Eight-Node Solid Element for Thick Shell Simulations

Transcription

1 Eigh-Node Solid Elemen for Thick Shell Simulaions Yong Guo Livermore Sofware Technology Corporaion Livermore, California, USA Absrac An eigh-node hexahedral solid elemen is incorporaed ino LS-DYNA o simulae hick shell srucure. The elemen formulaions are derived in a coroaional coordinae sysem and he srain operaor is calculaed wih a Taylor series expansion abou he cener of he elemen. Special reamens are made on he dilaaional srain componen and shear srain componens o eliminae he volumeric and shear locking. The use of consisen angenial siffness and geomeric siffness grealy improves he convergence rae in implici analysis. INTRODUCTION Large scale finie elemen analyses are exensively used in engineering designs and process conrols. For example, in auomobile crashworhiness, hundreds of housands of unknowns are involved in he compuer simulaion models, and in meal forming processing, ess in he design of new dies or new producs are done by numerical compuaions insead of cosly experimens. The efficiency of he elemens is of crucial imporance o speed up he design processes and reduce he compuaional coss for hese problems. Over he pas en years considerable progress has been achieved in developing fas and reliable elemens. In he simulaion of shell srucures, Belyschko-Lin-Tsay (Belyschko, 98a) and Hughes- Liu (Hughes, 98a and 98b) shell elemens are widely used. However, in some cases hick shell elemens are more suiable. For example, in he shee meal forming wih large curvaure, radiional hin shell elemens canno give saisfacory resuls. Also hin shell elemens canno give us deailed srain informaion hough he hickness. In LS-DYNA, he eigh-node solid hick shell elemen is sill based on he Hughes-Liu and Belyschko-Lin-Tsay shells (Hallquis, 998). A new eigh-node solid elemen based on Liu, 985, 99 and 998 is incorporaed ino LS-DYNA, inended for hick shell simulaion. The srain operaor of his elemen is derived from a Taylor series expansion and special reamens on srain componens are uilized o avoid volumeric and shear locking. The organizaion of his paper is as follows. The elemen formulaions are described in he nex secion. Several numerical problems are sudied in he hird secion, followed by he conclusions. ELEMENT FORMULATIONS Srain Operaor The new elemen is based on he eigh-node hexahedral elemen proposed and enhanced by Liu, 985, 99, 998. For an eigh-node hexahedral elemen, he spaial coordinaes, x i,andhe velociy componens, v i, in he elemen are approximaed in erms of nodal values, x ia and v ia, 8-7

2 by 8X x i = N a (ο; ; )x ia ; () a= 8X v i = N a (ο; ; )v ia ; i =; ; 3 () a= where he rilinear shape funcions are expressed as N a (ο; ; ) = 8 ( + ο aο)( + a )( + a ) (3) and he subscrips i and a denoe coordinae componens ranging from one o hree and he elemen nodal numbers ranging from one o eigh, respecively. The referenial coordinaes ο, and of node a are denoed by ο a, a and a, respecively. The srain rae (or rae of deformaion), _", is composed of six componens, _" =[ " xx " yy " zz " xy " yz " zx ] () and is relaed o he nodal velociies by a srain operaor, B, where B = 6 B xx B yy B zz B xy B yz B zx _" = B(ο; ; )v; (5) v =[ v x v y v z v x8 v y8 v z8 ]; (6) = 6 and B, B and B 3 are gradien vecors, B () 0 0 B (8) B () 0 0 B (8) B 3 () 0 0 B 3 (8) B () B () 0 B (8) B (8) 0 0 B 3 () B () 0 B 3 (8) B (8) B 3 () 0 B () B 3 (8) 0 B (8) B B B = N ;x(ο; ; ) N ;y (ο; ; ) N ;z (ο; ; ) (7) 3 5 : (8) Unlike sandard solid elemen where he srain operaor is compued by differeniaing he shape funcions, he srain operaor for his new elemen is expanded in a Taylor series abou he elemen cener up o bilinear erms as follows (Liu, 99, 998), B(ο; ; ) =B(0) +B ;ο (0) ο + B ; (0) + B ; (0) + B ;ο (0) ο + B ; (0) + B ; ο (0) ο Λ : (9) The firs erm on he righ-hand side of he above equaion (9) corresponds o he consan srain raes evaluaed a he cenral poin and he remaining erms are linear and bilinear srain rae erms. 8-8

