# （有限元分析与设计）磨削轴承钢52100引起相变和残余应力的数值模拟.pdf

dKeywords:Residual stressesGrindingstressandmaterial.The thermo-metallurgical and mechanical analysis has been performed using the commercial finiteelement software Abaquss/Standard with various user subroutines developed to model the thermal,processes,unitthe energytovarying internal stress/strain field. The evolution of temperaturesmaindis-usedtemperature. The FE model developed in this work takes intoContents lists available at SciVerse ScienceDirectFinite Elements in AnalysFinite Elements in Analysis and Design 61 (2012) 1–11Peclet (Pe) number, non-dimensional heat transfer coefficient (H)E-mail address: michel.coret@insa-lyon.fr (M. Coret).during grinding can be well predicted using models presented by account the transformation strains associated with martensitictransformation along with the temperature dependent materialproperties. The variations of the residual stresses and strains atintegration points have been considered, and the effects of the0168-874X/$-see front matter fax: þ33 4 72438913.occurs when the effective stress exceeds the yield strength. Allthese factors interact with each other and eventually lead to asteels in engineering components such as rolling bearings. More-over, its properties are easy to acquire over a wide range of[1–4], and therefore some variations of the material properties(like hardness, etc.). These phenomena may in fine play a role onthe surface integrity of the component. Phase transformations inmost steels introduce volumetric changes, transformation plasti-city and changes in mechanical properties. Local plastic flowpart of these phenomena. However, the influence of metallurgicaltransformations remains a problem still widely open. Theobjective of this work is to investigate the internal stresstribution and their evolution that occurs during grinding processof AISI-52100 steel which is one of the most commonlyuniform high temperatures, which may then result in solid-statephase transformations. The consequences of this rapid heatingand cooling include the formation of a heat affected zone (HAZ),the generation of residual stresses, possible shrinkage or crackingof the material, often chemical modifications of the materialstresses in the near-surface layer.Numerical simulation of such a problem requires modeling ofthree different types of phenomena: thermal, metallurgical, andmechanical, which are, mostly, fully coupled. Many researchers[10–14] have proposed various models which account for all or1. IntroductionCompared with other machiningan extremely high energy input persurface layer of the material. Most ofheat which is concentrated in the grindinginteracts with the workpiece. This leadsmetallurgical and mechanical behavior of the material. The heat generated during grinding process wasassumed as a moving heat flux with elliptical distribution. The effects of the Peclet number and heattransfer coefficient on the phase transformations and residual stresses have been analyzed. It was foundthat an optimal combination of grinding conditions could produce the desired magnitude ofcompressive residual stresses at the surface of the machined workpiece. It is also shown that omittingphase transformations could lead to a strong difference in the prediction of residual stresses.ð8ÞFinally the parameters ki(T) and ni(T) are obtained by solvingEq. (8):kiðTÞ¼C01ðtEÞniðTÞlnð1C00:99zeqÞð9ÞniðTÞ¼1ln tS=tEC0C1lnlnð1C00:01Þlnð1C00:99zeqÞC20C21ð10ÞThe martensitic transformation is a displacive transformationonly controlled by the temperature. The kinetics is given by theKoistinen–Marburger equation [25]:zM¼zg1C0exp C0bðMSC0TÞC2C3C8C9ð11Þwhere, zMand zgare the martensitic and austenitic phaseproportions, respectively, b is a material dependent coefficient;MSis the martensitic transformation start temperature, and T thetemperature.2.4. Mechanical modelAfter the thermo-metallurgical computations, temperaturesand volume phase fraction become inputs for the mechanicalsimulation. The temperature variations and the phase transfor-mations involve dilatational strains into the solid. The majorremaining difficulty is to obtain the mechanical behavior of themixture of phases. As a first assumption, the macroscopic beha-vior is supposed to follow an isotropic hardening with the vonMises criterion where the yield stress is obtained by a mixture ofthe yield stress of each phase. The flow stresses which take intoaccount hardening, thermal and viscous effects follow a Johnson–Cook model as initially proposed by Umbrello et al. [26] andadapted to multiphase materials.In the following, the total strain rate tensor is divided into arecoverable elastic part _eeijand an irrecoverable plastic one _epij:_eij¼ _eeijþ _epijð12Þ2.4.1. Elastic strainIn the case of coupling of mechanical field with temperatureand phase change, the relation for the elastic strain can beexpressed as:eeij¼1þvEsijC0vEskkdijþEthmdijð13ÞEthmT,zaðÞ¼1C0zaðÞethgTðÞþzaethaTðÞ ð14ÞethaTðÞ¼aaTC0TrefC2C3ð15ÞethgTðÞ¼agTC0TrefC2C3C0De25 