A NEW YIELD CRITERION FOR ORTHOTROPIC SHEET METALS UNDER PLANE-STRESS CONDITIONS

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1 In: Proc. of 7 th COLD METAL FORMING Conference ( Ed. D. Banaic) May -1 Cluj Naoca Romania ag (htt:// A NEW IELD CRITERION FOR ORTHOTROPIC SHEET METALS UNDER PLANE-STRESS CONDITIONS D. Banaic T. Balan D. S. Coma Technical Univerity of Cluj-Naoca Romania Atract The aer reent a new yield criterion for orthotroic heet metal under lane-tre condition. The criterion i derived from the one rooed y Barlat and Lian in Two additional coefficient have een introduced in order to allow a etter rereentation of the latic ehaviour of the orthotroic heet metal. The rediction of the new yield criterion are comared with the erimental data for two material. Keyword: heet metal aniotroy yield criterion 1 Content The comuter imulation of heet metal forming rocee need a quantitative decrition of latic aniotroy y the yield locu of the material. For taking into account the aniotroy the von Mie claical yield criterion mut e modified. A reentation of the hitorical develoment of the aniotroic yield criteria may e found in [1 ]. In thi work the recie decrition of comlex yielding ehaviour exhiited y heet metal i aroached from the viewoint. A new yield criterion comining the advantage of the Barlat and Karafilli-Boyce criteria i develoed. The aility of the new criterion to rereent the latic ehaviour of orthotroic heet metal i invetigated. Equation of the yield urface A yield urface i generally decried y an imlicit equation of the form Φ ( ): (1) i the equivalent tre and i a yield arameter. In ractice may e choen a one of the following arameter of the heet metal: (uniaxial yield tre along the rolling direction) 9 (uniaxial yield tre along the tranvere direction) 45 (uniaxial yield tre at 45 from the rolling direction) an average of 9 and 45 or (equi-iaxial yield tre). The equivalent tre i defined y the following relationhi:

2 In: Proc. of 7 th COLD METAL FORMING Conference ( Ed. D. Banaic) May -1 Cluj Naoca Romania ag (htt:// 1 k k k [ a( Γ cψ ) a( Γ cψ) ( 1 a)( cψ) ] k () a c and k are material arameter while G and? are function of the econd and third invariant of a fictitiou deviatoric tre tenor which will e decried later on. One may notice that the aove reion of the equivalent tre i derived from the one rooed y Barlat and Lian for orthotroic material under lane-tre tate [3]. Two additional arameter namely and c have een introduced in order to allow a etter rereentation of the latic ehaviour of the heet metal. The convexity of the yield urface decried y Eqn (1) and () i enured if a [ 1] and k i a trictly oitive integer numer. A we have already mentioned G and? are function of the econd and third invariant of a fictitiou deviatoric tre tenor. Thi tenor i related to the actual tre tenor y the Karafilli-Boyce linear tranformation [4]: 1 d g 1 e 1 g 1 e 3 f ( d e) ( e f ) d e f and g are alo material arameter. The comonent of the tre tenor in Eqn (3) are reed in the ytem of orthotroic axe (1 i the rolling direction - RD i the tranvere direction - TD and 3 i the normal direction - ND). The econd and third invariant of the deviatoric tenor have the following reion: 3 (3) J ( ) det J ( ) γγ 3 det γγ (4) the Greek indice take the value 1 and. The quantitie I γγ I 3 det (5) are not affected y the rotation that leave unchanged the third axi (ND). Thu in the cae of the lane-tre of heet metal we can ue I and I 3 intead of J and J 3 in order to define the function G and?. We have adoted the following reion for thee function: Γ I 3 (6) By uing Eqn (6) (5) and (3) we can re G and? a licit deendencie of the actual tre comonent: I Ψ I 1 ( P Q ) R 1 1 Γ M N Ψ (7) d e e f M d e N e f P Q R g (8) The aove equation how that the hae of the yield urface i defined y the material arameter a c d e f g and k. From thee arameter k ha a ditinct tatu. More

