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\title{On Probabilistic Yielding of (Geo--)Materials}


%\subtitle
%{Include Only If Paper Has a Subtitle}

%\author[Author, Another] % (optional, use only with lots of authors)
%{F.~Author\inst{1} \and S.~Another\inst{2}}
% - Give the names in the same order as the appear in the paper.
% - Use the \inst{?} command only if the authors have different
%   affiliation.
\pgfdeclareimage[height=0.15cm]{university-logo}{/home/jeremic/BG/amblemi/ucdavis_logo_blue_sm}

\author[Jeremi{\'c} and Sett] % (optional, use only with lots of authors)
{Boris~Jeremi{\'c} and Kallol Sett }
% - Give the names in the same order as the appear in the paper.
% - Use the \inst{?} command only if the authors have different
%   affiliation.


\institute[Computational Geomechanics Group \hspace*{0.3truecm}
\pgfuseimage{university-logo} \hspace*{-0.3truecm}] % (optional, but mostly needed)
{ Department of Civil and Environmental Engineering\\
  University of California, Davis }
% - Use the \inst command only if there are several affiliations.
% - Keep it simple, no one is interested in your street address.




\date
{WCCM8 / ECCOMAS2008 \\ 
1st. July 2008, \\
Venezia, Italia}
%\\
%June/July 2008}
%giugno/luglio 2008}
% %\date[] % (optional, should be abbreviation of conference name)
% %{
% %\small Graduate Student: Guanzhou Jie 
% %\\ ~\\
% %Funding: NSF--CMS--0324661, NSF--TeraGrid, NSF--EEC--9701568 \\
% %%\texttt{http://sokocalo.engr.ucdavis.edu/$\tilde{~}$jeremic/}
% }
% % - Either use conference name or its abbreviation.
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\subject{}
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%\logo{\pgfuseimage{university-logo}}

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% \logo{\pgfuseimage{university-logo}}




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  \titlepage
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  \tableofcontents
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% Structuring a talk is a difficult task and the following structure
% may not be suitable. Here are some rules that apply for this
% solution: 

% - Exactly two or three sections (other than the summary).
% - At *most* three subsections per section.
% - Talk about 30s to 2min per frame. So there should be between about
%   15 and 30 frames, all told.

% - A conference audience is likely to know very little of what you
%   are going to talk about. So *simplify*!
% - In a 20min talk, getting the main ideas across is hard
%   enough. Leave out details, even if it means being less precise than
%   you think necessary.
% - If you omit details that are vital to the proof/implementation,
%   just say so once. Everybody will be happy with that.





\section{Motivation}


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\subsection{}
%\subsection{Stochastic Systems: Historical Perspectives}
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% \begin{frame} 
%   
%   
%   \frametitle{History}
% 
%  
% 
% \begin{itemize}
% 
% 
% \item Probabilistic fish counting
% 
% \vspace*{0.2cm}
% 
% \vspace*{0.2cm}
% \item Williams' DEM simulations, differential displacement vortexes
% 
% %\vspace*{0.2cm}
% %\item Runesson's dilemma
% 
% \vspace*{0.2cm}
% \item SFEM round table
% 
% \vspace*{0.2cm}
% \item Kavvas' probabilistic hydrology 
% 
% 
% 
% 
% \end{itemize}
% 
%  
% 
% \end{frame}
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\begin{frame}
\frametitle{Uncertain (Geo--)materials} 

\begin{center}
%\vspace*{-0.5cm}
\includegraphics[height=4.3cm]{/home/jeremic/tex/works/Conferences/2007/GeoDenver/SFEM/Presentation/FrictionAngleProfile.jpg}
\hfill
\includegraphics[height=3.9cm]{/home/jeremic/tex/works/Conferences/2006/KragujevacSEECCM06/Presentation/MGMuzorak01.jpg}
\\
\small{
Mayne et al. (2000) 
\hspace*{4.3cm}
Sture et al. (1998) 
}
\end{center}
%\vspace*{-6.0cm}



% {%
% \begin{beamercolorbox}{section in head/foot}
% \usebeamerfont{framesubtitle}\tiny{
% Stein Sture, Nicholas C. Costes, Susan N. Batiste, Mark R. Langton, Khalid A. Al-Shibli, Boris
% Jeremi{\'c}, Roy A. Swanson and Melissa Frank. Mechanics of granular materials at low effective
% stresses. \textit{ASCE Journal of Aerospace Engineering}, vol. 11, No. 3, pages 67-72, 1998.}
% %\vskip2pt\insertnavigation{\paperwidth}\vskip2pt
% \end{beamercolorbox}%
% }

\end{frame}

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\begin{frame} 

\frametitle{Background}

 

\begin{itemize}



%
\item Brownian motion, Langevin equation (Einstein 1905)


\item Random forcing (Fokker-Planck, Kolmogorov 1941)

%\vspace*{0.2cm}
%\item  Approach  for  random  forcing  $\rightarrow$  relationship  between  the
%autocorrelation function and spectral density function (Wiener 1930)