1Cagð16Þwhere E is Young’s modulus and v Poisson’s ratio.2.4.2. Yield function and plastic strain rateThe yield function criterion is expressed in stress space asF sij,ep,Hm,T,ziC16C17¼0 ð17Þijwith sayand sbythe yield stresses of the ferrite and austenitephases, respectively.The equivalent plastic strain is given byepl¼Zt0_epldt ð21Þwith_epl¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ23_eplij_eplijrð22Þ2.4.3. Flow stressesFor an accurate simulation of grinding, the mechanical beha-vior must take into account the hardening, thermal, and viscouseffects. A usual way to do that is the use of the Johnson–Cookrelation, well adapted for severe loadings. In [26], the authorspropose a slight modification of Johnson–Cook relation fornumerical simulation of hard machining of AISI 52100 bearingsteel. The stresses are a product of three terms, representing thehardening curve at room temperature, the influence of tempera-ture and the last one the visco-plastic part (Eq. 23). In [26], thetemperature influence function zfactis interpolated as a 5th orderpolynomial function whose coefficients are given in Table 1. Thestrain hardening multiplier needs two other constants (m, A), alsogiven in Table 1.sgeq¼ CenC2C3zfactðTÞC2C31þlnð_eplÞmC0Ahið23Þwithzfact¼exp aT5þbT4þcT3þdT2þeTþfC16C17ð24ÞThe previous single phase model must be adapted for multiphasematerials. For sake of simplicity, the following two assumptions aremade:(1) the flow curve for each phase follows an exponential law:szi¼CziepC0C1nzi;Table 1Parameters used in Eqs. (23) and (24) for AISI52100 steel.Parameters Valuea 3.81C210–15b C04.29C210–12c C06.91C210–9d 5.50C210–6e C01.60C210–3f 2.44C210C02m 0.1259A 0.0567sgy¼and calculated by1C0zaðÞsgyepl,T,ziC16C17þzasayepl,T,ziC16C17ð20Þ(2) the mixture is calculated as a simple linear rule: s¼Pi ¼ 1,nzziszihiwhich is consistent with a Voigt hypothesison the strains. Table 2The stress–strain curves at room temperatures for each phasesare presented in Appendix A.Finally, the relation used issgeqe, _e,T,ziðÞ¼Xi ¼ 1,nzziszi“#zfactðTÞC2C31þlnð_eplÞmC0Ahið25Þwith szi¼Czienzi .3. Finite element simulationthe AISI 52100 steel is listed in Table 2 and the key physical andmechanical properties of the material are given in Appendix A3.3. Initial and boundary conditionsThe initial temperature considered for the workpiece is theroom temperature i.e. T(t¼0)¼20 1C.On lateral faces, heat flux is imposed as linear convectivetransfer law:qconv¼hconvðTC0T0Þð26Þwhere T and T0are the temperature of the semi-infinite solid andthe ambient temperature, respectively, and hconv(W/K m2) is theconvective heat transfer coefficient of the cooling media. Heat lossfrom the bottom surface was assumed to be zero i.e. q¼0. Thethermal boundary conditions are schematically shown in Fig. 6.lemS.M. Shah et al. / Finite Elements in Analysis and Design 61 (2012) 1–11 50.1m (200 eThe principle of numerical simulation of grinding entails theknow-how of a comprehensive database with reference to geo-metry, thermo-mechanical properties, initial conditions, bound-ary and loading conditions. A brief description of the mainfeatures of the FE model is given below.3.1. Finite element meshThe mesh density is generally defined by the applied loadingand/or boundary conditions. Since grinding processes involvehigh temperature gradient in and near the grinding zone, a veryfine mesh is required to capture the temperature distribution inthe contact area. As the temperature gradient becomes low faraway from the grinding zone, a relatively coarser mesh is theresufficient for the analysis.In this study the workpiece is considered as a 2D semi-infiniteplate of 0.1 m length and 0.03 m width. The finite element (FE)mesh (Fig. 5) consists of CPE4T (4-node plane strain thermallycoupled quadrilateral bilinear displacement and temperature)type elements totaling over 3216 nodes and 3000 elements withthe smallest element in the mesh as [(5C210C04)(8C210C05)] m2.3.2. Material: bearing steel AISI 52100AISI 52100 (also known as 100Cr6 in Europe) is a high carbon–chrome–manganese alloy steel which finds its applications inseveral rotating parts like anti-friction bearings, cams, crank shaft,etc. for its good resistance to corrosion and fatigue [27]. Com-pared to low-carbon steels, high-carbon steels can carry highercontact stresses, such as those encountered in point contactloading in rolling bearings [28]. The chemical composition ofFig. 