A volumetric and incremental energy based fatigue life calculation method for notched structures

Transcription

1 A volumetric and incremental energy based fatigue life calculation method for notched structures Jérôme Bénabès, Nicolas Saintier*, Thierry Palin-Luc & Francis Cocheteux 1 E.N.S.AM. CER de Bordeaux, Université Bordeaux 1, Laboratoire Matériaux Endommagement Fiabilité et Ingénierie des Procédés (LAMEFIP), EA 2727 Esplanade des Arts et Métiers, F Talence Cedex, France. 2 Agence d Essai Ferroviaire - SNCF 21, avenue du Président Allende, F Vitry-Sur-Seine, France. *corresponding author, nicolas.saintier@lamef.bordeaux.ensam.fr Résumé : Nous présentons ici une méthode de prévision de la durée de vie en fatigue applicable aux chargements multiaxiaux non-proportionnels d amplitude variable dans la gamme [ ] cycles. Cette méthode est basée sur le concept de courbe maîtresse permettant de regrouper l ensemble des résultats expérimentaux (ie quelque soit l état de contrainte) sur une courbe unique. Les calculs de durée de vie, réalisés avec la méthode proposée, sont confrontés aux résultats expérimentaux pour des chargements d amplitude constante et variable. L application au cas d une roue de train est également présentée. Abstract : Stress gradient effects due to stress-strain concentrations and load type effects are important for the applicability of fatigue models to real structures. In this paper, a fatigue life prediction method is proposed for variable amplitude non-proportional multiaxial loadings in the range 10 4 to 10 7 cycles. This method is based on the concept of a single fatigue crack initiation curve for any loading cases (master curve concept). The predictions of the proposal for constant and variable amplitude multiaxial loadings, are compared to experimental results. An application to a railway wheel is presented. Mots-clefs : High cycle multiaxial fatigue, Energy, Gradient 1 Introduction Designing metallic structures against fatigue is still a difficult task to handle for engineers. This is due to two major reasons : the full complexity of in service loadings (variable amplitude, non-proportional multiaxial loadings...) and the well known gradient effects. Several methods exist to take into account stress gradient effects : Papadopoulos and Panoskaltsis [1994], Sonsino et al. [1997] and Banvillet et al. [2003]. Among these approaches the last one is, according to the authors, the only one able to predict the difference between experimental endurance limits in tension and bending. The fatigue life calculation method presented here is based on this volumetric energy based approach. 2 A new fatigue life calculation method 2.1 An energy parameter Ellyin [1997] showed that the use of both the plastic and elastic strain work can be used as damage parameter in multiaxial fatigue. The LAMEFIP criterion (Banvillet et al. [2003]), 1

2 devoted to the field of endurance or limited endurance, uses for damage parameter, the volumetric density of the strain work given to the material per loading cycles after elastic shakedown (supposed to be reached after a few thousands cycles ). The proposal is based on two main hypothesis : (i) the strain work given to the material per loading cycle is considered as the driving force for fatigue crack initiation and (ii) it is calculated after macroscopic elastic shakedown. Many authors use cycle counting techniques to extract, from a random stress tensor sequence, cycles from which the damage could be estimated. These techniques have two main drawbacks : (i) the choice of the cycle counting algorithm influences the calculated fatigue life since the number of counted cycles is algorithm dependent (Dowling [1983]), and (ii) for multiaxial non-proportional stress states, in many approaches from the literature, the variable chosen for cycle counting differs from the damage parameter. To avoid such drawbacks an incremental model has been developed. The strain work density given at a point M is written in an incremental way as follows : 3 3 dw g (M, t) = σ ij (M, t). ε ij (M, t).h (σ ij (M, t). σ ij (M, t)) dt (1) i=1 j=1 - where ε ij (M, t) are the strain tensor components and ẋ = dx/dt, - σ ij (M, t) are the stress tensor components, - and H represents the Heaviside function : H(a) = 1 ifa 0 ; H(a) = 0 if a < 0. As underlined by Ellyin [1997], the strain work can be calculated as the sum of elastic and plastic strain works, so that : dw g (M, t) = dwg e (M, t) + dw g p (M, t) (2) The framework of this study being HCF and MCF, we choose to consider only the elastic part of the strain work (eq. 3) in the elastic shakedown state. The cumulated strain work on a time sequence of duration T is equivalent to the integral of dwg e (M, t) over T (4). Banvillet et al. [2003] has shown that for an uniaxial stress state W g is not shape dependent (sinus, triangle, square, etc...). 3 3 dwg e (M, t) = σ ij (M, t). ε e ij (M, t).h (σ ij(m, t). σ ij (M, t)) dt (3) i=1 j=1 W g (M, T) = dw g (M, t) (4) T 2.2 Multiaxial stress states To take into account the material sensitivity to the stress triaxiality, the triaxiality degree at a point M is defined by the ratio of the strain work associated with the spherical part of the stress tensor over the total given work Banvillet et al. [2003], but in an incremental way : dt(m, t) = withdw Sph g (M, t) = 1 3 Sph dwg (M, t) if dw g (M, t) 0 otherwisedt(m, t) = 0 (5) dw g (M, t) ( ) σ kk (M, t). ε e ll (M, t).h 3 σ kk (M, t). σ ll (M, t) dt (6) k=1 l=1 k=1 l=1 2

