Camilla de Andrade Gonçalves

Edgar Nobuo Mamiya

José Alexander Araújo

Comptes Rendus Mécanique (CRM)

/ent of therimentalgue’approchehe shearin crackC. R. Mecanique 332 (2004) 963–968http://france.elsevier.com/direct/CRAS2BA simple multiaxial fatigue criterion for metalsCamilla de Andrade Gonçalves, José Alexander Araújo, Edgar Nobuo Mamiya∗Departamento de Engenharia Mecânica, Universidade de Brasília, 70910-900 Brasília, DF, BrasilReceived 14 December 2003; accepted after revision 22 September 2004Presented by Évariste Sanchez-PalenciaAbstractA new simple multiaxial high-cycle fatigue endurance criterion, suitable for situations where the convex hull associated withthe stress path approximates well an ellipsod, is proposed. It considers, as measuresof fatigue solicitation: (i) a new definitionfor the equivalent shear stress amplitude; and (ii) the maximum principal stress along the stress history. Assessmresulting criterion for a wide range of in-phase and out-phase cyclic loads shows that it compares very well with expedata published in the literature.To cite this article: C.A. Gonçalves et al., C. R. Mecanique 332 (2004). 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.RésuméUn critère simple d’éndurance à la fatigue multiaxiale des métaux. On propose un critère simple d’endurance à la fatipolycyclique, applicable à des situations pour lesquelles l’envelope convexe associé à l’histoire des contraintes sbien d’une ellipse. Le critère considère, comme mesures de solicitation à la fatigue : (i) unenouvelle définition de l’amplitudede contrainte de cisaillement ; et (ii) la contrainte principale maximale au cours de l’histoire de chargement.Pour citer cetarticle : C.A. Gonçalves et al., C. R. Mecanique 332 (2004). 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.Keywords: Fatigue; Fatigue strength; Multiaxial high-cycle fatigue; Nonproportional loadingMots-clés : Fatigue ; Éndurance à la fatigue ; Fatigue multiaxiale polycyclique ; Chargement non-proportionel1. IntroductionThis note presents a new, simple multiaxial high-cycle fatigue endurance criterion, suitable for situations wherethe convex hull associated with the stress path approximates well an ellipsoid. The model considers tstresses as one of the driving forces of the fatigue process – since plasticity plays an important role* Corresponding author.E-mail address: mamiya@unb.br (E.N. Mamiya).1631-0721/$ – see front matter 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.doi:10.1016/j.crme.2004.09.003964 C.A. Gonçalves et al. / C. R. Mecanique 332 (2004) 963–968, inessorene couldtederalcriterionel,ting uponual to thee equiv-r, suchia-initiation – and the normal stress acting upon embryo-cracks – which has been shown by Sines [1] to affect thefatigue resistance. Models based on such assumptions are usually written as:f (S) + g(p)  λ (1)wheref (S) is a measure of the shear stress amplitude,S is the history of the deviatoric stress tensor,g(p) is afunction of the hydrostatic stressp observed along the stress history andλ is a material parameter. For instancethe criterion proposed by Crossland [3],f (S) is the√J2 radius of the sphere circumscribing the deviatoric strpath. Notice that proportional and nonproportional paths canbe circumscribed by the same sphere although a msevere solicitation is expected when the nonproportional stress history is considered. As an alternative, oconsider a quantity associated with the minimum ellipsoid circumscribing the stress path, as