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%\MyLogo{{\footnotesize September, 22 2004}}
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\begin{document}
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%\maketitle
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\vspace*{1.0truecm}
\begin{center}
{\huge \bf
%\vspace*{1.5truecm} \\
\rule[3mm]{0cm}{1.3cm} On Uncertainty of \\
\rule[3mm]{0cm}{1.3cm} ElasticPlastic Simulations \\
%\rule[3mm]{0cm}{1.3cm} Application to Performance based \\
%\rule[3mm]{0cm}{1.3cm} Computational Geomechanics \\
}
\end{center}
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\vspace*{4.0truecm}
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\begin{Large}
\begin{center}
{\bf Boris Jeremi{\'c}}\\
%\vspace*{0.20truecm}
%{\bf Kallol Sett and Lev Kavvas}
\end{center}
\end{Large}
\vspace*{0.20truecm}
%\begin{small}
\begin{center}
{Department of Civil and Environmental Engineering }\\
{University of California, Davis}\\
\end{center}
%\end{small}
\vspace*{2.0truecm}
\begin{small}
\begin{center}
Acknowledgment: Prof. Kavvas and Mr. Sett, support by NSF, Caltrans and PEER
\end{center}
\end{small}
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\slide{Talk Overview}
\begin{large}
\begin{itemize}
%\vspace*{1.0truecm}
\item Motivation: Uncertainty in (geo) material modeling and simulations
%\vspace*{1.0truecm}
\item Previous work
%\vspace*{1.0truecm}
\item Proposed formulation and solution (Forward Kolmogorov or FokkerPlanck equation)
%\vspace*{1.0truecm}
\item Select results and verifications
\begin{itemize}
\vspace*{0.08cm}
\item Elastic
\vspace*{0.08cm}
\item DruckerPrager Linear Hardening
\vspace*{0.08cm}
\item Cam Clay
\end{itemize}
\end{itemize}
\end{large}
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\slide{Motivation}
\begin{large}
\begin{itemize}
\vspace*{1.0truecm}
\item Material behavior is stochastic, both spatially and pointwise,
\vspace*{1.0truecm}
\item How is failure mechanics of solids and structures affected by that stochasticity?
\vspace*{1.0truecm}
\item Can the Stochastic approach to ElastoPlasticity offer more information
about the failure of a {\it particular} solid
\vspace*{1.0truecm}
\item Can the Stochastic approach to ElastoPlasticity offer more information (missing link)
about the failure of {\it general} solids (and structures)
\end{itemize}
\end{large}
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\slide{{Motivation: Typical Soil Profile}}
%\vspace*{1.0cm}
\begin{figure}[!hbpt]
%\nonumber
\begin{center}
%\begin{center}
%\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
\includegraphics[height=18cm]{/home/jeremic/tex/works/Conferences/2005/25th_Yugoslav_Congress_on_Theoretical_and_Applied_Mechanics/Presentation/TypicalSoilProfile.jpg}
% \vspace*{12cm}
% \end{flushleft}
% \begin{flushright}
% \includegraphics[width=14cm]{/home/jeremic/tex/works/Conferences/2005/25th_Yugoslav_Congress_on_Theoretical_and_Applied_Mechanics/Presentation/TypicalSoilCOV.jpg}
% %\vspace*{2.0cm}
% %\nonumber
% %\caption{DruckerPrager Associative Linear Hardening Response with Random $G$}
% %{\includegraphics[width=10.5cm]{QualSchematics4.jpg}}
% %\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
\end{center}
%\end{center}
\end{figure}
\vspace*{2.0cm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\slide{{Motivation: Pointwise Variation}}
\vspace*{2.0cm}
\begin{figure}[!hbpt]
%\nonumber
\begin{center}
% %\begin{center}
% %\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
% \includegraphics[height=12cm]{/home/jeremic/tex/works/Conferences/2005/25th_Yugoslav_Congress_on_Theoretical_and_Applied_Mechanics/Presentation/TypicalSoilProfile.jpg}
