MULTISCALE MODELING OF EQUAL CHANNEL ANGULAR EXTRUDED ALUMINIUM WITH STRAIN GRADIENT CRYSTAL PLASTICITY AND PHENOMENOLOGICAL MODELS L. Duchêne, M.G.D. Geer, W.A.M. Brekelman, E. Chen, B. Verlinden and A.M. Haraken ARGENCO Deartment, Univerity of Liège, Belgium Det. of Mechanical Engineering, Eindhoven Univerity of Technology, The Netherland Det. of Material Engineering, Katholieke Univeriteit Leuven, Belgium ABSTRACT The Equal Channel Angular Extruion roce i ued to modify the microtructure of an AA15 aluminum alloy in order to roduce an ultra fine grained material. Due to the evere latic deformation undergone y the material during the ECAE roce, the uequent ehavior of the material i non-conventional and difficult to model with claical contitutive law (e.g. ECAE aluminum reent a large initial ack-tre which mut e adequately incororated in the model). In thi tudy, the evolution of the ack-tre during the ECAE roce i analyzed. Two different numerical model were invetigated in thi reect. The firt one i a ingle crytal train gradient laticity model aed on dilocation denitie. The econd model i the Teodoiu and Hu hardening model, which i a microtructuraly aed henomenological model at the macrocale. The reult rovided y the two model are oviouly ditinct. Neverthele, ome common trend can e ointed out, among which the amlitude of the ack-tre that i imilar. In agreement with the cyclic deformation mode of the tudied route C ECAE roce, the evolution of the redicted ack-tre i alo cyclic in oth model. INTRODUCTION The Equal Channel Angular Extruion (ECAE) roce i ued in the reent tudy to roduce ultra fine grained aluminum. It i well-known that a decreae in the grain ize of a material i accomanied y an increae of the yield trength a can e rereented y the Hall- Petch relation [1]. Due to their articular mechanical roertie, an increaing interet i currently dedicated to the tudy of ultra fine grained material. However, the modeling of thee material i not traightforward. The material deformed y the ECAE roce i far from it virgin tate. Very large latic train (in the tudied cae, the latic equivalent train i 1.15 er a) are imoed to the material. In the framework of thi tudy, everal mechanical tet were erformed on the aluminum roduced y ECAE in order to ae it mechanical ehavior [2]. It aeared that a ignificant kinematic hardening wa oerved. Furthermore, the ECAE aluminum reented an initial acktre reulting from the deformation that occurred during the ECAE roce. Thi contriution aee the erformance of two different numerical model for an accurate modeling of ECAE roceed aluminum. A ingle crytal train gradient laticity model wa crutinized in thi reect. Thi model ue a internal variale the denitie of tatitically tored dilocation (SSD) and geometrically neceary dilocation (GND). The evolution of the SSD denitie i aed on a alance etween dilocation accumulation and annihilation rate deending on the li rate. The GND denitie on the other hand reult from the incomatiilitie in the crytal lattice due to gradient of the dilocation li. Both GND and
SSD denitie are taken into account for the iotroic hardening of the material. The GND denitie naturally induce a hyically aed kinematic hardening through their internal tree (i.e. the ack-tre, comuted a a function of the GND denitie gradient). In addition, a macrocoic henomenological hardening model wa invetigated. The Teodoiu and Hu' hardening model [3;4] i a hyically-aed microtructural model. Baically, it i ale to decrie oth kinematic and iotroic hardening, reflecting the influence of the dilocation tructure and their evolution, at a macrocoic cale. It ermit to decrie comlex hardening ehavior induced y train-ath change. The goal of thi reearch are multile. Firt, different numerical model were ued to redict the evolution of the ack-tre during an ECAE roce. Thi yield new inight and more accurate material arameter a ued in henomenological model adated to ECAE material (mainly a reaonale initial ack-tre). The modeling of the ECAE roce alo reult in an imroved knowledge of the mechanim involved during thi forming roce. Finally, thi reearch ermit the comarion of two romiing yet different model with very different aroache on a relevant alication. THE MODELS Strain gradient crytal laticity model The contitutive framework of thi model deart from the claical multilicative decomoition of the deformation gradient F into a latic art F and an elatic art F : e F= F F (1) F i aumed to e achieved only y dilocation li. include mall lattice deformation and (oily) large rigid ody rotation. The elatic ehavior of the material i determined y equation (2), where S i the