On Probabilistic Yielding of (Geo--)Materialssokocalo.engr.ucdavis.edu/~jeremic/...0 0.002 0.0216...

Motivation Uncertain Material Nonlinear Modeling Summary

On Probabilistic Yielding of (Geo–)Materials

Boris Jeremic and Kallol Sett

Department of Civil and Environmental EngineeringUniversity of California, Davis

WCCM8 / ECCOMAS20081st. July 2008,Venezia, Italia

Jeremic and Sett Computational Geomechanics Group



Outline

Motivation

Uncertain Material Nonlinear ModelingProbabilistic Elasto–PlasticityProbabilistic Yielding

Summary




Uncertain (Geo–)materials

Mayne et al. (2000) Sture et al. (1998)




Background

I Brownian motion, Langevin equation (Einstein 1905)I Random forcing (Fokker-Planck, Kolmogorov 1941)I Random coefficient (Hopf 1952), (Bharrucha-Reid 1968),

Monte Carlo methodI Epistemic uncertainty, lack of knowledgeI Aleatory uncertainty, inherent variationI Methods for treating epistemic uncertainty not well

developed, trade-off with aleatory uncertaintyI Ergodicity of geomaterials !(?)I Prof. Einav question




Probabilistic Elasto–Plasticity

Outline

Motivation


Summary





Uncertainty Propagation through Constitutive Eq.

I Incremental el–pl constitutive equation ∆σij = Dijkl∆εkl

Dijkl =

Del

ijkl for elastic

Delijkl −

DelijmnmmnnpqDel

pqkl

nrsDelrstumtu − ξ∗r∗

for elastic–plastic





Eulerian–Lagrangian FPK EquationLiouville continuity equation (Kubo 1963); ensemble average form (Kavvasand Karakas 1996); van Kampen’s Lemma (van Kampen 1976)

∂P(σ(xt , t), t)∂t

= − ∂

∂σ

»fiη(σ(xt , t), Del(xt), q(xt), r(xt), ε(xt , t))

fl+

Z t

0dτCov0

»∂η(σ(xt , t), Del(xt), q(xt), r(xt), ε(xt , t))

∂σ;

η(σ(xt−τ , t − τ), Del(xt−τ ), q(xt−τ ), r(xt−τ ), ε(xt−τ , t − τ)

–ffP(σ(xt , t), t)

–+

∂2

∂σ2

»Z t

0dτCov0

»η(σ(xt , t), Del(xt), q(xt), r(xt), ε(xt , t));

η(σ(xt−τ , t − τ), Del(xt−τ ), q(xt−τ ), r(xt−τ ), ε(xt−τ , t − τ))

– ffP(σ(xt , t), t)

–





Compact Form of Eulerian–Lagrangian FPK EquationI Advection-diffusion equation

∂P(σ, t)∂t

= − ∂

∂σ

[N(1)P(σ, t)− ∂

∂σ

{N(2)P(σ, t)

}]I Complete probabilistic description of responseI Solution PDF is second-order exact to covariance of time

(exact mean and variance)I It is deterministic equation in probability density spaceI It is linear PDE in probability density space → simplifies

the numerical solution process

B. Jeremic, K. Sett, and M. L. Kavvas, "Probabilistic Elasto–Plasticity: Formulation in 1–D", Acta Geotechnica, Vol.2, No. 3, 2007





Template Solution of E–L FPK EquationI FPK diffusion–advection equation is applicable to any

material model → only the coefficients N(1) and N(2) aredifferent for different material models

∂P(σ, t)∂t

= − ∂

∂σ

[N(1)P(σ, t)− ∂

∂σ

{N(2)P(σ, t)

}]I Initial condition random or deterministicI Solution for both

I probabilistic elastic (PEL) andI probabilistic elastic–plastic (PELPL) problems

K. Sett, B. Jeremic and M.L. Kavvas, "The Role of Nonlinear Hardening/Softening in Probabilistic Elasto–Plasticity",International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 31, No. 7, pp. 953-975, 2007




