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%\usepackage{subeqn}
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\begin{document}
%
%\title{Probabilistic ElastoPlasticity:
%Formulation of Evolution Equation of Probability Density Function}
\title{Probabilistic ElastoPlasticity: Solution and Verification in 1D}
%\subtitle{Do you have a subtitle?\\ If so, write it here}
\author{
Kallol Sett\inst{1}
\and
Boris Jeremi{\'c}\inst{2}
\and
M. Levent Kavvas\inst{3}
}
\institute{
Graduate Research Assistant, Department of Civil and Environmental
Engineering, University of California, Davis, CA 95616.
\and
Associate Professor, Department of Civil and Environmental Engineering,
University of California, Davis, CA 95616. \texttt{jeremic@ucdavis.edu}
\and
Professor, Department of Civil and Environmental Engineering, University of
California, Davis, CA 95616.
}
% \author{First author\inst{1} \and Second author\inst{2}% etc
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%\\
%and
%Younyee Kuhn%
%\thanks{Flourishing wife of same.},%
%\ Not a Member, ASCE
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{abstract}
In this paper, a solution is presented for evolution of Probability Density
Function (PDF) of elasticplastic stressstrain relationship for material
models with uncertain parameters. Developments in this paper are based on
already derived general formulation presented in the companion paper. The
solution presented is then specialized to a specific Drucker Prager
elasticplastic material model.
Three numerical problems are used to illustrate the developed solution.
The stress strain response (1D) is given as a PDF of stress as a function of
strain. The presentation of the stress strain response through the PDF differs
significantly from the traditional presentation of such results, which are
represented by a single, unique curve in stressstrain space. In addition to
that the numerical solutions are verified against closed form solutions where
available (elastic). In cases where the closed form solution does not exist
(elasticplastic), MonteCarlo simulations are used for verification.
\end{abstract}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
\label{Introduction}
An elasticplastic constitutive law can be represented by a set of linear or
nonlinear ordinary differential equations (ODEs), which relate rate
(increments) of stress with the rate of strain
through linear or nonlinear material modulus:
%
\begin{equation}
\label{eqno_I1}
\frac{d \sigma_{ij} (t)}{dt}=D^{ep}_{ijkl} \frac{d \epsilon_{kl} (t)}{dt}
\end{equation}
%
where $D^{ep}_{ijkl}$ could be linear or a nonlinear function of stresses,
strains and internal variables. If either the material modulus or the forcing term
(strain rate) becomes random, this set of linear or nonlinear ODEs becomes a
set of linear or nonlinear stochastic differential equations (SDEs).
%
%These types of equations are called SDEs with
%random coefficient and SDEs with random forcing respectively.
%
The uncertainty
associated with the coefficient (stiffness) term is generally attributed to the
inherent variability of the material. Geomaterials are particularly notorious for their
variability, sampling and testing errors and in general, uncertainty in their
properties.
%
The uncertainty in the
forcing term arises when the material is subjected to uncertain loads (usually
dynamic) like wind,
waves or earthquakes. Due to randomness in the parameter and/or forcing term
the response variable of the elasticplastic constitutive rate equation (stress)
will then be a random process. There exist several methods to estimate the
probabilistic characteristics of the response variable \citep{book:Gardiner}.
For the case where the material modulus is linear and deterministic and the
forcing is uncertain (Gaussian), the response is known to be Gaussian and can be estimated
by standard methods \citep{book:Gardiner}. General linear SDE with random forcing
can be solved by cumulant expansion method \citep{VanKampen:1976}.
%
One possible
way to solve nonlinear SDE with random forcing is to write its equivalent
FokkerPlanckKolmogorov (FPK) form. The
advantage of writing the FPK form is that it is linear and deterministic eventhough the
original equation is nonlinear and stochastic.
%
The general
solution method for FPK equation can be found in any standard textbook
(\citet{book:Gardiner}, \citet{book:Risken}). A solution scheme for FPK
equation with
application to structural reliability was presented by \citet{Langtangen:1991}
For the particular case where the forcing is deterministic and the
material linear elastic (but still uncertain), Eq.~\ref{eqno_I1} simplifies
to a linear set of algebraic equations of the form,
%
\begin{equation}
\label{eqno_I2}
\sigma_{ij}=D^{el}_{ijkl} \epsilon_{kl}
\end{equation}
%
where $D^{el}_{ijkl}$ is the elastic stochastic moduli tensor and hence the statistics of
the response variable (stress) can be easily obtained by transformation method
of random variable. For general linear SDEs with random coefficients cumulant
expansion method could be used \citep{VanKampen:1976}.
%
While the solution for the stochastic linear elastic stressstrain problem
is readily available, the nonlinear (elasticplastic) stochastic problem
presents itself as much harder to solve. This problems involves finding the
solution for a nonlinear SDEs with random coefficients.
%
Fortunately such solution recently developed by \citet{Kavvas:2003}.
%
The developed solution is presented as a generic EulerianLagrangian form of the
FokkerPlanckKolmogorov equation. That probabilistic solution is second
order exact for any stochastic nonlinear ODE or PDE with random coefficients
and random forcing.
