Nonlinear Soil Modeling for Seismic NPP Applications

Transcription

1 Nonlinear Soil Modeling for Seismic NPP Applications Boris and Federico Pisanò University of California, Davis Lawrence Berkeley National Laboratory, Berkeley LBNL Seminar, December 212

2 Outline Introduction Elastic-Plastic Models Summary

3 Outline Introduction Elastic-Plastic Models Summary

4 Motivation The Problem Seismic response of Nuclear Power Plants Soil modeling in particular Linear elastic Equivalent Linear Elastic (current state of practice/art) 3D elastic-plastic modeling Combine frictional (elastic-plastic, displacement proportional) and viscous (velocity proportional) energy dissipation in one model.

5 Micromechanical Origins of Elasto-Plasticity Granular Materials Pressure sensitive materials (particles) Usually assumed to be linear elastic (at least in the NPP field (?!)) Can be (rigorously) shown that as they can never be linear elastic (with normal and/or shear contact forces. R. D. Mindlin and H. Deresiewicz. Elastic spheres in contact under varying oblique forces. ASME Journal of Applied Mechanics, 53(APM-14): , September Izhak Etsion. Revisiting the Cattaneo-Mindlin concept of interfacial slip in tangentially loaded compliant bodies. Journal of Tribology, 132(2):281, 21.

6 Micromechanical Origins of Elasto-Plasticity Two Spheres

7 Micromechanical Origins of Elasto-Plasticity Contact Zone contact zone

8 Micromechanical Origins of Elasto-Plasticity Normal Stresses Hertz (1882) contact zone

9 Micromechanical Origins of Elasto-Plasticity Normal and Shear Stresses Cattaneo (1938), Mindlin (1949), Etsion (21) τ σ stick zone slip zone

10 Micromechanical Origins of Elasto-Plasticity Granular Materials (Summary) Particulate materials are never linear elastic

11 Outline Introduction Elastic-Plastic Models Summary

12 Classical Material Models von Mises perfectly plastic model f = 3 2 s ijs ij σ = undrained shear strength s u (s u = σ / 3)

13 Classical Material Models Drucker-Prager kinematic hardening model (sij ) ( ) f = pα ij sij pα ij 2 3 kp = ( ) ɛ p ij = λm f dev ij, m ij = 1 σ ij 3 Dδ ij, ( ) 2 dilatancy coefficient D = ξ 3 k d r mn r mn α ij = 2 3 h a ( ) ɛ p dev 2 ( ij cr α ij ɛ p ) dev ( 3 rs ɛ p ) dev rs

14 Classical Material Models Drucker-Prager kinematic hardening model q [kpa] ε vol [ ] k d =1. k d =.6 k d = ε dev [ ] ε dev [ ] 15

15 Classical Material Models Drucker-Prager Kinematic Hardening Model with Viscosity 1 8 from GP from reactions σ x σ y 6 21 σ z p 4 2 τ [kpa] 2 2 normal stress [kpa] γ [ ] x time [s]

16 Classical Material Models DK - Variation in Yield Stress G/Gmax [ ] D [ ] σ =5 kpa (GP) σ =5 kpa (react) σ =1 kpa (GP) σ =1 kpa (react) σ =2 kpa (GP) σ =2 kpa (react) σ =4 kpa (GP) σ =4 kpa (react) γ [%] γ [%]

17 Classical Material Models DK - Variation in Damping Coefficient ζ=.5 (GP) ζ=.5 (react) ζ=.1 (GP) ζ=.1 (react) ζ=.2 (GP) ζ=.2 (react) ζ=.5 (GP) ζ=.5 (react) G/Gmax [ ].6.4 D [ ] γ [%] γ [%]

18 Classical Material Models DK - Variation in Initial Confinement G/Gmax [ ] D [ ] p =5 kpa (GP) p =5 kpa (react) p =1 kpa (GP) p =1 kpa (react) p =2 kpa (GP) p =2 kpa (react) p =5 kpa (GP) p =2 kpa (react) γ [%] γ [%]

19 Classical Material Models Comparison with G/G max and Damping Curves 1 1 G/G max [ ] ζ [ ] GP RF Seed & Idriss.2 GP/RF Seed & Idriss γ [%] γ [%]

20 Classical Material Models Comparison with G/G max and Damping Curves G/G max [ ].6.4 GP/RF Seed & Idriss ζ [ ].6.4 GP RF Seed & Idriss γ [%] γ [%]

21 Classical Material Models Comparison with G/G max and Damping Curves 1 1 G/G max [ ] ζ [ ] GP RF Seed & Idriss.2 GP/RF Seed & Idriss γ [%] γ [%]

22 Pisanò- Material Model PJ Model: Assumptions Split stress into frictional and viscous components σ ij = σ f ij + σ v ij Remove the elastic region (limit analysis) Rotating kinematic hardening

23 Pisanò- Material Model PJ Model: Elasticity dσ ij = Dijhk e ( dɛhk dɛ p hk) ( ds ij = 2G max dehk de p ) hk dp = K ( dɛ vol dɛ p vol where p = σ kk /3, ɛ vol = ɛ kk, s ij = σij dev, e ij = ɛ dev ij, G max = E/2 (1 + ν) and K = E/3 (1 2ν) )

