Modeling of distortion induced by the nitriding process

Transcription

1 Modeling o distortion induced by the nitriding process P. Depouhon a,b, J.-M. Sprauel a, M. Mailhé b, E. Mermoz b a. Aix-Marseille Université, CNRS, ISM UMR 787, 388, Marseille cedex 9, France b. Eurocopter, Aéroport international Marseille Provence, 375 Marignane, France Résumé : Cette étude présente un outil de prédimensionnement capable de simuler les déormations résiduelles de pièces complexes engendrées au cours du traitement de nitruration. Ce procédé thermochimique est destiné à améliorer la tenue en service des pièces mécaniques. Le modèle est constitué d une première étape d identiication de la déormation volumique apparente engendrée par le traitement, dans le cas d une géométrie simple. Un problème de diusion équivalent produisant cette déormation est ensuite mis en place ain de s extraire de la géométrie des pièces. Enin, ce dernier est intégré à un code de calcul par éléments inis pour simuler des pièces de géométries complexes. Les résultats sont conrontés à des mesures expérimentales de déplacements sur une maquette de cas industriel. Abstract : This work deals with a pre-dimensioning tool able to simulate distortions o complex parts generated during nitriding process. Nitriding is a thermochemical process which is used to improve the atigue lie o structural parts. The irst step o the model consists in identiying apparent volumic strains generated by the treatment or a simple geometry. Then, an equivalent diusion problem producing this strain has been implemented in order to be independent rom the parts geometry. Finally, the latter is integrated into a inite elements code to simulate parts with complex geometries. The results are compared to experimental measurements on an industrial case. Mots cles : Nitriding process ; Modeling ; Distortion Introduction Thermochemical treatments such as nitriding are usually used to improve the atigue lie o structural parts. Beneits o residual stress generated by this kind o treatment is well known [,, 3]. Many studies have been perormed to describe the microstructure o nitrided layers. Several are available to model residual stress development [4, 5], but ew deal with the distortion o mechanical parts. However, in the aeronautic industry, the mass o structural parts tends to be constantly reduced. This speciicity implies the design o thinner parts which are more avorable to distortions. These distortions, usually neglected, have to be mastered in order to ensure the compliance o treated parts. The nitriding process involves complex multi-physic phenomena at the source o distortion. In this work, the case study is the gaseous nitriding o 3CrMoV3 low alloy steel treated during hours. In this kind o steel, nitrogen atoms diuse rom the surace and interact with iron and alloying elements. Nitrides are ormed rom initial carbides or alloying elements in iron solid solution. Carbon atoms released during the dissolution o initial carbides can diuse in the depth and orm new carbides such as cementite or be evacuated rom the steel, which involves decarburizing on the surace [6]. This microstructure evolution involves volume changes o nitrided layers, resulting in macroscopic residual stress and distortions [4].

2 This work ocuses on the mechanical aspect o the treatment. It allows to provide a pre-dimensioning tool, whose computational time is reduced while simulating complex geometrical parts. Volumic eigenstrain (accompanying precipitations) is considered as the source o residual stress and distortions. In the model, the volumic strain is identiied rom the curvature o a thin plate nitrided on one ace [7] using the radius curvature method. Once the volumic strain is identiied, it is imposed numerically to the studied structure thanks to an equivalent diusion problem. Volumic strain is considered as a dilatation accompanying the diusion o an equivalent quantity. The main assumption consists in considering that volumic strain generated by nitrogen and carbon diusion coupled with precipitation can be obtained by the diusion and the dilatation o a unique equivalent quantity. The beneit o this approach lies in its simplicity, its reduced computational time and its instant integration in usual inite elements methods. Mechanical approach o the treatment The aim o this study being the prediction o distortions, the behavior law is supposed to be linear elastic. Thereore, visco-plastic phenomena are neglected. The mechanical problem is to ind (u, σ)(x) suiciently regular at every point x o a domain Ω occupied by the studied solid ( Ω is its boundary) [8] such as : Kinematic compatibility : Equation o equilibrium : ε = ( ut + u) div(σ) = ε = ε e + ε Constitutive law : () ε = ε I σ = K ε e with u the displacement ield, σ the Cauchy stress tensor and K the Hooke tensor. ε is the total strain tensor, ε e the elastic strain, ε the macroscopic volumic apparent eingenstrain generated by nitriding. It is considered purely hydrostatic due to quasi-random distribution o nitrides in errite. Assuming the ininitesimal strain theory is veriied, the total strain can be separated in a summation o an elastic and apparent volumic strain [8]. Fig. schematically shows the generation o residual stress and distortion, caused by volumic strain mismatch. Beore nitriding Ater nitriding " = = = " = " " = " + " e Nitrided layer Sample core Compressive state Tensile state = K " e without strain compatibility veriying strain compatibility Figure Schematic approach o the generation o residual stress and distortion. 3 Apparent volumic strain identiication Goret used the curvature radius method to evaluate residual stresses [7]. A thin plate is nitrided on one ace in order to generate a bending moment which curves the plate. Its curvature radius