The probabilistic solution developed by \citet{Kavvas:2003} was used in the
companion paper \citet{Jeremic2005a} to develop the probability density function
(PDF) of a general localaverage form of elasticplastic constitutive rate
equation. This EulerianLagrangian FPK equation was then specialized to the
particular cases of pointlocation scale linear elastic and DruckerPrager
associative linear hardening elasticplastic constitutive rate equations to show
the applicability of the general formulation. In this paper the solution process
of those particular FPK equations will be presented.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{FokkerPlanckKolmogorov Equation for
Probabilistic Elasticity and ElastoPlasticity in 1D}
\label{FokkerPlanckEquations}
In the companion paper, \cite{Jeremic2005a} applied the EulerianLagrangian
form of
FokkerPlanckKolmogorov (FPK) equation to the description of the probabilistic
behavior of elastic and elasticplastic (DruckerPrager associative
linear hardening) 1D constitutive equations with random
material parameters and random strain rate.
By focusing attention to the
randomness of material properties only (i.e. assuming the forcing function
(strain rate) as deterministic), partial differential
equation (PDE) describing the evolution of probability density function (PDF) of stress
can be simplified. In particular, for 1D case, and for linear elastic material
(but still with probabilistic material properties, in this case shear modulus
$G$) one can write the following PDE
%(Eq. (23) of \cite{Jeremic2005a})
%
\begin{eqnarray}
\nonumber
\displaystyle \frac{\partial P(\sigma_{12}(t))}{\partial t}
=
&&
\left \displaystyle \frac{\partial P(\sigma_{12}(t))}{\partial \sigma_{12}}
\\ \nonumber
&+&
\left\{\int_{0}^{t} d \tau Cov_0
\left[G \displaystyle \frac{d\epsilon_{12}}{dt}; G
\displaystyle \frac{d \epsilon_{12}}{dt} \right] \right\}
\frac{\partial^2 P(\sigma_{12}(t))}{\partial \sigma_{12}^2}
\\
\label{eqno_FPE1}
\end{eqnarray}
Similarly, for elasticplastic state, again by neglecting the randomness in
strain rate, one can write the PDE for evolution of PDF of stress in 1D as
%zone for materials obeying DruckerPrager associative
%linear hardening type shear constitutive rate equations
%%(Eq. (35) of \cite{Jeremic2005a})
%
\begin{eqnarray}
& &
\frac{\partial P(\sigma_{12}(t))}{\partial t}
=
\left< \left(G^{ep}(t) \right) \frac{d\epsilon_{12}}{dt} \right> \
\displaystyle \frac{\partial P(\sigma_{12}(t))}
{\partial \sigma_{12}}
\nonumber \\
&+&
\left\{\int_{0}^{t} d \tau Cov_0
\left[
G^{ep}(t)
\frac{d\epsilon_{12}}{dt};
%\left(
G^{ep}(t\tau)
\displaystyle \frac{d\epsilon_{12}}{dt}
\right] \right \} \frac{\partial^2 P(\sigma_{12}(t))}{\partial \sigma_{12}^2}
\nonumber \\
\label{eqno_FPE2}
\end{eqnarray}
%
where $G^{ep}(a)$ is the probabilistic elasticplastic tangent stiffness,
(given
%(Eq. (34)
in \cite{Jeremic2005a})
%
\begin{equation}
G^{ep}(a)
=
G

\displaystyle
\frac{G^2}{G+9K\alpha^2+\displaystyle \frac{1}{\sqrt{3}}I_1(a)\alpha'}
\label{eqno_FPE2a}
\end{equation}
%
where in the previous equation (\ref{eqno_FPE2a}), $a$ assumes values $t$ or $t\tau$.
%It is important to note that the preyield, linear elastic
%probabilistic behavior of shear stress for this elasticplastic material
%is governed by Eq.~(\ref{eqno_FPE1}).
With appropriate initial and boundary
conditions as described in \cite{Jeremic2005a}, one can solve
Eqs.~(\ref{eqno_FPE1}) and ~(\ref{eqno_FPE2}) for evolution of PDF of shear
stress with shear strain.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Example Problem statements}
\label{ExampleProblemStatement}
The applicability of proposed FPK equations
(Eqs.~(\ref{eqno_FPE1}) and ~(\ref{eqno_FPE2})) in describing probabilistic
elastoplastic behavior, is verified using the following three example problems.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\bf Problem I.}
%
Assume the material is linear elastic, probabilistic, with probabilistic shear modulus ($G$)
given by a normal distribution at a pointlocation scale with mean of $2.5$~MPa
and standard deviation of $0.707$~MPa. The aim is to calculated the evolution of
PDF of shear stress ($\sigma_{12}$) with shear strain ($\epsilon_{12}$) for a
displacementcontrolled test with deterministic shear strain increment. The
other parameters are considered deterministic and are as follows: Poisson's
ratio ($\nu = 0.2$, and confining pressure $ I_1 =0.03$~MPa.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\bf Problem II.}
%
Assume elasticplastic material model, composed of linear elastic
component and DruckerPrager associative isotropic linear hardening
elasticplastic component. The probabilistic shear modulus ($G$) is given
through a normal distribution at a pointlocation scale with mean of $2.5$~MPa
and standard deviation of $0.707$~MPa. The aim is to calculate the evolution of the
PDF of shear stress ($\sigma_{12}$) with shear strain
($\epsilon_{12}$) for a displacementcontrolled test with deterministic shear
strain increment. The other parameters are considered deterministic and are as
follows: Poisson's ratio $ \nu = 0.2$, confining pressure $I_1 = 0.03$~MPa,
yield parameter\footnote{The yield parameter $\alpha$ is an internal variable
and is a function of friction angle $\phi$ given by ($\alpha = 2
\sin(\phi)/(\sqrt(3)(3sin \phi))$ (e.g. \citep{book:Chen})} $\alpha = 0.071$,
plastic slope\footnote{The plastic slope $\alpha'$ is a rate of change of friction angle
governing linear hardening.} $ \alpha' =5.5$.
