Elastic Plastic Behavior of Geomaterials: Modeling and Simulation Issues

Transcription

1 Elastic Plastic Behavior of Geomaterials: Modelig ad Simulatio Issues Boris Zhaohui Yag (UA), Zhao Cheg (EarthMechaics Ic.), Mahdi Taiebat (UBC) Departmet of Civil ad Evirometal Egieerig Uiversity of Califoria, Davis GheoMat Masseria Salamia Italy, Jue 2009

2 Outlie Before We Start Costitutive Level Small Deformatio Elasto Plasticity Explicit ad Implicit Costitutive Itegratios Fiite Elemet Level Formulatio Statics ad Dyamics Examples Piles Pile Groups Summary

3 Before We Start Motivatio Use well developed theory of elasto plasticity for modelig ad simulatig geomatarials Issues at the costitutive ad the fiite elemet levels Verificatio ad Validatio is very importat There is o limit to what problems oe ca address (ca umerically simulate)

4 Before We Start Loadig Process Stages Icremets Iteratios

5 Small Deformatio Elasto Plasticity Outlie Before We Start Costitutive Level Small Deformatio Elasto Plasticity Explicit ad Implicit Costitutive Itegratios Fiite Elemet Level Formulatio Statics ad Dyamics Examples Piles Pile Groups Summary

6 Small Deformatio Elasto Plasticity Small Deformatio 2500% shear strai 2000% 1500% 1000% 500% Φ E ij Error ε ij deformatio (arcta Φ) E ij = 1 2 ( ui,j + u j,i + u i,k u k,j ) ; ɛ ij = 1 2 (u i,j + u j,i )

7 Small Deformatio Elasto Plasticity Elasticity Hyperelasticity, = W / ɛ ij (where W is the strai eergy fuctio per uit volume) Hypoelasticity, direct modelig of oliear elastic deformatio, ot thermodyamically cosistet Liear ad oliear elastic models

8 Small Deformatio Elasto Plasticity Icremetal Elasto Plasticity Additive decompositio of strai ɛ ij = ɛ e ij + ɛ p ij Elastic relatioship (geeralized Hooke s law) = E ijkl ɛ e kl (o) Associated plastic flow rule ɛ p ij = λ Q/ = λ m ij (, q ) Hardeig/softeig (isotropic/aisotropic) law q = λ h (τ ij, q )

9 Small Deformatio Elasto Plasticity Karush Kuh Tucker Coditios Yield fuctio F (, q ) 0 Plastic cosistecy parameter λ 0 loadig uloadig coditio F λ = 0

10 Small Deformatio Elasto Plasticity Midpoit Itegratio Algorithm cross m ij +α m ij mij σ predictor ij Q=0 Q=0 cross σij Elastic regio +α σ 2 σ 1 σ 3 +α F=0 F=0 + α Q=0 F=0

11 Small Deformatio Elasto Plasticity Midpoit Itegratio Algorithm Rarely used (eve if for α = 0.5 it is secod order accurate) Explicit algorithm (α = 0.0) Implicit algorithm (α = 1.0)

12 Explicit ad Implicit Costitutive Itegratios Outlie Before We Start Costitutive Level Small Deformatio Elasto Plasticity Explicit ad Implicit Costitutive Itegratios Fiite Elemet Level Formulatio Statics ad Dyamics Examples Piles Pile Groups Summary

13 Explicit ad Implicit Costitutive Itegratios Explicit Itegratio Algorithm P σ ij Elastic regio σ 2 σ 1 σ 3 m ij 1 m ij F=0 F=0 FE

14 Explicit ad Implicit Costitutive Itegratios Explicit Itegratio Algorithm Icremets Elastic regio σ 2 σ 3 σ1 m ij 1 m ij F=0 F=0 P σij FE σ m = E mpq ɛ pq E mpq rs E rstu ɛ tu ab E abcd m cd ξ A h A m pq q A = ( m E mpq ɛ pq cros m E mpq cros m pq ξ A h A ) h A

15 Explicit ad Implicit Costitutive Itegratios Explicit Itegratio Algorithm P σij Taget stiffess Elastic regio σ 2 σ 3 σ1 m ij 1 m ij F=0 F=0 FE cot E ep pqm = E pqm E pqkl m kl ij E ijm ot E otrs m rs ξ A h A

16 Explicit ad Implicit Costitutive Itegratios Explicit Itegratio Algorithm P σij Relatively simple (first derivatives) m ij m ij Fast (sigle step) Elastic regio σ 2 σ1 1 F=0 FE Iaccurate (accumulates error) σ 3 F=0 Popular (most/all commercial codes) Works well with global explicit algorithm

