L6. Discretization and FE formulation, assignment 1

Transcription

1 L6. Dscretaton and FE formulaton, assgnment à Prelmnares ü Vrtual wor Deformaton gradent restated n terms of the dsplacement gradent based on the relaton ϕ = X + u,.e. F = + H wth H = u X where H s the dsplacement gradent. Let us now consder the vrtual wor representaton of statc momentum balance restated as (5) δf : P V = δh : P V = G (5) where G = δw Hρ bl V + δw t Γ and δf =δh =δw X Γ A more convenent alternatve s formulated as va the spatal dsplacement gradent h = u leadng to (5) and δh = Hδw L F =δl F (53) Hδl FL : P V = δl : τ V = δd : τ V = G (54) where t was used that δl F :P =δl :P F t =δl : HP F t L =δl : τ. Moverover, due to the smmetr of τ, n the last equalt the rate of deformaton tensor d was ntroduced defned as d = l sm = Hl + lt L (55) ü Lneared Vrtual wor Based on the lneared equlbrum dscussed n lecture 3, we recall the formulaton of the lneared vrtual wor expressed n Euleran tensors as δl : τ Ît V = δw Iρ bm V + δw t Γ Γ (56) where τ Ît s the nomnal stress rate whch s related to the smmetrc Oldrod stress rate τ = τ t va τ Ît = τ t + H τm : l E = E + τ (57) The nternal part of the lneared vrtual wor ma thus be wrtten as

2 organaton_lectures.nb δl : Jτ t + H τm : l N V = Iδd : τ + δl t : Hτ ll M V (58) where t was used that δl : HH τm : l L =δl : Hl τl = 8δl l τ =δl τ l t =δl t τ l t < =δl t : Hτ l t L We also have (from lecture 3) the relaton to the materal operators as (59) τ Ît = E : l; τ = τ t = E : l = E : d wth E = 4 HF FL : Hence, the lneared vrtual wor ma be wrtten as ψ C C : HFt Ft L (6) Hδd : E : d +δl t : Hτ lll V = δw Iρ bm V + δw t Γ Γ (6) à Dscretaton []: 7, []. Consder the reference confguraton B (=computatonal doman) dscretsed nto fnte elements B e,e=,.... It s assumed that each element has the nterpolaton NODE u = I= NODE N u I δu = NODE δl = I= I= where N are the element nterpolaton functons. N δu I δu I g I wth g I = G I F and G I = NI X In the present text, we shall restrct the dscusson to CST elements correspondng lnear dsplacement approxmaton wthn each trangle,.e. NODE = 3. To ths end, a local ξ, ξ -coordnate sstem s ntroduced, where the shape functons N ξ D are defned (wthn the trangle shown n Fg. x) as (6) N = Hξ +ξ L, N =ξ, N 3 = ξ Moreover, an soparametrc mappng of the ntal geometr s ntroduced,.e. (63) 3 X = N ξ D X I I= where X I are the nodal postons wthn B. It turns out that the materal gradent ma be specfed n terms of the ontra-varant bass vector E wrtten as (64) G I = NI ξ E wth E = J E (65)

3 organaton_lectures.nb 3 where the Jacoban J s defned as J = HJ L and J = E E = J E E E E E E E E N To see ths, let us frst ntroduce the (co-varant) dfferental dx = E dξ wth E = X ξ The relaton between the co- and ontra varant bass vectors ma now be establed from the relatonshps (66) (67) = X X = X ξ ξ X = E E = wth ξ X = E E = E HE E L = HE E L E E E =δ E = E = HE E L E = G E = G E E = E = E HE E L = G E = G = HG L The explct expressons for the nvolved transformatons ma be obtaned from the representaton whch gves leadng to X = ξ D X + ξ D X + N ξ D X 3 = = H Hξ +ξ LL X +ξ X +ξ X 3 E = X ξ = X X = X ; E = X ξ = X 3 X = X 3 (68) (69) and J = J = X X X X 3 ; X 3 X X 3 X 3 J = HJ L = D X3 X 3 X X 3 X 3 X X X (7) D = H X X L H X 3 X 3 L H X 3 X L In vew of (), the explct expressons for the materal gradents G I become: (7) where G I = NI E + NI E ξ ξ (7) E = J E + J E = D HH X3 X 3 L X H X X 3 L X 3 L E = J E + J E = D H H X3 X L X + H X X L X 3 L It s noted that the area of the element concdes wth the vector product A e = H X X 3 L E Z =.. = X Y 3 X 3 Y (73) (74)