%\vspace*{0.2cm}
\item Random coefficient (Hopf 1952), 
(Bharrucha-Reid 1968), Monte Carlo method


%\item Alternately, Monte Carlo method can be used for solution of Langevin equation  $\rightarrow$ computationally very expensive


\item Epistemic uncertainty, lack of knowledge

%    \begin{itemize}
%  
%       \item Can be reduced by \\ collecting more data
%  
%       \item Mathematical tools not well \\
%             developed, trade-off with \\
%             aleatory uncertainty
%  
%     \end{itemize}



%\vspace*{0.15cm}
\item Aleatory uncertainty, inherent variation 

%\vspace*{-1.9cm}
%\begin{figure}[!hbpt]
%\begin{flushright}
%\includegraphics[height=3.5cm,angle=-90]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Present/uncertain03.pdf}
%\end{flushright}
%\end{figure}


%    \begin{itemize}
% 
%       \item Can not be reduced
% 
%       \item Has highly developed mathematical tools 
% 
%     \end{itemize}


\item Methods for treating  
      epistemic uncertainty 
      not well developed, 
      trade-off with 
      aleatory uncertainty



\item Ergodicity of geomaterials !(?)

%    \begin{itemize}
%
%      \item Are geomaterials ergodic? Possibly  yes 
%
%%      \item Issues up for discussion (soil, concrete, rock, biomaterials...)
%
%    \end{itemize}


\item Prof. Einav question



\end{itemize}


\end{frame}



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% \begin{frame} 
% 
% \frametitle{Historical Overview}
% 
% 
%  
% 
% \begin{itemize}
% 
% %
% \item Brownian motion, Langevin equation $\rightarrow$  PDF governed by simple diffusion Eq. (Einstein 1905)
% 
% 
% %
% %  
% %  \begin{equation}
% %  \nonumber
% %  \frac{\partial f (x,t)}{\partial t} = D \frac{\partial^2 f(x,t)}{\partial x^2}
% %  \end{equation}
% %  
% 
% \vspace*{0.2cm}
% \item With external forces $\rightarrow$ Fokker-Planck-Kolmogorov (FPK) for the PDF (Kolmogorov 1941)
% 
% \vspace*{0.2cm}
% \item  Approach  for  random  forcing  $\rightarrow$  relationship  between  the
% autocorrelation function and spectral density function (Wiener 1930)
% 
% \vspace*{0.2cm}
% \item Approach for random coefficient $\rightarrow$ 
% Functional integration approach (Hopf 1952), Averaged equation approach (Bharrucha-Reid 1968), 
% Numerical approaches, Monte Carlo method
% 
% 
% %\item Alternately, Monte Carlo method can be used for solution of Langevin equation  $\rightarrow$ computationally very expensive
% 
% \end{itemize}
% 
% 
% \end{frame}
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\section{Uncertain Material Nonlinear Modeling}

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\subsection{Probabilistic Elasto--Plasticity}
%\subsection{PEP Formulations}
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\begin{frame} 

\frametitle{Uncertainty Propagation through Constitutive Eq.}

% 

\begin{itemize}

 


\item Incremental el--pl constitutive equation  
     $\Delta \sigma_{ij} = D_{ijkl} \Delta \epsilon_{kl}$

 
  
%\begin{normalsize}

%    
%    \begin{equation}
%    \nonumber
%    \frac{d\sigma_{ij}}{dt} = D_{ijkl} \frac{d\epsilon_{kl}}{dt}
%    \end{equation}
    
    
    \begin{eqnarray}
    \nonumber
     D_{ijkl} = \left\{\begin{array}{ll}
%
     D^{el}_{ijkl}
%
%
        \;\;\; & \mbox{\large{~for elastic}} \\
%
           \\
%
           D^{el}_{ijkl}
           -
           \frac{\displaystyle  D^{el}_{ijmn} m_{mn} n_{pq} D^{el}_{pqkl}}
           {\displaystyle n_{rs} D^{el}_{rstu} m_{tu} - \xi_* r_*}

           \;\;\; & \mbox{\large{~for elastic--plastic}}
%
           \end{array} \right.
           \end{eqnarray}

%\end{normalsize}

 



%\vspace{0.5cm}
% \item Non--linear coupling in the El--Pl modulus


%-  \item Focus on 1-D $\rightarrow$ a nonlinear ODE with random coefficient and random forcing
%-  
%-  
%- 
%-  \begin{eqnarray}
%-  \nonumber
%-  \frac{d\sigma(x,t)}{dt} &=& \beta(\sigma(x,t),D^{el}(x),q(x),r(x);x,t) \frac{d\epsilon(x,t)}{dt} \\
%-  \nonumber
%-  &=& \eta(\sigma,D^{el},q,r,\epsilon; x,t)  \mbox{\ \ \ \ with an I.C. $\sigma(0)=\sigma_0$}
%-  \end{eqnarray}
%- 
 