{\bf Problem III.}
%
Assume elasticplastic material model, with linear elastic component and
DruckerPrager associative isotropic linear hardening elasticplastic
component. The probabilistic yield parameter ($\alpha$) is given through a
normal distribution at a
pointlocation scale with mean of $0.52$ and standard deviation of $0.1$.
The aim is to calculate the evolution of the PDF of shear stress ($\sigma_{12}$) with
shear strain ($\epsilon_{12}$) for a displacementcontrolled test with
deterministic shear strain increment. The other parameters are considered
deterministic and are as follows: shear modulus $G = 2.5$~MPa, Poisson's ratio
$\nu = 0.2$, confining pressure $I_1 =0.03$~MPa, and the plastic slope
$\alpha' =5.5$.
The above three problems will be solved using the proposed FPK
equation approach. In addition to that, the solution will verified using
either variable transformation method, for linear elastic case or
repetitive Monte Carlo type simulations for elasticplastic case.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Numerical Scheme for Solving FokkerPlanckKolmogorov Equation}
\label{NumericalScheme}
For probabilistic elastic and elasticplastic constitutive rate equations,
the PDEs (Eqs.~(\ref{eqno_FPE1}) and (\ref{eqno_FPE2})) which describe the
evolution of probability densities of $ \sigma_{12} $ have the following general
form:
%
\begin{equation}
\label{eqno_NS1}
\frac{\partial P}{\partial t}
=

N_{(1)}\frac{\partial P}{\partial \sigma_{12}}
+
N_{(2)}\frac{\partial^2 P}{\partial \sigma_{12}^2}
\end{equation}
%
with an appropriate initial condition, which depends on the type of
problem, and boundary conditions in the form
%
\begin{equation}
\label{eqno_NS2}
\zeta(\infty,t)=\zeta(\infty,t)=0
\end{equation}
%
where $N_{(1)}$ and $N_{(2)}$ are called advection and diffusion
coefficients, respectively. The above PDE system (Eqs.~(\ref{eqno_NS1}) and
~(\ref{eqno_NS2}) with appropriate initial condition) were solved numerically by
Method of Lines \cite{book:Mathematica} using commercially available software
Mathematica \cite{software:Mathematica}.
%In order to cone must first define the
%domain of the state variables.
The stress (state) variable $ \sigma_{12} $ theoretically spans space from $
\infty$ to $+ \infty$. However, for simulation
(and practical) purposes, this theoretical domain is reduced to between
$0.1$~MPa and $+0.1$~MPa. This reduction is based on the material properties of
the example problems and span the practical range of shear stress,
$\sigma_{12}$. The FokkerPlanckKolmogorov PDE was semidiscretized
(Fig.~\ref{figure:StressDiscretization}) in stress ($ \sigma_{12} $) domain by
finite difference technique to obtain a set of linear simultaneous ODE systems.