% \vspace*{12cm}
% \end{flushleft}
% \begin{flushright}
\includegraphics[width=24cm]{/home/jeremic/tex/works/Conferences/2005/25th_Yugoslav_Congress_on_Theoretical_and_Applied_Mechanics/Presentation/TypicalSoilCOV.jpg}
%\vspace*{2.0cm}
%\nonumber
%\caption{DruckerPrager Associative Linear Hardening Response with Random $G$}
%{\includegraphics[width=10.5cm]{QualSchematics4.jpg}}
%\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
\end{center}
%\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\slide{Uncertainty in Geomechanics}
\begin{large}
\begin{itemize}
% \vspace*{0.08cm}
\item Uncertainty of geomaterial properties:
\begin{itemize}
% \vspace*{0.08cm}
\item[{\bf a}] Natural variability of soil deposits
% \vspace*{0.08cm}
\item[{\bf b}] Sampling error
% \vspace*{0.08cm}
\item[{\bf c}] Testing error
\end{itemize}
\item Aleatory uncertainty $\rightarrow$ inherent variation associated with the
physical system of the environment (variation in external excitation, material
properties...). Also know known as irreducible uncertainty, variability and
stochastic uncertainty. ({\bf a})
%\vspace*{1cm}
\item Epistemic uncertainty $\rightarrow$ potential deficiency in any phase of
the modeling process that is due to lack of knowledge (poor understanding of
mechanics...). Also known as reducible uncertainty, model form uncertainty and
subjective uncertainty. ({\bf b, c})
\end{itemize}
\end{large}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\slide{Previous Work}
\begin{large}
\begin{itemize}
\vspace*{1.08cm}
\item Linear algebraic relations (linear elastic) $\rightarrow$ analytical expressions:
\begin{itemize}
\item variable transformation (Montgomery and Runger 2003)
\item cumulant expansion method (Gardiner 2004)
\end{itemize}
\vspace*{1.08cm}
\item Nonlinear differential equations $\rightarrow$
\begin{itemize}
\item Monte Carlo analysis (Schueller 1997, De Lima et al, 2001, Mellah et al. 2000...)
\item Perturbation approach (Anders and Hori 2000, Kleiber and Hien 1992, Matthies et al. 2997, Mellah et al, 2000)
\end{itemize}
\end{itemize}
\end{large}
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\slide{Objectives of the Proposed Method}
\begin{large}
\begin{itemize}
\vspace*{1.58cm}
\item Overcome the disadvantages of the perturbation and Monte Carlo approaches,
\vspace*{1.58cm}
\item Capable of carrying out sensitivity analysis at a pointlocation scale,
when material parameter are modeled as random variables,
\vspace*{1.58cm}
\item Obtain probabilistic behavior of spatial average form (upscaled form) of
constitutive rate equation when material properties are modeled as random field.
\end{itemize}
\end{large}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\slide{Problem Statement: 3D}
\vspace*{1.57cm}
%\begin{normalsize}
%\begin{large}
\begin{itemize}
\item The general 3D constitutive rate equation  a nonlinear ODE system with random coefficient and random forcing
\vspace*{0.6cm}
\begin{equation}
\nonumber
\frac{d\sigma_{ij}(t)}{dt} = D_{ijkl} \frac{d\epsilon_{kl}(t)}{dt}
\end{equation}
\vspace*{0.6truecm}
\begin{eqnarray}
\nonumber
D_{ijkl} = \left\{\begin{array}{ll}
%
D^{el}_{ijkl}
%
%
\; & \mbox{when~} f < 0 \vee (f = 0 \wedge df < 0) \\
%
\\
%
D^{el}_{ijkl}

\displaystyle \displaystyle \frac{D^{el}_{ijmn}
\displaystyle \frac{\partial U}{\partial \sigma_{mn}}
\displaystyle \frac{\partial f}{\partial \sigma_{pq}}
D^{el}_{pqkl}}
{\displaystyle \frac{\partial f}{\partial \sigma_{rs}}
D^{el}_{rstu}
\displaystyle \frac{\partial U}{\partial \sigma_{tu}}

\displaystyle \frac{\partial f}{\partial q_*}r_*}
%
\; & \mbox{when~} f = 0 \vee df = 0
%
\end{array} \right.
