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\begin{document}
\CNM{1}{6}{00}{28}{00}

\runningheads{Jeremi{\'c} and Cheng}
{Equal Principal Stretches}

\title{Significance of Equal Principal Stretches in Computational Hyperelasticity}

\author{Boris Jeremi{\'c} and Zhao Cheng}

\address{Department  of  Civil  and  Environmental  Engineering,  University  of
California, Davis, CA, 95616, U.S.A. \\~\\Vol. 21, Issue 9, pp 477-486, September 2005}

\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--PEER ; NSF}{EEC-9701568; CMS-0337811}

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








\begin{abstract}

The  computational  significance  of  the  case  of two or three equal principal
stretches in large deformation analysis is investigated in this paper.
%
A  detailed analytical study shows that the previously suggested solutions,
based on numerical perturbations, are not adequate and might lead to erroneous results.
%
A number of examples are presented to illustrate the approach.
%


\end{abstract}

%  
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%  {\Large {\sc  Version: \today{}, \hhmm{} }}
%  \end{center}
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%  \tableofcontents

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\newpage
\section{INTRODUCTION}


An objective  computational  treatment of large deformation inelastic problems has
been developed recently by using multiplicative decomposition of the deformation
gradient   (eg.  Lee   and   Liu   \cite{Lee67}).  The  techniques  are based on
the hyperelastic--plastic approach and most of them stem from pioneering work by
Simo et al. \cite{local-12, Simo88a, Simo88b, Simo92}.
%
%
A  consistent  hyperelastic computational formulation forms    a    basis    of
all the developments for large deformation hyperelasto--plasticity.
Of   particular importance is that the computational formulation can be verified
on  a  complete stress and strain space. This requirement fits well with a
much wider goal of computational validation of the developed simulations tools
(eg. Oberkampf  et al. \cite{Oberkampf2002} and Roach \cite{Roach1998}).

In  this  paper a  peculiar (yet frequently present) case  when  two or all three principal
stretches  are  equal is explored. Of particular interest is the computational treatment of
the    consistent    tangent    stiffness    tensor and stress measures (second
Piola--Kirchhoff for example) for this case.
The  paper  is  organized  around  consistent and detailed  derivations  of large
deformation  measures,  large  deformation  stress  measures  and the resulting
consistent  tangent  stiffness  tensor. The special cases of equal stretches are
dealt    with    within  the  derivations,  following  the  general case of all
non--equal    stretches.      The  derivations  are  general  as they apply to any
hyperelastic   material   model,   including   those  that  are  deviatoric  only
(isochoric), volumetric only and those that can produce both types of responses.
%%

It  should  be noted that Simo and Taylor \cite{Simo91} did address the issue of
two  or  three  equal  principal  stretches but it was specifically done for the
Ogden  (deviatoric  only)  hyperelastic  material  model and in an Eulerian form.
%
It  will also be shown that the method of perturbations suggested later (eg.  Simo
\cite{Simo92}) can only be applied in a limited number of cases.
%
As  it turns out, a solution  for the treatment  of  equal  principal stretches
based  on  perturbations  can  lead  to  erroneous  results  and is dependent on
the numerical precision of computations.
%
Results  for  cases  of  two  or three equal principal stretches, obtained using
the present approach,  are shown.
%
In   order   to   provide   a  self--sufficient  treatment  of
hyperelasticity,    derivatives  used  to  obtain stress measures and consistent
tangent stiffness tensors for a  number  of  hyperelastic models are given in the
appendix.






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%\newpage
\section{HYPERELASTICITY}
\label{Hyperelasticity}



This section develops analytical forms for the deformation tensor, stress measures and
the  stiffness  tensor for general case (non--equal principal
stretches)  and  for  two  special  cases  with two  and/or  three  equal  principal
stretches.

