Simulation of Hot Forming Processes of Refractory Metals using Porous Metal Plasticity Models

Transcription

1 Simulation of Hot Forming Procee of Refractory Metal uing Porou Metal Platicity Model Erik Parteder *, Hermann Riedel +, Dong-Zi Sun + *Planee AG, Reutte, Tyrol, Autria + Fraunhofer Intitute for Mechanic of Material, Freiburg, Germany Summary In thi work two model for predicting the denification behavior of intered refractory metal during hot working operation are preented. It i known from experiment and cell model calculation that the pore hape change ha a ignificant influence on the denification behavior. Therefore thi effect hould be included in a continuum contitutive decription. The firt model preented i a phenomenological extenion of the Guron model, the econd one i the model of Gologanu, Leblond & Devaux, which wa implemented a a uer material model into the Finite-Element-code ABAQUS. The numerical reult are compared with the denity ditribution of a tapered dik made of pure molybdenum after the hot forming operation. Keyword Porou Metal Platicity, FE-Simulation, Hot Forming 1. Introduction Hot working of refractory metal after the intering procedure aim at achieving the deired hape and mechanical propertie and to eliminate the poroity. Therefore undertanding of the denification behavior of the porou metal during the hot forming operation i of great importance for proce deign and proce optimization. The preence of pore in intered part require that preure-dependent, dilatant contitutive law are ued to model hot forming operation on thee material. The poroity (or the relative denity), i treated a an internal variable.

2 In the pat a large number of phenomenological a well a micromechanic baed contitutive law for porou material were developed. The firt phenomenological yield function for porou media (Kuhn & Downey 1971, Shima & Oyane 1976) aume iotropic repone at yielding even for finite platic deformation. The only microtructural feature i the relative denity. Some model originate from the continuum damage theory of ductile material, the well-known Guron model (Guron 1977), extended by Tvergaard (1981), being one of them. Guron model aume that the pore hape i pherical at the beginning a well a throughout platic deformation, and therefore the mechanical behavior remain alo iotropic. The material behavior of the matrix i aumed to be perfectly platic. Iotropic contitutive law for linear and nonlinear vicou material were alo developed and applied to hot iotatic preing imulation (Duva & Hutchinon 1984). More recently, Gologanu et al. (1993, 1994) generalized the Guron model to include the evolution of the pore hape during platic deformation. The pore apect ratio i an internal variable, and the tre-train repone become aniotropic a the pore develop prolate or oblate hape. An alternative approach baed on a contitutive theory for particulate compoite wa propoed by Ponte Catañeda & Zaidman (1994). In the preent work two approache of continuum-model decribing the denification behavior of intered refractory metal are preented. The firt one i a modification of the Guron model (referred a model 1), by introducing evolution equation for the parameter q 1 and q 2 propoed by Tvergaard, with the retriction that the influence of the pore hape change on denification i incorporated in a phenomenological manner (the model remain iotropic). The parameter of the evolution equation for q 1 and q 2, are fitted to experimental reult (uniaxial tre tate) and axiymmetric unit cell calculation (tre tate with arbitrary triaxialitie). The advantage of thi model i that different microtructural feature which characterize the material (like the local ditribution of the pore within the matrix or the effect of different pore ize) can be taken into account in an overall manner. The diadvantage i that the model i applicable trictly only for proportional loading condition (no change of triaxiality during platic training). A a econd approach, we have implemented the model of Gologanu et al. (1993, 1994) a a uer defined material model in the FE-code ABAQUS (referred a model 2) (Andrieux et al. 2000). The advantage of thi model i that it decribe the evolution of the pore hape and of the aniotropic material repone conitently, baed on reliable model for void growth and void deformation. Hence a much wider range of validity i expected than for model 1. For example, data meaured in uniaxial