3 Le x = x = x x x 3 x x 5 x 6 x 7 x 8 Λ ; (0) x = y = y y y 3 y y 5 y 6 y 7 y 8 Λ ; () x 3 = z = z z z 3 z z 5 z 6 z 7 z 8 Λ ; () ο = Λ ; (3) = Λ ; () = Λ ; (5) he Jacobian marix a he cener of he elemen can be evaluaed as J(0) =[J ij ]= ο x ο y ο z x y z 8 x y z 3 5 ; (6) he deerminan of he Jacobian marix is denoed by j 0 and he inverse marix of J(0) is denoed by D D =[D ij ]=J (0): (7) The gradien vecors and heir derivaives wih respec o he naural coordinaes a he cener of he elemen are given as follows, b = N ;x (0) = 8 [D ο + D + D 3 ] ; (8) b = N ;y (0) = 8 [D ο + D + D 3 ] ; (9) b 3 = N ;z (0) = 8 [D 3ο + D 3 + D 33 ] : (0) b ;ο = N ;xο (0) = 8 [D fl + D 3 fl ] ; () b ;ο = N ;yο (0) = 8 [D fl + D 3 fl ] ; () b 3;ο = N ;zο (0) = 8 [D 3fl + D 33 fl ] ; (3) b ; = N ;x (0) = 8 [D fl + D 3 fl 3 ] ; () b ; = N ;y (0) = 8 [D fl + D 3 fl 3 ] ; (5) b 3; = N ;z (0) = 8 [D 3fl + D 33 fl 3 ] ; (6) b ; = N ;x (0) = 8 [D fl + D fl 3 ] ; (7) b ; = N ;y (0) = 8 [D fl + D fl 3 ] ; (8) b 3; = N ;z (0) = 8 [D 3fl + D 3 fl 3 ] ; (9) b ;ο = N ;xο (0) = 8 b ;ο = N ;yο (0) = 8 D3 fl Λ p x i bi;ο r x i bi; ; (30) Λ D3 fl p x i bi;ο r x i bi; ; (3) 8-9

4 b 3;ο = N ;zο (0) = 8 b ; = N ;x (0) = 8 b ; = N ;y (0) = 8 b 3; = N ;z (0) = 8 b ; ο = N ;x ο (0) = 8 b ; ο = N ;y ο (0) = 8 b 3; ο = N ;z ο (0) = 8 Λ D33 fl p 3 x i bi;ο r 3 x i bi; ; (3) Λ D fl q x i bi; p x i bi; ; (33) Λ D fl q x i bi; p x i bi; ; (3) Λ D3 fl q 3 x i bi; p 3 x i bi; ; (35) Λ D fl r x i bi; q x i bi;ο ; (36) Λ D fl r x i bi; q x i bi;ο ; (37) Λ D3 fl r 3 x i bi; q 3 x i bi;ο ; (38) where p i = D i h + D i3 h 3 ; (39) q i = D i h + D i h 3 ; (0) r i = D i h + D i3 h ; () fl ff = h ff h ff x i bi ; () and h h h 3 h = [ ]; (3) = [ ]; () = [ ]; (5) = [ ]: (6) In he above equaions h is he ο -hourglass vecor, h he -hourglass vecor, h 3 he οhourglass vecor and h he ο -hourglass vecor. They are he zero energy-deformaion modes associaed wih he one-poin-quadraure elemen which resul in a non-consan srain field in he elemen (Flanagan, 98, Belyschko, 98 and Liu, 98). The fl ff in equaions () (38) are he sabilizaion vecors. They are orhogonal o he linear displacemen field and provide a consisen sabilizaion for he elemen. The srain operaors, B(ο; ; ), can be decomposed ino wo pars, he dilaaional par, B dil (ο; ; ), and he deviaoric par, B dev (ο; ; ), boh of which can be expanded abou he elemen cener as in equaion (9) B dil (ο; ; ) = B dil (0) +B dil ;ο B dev (ο; ; ) = B dev (0) +B dev ;ο (0) ο + Bdil ; (0) + Bdil ; (0) + B dil ;ο (0) ο + Bdil ; (0) + Bdil ; ο (0) ολ ; (7) + B dev ;ο (0) ο + Bdev ; (0) + Bdev ; (0) (0) ο + Bdev ; (0) + Bdev ; ο (0) ολ : (8) To avoid volumeric locking, he dilaaional par of he srain operaors is evaluaed only a one quadraure poin, he cener of he elemen, i.e., hey are consan erms B dil (ο; ; ) =B dil (0): (9) 8-30