\end{itemize}

% 


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\end{frame}






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% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - \begin{frame} 
% - 
% - \frametitle{Previous Work}
% - 
% -  
% - 
% - \begin{itemize}
% - 
% - \item 
% - Linear algebraic or differential equations  $\rightarrow$ Analytical solution:
% - 
% -          \begin{itemize}
% -         \item Variable Transf. Method (Montgomery and Runger 2003)
% -         \item Cumulant Expansion Method (Gardiner 2004)
% -         \end{itemize}
% - 
% - \item 
% - Nonlinear differential equations (elasto-plastic/viscoelastic-viscoplastic):
% -      
% -       \begin{itemize}
% - 
% -       \item  Monte  Carlo Simulation (Schueller 1997, De Lima et al 2001, Mellah
% -       et al. 2000, Griffiths et al. 2005...) \\ $\rightarrow$ accurate, very costly
% - 
% -       \item  Perturbation  Method  (Anders and Hori 2000, Kleiber and Hien 1992,
% -       Matthies  et  al. 1997) \\ $\rightarrow$ first and second order Taylor series
% -       expansion  about mean - limited to problems with small C.O.V. and inherits
% -       "closure problem"
% - 
% -       \end{itemize}
% - 
% - % 
% - %  \item 
% - %  Monte Carlo method: accurate, very costly
% - %  
% - %  \item 
% - %  Perturbation method: 
% - 
% - \end{itemize}
% - 
% - 
% -  
% - 
% - 
% - 
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% - 
% - \end{frame}
% - 
% - 
% - 
% - 
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% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - \begin{frame} 
% - 
% - \frametitle{Problem Statement}
% -  
% - 
% - 
% -  
% -  
% - \begin{itemize}
% - 
% - \item Incremental 3D elastic-plastic stress--strain:
% - % 
% - %    
% - 
% -     \begin{equation}
% -     \nonumber
% -       \frac{ d\sigma_{ij}}{d t} = \left \{ 
% -     D^{el}_{ijkl}
% -            -
% -            \frac{\displaystyle  D^{el}_{ijmn} m_{mn} n_{pq} D^{el}_{pqkl}}
% -            {\displaystyle n_{rs} D^{el}_{rstu} m_{tu} - \xi_* r_*}
% -            \right \}  
% -            \frac{ d\epsilon_{kl}}{d t}
% -     \end{equation}
% -    
% - 
% - 
% -  
% - 
% -  \item  Focus  on  1D  $\rightarrow$  a  nonlinear  ODE  with random coefficient
% -  (material) and random forcing ($\epsilon$)
% - %
% - % 
% - %
% -  \begin{eqnarray}
% -  \nonumber
% -  \frac{d\sigma(x,t)}{dt} &=& \beta(\sigma(x,t),D^{el}(x),q(x),r(x);x,t) \frac{d\epsilon(x,t)}{dt} \\
% -  \nonumber
% -  &=& \eta(\sigma,D^{el},q,r,\epsilon; x,t)
% -  \end{eqnarray}
% - %
% - with initial condition $\sigma(0)=\sigma_0$
% - 
% -  \end{itemize}
% - 
% - \end{frame}
% - 
% - 
% - 
% - 
% - 
% - 
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% - \begin{frame} 
% - 
% - 
% - 
% - \frametitle{Evolution of the Density $\rho(\sigma,t)$ }
% - 
% -  
% - 
% - \begin{itemize}
% - 
% - \vspace*{1cm}
% -   \item  From each initial point in \\
% -          $\sigma$-space a trajectory \\
% -          starts out describing \\
% -          the corresponding solution \\
% -          of the stochastic process
% -   
% - \vspace*{0.3cm}
% -   \item  Movement of a cloud of initial\\
% -          points  described  by density \\
% -          $\rho(\sigma,0)$ in $\sigma$-space, \\
% -          is governed by the \\
% -          constitutive equation,
% - 
% - \end{itemize}
% - 
% - 
% - %\begin{figure}[!hbpt]
% - %\begin{center}
% - \vspace*{-5cm}
% - \hspace*{6cm}
% - \includegraphics[height=4.5cm,angle=-90]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Present/Cloud_of_Points.pdf}
% - %\end{center}
% - %\end{figure}
% -               
% - 
% - 
% - \end{frame}
% - 
% - 
% - 
% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% - \begin{frame} 
% - 
% - 
% - \frametitle{Stochastic Continuity (Liouville) Equation}
% - 
% -  
% - 
% - \begin{itemize}
% - 
% - 
% - \item phase density $\rho$ of $\sigma(x,t)$ varies in time according to a continuity 
% - Liouville equation (Kubo 1963): 
% - %
% - \begin{eqnarray}
% - \frac{\partial \rho (\sigma(x,t),t)}{\partial t}
% - = 
% - \nonumber
% - \\
% - -\frac{\partial  \eta (\sigma(x,t), D^{el}(x), q(x), r(x), \epsilon(x,t))  }{\partial \sigma}
% - \;\; 
% - \rho[\sigma(x,t),t]
% - \nonumber
% - \end{eqnarray}
% - 
% - \vspace{0.5cm}
% - \item with initial conditions $\rho(\sigma,0) = \delta(\sigma-\sigma_0)$
% - 
% - 
% - \end{itemize}
% -  
% - \end{frame}
% - 
% - 
% - 
% - 
% - 
% - 
% - 
% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - \begin{frame} 
% - 
% - 
% - \frametitle{{Ensemble Average form of Liouville Equation}}
% - 
% - 
% - 
% -  
% - 
% - 
% - 
% - \noindent
% - Continuity  equation  written  in 
% -  ensemble  average  form  (eg. cumulant
% - expansion method (Kavvas and Karakas 1996)):
% - 
% -  
% - %\vspace*{-0.5cm}
% - 
% - \begin{footnotesize}
% - 
% - \begin{eqnarray}
% - \nonumber
% - &&\displaystyle \frac{\partial \left < \rho(\sigma(x_t,t), t) \right >}{\partial t}=
% - - \displaystyle \frac{\partial}{\partial \sigma} \left[ \left\{\left< \vphantom{\int_{0}^{t} d\tau} \eta(\sigma(x_t,t), D^{el}(x_t),
% - q(x_t), r(x_t), \epsilon(x_t,t)) \right> \right. \right. \\
% - \nonumber
% - &+& \left. \left. \int_{0}^{t} d\tau Cov_0 \left[ \displaystyle \frac{\partial \eta(\sigma(x_t,t), D^{el}(x_t), q(x_t), r(x_t),
% - \epsilon(x_t,t))}{\partial \sigma}; \right. \right. \right. \\
% - \nonumber
% - & & \left. \left. \left. \eta(\sigma(x_{t-\tau},t-\tau), D^{el}(x_{t-\tau}), q(x_{t-\tau}), r(x_{t-\tau}),
% - \epsilon(x_{t-\tau},t-\tau) \vphantom{\int_{0}^{t} d\tau} \right] \right \} \left < \rho (\sigma(x_t,t),t) \right > \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(x_t,t), D^{el}(x_t), q(x_t), r(x_t), \epsilon(x_t,t)); \right. \right. \right. \\
% - \nonumber
% - & & \left. \left. \left. \eta (\sigma(x_{t-\tau},t-\tau), D^{el}(x_{t-\tau}), q(x_{t-\tau}), r(x_{t-\tau}),
% - \epsilon(x_{t-\tau},t-\tau)) \vphantom{\int_{0}^{t}} \right] \vphantom{\int_{0}^{t}} \right\} \left < \rho (\sigma (x_t,t),t) \right > \right] \\
% - \nonumber
% - \end{eqnarray}
% - 
% - \end{footnotesize}
% - 
% - 
% - 
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% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - 
% - 
% - \end{frame}
% - 
% - 
% - 



