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\begin{document}
\title{Soil Uncertainty and its Influence on Simulated $G/G_{max}$ and Damping Behavior}
\author{
Kallol Sett\thanks{
Department of Civil Engineering, The University of Akron, Akron, Ohio, U.S.A.},\ A.M. ASCE,
%
Berna Unutmaz\thanks{
Department of Civil Engineering, Kocaeli University, Kocaeli, Turkey},
%
Kemal \"{O}nder \c{C}etin\thanks{
Department of Civil Engineering, Middle East Technical University, Ankara, Turkey},\ M. ASCE, \\
%
Suzana Koprivica\thanks{
Department of Civil Construction Management, Union University, Belgrade, Serbia},
%
and
Boris Jeremi{\'c}\thanks{
Corresponding author; Department of Civil and Environmental Engineering, University of California, Davis, California, U.S.A.},\ M. ASCE
}
\maketitle
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\ \\
\noindent
\small{\bf To be published in the \\
ASCE Journal of Geotechnical and Geoenvironmental Engineering, 2010.}
\noindent \textbf{\uppercase{Abstract:}} In this paper, recently developed probabilistic
elastoplasticity is applied in simulating cyclic
behavior of clay. Simple von Mises elasticperfectly plastic material
model is used for simulation. Probabilistic soil parameters, elastic
shear modulus ($G_{max}$) and undrained shear strength
($s_u$), that are needed for the simulation are obtained from
correlations with SPT $N$value. It has been shown that the
probabilistic approach to geomaterial modeling captures some of
the important aspects  modulus reduction, material damping
ratio, and modulus degradation  of cyclic behavior of clay
reasonably well, even with the simple elasticperfectly
plastic material model.
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\section{\uppercase{Introduction}}
Behavior of geomaterials is inherently uncertain. This uncertainty stems from
natural soil variability, testing and transformation errors
\cite{Lacasse:1996,Kulhawy:1999A}.
%
Traditionally, geotechnical engineering community deals with
uncertainties in geomaterial by applying factor of
safety. However, use of (large) factors of safety not only result
in overexpensive design, but also, sometimes, in unsafe design
\cite{Duncan:2000}. Hence, in recent years,
geotechnical engineering practice has seen an increasing
emphasis on probabilistic treatment to data and subsequent
simulation/design.
Quantification or mathematical description of uncertainty is
usually done within the framework of probability theory,
although fuzzy sets \cite{Zadeh:1983}, convex models
\cite{book:Elishakoff}, and interval arithmetic \cite{book:Moore}
have also been used in the past to describe uncertainty
mathematically.
%
Under the framework of probability theory, uncertain spatial variability
of soil deposit is modeled as random field  a collection of
random variables, indexed from space continuum.
%
For complete characterization (up to second order) of a random field,
in addition to mean and variance, information on
autocovariance function and correlation
length or scale of fluctuation are also needed.
%
Thorough descriptions of random field modeling
techniques, with procedures for estimating correlation length
for geotechnical engineering applications, are given by \citeN{Baecher:1993}
and \citeN{Fenton:1999A}.
%
Testing and transformation
uncertainties, on the other hand, are
point uncertainties and are usually modeled as random
variables, which are completely characterized (up to
second order) by their respective means and variances.
%
%
Over the years researchers have quantified and collected typical
variations of different soil properties, ranging from consolidation parameters,
laboratory measured strength properties to insitu properties
\cite{Lumb:1966,Lacasse:1996,book:Christian,Kulhawy:1999A}, as well as testing
uncertainties, associated with the most commonly used test methods \cite{Hammitt:1966,Kulhawy:1999A,Marosi:2004},
and transformation uncertainties, associated with the most common
transformation relationships \cite{Kulhawy:1999B}.
%
A fair amount of work was also done
on subsequent probabilistic geotechnical design guidelines
\cite{book:Harr,Kulhawy:2002}, although
the existing geotechnical
LRFD codes still do not explicitly consider the soil properties
uncertainties.
%
The book by
\citeN{book:Christian} thoroughly
describes the current stateoftheart of probabilistic
geotechnical engineering design.
Modeling and simulation under uncertainty,
on the other hand, have received much less attention, mainly due to the concern about the the necessity,
usefulness, and tractability of probabilistic modeling in
geotechnical engineering, when geotechnical problems are
difficult to model even deterministically, unless advanced
modeling techniques are used.
%
However, published works on this subject \cite{Griffiths:1996,Rackwitz:2000,Manolis:2002,Griffiths:2002,Griffiths:2002a,DeLima:SFEMgeotechApplication,Griffiths:2003,borja:OpinionPaper,Griffiths:2005,Popescu:2005} show very promising
results, especially in quantifying our confidence in our simulation.
%
Most of the above works are based on Monte Carlo technique \cite{Ulam:1949} in tandem with (deterministic) finite element
method.
%
Monte Carlo technique relies on repeated random sampling (of, for example, soil properties) and because of this
repeatability, the computational cost associated with it could become extremely huge (and probably intractable), especially for
large scale problems, like dynamic soilstructure interaction analysis.