5 To remove shear locking, he deviaoric srain submarices can be wrien in an orhogonal coroaional coordinae sysem roaing wih he elemen as B dev xx B dev yy B dev zz B dev xy B dev yz B dev zx (ο; ; ) = Bdev xx (0) +Bdev xx;ο (0) ο + Bdev xx; (0) + Bdev xx; (0) + B dev xx;ο (0) ο + Bdev xx; (0) + Bdev xx; ο (0) ολ ; (50) (ο; ; ) = Bdev yy (0) +Bdev yy;ο (0) ο + Bdev yy; (0) + Bdev yy; (0) (ο; ; ) = Bdev zz + B dev yy;ο (0) ο + Bdev yy; (0) + Bdev yy; ο (0) ολ ; (5) (0) +Bdev zz;ο (0) ο + Bdev zz; (0) + Bdev zz; (0) + B dev zz;ο (0) ο + Bdev zz; (0) + Bdev zz; ο (0) ολ ; (5) (ο; ; ) = Bdev xy (0) +Bdev xy; (0) ; (53) (ο; ; ) = Bdev yz (0) +Bdev yz;ο (0) ο; (5) (ο; ; ) = Bdev zx (0) +Bdev zx; (0) : (55) Here, only one linear erm is lef for shear srain componens such ha he modes causing shear locking are removed. The normal srain componens keep all non-consan erms given in equaion (8). Summaion of equaion (9) and equaions (50) (55) yields he following srain submarices which can eliminae he shear and volumeric locking: B xx (ο; ; ) = B xx (0) +B dev xx;ο (0) ο + Bdev xx; (0) + Bdev xx; (0) + B dev xx;ο (0) ο + Bdev xx; (0) + Bdev xx; ο (0) ολ ; (56) B yy (ο; ; ) = B yy (0) +B dev yy;ο (0) ο + Bdev yy; (0) + Bdev yy; (0) + B dev yy;ο (0) ο + Bdev yy; (0) + Bdev yy; ο (0) ολ ; (57) B zz (ο; ; ) = B zz (0) +B dev zz;ο (0) ο + Bdev zz; (0) + Bdev zz; (0) + B dev zz;ο (0) ο + Bdev zz; (0) + Bdev zz; ο (0) ολ ; (58) B xy (ο; ; ) = B xy (0) +B dev xy; (0) ; (59) B yz (ο; ; ) = B yz (0) +B dev yz;ο (0) ο; (60) B zx (ο; ; ) = B zx (0) +B dev zx; (0) : (6) I is noed ha he elemens developed above can no pass he pach es if he elemens are skewed. To remedy his drawback, he gradien vecors defined in (8) (0) are replaced by he uniform gradien marices, proposed by Flanaga, 98, ~ b ~b ~b = V e ZΩ e B (ο; ; ) B (ο; ; ) B 3 (ο; ; ) where V e is he elemen volume and he sabilizaion vecors are redefined as 3 5 dv; (6) ~fl ff = h ff h ff x i ~b i : (63) 8-3

6 z y ^e e^ 3 g e^ g ζ=0 x Figure : Definiion of coroaional coordinae sysem The elemen using he srain submarices (56)-(6) and uniform gradien marices (6) wih four poin quadraure scheme is called HEXDS elemen. Coroaional Coordinae Sysem In elemens for shell/plae srucure simulaions, he eliminaion of he shear locking depends on he proper reamen of he shear srain. I is necessary o aach a local coordinae sysem o he elemen so ha he srain ensor in his local sysem is relevan for he reamen. The coroaional coordinae sysem deermined here is one of he mos convenien ways o define such a local sysem. A coroaional coordinae sysem is defined as a Caresian coordinae sysem which roaes wih he elemen. Le fx a ;y a ;z a g denoe he curren nodal spaial coordinaes in he global sysem. For each quadraure poin wih naural coordinaes (ο; ; ), we can have wo angen direcions on he midsurface ( =0)wihin he elemen (see Fig. )» @ο @ = N a;ο x a N a;ο y a N a;ο z a Λ(ο; ;0) ; (6) = N a; x a N a; y a N a; z a Λ(ο; ;0) : (65) The uni vecor ^e of he coroaional coordinae sysem is defined as he bisecor of he angle inerseced by hese wo angen vecors g and g ; he uni vecor ^e 3 is perpendicular o he midsurface and he oher uni vecor is deermined by ^e and ^e 3, i.e., ^e = which lead o he ransformaion marix g jg j + g jg j,fi fififi g jg j + g jg j fi fi fi fi ; (66) ^e 3 = g g jg g j ; (67) ^e = ^e 3 ^e ; (68) R = ^e ^e ^e : (69) 8-3