econd Piola-Kirchhoff tre tenor exreed in the tre-free intermediate configuration (the material deformed y only), 4 C i the fourth order aniotroic elaticity tenor (it i defined y three material arameter, which are three of it comonent: C 11,C 12,C 44 ). i the Green-Lagrange elatic train (work-conjugated to S ), otained y 1 Ee = Fe Fe I 2 E e T ( ). S= 4 C:E The latic deformation i due to dilocation glide on li ytem. Therefore, the latic velocity gradient tenor (related to the latic art of the deformation gradient y F = L F ) i exreed in the form: e F 12 L = γ P (3) = 1 In equation (3), γ i the li rate on li ytem, the non-ymmetric Schmid tenor are defined y P = n, with and n the li direction and the li lane normal ( ) ( ) ( ) ij i j for the li ytem (exreed in the undeformed configuration). A vico-latic li law relate the li rate to the effective tree τ : F e eff e (2)
1 m τ G τ kt eff eff γ = γ ex 1 ign ( τeff ) γ, m, G are material arameter; k i the Boltzmann contant and T i the aolute temerature. In equation (4), the exonential term rereent the thermally induced dilocation motion. The effective tree τ are otained from equation (5), where τ are the Schmid eff tree defined a the reolved art of the econd P-K tre on the li ytem τ =S:P. eff τ =τ τ (5) τ i the ack-tre related to the kinematic hardening included in thi model. It i the reolved art of the internal tree induced y the edge and crew dilocation denitie reent on all li ytem according to equation (6). int int ( ) τ = + σe σ :P (6) The reence of dilocation in the crytal create a deformation of the lattice. Auming (at thi tage only) that the material i an iotroic elatic continuum medium, the internal tree reulting from ditriution of dilocation denitie take the form of equation (7) for edge dilocation and equation (8) for crew dilocation. σ int σ int e 2 12 GRe = ρ + + ν 81 ν ( ) = 13 = 1 ( 3n n n nnn 4 n) GND 2 18 GR = ρ + + 4 ( n n n n ) GND = n (4) (7), with (8) In equation (7) and (8), G i the (iotroic) hear modulu, ν i Poion ratio, i the length of the Burger vector and Re and R are the radii of the fictitiou herical domain (around the oint where the internal tree are comuted) limiting the dilocation taken into conideration. It hould e noted at thi tage that the dilocation denitie are earated in two categorie: the geometrically neceary dilocation (GND) and the tatitically tored dilocation (SSD). The GND are generated during the latic deformation y dilocation glide in order to maintain the comatiility of the crytal lattice. They have a ecific orientation (deending on the li on each li ytem) and, they therefore have a net contriution to the internal tree. On the other hand, the SSD have a random orientation uch that their influence on the internal tree vanihe. It aear in equation (7) and (8) that the gradient of the GND denitie ρgnd i the governing quantity for the internal tree. The reent crytal laticity model conider face centered cuic metal having 12 li ytem and, conequently, 12 edge dilocation and 6 crew dilocation familie are conidered. In equation (4), the li reitance i till not defined. It i related to the local li ytem hardening. The li reitance evolve a a function of GND and SSD denitie through equation (9). 12 18 = ρ SSD + = 1 = 1 cg A A ρ (9) GND
c i a material contant and A are interaction arameter etween li ytem and dilocation. The evolution rule for the SSD denitie reult from a cometition etween an accumulation rate and an annihilation rate: with L 1 1 ρ SSD = 2y cρssd γ L = K 12 18 H ρ SSD + H ρgnd = 1 = 1 and an initial condition: SSD ( t ) ( SSD ) In equation (1) to (12), c (1) (11) ρ = = ρ (12) y, K and ( SSD ) ρ are material arameter; H i an interaction matrix (imilar to A ) and t i the time. The GND denitie evolution i aed on the geometrical comatiility of the crytal lattice during latic deformation. The GND are introduced to maintain the lattice continuity in the crytal in ite of the dilocation li. The evolution rule are: 1 ρ GND = ( ρgnd ) γ 1 1 2 2 ρ GND = ( ρ GND ) + ( γ + γ ) 1 for edge GND denitie (=1 12) and crew GND denitie (=13 18), reectively. Further detail aout the train gradient crytal laticity model can e found in [5-8]. Teodoiu and Hu hardening model Teodoiu and Hu hardening model i decried y 13 material arameter: C,C,C,C,C,f,n,n,r,Y,R,S,X R Sd SL X L P Sat at (15) and deend on four tate variale: PSX,,,R Variale P i a econd-order tenor that deict the olarity of the eritent dilocation tructure (PDS in [9]) and S i a fourth-order tenor that decrie the directional trength of the PDS. Scalar R rereent the iotroic hardening due to randomly ditriuted dilocation and the econd-order tenor X i the ack-tre. Thee tate variale evolve with reect to the latic train rate ε and the equivalent latic train rate a given y Y ( ε ) (13) (14) (16) Y = f Y, (17) A recie decrition of thee evolution equation can e found in [3;1-11]. It hould however e noticed that the fourth-order tenor S mut e decomoed into S D and S L according to Equation (18). S D i the trength of the dilocation tructure aociated with the currently
