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\begin{document}
\CNM{1}{6}{00}{28}{00}
\runningheads{Jeremi{\' c} and Sett}
{On Probabilistic Yielding of Materials}
\title{On Probabilistic Yielding of Materials}
\author{Boris Jeremi{\'c} and Kallol Sett}
\address{Department of Civil and Environmental Engineering, University of
California, Davis, CA, 95616, U.S.A.}
\corraddr{Boris Jeremi{\'c}, Department of Civil and Environmental Engineering, One Shields
Ave., University of California, Davis, CA, 95616, U.S.A. \texttt{jeremic@ucdavis.edu}}
\cgsn{NSF}{CMMI0600766}
\noreceived{}
\norevised{}
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\begin{abstract}
Uncertainty in material properties can have large effect on numerical modeling
of solids and structures. This is particularly true as all natural and man made
material exhibit spatial nonuniformity and pointwise uncertainty in material
behavior.
Presented is the methodology that accounts for probabilistic yielding of
elasticplastic materials. The recently developed EulerianLagrangian form of
FokkerPlanckKolmogorov equation is used to
obtain second order exact solution to elasticplastic constitutive differential
equations. In this paper that solution is used in deriving the weighted
probabilities of elastic, elasticplastic behavior and yielding. A number of
examples for two commonly used material models, von Mises and DruckerPrager,
illustrated findings.
\end{abstract}
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\section{INTRODUCTION}
Elasticplastic computations have so far been almost exclusively done in a
deterministic fashion. This was and is still done, despite inherent uncertainty
in material behavior. In particular, the elasticplastic (inelastic) behavior
of solids and structure is modeled using material parameters calibrated from a
number of tests. Proper testing procedure would require a (statistically
consistent) large number of tests, from which would then appropriate material
parameters be calibrated. However, the ostensible claims of good economy usually
dictate fewer tests, resulting in limited number of test results. Current state
of the art is that those (few) test results are used to calibrate material
parameters in a deterministic fashion, usually using mean of those few tests.
However, mechanical behavior of all engineering materials is inherently
uncertain. The uncertain response follows from either spatial nonuniformity of
material distribution or from inherent uncertainty of material behavior at the
constitutive level. The uncertainty of material behavior propagates through
numerical simulations of solids and structure. Deterministic simulations that
are currently almost exclusively performed rely on safety factors, trying to
take into account (material and modeling) uncertainties that were neglected.
%Use
%of (large) factors of safety does not always lead to safer structures (Duncan
%\cite{Duncan:2000}).
Recently, a number of methods were developed to deal with
simulating solids and structures made of materials with random fields of
material properties (Ghanem and Spanos \cite{book:Ghanem}, Matthies and Keese
\cite{Matthies2005}, Roberts and Spanos \cite{Roberts1986}, Zhu et al,
\cite{Zhu1994}, Soize \cite{Soize1994}). It should be noted that all of the
previously mentioned references deal exclusively with elastic materials.
Behavior of elasticplastic materials with uncertain properties has not
received much attention.
One of the earliest approaches to propagating randomness through the
elasticplastic constitutive equations (random Young's modulus) was presented
by Anders and Hori \cite{Hori:1999}. They based their approach on perturbation
expansion at the stochastic mean behavior and took advantage of bounding media
analysis in computing the mean response. However, because of the use of Taylor
series expansion in developments, this approach is limited to problems with
small coefficients of variation (Sudret and Der Kiureghian
\cite{DerKiureghian:2000ReportSFEM}). Another disadvantage of the perturbation
method is that it inherits the socalled "closure problem" (cf. Kavvas
\cite{Kavvas:2003}), where information on higherorder moments is always needed
to calculate lowerorder moments.
%
Similarly, Kaminski \cite{Kaminski:1999} used MonteCarlo
simulations with perturbation approach of stochastic finite element method
for stochastic porous plasticity in solids.
%
More recently, Fenton and Griffiths
\cite{Griffiths:2005} used a MonteCarlo method to propagate uncertainties
through elasticplastic $c\phi$ soil.
%
MonteCarlo approach to accounting for uncertainties might be computationally
expensive for elasticplastic simulations. The method requires a statistically
appropriate number of realizations per random variable in
order to satisfy statistical accuracy. The MonteCarlo simulation approach,
however, finds a great use in verification of analytical developments (cf.
Oberkampf et al, \cite{Oberkampf2002}).
%
A solution to overcoming drawbacks of MonteCarlo
technique and perturbation method is the use of general EulerianLagrangian form
of FokkerPlanckKolmogorov equation (FPKE) for the secondorder exact
probabilistic solution of nonlinear ODE with stochastic coefficient and
stochastic forcing (Kavvas \cite{Kavvas:2003}). Using the above mentioned
EulerianLagrangian form of FPKE, Jeremi{\'c} et al. \cite{Jeremic:2007} and
Sett et al. \cite{Sett:2007b,Sett:2007a} recently developed formulation and solution for
the general 1D elasticplastic constitutive equation with random material
properties and random strain rate.
