Nonlinear Soil Modeling for Seismic NPP Applicationssokocalo.engr.ucdavis.edu/~jeremic/ · I Seismic response of Nuclear Power Plants I Soil modeling in particular I Linear elastic

Introduction Elastic-Plastic Models Summary

Nonlinear Soil Modeling forSeismic NPP Applications

Boris Jeremic and Federico Pisanò

University of California, DavisLawrence Berkeley National Laboratory, Berkeley

LBNL Seminar, December 2012

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Nonlinear Soil Modeling for Seismic NPP Applications


Outline

Introduction

Elastic-Plastic Models

Summary

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Outline

Introduction


Summary

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Motivation

The Problem

I Seismic response of Nuclear Power Plants

I Soil modeling in particular

I Linear elastic

I Equivalent Linear Elastic (current state of practice/art)

I 3D elastic-plastic modeling

I Combine frictional (elastic-plastic, displacementproportional) and viscous (velocity proportional) energydissipation in one model.

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Micromechanical Origins of Elasto-Plasticity

Granular MaterialsI Pressure sensitive materials (particles)I Usually assumed to be linear elastic (at least in the NPP

field (?!))I Can be (rigorously) shown that as they can never be linear

elastic (with normal and/or shear contact forces.I R. D. Mindlin and H. Deresiewicz. Elastic spheres in

contact under varying oblique forces. ASME Journal ofApplied Mechanics, 53(APM-14):327-344, September1953.

I Izhak Etsion. Revisiting the Cattaneo-Mindlin concept ofinterfacial slip in tangentially loaded compliant bodies.Journal of Tribology, 132(2):020801, 2010.

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Two Spheres

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Contact Zone

contact zone

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Normal StressesHertz (1882)

contact zone

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Normal and Shear StressesCattaneo (1938), Mindlin (1949), Etsion (2010)

στ

stick zone

slip zone

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Granular Materials (Summary)

I Particulate materials are never linear elastic

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Outline

Introduction


Summary

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Classical Material Models

von Mises perfectly plastic model

I f =

√32

sijsij − σ0 = 0

I undrained shear strength su (su = σ0/√

3)

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Drucker-Prager kinematic hardening model

I f =√(

sij − pαij) (

sij − pαij)−√

23kp = 0

I εpij = λmij , mij =

(∂f∂σij

)dev

− 13

Dδij ,

dilatancy coefficient D = ξ

(√23

kd −√

rmnrmn

)

I αij =23

ha

(εpij

)dev− crαij

√23(εprs)dev (

εprs)dev

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Drucker-Prager kinematic hardening model

0 0.02 0.04 0.06 0.08 0.10

20

40

60

80

100

120

140

160

εdev

[−]

q [k

Pa]

0 0.02 0.04 0.06 0.08 0.1

−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

εdev

[−]

ε vol [−

]

kd=1.0

kd=0.6

kd=0.0

−0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08−150

−100

−50

0

50

100

150

εdev

[−]

q [k

Pa]

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Drucker-Prager Kinematic Hardening Model withViscosity

−10 −8 −6 −4 −2 0 2 4 6 8

x 10−3

−100

−80

−60

−40

−20

0

20

40

60

80

100

γ [−]

τ [k

Pa]

from GPfrom reactions

0 5 10 15 20 25 30130

140

150

160

170

180

190

200

210

220

230

time [s]

norm

al s

tres

s [k

Pa]

σx

σy

σz

p

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DK - Variation in Yield Stress

10−3

10−2

10−1

100

101

0

0.2

0.4

0.6

0.8

1

γ [%]

G/G

max

[−]

10−3

10−2

10−1

100

101

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

γ [%]

D [−

]

σ0=50 kPa (GP)

σ0=50 kPa (react)

σ0=100 kPa (GP)

σ0=100 kPa (react)

σ0=200 kPa (GP)

σ0=200 kPa (react)

σ0=400 kPa (GP)

σ0=400 kPa (react)

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DK - Variation in Damping Coefficient

10−3

10−2

10−1

100

101

0

0.2

0.4

0.6

0.8

1

γ [%]

G/G

max

[−]

10−3

10−2

10−1

100

101

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

γ [%]

D [−

]

ζ=0.005 (GP)ζ=0.005 (react)ζ=0.01 (GP)ζ=0.01 (react)ζ=0.02 (GP)ζ=0.02 (react)ζ=0.05 (GP)ζ=0.05 (react)

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DK - Variation in Initial Confinement