24 Pisanò- Material Model PJ Model: DP Yield and Bounding Surface f y = 3 2 ( sij pα ij ) ( sij pα ij ) k 2 p 2 = f B = 3 2 s ijs ij M 2 p 2 = σ 3 _ σ ij β(σ ij σ ij ) σ 2 σ ij σ ij bounding surface σ 1

25 Pisanò- Material Model PJ Model: Plastic Flow and Translation Rule Borrowed from Manzari and Dafalias (1997) ( dɛ p hk = dλ nij dev 1 ) 3 Dδ ij ) ( ) 2 D = ξ (α ij d α ij nij dev = ξ 3 k dnij dev α ij nij dev dα ij = dα ij n dev ij

26 Pisanò- Material Model PJ Model: Vanishing Elastic Region lim f y = lim s ij = pα ij ds ij = dα ij p + α ij dp k k n dev ij = ds ij α ij dp ds ij α ij dp σ 3 _ σ ij dα ij = 1 f pn dev dσ ij σ ij dα ij = 2 dq 3 p β(σ ij σ ij ) σ ij σ ij σ 1 bounding surface σ 2

27 Pisanò- Material Model PJ Model: Hardening Modulus and Plastic Multiplier dα ij = 2 Hdλ 3 p dλ = 2G max de ij + Kdɛ vol α ij nij dev 2G H KDα ijnij dev

28 Pisanò- Material Model PJ Model: Stress Projection, Hardening and Unloading s ij ds ij 2 dβ = (1 + β) 3 M2 pdp ( ) s ij s ij sij 2 3 M2 p ( p p ) > σ 3 _ σ ij β(σ ij σ ij ) σ 2 σ ij σ ij bounding surface σ 1

29 Pisanò- Material Model PJ Model: Triaxial Response 25 2 q [kpa] 15 1 k d =1.2 k =.4 d k = d ε ax [%] ε vol [%] k d = k d =.4 k d = ε [%] ax

30 Pisanò- Material Model PJ Model: Pure Shear Cyclic 5 frictional frictional+viscous τ [kpa] γ [%] 1 5 frictional frictional+viscous τ [kpa] γ [%]

31 Pisanò- Material Model PJ Model: Calibration for G/G max and Damping G/Gmax [ ] frict, frict+visc Seed & Idriss γ [%] max ζ [ ] frict+visc frict Seed & Idriss γ [%] max Figure: Comparison between experimental and simulated G/G max and damping curves (p =1 kpa, T=2π s, ζ =.3, G max = 4 MPa, ν=.25, M=1.2, k d =ξ=, h=g/(112p ), m=1.38)

32 Pisanò- Material Model PJ Model: Calibration for G/G max and Damping G/Gmax [ ] Seed & Idriss frict, frict+visc ζ [ ].2.1 frict+viscous frict γ max [%] Seed & Idriss γ [%] max Figure: Comparison between experimental and simulated G/G max and damping curves (p =1 kpa, T=2π s, ζ =.3, G max = 4 MPa, ν=.25, M=1.2, k d =ξ=, h=g max /(15p ), m=1)

33 Pisanò- Material Model PJ Model: Variation in Confining Pressure G/Gmax [ ] p =5kPa p =1kPa ζ [ ] p =5kPa p =1kPa p =2kPa.2 p =2kPa.1 p =3kPa p =3kPa γ max [%] γ max [%] Figure: Simulated G/G max and damping curves at varying confining pressure (T=2π s, G max = 4 MPa, ν=.25, M=1.2, k d =ξ=, h=g/(15p ), m=1)

34 Pisanò- Material Model PJ Model: Variation in Hardening Parameter h G/Gmax [ ] h=g/1.5p h=g/15p h=g/15p h=g/15p ζ [ ].2.1 h=g/15p h=g/15p h=g/15p γ max [%] h=g/1.5p γ max [%] Figure: Simulated G/G max and damping curves at varying h (p =1 kpa, T =2π s, G max = 4 MPa, ν=.25, M=1.2, k d =ξ=, m=1)

35 Pisanò- Material Model PJ Model: Variation in Hardening Parameter m 1.8 m=2.5.4 m=3 G/Gmax [ ].6.4 m=.5 m=1 m=3 ζ [ ].3.2 m= γ max [%] m=.5 m= γ max [%] Figure: Simulated G/G max and damping curves at varying m (p =1 kpa, T =2π s, G max = 4 MPa, ν=.25, M=1.2, k d =ξ=, h=g max /(15p ))

36 Pisanò- Material Model PJ Model: Variation in Viscous Damping G/Gmax [ ] ζ [ ] ζ =.5 ζ =.3 ζ =.1 ζ = γ [%] γ [%] max Figure: Damping curves simulated at varying ζ (p =1 kpa, T =2π s, G max = 4 MPa, ν=.25, M=1.2, k d =ξ=, h=g max /(15p ), m=1)

37 Outline Introduction Elastic-Plastic Models Summary

38 Summary Frictional and Viscous energy dissipation for granular materials Classical elastic-plastic models with addition of viscous damping Vanishing elastic region elastic-plastic material model Important for professional practice since G/G max and damping curves (all just in 1D!) is what is available, and this model calibrates well, and is full 3D model.