Fundamentals of Finite Elements. Mehrdad Negahban. W311 Nebraska Hall Department of Engineering Mechanics University of Nebraska-Lincoln

Transcription

1 EGM 98 unamentals f nte Elements Mehra egahban W ebraska Hall Deartment f Engneerng Mechancs Unversty f ebraska-lncln Phne: E-mal: mnegahban@unl.eu

2 Curse escrtn Objectve: Slutn f artal fferental euatns f engneerng mrtance usng EM. Emhaszng the smlarty f the evelment. Develment f cmmn mathematcal an numercal tls. Usng cnsstent an rbust meths. Develng an bject rente rgrammng structure fr EM. Aressng sme avance tcs. 4 Elements 7x77 r 9x99 e Elements 7x7 Element: 6 La stes 7 ewtn Iteratns 9x9 Element: 4 La stes 6 ewtn Iteratns u A v B u w v A B

3 Tcs Curse Object rente rgrammng n EM W Weghte Resual lmeths Lnear Prblems scalar an vectr fels mult hyscs Heat transfer elastcty an thermelastcty Cnstrants n-lnear Prblems nlnear elastcty large efrmatn Intal-Value Prblems Transent thermal analyss Vbratn f beams Bars Plates an Shells Knematcally efne cntnua umercal Imlementatn

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5 Shells Peanut Sunflwer Eggs Slanum cnereum germnatng Ccnut

6 Shells Ale Re bl cell Cell structure: Prcaryte Crnary artery

7 Shells Plant cells Plant cell wall Plar leaf cell Tallw tree Chlrlast: ste f htsynthess n lant cells

8 Tycal EM rmulatn Shell Knematcs Cnsttutve Characterstcs nte Element rmulatn nte Element Arxmatn Intal an Bunary Cntns umercal Slutn

9 nte Element rmulatn Shell Knematcs nte Element Arxmatn Varatnal rmulatn EM rmulatn Cnsttutve Characterstcs Intal an Bunary Cntns umercal Slutn

10 Shell Knematcs t ρ ζ Q ρ P Drectr Curvlnear Crnate t P Q M -surface s x O Reference rame

11 Shell Knematcs Ressner/Mnln Shell + s s x ς ς j j g g g + + s x g ς ς ς ρ ζ Q ρ P ρ P s x O ς ς

12 Shell Knematcs Makng the mangs nt materal mangs: X x JJ Current cnfguratn Γ Intal cnfguratn D Ω Γ Ω S J X s J x J g e J G e e e e O

13 Shell Knematcs Defrmatn Graent: Dslacement Graent: JJ H I Intal cnfguratn X Current cnfguratn x JJ Strans: D ε H + H T G R I T G L I T J X e e e S O s J x

14 Isarametrc shell x s + ς Current Cnfguratn nne s s 5 nne nne ς ς Isarametrc -D 5 Extrue -D 4 8 s O e e e e e e

15 Isarametrc shell X S + ς D Intal Cnfguratn nne S S nne D D nne ς ς 4 5 D 8 Isarametrc -D Extrue -D 5 e S e e e e e O

16 ρ s + e x J e X J Jacbans ζ s x + ρ s x + Δ Δ D S s s ρ ζ s x + ζ + s x ζ ζ ζ Δ Δ D x s ζ ζ + + Thckness unknwns + nf f a ζ ζ ζ + D S X Thckness base functns ζ ζ + + D D ζ

17 Thckness nterlatn ζ umber f nterlatn functns nf + ζ ζ a f f Legenre lynmals: Materal surface P + P s f P + P s even One term arxmatn: Δζ a f Δζ 0 M-surface

18 EM frmulatn Resual fr EM: mnal stress T J T T R u t Γ X u : T Ω + ρu bω ρu aω Γ t T T t T Ω Intal cnfguratn Γ D Ω X x JJ Ω Current cnfguratn al unknwns: Ω U j Δ s j j U j+ Δ j j U j+ 6 a j... nf j e e e S O J X s J x

