A generic method for modeling the behavior of. anisotropic metallic materials : application to. recrystallized zirconium alloys

Transcription

1 Submitted to Mehani of Material, 6 A generi method for modeling the behavior of aniotropi metalli material : appliation to rerytallized zironium alloy S. Lelerq,*, G. Rouelier, G. Cailletaud EDF R&D, Département Matériaux et Méanique de Compoant Le Renardière, 7788 Moret ur Loing Cedex, Frane Centre de Matériaux, ENSMP, UMR CNRS 7633, BP 87, 93 Evry, Frane Running title : Modeling of the behavior of aniotropi metalli material * Correponding author. Tel.: ; fax: addre: ylvain.lelerq@edf.fr

2 Abtrat A implified polyrytalline model (the o-alled RL model) i propoed to imulate the aniotropi vioplati behavior of metalli material. A generi method i preented that make it poible to build a implified aniotropi material texture, baed on the prinipal feature of the pole figure. The method i applied to a rerytallied zironium alloy, ued a lad material in the fuel rod of nulear power plant. An important data bae oniting in mehanial tet performed on Ziraloy tube i olleted. Only a mall number of tet (pure tenion, pure hear) are ued to identify the material parameter, and the texture parameter. It i hown that 6 rytallographi orientation (6 grain ) are uffiient to deribe the large aniotropy of uh hp alloy. The identified rytallographi orientation math the experimental pole figure of the material, not ued in the identifiation proedure. Speial attention i paid to the preditive ability of the model, i.e. it ability to imulate orretly experimental tet not belonging to the identifiation data bae. Thee preditive reult are good, thank to an identifiation proedure that enable to onider the ontribution of eah lip ytem in eah rytallographi orientation. Keyword: aniotropi material; ontitutive behaviour; rytal platiity; polyrytalline material; Zironium alloy.

3 . Introdution Sine the pioneering work of Hill (95), everal effort have been made to improve the deription of initial plati aniotropy aording to maroopi ontitutive equation (ee for intane, Gotoh, 977; Barlat et al., 99; Karafilli and Boye, 993; Bron and Beon, 4). Neverthele, thee maroopi model generally fail to aurately repreent non proportional and yli loading (Cailletaud and Pilvin, 994). Epeially, the imulation and predition of the yield urfae ditorion remain diffiult to ahieve with laial phenomenologial model, and more or le ompliated improvement of thee model do not give total atifation (Vinent et al., ; Kuroda and Tvergaard, ). Polyrytalline model generally give a good deription of the plati aniotropy and yield urfae ditorion, beaue they are baed intrinially on lip lattie a well a texture information (Cailletaud, 99; Pilvin 99; Calloh, 997). Neverthele, their main diadvantage i the large omputation time needed for parameter identifiation and finite element alulation. Indeed, thee model are built in uh a way that the number of internal variable to ompute may be a thouand time larger than the number of internal variable uing a phenomenologial model. Thu one undertand the relatively rare ue of polyrytalline model in indutrial alulation. In the field of nulear indutry, important work ha been done on the vioplati behavior of zironium alloy, uing phenomenologial model a well a polyrytalline approahe, both ombined to experimental multiaxial tet 3

4 performed on tube. Thee multiaxial tet (pure tenion, biaxial tenion, teniontorion) are neeary to invetigate the material repone to the omplex loading that are applied to the nulear fuel rod in Preurized Water Reator. It an be een that an aurate deription of the aniotropy and irradiation effet require quite a omplex et of equation in phenomenologial approahe (Robinet, 995; Shaeffler, 997; Rihard et al, 3) while the multi-ale approah allow muh impler development (Lebenohn and Tomé, 993 ; Geyer, 999 ; Onimu, 3). Neverthele, none of thee model i ued at preent time in really indutrial finite element alulation beaue of their omplexity or too large CPU time. In order to ombine the good preditive ability of polyrytalline approahe and the moderate CPU time neeary to ondut afety analye for the nulear indutry, we propoe in the preent paper a implified polyrytalline model (the o-alled RL model - Rouelier and Lelerq, 4) to imulate the aniotropi vioplati behavior of metalli material. A generi method i preented that make it poible to build a implified aniotropi material texture, baed on the prinipal feature of the pole figure. The method i applied to rerytallied zironium alloy. The model aim at repreenting the mehanial behavior of polyrytal with a minimum number (between 5 and ) of rytalline orientation ( grain ), and onider only 6 lip ytem by grain for deribing the inelati train tenor, whatever the rytallographi lattie of the material under tudy i. It mut be very lear that our goal i not to repreent aurately the material texture, but only it mehanial effet on the repone of a truture to a mehanial loading. Neverthele, we how in the following that our hoie of the rytallographi orientation i loely related to the real material texture, and 4

