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% does not look nice, try deleting the line with the fontenc.






%\title{NRC Staff Capacity Building: \\
%       Micromechanical Origins of Elasto-Plasticity }
\title{Nonlinear Soil Modeling for \\
       Seismic NPP Applications}


%\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=0.2cm]{university-logo}{/home/jeremic/BG/amblemi/ucdavis_logo_blue_sm}
\pgfdeclareimage[height=0.7cm]{lbnl-logo}{/home/jeremic/BG/amblemi/lbnl-logo}


\author[Jeremi{\'c}] % (optional, use only with lots of authors)
{Boris~Jeremi{\'c}  and Federico  Pisan{\`o}} 

%\institute[Computational Geomechanics Group \hspace*{0.3truecm}
\institute[\pgfuseimage{university-logo}\hspace*{0.1truecm}\pgfuseimage{lbnl-logo}] % (optional, but mostly needed)
%{ Professor, University of California, Davis\\
{ University of California, Davis\\
%  and\\ 
%  Faculty Scientist, Lawrence Berkeley National Laboratory, Berkeley }
  Lawrence Berkeley National Laboratory, Berkeley }
% - Use the \inst command only if there are several affiliations.
% - Keep it simple, no one is interested in your street address.


\date[] % (optional, should be abbreviation of conference name)
{\small LBNL Seminar, December 2012}

\subject{}
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\begin{document}

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

\begin{frame}
  \titlepage
\end{frame}

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

\begin{frame}
  \frametitle{Outline}
  \tableofcontents
  % You might wish to add the option [pausesections]
\end{frame}

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

% 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*!
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%   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{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Motivation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{The Problem}

\begin{itemize}
  

  \item Seismic response of Nuclear Power Plants


\vspace*{0.1cm}
  \item  Soil modeling in particular

\vspace*{0.1cm}
  \item Linear elastic

\vspace*{0.1cm}
  \item Equivalent Linear Elastic (current state of practice/art)

\vspace*{0.1cm}
  \item 3D elastic-plastic modeling

\vspace*{0.1cm}
  \item  Combine  frictional  (elastic-plastic,  displacement  proportional) and
  viscous (velocity proportional) energy dissipation in one model.
  

\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Micromechanical Origins of Elasto-Plasticity}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Granular Materials}

\begin{itemize}
  \item  Pressure sensitive materials (particles)

  \item Usually assumed to be linear elastic (at least in the NPP field (?!))

  \item Can be (rigorously) shown that  granular materials 
  can never be linear elastic (with
  normal and/or shear contact forces).
  

  \item  R.  D.  Mindlin  and  H.  Deresiewicz. Elastic spheres in contact under
         varying    oblique    forces.    ASME    Journal    of    Applied   Mechanics,
         53(APM-14):327-344, September 1953.



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


   
\end{itemize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Two Spheres}

\vspace*{-1.0cm}
\begin{figure}[!hbpt]
\begin{center}
\includegraphics[width=7cm]{/home/jeremic/tex/works/Conferences/2012/LBNL_Seminars/NuclearWasteProgram_2pm/Contact01.pdf}
\end{center}
\end{figure}

   
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Contact Zone}

%\vspace*{-1.0cm}
\begin{figure}[!hbpt]
\begin{center}
\includegraphics[width=8cm]{/home/jeremic/tex/works/Conferences/2012/LBNL_Seminars/NuclearWasteProgram_2pm/Contact02.pdf}
\end{center}
\end{figure}

   
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Normal Stresses}

Hertz (1882)

%\vspace*{-1.0cm}
\begin{figure}[!hbpt]
\begin{center}
\includegraphics[width=9cm]{/home/jeremic/tex/works/Conferences/2012/LBNL_Seminars/NuclearWasteProgram_2pm/Contact03.pdf}
\end{center}
\end{figure}

   
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Normal and Shear Stresses}

Cattaneo (1938), Mindlin (1949), Etsion (2010)

\vspace*{-0.7cm}
\begin{figure}[!hbpt]
\begin{center}
\includegraphics[width=11cm]{/home/jeremic/tex/works/Conferences/2012/LBNL_Seminars/NuclearWasteProgram_2pm/Contact04.pdf}
\end{center}
\end{figure}

   
\end{frame}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Granular Materials (Summary)}

\begin{itemize}

  \item Particulate materials are never linear elastic 
  

   
\end{itemize}
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Elastic-Plastic Models}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Classical Material Models}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{von Mises perfectly plastic model}

\begin{itemize}
\vspace*{0.1cm}
  \item $f= \sqrt{\dfrac{3}{2}s_{ij}s_{ij} } -\sigma_0 =0$
  