%
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.9\textwidth]{StressDiscretization.eps}
\caption{Stress Domain Discretization of FokkerPlanckKolmogorov PDE}
\label{figure:StressDiscretization}
\end{center}
\end{figure}
%
This set of linear simultaneous ODEs is solved using central difference
technique. By referring to Fig.~\ref{figure:StressDiscretization} a
semidiscretized form of Eq.~ (\ref{eqno_NS1}) can be written at any
intermediate node $ i $ as,
%
\begin{equation}
\frac{\partial P_i}{\partial t}
=
P_{i1} \left( \frac{N_{(1)}}{2 \Delta \sigma_{12}}
+
\frac{N_{(2)}}{\Delta \sigma_{12}^2}\right)

P_i \left( \frac{2N_{(2)}}{\Delta \sigma_{12}^2}\right)
+
P_{i+1} \left( \frac{N_{(1)}}{2 \Delta \sigma_{12}}
+
\frac{N_{(2)}}{\Delta \sigma_{12}^2}\right)
\label{eqno_NS3}
\end{equation}
%
Previous discretized system of equations forms an initial value problem in
the time dimension. By using forward
difference technique, one can introduce the boundary condition at the left
boundary (node 1 in Fig.~\ref{figure:StressDiscretization}) as,
%
\begin{equation}
\label{eqno_NS4}
N_{(1)}P_1N_{(2)}\frac{P_2P_1}{\Delta \sigma_{12}}=0
\end{equation}
%
or, after rearranging,
%
\begin{equation}
\label{eqno_NS5}
P_1
=
P_2\left( \displaystyle \frac{\displaystyle \frac{N_{(2)}}{\Delta \sigma_{12}}}{N_{(1)}
+
\displaystyle \frac{N_{(2)}}{\Delta \sigma_{12}}}\right)
=
0
\end{equation}
%
Similarly, using backward difference technique, one can introduce the boundary
condition at the right boundary (node $n$ in
Fig.~\ref{figure:StressDiscretization}) as,
%
\begin{equation}
\label{eqno_NS6}
N_{(1)}P_nN_{(2)}\frac{P_nP_{n1}}{\Delta \sigma_{12}}=0
\end{equation}
%
which, after rearranging becomes
%
\begin{equation}
\label{eqno_NS7}
P_n
=
P_{n1}\left( \displaystyle \frac{\displaystyle \frac{N_{(2)}}
{\Delta \sigma_{12}}}{\displaystyle \frac{N_{(2)}}{\Delta \sigma_{12}}

N_{(1)}}\right)
=
0
\end{equation}
The initial condition depends on the type of the problem and it could be
deterministic or random. For elastic constitutive rate equation with random
shear modulus (Problem I) and for preyield elasticplastic linear hardening
constitutive rate equation with random shear modulus (elastic part of Problem
II) the initial condition is deterministic. It will, therefor, be best
represented as Dirac delta function of the form,
%
\begin{equation}
\label{eqno_NS8}
P(\sigma_{12}) = \delta(\sigma_{12})
\end{equation}
For simulation purpose the Dirac delta initial condition was approximated with a
Gaussian function. That is, for Problem I, the initial condition was approximated with
a Gaussian function with mean of 0 and standard deviation of $0.00001$~MPa as
shown in Fig.~\ref{figure:ElasticInitialCondition}
%
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.9\textwidth]{ElasticInitCondview2}
\caption{Approximation of Dirac delta function, used as an initial condition
for Problem I}
\label{figure:ElasticInitialCondition}
\end{center}
\end{figure}
For postyield, probabilistic elasticplastic behavior (plastic part of
Problem II), the initial condition is random and it corresponds to the
probability density function of $\sigma_{12}$ just prior to yielding, that is
obtained from elastic part of Problem II. For ProblemIII, the preyield elastic
part is deterministic but initial condition for the postyield elasticplastic
response is random and corresponds to the assumed distribution in yield
strength.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Determination of Coefficients for FokkerPlanckKolmogorov Equation}
\label{CoefficientEstimation}
To solve Problems I, II, and III, the advection and
diffusion coefficients $ N_{(1)} $ and $ N_{(2)} $ must be determined for all
three problems. For sake of simplicity, a constant strain rate is assumed
and hence, terms containing $d \epsilon_{12} / dt$ in coefficients of Eqs.~ (\ref{eqno_FPE1}) and
~(\ref{eqno_FPE2}) can be substituted by a constant numerical value for the
entire simulation of the evolution of PDF. It should be noted that the
FPK equation
(Eqs.~(\ref{eqno_FPE1}) or ~(\ref{eqno_FPE2})) describes the evolution of PDFs
of stress with time, while, similarly, strain rate describes the evolution of strain with time.
Combining the two, the evolution of PDF of stress with strain can be obtained.
Time has been brought in this simulation as an intermediate dimension to help in
solution process, and hence, the numerical value of strain rate could be any
arbitrary value, which will cancel out once the time evolution of
PDF of stress is converted to strain evolution of PDF of stress. For simulation of all the
three example problems, an arbitrary value of strain rate of
$d \epsilon_{12} / dt = 0.054 1/s$ is assumed.
It should also be noted that since the material properties are
assumed as random
variables at a pointlocation scale, the covariance terms appearing within the
advection and diffusion coefficients become variances of random variables.
For estimations of means and variances of functions of random variables (e.g.
for Problems II and III) from basic random variables, commercially available
statistical software mathStatica \cite{book:Rose} was used.