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\begin{frame} 

\frametitle{Eulerian--Lagrangian FPK Equation}

%

%\begin{itemize}

\begin{footnotesize}

%\noindent
Liouville continuity equation (Kubo 1963); 
ensemble  average  form  
(Kavvas and Karakas 1996);
van  Kampen's  Lemma (van Kampen 1976)
% $\rightarrow$  $  <\rho(\sigma,t)>=P(\sigma,t) $, 
%ensemble average of phase density 
%%(in stress space here) 
%is the probability density;


\begin{eqnarray}
\nonumber
&&\displaystyle \frac{\partial P(\sigma(x_t,t), t)}{\partial t}=
- \displaystyle \frac{\partial}{\partial \sigma} \left[ \left\{\left< \vphantom{\int_{0}^{t} d\tau} \eta(\sigma(x_t,t), D^{el}(x_t),
q(x_t), r(x_t), \epsilon(x_t,t)) \right> \right. \right. \\
\nonumber
&+& \left. \left. \int_{0}^{t} d\tau Cov_0 \left[ \displaystyle \frac{\partial \eta(\sigma(x_t,t), D^{el}(x_t), q(x_t), r(x_t),
\epsilon(x_t,t))}{\partial \sigma}; \right. \right. \right. \\
\nonumber
& & \left. \left. \left. \eta(\sigma(x_{t-\tau},t-\tau), D^{el}(x_{t-\tau}), q(x_{t-\tau}), r(x_{t-\tau}),
\epsilon(x_{t-\tau},t-\tau) \vphantom{\int_{0}^{t} d\tau} \right] \right \} P(\sigma(x_t,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(x_t,t), D^{el}(x_t), q(x_t), r(x_t), \epsilon(x_t,t)); \right. \right. \right. \\
\nonumber
& & \left. \left. \left. \eta (\sigma(x_{t-\tau},t-\tau), D^{el}(x_{t-\tau}), q(x_{t-\tau}), r(x_{t-\tau}),
\epsilon(x_{t-\tau},t-\tau)) \vphantom{\int_{0}^{t}} \right] \vphantom{\int_{0}^{t}} \right\} P(\sigma (x_t,t),t) \right] \\
\nonumber
\end{eqnarray}

