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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}{CMMI-0600766}

\noreceived{}
\norevised{}
\noaccepted{}

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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 non-uniformity and point-wise uncertainty in material
behavior.

Presented    is  the  methodology that accounts for probabilistic yielding    of
elastic--plastic  materials. The recently developed Eulerian--Lagrangian form of
Fokker--Planck--Kolmogorov  equation    is used to
obtain second order exact solution to elastic--plastic constitutive differential
equations.  In  this  paper  that  solution  is  used  in  deriving the weighted
probabilities  of  elastic,  elastic--plastic behavior and yielding. A number of
examples  for  two commonly used material models, von Mises and Drucker--Prager,
illustrated findings.

\end{abstract}

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\section{INTRODUCTION}

Elastic--plastic  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 elastic--plastic (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 non-uniformity 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  elastic--plastic  materials  with  uncertain  properties  has  not
received much attention.


One  of  the  earliest  approaches  to propagating  randomness through    the   
elastic--plastic  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  so-called  "closure  problem"  (cf.  Kavvas
\cite{Kavvas:2003}), where information on higher--order moments is always needed
to   calculate   lower--order  moments. 
%
Similarly, Kaminski  \cite{Kaminski:1999}  used  Monte--Carlo
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  Monte--Carlo  method  to propagate uncertainties
through elastic--plastic $c-\phi$ soil.
%
Monte--Carlo  approach  to accounting for uncertainties might be computationally
expensive  for elastic--plastic simulations. The method requires a statistically
appropriate  number of realizations per random variable  in
order  to  satisfy  statistical  accuracy. The Monte--Carlo simulation approach,
however,  finds  a  great  use  in  verification of analytical developments (cf.
Oberkampf et al, \cite{Oberkampf2002}).
%


A solution  to  overcoming   drawbacks of Monte--Carlo
technique and perturbation method is the use of general Eulerian-Lagrangian form
of  Fokker--Planck--Kolmogorov  equation  (FPKE)  for  the  second--order  exact
probabilistic  solution  of  non-linear  ODE  with  stochastic  coefficient  and
stochastic    forcing   (Kavvas \cite{Kavvas:2003}). Using the above mentioned
Eulerian--Lagrangian  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  1-D  elastic-plastic  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 closed-form 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  FPKE-based  probabilistic elasto--plasticity 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)  elastic--plastic  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   elastic--plastic  behavior.  These  weighted
probabilities  are  used  to  develop probabilistic elastic--plastic response of
materials,   that   is   based  on  probabilistic  rather  then  mean  yielding.
Probabilistic  yielding  examples  using  von Mises and Drucker--Prager material
models are used to illustrate developed methodology.



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\section{FPKE-based Probabilistic Elasto--Plasticity}
\label{MasterKey}



The incremental form of spatial-average 3--D elastic-plastic
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,  non-linear  elastic-plastic
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_*$) could be scalar (for perfectly plastic and isotropic hardening
models),  or  second-order  tensor  (for  translational and rotational kinematic
hardening models), or fourth-order tensor (for distortional hardening models) or
any  combinations  of  the  above. The same classification applies to the random
direction of evolution of internal variables ($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  non-linear 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   Eulerian-Lagrangian  form  of
Fokker--Planck--Kolmogorov 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 non-linear, 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 Drucker--Prager Hardening Material Models}

The   general,  three  dimensional  PDF  for  stress  (Eq.~(\ref{eqno_S20}))  is
specialized to point-location 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  elastic--plastic  case  (using
elastic--plastic $G$):
%
%
%
\begin{equation}
\displaystyle \frac{\partial P(\sigma_{xy}(t))}{\partial t}  
= - 
\left<G \displaystyle \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 \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 elastic--plastic 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 elastic--plastic   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 elastic--plastic von--Mises  model (with hardening) are 
%
\begin{eqnarray}
\label{eqno_EPADC01}
N_{(1)}^{ep} =  \displaystyle \frac{d \epsilon_{xy}}{dt} 
\left<G-\displaystyle \
\frac{G^2}{G + \displaystyle \frac{1}{\sqrt{3}} c_u^{\prime}} \right> 
%\;\;\;\; \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 elastic--plastic Drucker--Prager model (with
hardening) are of the form
%
\begin{eqnarray}
\label{eqno_EPADC02}
N_{(1)}^{ep} =  \displaystyle \frac{d \epsilon_{xy}}{dt} 
\left<G-\displaystyle \
\frac{G^2}{G+9K\alpha^2+\displaystyle 
\frac{I_1\alpha'}{\sqrt{3}}} \right> 
\\
\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  Elastic--Plastic  Solution:  Probabilistic
Yielding}