Substituting the values of deterministic and random material properties and the
strain rate, coefficients $N_{(1)}$ and $N_{(2)}$ of the
FPK equations can be obtained for all problems:
%
\begin{description}
%
\item{Problem I}
%
\begin{eqnarray}
\nonumber
N_{(1)} &=& \left \\
\nonumber
&=& 2 \frac{d \epsilon_{12}}{dt} \left< G \right > \\
\nonumber
&=& 0.27 ~~\rm{MPa/s}
\end{eqnarray}
%
\begin{eqnarray}
\nonumber
N_{(2)} &=& \int_0^t d \tau Var\left[G \frac{d \epsilon_{12}}{dt}\right] \\
\nonumber
&=& 4 t \left(\frac{d \epsilon_{12}}{dt}\right)^2 Var[G] \\
\nonumber
&=& 0.0058 t ~~(\rm{MPa/s})^2
\end{eqnarray}
\item{Problem II}
For preyield linear elastic case, the coefficients $N_{(1)}$ and $N_{(2)}$
will be the same as those for Problem I. For postyield elasticplastic case the
coefficients are
\begin{eqnarray}
\nonumber
N_{(1)}
&=&
\left<\left(G\displaystyle
\frac{G^2}{G+9K\alpha^2+\displaystyle
\frac{1}{\sqrt{3}}I_1\alpha'}\right) \displaystyle
\frac{d \epsilon_{12}}{dt} \right>
\\ \nonumber
&=&
\displaystyle \frac{d \epsilon_{12}}{dt} \left
\\ \nonumber
&=&
0.147 ~~\rm{MPa/s}
\end{eqnarray}
%
\begin{eqnarray}
\nonumber
N_{(2)}
&=&
t \left(\displaystyle
\frac{d \epsilon_{12}}{dt}\right)^2 Var\left[G\displaystyle
\frac{G^2}{G+9K \alpha^2
+
\displaystyle \frac{1}{\sqrt{3}}I_1\alpha'}\right]
\\ \nonumber
&=&
0.00074t~~(\rm{MPa/s})^2
\end{eqnarray}
\item{Problem III}
%
For postyield elasticplastic simulation the coefficients $N_{(1)}$ and
$N_{(2)}$ are
%
\begin{eqnarray}
\nonumber
N_{(1)}
&=&
\left<\left(G\displaystyle \frac{G^2}{G+9K\alpha^2
+
\displaystyle \frac{1}{\sqrt{3}}I_1\alpha'}\right)
\displaystyle \frac{d \epsilon_{12}}{dt} \right>
\\ \nonumber
&=&
\displaystyle \frac{d \epsilon_{12}}{dt}
\left
\\ \nonumber
&=&
0.2365~~\rm{MPa/s}
\end{eqnarray}
%
\begin{eqnarray}
\nonumber
N_{(2)}
&=&
t \left(\displaystyle
\frac{d \epsilon_{12}}{dt}\right)^2 Var\left[G
\displaystyle \frac{G^2}{G+9K
\alpha^2+\displaystyle \frac{1}{\sqrt{3}}I_1\alpha'}\right]
\\ \nonumber
&=&
0.0001t~~(\rm{MPa/s})^2
\end{eqnarray}
\end{description}
It should be noted that for Problem III, since the shear modulus is deterministic, the
preyield elastic case is deterministic.
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%
%\input{Application2D}
%
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%
%\input{NumericalSchemeLinear}
%
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\section{Results and Verifications of Example Problems}
In this section results are presented for elastic and elasticplastic
probabilistic 1D problem. The results are obtained by using FPK equation
approach described in previous sections and in the companion paper
\citep{Jeremic2005a}. In addition to that, the Monte Carlo based verification of
developed solutions (results) is presented. The effort to verify developed
solutions (that are based on FPK approach) plays a crucial role in
presented development of probabilistic elastoplasticity as there are no
previously published solutions which could have been used for verification.
In addition to that, verification and validation efforts should always
be included in any modeling and simulations work \citep{Oberkampf2002}.
%
%
For linear elastic constitutive rate equations (ProblemI and preyield case
of ProblemII) the verification is performed by comparing solutions obtained
through the use of FPK equation approach with high accuracy (exact) solution,
using a transformation method of random variables \citep{book:Montgomery}. This
method is applicable as for rateindependent linear elastic case the 1D shear
constitutive equation simplify to a linear algebraic equation of the form,
%
\begin{equation}
\label{eqno_R1}
\sigma_{12}=G \epsilon_{12}=u(G,\epsilon_{12})
\end{equation}
%
Using the definition of strain rate, the above equation can be written
in terms of time $t$ as,
%
\begin{equation}
\label{eqno_R2}
\sigma_{12}=G (0.054 t)=v(G,t)
\end{equation}
%
where, $0.054$~1/s is the arbitrary strainrate assumed for this example
problem.
%
According to the transformation method of random variables
\citep{book:Montgomery}, and, given the continuous random variable (shear modulus)
$G$, with PDF $g(G)$ and Eqs.~(\ref{eqno_R1}) or ~(\ref{eqno_R2}) as
onetoone transformations between the values of random variables of $G$ and
$\sigma_{12}$, one can obtain the PDF of shear stress ($\sigma_{12}$),
$P(\sigma_{12})$ as,
%
\begin{equation}
\label{eqno_R3}
P(\sigma_{12})
=
g(u^{1}(\sigma_{12},\epsilon_{12}))\leftJ\right
\end{equation}
%
which will allow for predicting the evolution of PDF of $\sigma_{12}$ with
$\epsilon_{12}$ or,
%
\begin{equation}
\label{eqno_R4}
P(\sigma_{12})
=
g(v^{1}(\sigma_{12},t))\leftJ\right
\end{equation}
Eq.~(\ref{eqno_R4}) will predict the evolution of PDF of $\sigma_{12}$ with $t$.