\end{footnotesize}


\end{frame}







%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - \begin{frame} 
% - 
% - \frametitle{Eulerian--Lagrangian Format}
% - 
% - 
% - \vspace*{-0.5cm}
% -  
% - 
% - \begin{itemize}
% - 
% - %-  \item $Cov_0[\cdot]$ $\rightarrow$ Time-ordered covariance function
% - %-  %
% - %-  %
% - %-  \begin{eqnarray}
% - %-  \nonumber
% - %-  &&Cov_0[\eta(x,t_1);\eta(x,t_2)] \\
% - %-  && \ \ \ \ \ \ \ \ \ \ \ \ \ \ = \left < \eta(x,t_1)\eta(x,t_2) \right > - \left < \eta(x,t_1) \right > \cdot \left < \eta(x,t_2) \right >
% - %-  \nonumber
% - %-  \end{eqnarray}
% - %-  
% - \vspace*{0.5cm}
% - \item Real-space location (Lagrangian) $x_t$ is known but pull-back to Eulerian location $x_{t-\tau}$ is unknown 
% - 
% - \vspace*{0.5cm}
% - \item Can be related using strain rate $\dot \epsilon \ (=d\epsilon/dt)$
% - 
% - 
% - \begin{equation}
% - \nonumber
% - d\epsilon = \dot \epsilon \tau =\frac{x_t-x_{t-\tau}}{x_t}    \mbox{; \ \ \ \  or, \ \ \ }  x_{t-\tau}=(1-\dot \epsilon \tau)x_t 
% - \end{equation}
% - 
% - \end{itemize}
% - 
% - 
% - \end{frame}
% - 
% - 
% - 



%-- 
%--  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%--  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%--  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%--  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%--  \begin{frame} \frametitle{Equivalent It{\^o} Stochastic Differential Equation}
%--  
%--  \begin{itemize}  
%--  
%--   
%--  
%--  \item Equivalency between It{\^o} stochastic differential equation and FPK equation (Gardiner 2004):
%--  
%--   
%--  
%--  \end{itemize}
%--  
%--  
%--  
%--  
%--  
%--  
%--  \begin{normalsize}
%--  
%--  \begin{eqnarray}
%--  \nonumber
%--  \lefteqn{d\sigma (x,t)=
%--  \left\{ \left< \vphantom{\displaystyle \frac{\partial \eta_1 (\sigma(x_t,t), A(x_t,t), \epsilon(x_t,t))}{\partial
%--  \sigma}} \eta (\sigma(x_t,t), D(x_t), q(x_t), r(x_t), \epsilon(x_t,t)) \right> \right.} \\
%--  \nonumber
%--  &+& \left. \int_0^t d \tau Cov_0 \left[\displaystyle \frac{\partial \eta (\sigma(x_t,t), D(x_t), q(x_t), r(x_t), \epsilon(x_t,t))}
%--  {\partial \sigma}; \right. \right. \\
%--  \nonumber
%--  & & \left. \left. \eta (\sigma (x_{t-\tau},t-\tau), D(x_{t-\tau}), q(x_{t-\tau}), r(x_{t-\tau}), \epsilon (x_{t-\tau},
%--  t-\tau)) \vphantom{\displaystyle \frac{\partial \eta_1 (\sigma(x_t,t), A(x_t,t), \epsilon(x_t,t))}{\partial \sigma}}\right] \right\} dt
%--  + b(\sigma,t)dW(t)
%--  \end{eqnarray}
%--  %
%--  \noindent \flushleft{where},
%--  %
%--  \begin{eqnarray}
%--  \nonumber
%--  b^2(\sigma,t) &=& 2 \int_0^t d \tau Cov_0 \left[\vphantom{\int_0^t d \tau} \eta(\sigma(x_t,t), D(x_t), q(x_t), r(x_t),
%--  \epsilon(x_t,t)); \right. \\
%--  \nonumber
%--  & & \left. \eta(\sigma (x_{t-\tau},t-\tau), D (x_{t-\tau}), q(x_{t-\tau}), r(x_{t-\tau}), \epsilon
%--  (x_{t-\tau},t-\tau))
%--  \vphantom{\int_0^t d \tau} \right]
%--  \end{eqnarray}
%--  
%--  \end{normalsize}
%--  
%--  
%--  
%--  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%--  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%--  
%--  \end{frame}
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%--  
%--  