5 moreover that it an be identified thank to an optimization algorithm by omparion with the reult of experimental tet. In the next etion, we reall the major feature of the implified vioplati RL-polyrytalline model. In etion 3, we how how the model ha been applied to the imulation of the aniotropi behavior of a rerytallized zironium alloy. Partiular attention i paid to the deription of the parameter identifiation of the model uing everal multiaxial tet. An example of finite element alulation performed on a tube i preented in etion 4. Conluding remark are given in the lat etion of thi paper.. The implified vioplati RL-polyrytalline model The model i preented in the ae of mall deformation and mall rotation. We want to emphaize that the RL model doe not depart from laial polyrytalline model. Conequently, all exiting and future theoretial and oftware development apply eaily to the model, with regard for example to finite deformation, rotation of rytallographi orientation, loalization elf-onitent model, or lip ytem ontitutive equation. Let n be the normal vetor of a lip plane and l the unit lip diretion vetor. The orientation tenor of the lip ytem number i defined a: m = n l + l n ) / () ( 5

6 We onider only 6 univeral lip ytem orreponding to hear and extenion train rate in the loal frame of a given rytallographi orientation (Figure ): 3 x x x [ ] = n, [ ] = n, [ ] 3 = n [ ] = l, [ ] = l, [ ] 3 = l = m, = m, = 3 m [ ] / / 4 = n, [ ] / / 5 = n, [ ] / / 6 = n [ ] / 4 / = l, [ ] / / 5 = l, [ ] / / 6 = l = 4 m, = 5 m, = 6 m () The inompreible plati train rate of a given orientation or grain reult from the multiple lip: = = = γ γ γ γ γ γ γ γ γ γ γ γ γ ε & & & & & & & & & & & & & & p m (3) where γ& i the lip rate of the lip ytem. 6

7 For hp rytal, the deformation reult from 5 main lip familie (for intane, ee Geyer, 999; Staroleky and Anand, 3): baal B<a>, primati P<a>, pyramidal π<a>, π<+a> and π<+a> (Figure a). Only pyramidal lip <+a> ontribute to the train rate omponent p ε& 33. Aording to equation (3), the ytem 5 and 6 of the model will be aoiated with thee phyial lip ytem. For the ame reaon, the ytem and 3 of the model will be aoiated with p p baal lip that ontribute to the train rate omponent ε& and. Finally, the ytem and 4 of the model will be aoiated with primati lip that ontribute p p p to the train rate omponent ε&, ε&, and. So, for hp material, we hould onider 3 lip familie (,4), (,3) and (5,6) that will have the propertie repetively of primati, baal and pyramidal lip (Figure b). In order to take into aount the fat that ytem and 4 have not the ame hemati repreentation in Figure b, we hooe to plit the primati family into two part, and we aume that they have not the ame propertie, epeially in term of ritial reolved hear tre. We do not pretend that thee non phyial lip ytem aurately repreent the phyial one. Again it mut be lear that our goal i only to model the maroopi mehanial behavior of the material. Any ontitutive equation for the lip ytem an be ued with the RLpolyrytal model, for example the power law form of Aaro and Needleman (985) or equation depending on diloation denitie (for intane, ee Ho and Foret, ; Ho et al., 3). In the following, we will ue a phenomenologial model with two internal variable, iotropi r and kinemati X, a propoed by Cailletaud (99) : ε& 3 ε& 3 7

8 ( X & γ = v& Sign τ ) (4) n τ X r v& = Max, (5) K r 6 = + t= t [ exp( bt vt )] + Q[ exp( bv τ Q ] (6) X & = a & γ X v& (7) Coeffiient Qt are the omponent of the hardening moduli matrix Q. The diagonal term repreent the elf-hardening of eah ytem, and the non-diagonal term the latent hardening. Generally peaking, Q an be non-ymmetri. For impliity, in the following, we have hoen a ymmetri form for Q. Thi indue 8 different value to take into aount the 4 lip familie of the hp truture. There i no latent hardening for the kinemati variable. τ i the initial ritial value of the reolved hear tre τ on the lip ytem, given by: τ = σ : m (8) g where σ i the tre tenor in the grain number g (from to N ). Thi g tre tenor σ i evaluated uing loalization equation. In the ame way a for g the ontitutive equation of the lip ytem, everal hoie an be made. One 8