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\subsection{Deformation Tensor}


%
The                  deformation           tensor\footnote{Indicial  notation is
used  throughout this paper. Einstein tensor summation is assumed and applies to
all pairs of dummy indices. The upper case indices describe the undeformed while
lower  case  indices  describe  deformed configuration. It is also noted that no
summation  is  implied  over  indices  in parenthesis.} in material (Lagrangian)
description,  given  as $C_{IJ}^{m}\;(m=0,\pm1,\pm2,\cdots)$ can be expressed in
terms  of  its  eigenvalues  and eigenvectors (eg. Ting \cite{Ting85} and Morman
\cite{Morman86})   which   in   case  of  three  different  principal  stretches
($\lambda_{1} \ne \lambda_{2} \ne \lambda_{3}$) can be written as:
%
\begin{eqnarray}
C_{IJ}^{m} =
\lambda^{2m}_{1}N^{(1)}_{I}N^{(1)}_{J} +
\lambda^{2m}_{2}N^{(2)}_{I}N^{(2)}_{J} +
\lambda^{2m}_{3}N^{(3)}_{I}N^{(3)}_{J}
\label{Ccase1}
\end{eqnarray}
%
The explicit form for the cross product of deformation tensor's unit eigenvectors 
$N^{(A)}_{I}N^{(A)}_{J}$ can then be expressed as:
%
\begin{eqnarray}
N^{(A)}_{I}N^{(A)}_{J}
=
 \lambda_{(A)}^{2}
M_{IJ}^{(A)}
\label{Ncase12}
\end{eqnarray}
%
with
\begin{eqnarray}
D_{(A)} =
(\lambda_{(A)}^{2} - \lambda_{(B)}^{2})(\lambda_{(A)}^{2} - \lambda_{(C)}^{2})
=
2 \lambda_{(A)}^{4}
-
I_{1} \lambda_{(A)}^{2}
+
I_{3} \lambda_{(A)}^{-2}
\label{Ncase100}
\end{eqnarray}
%
\begin{eqnarray}
M_{IJ}^{(A)}
%=
%\lambda^{-2}_{A}N^{(A)}_{I}N^{(A)}_{J}
=
\frac
{C_{IJ}
-
\left(I_{1}-\lambda_{(A)}^2\right) \delta_{IJ}
+
I_{3} \lambda_{(A)}^{-2} C^{-1}_{IJ} }
{D_{(A)} }
\label{M001}
\end{eqnarray}
%



Here, obviously, $D_{(A)}$ and $M_{IJ}^{(A)}$ must satisfy:
%
\begin{eqnarray}
\sum_{A=1}^{3}{ \frac{1} {D_{(A)}} }
=
0
\;\;\;\;\;\;\;\; \mbox{;} \;\;\;\;\;\;\;\;
%
%
\sum_{A=1}^{3}{ \frac{\lambda_{(A)}^{2} } {D_{(A)}} }
=
0
\;\;\;\;\;\;\;\; \mbox{;} \;\;\;\;\;\;\;\;
%
%
\sum_{A=1}^{3}{ \frac{\lambda_{(A)}^{4} } {D_{(A)}} }
=
1
\nonumber  \\
\sum_{A=1}^{3}{ M_{IJ}^{(A)} }
=
C_{IJ}^{-1}
\;\;\;\;\;\;\;\; \mbox{;} \;\;\;\;\;\;\;\;
%
%
\sum_{A=1}^{3}{ \lambda_{(A)}^{2} M_{IJ}^{(A)} }
=
\delta_{IJ}
\;\;\;\;\;\;\;\; \mbox{;} \;\;\;\;\;\;\;\;
%
%
\sum_{A=1}^{3}{ \lambda_{(A)}^{4} M_{IJ}^{(A)} }
=
C_{IJ}
\label{M123}
\end{eqnarray}
%


In the case that two or three principal stretches are equal, the value of 
$D_{(A)}$  (from Equation \ref{Ncase100}) becomes zero.
%
This necessitates closer
inspection of equations which rely on $D_{(A)}$ in the denominator.
%




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{Special Case $\lambda = \lambda_{1} = \lambda_{2} \ne \lambda_{3}$.}
%

