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\begin{document}



\NAG{1}{31}{in~peer--review}{8}{09}

\runningheads{Sett and Jeremi{\'c}}
{Probabilistic Yielding and Cyclic Behavior of Geomaterials}

\title{Probabilistic Yielding and Cyclic Behavior of Geomaterials}

\author{Kallol~Sett\affil{1}, Boris~Jeremi{\'c}\affil{2}\corrauth}

\address{
\affil{1}\ Department  of  Civil Engineering,  The University  of Akron, Akron, OH 44325 \\
\affil{2}\ Department of Civil and Environmental Engineering, University of California, Davis, CA 95616
}
                                 

\corraddr{ Boris Jeremi{\'c}, Department of Civil and Environmental Engineering,
University of California, 
One Shields Avenue, Davis, CA 95616, \texttt{jeremic@ucdavis.edu}}

%\footnotetext[2]{Please ensure that you use the most up to date class file,
%available from the CFM Home Page at\\
%\texttt{http://www.interscience.wiley.com/jpages/1082-5010/}}

%\cgsn{NSF}{EEC-9701568}

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%\begin{abstract}
                        
\section{ABSTRACT}


In this paper, the novel concept of probabilistic yielding
is used for 1--D cyclic simulation of the constitutive behavior of geomaterials.
%
Fokker--Planck--Kolmogorov (FPK) equation based probabilistic
elastic--plastic constitutive framework is applied for obtaining the
complete probabilistic (probability density function) 
material response. 
%
Both perfectly plastic and hardening type material models are considered.
%
It is shown that when uncertainties in material parameters are
taken in consideration, even the simple, elastic--perfectly plastic model
captures some of the important features of geomaterial behavior, for example, modulus
reduction with cyclic strain, which, deterministically,  is only possible with more 
advanced constitutive models. 
%
Further, it is also shown that the use of isotropic and kinematic hardening
rules does not significantly improve the probabilistic material response.
%
%In addition to that, it seems that, rather than using   complex,  
%deterministic  material  modeling,  probabilistic elasto--plasticity approach 
%with  very  simple material models (extended in probabilistic space), is able to
%capture geomaterial behavior.
%


\keywords{Soil    uncertainty,    Elasto--plasticity,   Fokker-Planck-Kolmogorov
equation, Cyclic behavior}

%\end{abstract}

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\section{INTRODUCTION}

Modeling  of  geomaterials     is   inherently uncertain. This uncertainty stems
from  natural  variability  of  geomaterials  (spatial uncertainty), and testing
and    transformation    errors   (point   uncertainty)   (Lacasse   and   Nadim
\cite{Lacasse:1996}, Phoon and
Kulhawy  \cite{Kulhawy:1999a}).  These uncertainties not only affect the failure
characteristics  of  geomaterials,  but also the behavior of geostructures, made
with  geomaterials. Traditionally, geotechnical engineering community deals with
uncertainties in geomaterial by applying (large) factor of safety. However, use of
large  factors  of  safety  results not only in over-expensive design, but also,
sometimes,  in  unsafe  structures (cf.  Duncan \cite{Duncan:2000}). Hence, in
recent  years,  the  geotechnical  community  has seen an increasing emphasis on
probabilistic characterization of soil and subsequent reliability-based design.



One  of  the  important  aspects  of probabilistic geomechanics
simulation   that   has   received   less   attention   is  the
probabilistic  constitutive problem. Among the few published
papers    were    those    by    Fenton    and   Griffiths
(\cite{Griffiths:2002a},                 \cite{Griffiths:2003},
\cite{Griffiths:2005}) on probabilistic simulation of spatially
random  c-$\phi$ soil using Monte Carlo technique, and those by
Anders   and   Hori   (\cite{Hori:1999},  \cite{Hori:2000})  on
probabilistic simulation of von Mises elastic-perfectly plastic
material  using  perturbation  technique.  Both Monte Carlo and
perturbation   techniques   have   their   inherent   drawbacks
(Matthies et al. \cite{Matthies:1997reviewSFEM},   Keese \cite{keese:2003ReportSFEM})
and  in  dealing  with  those,  recently,  Jeremi\'{c}  et  al.
\cite{Jeremic:2007}   proposed   Eulerian--Lagrangian  form  of
Fokker--Planck--Kolmogorov equation (FPKE) approach (cf. Kavvas
\cite{Kavvas:2003})       to   modeling   and   simulation   for   probabilistic
elasto--plasticity. FPKE
approach   to   probabilistic  elasto--plasticity  not  only
overcomes  the  drawbacks  associated  with other probabilistic
simulation  techniques,  but also is fully compatible with the
incremental theory  of elasto--plasticity, and hence can
easily be applied to  probabilistic modeling and simulation of 
different   elastic--plastic   constitutive   models.  Solution
strategies for FPK partial differential equation, corresponding
to  elastic--plastic  constitutive  rate equation and simulated
probabilistic  stress-strain responses under monotonic loading,
assuming  mean  stress  yielding, were discussed by Sett et al.
(\cite{Sett:2007b},  \cite{sett:2007a})  for  both  linear  and
non-linear  hardening  models.  The  concept  of  probabilistic
yielding   was   introduced  and  its  effect  on  constitutive
simulation under monotonic loading was discussed by Jeremi\'{c}
and   Sett   \cite{Sett:2007c}.   It  was  shown  that  due  to
uncertainty  in  yield function (stress),  there is always a possibility,
depending  upon  the  magnitude  of  uncertainty,  that plastic
behavior  starts  at  very  very  low  strain  and influence of
elastic  behavior  continues  far into plastic domain (at large
strains)  and  hence,  the  ensemble  average (mean) of all the
possibilities  or  the  most probable (mode) possibility differ
from  deterministic  behavior.  In addition to that, a very  realistic, smooth
transition    between    elastic    and  plastic  domains  was observed even for
elastic perfectly plastic models.
Further,  nonlinear  behavior  was  observed  even for linear
hardening models.



In  this  paper,  the  concept  of  probabilistic  yielding  is
extended  to  1--D  cyclic  simulations  of  geomaterials. Both
elastic--perfectly plastic and hardening-type material model
are considered. The  numerical technique of solving FPKE cyclically with
probabilistic  yielding  is discussed. Simulated responses were
discussed  in  terms of probability density function (PDF) and
its statistical moments.















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\section{FOKKER--PLANCK--KOLMOGOROV APPROACH TO PROBABILISTIC ELASTO--PLASTICITY}
\label{FPKE}






The  Eulerian--Lagrangian  form  Fokker--Planck--Kolmogorov equation (cf. Kavvas
\cite{Kavvas:2003}) corresponding to generalized 1--D constitutive rate equation
can be written as (Jeremi\'{c} et al. \cite{Jeremic:2007}):


\begin{eqnarray}
\nonumber
\lefteqn{\displaystyle \frac{\partial P(\sigma(x_t,t), t)}{\partial t} =} \\ 
\nonumber
& & \displaystyle \frac{\partial}{\partial \sigma} \left[ \left\{\left< \vphantom{\int_{0}^{t}} \eta(\sigma, D, \epsilon; x_t,t) \right>
+ \int_{0}^{t} d\tau Cov_0 \left[\displaystyle \frac{\partial \eta(\sigma, D, \epsilon; x_t,t)} {\partial \sigma}; 
\eta (\sigma, D, \epsilon; x_{t-\tau}, t-\tau \vphantom{\int_{0}^{t}} \right] \right \} P(\sigma(x_t,t),t) \right] \\
\label{eqno_1}
& & + \displaystyle \frac{\partial^2}{\partial \sigma^2} \left[ \left\{ \int_{0}^{t} d\tau Cov_0 
\left[ \vphantom{\int_{0}^{t}} \eta(\sigma, D, \epsilon; x_t,t); \eta (\sigma, D, \epsilon; x_{t-\tau}, t-\tau)
\vphantom{\int_{0}^{t}} \right] \vphantom{\int_{0}^{t}} \right\} P(\sigma(x_t,t),t) \right]
\end{eqnarray}

\noindent 
where,  $P(\sigma(x_t,t),t)$  is the probability density of stress ($\sigma$) at
(pseudo)    time    $t$,    and    $\eta$  is the operator variable, obtained by
collecting together  all  the  operators  and  variables  on  the  r.h.s of the generalized
constitutive rate equation:

\begin{equation}
\label{eqno_2}
\displaystyle \frac{d \sigma (x_t,t)}{dt} = \eta(\sigma,D,\epsilon; x_t, t)
\end{equation}

\noindent 
In  Eq.~(\ref{eqno_2}),  $\epsilon$ is the strain, and $D$ is the tangent modulus, which
could be elastic or elastic--plastic:

\begin{eqnarray}
D = \left\{\begin{array}{ll}
%
           D^{el}
%
%
        \;\;\; & \mbox{elastic} \\
%
           \\
%
           D^{el}
           -
           \displaystyle \displaystyle \frac{D^{el}
           \displaystyle \frac{\partial U}{\partial \sigma}
           \displaystyle \frac{\partial f}{\partial \sigma}
           D^{el}}
           {\displaystyle \frac{\partial f}{\partial \sigma}
           D^{el}
           \displaystyle \frac{\partial U}{\partial \sigma}
           -
           \displaystyle \frac{\partial f}{\partial q_*}r_*}
%
        \;\;\; & \mbox{elastic-plastic}
%
           \end{array} \right.
\label{eqno_3}
\end{eqnarray}

\noindent  
where,    $D^{el}$,  $f$,  $U$,  $q_*$,  and $r_*$ are elastic  modulus,  yield 
surface,  plastic  potential  surface,  internal  variable(s),  and  rate(s)  of
evolution of internal variable(s) respectively.