9 an refer to the general preentation of the RL model (Rouelier and Lelerq, 4) to have an overview of what may be ued in the framework of a implified polyrytalline approah. In the preent appliation, we onider the loalization equation propoed by Cailletaud (987, 99), in whih auxiliary train variable β are memorized for eah grain : g σ g = Σ + α( B β ) with B = g N f g g= β (9) g α = μ( β ) and β = 5 4 5ν ν where μ and ν are the hear modulu and the Poion oeffiient, repetively. In equation (9), Σ i the maroopi tre tenor (over the Repreentative Volume of Element) and f g the volume fration of grain g. The evolution of β i given by a phenomenologial kinemati equation, g * with adjutable parameter D (Cailletaud, 987) and D (Pilvin, 99): & β g (& ε ) p * p p = & ε D β D ε J g g () g g 3 where ( T ) ( T : T / ) / J = and T i the deviator of a tenor T. d d d In the preent tudy, the parameter * D i hoen to be zero. Let u note the phenomenologial harater of the preent ale tranition rule. The idea behind the formulation i that the development of a plati aommodation will dereae the level of the reidual intergranular tree. The hape of the expreion i imported from the laial maroopi model. The original author (Pilvin and Cailletaud, 99) ugget to alibrate the parameter of the rule on finite element omputation of a repreentative material element. 9

10 Thi ha not been made in the preent tudy where, following the pirit of the model propoed in thi paper, we have deided to onider the variable in the et of parameter to be identified. Neverthele, the value found in the diret identifiation proe are in good agreement with laial value of thee parameter. 3. Appliation to the imulation of the rerytallized Ziraloy behavior 3.. Material The material under tudy onit in Ziraloy-4 tube, the weight hemial ompoition of whih i reported in Table. The material i teted in the α-rerytallized tate, i.e. after proeing, the tube are ubmitted to a heat treatment of 7 C ± 3 C during 4 to 5 hour. After rerytallization, the mirotruture onit in mall equiaxed grain ( to 5 μm) whih are bounded with larger one (5 to 8 μm) (Bouffioux, 4). The referene frame X X X 3 of the material i the frame rθ z of the tube. The texture of thee tube i trongly aniotropi, ee Figure 3 (Robinet, 995). It how θ z pole figure { }, i.e. baal pole orreponding to axi of the rytal or axi x, and pole figure { }, i.e. primati pole orreponding to 3 axi x in RL model. Note that the texture preented here reult in the ombination of two effet : a material one (due to the hp lattie, and to the heat treatment), and an effet due

11 to tube proeing. We have to point out that i texture i harateriti of tube made of rerytallized zironium alloy (tube made of old-word tre relieved material would have ignifiantly different pole figure). 3.. Experimental data bae and parameter identifiation Several experimental tet have been performed by Robinet (995) on tube peimen : tenion, tenion-hear (with everal biaxiality ratio E θ z E zz ), and tenion-internal preure (with everal biaxiality ratio E θθ Ezz ). The experiment are performed at room temperature. In order to proeed to the parameter identifiation and to the model validation, we deide to plit the data bae into two part. The identifiation data bae i built with three tet : two tenion tet at two different maroopi train 4 rate ( E & 7 = 6,6. and E & = 6,6. ), and one pure hear tet (equivalent 4 train rate E & = J ( E& ) = 6,6. ). The validation data bae (i.e. the data bae that i ued to demontrate the preditive ability of the model) i built with the remaining experimental tet. Uually three type of parameter are identified when one ue a polyrytalline approah : - the material parameter linked with the lip propertie of eah ytem (ritial reolved hear tre, vioity, hardening, ) ; - the matrix defining elf- and latent- hardening ( Q t ) ; - the parameter of the loalization equation ( D and α ).

12 We have onidered α to be a parameter, intead of uing it theoretial value α = μ( β ), Eq. (9). In the uual ae, the texture of the material i an input data whih i made of a ertain number of grain repreented by their rytallographi orientation. Generally, the number of grain i large. For example, Geyer (999) ued 4 orientation to deribe a Repreentative Elementary Volume of Ziraloy-4. In our ae, we aume that it i poible to repreent the material texture of the rerytallized zironium alloy under tudy with only 6 grain. It i worth to note that thi number of 6 grain mut be onidered a a minimum, in the pirit of the preent model to dratially redue the number of internal variable handled by finite element ode. Uing more grain (in agreement with the experimental evidene of the pole figure), would lead to improve the global repone of the model, a wa hown by Geyer (999) with 4 grain. But the prie to pay in term of CPU time i not worth ompared to the relatively low gain in term of mehanial repone. The Euler angle of our 6 grain are reported in Table. One an ee that three ontant are not aigned ( ψ, a ϕ b, ϕ ). Thi i an original feature of our approah to onider thee ontant a parameter, and to inorporate them into the parameter identifiation proedure. Of oure, the 6 grain deribed above are relevant for (and only for) rerytallyzed hp material that have pole figure imilar to thoe of Figure 3. For other material (e.g. old-work tre relieved material) with different type of