3 The material sensitivity to stress triaxiality is considered by using an empirical function F(dT, β) (eq. 7) depending on the material parameter β identified from two fully reversed fatigue limits (rotating bending and torsion). At any instant, for a multiaxial stress state, the strain work given to the material is corrected to evaluate an uniaxial equivalent strain work dw feq (M, t) (eq. 8). Under uniaxial stress state dw geq (M, t) = dw g (M, t). F(dT(M, t), β) = 2.3 Non-local damage parameter [ dt(m, t) β ln [ 1 + dt(m, t)(e β 1) ]] (7) dw geq (M, t) = dw g (M, t). F(dT uniax, β) F(dT(M, t), β) Palin-Luc et al. [1998] showed that a threshold stress, σ, can be defined below the conventional endurance limit σ D. σ is considered as a threshold stress of no damage initiation at the mesoscale, whereas the usual endurance limit σ D corresponds to a macrocrack initiation. σ can be identified from (R=-1) rotating bending and torsion fatigue limits. This threshold concept can also be expressed in terms of minimum strain work volumetric density (Banvillet et al. [2003]). It has to be noticed that this threshold is closely related to the concept of cycle (under constant amplitude Wg corresponds to a threshold per cycle). If the loading is of variable amplitude, the difficulty lies primarily in the calculation of the strain work part considered as damaging. By analogy with constant amplitude loading, we choose to consider the damaging part of the strain work given to the material as damage parameter in fatigue as follows : W geq,da (M) = W geq (M) α.w g (8) (9) where < A >= A if A 0 and A = 0 if A < 0, with W g = (σ ) 2 /E. The α parameter is closelly related to the temporal evolution of the stress/strain tensors and of the number of transitions within the time evolution of the cumulated strain work given. This is equal to 1 under constant amplitude cyclic loading (due to the length limitation of this paper, it is impossible to detail its computation hereafter, details are in Bénabès [2006]) Volume influencing fatigue crack initiation V* σ delimits a volume influencing fatigue crack initiation (Banvillet et al. [2003]). We consider that the potentially critical points C i (where a fatigue crack can occur) are the points where the damaging strain work W geq,da presents a local maximum. Around each one of these points C i, the influence volume V is defined by : V (C i ) = { points M(x,y,z) around C i so thatw geq,da (M) 0} (10) Volumetric damage parameter To consider the volumetric stress-strain distribution effect in fatigue, it is supposed that all the points of V have a significant influence in the fatigue damage process. The non-local damage parameter geq,da is defined as the volumetric average over V of the damaging strain work on the considered sequence : 3

4 geq,da (C i ) = 1 W V geq,da dv (11) (C i ) V (C i ) 2.4 Master curve In a multiaxial fatigue criterion, the damage parameter threshold is fixed for a long fatigue life. But fatigue criteria can be extended to a fatigue life calculation method by relating the value of the damage parameter to the fatigue life N. Figure 1 shows that the proposed damage parameter geq,da can be used both in MCF and HCF. Indeed, this figure illustrates that different S N curves corresponding to a wide variety of loading cases can be put on a unique geq,da N curve (master curve). Knowing one S-N curve equation σ a = f(n) it is easy to establish the master curve equation. For instance, by using as reference the Basquin model of the S-N curve (σ a = C/N b ) under fully reversed tension on smooth specimens, it is possible to show that the master curve equation is given by (eq. 12) where E is the Young modulus. By solving geq,da (C i ) = geq,da (N) it is possible to determine the fatigue life N for any loading cases and at any points C i of the structure. ( C/N b ) 2 (σ ) 2 geqda (N) = E (12) (a) (b) FIG. 1 W geq N curve and master curve geq,da N various loadings on ER7 steel. 3 Comparison tests results / predictions Fatigue tests experiments were carried out under load control, on smooth specimens manufactured from railway wheels. The material is a ferritic-pearlitic steel named ER7 steel (close to SAE1045). Its mechanical characteristics can be found in Bénabès et al. [2006]. The reference S-N curve is identified from fully reversed plane bending fatigue tests using a Basquin model. For this steel, the threshold stress σ is equal to 258 MPa and the material parameter β is Experiments on specimens Non-proportional combined plane bending and torsion fatigue tests were carried out under cyclic loadings. The bending moment was with a frequency f σ different from the torsion moment frequency f τ. Two frequency conditions were tested f σ /f τ = 8 and f σ /f τ = 1/8, in 4