7 Sress and Srain Measures Since he coroaional coordinae sysem roaes wih he configuraion, he sress defined in his coroaional sysem does no change wih he roaion or ranslaion of he maerial body and is hus objecive. Therefore, we use he Cauchy sress in he coroaional coordinae sysem, called he coroaional Cauchy sress, as our sress measure. The rae of deformaion (or velociy srain ensor), also defined in he coroaional coordinae sysem, is used as he measure of he srain rae, ^v def _" = ^d ^v def # + ; ^x where ^v def is he deformaion par of he velociy in he coroaional sysem ^x. If he iniial srain ^"(X; 0) is given, he srain ensor can be expressed as, ^"(X; )=^"(X; 0) + Z 0 ^d(x; fi) dfi: (7) The srain incremen is hen given by he mid-poin inegraion of he velociy srain ensor, ^" = Z n+ n ^d dfi : ^udef ^x n+ ^u ^x n+! 3 5 ; (7) where ^u def is he deformaion par of he displacemen incremen in he coroaional sysem ^x n+ referred o he mid-poin configuraion. Coroaional Sress and Srain Updaes For sress and srain updaes, we assume ha all variables a he previous ime sep n are known. Since he sress and srain measures defined in he earlier secion are objecive in he coroaional sysem, we only need o calculae he srain incremen from he displacemen field wihin he ime incremen [ n, n+ ]. The sress is hen updaed by using he radial reurn algorihm. All he kinemaical quaniies mus be compued from he las ime sep configuraion, Ω n,a = n and he curren configuraion, Ω n+,a = n+ since hese are he only available daa. Denoing he spaial coordinaes of hese wo configuraions as x n and x n+ in he fixed global Caresian coordinae sysem Ox, as shown in Fig., he coordinaes in he corresponding coroaional Caresian coordinae sysems, O^x n and O^x n+, can be obained by he following ransformaion rules: ^x n = R n x n ; (73) ^x n+ = R n+ x n+ ; (7) where R n and R n+ are he orhogonal ransformaion marices which roae he global coordinae sysem o he corresponding coroaional coordinae sysems, respecively. Since he srain incremen is referred o he configuraion a = n+, by assuming he velociies wihin he ime incremen [ n, n+ ] are consan, we have x n+ = (x n + x n+ ) ; (75) and he ransformaion o he coroaional sysem associaed wih his mid-poin configuraion, Ω n+,isgivenby ^x n+ = R n+ x n+ : (76) 8-33

8 Ω n+ Ω n+- ^ x n+ Ω n ^x n+- O ^ x n x Figure : Configuraions a imes n, n+ and n+ Similar o polar decomposiion, an incremenal deformaion can be separaed ino he summaion of a pure deformaion and a pure roaion (Belyschko, 973). Leing u indicae he displacemen incremen wihin he ime incremen [ n, n+ ], we wrie u = u def + u ro ; (77) where u def and u ro are, respecively, he deformaion par and he pure roaion par of he displacemen incremen in he global coordinae sysem. The deformaion par also includes he ranslaion displacemens which cause no srains. In order o obain he deformaion par of he displacemen incremen referred o he configuraion a = n+, we need o find he rigid roaion from Ω n o Ω n+ provided ha he mid-poin configuraion, Ω n+, is held sill. Defining wo virual configuraions, Ω0 and n Ω0 n+, by roaing he elemen bodies Ω n and Ω n+ ino he coroaional sysem O^x n+ (Fig. 3) and denoing ^x 0 n and ^x 0 n+ as he coordinaes of Ω 0 n and Ω 0 n+ in he coroaional sysem O^x n+,wehave ^x 0 n = ^x n; ^x 0 n+ = ^x n+: (78) We can see ha from Ω n o Ω 0 n and from Ω 0 n+ o Ω n+, he body experiences wo rigid roaions and he roaion displacemens are given by u ro = x 0 n x n = R n+ ^x 0 n x n = R n+ ^x n x n ; (79) u ro = x n+ x 0 n+ = x n+ R n+ ^x 0 n+ = x n+ R n+ ^x n+ : (80) Thus he oal roaion displacemen incremen can be expressed as u ro = u ro + u ro = x n+ x n R n+ (^x n+ ^x n ) = u R n+ (^x n+ ^x n ) : (8) 8-3