  \item undrained shear strength $s_u$ ($s_u=\sigma_0/\sqrt{3}$)
   
\end{itemize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Drucker-Prager kinematic hardening model }

\begin{itemize}
\vspace*{0.1cm}
  \item $f= \sqrt{\left(  s_{ij}-p\alpha_{ij}\right)  \left(  s_{ij}-p\alpha
_{ij}\right) } -\sqrt{\frac{2}{3}}kp=0$
  

  \item $\epsilon^p_{ij}=\dot{\lambda}m_{ij}$,
  $m_{ij}   =   \left(\dfrac{\partial  f}{\partial\sigma_{ij}}\right)^{dev}    -
  \dfrac{1}{3}D\delta_{ij}$,  \\
  dilatancy coefficient $D = \xi \left( \sqrt{\dfrac{2}{3}}k_d - \sqrt{r_{mn}r_{mn}}\right)$

\item $\dot{\alpha}_{ij}=\dfrac{2}{3}h_{a}\left(\epsilon_{ij}^p \right)^{dev}-c_{r}\alpha_{ij}\sqrt{\dfrac{2}{3}\left(\epsilon_{rs}^p \right)^{dev}\left(\epsilon_{rs}^p \right)^{dev}}$


\end{itemize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Drucker-Prager kinematic hardening model }

 
\begin{figure}  [!htb]
 \centering
 \includegraphics[width=10cm]{/home/jeremic/tex/works/Papers/2012/EP_Viscous_model/Figures/TX_dil.pdf}
% \caption{$q-p$ response and volumetric behavior predicted by the DPKH model for different $k_d$ values}
% \label{fig:DPKHvolumetric}
\end{figure}

\begin{figure}  [!htb]
 \centering
 \includegraphics [6cm] {/home/jeremic/tex/works/Papers/2012/EP_Viscous_model/Figures/TX_cycles.pdf}
% \caption{$q-p$ DPKH response at increasing deviatoric strain in $p$--constant cyclic tests}
% \label{fig:DPKHciclic_pconst}
\end{figure}




\end{frame}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Drucker-Prager Kinematic Hardening Model with Viscosity}



  
\begin{figure}  [!htb]
 \centering
 \includegraphics [width=1\textwidth] {/home/jeremic/tex/works/Papers/2012/EP_Viscous_model/Figures/DPKH4.pdf}
% \caption{test DPKH4}
% \label{fig:test DPKH4}
\end{figure}



\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{DK - Variation in Yield Stress}


\begin{figure}  [!htb]
 \centering
 \includegraphics [width=10cm] {/home/jeremic/tex/works/Papers/2012/EP_Viscous_model/Figures/VMs0.pdf}
% \caption{Parametric   analysis   at  varying  yield  stress  $\sigma_0$  (tests
% DPKHconf) (tests VMs0)}
 \label{fig:test VMs0}
\end{figure}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{DK - Variation in Damping Coefficient}

  
\begin{figure}  [!htb]
 \centering
 \includegraphics [width=10cm] {/home/jeremic/tex/works/Papers/2012/EP_Viscous_model/Figures/VMzeta.pdf}
% \caption{Parametric analysis at varying damping coefficient $\zeta$ (tests DPKHconf) (tests VMz)}
% \label{fig:test VMzeta}
\end{figure}

\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{DK - Variation in Initial Confinement}

  
    
\begin{figure}  [!htb]
 \centering
 \includegraphics [width=10cm] {/home/jeremic/tex/works/Papers/2012/EP_Viscous_model/Figures/DPKHconf.pdf}
% \caption{Parametric  analysis  at  varying  initial  confinement  $p_0$  (tests
% DPKHconf)}
 \label{fig:test DPKHconf}
\end{figure}




\end{frame}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Comparison with $G/G_{max}$ and Damping Curves}

   
\begin{figure}  [!htb]
 \centering
\includegraphics [width=10cm] {/home/jeremic/tex/works/Papers/2012/EP_Viscous_model/Figures/highrange.pdf}
\end{figure}



\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Comparison with $G/G_{max}$ and Damping Curves}

   
\begin{figure}  [!htb]
 \centering
\includegraphics [width=10cm] {/home/jeremic/tex/works/Papers/2012/EP_Viscous_model/Figures/smallrange.pdf}
\end{figure}