3 R is assumed to be constant. Succesives material removals are perormed using chemical etching and the evolution o the curvature radius R is measured using a 3D coordinate measuring machine (Fig. 3). The curvature χ in respect to the removed thickness e and the apparent volumic strain can be calculated in the case o a plate geometry (H its thickness, z the depth in the nitrided layer) : χ(e, ε ) = H H 6(H + e) ε (z) dz + z ε (z) dz R(e, ε ) = e e (H e) 3 () In order to calculate the curvature, the in-depth apparent volumic strain proile is assumed to be piecewise polynomial (Fig. ) such as : ε (z) = { az + bz + c z [; z M ] dz 3 + ez + z + g z [z M ; z F ] = ε (z, ε, ε M, z M, z F ) (3) where (a, b, c, d, e,, g) are constants which depend on (ε, ε M, z M, z F ). The choice made about the apparent volumic stress proile is based on phenomenological considerations. Indeed, the condition ε ε M comes rom the observation o decarburizing close to the surace [5] which involves an elementary volumic strain gradient. Maximal point (z m,ε M ) is directly linked to the maximal compression stress level observed around 3µm in depth [7]. At last, end o diusion layer is similar to a diusion ront which has a tangent to zero in z F. " M " (z) 3 4 Model (linear) Experiment [7] " Curvature (m ) z z M z F Figure Theoretical in-depth apparent volumic strain proile, assumed to be piecewise polynomial Material removal (µm) Figure 3 Curvature o a thin plate nitrided on one ace in respect to removed thickness [7]. The identiication o apparent volumic strain is to ind the parameters {ε, ε M, z M, z F } minimizing the cost unction I deined in the least squares sense by : I = N [χ exp (e k ) χ(e k, ε, ε M, z M, z F )] (4) k= with χ exp (e k ) the experimental curvature at e k depth o k st material removal and N the number o measurement points taken into account in the minimization problem. The simplex optimization algorithm is used to determine the our optimal parameters. 3

4 The experimental results (Fig. 3) show a remaining residual curvature even ater the total removal o the nitrided layer (depth over. mm). It means that plastic and visco-plastic eects appear in the non treated material during treatment. The considered model does not account or these phenomena, and expects a zero curvature once the nitrided layer is removed. It can explain the signiicant discrepancy between the experiment and the model prediction. Furthermore, the simulation is about prediction o distortions on mechanical parts which have not been chemically etched. This is why optimization o apparent volumic strain is only perormed on the irst measured point, which corresponds to a minimal material removal (N = ). Finally, ε (z) is an apparent volumic strain which is not only related to microstructure volume changes, but also accounts (in a global maner) or plastic and visco-plastic eects. 4 Equivalent diusion problem Once apparent volumic strain is calculated, it is considered like the solution o one dimensional equivalent diusion problem. A convection boundary condition is chosen. The diusion problem is to ind (C,q) suiciently regular in Ω = [; H] such as : Diusion equation : C(z, t) t Constitutive law : a C(z, t) t = (z, t) (Ω [; t ]) q(z, t) = λ C(z, t) z (z, t) (Ω [; t ]) (5) Boundary conditions : q(z Ω, t) = h(c ext (t) C(z Ω, t)) (z, t) ( Ω [; t ]) Solution at t time : ε di (z) = α C(z, t ) where C(z, t) is a diusing equivalent quantity and q its lux. Identiication is an inverse problem which consists in inding an unstationary external equivalent quantity C ext (t) which gives ε di (z) = α C(z, t ), or given parameters (a, λ, h, α) and at a given time t corresponding to the end o the simulation. 5 Identiication o an external equivalent quantity The one dimensional diusion problem is resolved using Crank-Nicolson numerical scheme. In order to reduce the solution search space, C ext (t) belongs to the amily o piecewise T-periodic sinusoidal unctions such as T = Ns t. N s is the number o sinusoidal segments on the time interval [; t ]. For each segment, the only unknown is the sinus amplitude Ci max where i [; N s ]. The optimization problem is to ind the parameters (t,n s,ci max ) which minimize the cost unction : J = H [ ε (z) ε di (z)] dz (6) Simplex algorithm is used in the same manner as or the identiication o apparent volumic strain. Fig. 4 shows the optimal external unstationary equivalent quantity. It is evaluated to best it the apparent volumic strain (at t = t, Fig. 5) which is experimentally identiied in the third section. 4