Eq.~(\ref{eqno_1}) is the most general form of elastic--plastic
constitutive  rate  equation,  written  in  probability density
space. This equation (Eq.~(\ref{eqno_1})) can be written in a more compact form:

\begin{equation}
\label{eqno_1a}
\displaystyle \frac{\partial P(\sigma(x_t,t), t)}{\partial t} =
\displaystyle \frac{\partial}{\partial \sigma} \left\{ N_{(1)} P(\sigma(x_t,t),t) \right\}
+ \displaystyle \frac{\partial^2}{\partial \sigma^2} \left\{ N_{(2)} P(\sigma(x_t,t),t) \right\}
\end{equation}


\noindent 
where,   $N_{(1)}$  and  $N_{(2)}$  are  advection  and  diffusion  coefficients
respectively,  and are material model specific. By specializing Eq.~(\ref{eqno_1a})
to (any) particular constitutive  model, the resulting FPKE can be solved to
obtain  the probability density function of stress response, given uncertainties
in  material  properties and driving strain. However, difference in material behavior
in  elastic and elastic-plastic regions necessities solution of FPKE twice - one
corresponding   to   elastic  constitutive  equation  (with  $N^{el}_{(1)}$  and
$N^{el}_{(2)}$,  the  advection  and  diffusion  coefficients  corresponding  to
elastic  constitutive  equation) and the other corresponding to elastic--plastic
constitutive equation (with $N^{ep}_{(1)}$ and $N^{ep}_{(2)}$, the advection and
diffusion coefficients corresponding to elastic--plastic constitutive equation).
The  switch from elastic to elastic--plastic region (solution) can be controlled
using mean stress yielding:

\begin{eqnarray}
\begin{array}{lll}
\mbox{if}~~~~~~~~~~~ & \left<f\right> < 0 \vee \left(\left<f\right> = 0 \wedge d\left<f\right> < 0 \right) ~~~~~ & \mbox{use elastic FPKE} \\
\mbox{or, if}~~~~~~~~~ & \left<f\right> = 0 \vee d\left<f\right> = 0 ~~~~~ & \mbox{use elastic--plastic FPKE}
\end{array}
\label{eqno_4}
\end{eqnarray}
  
\noindent  
However,  difficulty  arises  if  the material yield
parameter(s)    are    uncertain,    as  the  mean  yield  criteria then does not account
for the complete probabilistic yielding
of material. For example, such mean yielding will neglect the possibilities of
elastic--plastic behavior in the elastic region and vice versa.
The concept of probabilistic yielding overcomes this
limitation, as it solves Eq.~(\ref{eqno_1a}) once, with equivalent
advection   and   diffusion   coefficients,  $N_{(1)}^{eq}$  and  $N_{(2)}^{eq}$
(Jeremi\'{c} and Sett \cite{Sett:2007c}):

\begin{eqnarray}
\begin{array}{l}
N_{(1)}^{eq} (\sigma) = (1 - P[\Sigma_y \leq \sigma]) N_{(1)}^{el} + P[\Sigma_y \leq \sigma] N_{(1)}^{ep} \\
N_{(2)}^{eq} (\sigma) = (1 - P[\Sigma_y \leq \sigma]) N_{(2)}^{el} + P[\Sigma_y \leq \sigma] N_{(2)}^{ep}
\end{array}
\label{eqno_5}
\end{eqnarray}

\noindent 
where      $(1    -    P[\Sigma_y   \leq \sigma])$ represents the probability of
material   being   elastic,  while  $P[\Sigma_y  \leq  \sigma]$  represents  the
probability  of  material  being elastic--plastic. The probabilities of material
being  elastic  and  the  probabilities  of  material being elastic--plastic can
easily    be   calculated from the cumulative density function of yield function
(stress).  
%

It  is  worth noting that the probabilistic yield criterion (Eq.~(\ref{eqno_5}))
represents probabilistic restatement of the deterministic yield criteria.
%
The    probabilistic    yield  criteria  is  introduced (or, the deterministic  
yield criteria is written in probability  space) in order to properly model
uncertain (probabilistic) yield strength.
% %
% As a matter of fact,
% if  yield  strength  is uncertain, then there is a possibility,
% depending upon the degree of uncertainty, that the material may
% yield  at very small stress level (vanishing elastic region) or
% the  effect  of  elasticity  may  be  significant  way into the
% plastic  regime. 
% %
% Probabilistic yielding takes into account both
% the  possibilities and provides a realistic way to simulate the
% uncertain material behavior.

It is also very interesting to note that proposed approach  for  calculating 
equivalent advection and
diffusion   coefficients   is   similar  to  the  solution  strategy  of  famous
Black--Scholes    \cite{BlackScholes:1973}    equation  in
financial engineering modeling of European option, where
probabilities  of  exercise  of  the  (European)  option, obtained from cumulative
density  functions,  are multiplied with stock price and present value of option
strike price to calculate the option price.


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\section{ELASTIC--PERFECTLY PLASTIC MATERIAL}
\label{Perfectly_Plastic_Material}

In  this  section,  the  FPKE--approach,  along with the concept of probabilistic
yielding,  is  applied  to  simulate  1--D  (shear  stress--shear strain) cyclic
behavior  of  elastic--perfectly plastic material. 
%
Only von Mises
material model has been considered. It may, however, be noted that presented 
development  is general enough to be used with any material model and that
von Mises is just one such model we use for illustration purposes. 


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%\subsection{Probabilistic Elastic Perfectly Plastic von Mises Material Model}

The  von Mises yield criteria can be written as:
%
\begin{equation}
\label{eqno_00}
\sqrt{J_{2}} - k = 0
\end{equation}
%
where,   $k$ is a material parameter (yield strength like)  and   $J_{2}=3/2 s_{ij}s_{ij}$
is   the second  invariant  of  deviatoric  stress  tensor
$s_{ij}  =  \sigma_{ij}  -  {1}/{3}  \sigma_{kk} \delta_{ij}$. For 1--D shear,
Eq.~(\ref{eqno_00}) becomes:
%
\begin{eqnarray}
\label{eqno_12}
\left|\sigma \right| - \sigma_y = 0  \mbox{\;\;\;\; or \;\;\;\; } \sigma = \pm \sigma_y
\end{eqnarray}
%
The yielding occurs at a yield stress of $\pm \sigma_y$.
%
It, however, is important to note that both $\sigma_y$ and $\sigma$ are uncertain and are
described by their respective probability density functions.  
%
For  elastic--perfectly  plastic  material, the distribution of
yield  stress  ($\sigma_y$)  is  given  by  its  experimentally
measured initial distribution, and remains constant. The stress
($\sigma$),  however,  evolves  according to the governing FPKE
(Eq.~(\ref{eqno_1a}))  and its distribution is given by the
solution  of the governing FPKE (Eq.~(\ref{eqno_1a})). For 1--D von Mises
elastic--perfectly   plastic   shear  constitutive  model,  the
elastic   and  the  elastic--plastic  advection  and  diffusion
coefficients of the governing FPKE (Eq.~(\ref{eqno_1a})), becomes:

\begin{eqnarray}
\begin{array}{l}
N_{(1)}^{el} = \displaystyle \frac{d \epsilon_{xy}}{dt} \left< G \right >
%
\;\;\;\;\;\; \mbox{;} \;\;\;\;\;\;
%
%\label{eqno_E2}
N_{(2)}^{el}  =   t \left(\displaystyle \frac{d \epsilon_{xy}}{dt}\right)^2 Var[G] \\
N_{(1)}^{ep} = 0
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \mbox{;} \;\;\;\;\;\;
N_{(1)}^{ep} = 0 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\end{array}
\label{eqno_6}
\end{eqnarray}

\noindent 
where,  $G$  is  the  shear  modulus,  $d  \epsilon_{xy}$ is the (deterministic)
incremental shear strain, $t$ is the
pseudo   time,   $\left<\cdot\right>$   represents   expectation  operation  and
$Var[\cdot]$   represents  variance  operation.  The  equivalent  advection  and
diffusion    coefficients  (refer    Eq.~(\ref{eqno_5}))  for  von   Mises
elastic--perfectly plastic material, then, becomes:

\begin{eqnarray}
\begin{array}{l}
N_{(1)}^{eq} (\sigma) = (1 - P[\Sigma_y \leq \sigma]) \displaystyle \frac{d \epsilon_{xy}}{dt} \left< G \right >  \\
N_{(2)}^{eq} (\sigma) = (1 - P[\Sigma_y \leq \sigma]) t \left( \displaystyle \frac{d \epsilon_{xy}}{dt}\right)^2 Var[G] 
\end{array}
\label{eqno_7}
\end{eqnarray}
  

\noindent 
One  may  note  that, in deriving the elastic and elastic--plastic advection and
diffusion  coefficients (Eq.~(\ref{eqno_6})), it was assumed that spatial random
field  material properties ($G$, and $\sigma_y$) would be first discretized into
random variables, for example at Gauss points, by appropriate tools, for example
Karhunen--Lo\`{e}ve    expansion   (Karhunen   \cite{Karhunen:1947},   Lo\`{e}ve
\cite{Loeve:1948},  Ghanem  and  Spanos \cite{book:Ghanem}). In other words, the
solution   of   FPKE,   with  advection  and  diffusion  coefficients  given  by
Eq.~(\ref{eqno_7}),     represents     point--location     scale    von    Mises
elastic--perfectly   plastic  material  behavior,  and  not  the  local--average
material behavior. The local--average material behavior, if sought for, can then
be assembled using polynomial chaos expansion (Wiener \cite{Wiener:1938}, Ghanem
and Spanos \cite{book:Ghanem}).