3 In: Proc. of 7 th COLD METAL FORMING Conference ( Ed. D. Banaic) May -1 Cluj Naoca Romania ag (htt:// reciely it value i et in accordance with the crytallograhic tructure of the material [5]: k 3 for BCC alloy and k 4 for FCC alloy. The other arameter are etalihed in uch a way that the contitutive equation aociated to the yield urface reroduce a well a oile the latic ehaviour of the actual material. The rocedure ued for identifying the arameter a c d e f and g i decried in 4. 3 Flow rule The flow rule aociated to the yield urface reented in 1 i Φ λ α β 1 (9) are in-lane comonent of the latic train-rate tenor and? i a calar multilier. The value of the non-lanar comonent of the latic train-rate tenor are retricted y the lane-tre condition and the iochoric character of the latic deformation: 3 ε3 ε31 ε13 33 ( & ) ε & & & & ε Auming a urely iotroic hardening of the material only one calar tate arameter i needed in order to decrie the evolution of the yield urface. Thi arameter i the o-called equivalent latic train comuted a a time-integral of the equivalent latic train-rate: (1) ε t ε. dt () The equivalent latic train-rate i defined y equating the ower develoed y to the tre tenor and the ower aociated to the equivalent tre: ε. & ε Uing the homogeneity of the equivalent tre (ee Eqn ()) one can rove that the calar multilier? i in fact the equivalent latic train-rate. Thu the flow rule (9) take the following form: (1). Φ ε α β 1 (13) 4 Identification rocedure The arameter a c d e f and g in the reion of the equivalent tre are etalihed in uch a way that the contitutive equation aociated to the yield urface reroduce a well a oile the following characteritic of the orthotroic heet- metal: (yield tre otained y a uniaxial tenile tet along RD) 9 (yield tre otained y a uniaxial tenile tet along TD) 45 (yield tre otained y a uniaxial tenile tet along a direction equally

4 In: Proc. of 7 th COLD METAL FORMING Conference ( Ed. D. Banaic) May -1 Cluj Naoca Romania ag (htt:// inclined to RD and TD) (yield tre otained y an equi-iaxial tenile tet along RD and TD) r (coefficient of latic aniotroy aociated to RD) r 9 (coefficient of latic aniotroy aociated to TD) and r 45 (coefficient of latic aniotroy aociated to a direction equally inclined to RD and TD). There are a many condition a the material arameter in the reion of the equivalent tre. Thu it i oile to otain their value y olving a et of even nonlinear equation. But thi i a difficult aroach ecaue the et of equation ha multile olution. After everal trial and comarion with erimental data we have concluded that the et olution i to avoid the trict enforcement of the retriction mentioned aove. A more effective trategy of identification i to imoe the minimization of the following error function: F ( a c d e) r r r r r r (14) 9 45 r r 9 and r 45 are the uniaxial yield tree the equi-iaxial yield tre and the coefficient of latic aniotroy redicted y the contitutive equation. In order to ue the function defined y Eqn (14) in a minimization rocedure we need ome formula for calculating thee quantitie. 4.1 Prediction of the uniaxial yield tre Let f > e the yield tre otained y the uniaxial tenile tet of a ecimen cut at an angle f [ 9 ] with the rolling direction. In thi cae the non-zero comonent of the tre tenor (reed in the ytem of orthtroic axe) are given y the following relationhi: co in 1 1 in co (15) Eqn (1) () (7) and (15) allow the otention of a formula for evaluating the uniaxial yield tre at different angle with the rolling direction: 1 (16) k k k a A cb a A cb a cb [ ( ) ( ) ( 1 )( ) ] k A M co N in B P co N in R in co (17) 4. Prediction of the equiiaxial yield tre Let > e the yield tre otained y an equi-iaxial tenile tet along RD and TD. The in-lane comonent of the tre tenor are in thi cae a follow:

5 In: Proc. of 7 th COLD METAL FORMING Conference ( Ed. D. Banaic) May -1 Cluj Naoca Romania ag (htt:// 1 1 (18) Eqn (1) () (7) and (18) lead to a formula for evaluating the equiiaxial yield tre: 1 k k k [ a( A cb ) a( A cb ) ( 1 a)( cb ) ] k (19) A M N B P Q () 4.3 Prediction of the r-coefficient The coefficient of latic aniotroy aociated to a direction inclined at an angle f [ 9 ] with the rolling direction i defined a follow: r i the comonent of the latic train-rate tenor aociated to a direction erendicular to the longitudinal axi of the tenile ecimen and 33 i the comonent of the ame tenor aociated to DN. By uing the volume contancy condition we can rewrite Eqn(1) in the form (1) r & ε () on i the comonent of the latic train-rate tenor along the ecimen axi. Further may e written a & co in 1 in co ε (3) By uing Eqn () (3) (13) (1) and the homogeneity of the equivalent tre we arrive at the following reion of the latic aniotroy coefficient r Φ Γ Γ f i given y Eqn (19) and Γ Φ Ψ Ψ Ψ (4)