10−3

10−2

10−1

100

101

0

0.2

0.4

0.6

0.8

1

γ [%]

G/G

max

[−]

10−3

10−2

10−1

100

101

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

γ [%]

D [−

]

p0=50 kPa (GP)

p0=50 kPa (react)

p0=100 kPa (GP)

p0=100 kPa (react)

p0=200 kPa (GP)

p0=200 kPa (react)

p0=500 kPa (GP)

p0=200 kPa (react)

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Comparison with G/Gmax and Damping Curves

10−4

10−3

10−2

10−1

100

101

0

0.2

0.4

0.6

0.8

1

γ [%]

G/G

max

[−]

GP/RFSeed & Idriss

10−4

10−3

10−2

10−1

100

101

0

0.2

0.4

0.6

0.8

1

γ [%]

ζ [−

]

GPRFSeed & Idriss

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10−4

10−3

10−2

10−1

100

101

0

0.2

0.4

0.6

0.8

1

γ [%]

G/G

max

[−]

GP/RFSeed & Idriss

10−4

10−3

10−2

10−1

100

101

0

0.2

0.4

0.6

0.8

1

γ [%]

ζ [−

]

GPRFSeed & Idriss

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10−4

10−3

10−2

10−1

100

101

0

0.2

0.4

0.6

0.8

1

γ [%]

ζ [−

]

GPRFSeed & Idriss

10−4

10−3

10−2

10−1

100

101

0

0.2

0.4

0.6

0.8

1

γ [%]

G/G

max

[−]

GP/RFSeed & Idriss 1970

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Pisanò-Jeremic Material Model

PJ Model: Assumptions

I Split stress into frictional and viscous componentsσij = σf

ij + σvij

I Remove the elastic region (limit analysis)I Rotating kinematic hardening

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PJ Model: Elasticity

dσij = Deijhk(dεhk − dεphk

)dsij = 2Gmax

(dehk − dep

hk

)

dp = −K(dεvol − dεpvol

)where p = −σkk/3, εvol = εkk , sij = σdev

ij , eij = εdevij ,

Gmax = E/2 (1 + ν) and K = E/3 (1− 2ν)

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PJ Model: DP Yield and Bounding Surface

fy =32(sij − pαij

) (sij − pαij

)− k2p2 = 0

fB =32

sijsij −M2p2 = 0

σij

_

σij

β(σij−σ

ij0)

boundingsurface

σij0

σ2

σ3

σ1

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PJ Model: Plastic Flow and Translation Rule

Borrowed from Manzari and Dafalias (1997)

dεphk = dλ(

ndevij − 1

3Dδij

)

D = ξ(αd

ij − αij

)ndev

ij = ξ

(√23

kdndevij − αij

)ndev

ij

dαij = ‖dαij‖ndevij

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PJ Model: Vanishing Elastic Region

limk→0

fy = 0⇒ limk→0

sij = pαij ⇒ dsij = dαijp + αijdp

ndevij =

dsij − αijdp‖dsij − αijdp‖

‖dαij‖ =1

pNdev∂f∂σij

dσij

‖dαij‖ =

√23

dqp

σij

_

σij

β(σij−σ

ij0)

boundingsurface

σij0

σ2

σ3

σ1

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PJ Model: Hardening Modulus and Plastic Multiplier

‖dαij‖ =23

Hdλp

dλ =2Gmax‖deij‖+ Kdεvolαijndev

ij

2G +23

H − KDαijndevij

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PJ Model: Stress Projection, Hardening andUnloading

dβ = − (1 + β)sijdsij −

23

M2pdp

sij

(sij − s0

ij

)− 2

3M2p

(p − p0

) > 0

σij

_

σij

β(σij−σ

ij0)

boundingsurface

σij0

σ2

σ3

σ1Jeremic




PJ Model: Triaxial Response

0 5 10 15 200

50

100

150

200

250

εax

[%]

q [

kPa]

kd=0

kd=0.4

kd=1.2

0 5 10 15 20

−30

−20

−10

0

εax

[%]

ε vol [

%]

kd=0

kd=0.4

kd=1.2

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PJ Model: Pure Shear Cyclic

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2

−5

0

5

γ [%]

τ [k

Pa]

frictionalfrictional+viscous

−20 −15 −10 −5 0 5 10 15 20−100

−50

0

50

100

γ [%]

τ [k

Pa]

frictionalfrictional+viscous

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PJ Model: Calibration for G/Gmax and Damping

10−4

10−3

10−2

10−1

100

1010

0.2

0.4

0.6

0.8

1

γmax

[%]