\end{eqnarray}
\vspace{1truecm}
\end{itemize}
%\end{normalsize}
%\end{large}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\slide{Problem Statement: 1D}
%\begin{normalsize}
\vspace*{1.57cm}
\begin{itemize}
\item 1D  a nonlinear ODE, random coefficient and random forcing
\vspace{0.4truecm}
\begin{eqnarray}
\nonumber
\frac{d\sigma(t)}{dt}
=
\beta(\sigma,D,q,r;t) \frac{d\epsilon(t)}{dt}
=
\eta(\sigma,D,q,r,\epsilon; t)
\end{eqnarray}
\vspace{0.4truecm}
with an initial condition $\sigma(0)=\sigma_0$
\end{itemize}
%\end{normalsize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\slide{Stochastic Continuity Equation}
\begin{large}
\begin{itemize}
\vspace*{0.7truecm}
\item The 1D constitutive equation visualization: from each initial point in
$\sigma$space a trajectory starts out which describes the corresponding
solution of the stochastic process
\vspace*{0.1truecm}
\item Consider a cloud of initial points (described by density $ \rho
(\sigma,0) $ in $\sigma$space): movement of all these points is dictated by
the constitutive equation, the phase density $ \rho $ varies in time according
to a continuity equation (Liouville equation):
\vspace*{0.1truecm}
\begin{equation}
\nonumber
\frac{\partial \rho (\sigma(t),t)}{\partial t}=\frac{\partial}{\partial \sigma} \eta [\sigma(t), D, q, r, \epsilon(t)].\rho[\sigma(t),t]
\end{equation}
\vspace*{0.1truecm}
with initial condition
\vspace*{0.1truecm}
\begin{equation}
\nonumber
\rho(\sigma,0)=\delta(\sigma\sigma_0)
\end{equation}
\end{itemize}
\end{large}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\slide{FokkerPlanck Equation}
\vspace{1truecm}
\begin{itemize}
\begin{large}
\item Writing the continuity equation in ensemble average form and using Van
Kampen's Lemma ($ <\rho(h,t)>=P(h,t) $) yields the following FokkerPlanck
equation:
\end{large}
\begin{normalsize}
\vspace*{0.95truecm}
\begin{eqnarray}
\nonumber
\displaystyle \frac{\partial P(\sigma(t), t)}{\partial t}=
&& \displaystyle \frac{\partial}{\partial \sigma} \left[ \left\{\left< \vphantom{\int_{0}^{t} d\tau} \eta(\sigma(t), D,
q, r, \epsilon(t)) \right> \right. \right. \\
\nonumber
&+& \left. \left. \int_{0}^{t} d\tau Cov_0 \left[ \displaystyle \frac{\partial \eta(\sigma(t), D, q, r,
\epsilon(t))}{\partial \sigma}; \right. \right. \right. \\
\nonumber
& & \left. \left. \left. \eta(\sigma(t\tau), D, q, r,
\epsilon(t\tau) \vphantom{\int_{0}^{t} d\tau} \right] \right \} P(\sigma(t),t) \right] \\
\nonumber
&+& \displaystyle \frac{\partial^2}{\partial \sigma^2} \left[ \left\{ \int_{0}^{t} d\tau Cov_0 \left[ \vphantom{\int_{0}^{t}}
\eta(\sigma(t), D, q, r, \epsilon(t)); \right. \right. \right. \\
\nonumber
& & \left. \left. \left. \eta (\sigma(t\tau), D, q, r,
\epsilon(t\tau)) \vphantom{\int_{0}^{t}} \right] \vphantom{\int_{0}^{t}} \right\} P(\sigma (t),t) \right] \\
\nonumber
\end{eqnarray}
\end{normalsize}
\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\slide{Solution of FokkerPlanck Equation}
\begin{large}
\begin{itemize}
\vspace*{0.5truecm}
\item The FokkerPlanck equation $\rightarrow$ advectiondiffusion equation:
%
\vspace*{0.3truecm}
\begin{normalsize}
\begin{equation}
\nonumber
\frac{\partial P(\sigma,t)}{\partial t} = \frac{\partial}{\partial \sigma}\left[N_{(1)}P(\sigma,t)\frac{\partial}{\partial \sigma}