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\paragraph{Probability Density Function:}



The FPKE (Eq.~(\ref{eqno_1a})), with advection and diffusion coefficients given by Eq.~(\ref{eqno_7}),
was   solved  incrementally  with pseudo time steps using method of lines.  The 
stress domain of the Fokker--Planck--Kolmogorov PDE was
discretized  first  on  a  uniform  grid  by  central  differences,  and thereby
obtaining  a  series  of  ODE.  The  series  of  ODEs  was  then  solved,  after
incorporating  boundary  conditions,  simultaneously  and incrementally, with $n$
pseudo   time   steps,  using  a  standard  open--source  ODE  solver,  SUNDIALS
\cite{manual:SUNDIALS}, which utilizes ADAMS method and functional iteration.

The yield shear strength ($\sigma_y$) of the material was assumed to have a mean
value  of  60  kPa  with a COV of 30\%, values typical for clay (Federal Highway Administration 
\cite{FHWA:2002}, Lacasse and Nadim \cite{Lacasse:1996}). Also, the yield
shear   strength   was   assumed   to   be  either normal or Weibull (with shape parameter of 3.31 and 
scale parameter of 0.067) distribution 
as  shown  in Fig.~\ref{figure:YieldStress_PerfectlyPlastic_IJNAMG}.  
%
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.55\textwidth]{IJNAMG_Review02_Plot02-Edited.eps}
%\includegraphics[width=0.55\textwidth]{YieldStress_PerfectlyPlastic_IJNAMG-Ed.eps}
%\includegraphics[width=0.55\textwidth]{YieldStress_PerfectlyPlastic_IJNAMG.ps.02.eps}
\caption{Elastic--perfectly plastic probabilistic model: PDF of yield stress}
\label{figure:YieldStress_PerfectlyPlastic_IJNAMG}
\end{center}
\end{figure}
%
The  shear modulus ($G$)
was   also assumed to be either normal or Weibull distribution, but  with a mean value of 100 MPa and a
COV of 25\%. 
%
The cyclic probabilistic von Mises, elastic--perfectly plastic shear
stress--shear    strain  response (evolutionary  probability  density  function 
(PDF) of shear stress), for the case where both yield shear strength ($\sigma_y$) and shear modulus ($G$) are normally distributed,  is     shown     in
Fig.~\ref{figure:PDF_of_ShearStress_Cyclic_PerfectlyPlastic_IJNAMG-Ed}.     
%
%
%
\begin{figure}[!htbp]
\begin{center}
\mbox{a)}\includegraphics[width=0.95\textwidth]{PDF07_of_ShearStress_Cyclic_PerfectlyPlastic_IJNAMG-Ed.eps} \\
\mbox{b)}\includegraphics[width=0.95\textwidth]{PDF06_of_ShearStress_Cyclic_PerfectlyPlastic_IJNAMG-Ed.eps} 
\caption{Elastic--perfectly plastic  probabilistic model under cyclic loading: evolutionary PDF
of  shear  stress  
%
(a)  view from the junction of loading and unloading branches
(probability densities of shear stress are truncated at a value 1500 for clarity
of  the plot) and 
%
(b) view from the junction of unloading and reloading branches
(probability  densities  of  shear  stress  are  truncated at a value of 150 for
clarity of the plot)}
\label{figure:PDF_of_ShearStress_Cyclic_PerfectlyPlastic_IJNAMG-Ed}
\end{center}
\end{figure} 
%
%
Two
different    views    of    the  loading--unloading--reloading  cycle are shown,
focusing on the  transition  between  loading  and  unloading,  and  unloading and reloading
branches.            
%
As           can           be           seen           from
Fig.~\ref{figure:PDF_of_ShearStress_Cyclic_PerfectlyPlastic_IJNAMG-Ed},    PDF
for   initial   stress  (a  deterministic  Dirac  delta  function  at
stress--strain  origin) advected  and  diffused into the domain, governed by the
advection and diffusion
coefficients  (Eq.~(\ref{eqno_7})).  
%
%
It is very important to also note that, even--though the
deterministic  response  for  von  Mises  elastic--perfectly plastic material is
bi--linear,  due  to introduced  uncertainties  in  yielding, the probabilistic response is
non--linear  from the beginning.
%, with a vanishing linear region. 
%
%For example, in
%the    loading  branch,  at  the  beginning, the probabilistic response might be
%expected to be governed  by  the  elastic  advection  and  diffusion
%coefficients ($N_{(1)}^{el}$
%and  $N_{(2)}^{el}$,  refer  Eq.~(\ref{eqno_6}))  and  would  ideally be linear.
That is, due  to uncertainty in yield strength, there is a (small) possibility that the
material becomes elasto--plastic from the very beginning of loading. This possibility has
been  quantified from the PDF of the yield strength and taken into consideration
implicitly  during  simulation  using  the  equivalent  advection  and diffusion
coefficients  ($N_{(1)}^{eq}$  and  $N_{(2)}^{eq}$,  refer  Eq.~(\ref{eqno_7})).
%
Those coefficients assigns  probability weights to the realizations of stress response based
on  the probability of material being elastic or elastic--plastic. Initially,
in  the  loading  branch,  at  small  strains, the probability of material being
elastic--plastic  is  very small and  hence,  the  initial probabilistic stress response
(ensemble  of  all realizations) is closer (but not fully) to linear, elastic response. However,
as strain increases, the probability of elastic--plastic material behaving 
increases and the
probabilistic  stress  response  gradually becomes more elastic--plastic
(Fig.~\ref{figure:PDF_of_ShearStress_Cyclic_PerfectlyPlastic_IJNAMG-Ed}(a)).
%



Upon unloading,  the  material behaves as (mostly) elastic since elastic--plastic
probability   weights   from  the  governing  PDF  of  mirror image (negative)  of shear  strength
(Fig.~\ref{figure:YieldStress_PerfectlyPlastic_IJNAMG})       are      initially
very small.  
%
During  later stages of unloading (loading in the opposite direction),  and similar 
to   the  loading  branch,      the      elastic--plastic   probability  weights
increase  and      gradually     transition the  response  toward 
elasto--plasticity
(Fig.~\ref{figure:PDF_of_ShearStress_Cyclic_PerfectlyPlastic_IJNAMG-Ed}(b)).
%
Similar to this, in the subsequent reloading branch, the probability weights are again
governed        the       PDF       of       (positive, loading branch of)      
shear strength
(Fig.~\ref{figure:YieldStress_PerfectlyPlastic_IJNAMG}),     and    hence    the
probabilistic  response is again initially more linear, elastic, while gradually
it transitions to full elasto--plasticity.


%\begin{figure}[!htbp]
%\begin{center}
%\hspace*{-2.75truecm} \includegraphics[width=1.4\textwidth]{PDF07_of_ShearStress_Cyclic_PerfectlyPlastic_IJNAMG-Ed.eps} \\
%(a) \\
%\hspace*{-2.75truecm} \includegraphics[width=1.4\textwidth]{PDF06_of_ShearStress_Cyclic_PerfectlyPlastic_IJNAMG-Ed.eps} \\
%(b)
%\caption{Elastic--Perfectly Plastic Model under Cyclic Loading: Evolutionary PDF of Shear Stress (a) View from the 
%junction of loading and unloading branches (probability densities of shear stress are truncated at a value 1500 for clarity of the plot) and 
%(b) View from the junction of unloading and reloading branches (probability densities of shear stress are truncated at a value of 150 for
%clarity of the plot)}
%\label{figure:PDF_of_ShearStress_Cyclic_PerfectlyPlastic_IJNAMG-Ed}
%\end{center}
%\end{figure} 



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\paragraph{Case of Increasing Strain Loops:}
 
In
Fig.~\ref{figure:Mean_and_SD_of_ShearStress_Cyclic_IncreasingLoops_PerfectlyPlastic_IJNAMG},   
the   evolutionary   PDF   of   shear   stress   for   von  Mises
elastic--perfectly              plastic              material             (refer
Fig.~\ref{figure:PDF_of_ShearStress_Cyclic_PerfectlyPlastic_IJNAMG-Ed})       is
plotted  in  terms  of  its  statistical  moments -- the evolutionary mean
(Fig.~\ref{figure:Mean_and_SD_of_ShearStress_Cyclic_IncreasingLoops_PerfectlyPlastic_IJNAMG}(a)),               
and              standard              deviation
(Fig.~\ref{figure:Mean_and_SD_of_ShearStress_Cyclic_IncreasingLoops_PerfectlyPlastic_IJNAMG}(b))  
of  shear  stress  --  for  the  first  couple  of cycles with
increasing  strain   loops. 
%
The mean response, when both the yield shear strength ($\sigma_y$) and the shear modulus ($G$) are modeled as Weibull distribution,
is also shown in Fig.~\ref{figure:Mean_and_SD_of_ShearStress_Cyclic_IncreasingLoops_PerfectlyPlastic_IJNAMG}(a).       
%
The oscillations in the evolution of standard deviation of shear stress
with shear strain are due to step size issue, inherent to the forward Euler method that
has been used in solving the FPKE. 
%
Work  is  underway to implement linearly implicit mid-point rule for solving the
FPKE  corresponding  to  elastic-plastic  constitutive rate equation.
% and will be
%reported in future publications.