G/G

max

[−]

10−4

10−3

10−2

10−1

100

1010

0.1

0.2

0.3

0.4

γmax

[%]

ζ [−

]

Seed & Idriss

frict, frict+visc

frict

frict+visc

Seed & Idriss

Figure: Comparison between experimental and simulated G/Gmaxand damping curves (p0=100 kPa, T=2π s, ζ = 0.003, Gmax = 4 MPa,ν=0.25, M=1.2, kd=ξ=0, h=G/(112p0), m=1.38)

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PJ Model: Calibration for G/Gmax and Damping

10−4

10−3

10−2

10−1

100

1010

0.2

0.4

0.6

0.8

1

γmax

[%]

G/G

max

[−]

10−4

10−3

10−2

10−1

100

1010

0.1

0.2

0.3

0.4

γmax

[%]

ζ [−

]

Seed & Idriss

frict, frict+visc

Seed & Idriss

frict

frict+viscous

Figure: Comparison between experimental and simulated G/Gmaxand damping curves (p0=100 kPa, T=2π s, ζ = 0.003, Gmax = 4 MPa,ν=0.25, M=1.2, kd=ξ=0, h=Gmax /(15p0), m=1)

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PJ Model: Variation in Confining Pressure

10−4

10−3

10−2

10−1

100

1010

0.2

0.4

0.6

0.8

1

γmax

[%]

G/G

max

[−]

10−4

10−3

10−2

10−1

100

1010

0.1

0.2

0.3

0.4

γmax

[%]

ζ [−

]

p0=50kPa

p0=100kPa

p0=200kPa

p0=300kPa

p0=200kPa

p0=300kPa

p0=100kPa

p0=50kPa

Figure: Simulated G/Gmax and damping curves at varying confiningpressure (T=2π s, Gmax = 4 MPa, ν=0.25, M=1.2, kd=ξ=0,h=G/(15p0), m=1)

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PJ Model: Variation in Hardening Parameter h

10−4

10−3

10−2

10−1

100

1010

0.2

0.4

0.6

0.8

1

γmax

[%]

G/G

max

[−]

10−4

10−3

10−2

10−1

100

1010

0.1

0.2

0.3

0.4

γmax

[%]

ζ [−

]h=G/150p

0

h=G/15p0

h=G/1500p0

h=G/1.5p0

h=G/1.5p0

h=G/1500p0

h=G/150p0

h=G/15p0

Figure: Simulated G/Gmax and damping curves at varying h (p0=100kPa, T =2π s, Gmax = 4 MPa, ν=0.25, M=1.2, kd=ξ=0, m=1)

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PJ Model: Variation in Hardening Parameter m

10−4

10−3

10−2

10−1

100

1010

0.2

0.4

0.6

0.8

1

γmax

[%]

G/G

max

[−]

10−4

10−3

10−2

10−1

100

1010

0.1

0.2

0.3

0.4

0.5

γmax

[%]

ζ [−

]m=1

m=0.5

m=2

m=3

m=0.5

m=3

m=2

m=1

Figure: Simulated G/Gmax and damping curves at varying m (p0=100kPa, T =2π s, Gmax = 4 MPa, ν=0.25, M=1.2, kd=ξ=0, h=Gmax /(15p0))

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PJ Model: Variation in Viscous Damping

10−4

10−3

10−2

10−1

100

101

0

0.2

0.4

0.6

0.8

1

γ [%]

G/G

max

[−]

10−4

10−3

10−2

10−1

100

1010

0.1

0.2

0.3

0.4

γmax

[%]

ζ [−

]

ζ0=0

ζ0=0.003

ζ0=0.005

ζ0=0.001

Figure: Damping curves simulated at varying ζ0 (p0=100 kPa, T =2πs, Gmax = 4 MPa, ν=0.25, M=1.2, kd=ξ=0, h=Gmax /(15p0), m=1)

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Outline

Introduction


Summary

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Summary

I Frictional and Viscous energy dissipation for granularmaterials

I Classical elastic-plastic models with addition of viscousdamping

I Vanishing elastic region elastic-plastic material model

I Important for professional practice since G/Gmax anddamping curves (all just in 1D !) is what is available, andthis model calibrates well, and is full 3D model.

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Documents

Nonlinear Soil Modeling for Seismic NPP Applicationssokocalo.engr.ucdavis.edu/~jeremic/ · I Seismic response of Nuclear Power Plants I Soil modeling in particular I Linear elastic