\left\{N_{(2)} P(\sigma,t)\right\} \right] = \frac{\partial \zeta}{\partial \sigma}
\end{equation}
\end{normalsize}
\vspace*{0.4truecm}
%\vspace*{0.85truecm}
\vspace*{0.5truecm}
\item Initial condition  deterministic (Dirac delta function) or random
%
\vspace*{0.3truecm}
\begin{normalsize}
\begin{equation}
\nonumber
P(\sigma,0)=\delta(\sigma)
\end{equation}
\end{normalsize}
\vspace*{0.4truecm}
\vspace*{0.5truecm}
\item Boundary condition  reflecting (conserve probability mass or no probability current flow)
%
\vspace*{0.3truecm}
\begin{normalsize}
\begin{equation}
\nonumber
\zeta(\sigma,t)_{AtBoundaries}=0
\end{equation}
\end{normalsize}
\vspace*{0.4truecm}
%\vspace*{1.0truecm}
\vspace*{0.5truecm}
\item The FokkerPlanck equation solution $\rightarrow$
%\textit{Method of Lines} by
%semidiscretizing the stress domain using
\textit{Finite Difference Technique}
\end{itemize}
\end{large}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\slide{{Numerical Scheme}}
\begin{normalsize}
\begin{itemize}
%\vspace*{1.0truecm}
\item The FokkerPlanck equation was solved using \textit{Method of Lines} by semidiscretizing the stress domain using \textit{Finite
Difference Technique}
\begin{figure}[!hbpt]
%\nonumber
%\begin{flushleft}
\begin{center}
%\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
\includegraphics[width=20cm]{/home/jeremic/tex/works/Conferences/2005/25th_Yugoslav_Congress_on_Theoretical_and_Applied_Mechanics/Presentation/StressDiscretization.jpg}
%\vspace*{6cm}
%\end{flushleft}
%\begin{flushright}
%\includegraphics[height=13cm]{ContourLowOCR_RandomGm.pdf}
%\vspace*{2.0cm}
%\nonumber
%\caption{Low OCR Cam Clay Response with Random $G$}
%{\includegraphics[width=10.5cm]{QualSchematics4.jpg}}
%\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
%\end{flushleft}
\end{center}
%\end{flushright}
\end{figure}
\end{itemize}
\end{normalsize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\slide{{At Intermediate Node $i$}}
\begin{normalsize}
\begin{eqnarray*}
\nonumber
\frac{\partial P^{(i)}}{\partial t}
=
&+& P^{(i1)} \left[ \frac{N^{(i)}_{(1)}}{2 \Delta \sigma} + \frac{N^{(i)}_{(2)}}{\Delta \sigma^2}  \frac{1}
{\Delta \sigma} \frac{\partial N^{(i)}_{(2)}}{\partial \sigma} \right]
\nonumber \\
&& P^{(i)} \left[ \frac{\partial N^{(i)}_{(1)}}{\partial \sigma}
+ 2 \frac{N^{(i)}_{(2)}}{\Delta \sigma^2}  \frac{\partial^2 N^{(i)}_{(2)}}{\partial \sigma^2} \right] \\
\nonumber
&+& P^{(i+1)} \left [ \frac{N^{(i)}_{(1)}}{2
\Delta \sigma} + \frac{N^{(i)}_{(2)}}{\Delta \sigma^2} + \frac{1} {\Delta \sigma} \frac{\partial N^{(i)}_{(2)}}{\partial \sigma}
\right]
\end{eqnarray*}
\begin{itemize}
\item Not a very efficient scheme
\item Possible improvement through adaptivity
\item Also considering Reduced Order Modeling (ROM)
\end{itemize}
\end{normalsize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\slide{{Numerical Scheme}}
\begin{normalsize}
\begin{itemize}
\vspace{0.5truecm}
\item Introducing the BC at the left boundary
\begin{equation}
\nonumber
P^{(1)} = P^{(2)} \left[\displaystyle \frac{\displaystyle \frac{N^{(1)}_{(2)}}{\Delta \sigma}}{N^{(1)}_{(1)} + \displaystyle \frac{
N^{(1)}_{(2)}}{\Delta \sigma}  \displaystyle \frac{\partial N^{(1)}_{(2)}}{\partial \sigma}} \right]
\end{equation}
\vspace{1.5truecm}
\item and at the right boundary