3 compreion can be applied with greater confidence to forming operation where the tre triaxiality may be much higher. In it preent form the model neglect poible rotation of the principal axe of the pore. 2. Experiment Green compact with different green denitie were produced from pure molybdenum powder by applying different compaction preure. All ample were intered in one batch. The ame temperature hitory implie that the grain ize i the ame in all ample, and only the relative denity varie due to the different denitie of the green part. Thi mean, that the mechanical behavior after intering depend on the denity only. The intered ample were teted in uniaxial compreion at a temperature of 1000 C. One et of ample (diam. 5 mm and height 6 mm) were deformed up to a platic train of e»1 to evaluate the tre-train-behavior. The relative denitie of thi et range from D 0 = to The ample of the econd et (diam. 12 mm and height 14.4 mm) with initial relative denitie of D 0 = 092., 0.94 and 0.96 are compreed to different height, in order to meaure the relative denity after a certain amount of platic train with the Archimede method. Detail are decribed in Parteder et al. (1999). Additional compreion tet on tapered dik for verification of both model are decribed in ection The Guron model and it modification Guron yield function wa derived from an upper bound olution of the velocity field of a hollow phere or a hell with an incompreible hell material. The flow potential F ha the form 2 e F 3q2h F= + H G I 2 2q coh K J q3b1 Dg, (1) M 2 M where M denote the matrix flow tre, D i the relative denity defined a r r M, where r i the denity of the porou metal and r M i the denity of the matrix material, I e i the equivalent tre of the porou continuum and h i the hydrotatic tre. q 1, q 2 and q 3 are the parameter propoed by Tvergaard. He uggeted value of q 1 = 15., q 2 = 1 and q3 = q1 2. For all further calculation we alo ue the relation q = q. Note that for q = q = q = the

4 original Guron model i recovered and that for D = 1, which mean that the material i fully dene, the Mie yield condition i gained. The contitutive model i completed by the argument that normality of the matrix material implie normality of the porou continuum (Berg 1970) d e ij = d F l ij. (2) 3.1. Uniaxial tre tate A a firt tep the uniaxial tre tate i invetigated. The modification of Guron model i attained by conidering the parameter q 1 and q 2 a internal variable repreenting the influence of the pore hape in a manner which i not explicitly pecified. The evolution of q 1 and q 2 i linked to the equivalent platic train of the porou metal and the following equation are propoed q q F HG 1 A e = + e1 I K J n 1, q q F HG 1 A e = + e2 I K J n 2, (3) where A e i the equivalent platic train of the porou material, q 1 0 and q 2 0 are the initial parameter for zero platic train, e 1, e 2, n 1 and n 2 are contant which will be determined by a bet fit with the experimental reult, ee table 1 below. For uniaxial compreion the yield condition, F=0, with F from equation 1 give the yield tre y of the porou metal y = M 21 q b b 4 qq D + - D g g. (4) The evolution of the denity i related to the volumetric train increment, which i obtained from equation 2. For uniaxial compreion it follow dd =- 3Dqq qq b D D g de, (5) where e i the component of the platic train along the compreive axi. Equation 5, together with equation 3 i integrated numerically uing computer algebra. Thi give relation for the yield tre y = y( D, e) and for the evolution of the denity D = D() e. Figure 1 and 2 how the reult of the

5 yield tre σ y (N/mm 2 ) 300 ε = ε = 0.20 ε = 0.10 ε = 0.05 ε = 0.01 experiment model 1 0,90 0,92 0,94 0,96 0,98 1,00 relative denity D Figure 1. Yield tre of the porou metal veru relative denity for different level of platic train; experiment (uniaxial compreion) compared with the modified Guron model (model 1). 1,00 relative denity D 0,98 0,96 D 0 = D 0 = D 0 = ,94 experiment model 1 0,92 0,0 0,2 0,4 0,6 0,8 1,0 platic train ε Figure 2. Denity evolution in uniaxial compreion for different initial denitie; experiment compared with the modified Guron model (model 1).

6 integration adjuted to the experimental reult, and table 1 how the value of the parameter reulting from the bet fit. 0 q 1 = 466. e 1 = 0.1 n 1 = q 2 = 045. e 2 = 02. n 2 = 125. Table 1. Parameter of equation 3 for the uniaxial tre tate Arbitrary compreive tre tate For arbitrary compreive tree experimental data are hard to obtain. Intead, micromechanical cell model calculation are a uitable method to get reaonable reult for characterizing the denification behavior of the porou metal. In our cae imple axiymmetric cell model with ymmetric boundary condition were ued (Parteder 2000). Figure 3 how the implification of a porou olid, auming a regular cubic pore arrangement and the approximation of a hexaedron with a cylinder. For a given triaxiality, which i defined a X =h e, the ratio of the axial load z and the radial component r can be calculated a r z z r 3X + 1 = 3X - 2 3X + 2 = 1-3X " phere to crack", " phere to needle". Note that for axiymmetric tre tate we have to ditinct between the evolution of oblate and prolate pore hape. In the firt cae the full denified tate i a penny haped crack, tarting from a pherical pore, the evolution i reffered a phere to crack. The prolate pore hape evolve from phere to needle. The denification characteritic of the cell can be computed from the diplacement vector of the edge node of the cell. In order to fit the modified Guron model to thee cell model calculation the parameter q 1 and q 2 were aumed to depend on the tre triaxiality in addition to the dependence on the platic train pecified in equation 3 q1 = q1ba e, Xg, q2 q2b = A, Xg. (7) e (6)