One of the main advantages of FPKE based approach is that it
provides a second order accurate probability density function (PDF) of stress
(exact mean and variance) for given random material properties and/or random
strain. In addition to that, for cases (material models) where FPKE is not
solvable in closedform solution, the deterministic linearity of the FPKE (with
respect to the probability density of stress) considerably simplifies the
numerical solution process.
In the above mentioned FPKEbased probabilistic elastoplasticity papers
(Jeremi{\'c} et al. \cite{Jeremic:2007} and Sett et al. \cite{Sett:2007b,
Sett:2007a}) the FPKE was solved separately (twice) for (probabilistic) elastic
and then for (probabilistic) elasticplastic phase of loading. The mean of
yielding stress was used
to make the separation, to decide on when the yielding
occurs. However, since the material behavior is probabilistic, so should be the
point (or rather region) of yielding.
%
%
In this paper we derive the weighted
probabilities of elastic or elasticplastic behavior. These weighted
probabilities are used to develop probabilistic elasticplastic response of
materials, that is based on probabilistic rather then mean yielding.
Probabilistic yielding examples using von Mises and DruckerPrager material
models are used to illustrate developed methodology.
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\section{FPKEbased Probabilistic ElastoPlasticity}
\label{MasterKey}
The incremental form of spatialaverage 3D elasticplastic
constitutive rate equation can be written as:
%
\begin{equation}
\label{eqno_S2}
\frac{d\sigma_{ij}(x_t, t)}{dt}
=
D^{ep}_{ijkl}(\sigma_{ij}, D^{el}_{ijkl}, f, U, q_*, r_*;x_t,t)
\frac{d\epsilon_{kl}(x_t,t)}{dt}
\end{equation}
%
%
where, $D_{ijkl}^{ep}$ is the {\it random, nonlinear elasticplastic
coefficient tensor} which is a function of random stress tensor ($ \sigma_{ij}
$), random elastic moduli tensor ($D^{el}_{ijkl}$), random yield function ($f$),
random plastic potential function ($U$), random internal variables ($q_*$) and
random direction of evolution of internal variables ($r_*$). The random internal
variables ($q_*$) could be scalar (for perfectly plastic and isotropic hardening
models), or secondorder tensor (for translational and rotational kinematic
hardening models), or fourthorder tensor (for distortional hardening models) or
any combinations of the above. The same classification applies to the random
direction of evolution of internal variables ($r_*$). By denoting all the random
material parameters by a parameter tensor $D_{ijkl} = \left[D^{el}_{ijkl}, f, U,
q_*, r_*\right]$ and by introducing a random operator tensor, $\eta_{ij}$, one
can write Eq.~(\ref{eqno_S2}) and initial conditions as,
%
\begin{equation}
\label{eqno_S13}
\frac{d\sigma_{ij}(x,t)}{dt}
=
\eta_{ij}(\sigma_{ij},D_{ijkl}, \epsilon_{kl};x,t)
\;\;\;\;\; \mbox{;} \;\;\;\;\;
\sigma_{ij}(x,0)=\sigma_{{ij}_0}
\end{equation}
%
%
%In Eq.~(\ref{eqno_S13}), the stress tensor $\sigma_{ij}$ can be considered to
%represent a point in a $ 9 $dimensional stress ($\sigma$)space.
% and hence
% Eq.~(\ref{eqno_S13}) determines the velocity for the point in this space.
% It is
% possible to visualize this, by imagining an initial point in stress space, given
% by its initial condition $\sigma_{{ij}_0}$, with a trajectory starting out that
% describes the corresponding solution of the nonlinear stochastic ordinary
% differential equation (ODE) system (Eq.~(\ref{eqno_S13})).
% %
% Now, when one considers a cloud of initial points (described by the density
% $\rho(\sigma_{ij},0)$ in the $\sigma$space), with movements dictated by
% Eq.~(\ref{eqno_S13}), the phase density $\rho$ of $\sigma_{ij}(x,t)$ varies in
% time according stochastic Liouville equation (cf. Kubo \cite{Kubo:1963}), which
% is basically a continuity equation that expresses the conservation of all the
% points in the $\sigma$space.
% %
% % Expressing this continuity equation in mathematical terms, one obtains Kubo's
% % stochastic Liouville equation (Kubo \cite{Kubo:1963}):
% % %
% % \begin{equation}
% % \label{eqno_S15}
% % \displaystyle \frac{\partial \rho (\sigma_{ij}(x,t),t)}{\partial t}
% =
% \displaystyle
% \frac{\partial}{\partial \sigma_{mn}} \eta_{mn}
% \left(\sigma_{mn}(x,t), D_{mnpq}(x),\epsilon_{pq}(x,t)\right)
% \rho(\sigma_{ij}(x,t),t)
% \end{equation}
% %
% with initial condition,
% %
% \begin{equation}
% \label{eqno_S16}
% \rho(\sigma_{ij},0)=\delta(\sigma_{ij}\sigma_{{ij}_0})
% \end{equation}
%
%
%
%
%
% \noindent where $\delta(\cdot)$ is the Dirac delta function.