previously suggesby Deperrois [4], and later by Bin Li et al. [5]: the basicidea is to consider shear stress amplitudes in sevorthogonal directions, summing up their effects to provide a measure of the shear fatigue solicitation. Theproposed by Papadopoulos relies on the argument that the accumulated plastic deformations at mesoscopic levat each slip plane, are proportional to the resolved shear stress amplitude.The influence of the normal stress has been taken into account by many authors through its average acall the planes passing through the material point. As remarked by Papadopoulos [2], such an average is eqhydrostatic stress.Here we present an alternative fatigue endurance criterion based on new definitions for functionsf (S) andg(p)in expression (1).2. The equivalent shear stress amplitudeWe acknowledge the concept of the minimum circumscribing ellipsoid as an appropriate measure of thalent shear stress. In this setting, the termf (S) has been defined [4,5] as:f (S) :=√∑5i=1 λ2i (2)whereλi, i = 1, . . . ,5 are the semi-axes of the ellipsoid circumscribing the stress path. In general, howeveellipsoid and hence its semi-axes are difficult to determine.For the specific case where the ellipsoid is a good approximation for the convex hull associated with the devtoric part of the stress path (see Fig. 1), the result presented in what follows allows us to computef (S) in an almosttrivially way.Proposition 2.1. Given an ellipsoid E in Rm with centre located at the origin and an arbitrary orthonormalbasis {ni , i = 1, . . . ,m} of Rm, let P be a rectangular prism circumscribing E such that its faces are orthogonalto each one of the basis elements. If λi, i = 1, . . . ,m are the magnitudes of the principal semi-axes of E andai, i = 1, . . . ,m denote the distances of the centre of the ellipsoid to the faces of the rectangular prism, then:Fig. 1. (a) Convex hull of the stress path; (b) ellipsoid circumscribing the stress path.Fig. 1. (a) Enveloppe convexe de l’histoire des contraintes ;(b) ellipsoïde circonscrivant l’histoire des contraintes.C.A. Gonçalves et al. / C. R. Mecanique 332 (2004) 963–968 965dtudesaretting thess5∑i=1λ2i =5∑i=1a2i (3)Proof. Let Sm1 be the unit sphere inRm:Sm1 :={y ∈ Rm; ‖y‖ = 1} (4)where‖y‖ := (y21 + y22 + · · · + y2m)1/2 is the classical Euclidean norm inRm. The ellipsoidE can be characterizeas the set of points:E := {x ∈ Rm; x = L y, y ∈ Sm1 } (5)where L :Rm → Rm is a symmetric, positive semi-definite matrix with eigenvalues given by the magniλi, i = 1, . . . ,m of the semi-axes ofE . On the other hand, the distanceai , from the faces of the rectangulprism orthogonal to a basis elementni to the centre of the ellipsoid, can be expressed as:ai = supx∈E(x,ni ), i = 1, . . . ,m (6)where(x,y) = ∑mi=1 xi yi denotes the classical Euclidean inner product inRm. By considering the fact that thpointsx from the ellipsoidE satisfy (5), we can develop (6) as:ai = supx∈E(x,ni ) = supy∈Sm1(L y,ni ) = supy∈Sm1(y,L ni ) = ‖L ni‖ (7)since the supremum of(y,L ni ) among the pointsy from Sm1 is attained fory parallel toL ni . Now, let us representhe identity operator onRm as:I =m∑i−1ni ⊗ ni (8)where⊗ denotes the tensor product operator such that(a ⊗ b)u = (a,u)b. It follows that:L2 = L(m∑i−1ni ⊗ ni)L =m∑i−1L ni ⊗ L ni (9)Finally, since the Frobenius norm of the linear operatorL is given by‖L‖F = (∑mi=1 λ2i )1/2, from (7) and (9) weobtain:m∑i=1λ2i = ‖L‖2F = tr(L2) = tr(m∑i−1L ni ⊗ L ni)=m∑i−1(L ni ,L ni ) =m∑i−1‖L ni‖2 =m∑i−1a2i  (10)This statement precludes the need to determine the principal semi-axes of the ellipsoid when compumeasuref (S) of shear solicitation to fatigue. Let Dev3 denote the space of symmetric deviatoric tensors fromR3to R3 and let{Ni , i = 1, . . . ,5} be an arbitrarily chosen orthonormal basisfor such space. Any deviatoric strestateS(t) at a time instant can be written as:S(t) =5∑i−1si(t)Ni (11)For an appropriate basis of Dev3, the componentsi (t) of S(t) in this basis can be expressed as:s1(t) =√32 Sxx(t), s2(t) = 1√2(Syy(t) − Szz(t))s3(t) =√2Sxy(t), s4(t) =√2Sxz(t), s5(t) =√2Syz(t) (12)966 C.A. Gonçalves et al. / C. R. Mecanique 332 (2004) 963–968vexllythe mea-s [2], thesingideringe eigen-solicitationriale-dmetalslimits, theFrom (11), it is possible to describe the stress path in terms of a curve inR5, where each points(t) ∈ R5 can beexpressed as:s(t) := (s1(t) s2(t) . . . s5(t))T (13)Now, according to the aforementioned proposition, whenev r the loading conditions are such that the conhull of the generated stress path can be well approximated by an ellipsoid, the shear stress amplitudef (S) can besimply computed as:f (S) :=√∑5i=1 a2i (14)where, in the context of the present study,ai, i = 1, . . . ,5 are theamplitudes of the components si (t) of thedeviatoric stresses defined as:ai := 12{maxt[si (t)] − mint[si (t)]}, i = 1, . . . ,5 (15)3. The normal stressTensile normal stresses contribute to the fatigue degradation by acting (essentially in mode 1) upon eventuaexisting embryo-cracks in the material. Many fatigue endurance criteria consider the hydrostatic stress assure of the solicitation to fatigue produced by the normal stresses since, as remarked by Papadopoulohydrostatic stress is basically the quantity obtained by averaging the normal stresses over all the planes pasthrough a given material point. In this note, we claim that the worst situation – which corresponds to consthe existence of an embryo-crack oriented orthogonally to the maximum principal stress (among the threvalues of the stress tensor and along all the stress path) – should be considered rather than the averagegiven by the maximum hydrostatic stress.4. The resulting endurance criterionBased on the considerations developed along Sections 2.1 and 2.2, wepropose the following multiaxial highcycle fatigue endurance criterion:√∑5i=1 a2i + κ σpmax λ (16)whereai, i = 1, . . . ,5 are defined as in (15) andσpmax is the maximum principal stress acting upon the matepoint along the loading history, whileκ andλ are material parameters. Iff−1 andt−1 are the fatigue enduranclimits under alternate bending and alternate torsion solicitation, respectively, then the parametersκ andλ can becomputed as:κ = √2 f−1f−1 − t−1(t−1f−1− 1√3)and λ = √2 t−1f−1f−1 − t−1(1− 1√3)(17)5. Assessment of the criterionProportional and nonproportional multiaxil fatigue experiments for a number of different materials were considered to assess the proposed criterion. Experimental data presented in Tables 1 to 3 – obtained by Nishihara anKawamoto [6] (Table 1), Heidenreich et al. [7] (Table 2) and Froustey and Lassere [8] (Table 3) for hard(1.3  f−1/t−1 <√3) – describe biaxial normal/shear stress histories of the form:σ(t) = σm + σa sin(ω t), τ (t) = τm + τa sin(ω t − β) (18)corresponding to the maximum combination of stresses that the specimen can stand without failing, up to aof 106 cycles. In expressions (18),σ and τ describe respectively the normal and the shear stress historieC.A. Gonçalves et al. / C. R. Mecanique 