%
Due to the above drawback of Monte Carlo technique, in other fields of science and engineering,
stochastic differential equation approach \cite{book:Gardiner} or numerically, stochastic finite element method
\cite{book:Kleiber,book:Ghanem} is very popular. However, nonlinearities in soil properties prevent direct application of
those techniques for probabilistic simulations in geotechnical engineering.
The difficulty in propagating uncertainties in soil properties through
the elasticplastic constitutive equation lies in the high nonlinear
dependence of the elasticplastic modulus on stress.
%
Very few published literature exist on this subject.
%
In fact, the first attempt to propagate uncertainties
through elasticplastic constitutive rate equation was
published only recently \cite{Hori:1999}.
%
Anders and Hori \citeyear{Hori:1999,Hori:2000} used perturbation
approach  a linearized Taylor series expansion
with respect to mean  in propagating uncertainties
in modulus through the von Mises elasticperfectly
material model.
%
However, inherent to the Taylor series expansion, perturbation
technique suffers from small variance requirement \cite{Matthies:1997reviewSFEM}.
%
A rule of thumb restricts the coefficient of variance (COV) to 20\% \cite{DerKiureghian:2000ReportSFEM} to minimize the error
in perturbation approach.
%
This severely limits the applicability of perturbation approach to geotechnical
problems, where soil COVs are rarely less than 20\% (cf. Phoon and Kulhawy \citeyear{Kulhawy:1999A,Kulhawy:1999B}).
%
Perturbation approach also suffers from closure problem \cite{Kavvas:2003}, which means that
information about higherorder statistical moments are necessary to
calculate lowerorder statistical moments.
%
Griffiths and Fenton \citeyear{Griffiths:2002a,Griffiths:2003,Griffiths:2005}
used brute force Monte Carlo technique in propagating uncertainties through
elasticperfectly plastic MohrCoulomb model.
%
Recently, in circumventing the above drawbacks of Monte Carlo and
perturbation techniques, \citeN{Jeremic:2007} proposed EulerianLagrangian
FokkerPlanckKolmogorov equation (FPKE; \citeNP{Kavvas:2003}) based
probabilistic elastoplasticity.
%
Developed probabilistic elastoplasticity is compatible with the general
theory of (deterministic) plasticity, and hence can be used
for probabilistic simulation of a variety of elasticplastic models.
%
Solution strategies for probabilistic elastoplasticity were discussed by
Sett et al. \citeyear{Sett:2007b,sett:2007a} for both linear and nonlinear
hardening models. The concept of probabilistic yielding was
introduced by \citeN{Jeremic:2009b}, while \citeN{Sett:2010a}
discussed its effect on constitutive simulation under cyclic
loading.
%
It was shown that due to uncertainty in yield stress,
there is always a possibility that plastic behavior starts at very very low
strain and influence of elastic behavior continues far into the
plastic domain.
%
Because of this, the average (mean) constitutive response and the most probable
(mode) constitutive response show nonlinear behavior with a vanishing
linear region, even for the simplest elasticperfectly plastic
material model.
%
This in turn is significant since, by expanding into probability space, one
obtains very realistic soil behavior with simple constitutive models, requiring
very few soil parameters (and their distributions) that could be obtained
directly from insitu tests (e.g. SPT, CPT etc.).
%
In this paper, FPKE based probabilistic approach to geomaterial modeling
is applied in simulating $G/G_{max}$ and damping behavior of undrained
clay. Elasticperfectly plastic von Mises material model, which requires only two soil
parameters, the shear modulus ($G_{max}$) and the undrained shear
strength ($s_u$), is used.
%
Simulated responses
are then compared with published experimental measurements.
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\clearpage
\section{\uppercase{Probabilistic Framework for Constitutive Simulation}}
\label{PROBABILISTIC_FRAMEWORK}
The constitutive behavior of soil can be modeled by elasticplastic
constitutive rate equation, which, in 1D, can be written
as:
%
\begin{equation}
\label{eqno_1}
\displaystyle \frac{d \sigma}{dt} = D \displaystyle \frac{d \epsilon}{dt}
\end{equation}
%
where $\sigma$ is the stress, $\epsilon$ is the strain, $t$ is the pseudo time, and $D$ is the
stiffness modulus, that can be either 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_2}
\end{eqnarray}
%
where, $D^{el}$, $f$, $U$, $q_*$, and $r_*$ are
elastic modulus, yield function, plastic potential function,
internal variable(s), and rate(s) of evolution of internal
variable(s) respectively.
%
However, due to various uncertainties
associated with soil properties, as discussed in the previous section,
the modulus, $D$ in Eq.~(\ref{eqno_1}) becomes uncertain.
%
In traditional deterministic approach to elasticplastic geomaterial modeling, one typically
applies engineering judgment (qualitative) in obtaining the 'most probable' soil
parameters and substitute them in Eq.~(\ref{eqno_1}) in obtaining the 'most probable'
soil constitutive behavior.