% Eq.~(\ref{eqno_S16}) is the probabilistic restatement in the $\sigma$phase
% space of the original deterministic initial condition (Eq.~(\ref{eqno_S14})).
% One can then apply Van Kampen's Lemma (Van Kampen \cite{VanKampen:1976}) to
% obtain:
% %
% \begin{equation}
% \label{eqno_S17}
% <\rho(\sigma_{ij},t)>=P(\sigma_{ij},t)
% \end{equation}
% %
% where, the symbol $<\cdot>$ denotes the expectation operation, and
% $P(\sigma_{ij},t)$ denotes evolutionary probability density of the state
% variable tensor $\sigma_{ij}$ from the constitutive equations.
%
%
% Therefore, in order to obtain the multivariate probability density function
% (PDF), $P(\sigma_{ij},t)$, of the state variable tensor $\sigma_{ij}$, it is
% necessary to obtain the deterministic partial differential equation (PDE) of the
% $\sigma$space mean phase density $ < \rho (\sigma_{ij},t)> $ from the linear
% stochastic PDE system (Eqs.~(\ref{eqno_S15}) and ~(\ref{eqno_S16})). This
% necessitates the derivation of the ensemble average form of Eq.~(\ref{eqno_S15})
% for $ <\rho (\sigma_{ij},t)> $. The ensemble average form of
% Eq.~(\ref{eqno_S15}) was derived by Kavvas \cite{Kavvas:2003} as follows:
% %
% \begin{eqnarray}
% \nonumber
% \lefteqn{\frac{\partial \left<\rho (\sigma_{ij}(x_t,t), t)\right>}{\partial t} = \displaystyle \frac{\partial}{\partial \sigma_{mn}}
% \left[ \left\{\left< \vphantom{\int_{0}^{t}} \eta_{mn}(\sigma_{mn}(x_t,t), D_{mnrs}(x_t), \epsilon_{rs}(x_t,t))\right> \right. \right.}
% \\
% \nonumber
% && \left. \left. \int_{0}^{t} d\tau Cov_0\left[\vphantom{\frac{\partial}{\partial}}\eta_{mn}(\sigma_{mn}(x_t,t),D_{mnrs}(x_t),
% \epsilon_{rs}(x_t,t)); \right. \right. \right. \\
% \nonumber
% & & \left. \left. \left. \displaystyle \frac{\partial \eta_{ab} (\sigma_{ab}(x_{t\tau}, t\tau), D_{abcd}(x_{t\tau}),
% \epsilon_{cd}(x_{t\tau}, t\tau)}{\partial \sigma_{ab}}\right]\right\}\left<\rho (\sigma_{ij}(x_t,t), t)\right>\right] \\
% \nonumber
% &+& \displaystyle \frac{\partial}{\partial \sigma_{mn}}\left[ \left\{\int_{0}^{t} d\tau Cov_0\left[\eta_{mn}(\sigma_{mn}(x_t,t),
% D_{mnrs}(x_t), \epsilon_{rs}(x_t,t)); \right. \right. \right. \\
% \label{eqno_S18}
% & & \left. \left. \left. \eta_{ab} (\sigma_{ab}(x_{t\tau}, t\tau), D_{abcd}(x_{t\tau}), \epsilon_{cd}(x_{t\tau}, t\tau)) \right]
% \vphantom{\int_{0}^{t}} \right\} \displaystyle \frac{\partial \left<\rho (\sigma_{ij}(x_t,t), t)\right>}{\partial \sigma_{ab}} \right ]
% \end{eqnarray}
% %
% to exact second order (to the order of the covariance time of $ \eta $). In
% Eq.~(\ref{eqno_S18}), $ Cov_0[\cdot] $ is the time ordered covariance function
% defined by,
% %
% \begin{equation}
% \label{eqno_S19}
% Cov_0\left[\eta_{mn}(x,t_1), \eta_{ab}(x,t_2)\right]
% =
% \left<\eta_{mn}(x,t_1) \eta_{ab}(x,t_2)\right>\left<\eta_{mn}(x,t_1)\right>
% \cdot \left<\eta_{ab}(x,t_2)\right>
% \end{equation}
% %
%
% Using Van Kampen's Lemma (cf. Van Kampen \cite{VanKampen:1976}), which states
% that the ensemble average of phase density is the probability density, and after
Using developments described in more details in Jeremi{\'c} et al.
