Abstract: 1 Introduction. Walid NASRI a,b, Adinel GAVRUS a, Afia KOUADRI-DAVID a, Kacem SAI b

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1 ème Conrè Françai de Mécanique Lyon, 4 au 8 Août 15 Exerimental and numerical tudy concernin the aniotroic behavior of an AA4-T351 thick heet uin the unified formalim, a multi-mechanim model and a olycrytalline aroach Walid NASRI a,b, Adinel GAVRUS a, Afia KOUADRI-DAVID a, Kacem SAI b a. LGCGM EA3913, INSA, UEB, Av. de Butte de Coëme, 3578 Renne, France. walid.nari@ina-renne.fr, adinel.avru@ina-renne.fr, afia.kouadri-henni@ina-renne.fr b. LGPMM, Ecole Nationale d Inénieur de Sfax, BP , Sfax, Tuniie. kacemai@yahoo.fr Abtract: The uroe of thi work i to evaluate the aniotroic mechanical behavior of an AA4-T351 thick late uin three multi-cale aroache: a Macrocoic Unified formalim (HILL 48), a Multi- Mechanim Model and a Polycrytalline laticity model. The exerimental reult obtained from uniaxial tenile tet are firt decribed. Thereafter, the theoretical formulation and numerical identification of tudied elatolatic aroache are reented. Finally, finite element imulation of tenile tet and analyi of the thick heet tretchin formin roce are erformed. Keyword: Thick heet aniotroy, Multi-cale modellin, Numerical imulation, Aluminium alloy, Stretchin formin. 1 Introduction In the aeronautical manufacturin indutry, due to the imortant need for low-weiht, aircraft element made in aluminium alloy characterized by ood mechanical reitance and formability are ued. Generally thi material i obtained in a heet or late form by hot and cold rollin formin rocee which create latic aniotroy. Thi henomenon i the reult of the ditortion of the latic yield urface hae due to the material microtructural tate chane [1]. It i imortant to undertand the aniotroic mechanical behavior characterizin a thin [] or a thick heet and enure the reliability of the numerical model rediction in order to be interated in eneral enineerin otimal dein analyi or to be tronly ued for imrovement of heet formin rocee. Thi work reent an introduction of the three-dimenional aniotroy tudy of an aluminium alloy and tart with two ecific cae of a 3D latic aniotroy: normal and lanar. In normal aniotroy (ued for tudyin thin and thick heet), the roertie differ rincially in the thickne direction. Wherea, in lanar aniotroy (ued claically for tudyin thin heet), the roertie vary with the orientation in the lane of the heet. Three different elato-latic numerical aroache are in the focu of thi rooed tudy: a unified formalim, a multi-mechanim model and a olycrytalline one [3-7]. To demontrate their feaibility and to elect the mot accurate model decribin the latic aniotroy, comarion are realized with the exerimental data obtained from tenile tet erformed for ecimen with a thickne of 3 mm alon different direction to the rollin direction. Finite Element analye baed on

2 ème Conrè Françai de Mécanique Lyon, 4 au 8 Août 15 the three numerical model are ued for the imulation of uni-axial tenile tet and of a tretchin formin roce in order to evaluate their robutne on analyi of evere material formin condition. Exerimental tudy.1 The amle material and oeratin condition Uni-axial tenile tet were carried out on flat iece of mooth ection obtained by machinin from a rolled AA4-T351 thick late. The loadin axi of ecimen with a thickne of 3 mm coincide with different cuttin direction with reect to the rollin direction. Fiure 1 how the ued ecimen eometrical dimenion. 114 R1 3 5 Fiure 1: Plane ecimen dimenion (in mm) The exerimental tenile tet were conducted in the direction, 3, 45, 6, 75 and 9 with reect to the rollin direction uin an INSTRON machine of INSA Renne havin a maximum loadin caacity of 5 kn (fiure ), recordin both the load and the dilacement controllin a contant crohead eed of 5 mm/min correondin to an initial train rate of ε =x1. (a) (b) l b l b Fiure : INSTRON univeral tetin machine and the tak oint (A, B, C, D) oitioned on the tenile ecimen urface: (a) before and (b) after the tet.. Exerimental reult and claical analyi Axial load and dilacement in lonitudinal and tranvere direction wa recorded durin the exeriment uin a eed camera meaurin reectively two oint dilacement throuhout the material traction (fiure ). Then the camera follow two oint A and B alon the y axi of loadin (havin an initial ditance l between them) and two oint C and D alon the x axi (aced by an initial ditance b ).