In  this case, which applies, for example to a uniaxial stretching of materials,
one obtains for the deformation tensor
%
\begin{eqnarray}
C_{IJ}^{m} =
\lambda^{2m}(\delta_{IJ} - N^{(3)}_{I}N^{(3)}_{J}) +
\lambda^{2m}_{3}N^{(3)}_{I}N^{(3)}_{J}
\label{Ccase2}
\end{eqnarray}
%
while  the  tensor $M_{IJ}^{(3)}$, used to calculate the  product of deformation
tensor's eigenvectors ($N^{(A)}_{I}N^{(A)}_{J}=\lambda_{(A)}^{2} M_{IJ}^{(A)}$), is now
%
%
\begin{eqnarray}
M_{IJ}^{(3)}
 =
\frac
{C_{IJ}
-
\left(I_{1}-\lambda_{3}^2\right) \delta_{IJ}
+
I_{3} \lambda_{3}^{-2} C^{-1}_{IJ} }
{D_{(3)}}
\label{MdefC2}
\end{eqnarray}

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{Special Case $\lambda = \lambda_{1} = \lambda_{2} = \lambda_{3}$.}
%

For this special case, as found in hydrostatic stretching of materials, one obtains:
%
\begin{eqnarray}
C_{IJ}^{m} =
\lambda^{2m} \delta_{IJ}
\label{Ccase3}
\end{eqnarray}
%


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\subsection{Stress Measures}


The  second  Piola--Kirchhoff  stress  is  physically  defined  as the force per
undeformed area loaded on a surface in the undeformed reference configuration:
%
\begin{eqnarray}
S_{IJ}
=
2 \frac{\partial W }{\partial C_{IJ}}
=
\underbrace{w_{A} (M_{IJ}^{(A)})_{A}}_{\rm isochoric}
+
\underbrace{\frac{\partial       {}^{vol}\!W(J)}{\partial      J}      J      \;
C^{-1}_{IJ}}_{\rm volumetric}
%
\label{pk2stress}
\end{eqnarray}
%  %
where  $w_{A}$  can  be  expressed  as  (for detailed derivation see Jeremi{\'c}
\cite{Jeremic2000b} and Holzapfel \cite{Holzapfel2001})
%
\begin{eqnarray}
w_{A}
=
- \frac{1}{3} \;
\frac{\partial {}^{iso}\!W}{\partial \tilde{\lambda}_{B}} \;
\tilde{\lambda}_{B}    \;
+
\frac{\partial {}^{iso}\!W}{\partial \tilde{\lambda}_{(A)}} \;
\tilde{\lambda}_{(A)}
\label{SMA110}
\end{eqnarray}
%
It should be noted that   $\tilde{\lambda}  =  J^{-1/3}  \lambda$  is the
isochoric component of the principal stretch. The elastic potential function $W$
is used to specify material models.
%
The  isochoric  components of stress tensor will be affected by special cases of
two or three equal stretches.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{Special Case $\lambda = \lambda_{1} = \lambda_{2} \ne \lambda_{3}$.}
%
In  this  special  case,  the isochoric components of the second Piola Kirchhoff
stress is
%
\begin{eqnarray}
{}^{iso}\!S_{IJ}
&=&
w_{3}M_{IJ}^{(3)}
+
w_{1}(C^{-1}_{IJ} - M_{IJ}^{(3)})
\label{pk2isoC2}
\end{eqnarray}
%
where $M_{IJ}^{(3)}$ is given by the Equation (\ref{MdefC2}).



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{Special Case $\lambda = \lambda_{1} = \lambda_{2} = \lambda_{3}$.}
%
This  is actually a trivial case as there is no isochoric stress from volumetric
deformation
%
\begin{eqnarray}
{}^{iso}\!S_{IJ}
&=&
0
\label{pk2isoC3}
\end{eqnarray}
%
%



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

It is worth noting that other stress measures are easily obtained
once  the    Second  Piola--Kirchhoff  stress  is  known.  For example, the First
Piola--Kirchhoff  stress  is  simply  obtained from $P_{iJ}=F_{iK}S_{KJ}$, Cauchy
stress  is  $\sigma_{ij}={1}/{J}  \;  F_{iM}S_{MN}F_{jN}$,  Mandel stress is
$T_{IJ}=C_{IK}      S_{KJ}$,      and      the      Kirchhoff     stress    is  
$\tau_{ij}=F_{iA}(F_{jB})^{t}S_{AB}$.
%