In Eqs.~(\ref{eqno_R3}) and ~(\ref{eqno_R4}), functions
$G=u^{1}(\sigma_{12},\epsilon_{12})$ or $G=u^{1}(\sigma_{12},t)$ are the
inverse of functions
$\sigma_{12}=u(G,\epsilon_{12})$ or $\sigma_{12}=v(G,t)$ respectively and
$J=du^{1}(\sigma_{12},\epsilon_{12})/d\sigma_{12}$ and
$J=dv^{1}(\sigma_{12},t)/d\sigma_{12}$ are their respective Jacobians of
transformations.
For nonlinear elasticplastic constitutive rate equations (postyield cases of
Problems II and III) the verification is done using MonteCarlo simulation
technique by generating sample data for material properties from standard normal
distribution and by repeating solution of the deterministic elasticplastic constitutive
rate equation for each data generated above. The probabilistic
characteristics of resulting random stress variable for each time (or strain)
step are then easily computed. A relatively large number of data points
(1,000,000) were generated for each material constant random
variable for this simulation purpose.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Problem I}
The evolution of PDF of shear stress with time and shear strain is shown
in Figures~\ref{figure:ElasticPDF} and ~\ref{figure:ActualElasticPDF}.
%
Presented PDFs are for linear elastic material with random shear modulus, and
were obtained using \textit{FPE approach} (Fig.~~ref{figure:ElasticPDF}) and
\textit{transformation method} (Fig.~\ref{figure:ActualElasticPDF}).
%
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=\textwidth]{ElasticPDFm}
\caption{Evolution of PDF of shear stress versus strain (or time) for
linear elastic material model with random shear modulus (Problem I) obtained
using FPK equation approach.}
\label{figure:ElasticPDF}
\end{center}
\end{figure}
%
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=\textwidth]{ActualElasticPDFm}
\caption{Evolution of PDF of shear stress versus strain (or time) for linear
elastic material model with random shear modulus (Problem I) obtained using
transformation method.}
\label{figure:ActualElasticPDF}
\end{center}
\end{figure}
The contours of evolution of PDFs are compared in
Fig.~\ref{figure:ComparisonContourElastic_Actual_FPE}.
%
Similarly, comparison of the evolution of mean
and standard deviations are shown in
Fig.~\ref{figure:ComparisonElasticMeanStdDeviation_Actual_FPE}.
%
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=\textwidth]{ComparisonContourElastic_Actual_FPEm}
\caption{Comparison of Contours of Time (or Strain) Evolution of Probability Density Function for Shear Stress for Elastic Constitutive Rate Equation
with Random Shear Modulus (ProblemI) for FPE Solution and Variable Transformation Method Solution.}
\label{figure:ComparisonContourElastic_Actual_FPE}
\end{center}
\end{figure}
%
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=\textwidth]{ComparisonElasticMeanStdDeviation_Actual_FPEm}
\caption{Comparison of Mean and Standard Deviation of Shear Stress for Elastic Constitutive Rate Equation
with Random Shear Modulus (ProblemI) for FPE Solution and Variable Transformation Method Solution.}
\label{figure:ComparisonElasticMeanStdDeviation_Actual_FPE}
\end{center}
\end{figure}
%
It can be seen
from the comparison figure that eventhough the FPK approach predicted the mean behavior
exactly, it slightly overpredicted the standard deviation. This is because of
the approximation used to represent the Dirac delta function, which was used as
the initial condition for the FPK.
%
One may note that at $\epsilon_{12}=0$, the
probability of shear stress $\sigma_{12}$ should theoretically
be $1$ i.e. all the probability mass should theoretically be concentrated
at $\sigma_{12}=0$. As such, it would be best described by the Dirac delta function.
%
However,
for numerical simulation of FPK, Dirac delta function as initial condition was
approximated with a Gaussian function of mean zero and standard deviation of
$0.00001$~MPa, as shown in Fig.~\ref{figure:ElasticInitialCondition}.
%
This error in the initial condition advected and diffused into the domain with the
simulation of the evolution process. This error could be minimized by better
approximating the Dirac delta initial condition (but at higher
computational cost).
%
The effect of approximating the initial condition of the
PDF of shear stress at $\epsilon_{12}=0.0426~\%$ is shown in
Fig.~\ref{figure:CrudeComparisonAtYield_Actual_FPE}.
%
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=\textwidth]{CrudeComparisonAtYield_Actual_FPEm}
\caption{Effect of Approximating Function of Dirac Delta Initial Condition : PDF of Stress at Yield for Different Approximation of
Initial Condition with Actual (Variable Transformation Method) Solution).}
\label{figure:CrudeComparisonAtYield_Actual_FPE}
\end{center}
\end{figure}
%
In this figure the actual
PDF at $\epsilon_{12}=0.0426~\%$ obtained using the transformation method was
compared with the PDFs at $\epsilon_{12}=0.0426~\%$ obtained using the FPK
approach with three different approximate initial conditions  all having zero
mean but standard deviations of $0.01$~MPa, $0.005$~MPa and $0.00001$~MPa.