3 ème Conrè Françai de Mécanique Lyon, 4 au 8 Août 15 Thank to the diital imae of the ecimen, the evolution of each local ditance can be recorded and then to etimate the correondin total train ε t yy, t ε uin the followin formula: t l t b yy ln and l ln (1) b l, b are the meaured dilacement. The tenile load F wa continuouly meaured in order to evaluate the true tre: t ( ε ) F σ yy = ex yy () S S i the cro ection area of the ecimen. The correondin latic train i obtained from the artition of the total train by an elatic art and a latic one: ε + t e = ε ε or d ε + t e = dε dε (3) Accordin to the three orthotroic axe (x,y,z) the latic train can then be calculated a follow: ε υσyy σ t t yy = ε and ε yy = ε yy - (4) E E + Uin the volume conervation rincile ( tr( ) = ε can be etimated by: ( ε ε ) yy ε~ ), the latic train alon the ecimen thickne ε - + (5) Startin from the claical latic aniotroy theory of a thin heet, the Lankford coefficient, denoted r-value, wa etimated from different tenile direction with reect to the rollin one (, 45 and 9 ), uin the ratio between the latic width train and the thickne train [8]: ε r = ε = dε dε ε ε (6) It i hown that the r-value i equal to one for iotroic material and any other value indicate that latic aniotroy occur durin the material deformation. The normal aniotroy r and the lanar aniotroy r are alo ued to invetiate the formability and the earrin attern in circular cu drawin. Hiher value of r mean a le thinnin and a better formability. On the other hand, hiher lanar aniotroy r mean more aniotroic roertie. The two arameter can be comuted a: r r45 r9 r r45 r9 r and r (7) 4 Here the r,r45, r9 are the Lankford coefficient etimated by equation (6). For the, 45 and 9 tenile tet, the obtained exerimental true tre-true train curve before the onet of the ecimen neckin are hown in the fiure 3 (a). A it can be een, a cloe material behavior i obtained between the tenile direction 45 (DD) and 9 (TD). The latic behavior of the ecimen loaded at (RD) i lihtly different in intenity with a dozen of MPa. Some difference can be oberved in the fiure 3 (b) comarin to a thin AA4 heet []. Becaue the lanar aniotroy Δ r ha maller value (i.e.

4 ème Conrè Françai de Mécanique Lyon, 4 au 8 Août 15 Δ r = -.7 for 1% latic deformation), it i oible to conider that the conidered AA4 thick late ha rather a normal aniotroy, which i a articular cae of a 3D aniotroy. Moreover an averae Lankford coefficient lower than 1 ( r =.64 for 1% latic deformation) reflect a le ood formability of thi material. (a) (b) Fiure 3: Exerimental true tre true train curve alon the tenile direction, 45 and 9 of AA4-T351: (a) thick late (reent work) and (b) thin heet [] If a Hill 1948 quadratic latic aniotroic behaviour law i takin into account, the variation of the correondin coefficient with the latic train (F, G, H, L, M, N or dimenionle F, G, H, L, M, N one) eem to be in accordance with the claical analyi of the lane aniotroy (equation (9)) and normal aniotroy (equation (1)). Thee coefficient can be exreed in function of Lankford arameter uin the followin exreion: r F 1 1 F = ; F ' = F = G = ; F' = G' = r9(1+ r ) σ (1 + r ) σ (1 + r ) 1 G r r G = ; G ' = (9) H = ; H' = (1) (1 + r ) (1 + r ) σ (1 r ) σ + r H H = ; H ' = N = F + H ; N ' = F' + H' (1 + r ) σ ( r + r9 )(1 + r45 ) N N = ; N ' = r (1 + r ) 9 σ where σ i the equivalent tre correondin to the tenile behavior alon the rollin direction DL. The evolution of thee coefficient with the cumulated latic train i lotted in fiure 4. In a firt aroximation, etimatin all the r coefficient for.% of latic deformation, the dimenionle Hill coefficient F, G, H and N have contant value. Fiure 4: Variation of the comuted Hill coefficient a a function of the latic train for a lane aniotroy