13 texture, the Euler angle preented in Table are no more relevant and hould be re-etimated. Figure 4a preent the pole figure (axi x3-left and axi x-right) of the ix grain defined in Table, with ψ a, ϕ b, ϕ a parameter. One an thu figure out how the 3 angle at on the poition of the pole to define the material texture. Identifying thee parameter give u the opportunity to verify the ability of the model to math not only with maroopi tet, but alo with the real material texture. Thu, the parameter of the RL-polyrytal model have been identified uing the identifiation data bae defined above. The parameter value are reported in Table 3. A mentioned above, four lip familie are onidered (i.e.,, 4, -3 and 5-6), eah family having it own et of parameter. The elf- and latent- hardening matrix ha the following form : Q Q Q Q4 Q5 Q5 Q Q Q3 Q Q5 Q5 Q Q3 Q Q Q5 Q5 Q = () Q4 Q Q Q Q5 Q5 Q5 Q5 Q5 Q5 Q5 Q56 Q5 Q5 Q5 Q5 Q56 Q5 One ha to point out that, even if the RL lip familie are not phyial one, the identified value of the ritial reolved hear tree are in agreement with what i laially oberved on non-irradiated zironium alloy, i.e. τ primati < τ baal < τ pyramidal 3

14 The numerial value obtained for the whole et of parameter are onitent with the one obtained with tandard polyrytalline approahe on the ame type of material (ee for intane Geyer, 999). The omparion between experiment and imulation of the identifiation data bae are preented on Fig. 5 and 6. The imple line preent RL model imulation. The bold line orrepond to imulation performed with the ue of a laial maroopi vioplati model (uing one kinemati and one iotropi hardening, ee for intane Lemaitre and Chabohe, 985) in whih the yield urfae i defined by a Hill riterion, in order to take into aount the Ziraloy aniotropy. Of oure, thi model ha been identified with the ame data bae a the RL model. Ue of thi model i made in order to have a omparion bai between the approah propoed in thi paper and more laial model till widely ued. The imulated reult are good, and annot allow one to ditinguih between the two approahe. On Figure 4b, one an ee the pole figure (axi x3-left and axi x-right) of the 6 grain repreenting the texture of the material under tudy, after identifiation of the 3 ontant ψ a, ϕ b, ϕ. Thee pole figure are to be ompared with the one preented on Figure 3. The good agreement oberved here i a firt validation of our hoie to integrate 3 Euler angle into the parameter identifiation proedure (ee Table ). A eond validation of thi original hoie will follow with the tudy of the preditive ability of the RL model applied to the rerytallized Ziraloy-4. 4

15 3.3 Preditive ability of the model and diuion In order to invetigate it preditive ability, the model ha been teted on everal loading ituation whih had not been ued in the identifiation data bae. Thee multiaxial tet are onduted at the ame equivalent (in the von Mie ene) train rate ( E & 4 = 6,6. ). We preent here three tenion-hear tet with three different train biaxiality ratio ( E θ z E zz ), and two tenion-internal preure tet, with two different train biaxiality ratio ( E θθ Ezz are ompared with the experimental one on Figure 7 to. ). The imulated reult Several feature are to be pointed out: i. Firt of all, we have to preie that our goal wa to tet the RL model a onern it ability to repreent the aniotropi feature of the material in term of yield tre and flow diretion. To thi aim, we have ued only ontant train rate tet to identify the model parameter. A a matter of fat we do not, in the following, draw any onluion on the preditive ability of the repone of the model to other mehanial loading yli tet, reep or relaxation tet, Partiularly, we are aware that only two experimental tet are not uffiient enough to identify the model parameter aoiated with all the material harateriti. ii. A onern the RL model, agreement between imulation and experiment are good, for everal loading ituation and epeially the tenion-internal preure tet whih were not repreented in the identifiation data bae. Thi onfirm the ability of the model to repreent ituation whih have not been taken into aount in the identifiation 5