6 In: Proc. of 7 th COLD METAL FORMING Conference ( Ed. D. Banaic) May -1 Cluj Naoca Romania ag (htt:// Φ Γ Φ c Ψ Γ M Ψ P B k a A k a Γ N Ψ Q P co Q in B ( cb ) k a( A cb ) k ( A cb ) k a( A cb ) k ( 1 a)( cb ) P co Q in k (5) Eqn (16) (19) and (4) are ued in order to evaluate the quantitie involved in the error function F. We have adoted the downhill imlex method rooed y Nelder and Mead [6] for the numerical minimization ecaue it doe not need the evaluation of the gradient. The minimization rocedure ha een imlemented into a comuter rogramme written in the C language. The numerical reult reented in the next ection have een otained uing thi rogramme. 5 Comarion with eriment The rediction of the new yield criterion have een teted for two ort of heet metal: A6XXX-T4 and SPCE. The reult have een comared with the erimental data ulihed in [7 8]. Tale 1 how the erimental value needed a inut data y the comuter rogramme ued for the numerical identification of the material arameter involved in the reion of the yield criterion. Tale 1. Exerimental value ued a inut data for the numerical identification of the material arameter for A6XXX-T4 and SPCE heet metal 9 45 r r 9 r 45 k A6XXX-T SPCE Tale. Material arameter for A6XXX-T4 and SPCE heet metal otained y numerical identification A6XXX-T SPCE Tale how the value of the material arameter a c M N P Q and R otained y numerical identification for the two ort of heet metal. The correonding value of k and are alo reented for comletene. a c M N P Q R k

7 r r In: Proc. of 7 th COLD METAL FORMING Conference ( Ed. D. Banaic) May -1 Cluj Naoca Romania ag (htt:// y 5-5 y eriment x x eriment Fig. 1. ield urface for A6XXX-T4 Fig.. ield urface for SPCE eriment eriment ield tre 13 1 ield tre Angle with the rolling direction [ o ] Angle with the rolling direction [ o ] Fig. 3. Ditriution of the uniaxial yield tre for A6XXX-T4 Fig. 4. Ditriution of the uniaxial yield tre for SPCE.9.8 eriment.8 eriment Angle with the rolling direction [ o ] Angle with the rolling direction [ o ] Fig. 5. Ditriution of the r-coefficient for A6XXX-T4 Fig. 6. Ditriution of the r-coefficient for SPCE

8 In: Proc. of 7 th COLD METAL FORMING Conference ( Ed. D. Banaic) May -1 Cluj Naoca Romania ag (htt:// The yield urface redicted y the new yield criterion for the A6XXX-T4 and SPCE heet metal are reented in Figure 1 and reectively. The erimental data are alo lotted on the diagram. The redicted ditriution of the uniaxial yield tre with reect to the angle with the rolling direction i hown in Figure 3 and 4 for A6XXX-T4 and SPCE heet metal reectively. The redicted ditriution of the r-coefficient with reect to the angle with the rolling direction i hown in Figure 5 and 6 for A6XXX-T4 and SPCE heet metal reectively. 6 Concluion A new yield criterion derived from the one introduced y Barlat and Lian [3] ha een rooed. The new criterion ha an increaed flexiility due to the fact that it ue even coefficient in order to decrie the yield urface. The minimization of an error-function ha een ued for the numerical identication of the coefficient. The redicted yield urface for two material (A6XXX-T4 and SPCE) are in very good agreement with the erimental data ulihed y Kuwaara et al. [7 8]. The aociated flow rule redict very accurately the ditriution of the Lanckford coefficient and uniaxial yield tre reectively. Reference [1] Banaic D.; Müller W. and Pöhlandt K.: Exerimental determination of yield loci for heet metal Firt Conf. ESAFORM 98 Sohia Antioli France [] Banaic D.; Balan T.; Pohlandt K.: Analitical and erimental invetigation on aniotroic yield criteria 6 th Int. Conf. "ICTP99" Nuremerg [3] Barlat F. and Lian J.: Platic ehaviour and tretchaility of heet metal (Part I) A yield function for orthotroic heet under lane tre condition Int. J. of Platicity 5 (1989) [4] Karafilli A.P. and Boyce M.C.: A general aniotroic yield criterion uing ound and a tranformation weighting tenor J. Mech. Phy. Solid 41 (1993) [5] Hoford W.F.: A generalized iotroic yield criterion J.Al. Mech. 39(197)67-69 [6] Pre W.H. et al.: Numerical Recie in C. The Art of Scientific Comuting Camridge Univerity Pre Camridge 199. [7] Kuwaara T.; Ikeda S.; Kuroda K.: Meaurement and Analyi of Differential Work Hardening in Cold-Rolled Steel Sheet under Biaxial Tenion J. Mater. Proce. Technol (1998) [8] Kuwaara T.; Van Bael A.: Meaurement and Analyi of ield Locu of Sheet Aluminum Alloy 6XXX In: Proc. 4th Int. Conf. and Workho on Numerical Simulation of 3D Sheet Forming Procee Beançon (1999) 85-9.