%
However, one may note that due to the nonlinearity of soil behavior,
'most probable' soil parameters do not necessarily result in 'most probable'
constitutive behavior.
Recently, \citeN{Jeremic:2007} developed a probabilistic approach for
elasticplastic modeling of geomaterials. Proposed approach is based on the
extension of constitutive
rate equation (Eq.~(\ref{eqno_1})) into probability density space using
EulerianLagrangian FokkerPlanckKolmogorov approach \cite{Kavvas:2003}:
%
%
\begin{equation}
\label{eqno_3}
\displaystyle \frac{\partial P(\sigma(t), t)}{\partial t} =
\displaystyle \frac{\partial}{\partial \sigma} \left\{ N_{(1)} P(\sigma(t),t) \right\}
+ \displaystyle \frac{\partial^2}{\partial \sigma^2} \left\{ N_{(2)} P(\sigma(t),t) \right\}
\end{equation}
%
%
where $P(\sigma(t), t)$ is the probability density of
stress, $t$ is the pseudotime, while $N_{(1)}$ and $N_{(2)}$ are
advection and diffusion coefficients, respectively. The
advection and diffusion coefficients depend only on the material model
used for modeling.
%
It can be shown (see appendix of \citeNP{Jeremic:2007}), that the advection and
diffusion coefficients for von Mises elasticperfectly plastic constitutive
relationship (used here to model undrained behavior of clay) can be
written as:
%
%
\begin{eqnarray}
\begin{array}{l}
{}^{vM}\!N_{(1)}^{{el}} = \displaystyle \frac{d \epsilon}{dt} \left< G \right >
%
\;\;\;\;\;\; \mbox{;} \;\;\;\;\;\;
%
%\label{eqno_E2}
{}^{vM}\!N_{(2)}^{{el}} = t \left(\displaystyle \frac{d \epsilon}{dt}\right)^2 Var[G]
\\
{}^{vM}\!N_{(1)}^{{elpl}} = 0
\;\;\;\;\;\;\;\;\;\;\;\; \mbox{;} \;\;\;\;\;\;\;
{}^{vM}\!N_{(2)}^{{elpl}} = 0 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\end{array}
\label{eqno_4}
\end{eqnarray}
%
%%
In Eq.~(\ref{eqno_4}), $G$ is the elastic shear modulus and $\epsilon$ is
the shear strain. Furthermore, $\left< \cdot \right>$ represents the expectation
operator, while $Var\left[ \cdot \right]$ is the variance operator.
%
The superscripts $\cdot^{el}$ and $\cdot^{elpl}$ on the advection and
diffusion coefficients refer to preyield elastic region and postyield,
elasticplastic region.
%
%
However, for a heterogeneous material like soil, yield strength is quite
uncertain.
%
This is due to the fact that in a representative volume element (RVE;
\citeNP{local34}) of the heterogeneous
material, each of the large number of particle contacts has different yield
strengths and orientations.
%
Each of these particle contacts will yield differently, depending upon its yield strength.
%
Hence, for material with uncertain yield strength ($\Sigma_y$),
there exist possibilities that the material (RVE) starts yielding inside the elastic regime and/or
elastic behavior continues way into the plastic regime.
%
Under the framework of probability theory, these possibilities are
governed by the probability density function of yield strength ($\Sigma_y$), which
can be quantified by statistically analyzing the test results.
%
Hence, to realistically simulate the probabilistic material behavior, \citeN{Jeremic:2009b}
suggested probability weights, based on probability density function of yield
strength ($\Sigma_y$), to the elastic and plastic advection and diffusion
coefficients in obtaining equivalent advection and diffusion coefficients.
%
For von Mises elasticperfectly plastic soil with uncertain yield strength
($\Sigma_y$), the equivalent advection and diffusion coefficients
(${}^{vM}\!N_{(1)}^{{eq}}$ and ${}^{vM}\!N_{(2)}^{{eq}}$) would become (cf.
\citeNP{Sett:2010a}):
%
\begin{eqnarray}
%\begin{array}{l}
%\label{eqno_6a_VM}
\nonumber
{}^{vM}\!N_{(1)}^{{eq}} (\sigma)
&=& (1  P[\Sigma_y \leq \sigma]) {}^{vM}\!N_{(1)}^{{el}} + P[\Sigma_y \leq \sigma] {}^{vM}\!N_{(1)}^{{elpl}} \\
\label{eqno_6a_VM}
&=& (1  P[\Sigma_y \leq \sigma])
\displaystyle \frac{d \epsilon}{dt}
\left< G \right >
\ \\
\nonumber
\ \\
\nonumber
{}^{vM}\!N_{(2)}^{{eq}} (\sigma)
&=& (1  P[\Sigma_y \leq \sigma]) {}^{vM}\!N_{(2)}^{{el}} + P[\Sigma_y \leq \sigma] {}^{vM}\!N_{(2)}^{{elpl}} \\
\label{eqno_6b_VM}
&=& (1  P[\Sigma_y \leq \sigma])
t \left( \displaystyle \frac{d \epsilon}{dt}\right)^2 Var[G]
%\end{array}
\end{eqnarray}
%
The probability weight ($ P[\Sigma_y \leq \sigma]$) in the above equations
(Eqs.~(\ref{eqno_6a_VM}) and (\ref{eqno_6b_VM})), quantifies, at any given stress level,
the probability of the material RVE being elastic or elasticplastic. Using
the above equivalent advection and diffusion
coefficients (Eqs.~(\ref{eqno_6a_VM}) and (\ref{eqno_6b_VM})), one can solve the
constitutive rate equation, written in probability density space
(Eq.~(\ref{eqno_3})) to obtain the complete probabilistic description, in terms
of probability density function, of evolutionary stress response with
strain (pseudotime).