16 proedure, epeially a onern aniotropy. We would like to point out that getting thi reult wa not obviou, beaue of the mall number of grain ued to imulate the texture of the Ziraloy-4 tube. iii. An important feature, linked with the previou one, i that the RL model an be totally identified with only two experimental tet, repreenting two loading diretion (pure tenion zz, and pure hear θ z ). A onfirmation of thi an be found on Figure and 3, where the ritial tree of the 6 lip ytem are repreented for eah grain (from grain to grain 6), in the maroopi zz -θθ and zz - z θ plane. It an be een here that the ue of a tenion tet onduted up to minimum 5 MPa allow to ativate and thu identify the material parameter of lip ytem #, 4, 5 and 6. Note that in the ae of a tenile tet, lip ytem # and 3 are never ativated in the RL model. It i intereting to remember that thee two ytem are aoiated to baal lip, whih i known to be rarely oberved when tudying the experimental mehanim of unirradiated Ziraloy-4 (Geyer, 999). Uing a pure hear tet onduted up to 5 MPa allow one to ativate lip ytem #, 4 and 3. A the material parameter of lip ytem # are equal to thoe of lip ytem # 3, it i therefore demontrated that every lip ytem ha been ativated at leat one in the imulation of the identifiation data bae. Thi explain why thee two teting diretion are uffiient for the parameter identifiation, provided the experiment are onduted far enough in term of train (and thu tre) value. The ativation of many lip ytem at Σ θθ 5 MPa in Figure explain why the preure urve are o flat in Figure and, a feature that i not due to urve fitting but to the texture of the material and annot be predited by maroopi model. 6

17 iv. The omparion with the maroopi approah how the laial feature that the Hill yield urfae annot be identified without appropriate experimental tet in the three diretion. Typially, our identifiation data bae doe not inlude any internal preure tet, and thi reult in the impoibility of identifying any aniotropy between the rr, θθ, and zz diretion for the maroopi model, a hown on Fig. and. A a matter of fat, the identifiation data bae make it poible to identify only the hear ( θ z ) aniotropy for thi kind of model. v. The preent tudy how that, in the ae of an aniotropi material uh a Ziraloy, the parameter of the RL model an be identified knowing the general feature of the material texture, and with experimental tet performed in two diretion (pure tenion and pure hear) provided thee tet are onduted uffiiently far o a to ativate eah lip ytem. With thee information, it i poible to get material parameter that are robut enough to enure a good preditive ability, whih i not the ae for maroopi approahe. Of oure, enlarging the identifiation data bae with tet performed in other diretion allow to improve the parameter identifiation of both kind of model, but at a higher ot. One again, introdution of other experimental tet would be eential to identify parameter linked with peifi material feature uh a reep or yli effet, whih are not the onern of thi paper. Another way to verify the robutne of the material parameter (and of the propoed ontitutive equation) i to invetigate whether they are dependent on the identifiation data bae or not. In order to anwer thi quetion, we have added 7

18 to the identifiation data bae the equibiaxial tenion with internal preure tet E θθ E zz ( = ). With the ue of thi extended data bae, we have identified the material parameter of the RL model from the ame initial value a with the firt identifiation. The reult are preented in able 4. It i remarkable to point out how the value of thee new parameter are loe to thoe preented in Table 3. Thi feature i another proof that a relatively poor identifiation data bae i uffiient to get robut material parameter of the RL model, and that thi model i intrinially preditive. 4. A finite element alulation A the prinipal purpoe of thi paper i to propoe a material model baed on the laial polyrytalline approah and ompatible with finite element alulation (i.e., onuming reaonable CPU time), we preent in the following the imulation of a Ziraloy-4 tube ubmitted to an internal preure and an impoed axial diplaement. The truture i alulated uing both RL model and the Chabohe+Hill maroopi approah. We ompare the numerial reult in term of train-tre urve, a well a the CPU time needed to perform the alulation. The 3D meh of the finite element alulation i preented Figure 4. It onit in a ring, part of a Ziraloy-4 tube, mehed with quadrati element with node. The impoed boundary ondition are uh that the axial train of the ring i zero, and the hoop train rate i equal to 6, Two finite element alulation have been performed, with the Chabohe+Hill approah and the RL model. The maroopi train and tre tate in the inner part of the tube are 8