\begin{equation}
\nonumber
P^{(n)}= P^{(n1)} \left[\displaystyle \frac{\displaystyle \frac{N^{(n)}_{(2)}}{\Delta \sigma}}{N^{(n)}_{(1)} + \displaystyle \frac{
N^{(n)}_{(2)}}{\Delta \sigma}  \displaystyle \frac{\partial N^{(n)}_{(2)}}{\partial \sigma}} \right]
\end{equation}
\item The semidiscretized PDE (i.e. a set of simultaneous ODEs) was solved using ODE solver available in \textit{Mathematica}
\end{itemize}
\end{normalsize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\slide{{Elastic Response with Random $G$}}
\begin{large}
\begin{itemize}
\vspace*{0.5cm}
\item General form of elastic constitutive rate equation \\ ${d \sigma_{12}}/{dt}
= 2G {d \epsilon_{12}}/{dt} = \eta(G, \epsilon_{12};t)$
\vspace*{0.5cm}
\item The advection and diffusion coefficients of FPE are \\
$N_{(1)}=2{d \epsilon_{12}}/{dt}
\;\; \mbox{;} \;\;
N_{(2)}=4t\left(\displaystyle {d \epsilon_{12}}/{dt} \right)^2 Var[G]$
\end{itemize}
\end{large}
\vspace*{0.7cm}
\begin{figure}[!hbpt]
%\nonumber
%\begin{flushleft}
\begin{center}
%\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
\includegraphics[height=12cm]{/home/jeremic/tex/works/Conferences/2005/25th_Yugoslav_Congress_on_Theoretical_and_Applied_Mechanics/Presentation/PDFElastic_RandomGm.pdf}
%\vspace*{3cm}
%\end{flushleft}
%\begin{flushright}
%\includegraphics[height=9cm]{ReasonOfNotMatchingMonteCarlom.pdf}
%\vspace*{2.0cm}
%\nonumber
%\caption{DruckerPrager Associative Linear Hardening Response with Random $G$}
%{\includegraphics[width=10.5cm]{QualSchematics4.jpg}}
%\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
%\end{flushright}
\end{center}
\end{figure}
\vspace*{3.5cm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\slide{{Verification of Elastic Response \\ Variable Transformation}}
%\vspace*{1.0cm}
\begin{figure}[!hbpt]
%\nonumber
\begin{flushleft}
%\begin{center}
%\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
\includegraphics[height=8.5cm]{/home/jeremic/tex/works/Conferences/2005/25th_Yugoslav_Congress_on_Theoretical_and_Applied_Mechanics/Presentation/MonteCarlo_Elasticm.pdf}
\vspace*{4cm}
\end{flushleft}
\begin{flushright}
\includegraphics[height=10cm]{/home/jeremic/tex/works/Conferences/2005/25th_Yugoslav_Congress_on_Theoretical_and_Applied_Mechanics/Presentation/ReasonOfNotMatchingMonteCarlom.pdf}
%\vspace*{2.0cm}
%\nonumber
%\caption{DruckerPrager Associative Linear Hardening Response with Random $G$}
%{\includegraphics[width=10.5cm]{QualSchematics4.jpg}}
%\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
\end{flushright}
%\end{center}
\end{figure}
\vspace*{3.0cm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\slide{{DruckerPrager Associative Linear Hardening with Random $G$}}
\begin{normalsize}
\vspace{1truecm}
\begin{itemize}
\item The general form of DruckerPrager elasticplastic associative linear hardening constitutive rate equation
\vspace{1truecm}
\begin{eqnarray}
\nonumber
\frac{d \sigma_{12}}{dt} &=& G^{ep} \frac{d \epsilon_{12}}{dt} \\
\nonumber
&=& \eta(\sigma_{12}, D^{el}, q, r, \epsilon_{12};t)
\end{eqnarray}
\vspace{1truecm}
\item The advection and diffusion coefficients of FPE are
\vspace{0.5truecm}
\begin{equation}
\nonumber
N_{(1)}=\displaystyle \frac{d \epsilon_{12}}{dt} \left< 2G  \displaystyle \frac{G^2}{G+9K\alpha^2+\displaystyle \frac{1}{\sqrt{3}}I_1
