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\begin{document}
\NAG{1}{31}{in~peerreview}{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/10825010/}}
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%\begin{abstract}
\section{ABSTRACT}
In this paper, the novel concept of probabilistic yielding
is used for 1D cyclic simulation of the constitutive behavior of geomaterials.
%
FokkerPlanckKolmogorov (FPK) equation based probabilistic
elasticplastic 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, elasticperfectly 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 elastoplasticity approach
%with very simple material models (extended in probabilistic space), is able to
%capture geomaterial behavior.
%
\keywords{Soil uncertainty, Elastoplasticity, FokkerPlanckKolmogorov
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 overexpensive 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 reliabilitybased 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 elasticperfectly 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 EulerianLagrangian form of
FokkerPlanckKolmogorov equation (FPKE) approach (cf. Kavvas
\cite{Kavvas:2003}) to modeling and simulation for probabilistic
elastoplasticity. FPKE
approach to probabilistic elastoplasticity not only
overcomes the drawbacks associated with other probabilistic
simulation techniques, but also is fully compatible with the
incremental theory of elastoplasticity, and hence can
easily be applied to probabilistic modeling and simulation of
different elasticplastic constitutive models. Solution
strategies for FPK partial differential equation, corresponding
to elasticplastic constitutive rate equation and simulated
probabilistic stressstrain responses under monotonic loading,
assuming mean stress yielding, were discussed by Sett et al.
(\cite{Sett:2007b}, \cite{sett:2007a}) for both linear and
nonlinear 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 1D cyclic simulations of geomaterials. Both
elasticperfectly plastic and hardeningtype 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{FOKKERPLANCKKOLMOGOROV APPROACH TO PROBABILISTIC ELASTOPLASTICITY}
\label{FPKE}
The EulerianLagrangian form FokkerPlanckKolmogorov equation (cf. Kavvas
\cite{Kavvas:2003}) corresponding to generalized 1D 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 elasticplastic:
\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{elasticplastic}
%
\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 elasticplastic
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 elasticplastic 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 elasticplastic
constitutive equation (with $N^{ep}_{(1)}$ and $N^{ep}_{(2)}$, the advection and
diffusion coefficients corresponding to elasticplastic constitutive equation).
The switch from elastic to elasticplastic region (solution) can be controlled
using mean stress yielding:
\begin{eqnarray}
\begin{array}{lll}
\mbox{if}~~~~~~~~~~~ & \left < 0 \vee \left(\left = 0 \wedge d\left < 0 \right) ~~~~~ & \mbox{use elastic FPKE} \\
\mbox{or, if}~~~~~~~~~ & \left = 0 \vee d\left = 0 ~~~~~ & \mbox{use elasticplastic 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
elasticplastic 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 elasticplastic. The probabilities of material
being elastic and the probabilities of material being elasticplastic 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
BlackScholes \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{ELASTICPERFECTLY PLASTIC MATERIAL}
\label{Perfectly_Plastic_Material}
In this section, the FPKEapproach, along with the concept of probabilistic
yielding, is applied to simulate 1D (shear stressshear strain) cyclic
behavior of elasticperfectly 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 1D 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 elasticperfectly 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 1D von Mises
elasticperfectly plastic shear constitutive model, the
elastic and the elasticplastic 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
elasticperfectly 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 elasticplastic 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
KarhunenLo\`{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 pointlocation scale von Mises
elasticperfectly plastic material behavior, and not the localaverage
material behavior. The localaverage 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 FokkerPlanckKolmogorov 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 opensource 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_Plot02Edited.eps}
%\includegraphics[width=0.55\textwidth]{YieldStress_PerfectlyPlastic_IJNAMGEd.eps}
%\includegraphics[width=0.55\textwidth]{YieldStress_PerfectlyPlastic_IJNAMG.ps.02.eps}
\caption{Elasticperfectly 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, elasticperfectly plastic shear
stressshear 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_IJNAMGEd}.
%
%
%
\begin{figure}[!htbp]
\begin{center}
\mbox{a)}\includegraphics[width=0.95\textwidth]{PDF07_of_ShearStress_Cyclic_PerfectlyPlastic_IJNAMGEd.eps} \\
\mbox{b)}\includegraphics[width=0.95\textwidth]{PDF06_of_ShearStress_Cyclic_PerfectlyPlastic_IJNAMGEd.eps}
\caption{Elasticperfectly 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_IJNAMGEd}
\end{center}
\end{figure}
%
%
Two
different views of the loadingunloadingreloading 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_IJNAMGEd}, PDF
for initial stress (a deterministic Dirac delta function at
stressstrain 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, eventhough the
deterministic response for von Mises elasticperfectly plastic material is
bilinear, due to introduced uncertainties in yielding, the probabilistic response is
nonlinear 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 elastoplastic 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 elasticplastic. Initially,
in the loading branch, at small strains, the probability of material being
elasticplastic 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 elasticplastic material behaving
increases and the
probabilistic stress response gradually becomes more elasticplastic
(Fig.~\ref{figure:PDF_of_ShearStress_Cyclic_PerfectlyPlastic_IJNAMGEd}(a)).