active li ytem and S L i the latent art of S related to the eritent dilocation tructure aociated with the latent li ytem. where S = N N + S (18) SD ε ε L N ε i the latic train rate direction. Two ditinct evolution equation (with the form of Equation (17)) are alied to S D and S L. The yield condition i given y Equation (19). σ = σ = Y + R+ f S (19) y where σ i the equivalent tre, function of dev( σ) X, σ y i the current elatic limit, Y i the initial ize of the yield locu and R+ f S rereent the evolution of the iotroic hardening. The exreion of σ deend on the definition of the yield locu. MODELING THE ECAE PROCESS The ECAE roce i chematically rereented in Figure 1. In thi tudy, the material (one material oint i illutrated y the quare in Figure 1) wa aumed to e umitted to imle hear in the interection lane of the two channel. In thi reect, the numerical imulation were defined in a reference ytem attached to that lane. The material wa umitted to the ECAE roce following route C (18 rotation of the amle etween each a) for 4 ae. Thi route can e modeled y imle hear (with a hear train ranging from to 2 for the tudied ECAE geometry) for the firt a and revered hear for the econd a. The initial hae of the material i recovered every 2 ae. For the train gradient crytal laticity law, the orientation of the individual grain are required a inut data. The texture of the aluminum efore the ECAE roce wa ued to extract a et of 8 rereentative orientation. The grain with thee orientation were umitted to imle hear earately (Taylor analyi). The material arameter for the train gradient crytal ρ laticity model were largely extracted from [6] and [12]. Only the initial SSD denitie ( SSD ) were adjuted according to the actual hardening ehavior of the tudied aluminum. A value of 4 µm -2 wa adoted. For the Teodoiu and Hu model, the material arameter for the aluminum were otained from [13]. RD Figure 1. Schematic rereentation of the ECAE roce (RD i the rolling direction of the initial material). Y X
NUMERICAL RESULTS Figure 2 reent the evolution of the hear tre a a function of the hear train with the train gradient crytal laticity model during the ECAE roce following route C for 4 ae. The Bauchinger effect can e oerved at the load reveral etween ucceive ae. The kinematic hardening (linked to the ack-tre) i at the origin of thi effect in the model. Shear tre (MPa) 3 2 Pa n 3 1 Pa n 1-1 Pa n 2-2 Pa n 4-3.2.4.6.8 1 1.2 1.4 1.6 1.8 2 Shear train Figure 2. Shear tre hear train during ECAE roce for 4 ae (route C). The evolution of the ack-tre during the 4 ae i lotted in Figure 3. The ack-tre at the centre of each grain i next conidered in more detail. The mean value over the 8 grain of the rereentative et i hown. The maximum value for the ack-tre were otained after the firt and the third ae. Lower value were noticed after 2 and 4 ae. Thi reult i confirmed y the amlitude of the Bauchinger effect etween the ucceive ae in Figure 2. Due to the reetitive aect of the route C, it wa exected (and confirmed) that the material ehavior during the firt and the econd ae wa imilar to the one during the third and fourth ae. By comaring Figure 2 and 3, it aear that the amlitude of the ack-tre i a fraction (around a fifth) of the hear tre. Figure 4 reent the evolution of the ack-tre calculated with the Teodoiu and Hu hardening model. The overall amlitude of the ack-tre a redicted y Teodoiu and Hu model i imilar to the amlitude with the crytal laticity model. On the other hand, a very different evolution of each comonent wa oerved, i.e. the orientation of the ack-tre i very different. Again, a reetition (more trict in thi cae) of the two firt ae aeared during the two lat ae. Beide, an arut evolution of the ack-tre wa oerved during the firt tage of the revered hearing (eginning of ae 2 and 4). At the end of ae 2 and 4, only the hear comonent ( 12 ) of the ack-tre remain. CONCLUSIONS During thi tudy, two different model (microcoic and macrocoic) were invetigated for the numerical rediction of the ack-tre (kinematic hardening) during the ECAE roce following route C for 4 ae. The two model yield different reult in term of the evolution of the ack-tre while a imilar amlitude wa oerved.