17 Explicit ad Implicit Costitutive Itegratios Implicit Itegratio Algorithm P σ ij Elastic regio σ 2 σ 1 σ 3 m ij 1 m ij F=0 F=0 FE

18 Explicit ad Implicit Costitutive Itegratios Implicit Itegratio Algorithm Also based o elastic predictor plastic corrector = pred λ E ijkl m kl Tesor of residuals used i iteratios r ij = ( pred λ E ijkl m kl ) Iterative icremets d( λ) = ( old f T C old r ) / ( T C M ) { } dσm = C ( dq old r + d( λ)m ) B { } { m Eijkl m with =, m = kl ξ B h A Elastic regio σ 2 σ 3 σ1 m ij 1 m ij F=0 F=0 } { old }, old σ r = ij old r A P σij FE

19 Explicit ad Implicit Costitutive Itegratios Implicit Itegratio Algorithm Super-matrix C has differet formats depedig o a umber ad type of iteral variables [ C = Iijm s + λe ijkl m kl σ m [ I s m C = ijm + λe kl ijkl σ m λ h C = I s ijm + λe ijkl m kl σ m λ hz λ hα m ] 1 m λe kl ijkl q 1 λ h q m λe kl ijkl z m hz λ z m λ hα m z m I s ijm Elastic regio σ 2 σ 3 ] 1 σ1 m ij 1 m ij F=0 m λe kl ijkl α m λ hz α ij Iijm s λ hα m α ij F=0 P σij FE 1

20 Explicit ad Implicit Costitutive Itegratios Implicit Itegratio Algorithm P σij Cosistet (algorithmic) stiffess Elastic regio σ 2 σ 3 σ1 m ij 1 m ij F=0 F=0 FE { dσij dq A } = { C CmT C T C m } { Eijm dɛ pred m 0 }

21 Explicit ad Implicit Costitutive Itegratios Implicit Itegratio Algorithm Relatively complicated (first ad secod derivatives, iverse) Relatively slow (but improves global Newto iteratios) Accurate (cosistecy coditio satisfied at the ed, withi tolerace) Popular for research Upopular i commercial codes (except simple material models) Desiged to work with global Newto algorithm Elastic regio σ 2 σ 3 σ1 m ij 1 m ij F=0 F=0 P σij FE

22 Formulatio Outlie Before We Start Costitutive Level Small Deformatio Elasto Plasticity Explicit ad Implicit Costitutive Itegratios Fiite Elemet Level Formulatio Statics ad Dyamics Examples Piles Pile Groups Summary

23 Formulatio Priciple of Virtual Displacemets V δɛ ij dv = V ( f B i ρü i ) δu i dv + fi S δu i ds S

24 Formulatio Discretizatio u û a = H I ū Ia ɛ ab ê ab = 1 ) (ûa,b + û b,a = 2 = 1 ) ((H I ū Ia ) 2,b + (H I ū Ib ),a = = 1 (( ) ( )) HI,b ū Ia + HI,a ū Ib 2

25 Formulatio FEM Equatios (m) M IacJ ū Jc + (m) K IacJ ū Jc = (m) (m) m (m) F B Ia + m (m) F S Ia (m) M IacJ = H J δ ac ρ H I dv m V m ; (m) F B Ia = V m f B a H I dv m (m) K IacJ = H I,b E abcd H J,d dv m V m ; (m) F S Ia = S m f S a H I ds m

26 Statics ad Dyamics Outlie Before We Start Costitutive Level Small Deformatio Elasto Plasticity Explicit ad Implicit Costitutive Itegratios Fiite Elemet Level Formulatio Statics ad Dyamics Examples Piles Pile Groups Summary

27 Statics ad Dyamics Residual Force Equatio i Statics fi it r i (u j, λ) = fi it (u j ) λfi ext = 0 (u j ) are the iteral forces which are fuctios of the displacemets u j, f ext i is a fixed exteral loadig vector λ is a load level parameter Proportioal loadig

28 Statics ad Dyamics Advacig the Solutio Load λf λ 1 f ext λf 2 ext λf 3 ext δλ0f ext δλ 1 f ext δλ2f ext (u 1, λ 1 f ext or ) (up, λpf (u, λ f 2 2 ext) ext ) (u, λ f 3 3 ext) Equilibrium Path (u, λ f 0 0 ext) l Costrait Hypersurface f λ 0 ext δu 0 δu 1 δu 2 u u 1 u 0 2 u 3 Displacemet u