%
Upon unloading, the material behaves as (mostly) elastic since elasticplastic
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 elasticplastic probability weights
increase and gradually transition the response toward
elastoplasticity
(Fig.~\ref{figure:PDF_of_ShearStress_Cyclic_PerfectlyPlastic_IJNAMGEd}(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 elastoplasticity.
%\begin{figure}[!htbp]
%\begin{center}
%\hspace*{2.75truecm} \includegraphics[width=1.4\textwidth]{PDF07_of_ShearStress_Cyclic_PerfectlyPlastic_IJNAMGEd.eps} \\
%(a) \\
%\hspace*{2.75truecm} \includegraphics[width=1.4\textwidth]{PDF06_of_ShearStress_Cyclic_PerfectlyPlastic_IJNAMGEd.eps} \\
%(b)
%\caption{ElasticPerfectly 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_IJNAMGEd}
%\end{center}
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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
elasticperfectly plastic material (refer
Fig.~\ref{figure:PDF_of_ShearStress_Cyclic_PerfectlyPlastic_IJNAMGEd}) 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 midpoint rule for solving the
FPKE corresponding to elasticplastic 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_Plot01Edited.eps}
%\includegraphics[width=0.475\textwidth]{Mean_and_Mode_of_ShearStress_Cyclic_IncreasingLoops_PerfectlyPlastic_IJNAMGEd.eps}
\hspace*{0.1truecm}
\includegraphics[width=0.425\textwidth]{SD_of_ShearStress_Cyclic_IncreasingLoops_PerfectlyPlastic_IJNAMGEd.eps} \\
\hspace*{1.25truecm}(a)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(b)
\caption{Elasticperfectly 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, reloading, and reunloading 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_IJNAMGEd.eps}
%\caption{Elasticperfectly 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) reloading branch, and (d) reunloading 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 elasticperfectly 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 elasticplastic modeling into probability space seems to have
added significant new capabilities to modeling.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{Case of Constant Strain Loops:}
This von Mises elasticplastic 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_IJNAMGEd.eps}
\hspace*{0.1truecm}
\includegraphics[width=0.475\textwidth]{SD_of_ShearStress_Cyclic_AllEqualLoops03_01_PerfectlyPlastic_IJNAMGEd.eps} \\
\hspace*{1.0truecm}(a)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(b)
\caption{Elasticperfectly 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 elasticplastic 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 elasticperfectly 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 elasticperfectly 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 nonlinearly
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_02Ed.eps} \\
\hspace*{1.75truecm}(a)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(b)
\caption{Elasticperfectly 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_IJNAMGEd.eps}
% \hspace*{0.1truecm}
% \includegraphics[width=0.475\textwidth]{SD_of_ShearStress_Cyclic_IncreasingLoops_PerfectlyPlastic_IJNAMGEd.eps} \\
% \hspace*{1.25truecm}(a)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(b)
% \caption{Elasticperfectly 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_IJNAMGEd.eps}
% \caption{Elasticperfectly 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) reloading branch, and (d) reunloading 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_IJNAMGEd.eps}
% \hspace*{0.1truecm}
% \includegraphics[width=0.475\textwidth]{SD_of_ShearStress_Cyclic_AllEqualLoops03_01_PerfectlyPlastic_IJNAMGEd.eps} \\
% \hspace*{1.0truecm}(a)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(b)
% \caption{Elasticperfectly 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
elasticperfectly 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 elasticplastic 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 elasticplastic 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 elastoplasticity solves for
secondorder accurate evolutionary PDF of shear stress
(Fig.~\ref{figure:PDF_of_ShearStress_Cyclic_PerfectlyPlastic_IJNAMGEd}).
%
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 (squareroot of second moment of the
evolutionary PDF of shear stress (Fig.~\ref{figure:PDF_of_ShearStress_Cyclic_PerfectlyPlastic_IJNAMGEd})),
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, reloading,
reunloading, ...), 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 elasticplastic, the influence of uncertainty in shear modulus ($G$) decreases.