19 nlnear elastcty Defrmatn graent: general new new l + & Δt & l + U j Δ t U j l + U j Δ U j U j JJ U g U j U j j J U e J j J g e U g j x x J mnal stress: nnlnear elastc T T new new T T l l + T& Δt + & T : Δt T l + T : U j ΔU j rm shell knematcs T T + E : new l Materal secfc U U Δ j j U j Δ s j j U j+ Δ j j U j+ 6 a j j... nf Tangent mulus f nmnal stress: E T

20 nlnear elastcty Tw arameter nnlnear-elastc materal mel: mnal stress: Tangent mulus: Materal arameters: Shear an bulk mulus matche t lnear resnse

21 Examles frm nlnear Elastcty Cantlevere beam wth strbute en la 6 E. 0 ν 0.0 max h b 4 La stes 4 ewtn teratns L u t w t w v u

22 Examles frmnlnear Elastcty Cantlevere beam wth strbute en mment Sngle 9x-ne element 6 lang stes 75 ewtn teratns M E. 0 ν 0.0 L.0 h 0. b.0 6 M M 60 4 M M 60 w t Theretcal M M 60 u t w v M M 60 u

23 Examles frm nlnear Elastcty Cantlevere beam wth strbute en mment a -4x-60-6LIC-87I b -4x-60-6LIC-47I c 4-4x-60-6LIC-47I -5x-60-6LIC-50I e -5x-60-6LIC-50I f -5x-540-4LIC-4I g -6x-540-4LIC-I h -6x-70 -LIC-09I -7x-70 -LIC-9I

24 Examles frm nlnear Elastcty Bucklng n cmressn f a clumn u t w v u w t E.0 0 ν L 0.5 b h Crtcal bucklng la L

25 Examles frm nlnear Elastcty Transverse lang f an annular rng u w v R R A B A E 0 R 6.0 h 0.0 max ν 0.0 R 0.0 A B wa wb La Stes 4 ewtn stes 5 Elements 7x7-es er element B

26 Examles frm nlnear Elastcty Pnchng f hemshercal shell wth hle 4 Elements 7x7 r 9x9 e Elements 7x7 Element: 6 La stes 7 ewtn Iteratns 9x9 Element: 4 La stes 6 ewtn Iteratns u A v B z Hle rc Symmetrc x Symmetr A R 0 h E 6.85x0 7 ν 0. Hle 8 B y u w v A B

27 Examles frm nlnear Elastcty Pnchng f cylnrcal shell wth free ens 40 La Stes 74 ewtn stes 4 Elements 9x9-es er element w A v C v B A v B v C B w 6 v E ν 0.5 max u L 0.5 R 4.95 h R C L

28 Examles frm nlnear Elastcty Pnt la n a hnge cylnrcal rf R 540 L 54 θ 0. ra E 0.75 ν 0. Hnge ree ree Hnge L R θ

29 ur arameter elastc mel τ T κ G tanh I G τ G I + + B tr B I 5/ 5/ J G J I τ J G τ G G τ κ J et tr C J T C I * G γ

30 Large efrmatn elastc-lastc mel Cauchy stress: e e e tr B T κ J I + G B I 5 e J Over-stress: ΔT Yel functn: e e b T b 0 T T κ G E ν E + ν f : ΔS σ y ΔS ΔS Pwer-law harenng rule: ΔS ΔT tr ΔT I σ + n y σ y a & ΔT T : L

31 Tw mels n tensn ur Parameter Mel Large Defrmatn Elastc-Plastc Mel 000 Stress arameter Elastc-lastc κ G τ G κ G σ y a 0.6 n Stran

32 Beam n cmressn 000 ur Parameter Brck ur Parameter Shell 0000 La C Cmress n u Large Defrmatn Elastc-Plastc Brck L0 b h Dslacement

33 Beam n tensn La ur Parameter Brck ur Parameter Shell Large Defrmatn Elastc-Plastc Brck u L0 b h Dslacement

34 Cantlevere beam v u ur Parameter Brck ur arameter brck 00 Large Defrmatn Elastc-Plastc Brck Large Defrmatn Elastc-Plastc Brck ur Parameter Shell 000 ur Parameter Shell La u v L0 b h Dslacement

35 Mult-scale analyss Each ntegratn nt can have an RVE

36 abro Questns?