\alpha'} \right>
\end{equation}
\vspace{0.5truecm}
\begin{equation}
\nonumber
N_{(2)}=t\left(\displaystyle \frac{d \epsilon_{12}}{dt} \right)^2 Var\left[2G  \displaystyle \frac{G^2}{G+9K\alpha^2+\displaystyle
\frac{1}{\sqrt{3}}I_1 \alpha'} \right]
\end{equation}
\end{itemize}
\end{normalsize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\slide{{DruckerPrager Associative Linear Hardening with Random $G$}}
\vspace*{0.01cm}
\begin{figure}[!hbpt]
%\nonumber
\begin{flushleft}
%\begin{center}
%\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
\includegraphics[height=8cm]{/home/jeremic/tex/works/Conferences/2005/25th_Yugoslav_Congress_on_Theoretical_and_Applied_Mechanics/Presentation/PDFDruckerPrager_RandomGm.pdf}
\vspace*{5.5cm}
\end{flushleft}
\begin{flushright}
\includegraphics[height=13cm]{/home/jeremic/tex/works/Conferences/2005/25th_Yugoslav_Congress_on_Theoretical_and_Applied_Mechanics/Presentation/ContourDruckerPrager_RandomGm.pdf}
%\vspace*{2.0cm}
\nonumber
%\caption{DruckerPrager Associative Linear Hardening Response with Random $G$}
%{\includegraphics[width=10.5cm]{QualSchematics4.jpg}}
%\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
%\end{flushleft}
\end{flushright}
%
\vspace*{5.5cm}
\begin{flushleft}
\includegraphics[height=7.0cm]{/home/jeremic/tex/works/Conferences/2005/25th_Yugoslav_Congress_on_Theoretical_and_Applied_Mechanics/Presentation/MonteCarlo_DruckerPragerm.pdf}
\end{flushleft}
\vspace*{2.5cm}
%
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\slide{{Comparing Uncertainty in \\ Elastic and ElasticPlastic Response}}
\vspace*{1cm}
\begin{figure}[!hbpt]
%\nonumber
%\begin{flushleft}
\vspace*{18cm}
\begin{center}
\hspace*{3cm}
\includegraphics[width=24cm]{/home/jeremic/tex/works/Papers/2004/StochasticConstitutive1DJEMPart_2/Plastic_ExtendedElastic_PDFComparisonm.pdf}
%\vspace*{4cm}
%\end{flushleft}
%\begin{flushright}
%\includegraphics[height=11.5cm]{ContourDruckerPrager_RandomGm.pdf}
%\vspace*{2.0cm}
%\nonumber
%\caption{DruckerPrager Associative Linear Hardening Response with Random $G$}
%{\includegraphics[width=10.5cm]{QualSchematics4.jpg}}
%\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
%\end{flushleft}
%\end{flushright}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\slide{{Cam Clay Constitutive Model}}
\begin{normalsize}
\vspace{1truecm}
\begin{itemize}
\item The general form of Cam Clay 1D shear constitutive rate equation
\vspace{1truecm}
\begin{eqnarray}
\nonumber
\frac{d \sigma_{12}}{dt} &=& G^{ep} \frac{d \epsilon_{12}}{dt} \\
\nonumber
&=& \eta(\sigma_{12}, D^{el}, q, r, \epsilon_{12};t)
\end{eqnarray}
\vspace{1truecm}
where $\eta$ has the form:
\vspace{1truecm}
\begin{equation}
\nonumber
\eta = \left[2G  \displaystyle \frac{\left(36 \displaystyle \frac{G^2}{M^4} \right) \sigma_{12}^2}
{\displaystyle \frac{(1+e_0)p(2pp_0)^2}{\kappa} + \left(18 \displaystyle \frac{G}{M^4}\right) \sigma_{12}^2
+ \displaystyle \frac{1+e_0}{\lambda\kappa} p p_0 (2pp_0)} \right] \displaystyle \frac{d\epsilon_{12}}{dt}
\end{equation}
\vspace{1truecm}
\item The advection and diffusion coefficients of FPE are
\vspace{0.5truecm}
\begin{equation}
\nonumber
N_{(1)}^{(i)}=\left<\eta^{(i)}(t)\right> + \int_0^t d\tau cov\left[\displaystyle \frac{\partial \eta^{(i)}(t)}{\partial t};