19 reported Figure 5, ompared with the experimental data. The reult are onitent with the one hown on Figure, whih how the D (repreentative elementary volume) imulation of thi tet. It an be pointed out that, epeially for the tenile reult in Figure and 5, the maller train behavior i better predited with the Chabohe+Hill approah, and the RL model fit better for the larger train. One explanation for thi may be found in the fat that the Chabohe+Hill approah i written in a purely vioplati way (the vioplati train alway exit, even if it i very mall for mall total train). On the ontrary, the RL model i written in uh a way that vioplati lip i ativated on a given ytem if and only if the ritial value of the reolved hear tre i reahed. Thu non elati train i intrinially maller in the RL approah than in the laial one. At higher train, the problem i different : in the Chabohe+Hill model the 7 (one tenorial kinemati and one alar) hardening internal variable aturate rapidly (relatively low train hardening) and in the RL model the poible ativation of 36 (6 grain x 6 lip ytem) lip ytem make it poible to have a better deription of the global material behavior. For omplete undertanding, we have to note that the hoop tre gradient between the inner and the out part of the tube (tube thikne = 57 mirometer, inner radiu = 48 millimeter) i approximately 8 MPa. Thi onfirm that the tube may be onidered a a real truture, and not a a thin tube. Table V preent the CPU time of the two finite element alulation. One an ee that CPU time of RL model i not muh larger than the one of Chabohe+Hill model (ratio of approximately 5), ompared with the auray of the imulated reult. If one reall that the Chabohe+Hill model ued in the preent tudy need to integrate 3 internal variable (the elatiity tenor, one 9

20 tenorial kinemati hardening, one alar iotropi hardening, the equivalent vioplati train), and that the RL model require the integration of 48 internal variable, the ratio of 5 in the CPU time obtained in Table 5 an be eaily related to the ratio of the number of internal variable to integrate. It mut alo be pointed out that the ame finite element alulation performed with a tandard polyrytalline model (for intane Geyer, 999, with 76 internal variable to integrate) would have lat approximately time longer than the one with RL model. 5. Conluding remark In the preent paper, a implified polyrytalline model (the o-alled RL model) i propoed to imulate the aniotropi vioplati behavior of metalli material. A generi method i preented that make it poible to build a implified aniotropi material texture, baed on the prinipal feature of the pole figure. The method i applied to a rerytallied zironium alloy, ued a lad material in the fuel rod of nulear power plant. The objetive of thi tudy wa to propoe a robut, preditive but little CPU time onuming model for indutrial finite element alulation. In order to ahieve thi goal, everal implifiation of the laial polyrytalline approah have been performed : the definition of lip ytem and the hoie and number of rytalline orientation are the major apet developed here. A a matter of fat, thee implifiation lower the phyial bai of the approah propoed in thi paper, but it ha been hown that the main harateriti of the laial polyrytalline approah were kept : preditive

21 ability and intrini way to deribe the material aniotropy. Moreover, we have hown that the impliity of the RL model allow one to follow a parameter identifiation heme that an be dediated to eah lip ytem. The relatively mall CPU time (a ompared to tandard polyrytalline approahe), ombined with the poibility to follow eah lip ytem, make it poible to get robut material parameter with a minimum number of tet in the identifiation data bae. The finite element alulation performed with the RL model onfirm the promiing feature of thi model in term of good quality of the imulation, and CPU time of the ame order a with more laial maroopi model. The next tep, under progre, i to apply the material parameter of the model to another texture, obtained from rerytallized zironium alloy heet, of the ame hemial ompoition. The reult, ombined with the modeling of zironium alloy damage propertie, will be expoed in a forthoming paper. Aknowledgement The author would like to thank Dr. Jaque Beon (Centre de Matériaux, Eole de Mine de Pari) for hi fruitful help in the finite element alulation. Referene

22 Aaro, R.J., Needleman, A., 985. Texture development and train hardening in rate dependent polyrytal. Ata Metall. 33, Barlat, F., Lege, D. J., Brem, J. C, 99. A ix-omponent yield funtion for aniotropi material. Int. J. Platiity 7, Bouffioux, P., 4. Private ommuniation. Bron, F., Beon, J., 4. A yield funtion for aniotropi material - Appliation to aluminum alloy. Int. J. Platiity, Cailletaud, G., 987. Une approhe miroméanique phénoménologique du omportement inélatique de métaux. PhD thei, Pari Univerity (Pari 6). Cailletaud, G., 99. A miromehanial approah to inelati behavior of metal. Int. J. Platiity 8, Cailletaud, G., Pilvin, P., 994. Utiliation de modèle polyritallin pour le alul par élément fini. Revue européenne de élément fini 3, Calloh, S., 997. Eai triaxiaux non-proportionnel et ingéniérie de modèle de omportement. PhD thei, Eole Normale Supérieure de Cahan. Geyer, P., 999. Etude et modéliation du omportement élatovioplatique de tube en Ziraloy par une approhe polyritalline. PhD thei, Eole de Mine de Pari. Gotoh, M., 977. A theory of plati aniotropy baed on a yield funtion of fourth order (plane tre tate) I/II. Int. J. Meh. Si. 9, 55-5, Hill, R.,95. The Mathematial Theory of Platiity, Clarendon Pre. Ho, T., Foret, S.,. Polyrytal modeling of IF-Ti teel under omplex loading path. Int. J. Platiity 7,