%
One may note that the probabilistic framework
presented above is a pure constitutive level (pointlocation
scale) framework and assumes the input soil properties to be
random variables. The spatial average (localaverage)
probabilistic constitutive response, if sought for, for
example, in dealing with uncertain spatial variability of soil
properties, usually modeled as random fields, would
necessitate discretization of random fields into random
variables using appropriate technique, for example,
KarhunenLo\`{e}ve expansion \cite{Karhunen:1947,Loeve:1948,book:Ghanem}. Those random
variables could then be propagated through the pointlocation
scale constitutive framework presented above and spatial
average constitutive response could be assembled using a
stochastic finite element technique. Using polynomial chaos
expansion \cite{Wiener:1938,book:Ghanem} and
Galerkin technique, \citeN{Sett:2009b} proposed one
such stochastic finite element framework and applied the framework in
seismic wave propagation through spatially uncertain stochastic
soil.
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\section{\uppercase{Simulation Results and Discussion}}
\label{RESULTS_AND_DISCUSSION}
In this section, the constitutive behavior of normally
consolidated, high plasticity clay is simulated
probabilistically using the FPKE approach described in the
previous section. Elasticperfectly plastic von Mises material model is
used for clay. The model requires shear
modulus ($G_{max}$) and undrained shear strength ($s_u$) as
input soil parameters. Both the soil parameters are easily
obtainable through transformation from commonly used insitu tests.
%
In this paper, the above properties are obtained from SPT $N$value.
%
In this context, it is important to mention that the authors understand
the limitations of using SPT $N$value for (deterministic) estimation of $s_u$ and
$G_{max}$ for clay.
%
However, the authors' intent here is to demonstrate the power of a simple
constitutive model, but with uncertain soil
parameters, to simulate the actual response of soil.
%
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\subsection{Quantification of Uncertainties in Input Soil Parameters}
\label{UNCERTAINTY_QUANTIFICATION}
Transformation from measured insitu properties to mechanical properties usually
introduces uncertainty, which is currently (in traditional deterministic analysis)
accounted for by applying engineering judgment.
%
Alternatively, under the
framework of probability theory, one could quantify the transformation
uncertainty by modeling it as a random variable. For example, for alluvial clays
in Japan, \citeN{Kulhawy:1999B} proposed the following
relationship between SPT $N$value and undrained shear strength ($s_u$):
%
\begin{equation}
\label{eqno_RD1}
s_u = 0.29 \; p_a \; N^{0.72}
\end{equation}
%
where $p_a = 101,325~\rm{Pa}$ is the atmospheric pressure. The above relationship
(Eq.~(\ref{eqno_RD1})), along with the data \cite{Hara:1974} from
which the relationship is developed, is plotted in
FIG.~\ref{fig:data_ShearStrength}.
%
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.7\textwidth]{Fig01.eps}
\caption{Transformation relationship between SPT $N$value and undrained shear
strength, $s_u$}
\label{fig:data_ShearStrength}
\end{center}
\end{figure}
%
The datascatter in FIG.~\ref{fig:data_ShearStrength}
represents knowledge uncertainty in the above transformation
equation (Eq.~(\ref{eqno_RD1})), and under probability theory,
can be modeled as a random variable. To this end,
Eq.~(\ref{eqno_RD1}) can be written as:
%
\begin{equation}
\label{eqno_RD1a}
s_u = 0.29 \; p_a \; N^{0.72} + \chi
\end{equation}
%
where, $\chi$ is a zeromean random variable and
represent the dataresidual with respect to the deterministic
transformation equation. The histogram of the residual is
plotted in FIG.~\ref{fig:Histogram_ShearStrength}. Regarding
the model for the bestfit probability density function (PDF), a
Gaussian distribution can be ruled out as the histogram is skewed.
After trying few distributions, a Pearson IV type distribution, with Pearson
parameters of $0$, $2400$, $2.75 \times 10^5$, and $9 \times 10^8$, was found to best fit the
residual.
%
The fitted PDF is shown in FIG.~\ref{fig:Histogram_ShearStrength}.