5 each case the highest frequency was 48 Hz. In each case, two stress amplitude ratio σ a /τ a were tested : 1.61 and And a typical non-proportional multiaxial variable amplitude loading sequence encountered on railway wheels (Figure 2(a)) has been applied on smooth specimens under combined tension and torsion (see ERRI [2004]) and under combined plane bending and torsion (tests performed during this study). The reference signal amplitude was magnified by a factor k to reach different experimental fatigue lifes in the range 10 4 to cycles. Figure 2(b) illustrates the comparison between experimental and predicted fatigue lifes under all the previous test conditions differente frequency tests plane bending and torsion (variable ampl) tension and torsion (variable ampl) N predicted (sequences) Non conservative Conservative x 2 x N experiments (sequences) (a) (b) FIG. 2 (a) Variable amplitude combined tension-torsion load history and (b) Comparison of experimental and predicted fatigue lifes. Figure 2(b) shows that the predictions of the proposed fatigue life calculation method remain in an interval of more or less 2 times the experimental fatigue life. These predictions are very good for such non-proportional loadings which are known as discriminating. We can also note that the predictions are good for variable amplitude loadings with a relatively low number of sequence to fatigue crack initiation. 3.2 Tests in blocks on real wheel Uniaxial sinusoidal fatigue tests under block programmed loadings were carried out by SNCF-AEF on a railway wheel (ER7 steel) (see Bénabès [2006]) on which two holes (figure 3) were machined to create stress gradient and a local multiaxial stress state. The block programmed loading was established from in service recording. The wheel was tested on the SNCF-AEF fatigue test bench under a lateral load Fy (parallel to the wheel axis). The experimental and predicted number of sequences to fatigue crack initiation are in good agreement (Table 1). Figure 3 illustrates the very good correlation between the location of the experimental and computed fatigue crack areas. N exp N pred Comments Test n Test n 2 > No cracks after sequences TAB. 1 Experimental and predicted fatigue lifes (in sequences). 5

6 Crack initiation Critical points computed FIG. 3 Comparison between the experimental fatigue crack initiations areas and the critical points computed by the proposed method. 4 Conclusion This paper presents a high cycle multiaxial fatigue life calculation method depending on the stress-strain volumetric distribution. The proposal leads to good predictions for the presented various loading types. A post-processor for FEA software was developed in order to apply the proposal on a component with a complex geometry. Tests under service loadings on specimens and components have been carried out to validate the proposal. Références I.V. Papadopoulos and V.P. Panoskaltsis. Gradient dependent multiaxial high-cycle fatigue criterion. In Proc. 4th Int. Conf. Biaxial/Multiaxial Fatigue, volume 1, pages , St Germain en Laye, France, SF2M. C.M. Sonsino, H. Kaufmann, and V. Grubisic. Transferability of material data for the example of a randomly loaded forged trck stub axle. In SAE International, editor, SAE Tech. Paper Series, number in SAE technical paper, pages 1 22, Detroit, February SAE. A. Banvillet, T. Palin-Luc, and S. Lasserre. A volumetric energy based high cycle multiaxial fatigue criterion. Int. J. Fatigue,, 26(8) : , August F. Ellyin. Fatigue damage, crack growth and life prediction. Chapman and Hall, Edmonton, Canada, (469 p.). N.E. Dowling. Fatigue life prediction for complex load versus time histories. Transactions of the ASME,, 105 : , July T. Palin-Luc, S. Lasserre, and J-Y. Bérard. Experimental investigation on the significance of the conventional endurance limit of a spheroidal graphite cast iron. Fat. Fract. Engng. Mater. Struct.,, 21(3) : , J. Bénabès. Approche énergétique non locale du calcul de durée de vie de structures en fatigue multiaxiale sous chargements d amplitude vrariable. Application à une roue de train ferroviaire. PhD thesis, ENSAM CER de Bordeaux, J. Bénabès, N. Saintier, T. Palin-Luc, J-L. Charles, and F. Cocheteux. An energy based multiaxial fatigue life calculation method taking into account the stress-strain gradient effect. In Fatigue 2006, Atlanta, ERRI. Dimensionnement des roues - critères de fatigue multiaxiale. Technical report, ERRI, octobre B169/RP19. 6