%The authors are, however, working towards implementing
%a backward Euler scheme to improve the numerical solution technique.

\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.55\textwidth]{IJNAMG_Review02_Plot01-Edited.eps}
%\includegraphics[width=0.475\textwidth]{Mean_and_Mode_of_ShearStress_Cyclic_IncreasingLoops_PerfectlyPlastic_IJNAMG-Ed.eps}
\hspace*{0.1truecm}
\includegraphics[width=0.425\textwidth]{SD_of_ShearStress_Cyclic_IncreasingLoops_PerfectlyPlastic_IJNAMG-Ed.eps} \\
\hspace*{-1.25truecm}(a)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(b)
\caption{Elastic--perfectly  plastic  probabilistic  model under cyclic loading with increasing
strain  loops:  evolution  of (a) mean and (b) standard deviation of shear stress}
\label{figure:Mean_and_SD_of_ShearStress_Cyclic_IncreasingLoops_PerfectlyPlastic_IJNAMG}
\end{center}
\end{figure} 
%
%
%
%As        can       be       observed       from
%Fig.~\ref{figure:Mean_and_SD_of_ShearStress_Cyclic_IncreasingLoops_PerfectlyPlastic_IJNAMG}(a),  
%the  ensemble  average (mean) of all the realizations differs
%(but not  significantly) from the most probable (mode) of all the realizations.
%%
%This  is because the evolutionary PDF of shear stress is not Gaussian.
%
%, as can be
%observed    in   Fig.~\ref{figure:ShearStressEvolution_PerfectlyPlastic_IJNAMG},
%where  the  evolved  PDFs  of  shear stress at the beginning and end of loading,
%unloading,  re--loading,  and  re--unloading  branches  are plotted. 
%
%
%
%One may note that in Fig.~\ref{figure:ShearStressEvolution_PerfectlyPlastic_IJNAMG}(a), the 
%initial shear stress at the beginning of loading branch (refer to object A in Fig.~\ref{figure:ShearStressEvolution_PerfectlyPlastic_IJNAMG}(a)), 
%which is a Dirac delta function, is shown truncated for clarity of the figure. This Dirac delta function advected and
%diffused into the domain as the material was strained. The PDF of shear stress at the end of the
%loading branch is shown as object B in Fig.~\ref{figure:ShearStressEvolution_PerfectlyPlastic_IJNAMG}(a)). Once, the 
%strain was reversed, the PDF of shear stress at the end of the loading branch
%%
%\begin{figure}[!htbp]
%\begin{center}
%\includegraphics[width=\textwidth]{ShearStressEvolution_PerfectlyPlastic_IJNAMG-Ed.eps}
%\caption{Elastic--perfectly  plastic  probabilistic  model under cyclic loading with increasing
%loops: evolved PDF of shear stress at the beginning and
%end of (a) loading branch, (b) unloading branch, (c) re-loading branch, and (d) re-unloading branch}
%\label{figure:ShearStressEvolution_PerfectlyPlastic_IJNAMG}
%\end{center}
%
%
%

The   very    important      observation  that can be made using 
Fig.~\ref{figure:Mean_and_SD_of_ShearStress_Cyclic_IncreasingLoops_PerfectlyPlastic_IJNAMG}(a) 
is that, if one consider uncertainties in geomaterial properties,
even  the simplest elastic--perfectly model, captures some of the very important
features of geomaterial behaviors.
%
For example, reduction of (secant) modulus with cyclic
strain, commonly observed in soil (cf. Vucetic and Dobry \cite{Vucetic:1991}), is fairly nicely captured. 
%
If using deterministic models,
this  feature  can  only  be  somewhat  successfully modeled with fairly complex
models,  which  require many more parameters. It is important to remark that for
our probabilistic  modeling, (only) statistical distributions (probability
density functions) of shear modulus ($G$) and shear
strength ($\sigma_y$), are needed.
%
Expansion  of  elastic--plastic  modeling  into  probability space seems to have
added significant new capabilities to modeling.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{Case of Constant Strain Loops:}

This von Mises elastic--plastic material, however, didn't exhibit (secant) modulus
degradation,  commonly  observed  in  clay (cf. Vucetic and Dobry \cite{Vucetic:1988}),  when the material is cycled repeatedly 
at the same strain.
%
Fig.~\ref{figure:Mean_and_SD_of_ShearStress_Cyclic_AllEqualLoops_PerfectlyPlastic_IJNAMG}(a) 
shows such probabilistic response (mean of shear stress). The material was cycled repeatedly
up to $0.2$\% strain. Only first three cycles are shown.
%
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.475\textwidth]{Mean_of_ShearStress_Cyclic_AllEqualLoops03_01_PerfectlyPlastic_IJNAMG-Ed.eps}
\hspace*{0.1truecm}
\includegraphics[width=0.475\textwidth]{SD_of_ShearStress_Cyclic_AllEqualLoops03_01_PerfectlyPlastic_IJNAMG-Ed.eps} \\
\hspace*{-1.0truecm}(a)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(b)
\caption{Elastic--perfectly  plastic    probabilistic model under cyclic loading
with  all equal loops: evolution of (a) mean and (b) standard deviation of shear
stress}
\label{figure:Mean_and_SD_of_ShearStress_Cyclic_AllEqualLoops_PerfectlyPlastic_IJNAMG}
\end{center}
\end{figure}
%
It is important to note that the von Mises mean elastic--plastic material behavior 
is function of both the mean and standard deviation of both shear modulus ($G$) and yield shear 
strength ($\sigma_y$). The same von Mises elastic--perfectly plastic model with a different set of
material properties could, however, be able to capture the degradation of mean (secant)
shear modulus. For example, Japanese stiff clay, when modeled as von Mises elastic--perfectly plastic
material, exhibited modulus degradation with number of cycles (Sett et al. \cite{Sett:2008b}) 

\paragraph{Monotonic  Loading:} For completeness of comparison, the monotonic
behavior of  this  probabilistic  von  Mises perfectly
plastic  material  is  also  shown (refer Fig.~\ref{figure:Mean_and_SD_of_ShearStress_Monotonic_PerfectlyPlastic_IJNAMG}).  As  can  be  observed from
Fig.~\ref{figure:Mean_and_SD_of_ShearStress_Monotonic_PerfectlyPlastic_IJNAMG}(a),   the   mean   shear  stress  non--linearly
increases  with  shear  strain  before  reaching  the perfectly
plastic state.
%
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.7\textwidth]{Mean_and_SD_of_ShearStress_Monotonic_01_PerfectlyPlastic_IJNAMG_02-Ed.eps} \\
\hspace*{1.75truecm}(a)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(b)
\caption{Elastic--perfectly   plastic   probabilistic    model  under  monotonic
loading: evolution of (a) mean, (b) standard deviation, and (c)
mean $\pm$ standard deviation of shear stress}
\label{figure:Mean_and_SD_of_ShearStress_Monotonic_PerfectlyPlastic_IJNAMG}
\end{center}
\end{figure}



% moved up
% \begin{figure}[!htbp]
% \begin{center}
% \includegraphics[width=0.475\textwidth]{Mean_and_Mode_of_ShearStress_Cyclic_IncreasingLoops_PerfectlyPlastic_IJNAMG-Ed.eps}
% \hspace*{0.1truecm}
% \includegraphics[width=0.475\textwidth]{SD_of_ShearStress_Cyclic_IncreasingLoops_PerfectlyPlastic_IJNAMG-Ed.eps} \\
% \hspace*{-1.25truecm}(a)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(b)
% \caption{Elastic--perfectly plastic model under cyclic loading with increasing loops: Evolution of (a) mean, mode and (b) standard deviation of shear stress}
% \label{figure:Mean_and_SD_of_ShearStress_Cyclic_IncreasingLoops_PerfectlyPlastic_IJNAMG}
% \end{center}
% \end{figure} 

% moved up
% \begin{figure}[!htbp]
% \begin{center}
% \includegraphics[width=\textwidth]{ShearStressEvolution_PerfectlyPlastic_IJNAMG-Ed.eps}
% \caption{Elastic--perfectly plastic model under cyclic loading with increasing loops: Evolved PDF of shear stress at the beginning and
% end of (a) loading branch, (b) unloading branch, (c) re-loading branch, and (d) re-unloading branch}
% \label{figure:ShearStressEvolution_PerfectlyPlastic_IJNAMG}
% \end{center}
% \end{figure}