9 u Ω n+ P n+ u ro Ω n+ P n+ x^ n+ x^ n+- Ω n+- u def O P n Ω n P n ro u Ω n x^ n x Figure 3: Separaion of he displacemen incremen Then he deformaion par of he displacemen incremen referred o he configuraion Ω n+ is u def = u u ro = R n+ (^x n+ ^x n ) : (8) Therefore, he deformaion displacemen incremen in he coroaional coordinae sysem O^x n+ is obained as ^u def = R n+ u def = ^x n+ ^x n : (83) Once he srain incremen is obained by (7), he sress incremen, also referred o he mid-poin configuraion Ω n+, can be calculaed wih he radial reurn algorihm. The oal srain and sress can hen be updaed as ^" n+ = ^" n + ^"; (8) ^ff n+ = ^ff n + ^ff: (85) Noe ha he resulan sress and srain ensors are boh referred o he curren configuraion and defined in he curren coroaional coordinae sysem. By using he ensor ransformaion rule we can have he srain and sress componens in he global coordinae sysem. Tangen Siffness Marix and Nodal Force Vecors From he Hu-Washizu variaional principle, a boh νh and (ν +)h ieraion, we have Z ^Ω ν ffi ^" ν ij ^ffν ij dv = ffi^ßν ex ; (86) Z ffi ^" ν+ ^Ω ν+ ij ^ff ν+ ij dv = ffi^ß ν+ ex ; (87) where ffi^ß ex is he virual work done by he exernal forces. Noe ha boh equaions are wrien in he coroaional coordinae sysem defined in he νh ieraive configuraion given by x ν n+. The variables in his secion are wihin he ime sep [ n, n+ ] and superscrips indicae he number of ieraions. 8-35

10 Assuming ha all exernal forces are deformaion-independen, linearizaion of (87) gives (Liu, 99) Z Z ffi^u ν ^C ijkl ^u ν i;j k;l dv + ffi^u ν ^T ijkl ^u ν i;j k;l dv = ffi^ß ν+ ex ffi^ß ν ex ; (88) ^Ω ν ^Ω ν where he Green-Naghdi rae of Cauchy sress ensor is used, i.e., ^T ν ijkl = ffi ik ^ff ν jl : (89) The firs erm on he lef hand side of (88) denoes he maerial response since i is due o pure deformaion or sreching; he second erm is an iniial sress par resuling from finie deformaion effec. Taking accoun of he residual of he previous ieraion, equaion (87) can be approximaed as Z Z ^C ν ijkl + ^T ν ijkl ^u k;l dv = ffi^ß ν+ ex ffi ^" ν ij ^ffν ij dv: (90) ^Ω ν ^Ω ν ffi^u ν i;j If he srain and sress vecors are defined as we can rewrie (90) as Z " = " x " y " z " xy " yz " zx! xy! yz! zx Λ ; (9) ff = ff x ff y ff z ff xy ff yz ff zx Λ ; (9) ^Ω ν ffi ^" ν i Z ^C ν ij + ^T ν ij ffi ^" j dv = ffi^ß ν+ ex ffi ^" ν i ^ffν j dv; (93) ^Ω ν where ^C ν ij is he consisen angen modulus ensor corresponding o pure deformaion (see Secion 3..3) bu expanded o a 9 by 9 marix; ^T ν ij is he geomeric siffness marix which is given as follows (Liu (99)): T = 6 ff ff 0 0 ff 0 6 ff 0 ff6 ff ff 0 ff 5 0 ff ff 5 0 ff ff ff 6 0 ff5 ff 6 ff +ff symm: ff 6 ff +ff 3 ff 5 ff ff +ff 3 ff ff ff6 ff 5 ff +ff ff 6 ff 3 ff ff ff6 ff +ff 3 ff5 ff ff ff 3 ff5 ff ff 3+ff : (9) By inerpolaion u = N d; ffiu = Nffid; (95) " = B d; ffi" = Bffid; (96) where N and B are, respecively, he shape funcions and srain operaors defined in Secion. This leads o a se of equaions ^K ν ^d = ^r ν+ = ^f ν+ ex ^f ν in ; (97) 8-36