332 (2004) 963–968 967ionsTable 1Fatigue strength of hard steel (t−1 = 196.2 MPa,f−1 = 313.9 MPa): experimental data (Nishihara and Kawamoto [6]) and predictionsTableau 1L’intensité de fatigue d’acier dur (t−1 = 196,2 MPa,f−1 = 313,9 MPa) : données expérimentales (Nishihara et Kawamoto [6]) et prédictσa (MPa) σm (MPa) τa (MPa) τm (MPa) β (◦) Ia (%) Ib (%) Ic (%) Id (%)1-1 138.1 0 167.1 0 0 −2.27 −2.3 −2.28 −1.911-2 140.4 0 169.9 0 30 −2.60 −0.6 −0.64 −0.271-3 145.7 0 176.3 0 60 −3.61 3.1 3.10 3.491-4 150.2 0 181.7 0 90 −3.74 6.3 6.27 6.661-5 245.3 0 122.6 0 0 1.44 1.5 1.44 1.731-6 249.7 0 124.8 0 30 0.01 3.3 3.26 3.551-7 252.4 0 126.2 0 60 −8.35 4.4 4.39 4.691-8 258.0 0 129.0 0 90 −17.81 6.5 6.70 7.011-9 299.1 0 62.8 0 0 0.92 0.9 0.92 1.021-10 304.5 0 63.9 0 90 −2.99 2.7 2.74 2.83aCrossland,bPapadopoulos,cMamiya and Araújo,dCurrent model.Table 2Fatigue strength of 34Cr4 (t−1 = 256 MPa,f−1 = 410 MPa): experimental data (Heidenr ich et al. [7]) and predictionsTableau 2L’intensité de fatigue 34Cr4 (t−1 = 256 MPa,f−1 = 410 MPa) : données expérimentales (Heidenreich et al. [7]) et prédictionsσa (MPa) σm (MPa) τa (MPa) τm (MPa) β (◦) Ia (%) Ib (%) Ic (%) Id (%)2-1 314.0 0 157.0 0 0 −0.55 −0.6 −0.55 −0.272-2 315.0 0 158.0 0 60 −12.33 −0.1 −0.11 0.182-3 316.0 0 158.0 0 90 −22.93 0.1 0.08 0.372-4 315.0 0 158.0 0 120 −12.33 −0.1 −0.11 0.182-5 224.0 0 224.0 0 90 −8.38 5.2 5.15 5.552-6 380.0 0 95.0 0 90 −7.32 0.4 0.37 0.492-7 316.0 0 158.0 158.0 0 0.08 0.1 0.08 6.012-8 314.0 0 157.0 157.0 60 −12.69 −0.6 −0.54 5.342-9 315.0 0 158.0 158.0 90 −23.17 −0.1 −0.11 5.832-10 279.0 279.0 140.0 0 0 −6.38 −6.4 −6.38 −0.212-11 284.0 284.0 142.0 0 90 −25.5 −4.8 −4.83 1.452-12 212.0 212.0 212.0 0 90 −9.39 3.4 3.41 7.23aCrossland,bPapadopoulos,cMamiya and Araújo,dCurrent model.Table 3Fatigue strength of 30NCD16 (t−1 = 410 MPa,f−1 = 660 MPa): experimental data (Froustey and Lassere [8]) and predictionsTableau 3L’intensité de fatigue 30NCD16 (t−1 = 410 MPa,f−1 = 660 MPa) : données expérimentales (Froustey et Lassere [8]) et prédictionsσa (MPa) σm (MPa) τa (MPa) τm (MPa) β (◦) Ia (%) Ib (%) Ic (%) Id (%)3-1 485.0 0 280.0 0 0 1.77 1.8 1.77 2.073-2 480.0 0 277.0 0 90 −27.27 0.7 0.70 1.003-3 480.0 300.0 277.0 0 0 3.91 3.9 3.91 7.633-4 480.0 300.0 277.0 0 45 −3.36 3.9 3.91 7.633-5 470.0 300.0 270.0 0 60 −10.93 1.6 1.60 5.323-6 473.0 300.0 273.0 0 90 −25.12 2.5 2.45 6.173-7 590.0 300.0 148.0 0 0 0.11 0.1 0.11 4.323-8 565.0 300.0 141.0 0 45 −7.23 −4.1 −4.07 0.143-9 540.0 300.0 135.0 0 90 −14.97 −8.1 −8.15 −3.943-10 211.0 300.0 365.0 0 0 −0.68 −0.7 −0.68 1.86aCrossland,bPapadopoulos,cMamiya and Araújo,dCurrent model.968 C.A. Gonçalves et al. / C. R. Mecanique 332 (2004) 963–968e. A neg-has nothand, an errora-e same,ourstress isseeed morer shear orpredictionsoposedn of ourinngsource.hip, andativas”.Hill,orf00).83.. 29subscriptsa andm stand respectively for the amplitude and the mean value of stresses andβ accounts for the phasangle. To assess the quality of the results provided by our model, an error indexI is defined as:I = 1λ[(∑5i=1 a2i)1/2 + κ σpmax− λ] × 100 (%) (19)which gives a measure of how close the prediction of the criterion is with respect to the experimental dataative I yields a non-conservative fatigue strength prediction since it indicates that the stress solicitationattained a critical value while the experimental data are representative of limiting situations. On the otherpositiveI provides a conservative estimate. Application of our model to the experimental