One may also note that finer approximation of initial condition necessitates
finer discretization of stress domain close to (or at) $\sigma_{12}$ =0. The finite
difference discretization scheme adopted here uses the same fine discretization
uniformly all throughout the entire domain. It is noted that that fine,
uniform discretization is not needed (and is quite expensive) in later stages of
calculation of evolution of PDF, but is kept the same for simplicity sake.
%
In presented examples, to properly capture the approximate initial
condition (as shown in Fig.~\ref{figure:ElasticInitialCondition}),
the stress domain between $0.1$~MPa and $+0.1$~MPa was discretized with a
uniform step size of $0.000005$~MPa and hence there is a total of $40,000$~nodes.
%
This not only requires large computational effort
but is also very memory sensitive.
%
An adaptive discretization technique will be a
much better approach to solving this problem.
Current work is going on in formulating an
adaptive algorithm for the solution of this type of problem.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Problem II}
The solution to this problem involves the solving two FPK equations, one corresponding to
the preyield elastic part and the other corresponding to the postyield
elasticplastic part.
%
%
The elastic part of this problem is identical to
ProblemI. The initial condition for the postyield elasticplastic part of the
problem is random and is shown in Fig.~\ref{figure:PlasticInitialCondition}.
%
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=\textwidth]{PlasticInitCondm}
\caption{Initial condition for FPK equation for elasticplastic zone (ProblemII).}
\label{figure:PlasticInitialCondition}
\end{center}
\end{figure}
%
It may be noted that this initial condition corresponds to the PDF of shear
stress ($P(\sigma_{12}$)) at yield obtained from the solution of FPK equation of
the preyield elastic part.
%
A view of the surface of evolution of the PDF of shear stress versus shear strain
(time) is shown in Fig.~\ref{figure:ElasticPlasticPDFView1}. Another view to
the PDF of stressstrain surface is shown in
Fig.~\ref{figure:ElasticPlasticPDFView2}.
%
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=\textwidth]{ElasticPlasticPDFm}
\caption{Evolution of PDF of shear stress versus strain (time) for elasticplastic material with
random shear modulus (ProblemII). View 1.}
\label{figure:ElasticPlasticPDFView1}
\end{center}
\end{figure}
%
%
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=\textwidth]{ElasticPlasticPDFView2m}
\caption{Evolution of PDF of shear stress versus strain (time) for
elasticplastic material with random shear modulus (ProblemII). View 2.}
\label{figure:ElasticPlasticPDFView2}
\end{center}
\end{figure}
%
It is noted that the yielding of this material occurred at $t$=0.00789
second (which is equivalent to $\epsilon_{12}$= 0.0426 $\%$).
%
The evolution contours for PDF of shear stress versus strain (time) along with
the mean and standard deviations are shown in
Fig.~\ref{figure:ContourElasticPlasticPDFDetailed}.
%
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=\textwidth]{ContourElasticPlasticPDFDetailedm}
\caption{Contour of evolution of PDF for shear stress versus strain (time) for elasticplastic material
with random shear modulus (ProblemII).}
\label{figure:ContourElasticPlasticPDFDetailed}
\end{center}
\end{figure}
%
It can be seen from that figure that, as expected, the evolution of mean of
shear stress changes slope after the material yielded. Another interesting
aspect to note is the relative slope of the evolution of standard deviation with
respect to the evolution of mean. The relative slope in the preyield elastic
zone increases at a higher rate during the evolution process when compared with
that in the postyield elasticplastic zone.
%
In other words, in the evolution process the postyield elasticplastic
constitutive rate equation did not amplify the initial uncertainty as much as
the preyield elastic constitutive rate equation did. This can be easily viewed
from Fig.~\ref{figure:ComparisonPDFExtendedElasticElasticPlastic} where the
postyield elasticplastic evolution of PDF of shear stress was compared with
fictitious extension of elastic evolution of PDF.
%
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=\textwidth]{Plastic_ExtendedElastic_PDFComparisonm}
\caption{Comparison of evolution of PDF for elasticplastic material and
extended elastic material cases for random shear modulus.}
\label{figure:ComparisonPDFExtendedElasticElasticPlastic}
\end{center}
\end{figure}
%
Comparing the PDF of shear stress at $\epsilon_{12}=0.0804\%$ (which is
equivalent to $t=0.01489 s$), one can conclude that the variance of predicted
elasticplastic shear stress is much smaller (i.e. prediction is less uncertain)
as compared to the same if the material were modeled as completely elastic.
Fig.~\ref{figure:ComparisonElasticPlasticMeanStdDeviation_Actual_FPE} compares
the evolution of means and standard deviations of predicted shear stress
obtained using FPK equation approach and transformation method (preyield
behavior)/MonteCarlo approach (postyield behavior).
%
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=\textwidth]{ComparisonElasticPlasticMeanStdDeviation_Actual_FPEm}
\caption{Comparison of mean and standard deviation of shear stress for plastic constitutive rate equation
with random shear modulus (problemii) for FPK equation solution and Monte Carlo simulation solution.}
\label{figure:ComparisonElasticPlasticMeanStdDeviation_Actual_FPE}
\end{center}
\end{figure}
%
Although in the preyield
response the FPK equation approach overpredicted the evolution of standard deviations
because of reasons discussed earlier, in the postyield response it matched
closely at regions further from the yielding region. The somewhat larger
difference between FPK equation solution and the verification one (Monte Carlo solution) close to
the yielding region is attributed to the fact that the initial condition for solution
of postyield elasticplastic FPK equation was obtained from the solution of preyield
elastic FPK equation.