7 Figure 3. Sketch of regular pore arrangement and the extraction of a axiymmetric unit cell. The functional form of equation 7 i given in Parteder (2000). The above relation i only valid for proportional loading condition, becaue it i aumed that the change of the triaxiality during loading will influence the pore hape evolution and therefore the denity evolution. Figure 5 below how the denity evolution for variou triaxialitie a predicted by cell model, by the modified Guron model and by the Gologanu model. 4. Gologanu model and it implementation a a uer defined material model The model of Gologanu et al. i a generalization of the Guron model. In addition to the relative denity D, which i the only tate variable in the flow potential of the Guron model, the model of Gologanu et al. contain a pore hape variable S a a econd tate variable. If a and b are the emi-axe of a pheroidal pore (a i the emi-axi in axial direction, b i the emi-axi in radial direction), S i defined a S = ln( a b). Prolate pore have S > 0, oblate pore are characterized by S < 0, and pherical pore by S = 0. The material model exhibit an evolving aniotropy depending on the orientation of the axe of the pore and on the value of S. The functional form of the flow potential i given a *2 e F k F= H G I * h 2 * C q g g D coh K J - g + - q b 1gb 1 g b 1g bg + 1-Dg. (8) 2 1 M M

8 The parameter C, k, and g are dependent on the pore hape parameter S and the relative denity (g = 0 for prolate and pherical pore). e * i a modified equivalent tre, dependent on the internal variable. The Tvergaard parameter q 1 * in thi cae i alo dependent on the the pore hape. The evolution of the pore hape i given by the relation ' S = Saz + Sb h e e, (9) where the parameter S a and S b are function on the tre triaxiality and the pore hape. e ' z i the deviatoric part of the platic train rate in direction of the axi a (axial or z-direction) and e h i the volumetric train rate. The model of Gologanu et al. wa coded in Fortran for arbitrary uniform training hitorie (with no patial dependence) and a a uer ubroutine for the FE-code ABAQUS/Explicit. Special care mut be expended for pherical pore (when the pore hape parameter of the model i S = 0 and expreion of the form - occur), and for crack- or needle-like pore (when S =± ). Euler forward integration i ued throughout, including the yield condition, which i treated a df =0. Concern that the tre tate could drift away from the yield urface, if only the incremental form of the yield condition i ued, proved to be unneceary. With the mall time tep that are choen by an explicit code the deviation from the yield urface uually remained maller than 1% and i probably not larger than other inevitable inaccuracie of the time integration cheme. Figure 4 how the yield tre of the porou metal in dependence on the relative denity for different level of platic train. The reult for the modified Guron model (model 1) and the Gologanu model (model 2) are compared with the original Guron model (model 1 give the bet coincidence with the experimental reult). One can ee the increaing drop of the yield tre predicted by model 2 due to the evolving aniotropy. Figure 5 how the evolution of the denity for two different tre triaxialitie. For the uniaxial tre tate all three model are in good agreement, except during the lat tage of denification (during thi tage model 1 - which repreent the experimental reult - how a lower denification rate than the axiymmetric cell model and the Gologanu model). For the triaxiality X = -43 the Gologanu model give the highet denification rate.

9 yield tre σ y (N/mm 2 ) ε = 0.10 ε = 0.01 ε = 0.30 Guron-original model 1 model 2 0,92 0,94 0,96 0,98 1,00 relative denity D Figure 4. Yield tre of the porou metal veru relative denity for different level of platic train; original Guron model (with q1 = q2 = 1) compared with the modified Guron model (model 1) and the Gologanu model (model 2) relative denity D 0.96 X = -1/ X = -4/ axiymmetric unit cell 0.90 model 1 model equivalent platic train ε e Figure 5. Denity evolution for different triaxialitie; comparion between axiymmetric unit cell calculation, the modified Guron model (model 1) and the Gologanu model (model 2).

