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%\MyLogo{{\footnotesize September, 22 2004}}
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%{\footnotesize
%\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} Elastic-Plastic Simulations \\
%\rule[-3mm]{0cm}{1.3cm}   Application to Performance based    \\
%\rule[-3mm]{0cm}{1.3cm} Computational Geomechanics \\
}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace*{4.0truecm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\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 Fokker-Planck equation)

%\vspace*{1.0truecm}
\item  Select results and verifications

       \begin{itemize}

            \vspace*{0.08cm}
             \item Elastic

             \vspace*{0.08cm}
             \item Drucker-Prager Linear Hardening

             \vspace*{0.08cm}
             \item Cam Clay

        \end{itemize}

\end{itemize}

\end{large}

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\slide{Motivation}

\begin{large}

\begin{itemize}

\vspace*{1.0truecm}
\item Material behavior is stochastic, both spatially and point-wise,

\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 Elasto--Plasticity offer more information 
about the failure of a {\it particular} solid

\vspace*{1.0truecm}
\item Can the Stochastic approach to Elasto--Plasticity offer more information (missing link)
about the failure of {\it general} solids (and structures)


\end{itemize}

\end{large}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\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{Drucker-Prager 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: Point-wise 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{Drucker-Prager 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}
 
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\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 point--location 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 3-D 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 1-D -- 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  1-D  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{Fokker-Planck 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 Fokker-Planck
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 Fokker-Planck Equation}

\begin{large}

\begin{itemize}

\vspace*{-0.5truecm}
\item  The  Fokker-Planck equation $\rightarrow$ advection-diffusion 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  Fokker-Planck  equation solution $\rightarrow$
%\textit{Method of Lines} by
%semi-discretizing the stress domain using
\textit{Finite Difference Technique}


\end{itemize}

\end{large}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\slide{{Numerical Scheme}}

\begin{normalsize}

\begin{itemize}

%\vspace*{-1.0truecm}
\item The Fokker-Planck equation was solved using \textit{Method of Lines} by semi-discretizing 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_RandomG-m.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^{(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]  
\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^{(n-1)} \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 semi-discretized 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 \\
$N_{(1)}=2{d \epsilon_{12}}/{dt} <G>
\;\; \mbox{;} \;\;
N_{(2)}=4t\left(\displaystyle {d \epsilon_{12}}/{dt} \right)^2 Var[G]$

\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_RandomG-m.pdf}
%\vspace*{-3cm}
%\end{flushleft}
%\begin{flushright}
%\includegraphics[height=9cm]{ReasonOfNotMatchingMonteCarlo-m.pdf}
%\vspace*{2.0cm}
%\nonumber
%\caption{Drucker-Prager 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_Elastic-m.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/ReasonOfNotMatchingMonteCarlo-m.pdf}
%\vspace*{2.0cm}
%\nonumber
%\caption{Drucker-Prager 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}


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\slide{{Drucker-Prager Associative Linear Hardening with Random $G$}}

\begin{normalsize}

\vspace{-1truecm}

\begin{itemize}

\item The general form of Drucker-Prager elastic-plastic 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}

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\slide{{Drucker-Prager 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_RandomG-m.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_RandomG-m.pdf}
%\vspace*{2.0cm}
\nonumber
%\caption{Drucker-Prager 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_DruckerPrager-m.pdf}
\end{flushleft}
\vspace*{-2.5cm}
%
\end{figure}

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\slide{{Comparing Uncertainty in  \\ Elastic and Elastic--Plastic Response}}

\vspace*{1cm}
\begin{figure}[!hbpt]
%\nonumber
%\begin{flushleft}
\vspace*{-18cm}
\begin{center}
\hspace*{3cm}
\includegraphics[width=24cm]{/home/jeremic/tex/works/Papers/2004/StochasticConstitutive1D-JEM-Part_2/Plastic_ExtendedElastic_PDFComparison-m.pdf}
%\vspace*{-4cm}
%\end{flushleft}
%\begin{flushright}
%\includegraphics[height=11.5cm]{ContourDruckerPrager_RandomG-m.pdf}
%\vspace*{2.0cm}
%\nonumber
%\caption{Drucker-Prager 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}

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\slide{{Cam Clay Constitutive Model}}

\begin{normalsize}

\vspace{-1truecm}

\begin{itemize}

\item The general form of Cam Clay 1-D 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(2p-p_0)^2}{\kappa} + \left(18 \displaystyle \frac{G}{M^4}\right) \sigma_{12}^2
+ \displaystyle \frac{1+e_0}{\lambda-\kappa} p p_0 (2p-p_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}

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\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_RandomG-m.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-m.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}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\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_RandomM-m.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-m.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}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\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_Randomp0-m.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_Randomp0-m.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_Randomp0-m.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_Randomp0-m.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}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\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/CamClayPDFComparison-m.pdf}
%\vspace*{-6cm}
%\end{flushleft}
%\begin{flushright}
%\includegraphics[height=13cm]{ContourHighOCR_RandomG_RandomM-m.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_RandomM-m.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_RandomM-m.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}


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\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 1-D elastic-plastic constitutive rate equation.

\item  This  method  doesn't require repetitive use of computationally expensive
deterministic  elastic-plastic  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  non-linear.
%  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 3-D and incorporating
it to the formulation of stochastic elastic-plastic finite element method.

\end{itemize}

\end{large}

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\end{document}
\bye