data provided aindex which varied in the worst cases between−3.94% and 7.63% for all materials and loading conditions anlyzed. The results provided by both Papadopoulos [2] and by Mamiya and Araújo [9] were essentially thvarying between−8.15% and 6.7% while the Crossland criterion provided significantly poorer predictions. Inmodel, a shift of the error index towards the conservative region can be clearly observed whenever a meanpresent in the loading history.6. Discussion and conclusionA new multiaxial fatigue criterion has been proposed. Application of this criterion to a broad range of in-phaand out-of-phase loading conditions invol ing three different materials under multiaxial, in-phase and out-of-phasstates of stress yielded very good predictions of fatigue endurance. The proposed criterion always providconservative endurance estimates than all the other criteria considered in the present study, whenevenormal mean stresses were present in the loading history. When such mean stresses were absent, thewere essentially the same for all criteria with exception of Crossland. A very interesting feature of the prmodel which should be stressed is the great simplicity of its implementation. On the other hand, applicatiocriterion is restricted to cases where the shape of the convex hull circumscribing the microscopic loading pathDev3 approximates well an ellipsoid. Although this is a clear limitation, in practice there is a wide range of loadicases which fall within this condition, such as components under dynamic loadings caused by a singleStudies on a criterion suitable for more general loading situations are in course.AcknowledgementsThis project was supported by CNPq under contracts 520564/96-0, 300469/00-4 and a PIBIC scholarsby Centrais Elétricas do Norte do Brasil S. A. – Eletronorte under the contract “Fadiga de Máquinas RotThese supports are gratefully acknowledged.References[1] G. Sines, Behavior of metals under complex static and alternating stresses, in: G. Sines, L. Waisman (Eds.), Metal Fatigue, McGraw–New York, 1959, pp. 145–169.[2] I.V. Papadopoulos, P. Davoli, C. Gorla, M. Filippini, A. Bernasconi, A comparative study of multiaxial high-cycle fatigue criteria fmetals, Int. J. Fatigue 19 (1997) 219–235.[3] B. Crossland, Effect of large hydrostatic pressures on the torsional fatigue strength of an alloy steel, in: Proc. Int. Conf. on Fatigue oMetals, IMechE, London, 1956, pp. 138–149.[4] A. Deperrois, Sur le calcul des limites d’éndurance des aciers, Thèse de Doctorat, École Polytechnique, Paris, 1991.[5] B. Li, J.L. Santos, M. Freitas, A unified numerical approach for multiaxial fatigue limit evaluation, Mech. Structures Mach. 28 (2085–103.[6] T. Nishihara, M. Kawamoto, The strength of metals under combined alternating bendingand torsion with phase difference, in: MemCollege Engrg., vol. 11, Kyoto Imperial University, 1945, pp. 85–112.[7] R. Heidenreich, H. Zenner, I. Richter,Dauerschwingfestigkeit bei mehrachsiger Beanspruchungi, Forschungshefte FKM, Heft 105, 19[8] C. Froustey, S. Lassere, Multiaxial fatigue endurance of 30NCD16 steel, Int. J. Fatigue 11 (1889) 169–175.[9] E.N. Mamiya, J.A. Araújo, Fatigue limit under multiaxial loadings: on the definition of the equivalent shear stress, Mech. Res. Comm(2002) 141–151.

A simple multiaxial fatigue criterion for metals

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A simple multiaxial fatigue criterion for metals

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