5 ème Conrè Françai de Mécanique Lyon, 4 au 8 Août 15 The variation of the Hill coefficient (F', G', H', N ) exhibit a har fall before a latic train of 5%. Beyond thi level, thee coefficient decreae lowly. Commonly, contant value of the Hill coefficient comuted for a.% latic train are enerally ued in the henomenoloical aniotroic model. Nonethele thi aumtion, to ive a more accurate Hill 1948 criterion decribin the latic aniotroy of aluminum alloy, the aniotroy coefficient mut be etimated for each latic deformation deree and conequently variation of thee coefficient with the latic train will be analyzed tartin from obtained exerimental data. The evolution of the three Lankford coefficient ( r, r and 45 r 9 ) correondin to the lane aniotroy and of the averae Lankford coefficient r correondin to the normal aniotroy are lotted reectively in fiure 5 and fiure 6 toether with a olynomial interolation equation characterizin quantitatively their variation. Fiure 5: Four deree olynomial interolation of r, r45 and r 9 Lankford coefficient accordin to their variation with the latic train Fiure 6: Four deree olynomial interolation of normal aniotroy coefficient r accordin to it variation with the latic train The correondin curve rereentin the variation of the Hill coefficient comuted from equation (9) and (1) are lotted in fiure 7 for the normal aniotroy and in fiure 8 for the lane aniotroy. Fiure 7: Variation of the comuted Hill coefficient a a function of the latic train for a normal aniotroy

6 ème Conrè Françai de Mécanique Lyon, 4 au 8 Août 15 Fiure 8: Variation of the comuted Hill coefficient a a function of the latic train for a lane aniotroy It can be oberved that durin the material latic deformation the Hill coefficient have inificance variation with reect to the latic train. Then, a comared to the etimated value for.% or % of latic deformation, the coefficient have different value for 5%, 7% or 1% of latic train. Conequently thi oberved variation mut be takin into account by more realitic numerical imulation of formin rocee a the tretchin, dee drawin and tamin. 3 The contitutive equation Dealin with the elato-latic behavior of aluminum alloy dedicated to metal formin it i convenient to ue a finite train formulation by the mean of an udated Laranian formalim already imlemented in the FE oftware ZeBuLoN [9]. The material behavior decrition i baed on contitutive equation uin tenorial internal variable and on the Green Nahdi tranformation of the tre train roblem into an equivalent material referential. Thi tandard comutation allow the decomoition of the incremental total train into elatic and inelatic art i.e. d ε + t e = dε dε. The numerical tratey enerally ued to take into account a 3D latic aniotroy in a claical Macrocoic Unified model (Table 1) i to involve a tenor of fourth order into the criterion exreion. [3, 4, 7]. f( σ ~ )= 3 ( ~ - X ~ ) : B ~ :( ~ - Table 1: Contitutive Equation of the Unified Model X ~ ) - R X =( / 3)Cα ; R=R +Q[1-ex( -bλ )] f ε =λ =λn σ ; α=λ n-dα f F( σ σ ) G( σ σ ) ( σ ~ yy ) = + H( σ - σ ) + Lσ + Mσ + Nσ yy yz xz xy - R B : Fourth order tenor X : Back tre variable α : Hardenin variable R :Size of elatic domain R : Initial ize of R Q, b, C, D: Material arameter λ : Platic multilier λ : Rereent equivalent latic train. (F, G, H, L, M, N): Hill coefficient To take into account hyical latic henomena at a meocoic cale, a multi-mechanim model ha been develoed. In thi cae the inelatic train rate 1 ε and ε i lit into two comonent: 1 ε =ε +ε where ε are the local inelatic train rate of the mechanim 1 and reectively. Table ummarize the contitutive equation of the named M1C model ( mechanim, 1 criterion).