23 Ho, T., Crépin, J., Gélébart, L., Zaoui, A., 3. A proedure for identifying the plati behavior of ingle rytal from the loal repone of polyrytal. Ata Mater. 5, Karafilli, A. P., Boye, M. C., 993. A general aniotropi yield riterion uing bound and a tranformation weighting tenor. J. Meh. Phy. Solid 4, Kuroda, M., Tvergaard, V.,. A phenomenologial platiity model with nonnormality effet repreenting obervation in rytal platiity. J. Meh. Phy. Solid 49, Lebenohn, R. A., Tomé, C. N., 993. A elf-onitent aniotropi approah for the imulation of plati deformation and texture development of polyrital: appliation to zironium alloy. Ata Metall. Mater. 4, Lemaitre, J., Chabohe, J.-L., 985. Mehani of olid material. Cambridge Aademi Pre. Onimu, F., 3. Approhe expérimentale et modéliation miroméanique du omportement de alliage de zironium irradié, PhD thei, Eole Centrale Pari. Pilvin, P., 99. Approhe multiéhelle pour la préviion du omportement anélatique de métaux. PhD thei, Pari Univerity (Pari 6). Pilvin P., Cailletaud G., 99, Intergranular and trangranular hardening in vioplatiity, in : Zyzkowki M. (Ed), Creep in Struture, Springer Verlag, pp Rihard, F., Delobelle, P., Lelerq, S., Bouffioux, P., Rouelier, G., 3. Modeling of the old work tre relieved Ziraloy-4 ladding tube 3

24 mehanial behavior under PWR operating ondition, Tranation of SMiRT 7 (Prague) Augut 7-, 3. Robinet, P., 995. Etude expérimentale et modéliation du omportement vioplatique aniotrope du Ziraloly 4 dan deux état métallurgique. PhD thei, Franhe-Comté Univerity, Beançon. Rouelier, G., Lelerq, S., 6. A implified polyrytalline model for vioplati and damage finite element analye, Int. J. Platiity, Shaeffler, I., 997. Modéliation du omportement élato-vioplatique aniotrope de tube de gaine du rayon ombutible entre zero et quatre yle de fontionnement en réateur à eau preuriée. PhD thei, Franhe-Comté Univerity, Beançon. Staroleky, A., Anand, L., 3. A ontitutive model for hp material deforming by lip and twinning: appliation to magneium alloy AZ3B. Int. J. Platiity 9, Vinent, L., Calloh, S., Kurtyka, T., Marqui, D.,. An improvement of multiaxial rathetting modelling via yield urfae ditortion. J. Eng. Mat. Tehnol. 4,

25 Figure aption frame. Figure. Repreentation of the 6 lip ytem of the RL model in the loal Figure. a : the phyial lip familie of a hp rytal. b : the 4 lip familie, 4, (, 3), (5, 6) of a hp rytal repreented in the RL model frame. Figure 3. The experimental texture of the rerytallized Ziraloy-4. 3a : the θ z pole figure { }, i.e. baal pole orreponding to axi of the rytal. 3b : the θz pole figure { }, i.e. primati pole. Figure 4. The pole figure of the imulated material (axi x3-left and axi x-right) 4a : priniple of the definition of ψ a, identifiation of ψ a, ϕ b, ϕ. ϕ b, ϕ. 4b : the pole figure after Figure 5. Identifiation data bae : tenion tet. Experimental point, RL model (olid line), Chabohe+Hill (bold line). Figure 6. Identifiation data bae : pure hear tet. Experimental point, RL model (olid line), Chabohe+Hill (bold line). Figure 7. Simulation data bae : tenion-hear ( E θ = 3 4 ). z E zz Experimental point, RL model (olid line), Chabohe+Hill (bold line). Figure 8. Simulation data bae : tenion-hear ( E θ = 3 ). z E zz Experimental point, RL model (olid line), Chabohe+Hill (bold line). Figure 9. Simulation data bae : tenion-hear ( = 3 ). Experimental E θ z E zz point, RL model (olid line), Chabohe+Hill (bold line). Figure. Simulation data bae : tenion-internal preure ( E θθ = ). Experimental point, RL model (olid line), Chabohe+Hill (bold line). E zz 5