%
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.7\textwidth]{Fig02.eps}
\caption{Histogram of the residual (w.r.t the deterministic transformation
equation) undrained strength, along with fitted probability density function}
\label{fig:Histogram_ShearStrength}
\end{center}
\end{figure}
%
%
Similarly, for transformation between SPT $N$value and Young's modulus ($E$)
for alluvial clays in Japan, \citeN{Kulhawy:1999B}
proposed a transformation equation, which can be written in probabilistic form
as
%
\begin{equation}
\label{eqno_RD2}
E = 19.3 \; p_a \; N^{0.63} + \chi
\end{equation}
%
FIG.~\ref{fig:data_YoungsModulus} shows
experimental data \cite{Ohya:1982} along with the deterministic transformation
equation that in this case represents the mean trend.
%
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.7\textwidth]{Fig03.eps} \\
\caption{Transformation relationship between SPT $N$value and pressuremeter Young's modulus, $E$ }
\label{fig:data_YoungsModulus}
\end{center}
\end{figure}
%
The scatter with respect to the mean trend
(deterministic transformation equation), plotted as histogram,
is shown in FIG.~\ref{fig:Histogram_YoungsModulus}.
%
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.7\textwidth]{Fig04.eps}
\caption{Histogram of the residual (w.r.t the deterministic transformation equation) Young's modulus, along with fitted probability density function}
\label{fig:Histogram_YoungsModulus}
\end{center}
\end{figure}
%
A zeromean Gaussian random variable with a standard deviation
of 4041.8 kPa is found to best fit
(FIG.~\ref{fig:Histogram_YoungsModulus}) the scatter with respect to the
deterministic equation.
%
The above standard deviation is
obtained using maximum likelihood technique.
In this context, one may note that
the penetration test is a high strain test (to the order of
10\%; refer to Figure 65 in FHWA Geotechnical Engineering Circular No. 5 \cite{FHWA:2002}) and, the corresponding estimated
modulus (Eq.~\ref{eqno_RD2}) is a highstrain modulus.
%
Hence, to estimate the corresponding lowstrain modulus (needed as the input to
the von Mises material model, used in this paper), it is multiplied by $17.25$, assuming $(17.251)/17.25=94$~\%
reduction in modulus at 10\% strain, following \citeN{Idriss:1990} and presuming
that the reduction pattern of Young's modulus follows the same that of shear
modulus.
%
For example, at
$N=15$, a mean highstrain Young's modulus of $10.735$~MPa is
predicted by Eq.~(\ref{eqno_RD2}).
%
The corresponding mean
lowstrain Young's modulus used in this paper is
$10.735$~MPa $\times 17.25 = 185$~MPa.
%
%
The multiplying factor (low strain correction factor) for standard deviation
should, physically, be less than that of the mean. This is because, as the
soil is sheared (or, in other words, as soil plastifies), the
microstructure of soil changes and as a result, our knowledge uncertainty on it
increases. This was experimentally observed by \citeN{Stokoe:2004}.
Probabilistic simulations, published elsewhere by the authors \cite{Sett:2010a}, also show such increase
in uncertainty with strain.
%
Hence, in absence of experimental data for clay, following the probabilistic
$G/G_{max}$ versus shear strain curve, suggested by \citeN{Stokoe:2004}, a
multiplying factor (low strain correction factor) of $[1(0.23750.05)/0.05]
\times 17.25 = 10.7$ is used for standard deviation.
%
%
\noindent At $N$=15, a standard deviation of $4.04$~MPa was
estimated, using maximum likelihood technique, for highstrain
Young's modulus.
%
The corresponding standard deviation of
lowstrain Young's modulus, then, becomes
$4.04 \times 10.7 = 43.2$~MPa.
%
By assuming undrained condition, one could assume Poisson's ratio to be equal
to 0.5 (deterministic) and could transform the Young's modulus to shear modulus
as
%
\begin{equation}
\label{eqno_RD3}
G_{max}=\frac{E_{max}}{2(1+\nu)}
\end{equation}
%
One may note that, as the above equation (Eq.~(\ref{eqno_RD3})) that relates elastic
shear modulus and elastic Young's modulus is linear, the elastic shear modulus
($G_{max}$) would also be a Gaussian distribution.
%
Hence, the
statistical properties of the elastic shear modulus ($G_{max}$) can easily be
obtained using standard techniques. For example, at $N=15$, the mean and the
standard deviation of $G_{max}$ are $185{\rm MPa}/(2(1+0.5)) = 61.6$~MPa and
$43.2{\rm MPa}/(2(1+0.5)) = 14.4$~MPa respectively.
%
In this context, it is important to emphasize that the above estimation
of the lowstrain correction factors would become unnecessary if
smallstrain shear modulus ($G_{max}$) is measured directly
from geophysical tests or estimated through
direct correlation of geophysical testmeasured properties (for
example shear wave velocity, with SPT $N$value). The
transformation equations between SPT $N$value and shear wave
velocity, reported in the literature \cite{Hasancebi:2007,Jafari:2002,Pitilakis:1999,Imai:1977} have not been used in this
paper due to lack of reported data points for a meaningful statistical
analysis.
In addition to the transformation uncertainty, discussed above,
soil properties also include significant testing
uncertainties.
