Calibration of material parameters of anisotropic yield criterion with conventional tests and biaxial test

Transcription

1 Caliation of mateial paametes of anisotopi yield iteion with onventional tests and iaxial test hunying Zhang, Lionel Leotoing, Dominique Guines, andine Thuillie To ite this vesion: hunying Zhang, Lionel Leotoing, Dominique Guines, andine Thuillie. Caliation of mateial paametes of anisotopi yield iteion with onventional tests and iaxial test. 6th annual EAFORM Confeene on Mateial Foming, Ap, Aveio, Potugal. Tans Teh Puliations, , pp.-7,, <.48/ <hal-8758> HAL Id: hal umitted on 7 Ma 4 HAL is a multi-disiplinay open aess ahive fo the deposit and dissemination of sientifi eseah douments, whethe they ae pulished o not. The douments may ome fom teahing and eseah institutions in Fane o aoad, o fom puli o pivate eseah entes. L ahive ouvete pluidisiplinaie HAL, est destinée au dépôt et à la diffusion de douments sientifiques de niveau ehehe, puliés ou non, émanant des étalissements d enseignement et de ehehe fançais ou étanges, des laoatoies pulis ou pivés.

2 Caliation of mateial paametes of anisotopi yield iteion with onventional tests and iaxial test ZHANG hunying,a, LEOTOING Lionel,, GUINE Dominique,, THUILLIER andine,d LGCGM, INA, UEB, avenue des Buttes de Coësmes, 578 Rennes Cedex Fane LIMATB, Univesité de Betagne-ud, ue de aint Maudé, BP 96, 56 Loient Cedex Fane a shunying.zhang@insa-ennes.f, lionel.leotoing@insa-ennes.f, dominique.guines@insa-ennes.f, d sandine.thuillie@univ-us.f Keywods: Plasti anisotopy, Yield iteion, Biaxial test, Mateial paamete identifiation Astat. Bon and Besson yield iteion has een used to investigate the plasti anisotopi ehavio of an aluminum alloy AA586. The paametes of this anisotopi yield model have een identified y two diffeent methods: a lassial one, onsideing seveal homogeneous onventional eiments and an loatoy one, with only iaxial test. In this pape, the paamete identifiation with onventional eiments has een aied out with uniaxial tensile tests and simple shea tests in diffeent oientations to the olling dietion and with a hydauli ulge test. Fo ompaison s sae, Hill s 48 yield funtion has also een aliated analytially fom uniaxial tensile tests. Numeial simulation fo the oss iaxial test has een aied out with the anisotopi paametes identified fom the onventional tests. Fom this simulation, the piniple stains along a speified path in the gauge aea of the uifom speimen have een evaluated. A good ageement is oseved etween eimental and numeial values of pinipal stains fo a lage ange of stain paths. Intodution heet metal foming epesents a lass of impotant poesses widely used in the manufatue industy. heet metals usually exhiit a plasti anisotopy due to pevious poesses lie olling and annealing. To optimize the numeial simulation of these foming poesses, a peise desiption of the plasti ehavio is equied. Yield funtions ae fequently applied to desie the mateial ehavio and to tae into aount the initial anisotopy and hadening fo the susequent evolution with plasti stain. Yield funtions an ontain a high nume of paametes. The aliation of these paametes equies many mehanial tests with diffeent loading modes. The lassial appoah is to identify the yield iteion paametes fom initial yield stesses and anisotopy oeffiients. Balat and o-authos [] aliated the yield funtion Yld4-8p with uniaxial tensile tests and hydauli ulge test. To identify the paametes of BBC yield funtion, Banai et al [] used uniaxial tensile test and oss iaxial test. The uifom speimen is used to otain the uniaxial flow stess in olling dietion and tansvese dietion. The iaxial stess is otained y dividing the measued foe y an effetive aea. The limit of this appoah lies in the impeision of the detemination of the initial yield stess due to the diffiulty to define the effetive aea. Anothe appoah onsists in a paamete identifiation ove the tempoal evolution of eimental data. Zang et al [] onsideed a omination of stess level in uniaxial tension, simple shea test, Baushinge test and ulge test to identify the Bon and Besson yield funtion [4].