4 4 organaton_lectures.nb where E Z s a unt vector pontng n the out-of-plane drecton. ü dervatons à FE-formulaton []: 7, []. ü Voght matrx formulaton - Statc vrtual wor A crucal pont the fnte element formulaton of the statc vrtual wor concerns the matrx formulaton of the term δh : τ, representng the ntegrand n the nternal vrtual wor, cf. eq. (x). Based on the relaton δl : τ =δd : τ =δd τ wth δd = Hδl +δlt L we dentf the Voght matrx formulaton of the Krchhoff stress τˆ and the rate of deformaton tensor δ dˆ as (75) δd : τ =δd τ =δdˆt τˆ wth δ dˆ = Hδd, δd, δd 33,δd,δd 3,δd 3 L t and τˆ = H τ, τ, τ 33, τ, τ 3, τ 3 L t where the smmetr of both τ and δd was exploted,.e. (76) δd τ =δd τ +δd τ +δd 3 τ 3 +δd τ + δd τ +δd 3 τ 3 +δd 3 τ 3 +δd 3 τ 3 +δd 33 τ 33 = 8δd δd, δd 3 δd 3, δd 3 δd 3 < = δd τ +δd Hτ +τ L +δd τ + δd 3 Hτ 3 +τ 3 L +δd 3 Hτ 3 +τ 3 L +δd 33 τ 33 = 8τ τ, τ 3 τ 3, τ 3 τ 3 < =δd τ +δd τ + δd 33 τ 33 + δd τ + δd 3 τ 3 + δd 3 τ 3 Returnng to the present CST element, the rate of deformaton tensor ma now be expressed n matrx form from the dentfcaton (77) 3 δd = I= (wth summaton of the "I"s) HδuI g I + g I δu I L (78) δ dˆ = Hδu I L Hg I L δ dˆ = Hδu I L δ dˆ33 = Hδu I L 3 Hg I L 3 δ dˆ = Hδu I L + Hδu I L Hg I L (79) δ dˆ3 = Hδu I L Hg I L 3 + Hδu I L 3 Hg I L δ dˆ3 = Hδu I L Hg I L 3 + Hδu I L 3 The restrcton the plane stran mples that (f we choose the coordnates represented b sub-ndces and are assocated wth the n plane where the coodnate 3 s assocated wth the out-of-plane drecton)

5 organaton_lectures.nb 5 Hδu I L 3 = Hg I L def 3 = δ dˆ33 =δ dˆ3 =δ dˆ3 = We thus obtan δ dˆ = Bδ û where B s the usual stran-dsplacement matrx pertnent to plane stran condtons, defned va (8) Hδu L δ dˆ δ dˆ δ dˆ = Hg I L Hg I L Hg I L Hg I L Hg I L Hg I L where û s the dsplacement vector of the element wth the components ordered as δ û = HHδu L, Hδu L, Hδu L, Hδu L, Hδu 3 L, Hδu 3 L L t Hδu L Hδu L Hδu L Hδu 3 L Hδu 3 L (8) (8) ü Nodal FE forces of statc vrtual wor From the relaton δl : τ V = we now obtan the dscreted form e δ dˆt τˆ V = G (83) δ û t g e = δ û e f ext e D = where f ext e s the external load vector and b e s the nternal load vector obtaned as (84) b e = e B t τˆ V (85) In (), the assembl operator denotes the poston n a global vector of dsplacements where the contrbuton from element "e" fts. It s noted that n the case of plane stran s suffces to represent the Voght matrx τˆ n terms of the n-plane components,.e. τˆ = H τ, τ, τ L t (86) ü Voght matrx formulaton - Lneared Statc vrtual wor We are now concerned wth the fnte element representaton of the lneared vrtual wor. In partcular, we are concerned wth the matrx formulaton of the term δd : E : d +δl t : Hτ l t L, cf. eq. (x), appearng n the lneared nternal vrtual wor. Based on the relaton δd : E : d +δl t : Hτ l t L where the E -operator was dentfed for the consdered Neo-Hooe model n terms of the sochorc and volumetrc parts as (87)

6 6 organaton_lectures.nb E so = 3 GJ 3 JI JI + N H b + b LN; 3 E vol = KJ HH J L HJ L IL we dentf the Voght matrx formulaton (n the unrestrcted stran space) wth respect to the appearng structures n E,.e. (88) δd : I : d = δd : d = δ dˆt Î dˆ δd : : d = δ dˆt Iˆ ˆt M dˆ wth ˆ = H,,,,, L t δd : b : d = δ dˆt Iˆ bˆt M dˆ wth bˆt = Hb,b,b 33,b,b 3,b 3 L δd : b : d = δ dˆt Iˆ bˆt M dˆ where the matrx form of the dentt matrx becomes ˆ I = As a result we obtan δd : E : d = δ dˆ Ê dˆ wth Ê = Ê so + Ê vol As to the geometrc stffness contrbuton, t appears that ths term can be represented n matrx form as: δl t : Hτ ll =δ lˆt τˆgeo δ lˆ where δ lˆ s the Voght matrx representaton of the spatal velct gradent,.e. (89) (9) (9) (9) ˆ l = Hδl, δl, δl 33, δl, δl, δl 3, δl 3, δl 3, δl 3 L t and τˆgeo s the matrx representaton of the (geometrc) Krchoff stress n appearng n eq. () defned as τ ˆgeo = τ τ τ 3 τ τ τ 3 τ 33 τ 3 τ 3 τ τ τ 3 τ τ τ 3 τ 3 τ 3 τ 33 τ 3 τ τ τ 3 τ 3 τ 33 τ 3 τ τ where τˆgeo was obtaned from the expanson (93) (94) δl t : Hτ l t L = l τ l = δl τ l +δl τ l +δl τ 3 l 3 +