% moved up
% \begin{figure}[!htbp]
% \begin{center}
% \includegraphics[width=0.475\textwidth]{Mean_of_ShearStress_Cyclic_AllEqualLoops03_01_PerfectlyPlastic_IJNAMG-Ed.eps}
% \hspace*{0.1truecm}
% \includegraphics[width=0.475\textwidth]{SD_of_ShearStress_Cyclic_AllEqualLoops03_01_PerfectlyPlastic_IJNAMG-Ed.eps} \\
% \hspace*{-1.0truecm}(a)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(b)
% \caption{Elastic--perfectly plastic model under cyclic loading with all equal loops: Evolution of (a) mean and (b) standard deviation of shear stress}
% \label{figure:Mean_and_SD_of_ShearStress_Cyclic_AllEqualLoops_PerfectlyPlastic_IJNAMG}
% \end{center}
% \end{figure}

Physically, one may visualize the probabilistic soil constitutive
response as an ensemble of the behaviors of infinite number of soil particles
in a representative volume element (RVE), for example, a laboratory soil specimen.
%
The behavior of an individual soil particle in a RVE is governed, in case of 
elastic--perfectly plastic material, by its modulus and strength.
%
However, if the modulus and strength of each particle are different, for example,
governed by their respective PDF, then each particle would behave differently.
%
The PDF of the response behavior then represents the ensemble of all such
behaviors, with their respective probability weights.
%
The mean, on the other hand, represents the ensemble average of all such
behaviors.
%
In this context, it is important to note that the behaviors presented in this paper
do not take into account the correlation between soil particles (scale effect).
%
The scale effect can be accounted for, among others, using stochastic elastic--plastic finite element 
technique.
%
Sett \cite{Sett:2007f} proposed one such finite element method by extending
the spectral approach to stochastic finite element (cf. Ghanem and Spanos \cite{book:Ghanem}) 
to elastic--plastic problems by updating the material properties at Gauss integration points
using the FPKE approach, as the material plastifies.

Further to the  promise  of  an  alternate  approach  to geomaterial modeling,
probabilistic  approach also quantifies our confidence in the simulated behavior
of   geomaterials.   
%
FPKE  based  probabilistic  elasto--plasticity  solves  for
second-order      accurate      evolutionary     PDF     of     shear     stress
(Fig.~\ref{figure:PDF_of_ShearStress_Cyclic_PerfectlyPlastic_IJNAMG-Ed}).
%
Ability to obtain the PDF of stress accurately
is very important in failure simulation of geomaterials, as they often fail at low probabilities (tails of PDF).
A full PDF contains enormous amount of information. 
%
From the PDF, other than the statistical moments, other useful engineering information, 
for example, the probability of exceedance, most probable solution, as well as some derivative application like sensitivity analysis can be
easily obtained or derived. 
%
Figs.~\ref{figure:Mean_and_SD_of_ShearStress_Cyclic_IncreasingLoops_PerfectlyPlastic_IJNAMG}(b) and
\ref{figure:Mean_and_SD_of_ShearStress_Cyclic_AllEqualLoops_PerfectlyPlastic_IJNAMG}(b) show one of the important confidence
measuring parameters, the evolutionary standard deviation of shear stress (square-root of second moment of the
evolutionary PDF of shear stress (Fig.~\ref{figure:PDF_of_ShearStress_Cyclic_PerfectlyPlastic_IJNAMG-Ed})), 
for cyclic responses with increasing loops and all equal loops, respectively. 
%
As can be observed from the above figures
(Figs.~\ref{figure:Mean_and_SD_of_ShearStress_Cyclic_IncreasingLoops_PerfectlyPlastic_IJNAMG}(b) and     
\ref{figure:Mean_and_SD_of_ShearStress_Cyclic_AllEqualLoops_PerfectlyPlastic_IJNAMG}(b)), inside any branch (loading, unloading, re--loading,
re--unloading, ...), as well as in Fig.~\ref{figure:Mean_and_SD_of_ShearStress_Monotonic_PerfectlyPlastic_IJNAMG}(b), where the
monotonic response is shown,
the standard deviation, first increases and then decreases. 
%
This is because, initially, when the material is mostly
elastic, both the uncertainties in shear modulus ($G$) and yield strength ($\sigma_y$) are governing. As material becomes 
mostly elastic--plastic, the influence of uncertainty in shear modulus ($G$) decreases. 
%
%
%
%
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%\subsection{Influence of Varying Material Parameters Uncertainty}
%
%
%
%
However, it  is  important  to  note that this type of standard deviation response is not
generic to all von Mises elastic--perfectly plastic material.
%
The   standard   deviation  response  is  very  much  dependent  on  the  amount
uncertainties   present   in   both  shear  modulus  ($G$)  and  yield  strength
($\sigma_y$).
%
For example, 
Fig.~\ref{figure:Mean_and_SD_of_ShearStress_Cyclic_AllEqualLoops_VerLargeYieldCOV_PerfectlyPlastic_IJNAMG}(b), 
shows probabilistic response of cyclic behavior of the same material model,  except that
COV of yield strength ($\sigma_y$), is now assumed to be 300\%.
%
The standard deviation response shown here is always increasing which is completely different from what was 
observed in previous case 
(Figs.~\ref{figure:Mean_and_SD_of_ShearStress_Cyclic_IncreasingLoops_PerfectlyPlastic_IJNAMG}(b), 
\ref{figure:Mean_and_SD_of_ShearStress_Cyclic_AllEqualLoops_PerfectlyPlastic_IJNAMG}(b) and 
\ref{figure:Mean_and_SD_of_ShearStress_Monotonic_PerfectlyPlastic_IJNAMG}(b))).
%
%
This is because, for this material, the COV of shear modulus (assumed 30\%) is
non--significant, compared to the COV of yield strength (assumed 300\%), and hence, the standard deviation response
(Fig.~\ref{figure:Mean_and_SD_of_ShearStress_Cyclic_AllEqualLoops_VerLargeYieldCOV_PerfectlyPlastic_IJNAMG}(b)) is
predominantly influenced by the uncertainty in yield strength ($\sigma_y$). 
%
Similar standard deviation response can be observed in
Fig.~\ref{figure:Mean_and_Mode_and_SD_of_ShearStress_Monotonic_VeryLargeYieldCOV_PerfectlyPlastic_IJNAMG}(b), where
the material with large COV of yield strength was subjected to monotonic loading. 
 
% moved up
% \begin{figure}[!htbp]
% \begin{center}
% \hspace*{-1.2truecm} \includegraphics[width=1.2\textwidth]{Mean_and_SD_of_ShearStress_Monotonic_01_PerfectlyPlastic_IJNAMG-Ed.eps} \\
% \hspace*{1.75truecm}(a)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(b)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(c)
% \caption{Elastic--perfectly plastic model under monotonic loading: Evolution of (a) mean, (b) standard deviation, and (c)
% mean $\pm$ standard deviation of shear stress}
% \label{figure:Mean_and_SD_of_ShearStress_Monotonic_PerfectlyPlastic_IJNAMG}
% \end{center}
% \end{figure}

\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.475\textwidth]{Mean_of_ShearStress_Cyclic_AllEqualLoops_VeryLargeYieldCOV_PerfectlyPlastic_IJNAMG-Ed.eps}
\hspace*{0.1truecm}
\includegraphics[width=0.475\textwidth]{SD_of_ShearStress_Cyclic_AllEqualLoops_VeryLargeYieldCOV_PerfectlyPlastic_IJNAMG-Ed.eps} \\
\hspace*{-0.3truecm}(a)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(b)
\caption{Elastic--perfectly  plastic    probabilistic model under cyclic loading
with all equal loops (probabilistic model parameters are exactly the same as used for
simulation                                                                    in
Fig.~\ref{figure:Mean_and_SD_of_ShearStress_Cyclic_AllEqualLoops_PerfectlyPlastic_IJNAMG}, but with very large yield
uncertainty): evolution of (a) mean and (b) standard deviation of shear stress}
\label{figure:Mean_and_SD_of_ShearStress_Cyclic_AllEqualLoops_VerLargeYieldCOV_PerfectlyPlastic_IJNAMG}
\end{center}
\end{figure}

\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.7\textwidth]{Mean_and_Mode_and_SD_of_ShearStress_Monotonic_01_VeryLargeYieldCOV_PerfectlyPlastic_IJNAMG_02-Ed.eps} \\
\hspace*{1.75truecm}(a)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(b)
%\hspace*{0.25truecm}(a)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(b)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(c)
\caption{Elastic--perfectly plastic  probabilistic model under monotonic loading
(model parameters are exactly the same as used for simulation in
Fig.~\ref{figure:Mean_and_SD_of_ShearStress_Monotonic_PerfectlyPlastic_IJNAMG}, but with very large yield
uncertainty): evolution of (a) mean, mode, (b) standard deviation, and (c)
mean $\pm$ standard deviation of shear stress}
\label{figure:Mean_and_Mode_and_SD_of_ShearStress_Monotonic_VeryLargeYieldCOV_PerfectlyPlastic_IJNAMG}
\end{center}
\end{figure}


It is also interesting to compare Figs.~\ref{figure:Mean_and_SD_of_ShearStress_Cyclic_AllEqualLoops_PerfectlyPlastic_IJNAMG}(a) 
and \ref{figure:Mean_and_SD_of_ShearStress_Cyclic_AllEqualLoops_VerLargeYieldCOV_PerfectlyPlastic_IJNAMG}(a). 
%
Both are
mean  responses of von Mises elastic--perfectly plastic material model with same
material parameters, except with different COV
of yield strength. 
%
COV of yield strength  
for simulation in Fig.~\ref{figure:Mean_and_SD_of_ShearStress_Cyclic_AllEqualLoops_PerfectlyPlastic_IJNAMG}(a)
was 30\% and that for simulation in 
Fig.~\ref{figure:Mean_and_SD_of_ShearStress_Cyclic_AllEqualLoops_VerLargeYieldCOV_PerfectlyPlastic_IJNAMG}(a)
was 300\%. 
%
It  is observed that  a completely different responses were obtained. The effect
of  COV of yield strength on monotonic mean behavior can, similarly, be compared
in
Figs.~\ref{figure:Mean_and_SD_of_ShearStress_Monotonic_PerfectlyPlastic_IJNAMG}(a)                                                                           and
\ref{figure:Mean_and_Mode_and_SD_of_ShearStress_Monotonic_VeryLargeYieldCOV_PerfectlyPlastic_IJNAMG}(a).
 