5. Finite element3.4. Imposed heat sourcesThe thermal loading consists in applying a surface heat fluxthrough a moving heat source. Jaeger [29] and Carslaw and Jaeger[5] have presented solutions for uniform moving rectangular heatsources and a uniform stationary heat source using the heatsource method. The temperature distribution in a sliding contactwas then estimated by several authors based on Jaeger’s theory[30–34]. There are differing views among researchers on whichdistribution of heat flux is best to use for grinding. Some [7,35,36]have used a rectangular (uniform) distribution, so as to simplifysubsequent calculations. However, due to the localized ‘‘spike’’temperatures during a very short time, others [4,37,38] haveargued that the assumption of a uniform heat flux field may notlead to accurate predictions. Keeping in view the contact origin ofthe heat source, theoretically the pressure and correspondingheat flux distribution – if one assumes a uniform friction coeffi-cient in the contact area – should be modeled according to asliding/rolling contact approach. Also by recalling that the Hertzcontact pressure distribution between a cylinder and a plane iselliptical in shape, it seems reasonable to assume an ellipticaldistribution of the heat flux. Fig. 6 shows the presence of aschematic heat source moving with velocity Vwon the top surfaceof the FE model. Here, the length of the heat source is equal to thecontact length (2a¼Lc) between the grinding wheel and theworkpiece. The heat flux distribution entering the work piece istherefore given byqxðÞ¼2Qpaﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1C0x2a2C18C19sð27Þwhere Q is the total heat per unit length (in W/m).ents) mesh.Fig. 7. Temperature distribution at the surface of the workpiece for different heatsource shapes.0200400600800100012000Distance 2x/LcTemperature [°C]2y/Lc=02y/Lc=0.1582y/Lc=0.3422y/Lc=0.5562y/Lc=1.092y/Lc=1.42Q = 400 W/mLc = 0.003mH = 0.01Pe = 1.0t΄=33.61xyNodal TemperatureAc1Ac35 10152025Fig. 8. Temperature profile along the x-axis at various depths.S.M. Shah et al. / Finite Elements in Analysis and Design 61 (2012) 1–116In Abaquss/Standard, the moving heat source is integrated inthe finite element model through a FORTRAN subroutine, calledDFLUX. The heat source is moving along the horizontal (x-) axis.Before carrying out a complete thermo-mechanical analysis, afew heat-transfer simulations were run to perform a sensitivityanalysis of various types of heat source distributions. An elliptical,a triangular, and a uniform heat source were used. The resultingtemperature distributions can be compared in Fig. 7. It was foundthat the peak surface temperatures in all three cases are veryclose, however, the distribution of temperatures over the surfacevaries to some extent. An interesting observation was that theflux distribution from elliptical source lies almost midwaybetween the triangular and the uniform heat sources.3.5. Implementation of the mechanical behavior of a multiphasematerialAs shown in Fig. 3, the mechanical behavior depends on thephase proportion but this coupling is not directly available withInsulation (q=0)WORKPIECE Fig. 6. Thermal loading and boundary conditions.Table 2Chemical composition of AISI 52100 (100Cr6) bearing steel.Element C Si MnMass (%) 0.95–1.10 0.15–0.35 0.20–0.40Convective heat transfer (q )Heat source VAbaquss/Standard. Therefore a UMAT subroutine has been devel-oped to go beyond this difficulty. The UMAT subroutine calls threeother subroutines: PHASE, PROP and UEXPAN. For a given tem-perature field, PHASE compute the austenitic phase proportionformed during heating, and the martensitic phase proportionformed during cooling (see Section 2.3). Knowing the phaseproportion, UEXPAN subroutine gives the expansion coefficientfor the mixed material at a given temperature (see Eqs. (15) and(16)). Finally, PROP subroutine compute the material properties ata given temperature based on the fraction of phases. A linearmixture rule was used for the identification of the multiphasematerial properties.4. Results and discussion4.1. Temperature distributionThe temperature variation in time (or profile along thehorizontal axis) calculated at a given time instance (t0¼33.61)and at various dimensionless depth (y/a¼2y/Lc) is shown in Fig. 8for a specific set of parameters Q, Lc, H and Pe. From the values ofpeak temperatures, it is found that up to a certain dimensionlessdepth (here 2y/Lc¼0.30) the temperature goes beyond Ac1andS Cr Mo Pr0.025 1.35–1.60 r0.10 r0.03003006009001200-20Distance related to start of Flux [mm]Max. Temperature [°C] Elliptical Heat distributionTriangualr Heat DistributionUniform Heat DistributionQ=400W/m Lc=0.003m Pe=1.0-15 -10 -5 0 5 10Ac3(750 1C and 815 1C [39] for AISI 52100, respectively). Duringcooling the temperature at these points will quickly drops belowMs(250 1C [39]). It means that at high cooling the transformationof austenite to martensite will occur at the top most surface.In Fig. 9 the maximum surface temperature as a function of thePeclet number is plotted for a given set of grinding parameters Qand Lcas specified. It is shown that the peak temperaturedecreases when increasing the Peclet number. A comparison withthe analytical solution of Blok [34] is also provided and a goodagreement is found, which validates the numerical model. Theeffect of the variation of the dimensionless heat transfer coeffi-cient, H, or the contact length, Lc, is illustrated in Fig. 10. It can beobserved that an increase of the heat transfer coefficientdecreases the maximum temperature as an increase in thecontact ar