%
One way to better predict the overall probabilistic elasticplastic behavior,
would probably be to obtain the preyield elastic behavior through the
transformation method and then use the FPK approach to predict postyield
elasticplastic behavior.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Problem III}
In this problem, the preyield linear elastic part is deterministic, however, at yield
there is a distribution (with very small standard deviation) in shear stress due to assumed distribution in yield
parameter $\alpha$. The distribution in shear stress corresponds to the PDF of
the random variable $\alpha I_1$ (first invariant of the
stress tensor or mean confining stress) and is assumed to be
deterministic. This PDF of shear stress at yield was assumed to be the initial
condition for the solution of postyield elasticplastic FPK equation and is shown in
Fig.~\ref{figure:RandomYieldInitialCondition}.
%
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=\textwidth]{RandomYieldInitialConditionm}
\caption{Initial condition for FPK equation for elasticplastic material with random yield strength (ProblemIII).}
\label{figure:RandomYieldInitialCondition}
\end{center}
\end{figure}
The evolution of PDF for shear stress versus strain (time) is shown in
Fig.~\ref{figure:RandomYieldPlasticPDFView1}. In addition to that the contours (including mean and standard deviation) of the evolution of
PDF for shear stress versus strain (time) are shown in
Fig.~\ref{figure:RandomYieldContourPlasticPDFDetailed}.
%
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=\textwidth]{RandomYieldPlasticPDFm}
\caption{Evolution of PDF for shear stress for elasticplastic material with
random yield strength (ProblemIII) (only plastic zone is shown).}
\label{figure:RandomYieldPlasticPDFView1}
\end{center}
\end{figure}
%
%
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=\textwidth]{RandomYieldContourPlasticPDFDetailedm}
\caption{Contour PDF for shear stress versus strain (time) for
elasticplastic material with random yield strength (ProblemIII).}
\label{figure:RandomYieldContourPlasticPDFDetailed}
\end{center}
\end{figure}
Looking at Fig.~\ref{figure:RandomYieldContourPlasticPDFDetailed} and
comparing the slopes of evolution of mean and standard deviation, one can
conclude that the elasticplastic evolution process didn't amplify the initial
uncertainty in yield strength significantly. The initial (at yield) probability
density function of shear stress just advected into the domain during the
elasticplastic evolution process without diffusing much.
Fig.~\ref{figure:RandomYieldPlasticPDFView1} clearly shows this advection
process. The evolution of mean and standard deviations of shear stress obtained
from the FPK equation approach was compared with those obtained from the Monte
Carlo simulation and is shown in
Fig.~\ref{figure:ComparisonMeanStdDeviation_Actual_FPE}.
%
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=\textwidth]{RandomYieldComparisonElasticMeanStdDeviation_Actual_FPEm}
\caption{Comparison of Mean and Standard Deviation of Shear Stress for ElasticPlastic Constitutive Rate Equation
with Random Yield Strength (ProblemIII) for FPE Solution and Monte Carlo Simulation Solution}
\label{figure:ComparisonMeanStdDeviation_Actual_FPE}
\end{center}
\end{figure}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusions}
In this paper a solution was presented for the evolution of the Probability
Density Function (PDF) of elasticplastic stressstrain relationship, in 1D.
The solution was based on expressions developed in a companion paper and
specialized to the Drucker Prager elastoplastic material model with linear
isotropic hardening.
Three numerical problems were used to illustrate the developed solution and
discuss the general behavior of elasticplastic materials which exhibit
uncertainty in material parameters. The solutions to numerical problems were
verified against closed, analytical forms, where available, while MonteCarlo
simulations were used for all other verifications.
The approach to solving probabilistic elasticplastic problems presented here is
quite unique and shows great promise in dealing with general 3D probabilistic
constitutive problems. Subsequently, the developed methodology is to be used in
solving general, probabilistic elasticplastic boundary value problems using
the finite element method. Current work is progressing in that direction.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\section*{Acknowledgment}
The work presented in this paper was supported in part by a number of Agencies listed below:
%
Civil, Mechanical and Manufacturing Innovation program, Directorate of Engineering of the National
Science Foundation, under Award NSFCMMI0600766 (cognizant program director Dr.
Richard Fragaszy);
%
Civil and Mechanical System program, Directorate of Engineering of the National
Science Foundation, under Award NSFCMS0337811 (cognizant program director Dr.
Steve McCabe);
%
Earthquake Engineering Research Centers Program of the National Science
Foundation under Award Number NSFEEC9701568 (cognizant program director Dr. Joy
Pauschke);
%
California Department of Transportation (Caltrans) under Award \# 59A0433,
(cognizant program director Dr. Saad ElAzazy)
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\end{document}
\bye