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\section{HARDENING MATERIAL}
\label{Hardening_Rules}


In  this section, the influence of probabilistic yielding is evaluated on cyclic
responses of isotropic and kinematic hardening materials. 
%
To    this    end,   the   same   example,   as   discussed   in   the previous   section
(Section~\ref{Perfectly_Plastic_Material})   is   used   but   with  appropriate
hardening rule -- isotropic or kinematic.


The     main     difference     between     the     simulations     shown     in
Section~\ref{Perfectly_Plastic_Material} for elastic--perfectly plastic material
is that for a hardening  material  the internal variables
($q_*$,  refer  Eq.~(\ref{eqno_3}))  will evolve as the material plastifies.
%
Such  evolution  (change)  of  internal  variables  is here assumed to be a
function of  plastic strain.
% 
The FPKE that govern
the  probabilistic  evolution of internal variable ($q$) can be written, in most the
general form, as:
%
\begin{equation}
\label{eqno_8}
\displaystyle \frac{\partial P(q(x_t,t), t)}{\partial t} =
\displaystyle \frac{\partial}{\partial q} \left\{ N^{eq}_{(1)_{IV}} P(q(x_t,t),t) \right\}
+ \displaystyle \frac{\partial^2}{\partial q^2} \left\{ N^{eq}_{(2)_{IV}} P(q(x_t,t),t) \right\}
\end{equation}
%
where,  $N^{eq}_{(1)_{IV}}$ and $N^{eq}_{(2)_{IV}}$ are the equivalent advection
and diffusion coefficients, respectively, for the internal variable.
%
As   explained   for   the   case   of   probabilistic   stress   response   for
elastic--perfectly              plastic              material             (refer
Section~\ref{Perfectly_Plastic_Material}), since point--location scale FPKE will
be  solved, the equivalent advection and diffusion coefficients for the internal
variable, $N^{eq}_{(1)_{IV}}$ and $N^{eq}_{(2)_{IV}}$, can be written as:
%
\begin{eqnarray}
\begin{array}{c}
N_{(1)_{IV}}^{eq} (q) = P[\Sigma_y \leq \sigma(q)] \displaystyle \frac{d \epsilon_{xy}}{dt} 
\left< \displaystyle \frac{G r}{G + \displaystyle \frac{1}{\sqrt{3}}r} \right >  \\
N_{(2)_{IV}}^{eq} (q) = P[\Sigma_y \leq \sigma(q)] t \left( \displaystyle \frac{d \epsilon_{xy}}{dt}\right)^2 
Var\left[ \displaystyle \frac{G r}{G + \displaystyle \frac{1}{\sqrt{3}}r} \right]
\end{array}
\label{eqno_9}
\end{eqnarray}
%
where,  $r$  is  the  rate  of evolution of internal variable ($q$) with plastic
strain.  
%
One    may    note    that    in   the  above equivalent advection and diffusion
coefficients (Eq.~(\ref{eqno_9})), the contributions of probability weights that
the material being elastic are absent.
%
This is because the evolution rule of internal  variable  is  governed  by  the 
plastic component of strain only.
%
The   equivalent   advection   and   diffusion  coefficients  for  shear  stress
($N^{eq}_{(1)}$  and  $N^{eq}_{(2)}$)  for  hardening--type materials, will have
contributions   from   both  elastic  and  plastic  components,  just  like  the
elastic--perfectly plastic case.
%
However,  unlike  the  elastic--perfectly plastic case,  those  ($N^{eq}_{(1)}$ 
and $N^{eq}_{(2)}$) will contain the hardening terms:
%  
\begin{eqnarray}
\begin{array}{c}
N_{(1)}^{eq} (\sigma) = \displaystyle \frac{d \epsilon_{xy}}{dt} \left[ (1 - P[\Sigma_y \leq \sigma]) \left< G \right >  +
P[\Sigma_y \leq \sigma] \left<G - \displaystyle \frac{G^2}{G + \displaystyle \frac{1}{\sqrt{3}}r} \right > \right] \\
%
N_{(2)}^{eq} (\sigma) = t \left( \displaystyle \frac{d \epsilon_{xy}}{dt}\right)^2 \left[(1 - P[\Sigma_y \leq \sigma]) Var[G] + 
P[\Sigma_y \leq \sigma] Var\left[G - \displaystyle \frac{G^2}{G + \displaystyle \frac{1}{\sqrt{3}}r} \right] \right]
\end{array}
\label{eqno_10}
\end{eqnarray}


To  obtain  the probabilistic response of von Mises hardening material, the FPKE
for  probabilistic  evolution  of  internal  variable  (Eq.~(\ref{eqno_8}), with
advection  and  diffusion  coefficients given by Eq.~(\ref{eqno_9})) needs to be
solved  incrementally.
%
This solution needs to be done simultaneously with the FPKE for probabilistic evolution
of  shear stress (Eq.~(\ref{eqno_1a}), with advection and diffusion coefficients
given by Eq.~(\ref{eqno_10})). 
%
Those, in turn, need also to be solved incrementally, with
the  yield  strength  random  variable  ($\Sigma_y$)  in Eqs.~(\ref{eqno_9}) and
(\ref{eqno_10}) being updated after each incremental step.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
%\clearpage
\subsection{Isotropic Hardening} 
\label{Isotropic_Hardening}



%The     von  Mises  yield  criteria, with isotropic hardening can be written the
%same  way  as  in  \ref{eqno_11} except that hardening material parameter (yield
%strength like) can now evolve, in this case linearly.
%
For  von  Mises isotropic hardening material, the yield strength ($\sigma_y$) is
the internal variable.
%
Yield  strength  will  evolve  probabilistically  with plastic strain, following
Eq.~(\ref{eqno_8}),   with   advection   and  diffusion  coefficients  given  by
Eq.~(\ref{eqno_9}).
%
The   shear   stress,   on   the   other   hand,   evolves  in  accordance  with
Eq.~(\ref{eqno_1a}),   with   advection  and  diffusion  coefficients  given  by
Eq.~(\ref{eqno_10}).



Fig.~\ref{figure:Mean_and_SD_of_ShearStress_Cyclic_IncreasingLoops_IsotropicHardening_IJNAMG} 
shows the evolutionary mean and standard deviation of shear stress
during  first  couple  of  loading--unloading  cycles  for  von  Mises isotropic
hardening  material  with  a  non--dimensional rate  of  evolution  of  internal  variable (yield
strength, in this case) of 10.
%
All  other material parameters are assumed to be the same as used for simulation
of    elastic--perfectly    plastic    material    in   the   previous   section
(Section~\ref{Perfectly_Plastic_Material}).
%
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.475\textwidth]{Mean_of_ShearStress_Cyclic_IncreasingLoops_IsotropicHardening_IJNAMG-Ed.eps}
\hspace*{0.1truecm}
\includegraphics[width=0.475\textwidth]{SD_of_ShearStress_Cyclic_IncreasingLoops_IsotropicHardening_IJNAMG-Ed.eps} \\
\hspace*{-1.25truecm}(a)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(b)
\caption{Isotropic  hardening probabilistic  model under cyclic loading with increasing loops:
evolution of (a) mean and (b) standard deviation of shear stress}
\label{figure:Mean_and_SD_of_ShearStress_Cyclic_IncreasingLoops_IsotropicHardening_IJNAMG}
\end{center}
\end{figure}