\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Comparison with $G/G_{max}$ and Damping Curves}

   
\begin{figure}  [!htb]
 \centering
\includegraphics[width=10cm]{/home/jeremic/tex/works/Papers/2012/EP_Viscous_model/Figures/wholerange.pdf}
\end{figure}



\end{frame}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Pisan{\`o}-Jeremi{\'c} Material Model}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{PJ Model: Assumptions}

\begin{itemize}

  \item    Split    stress    into    frictional    and    viscous    components
  $\sigma_{ij}=\sigma_{ij}^{f} + \sigma_{ij}^{v}$

  \item Remove the elastic region (limit analysis)

  \item Rotating kinematic hardening
 
   
\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{PJ Model: Elasticity}

        
\begin{equation}
\label{eq:elastic}
d\sigma_{ij}=D_{ijhk}^e\left ( d\epsilon_{hk}-d\epsilon_{hk}^p \right )
\nonumber
\end{equation}
%
%
\begin{equation}
\label{eq:elastic_dev}
ds_{ij}=2G_{max}\left ( de_{hk}-de_{hk}^p \right )
\nonumber
\end{equation}
%

%
\begin{equation}
\label{eq:elastic_vol}
dp=-K\left ( d\epsilon_{vol}-d\epsilon_{vol}^p \right )
\nonumber
\end{equation}
%

where $p=-\sigma_{kk}/3$,  $\epsilon_{vol}=\epsilon_{kk}$, 
$s_{ij}=\sigma_{ij}^{dev}$, $e_{ij}=\epsilon_{ij}^{dev}$, 
$G_{max}=E/2\left(1+\nu\right)$ and $K=E/3\left(1-2\nu\right)$ 


\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{PJ Model: DP Yield and Bounding Surface}

            
\begin{equation}
\label{eq:yieldlocus}
f_y=\dfrac{3}{2} \left(s_{ij}-p\alpha _{ij}\right)\left(s_{ij}-p\alpha _{ij}\right)-k^2 p^2=0
\nonumber
\end{equation}
%

 
\begin{equation}
\label{eq:boundingsurface}
f_B=\dfrac{3}{2} s_{ij}s_{ij}-M^2p^2=0
\nonumber
\end{equation}\\
%

 
 \vspace*{-0.6cm}
 \hspace*{-1cm}
\begin{figure}  [!htb]
 \centering
\includegraphics [width=4.8cm] {/home/jeremic/tex/works/Papers/2012/Bounding_Surface_Frictional_Model/Figures/projection.pdf}
% \caption{Representation of the stress projection onto the bounding surface}
% \label{fig:projection}
\end{figure}



\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{PJ Model: Plastic Flow and Translation Rule}

 Borrowed from Manzari and Dafalias (1997) 
\begin{equation}
\label{eq:flow_rule}
d\epsilon_{hk}^p=d\lambda\left(n^{dev}_{ij} - \dfrac{1}{3} D\delta_{ij} \right)
\nonumber
\end{equation}
%

%
\begin{equation}
\label{eq:dil_coeff}
D = \xi \left(\alpha^d_{ij}-\alpha_{ij}\right)n^{dev}_{ij}=\xi \left(\sqrt{\dfrac{2}{3}}k_dn^{dev}_{ij}-\alpha_{ij}\right)n^{dev}_{ij}
\nonumber
\end{equation}
          
 
\begin{equation}
\label{eq:prager_rule}
d\alpha_{ij}=\Vert d\alpha_{ij}\Vert n_{ij}^{dev}
\nonumber
\end{equation}
%

\end{frame}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{PJ Model: Vanishing Elastic Region}


    
\begin{flushleft}
\begin{equation}
\label{eq:vanish_locus}
\lim_{k\rightarrow 0} f_y= 0        
\Rightarrow
\lim_{k\rightarrow 0}s_{ij}=p\alpha_{ij}
\Rightarrow 
ds_{ij}=d\alpha_{ij}p+\alpha_{ij}dp
\nonumber
\end{equation}

      
\begin{equation}
n^{dev}_{ij}=\dfrac{ds_{ij}-\alpha_{ij}dp}{\Vert ds_{ij}-\alpha_{ij}dp\Vert}
\nonumber
\end{equation}
%

\begin{equation}
\label{eq:alpha_incr}
\Vert d\alpha_{ij}\Vert=\dfrac{1}{pN^{dev}}\dfrac{\partial f}{\partial \sigma_{ij}} d\sigma_{ij}
\nonumber
\end{equation}
%
   
\begin{equation}
\label{eq:alpha_incr_rad}
\Vert d\alpha_{ij}\Vert=\sqrt{\dfrac{2}{3}}\dfrac{dq}{p}
\nonumber
\end{equation}
%
\end{flushleft}