MPa 5 4 3 2 X11 X22 X33 X12 X13 X23 1-1 -2.5 1 1.5 2 2.5 3 3.5 4 Pa numer Figure 3. Evolution of the ack-tre (X) during the ECAE roce for 4 ae (route C) with the train gradient crytal laticity model. The comonent of X are exreed in the reference frame of Fig. 1. MPa 4 3 2 1 X11 X22 X33 X12 X13 X23-1 -2-3 -4.5 1 1.5 2 2.5 3 3.5 4 Pa numer Figure 4. Evolution of the acktre (X) during the ECAE roce for 4 ae (route C) with the Teodoiu and Hu hardening model. The comonent of X are exreed in the reference frame of Fig. 1.
The difference etween the two model can e artly exlained y the microcoic and macrocoic aect of thee model. On the one hand, for the train gradient crytal laticity model, the ack-tre originate from a gradient of the GND denitie inide each grain. Thi gradient i mainly due to the reence of the grain oundarie. On the other hand, a homogeneou ehavior i aumed with the Teodoiu and Hu model. More conitently, oth model redicted a larger overall ack-tre after ae 1 and 3 and a lower value after 2 and 4 ae. Thi oervation hould e checked exerimentally. ACKNOWLEDGEMENTS The author acknowledge the Interuniverity Attraction Pole Programme - Belgian State Belgian Science Policy (Contract P6/24). A.M.H. and L.D. alo acknowledge the Belgian Fund for Scientific Reearch FRS-FNRS for it uort. REFERENCES 1. Hall, E.O., The deformation and aging of mild teel: iii. Dicuion of reult. Proc. Phyical Society of London, B64, 747-753 (1951). 2. Poortman, S. and Verlinden, B., Mechanical roertie of fine-grained AA15 after ECAP. Material Science Forum 53-54, 847-852 (26). 3. Teodoiu, C., Hu, Z., Evolution of the intragranular microtructure at moderate and large train: modeling and comutational ignificance. In Proc. NUMIFORM 95, edited y Shen, S.F. and Dawon, P.R.,. 173-182 (1995). 4. Bouvier, S., Teodoiu, C., Haddadi, H. and Taacaru, V., Aniotroic Work-Hardening Behaviour of Structural Steel and Aluminium Alloy at Large Strain. In: Proc. of the Sixth Euroean Mechanic of Material Conference (EMMC6), S. Cecotto Ed, 329-336 (22). 5. Ever, L.P., Brekelman, W.A.M., Geer, M.G.D., Non-local crytal laticity model with intrinic SSD and GND effect. J. of the Mechanic and Phyic of Solid, 52, 2379-241 (24). 6. Ever, L.P., Brekelman, W.A.M., Geer, M.G.D., Scale deendent crytal laticity framework with dilocation denity and grain oundary effect. International Journal of olid and tructure, 41, 529-523 (24). 7. Bayley, C.J., Brekelman, W.A.M., Geer, M.G.D., A comarion of dilocation induced ack tre formulation in train gradient crytal laticity. International Journal of Solid and Structure, 43, 7268-7286 (26). 8. Bayley, C.J., Brekelman, W.A.M., Geer, M.G.D., A three-dimenional field crytal laticity aroach alied to miniaturized tructure. Philoohical Mag., 87 (8-9), 1361-1378 (27). 9. Li, S., Hoferlin, E., Van Bael, A., Van Houtte, P., Teodoiu, C., Int. J. Plat. 19, 647-674 (23). 1. Bouvier, S., Alve, J.L., Oliveira, M.C., Meneze, L.F., Com. Mater. Sci. 32, 31-315 (25). 11. Alve, J.L., Oliveira, M.C., Meneze, L.F., An advanced contitutive model in heet metal forming imulation: the Teodoiu microtructural model and the Cazacu Barlat yield criterion. In Proc. NUMIFORM, edited y Ghoh, S. et al., AIP Conf. Proc. 712,. 1645-165 (24). 12. Fülö, T., Brekelman, W.A.M., Geer, M.G.D., Size effect from grain tatitic in ultrathin metal heet. Journal of Material Proceing Technology 174, 233-238 (26). 13. Flore, P., Duchêne, L., Bouffioux, C., Lelotte, T., Henrard, C., Pernin, N., Van Bael, A., He, S., Duflou, J., Haraken, A.M., Model Identification and FE Simulation: Effect of Different Yield Loci and Hardening Law in Sheet Forming. Int. J. of Platicity, 23 (3), 42-449 (27).