10 5. Tapered dik compreion 5.1. Forging experiment In order to ubject the two model preented in thi paper to a critical tet, tapered dik compreion experiment were carried out. The initial pecimen geometry i hown in figure 6. Thi tet configuration provide a wide range of tre tate: in the center of the dik high hydrotatic compreive tree reult in a rapid local denification, wherea tenile tree at the outer part lead to a decreae in relative denity. Two dik of intered molybdenum (initial denity D 0 = 095. ) were deformed with a crew-pre. The firt troke reduced the pecimen height to h 1 = 50 mm, the econd one to h 2 = 40 mm. The temperature at the beginning of deformation wa T = 1000 C. Sample were cut from each dik after deformation to determine the local denity ditribution with the Archimede method. The finite element meh i depicted in figure 7 together with the contour plot of the experimental and numerical reult for the denity ditribution Numerical reult An iothermal imulation wa carried out uing axiymmetric condition with 4 node linear element. The Coulomb friction factor wa taken to be 0.05 between the dik and the rigid die. Thi value wa obtained from ring upetting tet. The flow curve for variou temperature and train rate were alo obtained experimentally (Parteder 2000). The parameter of model 1 (equation 7) were defined by the option USER DEFINED FIELD in the ABAQUS input deck, for model 2 a UMAT wa developed, a decribed above. In figure 7 the experimental reult can be compared with the prediction of the two model and the original Guron model (with contant parameter q 1 = 15. and q 2 = 1). On the left column the reult for h 1 = 50 mm (firt troke) are hown, on the right ide reult for h 2 = 40 mm (econd troke) are depicted. A comparion of the experimental reult and the three model how, that the modified Guron model with parameter adjuted to reult of hot upetting tet and cell model calculation (model 1) give the bet conitency with the experiment. Epecially in the center of the ample the predicted value are quite reaonable (the relative denity in the center obtained from the experiment i» %, which i alo predicted with model 1). Thi i not urpriing ince the triaxiality in the center of the ample

11 i about X»-0. 4, which i cloe to uniaxial compreion for which the parameter were fitted. Figure 6. Sketch of the tapered dik ample with finite element meh. Alo the Gologanu model (model 2) give denity ditribution that are generally in good agreement with the experiment. However, the model predict that the material i completely denified in the center of the ample. Thi overetimation can be explained looking at figure 5, where the denification in the lat tage i more rapid compared with the experimental value. Contrary to the modified Guron model the Gologanu model contain no adjutable parameter (except the hardening curve of the bae material). On the other hand, the original Guron model give denity ditribution that are far le accurate. Another comparion can be made with the model of Ponte Catañeda (Kailaam et al. 2000). In thi cae the ame tapered dik ample wa ued and the firt troke wa imulated, however with an initial denity of D 0 = In the center of the ample the ame tendency a with Gologanu model can be een: the final denity i overetimated compared to the experimental reult. In figure 8 the pore hape parameter S i plotted for the firt troke calculated with model 2 together with micrograph on four pecific poition. In the center (point 1) a value of S =-673. i predicted, which mean a very low pore apect ratio and therefore an almot fully dene material. The micrograph

12 relative denity -INFINITY +9.41E E E E E E E E+00 relative denity -INFINITY +9.37E E E E E E E E+00 A1 A2 B1 B2 C1 C2 D1 D2 Figure 7. Denity ditribution for the firt troke (left ide: A1 - D1) and for the econd troke (right ide: A2 - D2). Comparion of the meaured value (A1 and A2), the modified Guron model (model 1: B1 and B2), the Gologanu model (model 2: C1 and C2) and the original Guron model (with q 1 = 15. and q 2 = 1) (D1 and D2). at thi poition how little poroity, correponding to the relative denity in figure 7, A1 (experimental value). The apect ratio calculated at the outer ide (point 2) i approximately 1 and i in good accordance to the micrographical

13 obervation. At point 3 a value of 0.51 and 0.59 at point 4 rep. i predicted. Alo thee value are in good coincidation with the micrograph. SDV2 VALUE -6.73E E E E E E E E E μm Figure 8. Pore hape parameter S for the firt troke calculated with the Gologanu model (model 2), together with four optical micrograph. 6. Concluion The accuracy of predicted denity ditribution during forming procee of porou metal with cloed poroity (up to level of 10%) can be increaed ignificantly by incorporating the pore hape evolution within the continuum model. Two poibilitie were preented in thi paper, the firt one wa a phenomenological modification of the Guron model, the econd one wa the model of Gologanu et al. Both model lead to much better reult for the

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