3 In the pesent atile, the plasti anisotopy of an AA586 is haateized fom onventional eiments: uniaxial tensile test, ulge test and simple shea test. Fom these onventional esults, yield funtions have een identified: the well nown Hill s 948 yield funtion [5] and a moe sophistiated one, the Bon and Besson yield funtion. Finally, a iaxial tensile test on a uifom speimen is poposed. The identified Bon and Besson yield funtion is then intodued in the FE model of the iaxial tensile test. Compaison etween eimental and numeial esults of pinipal stains along a speified path is pefomed. It is shown that the oss-iaxial test involves a lage ange of stain paths. Mehanial ehavio of AA586 sheets of mm thiness Assuming othotopi symmety, ( x, y, z ae espetively the olling dietion (RD, the tansvese dietion (TD and the nomal dietion (ND. In the fame of a uniaxial tensile test, ( x,, z ae espetively the elongation dietion, the tansvese dietion and the nomal dietion. Hill s 948 yield funtion. Hill s 948 othotopi yield funtion is witten in the following fom: ψ F( + G( + H ( + L + M + N Y ( zz zz yz xz xy whee ψ denotes the yield funtion, is the equivalent stess, Y a efeene yield stess of the mateial, F, G, H, L, M and N ae mateial paametes. With the ondition: G + H, and plane stess ondition (, thee independent anisotopi paametes have to e zz xz yz identified ( F, G and N : F, G + + and ( + ( + N 9 ( 9 ( + Hill s paametes an e alulated y thee anisotopi oeffiients (,, 9 z z ( Bon and Besson yield funtion. Bon and Besson poposed a 6 paametes yield funtion whih is in the fom: a a a ( defined y: ψ α Y (4 The yield funtion ψ onsists in two onvex funtions. α ae positive oeffiients, the sum of whih is equal to. ae essed in the fom: + + ( whee a,, and α ( α α ae fou isotopi paametes whih define the shape of the yield sufae. i ae the pinipal values of the tansfomed stess deviatos s, defined y: s L (

4 with L whee i ae paametes whih ae elated to the anisotopy of the mateial. Thee ae a total of 6 paametes in Bon and Besson yield model. In plane stess ondition, the paamete nume edues to with. 5 6 Mateial data. In this wo, the paametes of the Bon and Besson yield funtion desied aove have een aliated y: thee yield stesses (, and 9 and thee anisotopi oeffiients (, and 9 otained fom the uniaxial tensile tests in diffeent oientations; thee simple shea stesses (, and 9 fom thee simple shea tests in diffeent oientations; an equi-iaxial yield stess ( and a iaxial oeffiient ( fom iaxial tension (ulge test. Mateial paamete identifiation. In the plane stess ondition,,, xy and yx ae the only non-zeo stess omponents. With the assoiated flow ule: λ (7 tain inement an e witten in the fom of yield funtion gadient: xy λ (8 yx ( + uppose that the uniaxial tensile test is pefomed along a dietion defined y an oientation angle fom the olling dietion. Then the anisotopi oeffiient whih is assoiated with oientation angle an e alulated y: sin ( sin ( os ( + xy (9 z z ( + ( The iaxial oeffiient is alulated in the fom: zz (6 (

5 The identifiation poess an e aied out y minimizing the following ost funtion: δ ( a, i, α, i Y j ( Y j + ( Y + ( Y + + Y ( Y j whee Y is the efeene yield stess detemined y the flow stess along olling dietion; ae espetively measued anisotopi oeffiients and iaxial oeffiient; j ( j j j ( and j (Eq. 9 and j and ae (Eq. ae pedited anisotopi oeffiients and iaxial oeffiients; j, equivalent stesses otained y the definition of yield funtion (Eq. 4 and essed in tems of the measued yield stess (,iaxial stess ( and shea stess (. A weight oeffiient is given j to the shea stess sine the unetainty of the detemination of the initial shea stess. The majo tas lies in the optimization of the anisotopi paametes to minimize the ost funtion. The algoithm Boyden Flethe Goldfa hanno (BFG is pefeed hee. BFG is an appoximate Newton's method, whih is a hill-liming optimization. Appliation fo AA586. Expeimental mateial data fo AA586 is given in Tale. The initial yield stess fo uniaxial tensile test is detemined at the onset of a non-lineaity on the equivalent stess vesus equivalent stain uve. Fo simple shea test and hydauli ulge test, the initial yield point is detemined y the offset equivalent plasti stain.%. The anisotopi oeffiients (,, 9 ae defined as the aveage value in an equivalent plasti stain ange fom 5% to 5%. Y 9 Tale. Mateial data The paametes of Bon and Besson yield funtion alulated fom the eimental data pesented aove ae given in Tale. Tale. Anisotopi paametes of Bon and Besson yield funtion Bon and Besson yield funtion α a As a ompaison, Hill s 48 yield iteion has een aliated y the thee (,, 9 values : Tale. Anisotopi paametes of Hill s 948 yield funtion Hill s 948 yield funtion F G H N Figs. to 4 show the esults fo oth Bon and Besson and Hill s 948 yield funtions. Both funtions well pedit the anisotopi oeffiients. Fo the uniaxial yield stesses, Bon and Besson yield funtion desies them well, while Hill s 948 yield model has not the aility to adjust its 4 4