\eta^{(i)} (t\tau)\right]
\end{equation}
\vspace{0.5truecm}
\begin{equation}
\nonumber
N_{(2)}^{(i)} = \int_0^t d\tau cov\left[\eta^{(i)}(t); \eta^{(i)} (t\tau)\right]
\end{equation}
\end{itemize}
\end{normalsize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\slide{{Low OCR Cam Clay with Random $G$}}
%\vspace*{2.0cm}
\begin{figure}[!hbpt]
%\nonumber
\begin{flushleft}
%\begin{center}
%\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
\includegraphics[height=8cm]{/home/jeremic/tex/works/Conferences/2005/25th_Yugoslav_Congress_on_Theoretical_and_Applied_Mechanics/Presentation/PDFLowOCR_RandomGm.pdf}
\vspace*{6cm}
\end{flushleft}
\begin{flushright}
\includegraphics[height=13cm]{/home/jeremic/tex/works/Conferences/2005/25th_Yugoslav_Congress_on_Theoretical_and_Applied_Mechanics/Presentation/ContourLowOCR_RandomGm.pdf}
%\vspace*{2.0cm}
%\nonumber
%\caption{Low OCR Cam Clay Response with Random $G$}
%{\includegraphics[width=10.5cm]{QualSchematics4.jpg}}
%\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
%\end{flushleft}
\end{flushright}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\slide{{Low OCR Cam Clay Response with Random $G$ and Random $M$}}
%\vspace*{2.0cm}
\begin{figure}[!hbpt]
%\nonumber
\begin{flushleft}
%\begin{center}
%\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
\includegraphics[height=8cm]{/home/jeremic/tex/works/Conferences/2005/25th_Yugoslav_Congress_on_Theoretical_and_Applied_Mechanics/Presentation/PDFLowOCR_RandomG_RandomMm.pdf}
\vspace*{6cm}
\end{flushleft}
\begin{flushright}
\includegraphics[height=13cm]{/home/jeremic/tex/works/Conferences/2005/25th_Yugoslav_Congress_on_Theoretical_and_Applied_Mechanics/Presentation/ContourLowOCR_RandomG_RandomMm.pdf}
%\vspace*{2.0cm}
%\nonumber
%\caption{Low OCR Cam Clay Response with Random $G$ and Random $M$}
%{\includegraphics[width=10.5cm]{QualSchematics4.jpg}}
%\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
%\end{flushleft}
\end{flushright}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\slide{{Low OCR Cam Clay Response with Random $G$ and Random $p_0$}}
%\vspace*{2.0cm}
\begin{figure}[!hbpt]
%\nonumber
\begin{flushleft}
%\begin{center}
%\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
\includegraphics[height=8cm]{/home/jeremic/tex/works/Conferences/2005/25th_Yugoslav_Congress_on_Theoretical_and_Applied_Mechanics/Presentation/PDFLowOCR_RandomG_Randomp0m.pdf}
\vspace*{6cm}
\end{flushleft}
\begin{flushright}
\includegraphics[height=13cm]{/home/jeremic/tex/works/Conferences/2005/25th_Yugoslav_Congress_on_Theoretical_and_Applied_Mechanics/Presentation/ContourLowOCR_RandomG_Randomp0m.pdf}
%\vspace*{2.0cm}
%\nonumber
%\caption{Low OCR Cam Clay Response with Random $G$ and Random $p_0$}
%{\includegraphics[width=10.5cm]{QualSchematics4.jpg}}
%\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
%\end{flushleft}
\end{flushright}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\slide{{Low OCR Cam Clay Response with Random $G$, Random $M$ and Random $p_0$}}
%\vspace*{2.0cm}
\begin{figure}[!hbpt]
%\nonumber
\begin{flushleft}
%\begin{center}
%\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
\includegraphics[height=8cm]{/home/jeremic/tex/works/Conferences/2005/25th_Yugoslav_Congress_on_Theoretical_and_Applied_Mechanics/Presentation/PDFLowOCR_RandomG_RandomM_Randomp0m.pdf}