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Constitutive Tangent Tensor}

Material tangent stiffness relation is defined from:
%
\begin{eqnarray}
d S_{IJ}
=
\frac{1}{2} \; {\cal L}_{IJKL}
 \; d C_{KL}
\;\;\;\; \mbox{;} \;\;\;\;
{\cal L}_{IJKL}
=
4 \; \frac{\partial^2 \left( {}^{vol}\!W \right) } {\partial C_{IJ} \; \partial C_{KL}}
+
4 \; \frac{\partial^2 \left( {}^{iso}\!W \right) } {\partial C_{IJ} \; \partial C_{KL}}
\label{TSO010}
\end{eqnarray}
%
with the definition of volumetric component of the stiffness tensor
%
%
\begin{eqnarray}
{}^{vol}{\cal L}_{IJKL}
=
(
J^2
\frac{\partial^2 {}^{vol}\!W(J)}{\partial J  \partial J }
+
\; J
\frac{\partial{}^{vol}\!W(J)}{\partial J}
)
 C^{-1}_{IJ}  C^{-1}_{KL}
+
\; 2  J
\frac{\partial{}^{vol}\!W(J)}{\partial J}
I_{IJKL}^{C^{-1}}
\label{TSO060}
\end{eqnarray}
%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{General case $\lambda_{1} \ne \lambda_{2} \ne \lambda_{3}$.}