\cite{Jeremic:2007} and Sett et al. \cite{Sett:2007b}, it can be shown that the
probability density function (PDF) of stress $P(\sigma_{ij}(x_t,t), t)$, obeying
Eq.~(\ref{eqno_S13}), is governed by the EulerianLagrangian form of
FokkerPlanckKolmogorov equation of the form:
%
\begin{eqnarray}
\nonumber
\lefteqn{\displaystyle \frac{\partial P(\sigma_{ij}(x_t,t), t)}{\partial t} = \displaystyle \frac{\partial}{\partial \sigma_{mn}}
\left[ \left\{\left< \vphantom{\int_{0}^{t}} \eta_{mn}(\sigma_{mn}(x_t,t), D_{mnrs}(x_t), \epsilon_{rs}(x_t,t))\right> \right. \right.} \\
\nonumber
&+& \left. \left. \int_{0}^{t} d\tau Cov_0 \left[\displaystyle \frac{\partial \eta_{mn}(\sigma_{mn}(x_t,t), D_{mnrs}(x_t),
\epsilon_{rs}(x_t,t))} {\partial \sigma_{ab}}; \right. \right. \right. \\
\nonumber
& & \left. \left. \left. \eta_{ab} (\sigma_{ab}(x_{t\tau}, t\tau), D_{abcd}(x_{t\tau}), \epsilon_{cd}(x_{t\tau}, t\tau)
\vphantom{\int_{0}^{t}} \right] \right \} P(\sigma_{ij}(x_t,t),t) \right] \\
\nonumber
&+& \displaystyle \frac{\partial^2}{\partial \sigma_{mn} \partial \sigma_{ab}} \left[ \left\{ \int_{0}^{t} d\tau Cov_0 \left[
\vphantom{\int_{0}^{t}} \eta_{mn}(\sigma_{mn}(x_t,t), D_{mnrs}(x_t), \epsilon_{rs}(x_t,t)); \right. \right. \right. \\
& & \left. \left. \left. \eta_{ab} (\sigma_{ab}(x_{t\tau}, t\tau), D_{abcd}(x_{t\tau}), \epsilon_{cd}(x_{t\tau}, t\tau))
\vphantom{\int_{0}^{t}} \right] \vphantom{\int_{0}^{t}} \right\} P(\sigma_{ij}(x_t,t),t) \right]
\label{eqno_S20}
\end{eqnarray}
%
The above equation is second order accurate (mean value and variance) in
probability density of stress state $P(\sigma_{ij},t)$. The solution of
this deterministic linear FPKE (Eq.~(\ref{eqno_S20})) in terms of
$P(\sigma_{ij},t)$ under appropriate initial and boundary conditions will yield
the PDF of the state variable tensor $ \sigma_{ij} $. It is important to note
that while the original equation (Eq.~(\ref{eqno_S2})) is nonlinear, the FPKE
(Eq.~(\ref{eqno_S20})) is linear in terms of its unknown, the probability
density $ P(\sigma_{ij},t) $ of the state variable tensor $ \sigma_{ij} $. This
linearity, in turn, provides significant advantages in probabilistic solution of
the constitutive rate equation.
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\subsection{Specialization to 1D, von Mises and DruckerPrager Hardening Material Models}
The general, three dimensional PDF for stress (Eq.~(\ref{eqno_S20})) is
specialized to pointlocation scale (where the uncertain material parameters are
random variables) one dimensional case of shearing rate
equations ($d \sigma_{xy}/dt = G d\epsilon_{xy}/dt$) and is written for both
linear elastic (using elastic $G$) and for elasticplastic case (using
elasticplastic $G$):
%
%
%
\begin{equation}
\displaystyle \frac{\partial P(\sigma_{xy}(t))}{\partial t}
= 
\left
\displaystyle \frac{\partial P(\sigma_{xy}(t))}{\partial \sigma_{xy}}
+
\left\{\int_{0}^{t} d \tau Cov_0
\left[G \displaystyle \frac{d\epsilon_{xy}}{dt}; G
\displaystyle \frac{d \epsilon_{xy}}{dt} \right] \right\}
\frac{\partial^2 P(\sigma_{xy}(t))}{\partial \sigma_{xy}^2}
\label{eqno_FPE1}
\end{equation}
%
%
% and for the elasticplastic case
% %
% \begin{equation}
% \frac{\partial P(\sigma_{xy}(t))}{\partial t}
% =
% \left< G^{ep} \frac{d\epsilon_{xy}}{dt} \right> \
% \displaystyle \frac{\partial P(\sigma_{xy}(t))}
% {\partial \sigma_{xy}}
% +
% \left\{\int_{0}^{t} d \tau Cov_0
% \left[
% G^{ep}
% \frac{d\epsilon_{xy}}{dt};
% %\left(
% G^{ep}
% \displaystyle \frac{d\epsilon_{xy}}{dt}
% \right] \right \} \frac{\partial^2 P(\sigma_{xy}(t))}{\partial \sigma_{xy}^2}
% \label{eqno_FPE2}
% \end{equation}
% %
%
Previous equations can be written in generalized form describing evolution of
stress (with appropriate initial and boundary conditions) as
%
\begin{equation}
\label{eqno_NS1}
\frac{\partial P}{\partial t}
=

N_{(1)}\frac{\partial P}{\partial \sigma_{xy}}
+
N_{(2)}\frac{\partial^2 P}{\partial \sigma_{xy}^2}
\;\;\;\;\;\;\;\;\;\;\;\; \mbox{;} \;\;\;\;\;\;\;\;\;\;\;\;
\zeta(\infty,t)=\zeta(\infty,t)=0
\end{equation}
%
%with an appropriate initial condition 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, and $\zeta$ is the probability current.