The  evolved  PDFs  of yield strength after each branch
(loading,    unloading,   re--loading,   and   re--unloading)   are   shown   in
Fig.~\ref{figure:YieldStressEvolution_IsotropicHardening_IJNAMG}.  
%
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=\textwidth]{YieldStressEvolution_IsotropicHardening_IJNAMG-Ed.eps}
\caption{Isotropic  hardening    probabilistic  model  under cyclic loading with
increasing loops: evolved PDF of yield stress after (a) loading branch,
(b) unloading branch, (c) re--loading branch, and (d) re--unloading branch}
\label{figure:YieldStressEvolution_IsotropicHardening_IJNAMG}
\end{center}
\end{figure}
%
The  initial
PDFs    of    yield    strength  (positive for loading branch  and  negative for
unloading branch) are the same as assumed for
elastic--perfectly  plastic material in Section~\ref{Perfectly_Plastic_Material}
(refer  Fig.~\ref{figure:YieldStress_PerfectlyPlastic_IJNAMG}). 
%
As expected (and prescribed by the isotropic hardening model), 
the  yield  strength  evolved (grew) isotropically. 
%
However, it is interesting to note the change in 
probability  distributions  of  yield strength. 
%
The normally distributed initial
PDFs  of  yield strength (Fig.~\ref{figure:YieldStress_PerfectlyPlastic_IJNAMG})
evolved  into  much dispersed non-Gaussian distributions having low kurtosis. 
%
%
%\noindent
%%
%\rule{\textwidth}{1mm}
%
%KALLOL: see if my explanation of this increase in uncertainty (below) can be improved 
%
%\noindent
%%
%\rule{\textwidth}{1mm}
%
In
other words, when the material is cycled through loading--unloading cycles, the
uncertainty  in  yield strength increases. 
%%
%This  increase in uncertainty can be explained both from modeling (mathematical)
%and from physical points of view.
%
Mathematically, increase in uncertainty of shear strength is due to the 
nonlinearity in formulation of probabilistic yielding , that is,
the  state  variable $q$ appears in both advection and diffusion equations (refer
Eq.~(\ref{eqno_9})),  and in  the  evolution  equation  for  internal  variable
(Eq.~(\ref{eqno_8})).  
%%
%%And,  this  'non--linearity'\footnote{One  may note that,
%%deterministically,   a  linear--hardening  model  is  solved  here}  stems  from
%%probabilistic yielding.
%% 
%Physically,        initial,  uncertain   shear  strength, is obeying increase in
%entropy,and is 
%uncertain, is  


%The  evolved PDFs of shear stress are also non--Gaussian, as can be observed in
%Fig.~\ref{figure:ShearStressEvolution_IsotropicHardening_IJNAMG}, where the PDFs
%of  shear  stress  at  the  beginning and end of each cycle (loading, unloading,
%re--loading, and re--unloading) are plotted.
%
%\begin{figure}[!htbp]
%\begin{center}
%\includegraphics[width=\textwidth]{ShearStressEvolution_IsotropicHardening_IJNAMG-Ed.eps}
%\caption{Isotropic  hardening    probabilistic  model  under cyclic loading with
%increasing loops: evolved PDF of shear stress at the beginning and
%end of (a) loading branch, (b) unloading branch, (c) re--loading branch, and (d) re--unloading branch}
%\label{figure:ShearStressEvolution_IsotropicHardening_IJNAMG}
%\end{center}
%\end{figure}







When comparison is made between 
Figs.~\ref{figure:Mean_and_SD_of_ShearStress_Cyclic_IncreasingLoops_IsotropicHardening_IJNAMG} 
and 
\ref{figure:Mean_and_SD_of_ShearStress_Cyclic_IncreasingLoops_PerfectlyPlastic_IJNAMG},
one can clearly see that, in simulating cyclic behaviors of geomaterials, 
isotropic hardening model 
(Fig.~\ref{figure:Mean_and_SD_of_ShearStress_Cyclic_IncreasingLoops_IsotropicHardening_IJNAMG})
performed, as expected, poorly. That is, the elastic--perfectly plastic probabilistic model 
(Fig.~\ref{figure:Mean_and_SD_of_ShearStress_Cyclic_IncreasingLoops_PerfectlyPlastic_IJNAMG})  
captures  (PDF  of) stress--strain loops in a much more realistic way.
%
However, for completeness of 
comparison, the behavior of isotropic hardening material, when it was cycled to same level
(Fig.~\ref{figure:Mean_and_SD_of_ShearStress_Cyclic_AllEqualLoops_IsotropicHardening_IJNAMG}) 
and when loaded monotonically 
(Fig.~\ref{figure:Mean_and_SD_of_ShearStress_Monotonic_IsotropicHardening_IJNAMG}) 
are also shown.
%
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.475\textwidth]{Mean_of_ShearStress_Cyclic_AllEqualLoops03_IsotropicHardening_IJNAMG-Ed.eps}
\hspace*{0.1truecm}
\includegraphics[width=0.475\textwidth]{SD_of_ShearStress_Cyclic_AllEqualLoops03_IsotropicHardening_IJNAMG-Ed.eps} \\
\hspace*{-0.5truecm}(a)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(b)
\caption{Isotropic  hardening    probabilistic  model  under cyclic loading with
equal loops: evolution of (a) mean and (b) standard deviation of shear stress}
\label{figure:Mean_and_SD_of_ShearStress_Cyclic_AllEqualLoops_IsotropicHardening_IJNAMG}
\end{center}\end{figure}


It   is   noted  that  monotonic  loading  curves  for  both  perfectly  plastic
probabilistic model 
(Fig.~\ref{figure:Mean_and_SD_of_ShearStress_Monotonic_PerfectlyPlastic_IJNAMG})
and linear isotropic hardening probabilistic model
(Fig.~\ref{figure:Mean_and_SD_of_ShearStress_Monotonic_IsotropicHardening_IJNAMG}) 
do look similar (with a noted difference of more pronounced hardening for a
hardening  model),  but  the  real difference in stress--strain predictions with
both probabilistic models becomes obvious in the case of cyclic loading.
%
%
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.7\textwidth]{Mean_and_SD_of_ShearStress_Monotonic_IsotropicHardening_IJNAMG_02-Ed.eps} \\
\hspace*{1.75truecm}(a)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(b)
\caption{Isotropic  hardening    probabilistic  model  under  monotonic loading:
evolution of (a) mean, (b) standard deviation, and (c)
mean $\pm$ standard deviation of shear stress}
\label{figure:Mean_and_SD_of_ShearStress_Monotonic_IsotropicHardening_IJNAMG}
\end{center}
\end{figure}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
%\clearpage
\subsection{Kinematic Hardening} 
\label{Kinematic_Hardening}

Expanding  on  elastic--plastic  hardening  probabilistic  models, we now focus on a
simple linear  kinematic hardening rule based on evolution of  back  stress  ($\alpha$).
By  introducing  back  stress  ($\alpha$)  to  von Mises yield criteria, one can
write:
%
\begin{equation}
\label{eqno_11}
\sqrt{J_{\alpha}} - k = 0
\end{equation}
%
where,     $k$ is again material parameter (yield strength like)  and   $J_{\alpha}=
3/2 (s_{ij} - \alpha_{ij})(s_{ij} - \alpha_{ij})$
is   the   $\alpha$--modified  second  invariant  of  deviatoric  stress  tensor
($s_{ij}$). For 1--D shear, Eq.~(\ref{eqno_11}) becomes:

\begin{eqnarray}
\label{eqno_12a}
\left|\sigma -\alpha \right| - \sigma_y = 0  
\mbox{\;\;\;\; or \;\;\;\;}
\sigma = \alpha \pm \sigma_y
\end{eqnarray}
%
Hence,  for  kinematic  hardening  material,  the yielding occurs at a stress of
$\alpha \pm \sigma_y$, termed in the following as the equivalent yield stress.
%
Initially,  $\alpha$  is zero, and $\sigma_y$ is assumed to have a mean value of
60 kPa with a standard deviation of 20 kPa, resulting in equivalent yield stress
of 60 kPa with a COV of 30\%, same as the assumed yield stress for the elastic--perfectly plastic
material in 
Section~\ref{Perfectly_Plastic_Material}  and  isotropic  hardening  material in
Section~\ref{Isotropic_Hardening}.
%
However, the same distribution of equivalent yield stress will be obtained, if one transfers the initial uncertainty from
$\sigma_y$ to $\alpha$. 
%
In  other words, a deterministic $\sigma_y$ of 60 kPa, and an uncertain $\alpha$
of  zero  mean  and  a  standard  deviation  of  20  kPa will result in the same
equivalent yield stress.
%
The  advantage  of  keeping $\sigma_y$  deterministic  is  that it will simplify
the  probabilistic addition/subtraction   in   Eq.~(\ref{eqno_12a}),   while  estimating  the
equivalent yield stress after each incremental step of the governing FPKEs, once
the  back  stress  ($\alpha$),  the  internal  variable  for kinematic hardening
material, starts evolving.