29 Statics ad Dyamics Hyper spherical Costrait s = ds where ds = or, i icremetal form: ( a = ( s) 2 ( l) 2 = ψ 2 u u 2 ref ψ 2 u u 2 ref du i S ij du j + dλ 2 ψf 2 ) u i S ij u i + λ 2 ψf 2 ( l) 2

30 Statics ad Dyamics Specializatios Load λf ψ f = ψ u ψ f < ψ u ψ f <<ψ u Load λf ψ f >>ψ u ψ f = ψ u ψ f > ψ u Equilibrium Path Equilibrium Path l Costrait Hypersurfaces l Costrait Hypersurfaces Displacemet u Displacemet Coefficiets ψ u ad ψ f may ot be simultaeously zero If S ij = I ij ad u ref = 1 arclegth method If S ij = Kij t ad ψ f 0 exteral work costrait If ψ u 0 ad ψ f 1 load cotrol If ψ u 1, ψ f 0 ad S ij = I ij geeralized displacemet cotrol u

31 Statics ad Dyamics Followig the Equilibrium Path i Statics Family of Newto methods (full, iitial stress, modified...) Traversig equilibrium path i positive sese (positive exteral work criterio; agle criterio) Accuracy cotrol Numerical stability Automatic icremets Covergece criteria (absolute, relative, force ad/or displacemet ad/or eergy based)

32 Statics ad Dyamics Trasiet Itegratio Algorithms Fiite differeces, simple, but iacurate Wilso θ = 1.37, too much umerical dampig Newmark, cotrollable umerical dampig, period elogatio γ 1/2, β = 1/4(γ + 1/2) 2 Hilber Hughes Taylor, extesio of Newmark with better dampig 1/3 α 0, γ = 1/2(1 2α), β = 1/4(1 α) 2

33 Statics ad Dyamics Dyamic Aalysis Stability (artificial itroductio of higher frequecies by discretizatio process) Accuracy, coservatio of eergy ad period Time step choice (the shorter the better, uless too may (artificial) high frequecies are preset). multiple DOF type systems (u-p-u, structural elemets...)

34 Piles Outlie Before We Start Costitutive Level Small Deformatio Elasto Plasticity Explicit ad Implicit Costitutive Itegratios Fiite Elemet Level Formulatio Statics ad Dyamics Examples Piles Pile Groups Summary

35 Piles Sigle Pile i Layered Soils: Model Case 1&2: Clay Case 3&4: Sad Case 1&4: Clay Case 2&3: Sad Case 1 & 2: Clay Case 3 & 4: Sad Iterface 0.429m 2.40m 1.72m 1.72m 7.86m

36 Piles Available Data for Prototype Sad: frictio agle φ of 37.1 o, Shear modulus at a depth of 13.7 m of 8960 kpa (E o = kpa), Poisso ratio of 0.35 Uit weight of kn/m 3. Dilatio agle 0 o Clay (made up) Shear stregth 21.7 kpa Youg s modulus kpa Poisso ratio 0.45 Uit weight 13.7 kn/m 3

37 Piles Sigle Pile i Sad: M, Q, p

38 Piles Sigle Pile i Clay: M, Q, p

39 Piles Sigle Pile i Sad with Clay Layer: M, Q, p

40 Piles Sigle Pile i Clay with Sad Layer: M, Q, p

41 Piles Sigle Pile i Sad: p y Respose

42 Piles Sigle Pile i Clay: p y Respose

43 Piles Sigle Pile i Sad with Clay Layer: p y Respose

44 Piles Sigle Pile i Clay with Sad Layer: p y Respose

45 Piles p y Pressure Ratio Reductio for Layered Soils

46 Pile Groups Outlie Before We Start Costitutive Level Small Deformatio Elasto Plasticity Explicit ad Implicit Costitutive Itegratios Fiite Elemet Level Formulatio Statics ad Dyamics Examples Piles Pile Groups Summary

47 Pile Groups Pile Group Simulatios

48 Pile Groups Bedig Momets

49 Pile Groups Out of Plae Effects

50 Pile Groups Load Distributio per Pile Lateral Load Distributio i Each Pile (%) Trail Row, Side Pile Third Row, Side Pile Secod Row, Side Pile Lead Row, Side Pile Trail Row, Middle Pile Third Row, Middle Pile Secod Row, Middle Pile Lead Row, Middle Pile Displacemet at Pile Group Cap (cm)

51 Pile Groups Piles Iteractio at -2.0m (p y)

52 Summary Importace of cosistet formulatio, material modelig ad implemetatio Verified, validate models ad simulatios tools used for predictio of behavior Program ad examples available i public domai (Author s web site)