Probabilistic Yielding

Outline

Motivation


Summary





Probabilistic Yielding: Weighted CoefficientsI Probabilistic yielding: take both PEL and PELPL into

account at the same time with some (un–)certainty

I Weighted elastic and elastic–plastic Solution∂P(σ, t)/∂t = −∂

(Nw

(1)P(σ, t)− ∂(

Nw(2)P(σ, t)

)/∂σ

)/∂σ

I Weighted advection and diffusion coefficients are thenNw

(1,2)(σ) = (1− P[Σy ≤ σ])Nel(1) + P[Σy ≤ σ]Nel−pl

(1)

I Similar to European Pricing Option in financial simulations(Black–Scholes options pricing model ’73, Nobel prize forEconomics ’97)

B. Jeremic and K. Sett. On Probabilistic Yielding of Materials. in print in Communications in Numerical Methods inEngineering, 2008.





Probability of Yielding

I Probabilistic von MisesI Probability of yielding at

σ = 0.0012 MPaP[Σy ≤ (σ = 0.0012 MPa)]= 0.8

I Equivalent advection anddiffusion coefficients are

Neq(1)|σ=0.0012 MPa = (1− 0.8)Nel

(1) + 0.8Nep(1)

Neq(2)|σ=0.0012 MPa = (1− 0.8)Nel

(2) + 0.8Nep(2)

0.00005 0.0001 0.00015Σy HMPaL

0.2

0.4

0.6

0.8

1F@SyD = P@Sy£ΣyD

(a)(b)





Bi–Linear von Mises Response

0 0.0108 0.0216 0.0324 0.0432 0.0540

0.00005

0.0001

0.00015

0.0002

0.00025

0.0003S

tres

s (M

Pa)

Strain (%)

Mode

Solution

Mean

Deterministic

Mean ± Std. Deviation





Bi–Linear von Mises Response

0

0

0.0001

0.0002

0.0003

0.0004

0200040006000

8000

10000

P[St

ress

]

Strain (%)

Stre

ss (M

Pa)

0

0.01080.0216

0.0324

0.0432

0.054Jeremic and Sett Computational Geomechanics Group




von Mises, Certain cu, Uncertain G

0 0.0108 0.0216 0.0324 0.0432 0.0540

0.00005

0.0001

0.00015

0.0002

0.00025

0.0003

Mean

Mode

DeterministicSolution

Strain (%)

Str

ess

(MP

a)






von Mises, Uncertain cu, Certain G

0.0108 0.0216 0.0324 0.0432 0.054

0.00005

0.0001

0.00015

0.0002

0.00025

Strain (%)

MeanMode

Str

ess

(MP

a)

Deterministic Solution






Drucker Prager: Uncertain G, Certain φ

a)0 0.0108 0.0216 0.0324 0.0432 0.054

0

0.00005

0.0001

0.00015

0.0002

0.00025

0.0003

Strain (%)


Mode

Mean

Str

ess

(MP

a)






Drucker Prager: Certain G, Uncertain φ

b)0 0.002 0.0216 0.0324 0.0432 0.054

0

0.00005

0.0001

0.00015

0.0002

0.00025

0.0003

0.00035S

tres

s (M

Pa)


MeanMode

Strain (%)





Summary

I Second-order (mean and variance) exact, method toaccount for probabilistic elastic–plastic material simulation

I Probabilities of material yielding, probably governunderlying mechanics of elasto–plasticity

I Probabilities of material yielding (spatial distributions) alsoprobably govern underlying mechanics of failure,localization of deformation...

I Probably numerous applications



On Probabilistic Yielding of (Geo--)Materialssokocalo.engr.ucdavis.edu/~jeremic/...0 0.002 0.0216...

Documents

Transcript of On Probabilistic Yielding of (Geo--)Materialssokocalo.engr.ucdavis.edu/~jeremic/...0 0.002 0.0216...