6 vaiation. Fo the shea stesses, ompaing with eimental values, Bon and Besson yield funtion shows a little oveestimation ut has a good desiption of thei vaiation. The assoiated yield sufaes fo the two yield funtions pesent signifiant diffeenes Bon and Besson Expeiment Hill Angle aoding to olling dietion Fig. Expeimental and pedited values Bon and Besson Expeiment Hill Angle aoding to olling dietion Fig. Expeimental and pedited shea stess Bon and Besson Expeiment Hill Angle aoding to olling dietion Fig. Expeimental and pedited uniaxial stess Y Bon et Besson -. Hill Y Fig. 4 Pedited yield sufae ontous Biaxial tensile test To simplify oth the design of the eimental dataase and the aliation of anisotopi paametes of the yield funtion, a seond method ased on a oss iaxial tensile test is uently investigated. This test seems patiulaly inteesting sine diffeent stain paths an e otained simultaneously with a unique speimen. Up to now, few wos have een foused on the paamete identifiation of yield funtion fom the iaxial tensile test. Geen et al [6] have pefomed oss iaxial test with seven diffeent popotional paths to identify the paametes of the yield funtions whih ouldn t e identified y uniaxial tensile test. Teaa et al [7] poposed to identify the FMM yield funtion [8] in omining esults of uniaxial tensile tests and oss iaxial test. Only two paametes of the yield model wee aliated fom a stain analysis of the ental pat of the uifom speimen. In this pape, the potential of the iaxial tensile test on a uifom speimen is analyzed. A uifom speimen shape has een designed and is shown in Fig. 5. Expeiments on a sevohydauli testing mahine have een pefomed with a onstant veloity atio v / mm/s imposed on the fou ams of the uifom speimen. The ental zone of the speimen is filmed y a amea and a digital image oelation (DIC tehni is used to ompute the in-plane stains. x v y

7 Fig. 5 Geomety of quate uifom speimen Fig. 6 FE ounday onditions Finite element simulations of the iaxial test have een aied out with the ommeial softwae ABAQU. The anisotopi ehavio of the mateial is modeled y Bon and Besson yield funtion implemented though the use suoutine UMAT with the paametes alulated aove (Tale. Due to the symmety of the uifom speimen, only a quate of the speimen is modeled. Displaements d d mm ae imposed on two ams of the uifom speimen duing the x y 4 simulation poess. As pesented in Fig. 7, the pinipal stains at the nodes along the diagonal path OA (Fig. 5 in the gauge aea of the speimen have een otained and ompaed with the eimental values. The pedited pinipal stains ae quite lose to the eimental ones. Fig. 8 pesents the eimental and pedited atio of the pinipal stains along the path OA. As an e seen, the gauge aea of the uifom speimen ontains a lage ange of stain path: fom equi-iaxial to uniaxial stain path, whih an e patiulaly inteesting fo the paamete identifiation instead of the aove tests used in the onventional appoah. The paamete identifiation poess with only oss iaxial tensile test will e aied out in next eseah wos y minimizing the eimental and FE simulated pinipal stains along diffeent paths. Majo stain Majo stain_bon and Besson Mino stain_bon and Besson Majo stain_expeiment Mino stain_expeiment 5 5 Initial distane fom ente Fig. 7 Expeimental and pedited pinipal stains along a diagonal path E/E Pedited piniple stain atio E/E Expeimental piniple stain atio E/E Initial distane fom ente Fig. 8 Expeimental and pedited pinipal stain atio along a diagonal path Conlusion This eseah wo fouses on the anisotopi paamete aliation of Bon and Besson yield funtion used to investigate the plasti anisotopi ehavio of AA586 sheet. The paametes of the yield funtion have een alulated fom a lassial appoah ased on thee uniaxial tensile tests, thee simple shea tests and a hydauli ulge test. Numeial simulation of the oss iaxial test has een pefomed with the paametes otained y this lassial appoah. The pedited pinipal stains along a speified path in the gauge aea of the uifom speimen ae in good ageement

8 with eimental ones. Based on this oss iaxial tensile test, a new identifiation poedue of the anisotopi paametes of Bon and Besson yield funtion will e poposed in futue wos. Refeenes [] F. Balat, H. Aetz, J. W. Yoon, M. E. Kaain, J. C. Bem, and R. E. Di, Linea tansfomation-ased anisotopi yield funtions, Intenational Jounal of Plastiity. ( [] D. Banai, H. Aetz, D.. Comsa, and L. Paaianu, An impoved analytial desiption of othotopy in metalli sheets, Intenational Jounal of Plastiity. ( []. L. Zang,. Thuillie, A. Le Pot, and P. Y. Manah, Pedition of anisotopy and hadening fo metalli sheets in tension, simple shea and iaxial tension, Intenational Jounal of Mehanial ienes. 5 ( [4] F. Bon and J. Besson, A yield funtion fo anisotopi mateials Appliation to aluminum alloys, Intenational Jounal of Plastiity. ( [5] R. Hill, A Theoy of the Yielding and Plasti Flow of Anisotopi Metals, Poeedings of the Royal oiety of London. eies A, Mathematial and Physial ienes. 9 ( [6] D. E. Geen, K. W. Neale,. R. MaEwen, A. Mainde, and R. Pein, Expeimental investigation of the iaxial ehaviou of an aluminum sheet, Intenational Jounal of Plastiity. ( [7] M. Teaa, I. Chapentie, M. Matiny, and G. Feon, Identifiation of sheet metal plasti anisotopy using heteogeneous iaxial tensile tests, Intenational Jounal of Mehanial ienes. 5 ( [8] G. Feon, R. Maou, and J. Moeale, A paameti desiption of othotopi plastiity in metal sheets, Intenational Jounal of Plastiity. (