 \vspace*{-4.6cm}
% \hspace*{-1cm}
\begin{figure}  [!htb]
\begin{flushright}
\includegraphics [width=4.8cm] {/home/jeremic/tex/works/Papers/2012/Bounding_Surface_Frictional_Model/Figures/projection.pdf}
\end{flushright}
% \caption{Representation of the stress projection onto the bounding surface}
% \label{fig:projection}
\end{figure}



\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{PJ Model: Hardening Modulus and Plastic Multiplier}

  
\begin{equation}
\Vert d{\alpha}_{ij} \Vert=\dfrac{2}{3}\frac{Hd{\lambda}}{p}
\nonumber
\end{equation}
%

 
\begin{equation}
\label{eq:plast_mult}
d{\lambda}=\dfrac{ 2G_{max}\Vert d{e}_{ij}\Vert
+Kd{\epsilon}_{vol}\alpha_{ij}n_{ij}^{dev}  }{2G + \dfrac{2}{3}H -KD\alpha_{ij}n_{ij}^{dev} }
\nonumber
\end{equation}
%
   

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{PJ Model: Stress Projection, Hardening and Unloading}


 
\begin{equation}
\label{eq:unloading}
d{\beta}=-\left    (    1+\beta    \right   )\dfrac{   \bar{s}_{ij}d{s}_{ij}   -
\dfrac{2}{3}M^2\bar{p}d{p}}{  \bar{s}_{ij}\left  (  s_{ij}-s_{ij}^0  \right  ) -
\dfrac{2}{3}M^2\bar{p}\left ( p-p^0 \right )}>0
\nonumber
\end{equation}
%

 \vspace*{-0.0cm}
% \hspace*{-1cm}
\begin{figure}  [!htb]
\begin{center}
\includegraphics [width=4.0cm] {/home/jeremic/tex/works/Papers/2012/Bounding_Surface_Frictional_Model/Figures/projection.pdf}
\end{center}
% \caption{Representation of the stress projection onto the bounding surface}
% \label{fig:projection}
\end{figure}

    

\end{frame}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{PJ Model: Triaxial Response}


\begin{figure}  [!htb]
 \centering
\includegraphics [width=6cm] {/home/jeremic/tex/works/Papers/2012/Bounding_Surface_Frictional_Model/Figures/res1a.pdf}
\\
\includegraphics [width=6cm] {/home/jeremic/tex/works/Papers/2012/Bounding_Surface_Frictional_Model/Figures/res1b.pdf}
% \caption{Predicted  triaxial  responses  for  different  dilatancy  surfaces  (
% $p_0$=100   kPa,   $G_{max}$   =   4   MPa,   $\nu$=0.25,   $M$=1.2,   $\xi$=1,
% $h$=$G$/(1.5$p_0$), $m$=1)}
 \label{fig:res1}
\end{figure}


    

\end{frame}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{PJ Model: Pure Shear Cyclic}

   
\begin{figure}  [!htb]
 \centering
\includegraphics [width=6cm] {/home/jeremic/tex/works/Papers/2012/Bounding_Surface_Frictional_Model/Figures/res2a.pdf}
\\
\includegraphics [width=6cm] {/home/jeremic/tex/works/Papers/2012/Bounding_Surface_Frictional_Model/Figures/res2b.pdf}
% \hfill
% \caption{Predicted pure shear response at two different shear strain amplitudes ($p_0$=100 kPa, $T$=2$\pi$ s, $\zeta_0$ = 0.003,
%$G_{max}$= 4 MPa, $\nu$=0.25, $M$=1.2, $k_d$=$\xi$=0, $h$=$G$/(1.5$p_0$), $m$=1)}
% \label{fig:res2}
\end{figure}


    

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{PJ Model: Calibration for $G/G_{max}$ and Damping}


  
     
\begin{figure}  [!htb]
 \centering
\includegraphics [width=0.8\textwidth] {/home/jeremic/tex/works/Papers/2012/Bounding_Surface_Frictional_Model/Figures/res3.pdf}
 \hfill
 \caption{Comparison  between  experimental and simulated $G/G_{max}$
 and  damping  curves ($p_0$=100 kPa, T=2$\pi$ s, $\zeta$ = 0.003, $G_{max}$ = 4
 MPa, $\nu$=0.25, $M$=1.2, $k_d$=$\xi$=0, $h$=$G$/(112$p_0$), $m$=1.38)}
 \label{fig:res3}
\end{figure}
    