In a general case, when the principal stretches are all different, one can write
the isochoric part of the stiffness tensor as:
%
\begin{eqnarray}
%
{\cal L}_{IJKL}^{iso}
=
Y_{AB} \;
(M_{KL}^{(B)})_{B} \;
(M_{IJ}^{(A)})_{A}
+
2 \;
w_{A} \; ({\cal M}_{IJKL}^{(A)})_{A}
\label{LC01}
\end{eqnarray}
%
%
where  tensor  $Y_{AB}$  can  be  expressed  as  (for  detailed  derivation  see
Jeremi{\'c} \cite{Jeremic97g})
%
%
\begin{eqnarray}
Y_{AB}
%&=&
=
\frac{\partial {}^{iso}\!W}
{\partial \tilde{\lambda}_{(A)}} \;
\delta_{(A)(B)}
\; \tilde{\lambda}_{(B)}
+
\frac{\partial^2 {}^{iso}\!W}
{\partial \tilde{\lambda}_{(A)} \partial \tilde{\lambda}_{(B)} } \;
\tilde{\lambda}_{(A)}
\; \tilde{\lambda}_{(B)}
%\nonumber \\
%&-&
-
\frac{1}{3} \;
\left(
\frac{\partial {}^{iso}\!W}
{\partial \tilde{\lambda}_{(A)}} \;
\tilde{\lambda}_{(A)}
+
\frac{\partial {}^{iso}\!W}
{\partial \tilde{\lambda}_{(B)} }
\; \tilde{\lambda}_{(B)}
\right) 
\nonumber \\
%&-&
-
\frac{1}{3} \;
\left(
\frac{\partial^2 {}^{iso}\!W}
{\partial \tilde{\lambda}_{(A)} \partial \tilde{\lambda}_{D} } \;
\tilde{\lambda}_{(A)}
\tilde{\lambda}_{D}
+
\frac{\partial^2 {}^{iso}\!W}
{\partial \tilde{\lambda}_{C} \partial \tilde{\lambda}_{(B)} } \;
\tilde{\lambda}_{C}
\; \tilde{\lambda}_{(B)}
\right) 
%\nonumber \\
%&+&
+
\frac{1}{9}
\left(
\frac{\partial^2 {}^{iso}\!W}
{\partial \tilde{\lambda}_{C} \partial \tilde{\lambda}_{D} } \;
\tilde{\lambda}_{C} \;
\tilde{\lambda}_{D}
+
\frac{\partial {}^{iso}\!W}
{\partial \tilde{\lambda}_{D} } \;
\tilde{\lambda}_{D}
\right)
\label{YBA090}
\end{eqnarray}
%
The   Simo--Serrin   fourth   order  tensor    ${\cal    M}_{IJKL}  =  {\partial
M_{IJ}^{(A)}}/ {\partial C_{KL}  }$  (eg.
Morman \cite{Morman86} and  Simo and Taylor \cite{Simo91}) is defined as:
%
\begin{eqnarray}
{\cal M}_{IJKL}^{(A)}
&=&
\frac{1}{D_{(A)}}\;
(I_{IJKL}
-
\delta_{IJ}\delta_{KL}
+
\lambda_{(A)}^{2}
(\delta_{IJ}M_{KL}^{(A)} + M_{IJ}^{(A)}\delta_{KL})
\nonumber \\
&+&
I_{3}\lambda_{(A)}^{-2}
(C^{-1}_{IJ}C^{-1}_{KL} + I_{IJKL}^{C^{-1}}
-
C^{-1}_{IJ}M_{KL}^{(A)} - M_{IJ}^{(A)}C^{-1}_{KL}) 
%\nonumber \\
%&-&
-
d_{A}M_{IJ}^{(A)}M_{KL}^{(A)})
\label{calM}
\end{eqnarray}
%
The associated fourth order tensors $I_{IJKL}^{C^{-1}}$ and $I_{IJKL}$ are defined as
%
\begin{eqnarray}
I_{IJKL}^{C^{-1}}
=
- \frac{1}{2} \left( C^{-1}_{IK} C^{-1}_{JL} + C^{-1}_{IL} C^{-1}_{JK}  \right)
\;\;\;\; \mbox{;} \;\;\;\;
I_{IJKL}
=\frac{1}{2} \left( \delta_{IK}\delta_{JL} + \delta_{IL}\delta_{JK} \right)
\label{SSFA030}
\end{eqnarray}
%
By using equations (\ref{M123}), one observes that ${\cal {M}}_{IJKL}^{(A)}$ satisfies 
%
\begin{eqnarray}
\sum_{A=1}^{3}{ {\cal M} ^{(A)}_{IJKL} }
=
I_{IJKL}^{C^{-1}}
\nonumber
\;\;\;\; \mbox{;} \;\;\;\;
%
%
\sum_{A=1}^{3}{\lambda^{2}_{(A)} {\cal M} ^{(A)}_{IJKL} }
+
\sum_{A=1}^{3}{\lambda^{2}_{(A)} {M}^{(A)}_{IJ}{M}^{(A)}_{KL} }
=
0
\nonumber \\
%
%
\sum_{A=1}^{3}{\lambda^{4}_{(A)} {\cal M} ^{(A)}_{IJKL} }
+
2\sum_{A=1}^{3}{\lambda^{4}_{(A)} {M}^{(A)}_{IJ}{M}^{(A)}_{KL} }
=
I_{IJKL} 
\label{CapM123}
\end{eqnarray}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{Special case  $\lambda = \lambda_{1} = \lambda_{2} \ne \lambda_{3}$.}



In this special case the stiffness tensor is obtained as:
%
%
\begin{eqnarray}
{\cal L}_{IJKL}^{iso}
&=&
Y_{33}M_{IJ}^{(3)}M_{KL}^{(3)}
+
Y_{11}\left(C^{-1}_{IJ} - M_{IJ}^{(1)}\right)\left(C^{-1}_{KL} - M_{KL}^{(1)}\right) \nonumber \\
&+&
Y_{13}\left(C^{-1}_{IJ} - M_{IJ}^{(1)}\right)M_{KL}^{(3)}
+
Y_{31}M_{IJ}^{(3)}\left(C^{-1}_{KL} - M_{KL}^{(1)}\right) \nonumber \\
&+&
2w_{3}{\cal M}_{IJKL}^{(3)} + 2w_{1}(I_{IJKL}^{C^{-1}} - {\cal M}_{IJKL}^{(3)})
\label{LC02}
\end{eqnarray}
%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{Special case  $\lambda = \lambda_{1} = \lambda_{2} = \lambda_{3}$.}