11 where he angen siffness marix, ^K ν, and he inernal nodal force vecor, ^f ν in,are ^K ν = Z ^f in ν = Z ^Ων ^B ^C ν + ^T ν ^B dv; (98) ^Ω ν ^B ^ff ν dv: (99) The angen siffness and nodal force are ransformed ino he global coordinae sysem ensorially as K ν = R ν ^K ν R ν ; (00) r ν+ = R ν ^r ν in ; (0) where R ν is he ransformaion marix of he coroaional sysem defined by x ν n+. Finally we ge a se of linear algebraic equaions K ν d ν+ = r ν+ : (0) NUMERICAL EXAMPLES To invesigae he performance of he elemen inroduced in his paper, a variey of problems including linear elasic and nonlinear elasic-plasic/large deformaion problems are sudied. Since he elemen is developed o avoid locking, he applicabiliy o problems of hin srucures is sudied by solving he sandard es problems including pinched cylinder and Scordelis-Lo roof, which are proposed by MacNeal, 985 and Belyschko, 98b. Also a shee meal forming problem is solved o es and demonsrae he effeciveness and efficiency of his elemen. Timoshenko Canilever Beam The firs problem is a linear, elasic canilever beam wih a load a is end as shown in Fig., where M and P a he lef end of he canilever are reacions a he suppor. The analyical soluion from Timoshenko, 970 is where u x (x; Py y) = 6EI [(6L 3x)x +(+ν)(y D )]; (03) u y (x; P y) = 6EI [3νy (L x) + (+5ν)D x +(3L x)x ]; (0) I = D3 ; E = ρ E; E=( ν ); ρ ν; ν = ν=( ν); for plane sress; for plane srain: The displacemens a he suppor end, x =0, D» y» D are nonzero excep a he op, boom and midline (as shown in Fig. 5). Reacion forces are applied a he suppor based on he sresses corresponding o he displacemen field a x =0, which are ff Py xx = I (L x); ff yy =0; ff P xy = I ( D y ): (05) The disribuion of he applied load o he nodes a x = L is also obained from he closed-form sress fields. 8-37

12 y M P P x D L Figure : Timoshenko canilever beam. y _D P/ x y L (a) Regular mesh θ P. A _D P/ x L/ (b) Skewed mesh P. A Figure 5: Top half of anisymmeric beam mesh The parameers for he canilever beam are: L =:0, D =0:0, P =:0, E = 0 7 ;and wo values of Poisson s raio: () ν =0:5,()ν =0:999. Since he problem is anisymmeric, only he op half of he beam is modeled. Plane srain condiions are assumed in he z-direcion and only one layer of elemens is used in his direcion. Boh regular mesh and skewed mesh are esed for his problem. Normalized verical displacemens a poin A for each case are given in Tables. Tables a and b show he normalized displacemen a poin A for he regular mesh. There is no shear or volumeric locking for his elemen. For he skewed mesh, wih he skewed angle increased, we need more elemens o ge more accurae soluion (Table c). Pinched Cylinder Figure 6 shows a pinched cylinder subjeced o a pair of concenraed loads. Two cases are sudied in his example. In he firs case, boh ends of he cylinder are assumed o be free. In he second case, boh ends of he cylinder are covered wih rigid diaphragms so ha only he displacemen in he axial direcion is allowed a he ends. The parameers for he firs case 8-38

13 Table : Normalized displacemen a poin A of canilever beam (a) ν =0:5, regular mesh analyical soluion w A =9: Mesh 8 8 HEXDS (b) ν =0:999, regular mesh analyical soluion w A =7:50 0 Mesh 8 8 HEXDS (c) ν =0:5,skewedmesh ffi 5 ffi 0 ffi P free or wih diaphragm symmeric R P L P symmeric symmeric Figure 6: Pinched cylinder and he elemen model (wihou diaphragms) are E =: ; ν =0:35; L=0:35; R =:0; =0:09; P= 00:0; while for he second case (wih diaphragms), he parameers are se o be E =3 0 6 ; ν =0:3; L = 600:0; R = 300:0; =3:0; P =:0: Due o symmery only one ocan of he cylinder is modeled. The compued displacemens a he loading poin are compared o he analyic soluions in Table. HEXDS elemen works well in boh cases, indicaing ha his elemen can avoid no only shear locking bu also membrane locking; his is no unexpeced since membrane locking occurs primarily in curved elemens (Solarski, 983). Scordelis-Lo Roof Scordelis-Lo roof subjeced o is own weigh is shown in Figure 7. Boh ends of he roof 8-39

