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\NameTag{Sett et al.}
%
\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
elasto--plasticity is applied in simulating cyclic 
behavior of clay. Simple von Mises elastic--perfectly 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 elastic--perfectly
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  over-expensive  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    in-situ    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   state-of-the-art  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 soil-structure 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, non-linearities 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 elastic-plastic constitutive equation lies in the high non-linear
dependence of the elastic-plastic modulus on stress.
%
Very  few published literature exist on this subject.
%
In fact, the first attempt to propagate uncertainties
through elastic-plastic 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 elastic--perfectly
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 higher-order statistical moments are necessary to
calculate lower-order statistical moments. 
%
Griffiths and Fenton \citeyear{Griffiths:2002a,Griffiths:2003,Griffiths:2005}
used brute force Monte Carlo technique in propagating uncertainties through 
elastic-perfectly plastic Mohr-Coulomb model.
%
Recently, in circumventing the above drawbacks of Monte Carlo and
perturbation techniques, \citeN{Jeremic:2007} proposed Eulerian--Lagrangian
Fokker--Planck--Kolmogorov  equation (FPKE; \citeNP{Kavvas:2003}) based
probabilistic   elasto--plasticity.
%
Developed probabilistic   elasto--plasticity is  compatible with the general
theory of (deterministic) plasticity, and hence can be used 
for   probabilistic   simulation of a variety of elastic--plastic   models. 
%
Solution strategies for probabilistic elasto--plasticity were discussed by 
Sett et al. \citeyear{Sett:2007b,sett:2007a} for both linear and non-linear
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 elastic--perfectly 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 in--situ 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.  Elastic--perfectly  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  elastic--plastic
constitutive rate equation, which, in 1--D,  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 elastic--plastic:
%
\begin{eqnarray}
D = \left\{\begin{array}{ll}
%
           D^{el}
%
%
        \;\;\; & \mbox{elastic} \\
%
           \\
%
           D^{el}
           -
           \displaystyle \displaystyle \frac{D^{el}
           \displaystyle \frac{\partial U}{\partial \sigma}
           \displaystyle \frac{\partial f}{\partial \sigma}
           D^{el}}
           {\displaystyle \frac{\partial f}{\partial \sigma}
           D^{el}
           \displaystyle \frac{\partial U}{\partial \sigma}
           -
           \displaystyle \frac{\partial f}{\partial q_*}r_*}
%
        \;\;\; & \mbox{elastic-plastic}
%
           \end{array} \right.
\label{eqno_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 elastic--plastic geo-material 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 non-linearity 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
elastic--plastic  modeling  of  geomaterials.  Proposed approach is based on the
extension of constitutive
rate equation (Eq.~(\ref{eqno_1})) into probability density space using
Eulerian--Lagrangian Fokker--Planck--Kolmogorov 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  pseudo-time, 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  elastic-perfectly 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)}^{{el-pl}} = 0
\;\;\;\;\;\;\;\;\;\;\;\; \mbox{;} \;\;\;\;\;\;\;
{}^{vM}\!N_{(2)}^{{el-pl}} = 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    $\cdot^{el-pl}$  on  the  advection and
diffusion  coefficients  refer  to  pre--yield  elastic  region  and post-yield,
elastic--plastic 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{local-34}) 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  elastic--perfectly  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)}^{{el-pl}} \\
\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)}^{{el-pl}} \\ 
\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 elastic--plastic. 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 (pseudo-time).
% 


One may  note  that  the  probabilistic framework
presented  above  is  a pure constitutive level (point-location
scale)  framework  and  assumes the input soil properties to be
random   variables.   The   spatial   average   (local-average)
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,
Karhunen--Lo\`{e}ve      expansion  \cite{Karhunen:1947,Loeve:1948,book:Ghanem}.   Those   random
variables  could  then be propagated through the point-location
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. Elastic--perfectly 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 in-situ 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.
%


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

\subsection{Quantification of Uncertainties in Input Soil Parameters}
\label{UNCERTAINTY_QUANTIFICATION}

Transformation  from  measured  in-situ 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 data-scatter 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  zero-mean random variable and
represent    the    data-residual   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  best-fit  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 pressure-meter 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 zero-mean  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 high-strain modulus. 
%
Hence,  to estimate the corresponding low-strain modulus (needed as the input to
the  von  Mises  material  model, used in this paper), it is multiplied by $17.25$, assuming $(17.25-1)/17.25=94$~\%
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  high-strain  Young's modulus of $10.735$~MPa is
predicted   by  Eq.~(\ref{eqno_RD2}).  
%
The  corresponding  mean
low-strain    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
micro-structure 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 $[1-(0.2375-0.05)/0.05]
\times 17.25 = 10.7$ is used for standard deviation.
%
%
\noindent At  $N$=15,  a  standard  deviation  of  $4.04$~MPa  was
estimated, using  maximum  likelihood  technique, for high-strain
Young's   modulus.  
%
The  corresponding  standard  deviation  of
low-strain       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 low-strain correction factors would become unnecessary if
small-strain  shear  modulus  ($G_{max}$) is measured  directly   
from   geophysical   tests   or  estimated through
direct correlation   of   geophysical  test-measured  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 15-45\%. 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 elastic-plastic 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 point-location (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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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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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 elastic--plastic 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, elastic--perfectly plastic von Mises material model, 
but extended into probability space.

\subsubsection{Probabilistic elastic--plastic stress--strain 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 elastic-perfectly 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   Fokker--Planck--Kolmogorov   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
elastic--perfectly plastic model) simple bi-linear response would result. 
%
Such (deterministic) bi-linear 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  non--linear 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  
elastic--plastic 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{local-34})),  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 micro-mechanical simulation.
%
It may be noted
that the point-location 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
elastic--plastic finite  element method  in obtaining local-average constitutive
behavior.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Evolution of secant shear modulus}

Elastic--plastic 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
elastic--perfectly  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  two-parameter  
elastic-perfectly plastic model, used here, cannot model such changes accurately.
% is not accurate enough to model that.
%
The  elastic-plastic probabilistic solution (advection-diffusion 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 stress-strain
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 
%  (point-location 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 elastic-perfectly plastic (two-parameter)
material model.
%
This is particularly significant since in geotechnical engineering practice, due to various
constraints, advanced laboratory tests are rarely  performed,
%
while in--situ 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,
elastic--perfectly  plastic  model  but extended into probability space, by
using number of in--situ tests. Such calibrated probabilistic elastic--perfectly
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 $NSF-CMMI-0600766$ (cognizant program director Dr. Richard
Fragaszy).

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

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