%
The advection and diffusion coefficients $N_{(1)}$ and $N_{(2)}$ can be derived
for any elastic and/or elasticplastic material model. For
example, for linear elastic material model, these coefficients are
%
\begin{eqnarray}
\label{eqno_EADC01}
N_{(1)}^{el} = \frac{d \epsilon_{xy}}{dt} \left< G \right >
%
\;\;\;\;\;\; \mbox{;} \;\;\;\;\;\;
%
%\label{eqno_E2}
N_{(2)}^{el} = t \left(\frac{d \epsilon_{xy}}{dt}\right)^2 Var[G]
\end{eqnarray}
%
Similarly, using equation (\ref{eqno_FPE1}), advection and
diffusion coefficients for elasticplastic vonMises model (with hardening) are
%
\begin{eqnarray}
\label{eqno_EPADC01}
N_{(1)}^{ep} = \displaystyle \frac{d \epsilon_{xy}}{dt}
\left
%\;\;\;\; \mbox{;} \;\;\;\;
\\
\label{eqno_EPADC01a}
N_{(2)}^{ep}
=
t \left(\displaystyle
\frac{d \epsilon_{xy}}{dt}\right)^2
Var\left[G\displaystyle
\frac{G^2}{G + \displaystyle \frac{1}{\sqrt{3}} c_u^{\prime}}\right]
\end{eqnarray}
%
%
while these coefficients for elasticplastic DruckerPrager model (with
hardening) are of the form
%
\begin{eqnarray}
\label{eqno_EPADC02}
N_{(1)}^{ep} = \displaystyle \frac{d \epsilon_{xy}}{dt}
\left
\\
\label{eqno_EPADC02a}
N_{(2)}^{ep}
=
t \left(\displaystyle
\frac{d \epsilon_{xy}}{dt}\right)^2
Var\left[G\displaystyle
\frac{G^2}{G+9K \alpha^2
+
\displaystyle \frac{I_1\alpha'}{\sqrt{3}}}\right]
\end{eqnarray}
%
Once the coefficients are derived, the solution to Eq.~(\ref{eqno_NS1}) can
proceed using a number of different methods, including simple finite difference
scheme.
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\subsection{Weighted Elastic and ElasticPlastic Solution: Probabilistic
Yielding}
The above development needs solutions of two equations  one for the elastic
(preyield) region, governed by $N_{(1)}^{el}$ and $N_{(2)}^{el}$ and the other
for the elasticplastic (postyield) region, governed by $N_{(1)}^{ep}$ and
$N_{(2)}^{ep}$  for complete simulation of probabilistic constitutive
behavior. However, if yield surface (yield point in 1D) is uncertain,
uncertainty propagates into separation of elastic and elasticplastic regions.
That is, depending on the degree of uncertainty of yield surface, stress points
can only have certain probability of being in elastic or elasticplastic state.
%
% One possible simplification is to use the mean stress yield criteria i.e. when the mean
% of elastic solution exceeds the mean of yield stress yielding
% occurs and switch is made from elastic equation ($N_{(1)}^{el}$
% and $N_{(2)}^{el}$) to elasticplastic equation
% ($N_{(1)}^{ep}$ and $N_{(2)}^{ep}$) in solving the FPKE.
% %
% However, this yielding criteria neglects the higherorder statistical moments
% of the yield stress random variable and hence doesn't consider
% the possibility of elasticplastic equation coming into play
% in the preyield region and vice versa.
% %
A solution to this problem of uncertain yielding is to
assign weights to the elastic and
elasticplastic advection ($N_{(1)}^{el}$ and $N_{(1)}^{ep}$) and diffusion
($N_{(2)}^{el}$ and $N_{(2)}^{ep}$)
coefficients based on the cumulative probability
density function (CDF) of the yield function (or stress $\Sigma_y$ in 1D) random
variable. That combined (weighted) equation, which now has both elastic and
elasticplastic coefficient, appropriately weighted, can then be used to obtain the complete
constitutive behavior with equivalent advection and diffusion
coefficients. In other words, while solving the FPK partial
differential equation, for each stress ($\sigma$) in the stress
domain, probability weight will be assigned to the elastic and
elasticplastic advection and diffusion coefficients
corresponding to that stress. Mathematically, the equivalent (weighted)
advection and diffusion coefficients ($N_{(1)}^{eq}$ and
$N_{(2)}^{eq}$) can be written as:
%
\begin{eqnarray}
\label{eqno_PY01}
&&N_{(1)}^{eq} (\sigma)
=
(1  P[\Sigma_y \leq \sigma]) N_{(1)}^{el} + P[\Sigma_y \leq \sigma] N_{(1)}^{ep} \\
\label{eqno_PY02}
&&N_{(2)}^{eq} (\sigma)
=
(1  P[\Sigma_y \leq \sigma]) N_{(2)}^{el} + P[\Sigma_y \leq \sigma] N_{(2)}^{ep}
\end{eqnarray}
%
where $(1  P[\Sigma_y \leq \sigma])$ represents the probability of material
being elastic, while $P[\Sigma_y \leq \sigma]$ represents the probability of
material being elasticplastic.