%
For example, in SPT, the testing uncertainty
arises from equipment, procedure and operator errors. \citeN{Kulhawy:1999A}
proposed typical range of COV
for SPT as 1545\%. In this paper, an equivalent of 45\% COV is
added to the undrained shear strength ($s_u$) and 15\% COV is added
to the elastic shear modulus ($G_{max}$) to account for SPT testing
uncertainties. Larger uncertainty is used for
undrained shear strength ($s_u$) because the shear strength is not unique but depends on
many factors, e.g. direction of loading, strain rate, boundary conditions, stress
level, and sample disturbance effects \cite{Ladd:1991,Mayne:2007}.
Uncertain spatial variability represents the other important source of
uncertainty in soil property.
%
This uncertainty is present because soil properties are measured
at few locations and then
extrapolated/interpolated to all (some) other points
of the soil continuum.
%
In other words, in 'estimating' soil properties
between two adjacent boreholes, uncertain spatial
variability is incurred.
%
This uncertain spatial variability is traditionally accounted for by applying
engineering judgment.
%
Probability theory, on the other hand, deals with uncertain spatial
variability through random field modeling \cite{Baecher:1993,Fenton:1999A,Fenton:1999B}.
%
Random field modeling characterizes the uncertain spatial variability in terms
of standard deviation and correlation structure.
%
The standard deviation is usually added to the uncertainties arising from transformation equation
and testing method, while the correlation structure can be
accounted for, among others, through stochastic elasticplastic finite element
method \cite{Sett:2009b,Sett:2010c}.
%
In this paper, the uncertain spatial variability has not been explicitly
accounted for as the focus of this paper is on pointlocation (constitutive)
behavior.
%
However, one may note that the data of SPT $N$value versus
undrained shear strength (FIG.~\ref{fig:Histogram_ShearStrength}) and SPT
$N$value versus
Young's modulus (FIG.~\ref{fig:data_YoungsModulus}) contain some effects of spatial variability
as SPT is performed at approximately every $30$~cm (1 foot) and the blow counts
obtained in such way represent average values over that length.
%
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\subsection{Simulation of $G/G_{max}$ and Damping Behavior}
\label{GOVERGMAX_DAMPING}
The above described uncertain data set is used to analyze, using
the probabilistic elasticplastic constitutive framework (described in Section~\ref{PROBABILISTIC_FRAMEWORK}),
the undrained cyclic (shear) behavior of clay.
%
Of particular interest is the performance evaluation of a simple, elasticperfectly plastic von Mises material model,
but extended into probability space.
\subsubsection{Probabilistic elasticplastic stressstrain response}
FIG.~\ref{fig:HysteresisLoop_01}(a) shows the mean shear stress
versus shear strain hysteresis loop for an undrained clay, simulated using
probabilistic von Mises elasticperfectly plastic material model.
%
The input to the model were the statistics of the soil properties  elastic shear modulus, $G_{max}$ and undrained shear strength, $s_u$ 
corresponding to SPT $N$value of 15.
%
The uncertain clay material is cyclically sheared to $ \pm 1.026$\% shear strain.
%
The
cyclic evolution of standard deviation of shear stress is shown
in FIG.~\ref{fig:HysteresisLoop_01}(b).
%
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.47\textwidth]{Fig05a.eps}
\hspace*{0.5truecm}
\includegraphics[width=0.43\textwidth]{Fig05b.eps}
~~~~~~~~~~~~~~~~~~~~~(a)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(b)
\caption{Simulated hysteresis shear stress versus shear strain loop at $\pm$
1.026\% shear strain: (a) mean and (b) standard deviation behavior}
\label{fig:HysteresisLoop_01}
\end{center}
\end{figure}
%
Results shown in Figures~\ref{fig:HysteresisLoop_01}(a)~and~(b) are obtained by
solving the FokkerPlanckKolmogorov equation (FPKE; Eq.~(\ref{eqno_3}), with advection and diffusion coefficients
given by Eqs.~(\ref{eqno_6a_VM}) and (\ref{eqno_6b_VM})) numerically, with appropriate
initial and boundary conditions. The solution to the FPKE, the evolutionary PDF of shear stress with shear strain, is then integrated by standard
techniques to obtain the evolutionary mean and standard
deviation behavior.
%
The details of the solution technique for governing
FPKE can be found in \citeN{Sett:2010a}.
%
It is important to note that simulation results shown in
FIG.~\ref{fig:HysteresisLoop_01} are obtained using elastic  perfectly
plastic von Mises material model and require only two probabilistic soil
parameters (their probability distribution)  elastic shear modulus
($G_{max}$) and undrained shear strength ($s_u$).
%
If probability distribution of material parameters were neglected and only mean
values were used (thus simplifying to deterministic von Mises
elasticperfectly plastic model) simple bilinear response would result.
%
Such (deterministic) bilinear response is also shown in FIG.~\ref{fig:HysteresisLoop_01}(a).
%
In FIG.~\ref{fig:HysteresisLoop_01}(a), it is also interesting to observe that the
mean response (of the full probability response described by the stress PDF)
is nonlinear even at very small strain, although, the deterministic
model assumes linearity till yielding and then behaves as perfectly
plastic material.