14 Table : Normalized displacemen a loading poin of pinched cylinder (a) Firs case wihou diaphragms Analyical soluion w max =0:37 Mesh HEXDS (a) Second case wih diaphragms Analyical soluion w max =: Mesh HEXDS L θ R Figure 7: Scordelis-Lo roof under self weigh are assumed o be covered wih rigid diaphragms. The parameers are seleced o be: E = :3 0 8, ν = 0:0, L = 50:0, R = 5:0, = 0:5, = 0 ffi, and he graviy is per volume. Due o symmery only one quarer of he roof is modeled. The compued displacemen a he midpoin of he edge is compared o he analyic soluion in Table 3. In his example he HEXDS elemen can ge good resul wih 00 elemens. Table 3: Normalized displacemen a mid-edge of Scordelis-Lo roof Analyical soluion w max =0:30 Mesh HEXDS Circular Shee Sreched wih a Tigh Die A circular shee is sreched under a hemisphere punch and a igh die wih a small corner radius (Fig. 8). The maerial is elasoplasic wih nonlinear hardening rule. The elasic maerial con- 8-0

15 R 0 =50.8mm hemisphere punch =mm R d =5mm R =56mm igh die r=mm Figure 8: Circular shee sreched wih a igh die sans are: E = 06 GPa and ν = 0:3. In he plasic range, he uniaxial sress-srain curve is given by ff = K" n ; where K = 509:8 MPa, n =0:, ff is Cauchy sress and " is naural srain (logarihmic srain). The iniial yield sress is obained o be ff 0 = 03:05 Mpa and he angen modulus a he iniial yield poin is E =0: MPa. Because of he small corner radius of he die, he same difficulies as in he problem of shee srech under he rigid cylinders lead he shell elemens o failure in his problem. Three dimensional solid elemens are needed and fine meshes should be pu in he areas near he cener and he edge of he shee. One quarer of he shee is modeled wih 00 HEXDS elemens due o he double symmeries. The mesh is shown in Fig. 9. Two layers of elemens are used in he hickness. Around he cener and near he circular edge of he shee, fine mesh is used. The nodes on he edge are fixed in x- andy-direcions and he boom nodes on he edge are prescribed in hree direcions. No fricion is considered in his simulaion. For comparison, he axisymmeric four-node elemen wih reduced inegraion (CAXR) is also used and he mesh for his elemen is he same as shown in he op of Figure 9. The resuls presened here are afer he punch has raveled down 50 mm. The profile of he circular shee is shown in Figure 0 where we can see ha he shee under he punch experiences mos of he sreching and he hickness of he shee above he die changes a lo. The deformaion beween he punch and he die is small. However, he shee hickness obained by he CAXR elemen is less han ha by he HEXDS elemen and here is sligh difference above he die. These observaions can be verified by he srain disribuions in he shee along he radial direcion (Figures ). The direcion of he radial srain is he angen of he mid-surface of he elemen in he rz plane and he hickness srain is in he direcion perpendicular o he mid-surface of he elemen. The uni vecor of he circumferenial srain is defined as he crossproduc of he direcional cosine vecors of he radial srain and he hickness srain. We can see ha he CAXR elemen yields larger srain componens in he area under he punch han he HEXDS elemen. The main difference of he srain disribuions in he region above he die is ha he CAXR elemen gives zero circumferenial srain in his area bu he HEXDS elemen 8-

16 z r 3mm 0x 8mm 0x 36mm 5x elemens 7mm 0x mm 0x y 0 elemens x elemens 0 0 Figure 9: Mesh for circular shee sreching 8-

17 Lower side HEXDS Upper side HEXDS Lower side CAXR Upper side CAXR Punch Die z (m) r (m) Figure 0: Deformed shape of a circular shee wih punch ravel 50 mm HEXDS CAXR 5 Reacion Force (kn) Punch Travel (mm) Figure : Reacion force vs punch ravel for he circular shee 8-3

18 HEXDS CAXR Radial Srain r (m) 0.8 (a) Radial srain disribuion Circumferenial Srain HEXDS CAXR r (m) 0 (b) Circumferenial srain disribuion -0. HEXDS CAXR -0. Thickness Srain r (m) (c) Thickness srain disribuion Figure : Srain disribuions in circular shee wih punch ravel 50 mm 8-