For this special case the stiffness tensor collapses to the following form:
%
\begin{eqnarray}
{\cal L}_{IJKL}^{iso}
=
2G\lambda^{-4}\left( I_{IJKL} - \frac{1}{3}\delta_{IJ}\delta_{KL} \right)
\label{LC03}
\end{eqnarray}
%


\paragraph{Remark.}
For the small deformation case (where 
$\lim_{\lambda  \rightarrow  1} C_{ij}=\delta_{ij}  \; \mbox{;} \; \lim_{\lambda
\rightarrow 1} J=1$) the stiffness tensor collapses to 
%
\begin{eqnarray}
{\cal L}_{IJKL}  \rightarrow E_{ijkl} =
\left( K_{b} - {2}/{3}\; G \right) \delta_{ij}\delta_{kl}
+
(2G) I_{ijkl}
\label{SDE}
\end{eqnarray}
%
This is exactly the small deformation linear elastic stiffness tensor 
in terms of bulk modulus $K_{b}$ and shear modulus $G$.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Perturbation Technique.}

One    of  the  suggested  techniques  for resolving cases of two or three equal
principal  stretches  is  based  on perturbations (eg. Simo \cite{Simo92}). This
technique    should  be carefully utilized as in some cases inconsistent results
are obtained.
%
This is illustrated on a set of analytical experiments.


Let us assume an initial state of deformation given by
%
\begin{equation*}
F_{iJ}=C_{IJ}=\delta_{ij}
\;\;\;\;
\Rightarrow
\;\;\;\;
\lambda_{1}=\lambda_{2}=\lambda_{3}=1
\end{equation*}
%
Since        this       state  represents  total  lack of deformation, the large
deformation  constitutive  stiffness  tensor  should be the same as that one for
small deformations.

A  perturbation  $\chi  = \alpha  \sqrt{M}$ is introduced  to the principal stretches
$\lambda_{1}$,  $\lambda_{2}$  and  $\lambda_{3}$  to  distinctly separate their
values.
%
Here $M$ is the machine  precision\footnote{
%
Machine  precision  or  machine  epsilon ($macheps$) is the smallest
distinguishable  positive  number  (in  a given precision, i.e. float (32 bits),
double  (64  bits)  or  long  double (80 bits), such that $1.0 + macheps > 1.0 $
yields true on the given computer platform.
%
The  ANSI/IEEE Standard 754-1985, defines $macheps$ for a number of floating point
precisions.
For   example,  double  precision  $macheps  =  2.22\times10^{-16}$, while the long double
$macheps = 1.08\times10^{-19}$. Curiously enough,
Intel  80x86  platform  promotes  all  the float, double and long double
precision  numbers  to  long  number inside the floating point unit (FPU) and all the
computations  are  performed  using  80  bits    of precision. Results are then
truncated to predefined predefined precision upon exiting the FPU. Consequence
is  that  one  is  forced  to  use the highest precision macheps on Intel
80x86 even if it might not be portable.}
%
and $\alpha$  is  a  factor. 
%
%%%%%%%%%%%%%%%%%%%%%%
%
On    our    computer   platform (Intel Pentium CPU), the square root of machine
precision is $\sqrt{M} = 3.293\times10^{-10}$.