For example, for a given CDF of yield stress (shown in
Fig.~\ref{vonMises_Yield_CDF}(a)), the probability of yielding happening at
$\sigma=0.0012~{\rm MPa}$\footnote{Mathematically we will write this probability as
$P[\Sigma_y \leq (\sigma=0.0012~{\rm MPa})]$} is $P[\Sigma_y \leq (\sigma=0.0012~{\rm
MPa})] = 0.8$
so that the equivalent advection and diffusion coefficients are
%
\begin{equation}
N_{(1)}^{eq}_{\sigma=0.0012~MPa}
=
(10.8)N_{(1)}^{el}+0.8N_{(1)}^{ep}
\nonumber
\end{equation}
%
\begin{equation}
N_{(2)}^{eq}_{\sigma=0.0012~MPa}
=
(10.8)N_{(2)}^{el}+0.8N_{(2)}^{ep}
\nonumber
\end{equation}
%
where linear elastic coefficients $N_{(1)}^{el}$ and $N_{(2)}^{el}$ are given by
Eq.~(\ref{eqno_EADC01}), while coefficients $N_{(1)}^{ep}$ and $N_{(2)}^{ep}$
are given by
Eqs.~(\ref{eqno_EPADC01}) and (\ref{eqno_EPADC01a}) for von Mises and by
Eqs.~(\ref{eqno_EPADC02}) and (\ref{eqno_EPADC02a}) for DruckerPrager material
models.
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%\newpage
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\section{Examples}
\label{Results}
In this section the developed concept is applied to two elasticplastic
linear hardening material models. The models are represented by von Mises and
DruckerPrager yield and plastic potential functions, respectively, with linear
isotropic hardening.
%
For both models shear modulus and yield parameter (shear
strength ($c_u$) for von Mises model and friction coefficient ($\alpha$) for
DruckerPrager model) are considered independent, normally distributed random
variables.
%
Three examples are shown for each model. The first example for each material
model presents a case where shear modulus and yield
parameter are very uncertain. The other examples for each material model
are limiting cases  one where shear modulus is fairly certain while
yielding parameter is very uncertain, and the
other where shear modulus is very certain while yield parameter is fairly certain.
%
It is important to note that for all the examples, only one equation is solved,
using equivalent advection and diffusion coefficients, to obtain both
uncertain elastic and uncertain elasticplastic behavior.
The governing FPK partial differential equation (Eq.~(\ref{eqno_NS1})), with
equivalent advection and diffusion coefficients (Eqs.~(\ref{eqno_PY01}) and
(\ref{eqno_PY02})), was solved numerically using the \textit{method of lines}.
To this end, the FPK partial differential equation was first semidiscretized
in the stress domain on a uniform grid by \textit{central differences} to
obtain a series of ordinary differential equation (ODE). These ODEs are then
solved simultaneously, after incorporating boundary conditions, using a standard
ODE solver which utilizes \textit{ADAMS method} and \textit{functional
iteration}. Open source code SUNDIALS \cite{manual:SUNDIALS} was used for this
purpose.
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\subsection{von Mises Associative, Linear Hardening ElasticPlastic Model}
The yielding probability weights assigned to the advection ($N_1$) and
diffusion ($N_2$) coefficients are based on the cumulative density function
(CDF) of the yield stress, which in this case corresponds to CDF of the shear
strength ($c_u$)). The CDF of yield stress is shown in
Fig.~\ref{vonMises_Yield_CDF}.
%
\begin{figure}[!htbp]
\begin{center}
\includegraphics[height=4cm]{vonMises_YieldCDF_Combinededited.eps}
\vspace*{0.8cm}
\caption{CDF of shear strength for von Mises model: (a) very uncertain case,
(b) fairly certain case.}
\label{vonMises_Yield_CDF}
\end{center}
\end{figure}
Fig.~\ref{vonMises_G_and_cu_very_uncertain}(a) shows the
evolution of probability density function (PDF) of shear stress
with respect to shear strain for probabilistic von Mises associative plasticity
model.
%
The shear modulus ($G$) is modeled as normally distributed
random variable with a mean of $2.5$~MPa and coefficient of variation (COV) of
$20$~\%.
%
The shear strength ($c_u$) is also assumed to
be normally distributed with a mean of $0.0001$~MPa and COV of
$20$~\%.
%
The hardening parameter ($c_u^{\prime}$, representing the rate of
evolution of $c_u$ with plastic strain) is considered
deterministic with an assumed value of $0.3$~MPa.