%
Such nonlinear mean response is due to the uncertainty in the
yield stress, as there is always a probability (however small) that
elasticplastic response starts at a very small strain.
%
In addition to that, there also exist a probability that material is elastic
at strains past the (mean) yield point, and since the mean solution is an ensemble average of
all the possibilities, such probable elastic influence is extended into
plastic range as well.
%
One can visualize this probabilistic yielding effect by observing that within a
laboratory specimen (considered generally as a representative
volume element (RVE; \citeNP{local34})), each of large (infinite) number of particle
contacts has different and unknown yield strengths.
%
%
At a given strain, some of those particle contacts might be elastic
while others might be fully yielding.
%
What is observed in any laboratory experiment is the ensemble
average (mean) behavior of all the particle contacts. Similar conclusion was
developed by \citeN{Einav:2008}, using probabilistic micromechanical simulation.
%
It may be noted
that the pointlocation scale constitutive simulation presented
in this paper doesn't account for the scale effect.
%
Such scale effects could be accounted for by quantifying the uncertain spatial
variability (for example, through random field modeling) of soil and accounting
for it in our simulation. This can be done, for example, through stochastic
elasticplastic finite element method in obtaining localaverage constitutive
behavior.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Evolution of secant shear modulus}
Elasticplastic constitutive simulation, described above, is used to obtain the evolution of
secant shear modulus with shear strain.
%
The deterministic evolution of the secant shear modulus is shown in FIG.~\ref{fig:G}(a).
%
The deterministic shear modulus remains constant, equal to
$G_{max}=61.6$~MPa until $\approx 0.3$\% strain, representing
deterministic yield point, before suddenly dropping after yield point.
%
FIG.~\ref{fig:G}(a) also shows mean and mean$\pm$standard deviation of the evolutionary
(probabilistic) secant shear modulus.
%
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.441\textwidth]{Fig06a.eps}
\hspace*{0.5truecm}
\includegraphics[width=0.459\textwidth]{Fig06b.eps}
~~~~~~~~~~~~~~~~~~~~~(a)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(b)
\caption{Simulated (a) probabilistic reduction and (b) evolution of coefficient
of variation (COV) of secant shear modulus with cyclic shear strain.}
\label{fig:G}
\end{center}
\end{figure}
%
Compared with the deterministic evolution of secant shear
modulus, the mean solution predicts a realistic reduction with cyclic shear strain.
% %
The region between mean and mean$\pm$standard deviation contains the
most likely values of evolutionary shear modulus.
%
Coefficient of Variation (COV), which can also be used to visualize
uncertainty, is shown in FIG.~\ref{fig:G}(b).
%
The initial COV of secant shear modulus was [($14.4$~MPa $+$ $0.15$ $\times$
$61.6$ MPa)$/61.6$~MPa]$\times 100$\% = $38.3$\%. It was calculated from the
mean and standard deviation values for $G_{max}$, obtained earlier
as $61.6$~MPa and $14.4$~MPa, respectively. The second term in the numerator ($0.15$ $\times$
$61.6$ MPa) represents the contribution of the testing uncertainty, which was assumed to
have a COV of 15\%.
%
%
It is interesting to observe that COV of
secant shear modulus increases with cyclic shear strain.
%
This increase in uncertainty comes from the fact that as the material
plastifies,
%(breaking of internal structure between soil particle),
this simple two parameter
model becomes less and less accurate. In other words, more
detailed investigation of the soil (micro) structure is needed and more advanced
modeling technique needs to be used if one wishes to reduce
such uncertainty.
The above probabilistic evolution of secant shear modulus
(FIG.~\ref{fig:G}) is shown in FIG.~\ref{fig:GoverGmax} in a
more common form, in terms of variation of $G/G_{max}$ versus
shear strain.
%
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.7\textwidth]{Fig07.eps}
\caption{Simulated probabilistic $G/G_{max}$ behavior}
\label{fig:GoverGmax}
\end{center}
\end{figure}
%
It is important to note that, in FIG.~\ref{fig:GoverGmax}, the
normalizations of evolutionary mean and mean$\pm$standard
deviation are done by dividing each of those by the mean of elastic
shear modulus (Mean[$G_{max}$]).
%
In other words, the upper and
lower limits of normalized secant shear modulus, shown in
FIG.~\ref{fig:GoverGmax} represent (Mean[G]$\pm$Standard
Deviation[G])/Mean[$G_{max}]$, rather than $G/G_{max} \pm$standard
deviation.
%
The probabilistic evolution of
material damping ratio versus shear strain is shown in
FIG.~\ref{fig:Damping}.
%
The mean damping ratio, shown in FIG.~\ref{fig:Damping}
is obtained from the hysteresis loop of mean shear stress versus
shear strain.
%
Likewise, the upper and lower bounds of damping
ratio in FIG.~\ref{fig:Damping}, are obtained from the hysteresis
loops of
mean$\pm$standard deviation of shear stress versus shear strain.