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{PJ Model: Calibration for $G/G_{max}$ and Damping}

        
\begin{figure}  [!htb]
 \centering
\includegraphics [width=0.8\textwidth] {/home/jeremic/tex/works/Papers/2012/Bounding_Surface_Frictional_Model/Figures/res4.pdf}
 \hfill
 \caption{Comparison  between  experimental and simulated $G/G_{max}$
 and  damping  curves ($p_0$=100 kPa, T=2$\pi$ s, $\zeta$ = 0.003, $G_{max}$ = 4
 MPa, $\nu$=0.25, $M$=1.2, $k_d$=$\xi$=0, $h$=$G_{max}$/(15$p_0$), $m$=1)}
 \label{fig:res4}
\end{figure}

   

\end{frame}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{PJ Model: Variation in Confining Pressure}


 
\begin{figure}  [!htb]
 \centering
\includegraphics [width=0.8\textwidth] {/home/jeremic/tex/works/Papers/2012/Bounding_Surface_Frictional_Model/Figures/res7.pdf}
 \hfill
 \caption{Simulated  $G/G_{max}$  and  damping  curves  at  varying
 confining  pressure  (T=2$\pi$  s,  $G_{max}$  =  4  MPa,  $\nu$=0.25, $M$=1.2,
 $k_d$=$\xi$=0, $h$=$G$/(15$p_0$), $m$=1)}
 \label{fig:res5}
\end{figure}


    

\end{frame}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{PJ Model: Variation in Hardening Parameter $h$}


 
\begin{figure}  [!htb]
 \centering
\includegraphics [width=0.8\textwidth] {/home/jeremic/tex/works/Papers/2012/Bounding_Surface_Frictional_Model/Figures/res5.pdf}
 \hfill
 \caption{Simulated  $G/G_{max}$ and  damping  curves  at  varying h
 ($p_0$=100   kPa,  $T$=2$\pi$  s,  $G_{max}$  =  4  MPa,  $\nu$=0.25,  $M$=1.2,
 $k_d$=$\xi$=0, $m$=1)}
 \label{fig:res6}
\end{figure}


    

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{PJ Model: Variation in Hardening Parameter $m$}


         
\begin{figure}  [!htb]
 \centering
\includegraphics [width=0.8\textwidth] {/home/jeremic/tex/works/Papers/2012/Bounding_Surface_Frictional_Model/Figures/res6.pdf}
 \hfill
 \caption{Simulated  $G/G_{max}$  and  damping  curves  at  varying m
 ($p_0$=100   kPa,  $T$=2$\pi$  s,  $G_{max}$  =  4  MPa,  $\nu$=0.25,  $M$=1.2,
 $k_d$=$\xi$=0, $h$=$G_{max}$/(15$p_0$))}
 \label{fig:res7}
\end{figure}


    

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{PJ Model: Variation in Viscous Damping}


         
\begin{figure}  [!htb]
 \centering
\includegraphics [width=0.4\textwidth] {/home/jeremic/tex/works/Papers/2012/Bounding_Surface_Frictional_Model/Figures/res9.pdf}
% \hfill
 \caption{Damping   curves   simulated  at  varying  $\zeta_0$  ($p_0$=100  kPa,
 $T$=2$\pi$   s,   $G_{max}$   =  4  MPa,  $\nu$=0.25,  $M$=1.2,  $k_d$=$\xi$=0,
 $h$=$G_{max}$/(15$p_0$), $m$=1)}
 \label{fig:res9}
\end{figure}


    

\end{frame}

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\section{Summary}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
  \frametitle{Summary}
  
  \begin{itemize}


%  \item  High fidelity
%  modeling and simulations for performance assessment of infrastructure systems


  \item Frictional and Viscous energy dissipation for granular materials    
  
\vspace*{0.2cm}
  \item Classical elastic-plastic models with addition of viscous damping

\vspace*{0.2cm}
  \item Vanishing elastic region elastic-plastic material model
 
\vspace*{0.2cm}
  \item Important for professional practice since $G/G_{max}$ and damping curves
  (all just in 1D  !)   is what is available, and this model calibrates well, and is full 3D
  model.
 
  
\end{itemize}


\end{frame}
%



\end{document}