26 Figure. Simulation data bae : tenion-internal preure ( E θθ = ). Experimental point, RL model (olid line), Chabohe+Hill (bold line). Figure. Critial tree of the 6 lip ytem repreented for the material texture, in the maroopi zz -θθ plane. Figure 3. Critial tree of the 6 lip ytem repreented for the material texture, in the maroopi zz -θ z plane. Figure 4. 3D meh of a ring ubmitted to tenion and internal preure ( E θθ = ). E zz Figure 5. Maroopi train and tre tate at the outer part of the ring. Experimental point, RL model (olid line), Chabohe+Hill (bold line). E zz 6

27 Table. Weight hemial ompoition (%) of the Ziraloy-4 under tudy. Cr Fe Sn O Zr,,,3,35 balane Al (ppm) C (ppm) H (ppm) Hf (ppm) N (ppm) Si (ppm)

28 Table. Euler angle of the 6 grain repreenting the rerytallized Ziraloy-4 texture. g ψ a π ψ a π π ψ g φ g π π π π π π ϕ g ϕ b ϕ b ϕ ϕ 8

29 Table 3. Material parameter of the RL model identified on the rerytallized Ziraloy-4, with the initial identifiation data bae. ϕ b ϕ α (MPa) D , E (MPa) ν Q (MPa) Q (MPa) Q 5 (MPa) ,4 88,8 78 Q (MPa) Q 4 (MPa) Q 5 (MPa) Q 3 (MPa) Q 56 (MPa) 98,7 6,3 47,7 75 Q (MPa) b 9,8,4 τ (MPa) n K (MPa) τ (MPa) n K (MPa) 3,4, ,6, τ 4 (MPa) n 4 K 4 (MPa) τ 5 (MPa) n 5 K 5 (MPa) 46,, 396,3 3,7,7 3463,5 a (MPa) b a (MPa) b 54,6 5,8 9,89 358,34 7,77 64,8 a 4 (MPa) b 4 4 a 5 (MPa) b ,46 6,33 765,4 65,6,74 97,6 ψ a 9

30 Table 4. Material parameter of the RL model identified on the rerytallized Ziraloy-4, with the extended identifiation data bae. ϕ b ϕ α (MPa) D 57 66,7 67, E (MPa) ν Q (MPa) Q (MPa) Q 5 (MPa) 5,36 78,4 89,5 76,4 Q (MPa) Q 4 (MPa) Q 5 (MPa) Q 3 (MPa) Q 56 (MPa) 95,8 6,3 49 7,3 Q (MPa) b 9,4, τ (MPa) n K (MPa) τ (MPa) n K (MPa) 3,4, ,7, τ 4 (MPa) n 4 K 4 (MPa) τ 5 (MPa) n 5 K 5 (MPa) 4,, 394, 4,,7 347,5 a (MPa) b a (MPa) b 545, 5,7 98,88 343,9 7,8 68,53 a 4 (MPa) b 4 4 a 5 (MPa) b ,83 6,4 765,9 664,55,69 97,6 ψ a 3

31 Table 5. CPU time (PC/Linux) obtained with Chabohe+Hill and RL model. Model Hill RL CPU time ()

32 Figure 3

33 Figure 33

34 z z θ θ Figure 3 34

35 Figure 4a Figure 4b 35

36 6 Σzz (MPa) Tenion 6, Tenion 6,6.-7 -,,4,6,8,,,4,6 Ezz Figure 5 36

37 3 Σzθ (MPa) Ezθ,5,,5,,5,3 Figure 6 37

38 5 45 Σ (MPa) Shear Tenion 5 5,5,,5,,5,3 E Figure 7 38

39 45 4 Σ (MPa) Shear Tenion 5 5,5,,5,,5,3 E Figure 8 39

40 3 Σ (MPa) 5 5 Tenion Shear 5,5,,5,,5,3 E Figure 9 4

41 7 Σ (MPa) Internal Preure Tenion,5,,5,,5 E Figure 4

42 7 Σ (MPa) Tenion Internal Preure E,5,,5,,5,3,35 Figure 4

43 Figure 43

44 Figure 3 44

45 Figure 4 45

46 7 6 Σ (MPa) Tenion Internal Preure 5 4 3,5,,5,,5 E Figure 5 46