%--  
%--  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%--  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%--  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%--  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%--  \begin{frame} \frametitle{{3D FPK Equation}}
%--  
%--  \begin{footnotesize}
%--  
%--  \begin{eqnarray}
%--  \nonumber
%--  \lefteqn{\displaystyle \frac{\partial P(\sigma_{ij}(x_t,t), t)}{\partial t} = \displaystyle \frac{\partial}{\partial \sigma_{mn}} 
%--  \left[ \left\{\left< \vphantom{\int_{0}^{t}} \eta_{mn}(\sigma_{mn}(x_t,t), D_{mnrs}(x_t), \epsilon_{rs}(x_t,t))\right> \right. \right.} \\
%--  \nonumber
%--  &+& \left. \left. \int_{0}^{t} d\tau Cov_0 \left[\displaystyle \frac{\partial \eta_{mn}(\sigma_{mn}(x_t,t), D_{mnrs}(x_t), 
%--  \epsilon_{rs}(x_t,t))} {\partial \sigma_{ab}}; \right.  \right. \right. \\
%--  \nonumber
%--  & & \left. \left. \left. \eta_{ab} (\sigma_{ab}(x_{t-\tau}, t-\tau), D_{abcd}(x_{t-\tau}), \epsilon_{cd}(x_{t-\tau}, t-\tau) 
%--  \vphantom{\int_{0}^{t}} \right] \right \} P(\sigma_{ij}(x_t,t),t) \right] \\
%--  \nonumber
%--  &+& \displaystyle \frac{\partial^2}{\partial \sigma_{mn} \partial \sigma_{ab}} \left[ \left\{ \int_{0}^{t} d\tau Cov_0 \left[ 
%--  \vphantom{\int_{0}^{t}} \eta_{mn}(\sigma_{mn}(x_t,t), D_{mnrs}(x_t), \epsilon_{rs}(x_t,t)); \right. \right. \right. \\
%--  \nonumber
%--  & & \left. \left. \left. \eta_{ab} (\sigma_{ab}(x_{t-\tau}, t-\tau), D_{abcd}(x_{t-\tau}), \epsilon_{cd}(x_{t-\tau}, t-\tau)) 
%--  \vphantom{\int_{0}^{t}} \right] \vphantom{\int_{0}^{t}} \right\} P(\sigma_{ij}(x_t,t),t) \right]
%--  \end{eqnarray}
%--  
%--   
%--  \end{footnotesize}
%--  
%--  
%--  
%--  %-- \begin{itemize}
%--  %-- 
%--  %-- 
%--  %-- 
%--  %-- \item 6 equations
%--  %-- 
%--  %-- \item Complete description of 3-D probabilistic stress-strain response
%--  %-- 
%--  %-- \end{itemize}
%--  %-- 
%--  %--  
%--  
%--  
%--  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%--  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%--  
%--  \end{frame}
%--  
%--  
%--  
%--  


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame} 

\frametitle{Compact Form of Eulerian--Lagrangian FPK Equation}


 

\begin{itemize}

\item Advection-diffusion equation 
%
\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]
\end{equation}

%

\item Complete probabilistic description of response 


\item Solution PDF is second-order exact to covariance of time (exact mean and variance)


\item It is deterministic equation in probability density space

\item It is linear PDE in probability density space 
$\rightarrow$ simplifies the numerical solution process

%\vspace*{0.2truecm}

\end{itemize}


\vspace*{0.5cm}
{%
\begin{beamercolorbox}{section in head/foot}
\usebeamerfont{framesubtitle}\tiny{B. Jeremi\'{c}, K. Sett, and M. L. Kavvas, "Probabilistic
Elasto--Plasticity: Formulation in 1--D", \textit{Acta Geotechnica}, Vol. 2, No. 3, 2007}
%\vskip2pt\insertnavigation{\paperwidth}\vskip2pt
\end{beamercolorbox}%
}



\end{frame}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame} 



\frametitle{Template Solution of E--L FPK Equation}

 

\begin{itemize}




\item FPK diffusion--advection equation is applicable to any material model $\rightarrow$ 
only the coefficients $N_{(1)}$ and $N_{(2)}$ are different for different material models
%
%
%
%\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]
%%\nonumber
% = -\frac{\partial \zeta}{\partial \sigma}
\end{equation}
%\end{normalsize}

%

\item Initial condition random or deterministic 
%  
%  \begin{itemize}
%  
%  \item Deterministic $\rightarrow$ Dirac delta function $\rightarrow$ $ P(\sigma,0)=\delta(\sigma) $
%  
%  \item Random $\rightarrow$ Any given distribution
%  
%  \end{itemize}


%
%\item Boundary condition: Reflecting BC $\rightarrow$ conserves probability mass
%$\zeta(\sigma,t)|_{At \ Boundaries}=0$
%
%\item Finite Differences used for solution (among many others)
%

\item Solution for both 
\begin{itemize} 
  \item  probabilistic elastic (PEL) and
  \item probabilistic elastic--plastic (PELPL) problems
\end{itemize} 




\end{itemize}

 
  
\vspace*{1.0cm}
{%
\begin{beamercolorbox}{section in head/foot}
\usebeamerfont{framesubtitle}\tiny{K. Sett, B. Jeremi{\'c} and M.L.  Kavvas, "The Role of Nonlinear 
Hardening/Softening in Probabilistic Elasto--Plasticity", \textit{International Journal for Numerical 
and Analytical Methods in Geomechanics}, Vol. 31, No. 7, pp. 953-975, 2007}
%\vskip2pt\insertnavigation{\paperwidth}\vskip2pt
\end{beamercolorbox}%
}


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


\end{frame}






%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%==%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Probabilistic Yielding}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame} 




\frametitle{Probabilistic Yielding: Weighted Coefficients}

\begin{itemize}

%

%\vspace*{0.5cm}
\item  Probabilistic  yielding: take both PEL and PELPL into account at the same
time with some (un--)certainty