In  this  study,  the  back  stress ($\alpha$) is assumed to evolve with plastic
strain  and  hence,  it  would evolve probabilistically similar to probabilistic
evolution of the yield strength for isotropic hardening material.
%
Probabilistic evolution of the back stress will occur according to
Eq.~(\ref{eqno_8}), with advection and diffusion coefficients 
given by Eq.~(\ref{eqno_9}).
%
Shear stress evolves according to Eq.~(\ref{eqno_1a}), 
with advection and diffusion coefficients given by Eq.~(\ref{eqno_10}). 
%
One  may note that the yield strength random variable ($\Sigma_y$), appearing in
Eqs.~(\ref{eqno_9}) and (\ref{eqno_10}), is the equivalent yield strength and is
given by Eq.~(\ref{eqno_12a}). 
%
Fig.~\ref{figure:Mean_and_SD_of_ShearStress_Cyclic_IncreasingLoops_KinematicHardening_IJNAMG} 
shows  the  probabilistic  evolution of shear stress in terms of mean, mode, and
standard   deviation,   when   a   kinematic   hardening  material\footnote{with
non--dimensional  rate  of evolution of back stress with plastic strain of 10.},
was cycled couple of times with increasing strain loops.
%
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.475\textwidth]{Mean_and_Mode_of_ShearStress_Cyclic_IncreasingLoops_KinematicHardening_IJNAMG-Ed.eps}
\hspace*{0.1truecm}
\includegraphics[width=0.475\textwidth]{SD_of_ShearStress_Cyclic_IncreasingLoops_KinematicHardening_IJNAMG-Ed.eps} \\ 
\hspace*{-1.25truecm}(a)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(b)
\caption{Kinematic  hardening    probabilistic  model  under cyclic loading with
increasing  loops:  evolution  of  (a)  mean, mode and (b) standard deviation of
shear stress}
\label{figure:Mean_and_SD_of_ShearStress_Cyclic_IncreasingLoops_KinematicHardening_IJNAMG}
\end{center}
\end{figure}
%
All   other  material  parameters  are  assumed  to  be  the  same  as  for  the
elastic--perfectly plastic material in Section~\ref{Perfectly_Plastic_Material}.
%
The  evolved PDFs of the back stress ($\alpha$) at the beginning and end of each
branch  (loading,  unloading,  re--loading,  and  re--unloading)  are  shown  in
Fig.~\ref{figure:BackStressEvolution_KinematicHardening_IJNAMG}.
%
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=\textwidth]{BackStressEvolution_KinematicHardening_IJNAMG-Ed.eps}
\caption{Kinematic  hardening    probabilistic  model  under cyclic loading with
increasing loops: evolved PDF of back stress at the beginning and
end of (a) loading branch, (b) unloading Branch, (c) re--loading branch, and (d) re--unloading branch}
\label{figure:BackStressEvolution_KinematicHardening_IJNAMG}
\end{center}
\end{figure}
%
The  evolved  PDFs of equivalent yield stress (refer Eq.~(\ref{eqno_12a})) after
each           loading           branch           are          shown          in
Fig.~\ref{figure:EqYieldStressEvolution_KinematicHardening_IJNAMG}.
%
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=\textwidth]{EqYieldStressEvolution_KinematicHardening_IJNAMG-Ed.eps}
\caption{Kinematic  hardening    probabilistic  model  under cyclic loading with
increasing  loops:  evolved  PDF  of  equivalent  yield stress after (a) loading
branch,
(b) unloading branch, (c) re--loading branch, and (d) re--unloading branch}
\label{figure:EqYieldStressEvolution_KinematicHardening_IJNAMG}
\end{center}
\end{figure}
%
%
%
%
Similar  to  the  isotropic hardening case the uncertainty in (equivalent) yield
strength  increased  as  the  material  was cycled through, but unlike the isotropic
hardening  model,  kinematic  hardening  model resulted in high kurtosis PDFs of
(equivalent) yield strength.
%
%
%
%
It is noted that the cyclic shear stress response of kinematic hardening material 
(Fig.~\ref{figure:Mean_and_SD_of_ShearStress_Cyclic_IncreasingLoops_KinematicHardening_IJNAMG}), 
was more realistic 
than isotropic hardening material 
(Fig.~\ref{figure:Mean_and_SD_of_ShearStress_Cyclic_IncreasingLoops_IsotropicHardening_IJNAMG}), 
however, it didn't differ much from elastic--perfectly plastic material response 
(Fig.~\ref{figure:Mean_and_SD_of_ShearStress_Cyclic_IncreasingLoops_PerfectlyPlastic_IJNAMG}).
%
Qualitatively,  those,  the  elastic--perfectly  plastic and the kinematic hardening
responses, are similar.
%
Like    the    elastic--perfectly  plastic  material,  for  kinematic  hardening
material, the mean and mode of the evolutionary shear stress
(refer Fig.~\ref{figure:Mean_and_SD_of_ShearStress_Cyclic_IncreasingLoops_KinematicHardening_IJNAMG})
are different, although not significantly.
%
%In addition to that the evolved PDFs of shear stress at any particular strain, as the material was
%cycled through, are non-Gaussian, as seen in  Fig.~\ref{figure:ShearStressEvolution_KinematicHardening_IJNAMG}, 
%where  the  evolved PDFs of shear stress after each branch of loading--unloading
%cycles are shown.
%%
%\begin{figure}[!htbp]
%\begin{center}
%\includegraphics[width=\textwidth]{ShearStressEvolution_KinematicHardening_IJNAMG-Ed.eps}
%\caption{Kinematic  hardening    probabilistic  model  under cyclic loading with
%increasing loops: evolved PDF of shear stress at the beginning and
%end of (a) loading branch, (b) unloading branch, (c) re--loading branch, and (d) re--unloading branch}
%\label{figure:ShearStressEvolution_KinematicHardening_IJNAMG}
%\end{center}
%\end{figure}
%


Similarly,   when  one  compares  response (mean and standard deviation of shear
stress) for loading cycles to the same strain
level, for 
(i) elastic--perfectly plastic, 
(Fig.~\ref{figure:Mean_and_SD_of_ShearStress_Cyclic_AllEqualLoops_PerfectlyPlastic_IJNAMG}),
(ii) isotropic linear hardening 
(Fig.~\ref{figure:Mean_and_SD_of_ShearStress_Cyclic_AllEqualLoops_IsotropicHardening_IJNAMG}), 
and (iii) linear kinematic hardening 
(Fig.~\ref{figure:Mean_and_SD_of_ShearStress_Cyclic_AllEqualLoops_KinematicHardening_IJNAMG}), 
probabilistic material models, one can easily
observe  the  qualitative  similarity between elastic--perfectly plastic (i) and
kinematic hardening responses (iii). 
%
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.475\textwidth]{Mean_of_ShearStress_Cyclic_AllEqualLoops03_KinematicHardening_IJNAMG-Ed.eps}
\hspace*{0.1truecm}
\includegraphics[width=0.475\textwidth]{SD_of_ShearStress_Cyclic_AllEqualLoops03_KinematicHardening_IJNAMG-Ed.eps} \\
\hspace*{-0.5truecm}(a)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(b)
\caption{Kinematic  hardening  probabilistic    model  under cyclic loading with
equal loops: evolution of (a) mean and (b) standard deviation of shear stress}
\label{figure:Mean_and_SD_of_ShearStress_Cyclic_AllEqualLoops_KinematicHardening_IJNAMG}
\end{center}
\end{figure} 



Monotonic loading cases, however, for all probabilistic material models 
((i) elastic--perfectly plastic, 
(Fig.~\ref{figure:Mean_and_SD_of_ShearStress_Monotonic_PerfectlyPlastic_IJNAMG}),
(ii) isotropic linear hardening 
(Fig.~\ref{figure:Mean_and_SD_of_ShearStress_Monotonic_IsotropicHardening_IJNAMG}), 
and (iii) linear kinematic hardening 
(Fig.~\ref{figure:Mean_and_SD_of_ShearStress_Monotonic_KinematicHardening_IJNAMG})),
are qualitatively similar, with expected differences in rate of hardening.
%
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.7\textwidth]{Mean_and_SD_of_ShearStress_Monotonic_KinematicHardening_IJNAMG_02-Ed.eps} \\
\hspace*{1.75truecm}(a)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(b)
\caption{Kinematic  hardening    probabilistic  model  under  monotonic loading:
evolution  of  (a)  mean,  (b)  standard  deviation, and (c) mean $\pm$ standard
deviation of shear stress}
\label{figure:Mean_and_SD_of_ShearStress_Monotonic_KinematicHardening_IJNAMG}
\end{center}
\end{figure}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{DISCUSSIONS AND CONCLUDING REMARKS}


In   this  paper,  a  probabilistic  framework  for  macroscopic  simulation  of
geomaterials'  behavior  is  presented,  including  novel probabilistic yielding
concept.
% 
It has been shown that, if uncertainties in
material parameters are taken into account, a realistic cyclic material behavior could
be  obtained even with the simple elastic--perfectly plastic probabilistic model. 
%
It is also shown that isotropic or
kinematic  hardening rule did not significantly improve (if at all) the qualitative nature of
the  simulated cyclic  geomaterial  response.  
%
These findings seem to nicely support the probabilistic
micromechanical simulation results by Einav and Collins \cite{Einav:2008}.


In  authors'  opinion,  probabilistic  approach to geomaterial modeling could be
very   significant   in   geotechnical   engineering.  Other  than  providing  a
mathematical  tool to quantify our confidence in our simulation of geomaterials'
behavior,  presented approach  promises  an  alternate avenue to  geomaterial  modeling.  By
expanding  material modeling into  probability  space,  
one  could  simulate realistic geomaterial
behavior  using  simple  constitutive models, 
(elastic--perfectly plastic for example)
requiring very few soil parameters
that  could  be easily obtained from in--situ tests, very common in geotechnical
practice. 

% Realistic  deterministic  simulation  of  soil behavior, on the other
%hand,   require  advanced  constitutive  models  that  require  many  more  soil
%parameters which can only be obtained through advanced laboratory tests.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{ACKNOWLEDGMENT}

The  work presented in this paper was supported by  a grant from 
Civil, Mechanical and Manufacturing Innovation program, Directorate of Engineering of the National
Science  Foundation, under Award NSF--CMMI--0600766 (cognizant program director Dr.
Richard Fragaszy). 


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% 
%  %\bibliography{/home/kallol/publication/Dissertation/SFEM,/home/kallol/publication/Dissertation/SoilProperties}
%  \bibliography{SFEM,SoilProperties}
%  %\bibliography{SFEM}
%  \bibliographystyle{plain}
%  

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