7 organaton_lectures.nb 7 δl τ l +δl τ l +δl τ 3 l 3 + δl 3 τ l 3 +δl 3 τ l 3 +δl 3 τ 3 l 33 + δl τ l +δl τ l +δl τ 3 l 3 + δl τ l +δl τ l +δl τ 3 l 3 + δl 3 τ l 3 +δl 3 τ l 3 +δl 3 τ 3 l 33 + δl 3 τ 3 l +δl 3 τ 3 l +δl 3 τ 33 l 3 + δl 3 τ 3 l +δl 3 τ 3 l +δl 3 τ 33 l 3 + δl 33 τ 3 l 3 +δl 33 τ 3 l 3 +δl 33 τ 33 l 33 We also need to formulate the relaton between the spatal veloct gradent and the dsplacement (smlar to δd),.e. from 3 δl = δu I g I I= we dent the 9 components of δl (wth the summaton of I) as (96) δ lˆ = Hδu I L Hg I L δ lˆ = Hδu I L δ lˆ33 = Hδu I L 3 Hg I L 3 δ dˆ = Hδu I L δ dˆ = Hδu I L Hg I L (97) δ lˆ3 = Hδu I L Hg I L 3 δ lˆ3 = Hδu I L 3 Hg I L δ lˆ3 = Hδu I L Hg I L 3 δ lˆ3 = Hδu I L 3 It s noted once agan that plane stran,.e. Hδu I L 3 = Hg I L def 3 = mples the smplfcaton that δ lˆ33 =δ lˆ3 =δ lˆ3 =δ lˆ3 =δ lˆ3 = whereb we obtan the matrx representaton δ lˆ = C û where C s a plane stran -dsplacement matrx smular to B, defned va (98) δ lˆ Hg L Hg L Hg 3 L Hδu L Hδu L δ lˆ δ lˆ δ lˆ = Hg L Hg L Hg 3 L Hg L Hg L Hg 3 L Hg L Hg L Hg 3 L Hδu L Hδu L Hδu 3 L Hδu 3 L Of course, the Voght matrx representaton of the matrx τˆgeo assocated wth plane stran C-matrx also becomes reduced n vew of δ lˆ33 =δ lˆ3 =δ lˆ3 =δ lˆ3 =δ lˆ3 = as (99)

8 8 organaton_lectures.nb ˆgeo τ pl. st. = τ τ τ τ τ τ τ τ (3) ü dervatons ü Newton-Raphson soluton procedure We propose soluton procedure for the dscreted problem δ û t g = δ û e f ext e D = (34) defned b the local Newton teraton, gven nodal dsplacements û HL from an terate HL, the mproved soluton u ˆH+L s obtaned as u ˆH+L = û HL + ξˆ where the mprovement ξˆ s solved from the lneared problem wth respect to the state û HL at the terate HL,.e. û t Hb e f ext e L +δû t K e ξˆd = The element stffness K e s defned from the lneared vrtual wor relaton wth the assumpton of conservatve loadng (no follower load,.e. t = ) as (36) Hδd : E : d +δl t : Hτ lll V = Iδ dˆt Ê dˆ +δl B t τˆ lˆm V = δ û t Ê B + C t τˆ CD û e Note the subdvson of the element stffness K e n a materal and a geometrc contrbuton defned as (37) δ û t K e û = (38) wth K e = K e mat + K e geo = e B t Ê B V + e C t τˆ C V à Assgnment : Plane stran tenson test Consder the plane stran tenson test n the Fgure below.the materal of the specement s assumed to be sotropc hperelastc obeng the dscussed Neo-Hooe model. The specmen s clamped at both ends and s subected to a prescrbed dsplacement at the rght end.

9 organaton_lectures.nb 9 Fgure 4 Analse the relaton between the prescrbed dsplacement r and the correspondng reacton force R, when the materal ranges from beng elastcall compressble to elastcall ncompressble. For example, the materal parameters ma be taen as E G = H +νl ; K = E 3 H νl wth E =.985 MPaand the value of Posson's rato vares n the range.3 <ν<.5. As tpe tpcal R versus r relatonshp s shown n the Fgure below. The underlng bass for the analss, fnte element mplementaton and results shall be presented n report format. Fgure 5