\vspace*{0.3cm}
\item Weighted elastic and elastic--plastic Solution
${\partial P(\sigma,t)}/{\partial t} 
= 
-{\partial \left(N^w_{(1)}P(\sigma,t)
-{\partial \left(N^w_{(2)} P(\sigma,t)\right) }/{\partial \sigma} \right)}/
{\partial \sigma}$


%\vspace*{1.0cm}
\vspace*{0.3cm}
\item Weighted advection and diffusion coefficients are then 

$N_{(1,2)}^{w} (\sigma) 
= 
(1 - P[\Sigma_y \leq \sigma]) N_{(1)}^{el} + P[\Sigma_y \leq \sigma] N_{(1)}^{el-pl} $

%%\vspace*{0.5cm}
%\item   
%%\vspace*{-10cm}
%Cumulative Probability Density  function  (CDF) of the yield function 
%\vspace*{-0.5cm}
%\begin{figure}[!h]
%%\hspace*{7cm}
%\includegraphics[width=4cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/vonMises_YieldCDF_Combined-edited.pdf} 
%\end{figure}
%\vspace*{-0.5cm}

\vspace*{0.3cm}
\item Similar to European Pricing Option in financial simulations (Black--Scholes options 
 pricing model '73, Nobel prize for Economics '97)



\end{itemize}

\vspace*{0.5cm}
{%
\begin{beamercolorbox}{section in head/foot}
\usebeamerfont{framesubtitle}\tiny{B.  Jeremi{\'c} and K. Sett. On Probabilistic
Yielding  of  Materials.  in print in  \textit{Communications in Numerical Methods in
Engineering}, 2008.}
%\vskip2pt\insertnavigation{\paperwidth}\vskip2pt
\end{beamercolorbox}%
}




\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}



\frametitle{Probability of  Yielding}



\vspace*{-1.00cm}

\begin{itemize}

\item Probabilistic von Mises

\item Probability    of yielding at \\
$\sigma=0.0012~{\rm MPa}$\\ 
$P[\Sigma_y \leq (\sigma=0.0012~{\rm MPa})]$
\\
$ = 0.8$

\item Equivalent advection and \\
 diffusion coefficients are
%                              
\begin{equation}
N_{(1)}^{eq}|_{\sigma=0.0012~MPa}
=
(1-0.8)N_{(1)}^{el}+0.8N_{(1)}^{ep}
\nonumber
\end{equation}
%
\begin{equation}
N_{(2)}^{eq}|_{\sigma=0.0012~MPa}
=
(1-0.8)N_{(2)}^{el}+0.8N_{(2)}^{ep}
\nonumber 
\end{equation}
%



\end{itemize}


\vspace*{-7.0cm}
\begin{figure}[!htbp]
\begin{flushright}
\includegraphics[height=4.2cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/vonMises_YieldCDF_Combined-edited.pdf}
%\vspace*{-0.8cm}
%\caption{CDF  of  shear  strength for von Mises  model: (a) very uncertain case,
%(b) fairly certain case.}
\label{vonMises_Yield_CDF}
\end{flushright}
\end{figure}




%
%
%\begin{figure}[!hbpt]
%\begin{center}
%\includegraphics[width=8cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/figures/vonMises_G_and_cu_very_uncertain/Contour_PDF-edited.pdf}
%\end{center}
%\end{figure}                                                          
%
%\vspace*{-0.3cm}
%linear elastic -- linear hardening plastic von Mises
%

\end{frame}



%--  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}



\frametitle{Bi--Linear von Mises Response}



\vspace*{0.20cm}
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=8.0cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/Final/vonMises_G_and_cu_very_uncertain_Contour_PDF-edited.pdf}
% %\vspace*{-0.5cm}
% \caption{von  Mises  associative  plasticity  model  with uncertain shear modulus and
% shear strength (yield parameter): (a) Evolution of PDF of stress with 
% strain (PDF=10000 was used as a cutoff for surface plot) and 
% (b) Contours of evolution of stress PDF with strain.}
% \label{vonMises_G_and_cu_very_uncertain}
\end{center}
\end{figure}
%



\end{frame}



%--  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}



\frametitle{Bi--Linear von Mises Response}



\vspace*{0.50cm}
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=9.5cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/Final/vonMises_G_and_cu_very_uncertain_PDF-edited.pdf}
% %\vspace*{-0.5cm}
% \caption{von  Mises  associative  plasticity  model  with uncertain shear modulus and
% shear strength (yield parameter): (a) Evolution of PDF of stress with 
% strain (PDF=10000 was used as a cutoff for surface plot) and 
% (b) Contours of evolution of stress PDF with strain.}
% \label{vonMises_G_and_cu_very_uncertain}
\end{center}
\end{figure}
%



\end{frame}



%--  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}



\frametitle{von Mises, Certain $c_u$, Uncertain $G$}


\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=8.5cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/Final/vonMises_G_very_uncertain_but_cu_fairly_certain_Contour_PDF-edited.pdf}
\end{center}
\end{figure}
%