7 ème Conrè Françai de Mécanique Lyon, 4 au 8 Août 15 Table : Multi-Mechanim model (M1C) 1 ε =ε +ε f : Yield criterion function f = ( J (σ ~ X ~ )) + ( J (σ ~ X ~ 1 ε : Tenor of latic train rate )) L - L - - R I ε : Tenor of local inelatic train rate 3 J = J (σ ~ - X ~ I )= ( ~ - X ~ I ) : L ~ : ( ~ - X ~ I of the mechanim I LI L ) ; I=1, I X : Back tre tenor of each R=R +Q[1-ex( -bλ )] mechanim I ~ 1 1 C11 C1 ~ I L ~ X ~X ; L: I 3 -X ~ : Fourth order tenor n C 1/ 11,C 1, C, D 1, D : Kinematic hardenin 3 C1 C J LI +J LI moduli 3 D α ~ 1 λ( n ~ 1 1 = - X ~ 1 ) C ; 3 D α ~ λ( n ~ = - X ~ I ) n : Normal to the yield urface for each mechanim 11 C I α : Kinematic internal variable For a multi-cale aroach takin into account the laticity a a microtructural cale, the olycrytalline laticity model eem to be the mot aroriate. In thi cae the latic deformation i the reult of the dilocation lide alon the active direction of crytallorahic ytem definin the metallic crytal tructure (FCC for AA4). Σ ~ REV Ε ~ Macrocoic Scale σ ~ ε~ Grain Scale τ γ Microcoic Scale: Sli ytem level Fiure 9: Diaram of the Polycrytalline model uin REV method Table 3: Equation of the Polycrytalline model σ ~ = Σ ~ +C( β ~ - β ~ ); β ~ = ~ ε - Dβ ~ ε~ E : Macro latic train rate Σ : Macro tre τ = σ ~ : m ~ 1 with m ~ n ~ l l n ~ σ : Local rain tre β ~ : Hardenin variable for each rain n τ - x - r C and D: Material arameter γ = in(τ - x ) τ : Local hear tre k m : Orientation tenor x = cγ - r v dx r r r Q H (1 b ) with 1 ε~ = m ~ γ =1 E ~ = ε~ r ν = γ r γ : Shear rate k and n : Material coefficient r : Variable of iotroic hardenin x : Variable of kinematic hardenin r : Initial critical hear tre b: Rate of aturation c and d : Material arameter Q : Hardenin caacity H r : Hardenin matrix

8 ème Conrè Françai de Mécanique Lyon, 4 au 8 Août 15 In a numerical oint of view the olycrytalline model ue the concet of a rereentative elementary volume (REV) which can be couled to a finite element method. Fiure 9 how a imlified diaram of the Polycrytalline model aroach detailled in Ref. [5] and adoted in thi work (Table 3). All the three above material contitutive model: the macrocoic unified, the multimechanim (MM M1C) and the olycrytalline laticity one are imlemented into the material library ZMaT and the ubroutine ZSeT of the finite element code ZeBuLoN [9] uin a θ-method olver via an imlicit Newton cheme alied for the local interation. 4 Finite element modelin and arameter identification Finite element imulation of each reented model require a well known of contitutive arameter value. Thee one can be carried in two tae. Firtly an initial etimation of the arameter value are obtained via an interactive-rahic method baed on manual comarion between the exerimental reult and thoe obtained from numerical imulation. Final identified value are obtained uin the otimization module of ZeBuLoN to imrove the olution by an automatic nonlinear rereion. Fiure 1 how the rincile of the numerical otimization technique ued to identify the arameter of a defined behavior law or contitutive model. Exerimental data Simulated curve Z-ot New et of arameter E, ʋ, K, n, External oftware Z-im Fiure 1: Otimization rincile in the ubroutine ZSeT A imilar tenile ecimen a the exerimental one i ued for the numerical imulation and it finite element meh i reented in fiure 11. Contant dilacement alied in the end node on y-direction Fixin in the node Fine meh Fiure 11: Finite Element mehe for the tenile ecimen y x

9 ème Conrè Françai de Mécanique Lyon, 4 au 8 Août 15 Fiure 1 how a numerical imulation of a tenile tet alon the rollin direction uin a Voce law decribin an iotroic hardenin (without kinematic hardenin) identified from the exerimental tenile tet at and defined by: R = 31+ 5[1- ex( -18λ )]. Fiure 1: Numerical imulation of the tenile tet It can be oberved that a homoeneou tate of train occur in the active area of the tenile ecimen which confirm the hyothei of the homoeneity ued by the claical comutation method to etimate all variable decribin the uni-axial tet, the mechanical behavior and eecially the aniotroy. Macrocoic Unified Model In the cae of the macrocoic unified model, the identification reult of all the arameter are reented in table 4, uin a lot of FE imulation. Even if variation of Hill coefficient are not taken into account it i obtained a ood areement between the exeriment and imulation of a imle tenile tet (Fiure 13). Fiure 13: Comarion between exerimental and imulated tenile tet for the macrocoic unified model without kinematic hardenin Table 4: Identified arameter of the aniotroic unified model without kinematic hardenin (MPa) Elaticity Iotroic hardenin Kinematic hardenin E υ R Q b C D Hill 48 arameter Plane aniotroy F G H L M N Normal aniotroy F G H L M N