5 Amplitude C ext(t) external equivalent quantity 3 Apparent volumic strain (%) Objective ε (z) rom experiment ε di (z) rom diusion at t = t.5.5 Time (s) 3 Deep (mm) Figure 4 External equivalent quantity which is the boundary condition numerically applied to the surace layer o the treated material. Figure 5 Comparison o in-depth apparent volumic strain resulting rom numerical diusion at t = t and rom experimental identiication. 6 Results Once the external equivalent quantity C ext (t) identiied, it is numerically applied on each nitrided surace o a inite element model. In order to validate the model, an industrial complex geometry case (Fig. 6) is simulated using an industrial inite element code (Fig. 7), then compared (Fig. 8) to experimental measurements acquired by a 3D coordinate measuring machine Figure 6 Industrial case or validation. All suraces are nitrided. Landmarks to 7 correspond to the measurement results shown in Fig. 8 (Design details are suppressed). Figure 7 Theoretical distortion. Blue section is the undeormed geometry beore treatment ; in color, the geometry ater treatment ( ). Axial and radial displacements are shown in Fig. 8. Seven measurements o distortions part geometry beore and ater nitriding have been carried out in dierent zones (see landmark Fig. 6). Fig. 8 compares these experiments with the simulation results. Measures and correspond to axial displacements. The maximal absolute discrepancy is around 34 µm, and the relative one around 3 %. The next measurements correspond to radial displacements, with a maximal absolute discrepancy around 3 µm. The relative discrepancy becomes not relevant because o the low level o displacements compared to the measurement uncertainty. 5

6 Distortion (µm) Model Experiment Measurement number Figure 8 Comparison o model predictions and experimental distortions measured at 7 points. The numbers above model bars are the absolute discrepancy between experiments and model (µm). 7 Conclusions This work presented a simpliied model or the prediction o structural parts distortions generated by nitriding. Our results showed that these deormations can be accurately related to apparent volumic eigenstrains. These eigenstrains - involved during nitriding - were assumed to be the only source o distortions. They could be identiied rom a nitrided plate and then simulated thanks to an equivalent diusion problem. Furthermore, the boundary condition identiied in the latter diusion problem was supposed to characterize the distortions generated by a given set o nitriding parameters applied to a ixed material. Thus, only one nitrided plate was needed to identiy the boundary condition to be imposed to uture simulations. It allows to predict distortions or any complex geometry or the same ixed material and nitriding parameters. The results o such simulations were compared to experimental measurements and gave a good estimation o the real distortions. Thus, our simulation method proved to be a good pre dimensioning tool to anticipate distortions. Moreover, the presented approach is not limited to nitriding because there is no description o nitriding speciic micro structural transormation but could enable to model distortions o any phenomena generating volume changes such that the macroscopic ree strain is hydrostatic. Réérences [] Rozendaal, H.C.F. and Mittemeijer, EJ and Colijn, PF and Van Der Schaa, PJ 983 The development o nitrogen concentration proiles on nitriding iron. Metallurgical and Materials Transactions A [] Li, C.X. and Sun, Y. and Bell, T. 999 Consideration o retting atigue properties o plasma nitrided steel : second prize. Surace Engineering pp , Maney Publishing. [3] Aissat, Sahraoui and Iost, Alain and Guillemot, Gildas and Benarioua, Younes and Mechmeche, Mohamed Caractérisation des propriétés mécaniques par nanoindentation d un traitement de diusion et d un revêtement pour l amélioration de la résistance à l usure des aciers à bas carbone. Mécanique & Industries pp , Cambridge Univ Press. [4] Somers, M.A.J. and Mittemeijer, EJ 99 Development and relaxation o stress in surace layers ; Composition and residual stress proiles in γ -Fe 4 N x layers on α-fe substrates. Metallurgical and Materials Transactions A pp. 89 4, Springer. [5] Jegou, S. and Barrallier, L. and Kubler, R. Phase transormations and induced volume changes in a nitrided ternary Fe 3% Cr.345% C alloy. Acta Materialia 58 pp , Elsevier. [6] Jegou, S. 4 Inluence des éléments d alliage sur la genèse des contraintes résiduelles d aciers nitrurés. Thèse de Doctorat [7] Barrallier, L. and Goret,V. and Vardon,P. and Deloison, D. 9 Residual stresses and distortion simulation o nitrided disc. JCPDS-International Centre or Diraction Data [8] Lemaitre, J. and Chaboche, J.L. and Benallal, A. and Desmorat, R. 9 Mécanique des matériaux solides-3ème édition. Dunod 6