\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}



\frametitle{von Mises, Uncertain $c_u$, Certain $G$}


\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=9.5cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/Final/vonMises_G_fairly_certain_but_cu_very_uncertain_Mean_SD_Mode_DeterSol-edited.pdf}
\end{center}
\end{figure}
%


\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - \begin{frame}
% - 
% - 
% - 
% - \frametitle{Drucker Prager}
% - 
% - 
% - \begin{figure}[!htbp]
% - \begin{center}
% - \includegraphics[height=4cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/Final/DruckerPrager_YieldCDF_Combined-edited.pdf}
% - \vspace*{-0.8cm}
% - \caption{CDF of yield stresses for Drucker-Prager model:
% - (a) very uncertain and (b) fairly certain}
% - \label{DruckerPrager_Yield_CDF}
% - \end{center}
% - \end{figure}
% - 
% - 
% - 
% - \end{frame}
% - 
% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - \begin{frame}
% - 
% - 
% - 
% - \frametitle{Drucker Prager}
% - 
% - %
% - \begin{figure}[!htbp]
% - \begin{center}
% - \mbox{a)}\includegraphics[width=4.5cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/Final/DruckerPrager_G_and_alpha_very_uncertain_PDF-edited.pdf}
% - \hfill
% - \mbox{b)}\includegraphics[width=4.5cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/Final/DruckerPrager_G_and_alpha_very_uncertain_Contour_PDF-edited.pdf}
% - %\vspace*{-0.5cm}
% - \caption{Drucker-Prager  associative  elastic--plastic  model  with  uncertain shear
% - modulus and frictional coefficient: (a) Evolution of probability density function (PDF) of stress with 
% - strain (PDF=10000 was used as a cutoff for surface plot)   and (b) Contours of evolution of PDF with strain}
% - \label{DruckerPrager_G_and_cu_very_uncertain}
% - \end{center}
% - \end{figure}
% - 
% - 
% - 
% - 
% - \end{frame}
% - 
% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}



\frametitle{Drucker Prager: Uncertain $G$, Certain $\phi$ }

%
\begin{figure}[!htbp]
\begin{center}
\mbox{a)}\includegraphics[width=8.5cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/Final/DruckerPrager_G_very_uncertain_but_alpha_fairly_certain_Contour_PDF-edited.pdf}
\hfill
\end{center}
\end{figure}




\end{frame}

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

\begin{frame}



\frametitle{Drucker Prager: Certain $G$, Uncertain $\phi$ }

%
\begin{figure}[!htbp]
\begin{center}
\mbox{b)}\includegraphics[width=8.5cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/Final/DruckerPrager_G_fairly_certain_but_alpha_very_uncertain_Contour_PDF-edited.pdf}
\end{center}
\end{figure}




\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Summary}

\begin{frame}



\frametitle{Summary}

\begin{itemize}

\item   Second-order (mean and variance) exact, method 
to account for probabilistic elastic--plastic material simulation

\vspace*{0.3cm}
\item  Probabilities  of material yielding, probably govern underlying mechanics
of elasto--plasticity

%\vspace*{0.3cm}
\vspace*{0.3cm}
\item  Probabilities  of material yielding (spatial distributions) also probably
govern underlying mechanics of failure, localization of deformation...


%\vspace*{0.3cm}

\vspace*{0.3cm}
\item Probably numerous applications

\end{itemize}



\end{frame}

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% \begin{frame}
%
%\begin{itemize}
%\item Language used by Beamer: L\uncover<2->{A}TEX
%\item Language used by Beamer: L\only<2->{A}TEX
%\end{itemize}
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%\end{frame}
%
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\begin{frame}



\begin{center}
%\only<1>{\includegraphics[width=4.0cm]{/home/jeremic/tex/works/ThankYou.jpg}} 
%\only<2>{\includegraphics[width=4.0cm]{/home/jeremic/tex/works/ThankYou01.jpg}}
%\only<3>
{\includegraphics[width=6.0cm]{/home/jeremic/tex/works/ThankYou02.jpg}}
\end{center}

\end{frame}

% All of the following is optional and typically not needed. 
%\appendix
%\section<presentation>*{\appendixname}
%\subsection<presentation>*{For Further Reading}
%
%\begin{frame}[allowframebreaks]
%  \frametitle<presentation>{For Further Reading}
%    
%  \begin{thebibliography}{10}
%    
%  \beamertemplatebookbibitems
%  % Start with overview books.
%
%  \bibitem{Author1990}
%    A.~Author.
%    \newblock {\em Handbook of Everything}.
%    \newblock Some Press, 1990.
% 
%    
%  \beamertemplatearticlebibitems
%  % Followed by interesting articles. Keep the list short. 
%
%  \bibitem{Someone2000}
%    S.~Someone.
%    \newblock On this and that.
%    \newblock {\em Journal of This and That}, 2(1):50--100,
%    2000.
%  \end{thebibliography}
%\end{frame}

\end{document}