\vspace*{6cm}
\end{flushleft}
\begin{flushright}
\includegraphics[height=13cm]{/home/jeremic/tex/works/Conferences/2005/25th_Yugoslav_Congress_on_Theoretical_and_Applied_Mechanics/Presentation/ContourLowOCR_RandomG_RandomM_Randomp0m.pdf}
%\vspace*{2.0cm}
%\nonumber
%\caption{Low OCR Cam Clay Response with Random $G$, Random $M$ and Random $p_0$}
%{\includegraphics[width=10.5cm]{QualSchematics4.jpg}}
%\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
%\end{flushleft}
\end{flushright}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\slide{{Low OCR Cam Clay Predictions at $\epsilon = 1.62$ \%}}
\vspace*{2.0cm}
\begin{figure}[!hbpt]
%\nonumber
%\begin{flushleft}
\begin{center}
%\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
\includegraphics[height=13cm]{/home/jeremic/tex/works/Conferences/2005/25th_Yugoslav_Congress_on_Theoretical_and_Applied_Mechanics/Presentation/CamClayPDFComparisonm.pdf}
%\vspace*{6cm}
%\end{flushleft}
%\begin{flushright}
%\includegraphics[height=13cm]{ContourHighOCR_RandomG_RandomMm.pdf}
%\vspace*{2.0cm}
%\nonumber
%\caption{High OCR Cam Clay Response with Random $G$ and Random $M$}
%{\includegraphics[width=10.5cm]{QualSchematics4.jpg}}
%\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
%\end{flushleft}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\slide{{High OCR Cam Clay Response with Random $G$ and Random $M$}}
%\vspace*{2.0cm}
\begin{figure}[!hbpt]
%\nonumber
\begin{flushleft}
%\begin{center}
%\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
\includegraphics[height=8cm]{/home/jeremic/tex/works/Conferences/2005/25th_Yugoslav_Congress_on_Theoretical_and_Applied_Mechanics/Presentation/PDFHighOCR_RandomG_RandomMm.pdf}
\vspace*{6cm}
\end{flushleft}
\begin{flushright}
\includegraphics[height=13cm]{/home/jeremic/tex/works/Conferences/2005/25th_Yugoslav_Congress_on_Theoretical_and_Applied_Mechanics/Presentation/ContourHighOCR_RandomG_RandomMm.pdf}
%\vspace*{2.0cm}
%\nonumber
%\caption{High OCR Cam Clay Response with Random $G$ and Random $M$}
%{\includegraphics[width=10.5cm]{QualSchematics4.jpg}}
%\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
%\end{flushleft}
\end{flushright}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\slide{Summary}
\vspace{1truecm}
\begin{large}
\begin{itemize}
\vspace*{1.5cm}
\item Expression for evolution of probability densities of stress was derived
for any general 1D elasticplastic constitutive rate equation.
\item This method doesn't require repetitive use of computationally expensive
deterministic elasticplastic model and doesn't suffer from 'closure problem'
associated with regular perturbation approach.
\item Furthermore, the developed expression is linear and deterministic PDE
whereas the constitutive rate equation is random and nonlinear.
% This
%deterministic linearity provides great simplicity in solving the PDE for
%probability densities and subsequently mean and variance behavior of stress
%obeying any constitutive model
\item Current work is going on in extending this method to 3D and incorporating
it to the formulation of stochastic elasticplastic finite element method.
\end{itemize}
\end{large}
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\end{document}
\bye