The   above  development needs solutions of two equations -- one for the elastic
(pre--yield) region, governed by $N_{(1)}^{el}$ and $N_{(2)}^{el}$ and the other
for  the  elastic--plastic  (post--yield) region, governed by $N_{(1)}^{ep}$ and
$N_{(2)}^{ep}$   --   for  complete  simulation  of  probabilistic  constitutive
behavior.  However,  if  yield  surface  (yield point in 1--D) is uncertain,
uncertainty  propagates into separation of elastic and elastic--plastic regions.
That  is, depending on the degree of uncertainty of yield surface, stress points
can only have certain probability of being in elastic or elastic--plastic 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     elastic--plastic    equation
% ($N_{(1)}^{ep}$  and  $N_{(2)}^{ep}$)  in solving the FPKE. 
% %
% However, this yielding criteria neglects the higher-order statistical moments
% of the yield stress random variable and hence doesn't consider
% the possibility of elastic--plastic equation coming into play
% in the pre-yield region and vice versa.
% %

A solution to this problem of uncertain yielding is to 
assign weights to the elastic and
elastic--plastic    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
elastic--plastic 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
elastic--plastic    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 elastic--plastic.

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}
=
(1-0.8)N_{(1)}^{el}+0.8N_{(1)}^{ep}
\nonumber
\end{equation}
%
\begin{equation}
N_{(2)}^{eq}|_{\sigma=0.0012~MPa}
=
(1-0.8)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 Drucker-Prager 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 elastic--plastic
linear  hardening  material  models. The models are represented by von Mises and
Drucker--Prager 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
Drucker--Prager  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 elastic--plastic 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 semi-discretized
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 Elastic--Plastic 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_Combined-edited.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_PDF-edited.eps}
\hfill
\mbox{b)}\includegraphics[width=6.5cm]{vonMises_G_and_cu_very_uncertain_Contour_PDF-edited.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  elasto--plastic  model.    On    the  other hand, the
deterministic von Mises elastic--plastic model shows an expected sharp change in
stiffness at the boundary between elastic and elastic--plastic 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 elasto--plasticity 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 pre-yield
%region   (probabilistic   elastic)  and  the  second  time  for  the  post-yield
% region      (probabilistic  elastic--plastic).  For the mean yielding approach,
% the transition between probabilistic elastic and probabilistic elastic--plastic
% 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) elastic--plastic) 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_PDF-edited.eps}
\hfill
\mbox{b)}\includegraphics[width=6.5cm]{vonMises_G_fairly_certain_but_cu_very_uncertain_Mean_SD_Mode_DeterSol-edited.eps}
%\vspace*{-0.5cm}
\caption{von  Mises  elastic--plastic  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
%  elastic-plastic equation [d(sigma) = G-G^2/(G+cu') d(epsilon)] for von
%  Mises model. This is in contrast with DP model, where alpha appears in
%  elastic-plastic 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 elastic-plastic cases are very small,
%  especially for elastic-plastic 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{Drucker-Prager Associative Plasticity Model}

Similar  to the von Mises case, presented are results
of  probabilistic Drucker-Prager 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 Drucker--Prager probabilistic elastic--plastic
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_Combined-edited.eps}
\vspace*{-0.8cm}
\caption{CDF of yield stresses for Drucker-Prager 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_PDF-edited.eps}
\hfill
\mbox{b)}\includegraphics[width=6.5cm]{DruckerPrager_G_and_alpha_very_uncertain_Contour_PDF-edited.eps}
%\vspace*{-0.5cm}
\caption{Drucker-Prager  associative  elastic--plastic  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_PDF-edited.eps}
\hfill
\mbox{b)}\includegraphics[width=6.5cm]{DruckerPrager_G_fairly_certain_but_alpha_very_uncertain_Contour_PDF-edited.eps}
%\vspace*{-0.5cm}
\caption{Drucker-Prager elastic--plastic 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  
elastic--plastic  materials. 
%
Use was made of the recently developed second order
accurate  solution to the probabilistic elastic--plastic problem (which is based
on  solution  of  the  Eulerian--Lagrangian  form of Fokker--Planck--Kolmogorov 
equation).
%
The  derived  weighted  probabilities  of  elastic and elastic--plastic response
were  used  in  modeling  and simulating probabilistic behavior of von Mises and
Drucker--Prager 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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%

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\small\normalsize  % trick from Kopka and Daly p47.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\bibliography{refmech,refcomp,aux}

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