%
%
%
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.7\textwidth]{Fig08.eps}
\caption{Simulated probabilistic material damping behavior}
\label{fig:Damping}
\end{center}
\end{figure}
In both FIGs.~\ref{fig:GoverGmax} and
\ref{fig:Damping}, the deterministic solutions are also
plotted. Though the deterministic solutions fail to predict
realistic soil behavior, probabilistic solutions, even with the simplest
elasticperfectly plastic model, are comparable to the
experimental observations reported in the literature. For
example, the probabilistic $G/G_{max}$ and damping ratio
curves, presented
in FIGs..~\ref{fig:GoverGmax} and
\ref{fig:Damping}, compared well with the experimental data
reported by \citeN{Vucetic:1991} and \citeN{Stokoe:2004} for high
plasticity clay.
In addition to modulus reduction, the probabilistic approach
also captures modulus degradation when the clay material is
cyclically sheared repeatedly. The simulated
hysteresis loops when the clay material is sheared repeatedly to $\pm$0.1026\% strain, is shown in
FIG.~\ref{fig:HysteresisLoop_02}.
%
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.45\textwidth]{Fig09a.eps}
\hspace*{0.5truecm}
\includegraphics[width=0.45\textwidth]{Fig09b.eps}
~~~~~~~~~~~~~~~~~~~~~(a)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(b)
\caption{Simulated hysteresis shear stress versus shear strain loop, when sheared repeatedly at $\pm$ 0.1026\% strain: (a) mean and (b) standard deviation behavior; First four
cycles are shown}
\label{fig:HysteresisLoop_02}
\end{center}
\end{figure}
%
Only the first four loops are
shown in FIG.~\ref{fig:HysteresisLoop_02} for clarity. The
absolute values of mean and standard deviation of secant shear
modulus after each cycle are plotted in
FIGs.~\ref{fig:ModulusDegradation}(a) and (b), respectively.
%
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.45\textwidth]{Fig10a.eps}
\hspace*{0.75truecm}
\includegraphics[width=0.45\textwidth]{Fig10b.eps}
~~~~~~~~~~~~~~~~~~~~~(a)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(b)
\caption{Simulated probabilistic degradation of shear modulus, when sheared repeatedly at $\pm$ 0.1026\% strain: (a) mean and (b) standard deviation behavior}
\label{fig:ModulusDegradation}
\end{center}
\end{figure}
%
The
mean shear modulus degrades 8\% after 10 cycles at 0.1026\%
strain. The rate of degradation of mean secant shear modulus
is higher initially, but stabilizes as the number of cycles
increases. The standard deviation of secant shear modulus, on
the other hand, increases (275\% increase after 10 cycles at
0.1026\% strain) with number of cycles. It, however, also
stabilizes as the number of cycles increases.
%
%
The explanation for increased uncertainty in the secant shear modulus is
based on mechanics.
%
With repeated shearing, soil (micro) structure is continuously disturbed
and hence, our knowledge uncertainty on it increases.
%
In other words, after repeated shearing, simplistic twoparameter
elasticperfectly plastic model, used here, cannot model such changes accurately.
% is not accurate enough to model that.
%
The elasticplastic probabilistic solution (advectiondiffusion equation) aptly
captures that fact.
%
The diffusion component, which controls the spread of the response (stress)
probability density function, keeps evolving continuously with strain, irrespective
of the direction of loading (shearing) until plasticity is fully mobilized with
100\% probability, when the diffusion coefficient becomes zero.
%
The advection component, on the other hand, controls the translation
of the response (stress) PDF in the stressstrain
domain.
%
This component is a function of loading (shearing) direction, advection coefficient and
initial condition at the beginning of each loading direction, which in turn,
is a function of the uncertainty present in the system at that state of
strain/shearing.
%
The advection component hence gives rise to the degraded modulus after each
cycle, until plasticity is fully mobilized with 100\% probability.
%
The modulus degradation is, therefore, appearing as a direct consequence of
probabilistic yielding of material (clay).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{\uppercase{Conclusions}}
Presented in this paper was a probabilistic approach to
constitutive
% (pointlocation scale)
simulation of undrained clay behavior.
It had been shown that probabilistic approach allowed for
not only quantification of our confidence in numerical
prediction, but also modeling modulus reduction, modulus
degradation and damping behavior with simple elasticperfectly plastic (twoparameter)
material model.
%
This is particularly significant since in geotechnical engineering practice, due to various
constraints, advanced laboratory tests are rarely performed,
%
while insitu tests are usually preferred, data from
which can be used to calibrate probabilistic material models, one of which was
presented here.
%
In particular, shown here was (probabilistic) calibration of a simple,
elasticperfectly plastic model but extended into probability space, by
using number of insitu tests. Such calibrated probabilistic elasticperfectly
plastic model was then used to predict various aspects of undrained shear behavior of clay.
%
It was shown that simulation results compared well with published data (within
mean $\pm$ standard deviation).
%
More importantly, as results contained full PDFs of
the responses, they might have many (other) uses in research and practice.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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