%
%
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\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 elasticperfectly 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
nonsignificant, 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_IJNAMGEd.eps} \\
% \hspace*{1.75truecm}(a)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(b)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(c)
% \caption{Elasticperfectly 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_IJNAMGEd.eps}
\hspace*{0.1truecm}
\includegraphics[width=0.475\textwidth]{SD_of_ShearStress_Cyclic_AllEqualLoops_VeryLargeYieldCOV_PerfectlyPlastic_IJNAMGEd.eps} \\
\hspace*{0.3truecm}(a)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(b)
\caption{Elasticperfectly 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_02Ed.eps} \\
\hspace*{1.75truecm}(a)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(b)
%\hspace*{0.25truecm}(a)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(b)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(c)
\caption{Elasticperfectly 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 elasticperfectly 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).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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 elasticperfectly 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
elasticperfectly plastic material (refer
Section~\ref{Perfectly_Plastic_Material}), since pointlocation 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 hardeningtype materials, will have
contributions from both elastic and plastic components, just like the
elasticperfectly plastic case.
%
However, unlike the elasticperfectly 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 \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 loadingunloading cycles for von Mises isotropic
hardening material with a nondimensional 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 elasticperfectly 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_IJNAMGEd.eps}
\hspace*{0.1truecm}
\includegraphics[width=0.475\textwidth]{SD_of_ShearStress_Cyclic_IncreasingLoops_IsotropicHardening_IJNAMGEd.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, reloading, and reunloading) are shown in
Fig.~\ref{figure:YieldStressEvolution_IsotropicHardening_IJNAMG}.
%
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=\textwidth]{YieldStressEvolution_IsotropicHardening_IJNAMGEd.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) reloading branch, and (d) reunloading 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
elasticperfectly 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 nonGaussian 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 loadingunloading 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 'nonlinearity'\footnote{One may note that,
%%deterministically, a linearhardening 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 nonGaussian, 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,
%reloading, and reunloading) are plotted.
%
%\begin{figure}[!htbp]
%\begin{center}
%\includegraphics[width=\textwidth]{ShearStressEvolution_IsotropicHardening_IJNAMGEd.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) reloading branch, and (d) reunloading 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 elasticperfectly plastic probabilistic model
(Fig.~\ref{figure:Mean_and_SD_of_ShearStress_Cyclic_IncreasingLoops_PerfectlyPlastic_IJNAMG})
captures (PDF of) stressstrain 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_IJNAMGEd.eps}
\hspace*{0.1truecm}
\includegraphics[width=0.475\textwidth]{SD_of_ShearStress_Cyclic_AllEqualLoops03_IsotropicHardening_IJNAMGEd.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 stressstrain 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_02Ed.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 elasticplastic 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 1D 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 elasticperfectly 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
nondimensional 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_IJNAMGEd.eps}
\hspace*{0.1truecm}
\includegraphics[width=0.475\textwidth]{SD_of_ShearStress_Cyclic_IncreasingLoops_KinematicHardening_IJNAMGEd.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
elasticperfectly 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, reloading, and reunloading) are shown in
Fig.~\ref{figure:BackStressEvolution_KinematicHardening_IJNAMG}.
%
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=\textwidth]{BackStressEvolution_KinematicHardening_IJNAMGEd.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) reloading branch, and (d) reunloading 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_IJNAMGEd.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) reloading branch, and (d) reunloading 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 elasticperfectly plastic material response
(Fig.~\ref{figure:Mean_and_SD_of_ShearStress_Cyclic_IncreasingLoops_PerfectlyPlastic_IJNAMG}).
%
Qualitatively, those, the elasticperfectly plastic and the kinematic hardening
responses, are similar.
%
Like the elasticperfectly 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 nonGaussian, as seen in Fig.~\ref{figure:ShearStressEvolution_KinematicHardening_IJNAMG},
%where the evolved PDFs of shear stress after each branch of loadingunloading
%cycles are shown.
%%
%\begin{figure}[!htbp]
%\begin{center}
%\includegraphics[width=\textwidth]{ShearStressEvolution_KinematicHardening_IJNAMGEd.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) reloading branch, and (d) reunloading 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) elasticperfectly 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 elasticperfectly plastic (i) and
kinematic hardening responses (iii).
%
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.475\textwidth]{Mean_of_ShearStress_Cyclic_AllEqualLoops03_KinematicHardening_IJNAMGEd.eps}
\hspace*{0.1truecm}
\includegraphics[width=0.475\textwidth]{SD_of_ShearStress_Cyclic_AllEqualLoops03_KinematicHardening_IJNAMGEd.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) elasticperfectly 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_02Ed.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 elasticperfectly 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,
(elasticperfectly plastic for example)
requiring very few soil parameters
that could be easily obtained from insitu 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 NSFCMMI0600766 (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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%\include{Appendix}
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\end{document}
\bye