell and R. Ellis). American

    method has the same accuracy and reliability in the Society for Testing and Materials, Philadelphia, PA,multiaxial fatigue assessment as the Papadopoulos cri- pp. 120130.terion but offers two advantages: first, the analytical 14 Papadopoulos, I. V. (1987) Fatigue polycyclique des metaux: une

    novelle approche.The`se de Doctorat, Ecole Nationale des Pontsformulation is the same for hard and mild metals; second,et Chaussees, Paris, France.when a generic multiaxial load history is considered, the

    15 Papadopoulos, I. V. (1995) A. high-cycle fatigue criterionnew method results in a reduction of the calculationapplied in biaxial and triaxial out-of-phase stress conditions.

    time with respect to the time required to determine Fatigue Fract. Engng. Mater. Struct. 18, 7991.Papadoupolos parameters M

    s and T

    s, at least if one 16 McDiarmid, D. L. (1991) A. general criterion for high-cycle

    takes advantage of some skills already proposed by multiaxial fatigue failure. Fatigue Fract. Engng. Mater. Struct.14, 429153.Weber et al.

    17 McDiarmid, D. L. (1994) A. shear stress based critical-planeFinally, the accuracy of the method has been checkedcriterion of multiaxial fatigue failure for design and life predic-in the case of components weakened by quite blunttion.Fatigue Fract. Engng. Mater. Struct. 17, 14751484.

    notches (their fatigue strength reduction factors being18 Matake, T. (1977) An. explanation on fatigue limit under

    always less than 2.5). The method has been seen to be combined stress.Bull. JSME20, 257263.more precise than Goughs criterion and a more recent 19 Findley, W. N. (1959) A. theory for the effect of mean stress

    on fatigue under combined torsion and axial load or bending.criterion due to McDiarmid.Trans. ASME, Series B 81, 301306.

    20 Carpinteri, A. N. and Spagnoli, A. (2001) Multiaxial high-cycleR E F E R E N C E S fatigue criterion for hard metals. Int. J. Fatigue23, 135145.

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    Press, Cambridge, UK.fatigue of Inconel 718 including mean stress effects. In:

    22 Papadopoulos, I. V. (1998) Critical plane approaches in high-Multiaxial Fatigue,ASTM STP 853, (Edited by M. W. Brown

    cycle fatigue. on the definition of the amplitude and mean valueand K. J. Miller). American Society for Testing and Materials,

    of the shear stress acting on the critical plane. Fatigue Fract.Philadelphia, PA, pp. 463481.

    Engng. Mater. Struct. 21, 269285.2 Socie, D. F. (1987) Multiaxial fatigue damage models. Trans.

    23 Miller, K. J. (1993) The two thresholds of fatigue behaviour.ASME J. Engng Mater. Technol.109, 293298. Fatigue Fract. Engng. Mater. Struct. 16, 931939.

    3 Fatemi, A. and Socie, D. F. (1988) A. critical plane approach to 24 Miller, K. J. (1999) The fatigue limit and its elimination.Fatiguemultiaxial fatigue damage including out-of-phase loading. Fract. Engng. Mater. Struct. 22, 545557.

    Fatigue Fract. Engng Mater Struct. 11, 149165. 25 Ellyin, F. (1997) Fatigue Damage, Crack Growth and Life4 Socie, D., Kurath, F. P. and Koch, J. (1989) A multiaxial fatigue Prediction. Chapman & Hall, London.

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    (Edited by M. W. Brown and K. J. Miller). Mechanical 27 Papadopoulos, I. V. (1997) Exploring the high-cycle fatigueEngineering Publications, London, pp. 535550. behaviour of metals from the mesoscopic scale. In: Notes of the5 Brown, M. W. and Miller, K. J. (1973) Proc. Inst. Mech. Engrs. CISM Seminar,Udine (Italy).

    187, 754755. 28 Morel, F. (1998) A fatigue life prediction method based on a6 Wang, C. H. and Brown, M. W. (1993) A. path-independent mesoscopic approach in constant amplitude multiaxial loading.

    parameter for fatigue under proportional and non-proportional Fatigue Fract. Engng. Mater. Struct.21, 241256.loading. Fatigue Fract. Engng. Mater. Struct. 16, 12851298. 29 Morel, F., Palin-Luc, T. and Frosustey, C. (2001) Comparative

    7 Ellyin, F. (1989) Cyclic strain energy density as a criterion for study and link between mesoscopic and energetic approachesmultiaxial fatigue failure. In.Biaxial and Multiaxial Fatigue, EGF in high cycle multiaxial fatigue.Int. J. Fatigue 23, 317327.3 (Edited by M. W. Brown and K. J. Miller), Mechanical 30 Socie, D. and Bannantine, J. (1988) Bulk deformation fatigue

    damage models.Mater. Sci. EngngA103, 313.Engineering Publications London. 571583.

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