%
The contours of the evolution of PDF of shear stress with shear
strain are shown in
Fig.~\ref{vonMises_G_and_cu_very_uncertain}(b), along with
mean, mode (most probable solution), and the standard deviations.
%
\begin{figure}[!htbp]
\begin{center}
\mbox{a)}\includegraphics[width=6.5cm]{vonMises_G_and_cu_very_uncertain_PDFedited.eps}
\hfill
\mbox{b)}\includegraphics[width=6.5cm]{vonMises_G_and_cu_very_uncertain_Contour_PDFedited.eps}
%\vspace*{0.5cm}
\caption{von Mises associative plasticity model with uncertain shear modulus and
shear strength (yield parameter): (a) Evolution of PDF of stress with
strain (PDF=10000 was used as a cutoff for surface plot) and
(b) Contours of evolution of stress PDF with strain.}
\label{vonMises_G_and_cu_very_uncertain}
\end{center}
\end{figure}
%
The deterministic solution obtained using mean values of shear
modulus ($2.5$~MPa) and shear strength ($0.0001$~MPa) is also shown.
%
It is interesting to note the smooth response for
probabilistic von Mises elastoplastic model. On the other hand, the
deterministic von Mises elasticplastic model shows an expected sharp change in
stiffness at the boundary between elastic and elasticplastic solutions.
%
In addition to that, the most likely stress response (mode) is different than
the mean and the deterministic stresses. This difference in mode, mean and
deterministic stress responses observed for linear hardening material model is a
novel feature that deserves some attention. This, more realistic yielding
response contrasts earlier observation of equivalence of mode, mean and
deterministic solutions for probabilistic elastoplasticity with mean stress
yielding (Sett et al. \cite{Sett:2007b}).
%
%
%It may be noted that in Sett et al.
%\cite{Sett:2007b}, the FPK equation was solved twice, first time for the preyield
%region (probabilistic elastic) and the second time for the postyield
% region (probabilistic elasticplastic). For the mean yielding approach,
% the transition between probabilistic elastic and probabilistic elasticplastic
% made based on the concept of mean yield criteria, that is the mean of the yield
% stress is used to decide on separation of (probabilistic) elastic and
% (probabilistic) elasticplastic) response.
In the approach presented here, involving probabilistic yielding, the
difference between mode, mean and deterministic solutions is present even for
linear elastic hardening cases (which will involve perfectly plastic material
model as well).
The probabilistic solution for one of the limiting cases, where shear
modulus is very uncertain (statistical properties are same as
the above example), while shear strength is fairly certain (mean of
$0.0001$~MPa and COV of $1$~\%) is shown in
Fig.~\ref{vonMises_limiting_cases}(a).
%
The CDF of the fairly certain yield stress is shown in
Fig.~\ref{vonMises_Yield_CDF}.
%
Note that, since
yielding is fairly certain, the mean, mode and deterministic
behaviors are very similar in the yielding region.
%
The other limiting case that was considered has a fairly
certain shear modulus (mean of $2.5$~MPa and COV of $1$~\%), however shear
strength is very uncertain (mean of $0.0001$~MPa and COV of $20$~\%). The simulation
result is shown in
Fig.~\ref{vonMises_limiting_cases}(b).
%
\begin{figure}[!htbp]
\begin{center}
\mbox{a)}\includegraphics[width=6.5cm]{vonMises_G_very_uncertain_but_cu_fairly_certain_Contour_PDFedited.eps}
\hfill
\mbox{b)}\includegraphics[width=6.5cm]{vonMises_G_fairly_certain_but_cu_very_uncertain_Mean_SD_Mode_DeterSoledited.eps}
%\vspace*{0.5cm}
\caption{von Mises elasticplastic model: (a) Shear modulus: very
uncertain; shear strength: fairly certain,
(b) Shear modulus: fairly certain; shear strength: very uncertain.}
\label{vonMises_limiting_cases}
\end{center}
\end{figure}
%
Note that, as shear
modulus is fairly certain, the standard deviations lines
almost match the mean and mode response line. The response is still probabilistic,
if very sharp (in stress PDF  strain space) so instead of showing contours
(most of which overlap) shown are only the mean, mode, standard deviation and
the deterministic response. The certainty of response as shown in
Fig.~\ref{vonMises_limiting_cases}(b) is attributed to the lack of shear strength
parameter $c_u$ (which is the only very uncertain parameter here) in either of
the coefficients $N_{(1)}^{ep}$ and $N_{(2)}^{ep}$ in equations
(\ref{eqno_EPADC01}) and (\ref{eqno_EPADC01a}).
%
It is important to note, however, that the deterministic
response is still quite a bit different from mode and mean response.
%
%
% cu appears neither in elastic [d(sigma) = G d(epsilon)] nor in
% elasticplastic equation [d(sigma) = GG^2/(G+cu') d(epsilon)] for von
% Mises model. This is in contrast with DP model, where alpha appears in
% elasticplastic equation.
%
% However, the deterministic
% response is still quite a bit different.