10 ème Conrè Françai de Mécanique Lyon, 4 au 8 Août 15 Multi-Mechanim Model (M1C) For the multi-mechanim model (M1C) the identified aniotroic material arameter are hown in table 5 (the iotroic arameter L e and L f are not identified here). The numerical imulation are in very ood areement with the exeriment and more accurate than the reult iven by the reviou unified model. The comarion between the tre-train curve imulation and exerimental data were illutrated by fiure 14. Fiure 14: Comarion between exerimental and imulated tenile tet for the Multi-Mechanim (M1C) model Polycrytalline Model Table 5: Identified arameter of the aniotroic M1C model E υ R Q C 11 C C b D 1 D L a L b L c L d For the olycrytalline model of an aluminum alloy, the crytallorahic texture reorted in thi tudy i taken from the literature and ha been alo ued to characterize the AA4 material. An initial {111} ole fiure aroximated by a et of 1 orientation correondin to an initial material tate and contructed uin a numerical tereorahic rojection code (Projtern) i hown in fiure 15 (a). The identified arameter value obtained from the bet fit between the exerimental data and thoe obtained from FE imulation (fiure 15 (b)) are iven in Table 6. (a) (b) Fiure 15: (a) {111} Pole fiure and (b) comarion between exerimental and imulated tenile curve for the olycrytalline model.

11 ème Conrè Françai de Mécanique Lyon, 4 au 8 Août 15 Table 6: Identified arameter of the olycrytalline model (MPa). Elaticity Iotroic hardenin Norton arameter Kinematic hardenin E υ r Q b k n C D c d Alication to a thick heet tretchin formin roce In the reent invetiation, and due to the friction between the blank holder/blank which add an additional arameter durin the numerical imulation of the dee drawin of a thick heet, the tudy i limited to a tretch formin roce. The tretchin formin i a metal formin roce in which a iece of heet metal i tretched over a die in order to form lare contoured art and to reach the latic area o that the elonation kee ermanent. Fiure 16 how the rincile of the dee drawin roce and of the correondin device. Blank- Holder Punch Blank Die r d d d (a) (b) Fiure 16: (a) 3D illutration of the dee drawin roce and (b) The eometry of the dee drawin device. Concernin the tretchin formin thi roce i erformed on a tretch re, in which a iece of heet metal i ecurely ried alon it ede by riin jaw. The table 7 ive the characteritic dimenion of the tretchin formin device. The main advantae of the tretch formin roce i the roduction of lare art, uually made in aluminum, with lower toolin cot a comared to the dee drawin. Generally a inle a on a inle tool i required to form the heet and motly thi roce reduce the rin-back henomenon. The axiymmetric D imulation of the tretchin formin oeration i erformed uin the finite element code ZeBuLoN. The eometry of the tool and the blank i defined in fiure 17. The blank, with a thickne of 3 mm, wa modeled with 4-node linear axiymmetrical element. The imulation are erformed without a blank holder, coniderin that the heet i laced on the die and clamed on

12 ème Conrè Françai de Mécanique Lyon, 4 au 8 Août 15 it ede. The tretchin formin imulation wa conducted under dilacement control to a relative unch-dilacement initially of 8 mm. The clearance between the unch and the die i 3 mm and the friction of contact between unch/blank and blank/die were defined with a coulomb model via a friction coefficient equal to.1. d Punch y x r Blank 3 Table 7: Characteritic dimenion of the Stretchin formin device (mm) d r D d r d d Blank r d D d 8 r d Die d blank Fiure 17: The eometry and the meh ued in a finite element tretch formin roce imulation (all dimenion in mm) Fiure 18 how the evolution of the numerical unch load accordin to it movement durin the cuin for the three reented model uin numerical imulation made with ZeBuLoN code. Fiure 18: Numerical load-dilacement curve obtained for RD, TD and DD tretchin formin tet and the three model (ZeBuLoN): Macrocoic Unified, MM (M1C) and Polycrytalline Nearly imilar hae and maximum load are obtained for unified and M1C model a comared to the olycrytalline one which eem to overetimate the drawin force.