19 yields non-zero srain. The value of he reacion force shown in he Figure is only one quarer of he oal punch reacion force since only one quarer of he shee is modeled. From his figure we can see ha he shee begins sofening afer he punch ravels down abou 5 mm, indicaing ha he shee may have necking hough his canno be seen clearly from Figure 0. CONCLUSIONS A new eigh-node hexahedral elemen is implemened for he large deformaion elasic-plasic analysis. Formulaed in he coroaional coordinae sysem, his elemen is shown o be effecive and efficien and can achieve fas convergence in solving a wide variey of nonlinear problems. By using a coroaional sysem which roaes wih he elemen, he locking phenomena can be suppressed by omiing cerain erms in he generalized srain operaors. In addiion, he inegraion of he consiuive equaion in he coroaional sysem akes he same simple form as small deformaion heory since he sress and srain ensors defined in his coroaional sysem are objecive. Radial reurn algorihm is used o inegrae he rae-independen elasoplasic consiuive equaion. The angen siffness marix consisenly derived from his inegraion scheme is crucial o preserve he second order convergence rae of he Newon s ieraion mehod for he nonlinear saic analyses. Tes problems sudied in his paper demonsrae ha he elemen is suiable o coninuum and srucural numerical simulaions. In meal shee forming analysis, his elemen has advanages over shell elemens for cerain problems where hrough he hickness deformaion and srains are significan. REFERENCES Belyschko, T. and Hsieh, B.J. (973), Nonlinear Transien Finie Elemen Analysis wih Conveced Coordinaes, In. J. Num. Mehs. Engrg., Vol.7, pp Belyschko, T., Lin, J.I. and Tsay, C.S. (98a), Explici Algorihms for he Nonlinear Dynamics of Shells, Compu. Mehs. Appl. Mech. Engrg., Vol., pp Belyschko, T., Ong, J. S.-J., Liu, W.K. and Kennedy, J.M. (98b), Hourglass Conrol in Linear and Nonlinear Problems, Compu. Mehs. Appl. Mech. Engrg., Vol.3, pp Flanagan, D.P. and Belyschko, T. (98), A Uniform Srain Hexahedron and Quadrilaeral wih Orhogonal Hourglass Conrol, In. J. Num. Mehs. Engrg., Vol.7, pp Hallquis, J.O. (998), LS-DYNA Theoreical Manual, May, 998. Hughes, T.J.R. and Liu, W.K. (98a), Nonlinear Finie Elemen Analysis of Shells: Par I. Three-dimensional Shells, Compu. Mehs. Appl. Mech. Engrg., Vol.6, pp Hughes, T.J.R. and Liu, W.K. (98b), Nonlinear Finie Elemen Analysis of Shells: Par II. Two-dimensional Shells, Compu. Mehs. Appl. Mech. Engrg., Vol.7, pp Liu, W.K. and Belyschko, T. (98), Efficien Linear and Nonlinear Hea Conducion wih a Quadrilaeral Elemen, In. J. Num. Mehs. Engrg., Vol.0, pp

20 Liu, W.K., Belyschko, T., Ong, J.S.J. and Law, E. (985), Use of Sabilizaion Marices in Nonlinear Finie Elemen Analysis, Engineering Compuaions, Vol., pp Liu, W.K. (99), Lecure Noes on Nonlinear Finie Elemen Mehods, Norhwesern Universiy, 983, revised in 99. Liu, W.K., Hu, Y.K., and Belyschko, T. (99), Muliple Quadraure Underinegraed Finie Elemens, In. J. Num. Mehs. Engrg., Vol. 37, pp Liu, W.K., Guo, Y., Tang, S. and Belyschko T. (998), A Muliple-quadraure Eigh-node Hexahedral Finie Elemen for Large Deformaion Elasoplasic Analysis, Compu. Mehs. Appl. Mech. Engrg., Vol.5, pp MacNeal R.H. and Harder R.L. (985), A Proposed Sandard Se of Problems o Tes Finie Elemen Accuracy, Finie Elemens Anal. Des., Vol., pp Solarski, H. and Belyschko, T. (983), Shear and Membrane locking in curved elemens, Compu. Mehs. Appl. Mech. Engrg., Vol., pp