%
% In the simulation in Fig 3(b), the diffusion coefficients (function of
% variance) in both the elastic and elasticplastic cases are very small,
% especially for elasticplastic case (square of small variance is even
% smaller). That's the reason it didn't diffuse. But it did advect  moved
% with time.
%
%
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\subsection{DruckerPrager Associative Plasticity Model}
Similar to the von Mises case, presented are results
of probabilistic DruckerPrager material model simulation, for
the case where both shear modulus and yield parameter (frictional
coefficient, $\alpha$) are considered as very uncertain.
%
The assumed mean and COV of normally distributed shear modulus ($G$)
are $2.5$~MPa and $20$~\%, respectively.
%
The frictional coefficient ($\alpha$) has a mean and COV of $0.1$
and $20$~\%, respectively.
%
The other parameters needed for DruckerPrager probabilistic elasticplastic
simulations, were assumed deterministic and were as follows: the bulk modulus
$K=3.33$~MPa, the rate of evolution of $\alpha$ with plastic strain
$\alpha^{\prime}=5.5$, and the confinement pressure $I_1=0.01$~MPa.
%
For a given confinement pressure ($I_1$) and (probabilistic) frictional
coefficient ($\alpha$), the CDF of a yield stress ($\sigma_y~=~I_1\alpha$) is calculated
and shown in Fig.~\ref{DruckerPrager_Yield_CDF}(a).
%
\begin{figure}[!htbp]
\begin{center}
\includegraphics[height=4cm]{DruckerPrager_YieldCDF_Combinededited.eps}
\vspace*{0.8cm}
\caption{CDF of yield stresses for DruckerPrager model:
(a) very uncertain and (b) fairly certain}
\label{DruckerPrager_Yield_CDF}
\end{center}
\end{figure}
Fig.~\ref{DruckerPrager_G_and_cu_very_uncertain}(a) shows the
evolution of PDF of shear stress with strain.
%
Same results are presented as contours of evolution of PDF of shear stress with shear
strain along with the mean, mode, and the deterministic
solution in Fig.~\ref{DruckerPrager_G_and_cu_very_uncertain}(b).
%
\begin{figure}[!htbp]
\begin{center}
\mbox{a)}\includegraphics[width=6.5cm]{DruckerPrager_G_and_alpha_very_uncertain_PDFedited.eps}
\hfill
\mbox{b)}\includegraphics[width=6.5cm]{DruckerPrager_G_and_alpha_very_uncertain_Contour_PDFedited.eps}
%\vspace*{0.5cm}
\caption{DruckerPrager associative elasticplastic model with uncertain shear
modulus and frictional coefficient: (a) Evolution of probability density function (PDF) of stress with
strain (PDF=10000 was used as a cutoff for surface plot) and (b) Contours of evolution of PDF with strain}
\label{DruckerPrager_G_and_cu_very_uncertain}
\end{center}
\end{figure}
The limiting cases, where shear modulus ($G$) is very
uncertain (COV = $20$~\%), while frictional coefficient ($\alpha$) is
considered fairly certain (COV = $5$~\%) is shown in
Figs.~\ref{DruckerPrager_limiting_cases}(a). This case is to be contrasted with
the case where shear modulus ($G$) is
fairly certain (COV = $1$~\%), while frictional coefficient ($\alpha$)
is very uncertain (COV = 20~\%), shown in
Figs.~\ref{DruckerPrager_limiting_cases}(b).
%
\begin{figure}[!htbp]
\begin{center}
\mbox{a)}\includegraphics[width=6.5cm]{DruckerPrager_G_very_uncertain_but_alpha_fairly_certain_Contour_PDFedited.eps}
\hfill
\mbox{b)}\includegraphics[width=6.5cm]{DruckerPrager_G_fairly_certain_but_alpha_very_uncertain_Contour_PDFedited.eps}
%\vspace*{0.5cm}
\caption{DruckerPrager elasticplastic model: (a) Shear modulus: very
uncertain; frictional coefficient: fairly certain,
(b) Shear modulus: fairly certain; frictional coefficient: very uncertain.}
\label{DruckerPrager_limiting_cases}
\end{center}
\end{figure}
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\section{Summary}
In this paper we presented the methodology that accounts for probabilistic yielding of
elasticplastic materials.
%
Use was made of the recently developed second order
accurate solution to the probabilistic elasticplastic problem (which is based
on solution of the EulerianLagrangian form of FokkerPlanckKolmogorov
equation).
%
The derived weighted probabilities of elastic and elasticplastic response
were used in modeling and simulating probabilistic behavior of von Mises and
DruckerPrager material models.
%
Results show that the most likely response (mode) is different than the mean
and/or deterministic solutions. In addition to that, smooth response curves
(mode and mean) were observed for material models with linear hardening, even if
deterministic response was characterized with a sudden change in stiffness.
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%\newpage
%\clearpage
\renewcommand{\baselinestretch}{1.2}
\small\normalsize % trick from Kopka and Daly p47.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\bibliography{refmech,refcomp,aux}
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