13 ème Conrè Françai de Mécanique Lyon, 4 au 8 Août 15 Numerical imulation for the macrocoic unified model were erformed via both Abaqu and ZeBuLoN code (Fiure 19) uin an iotroic hardenin Voce law [1] defined by: R = 31+ 5[1- ex( -18λ )]. Fiure 19: Numerical load-dilacement curve obtained for RD, TD and DD drawin tet uin the Macrocoic Unified model (Abaqu and ZeBuLoN). The contour lot of the equivalent tre after 8 mm of unch dilacement i hown in fiure. (RD) 45 (DD) 9 (TD) (a) (RD) 45 (DD) 9 (TD) (b) (RD) 45 (DD) 9 (TD) (c) Fiure : Contour lot of the equivalent tre at 8 mm deth: (a) Macrocoic Unified model, (b) MM (M1C) model and (c) Polycrytalline model. It can be oberved that the macrocoic unified formalim and the M1C model lead to imilar ditribution of the equivalent tre. On the other ide, the olycrytalline model ive different reult rincially due to the imortant influence of the material texture evolution. It can be concluded that for more ronounced latic deformation the local aniotroy become more imortant that the lobal aniotroy taken into account by the firt two model.

14 ème Conrè Françai de Mécanique Lyon, 4 au 8 Août 15 Concluion In thi aer exerimental analyi and three multi-cale aroache have been ued to decribe the latic aniotroy for an AA4-T351 thick heet. Startin form the claical analyi of the latic aniotroy ha been rooed an analytical etimation of the variation of Lankford and Hill coefficient with the latic train. Accordin to the three rooed contitutive model and tartin from imulation of a imle tenile tet, a ood areement i obtained between the numerical reult and the exerimental data in term of tre-train behavior. However it can be een that the Multi- Mechanim (MM) and Polycrytalline model are more accurate to imulate the aniotroy of the tudied material comared to the macrocoic unified model. Finally the numerical imulation of the AA4 thick heet tretch formin roce in different direction of the aniotroy are reented and comared for the three aroache. The load-dilacement curve and the ditribution of the equivalent tre of thi tye of heet formin roce (tretchin formin) how the feaibility of all rooed model, hihlihtin however the main advantae of the Multi-Mechanim and of the Polycrytalline model a comared to the claical unified Hill formalim uin contant value of Hill arameter. In a future will be make more comarion with exerimental reult uin channel die comreion, biaxial tenile and imle hear tet in order to obtain a comlete decrition of tridimenional latic aniotroy correondin to lare deformation. Moreover a lot of exerimental and numerical analyi of a dee drawin roce alied to a thick heet will be conidered. Reference [1] Banabic D., Bune H.-J., Pohlandt K., Tekkaya A.E.: Formability of metallic material, Sriner- Verla, Berlin Heidelber New York,. [] Seidt, J.D., Platic Deformation and Ductile Fracture of 4-T351 Aluminum under Variou Loadin Condition, Ph.D. Thei, Graduate School of the Ohio State Univerity, 1. [3] Hill, E., A theory of the yieldin and latic flow of aniotroic metal, Proc. Roy. Soc., London, [4] Zhan, S., Francillette, H., Gavru, A., Analyi of the aniotroic behavior and the formability atitude for an AA4 alloy uin channel die comreion tet and the imle tenion tet, Key En. Mat., 54-56,.3-8, 1. [5] Cailletaud, G., Saï, K., A olycrytalline model for the decrition of ratchettin: Effect of interranular and intraranular hardenin, Mat. Sci. En., 48,. 4-39, 8. [6] Saï, K., Multi-mechanim model: Preent tate and future trend, Int. J. Plat, 7,. 5-81, 11. [7] Nari, W., Gavru, A., Kouadri, A., Saï, K., Alication of multi-cale model to numerical imulation and exerimental analyi of aniotroic elatolatic behavior of metallic heet, Key En. Mat., , , 14. [8] W. Hoford, Mechanical Behavior of Material, Cambride Univerity Pre, New York, 5. [9] J. Beon, R. Leriche, R. Foerch, G. Cailletaud, Rev. Eur. Elem. Fini 7 (1998) [1] Voce, E., A ractical train hardenin function, Metalluria, 51 (37),. 19-6, 1955.