A perturbation of equal principal stretches of the form 
%
\begin{equation*}
\lambda_{1}=1+\chi
\;\;\;\; \mbox{;} \;\;\;\;
\lambda_{2}=1
\;\;\;\; \mbox{;} \;\;\;\;
\lambda_{3}=1-\chi
\end{equation*}
%
is  assumed  and  used for calculation of the constitutive tensor. This state of
principal  stretches  is  essentially a small deformation configuration and both
large and small deformation theories should give the same result.
%
Given    bulk  modulus    $K_b  =  1.9717~{\rm  MPa}$  and  shear   modulus $G =
0.4225~{\rm  MPa}$,  and  using  the definition for small deformation linear elastic
stiffness  tensor in equation (\ref{SDE}), a few illustrative stiffness values are
given in Table (\ref{stiffness}).
%
\begin{table}[!htbp]
\caption{\label{stiffness}      Selected    small   deformation (SD) and  large 
deformation  (LD) stiffness components for different perturbation numbers. Units
are Pascals [Pa].}
\begin{center}
\begin{small}
{%\footnotesize
\begin{tabular}{|r||r|r|r|} \hline
%                                                                                                    
SD Stiffness component (Eq. (\ref{SDE})) & $E_{1111}$             & $E_{1122}$             & $E_{1212}$            \\ \hline\hline
 (reference values)                                       &  $2.530 \times 10^{6}$ & $1.690 \times 10^{6}$  & $4.225 \times 10^{5}$ \\ \hline \hline
%                                                                                                    
%                                                                                                    
Perturbation  / LD Stiffness component & ${\cal L}_{1111}$     & ${\cal L}_{1122}$      & ${\cal L}_{1212}$     \\ \hline\hline
%
$\alpha=10^{0},        10^{1}$      &    \texttt{NaN} &\texttt{NaN} & \texttt{NaN} \\ \hline
%                                                                                                              
$\alpha=10^{2}$                     & $ 2.710 \times 10^{6}$ &  $2.031 \times 10^{6}$ &  $3.396 \times 10^{5} $  \\ \hline
%                                                                                                              
$\alpha=10^{3}, 10^{4}, 10^{5}$     & $ 2.711 \times 10^{6}$ &  $2.035 \times 10^{6}$ &  $3.396 \times 10^{5} $  \\ \hline \hline
%                                                                                                    
%                                                                                                    
%
\end{tabular}
}
\end{small}
\end{center}
\end{table}

%
The      reason      for      the    differences  and  in  some  cases  lack  of
results\footnote{The \texttt{NaN} means Not a
Number  and  is essentially a result of a numerical value overflowing the largest
accurate  representation  of any number in given accuracy} can be  attributed to
the  fact  that  $C_{IJ}$,    $C^{-1}_{IJ}$    and   $I^{C^{-1}}_{IJKL}$  remain
unperturbed,
while  $\lambda_{1}$,  $\lambda_{2}$,  and  $\lambda_{3}$  are perturbed. This
inconsistency produces a difference, and may result in erroneous results.
%
A  simple  check  why  the  perturbation  method  does  not  work is obtained by
verifying that identities given by
equations  (\ref{M123})  and  (\ref{CapM123})  are  not satisfied
after perturbation.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage
\section{NUMERICAL EXAMPLES}
\label{Results}


%
A  set  of  two  representative  examples is  provided to illustrate closed form
solutions derived above.
%
The examples are related to  
uniaxial tension and volumetric deformation, since those cases are usually found
in formulation verifications and model validations.
%
It  is  noted  that  the examples  described  and  analyzed below will fail to yield
results      (numerical  overflow)  or  will  yield  wrong  results  (see  Table
\ref{stiffness}) if the perturbation technique is used.
%
The    assumed  material  properties  (for  small deformation model) are Young's
modulus   $E = 1.183\times10^{6}~{\rm Pa}$ and Poisson's Ratio $\nu = 0.4$ (which gives
bulk  modulus    $K_b  =  1.9717~{\rm  MPa}$ and shear  modulus $G = 0.4225~{\rm
MPa}$).
%
Related  material  constants  for  the Ogden  material model (using  the  parameters
from Ogden \cite{Ogden84}) are: $N = 3$, $\mu_{1} = 1.3$,
$\mu_{2}  =  5.0$,  $\mu_{3}  =  -2.0$,  $c_{1}  =  6.3\times10^{5}~{\rm Pa}$, $c_{2} =
1.2\times10^{3}~{\rm Pa}$,  $c_{1}  =  -1.0\times10^{4}~{\rm Pa}$ .
%
The   parameters   for  the Mooney--Rivlin   material   model   (as   suggested  by
Anand \cite{Anand86}) are: $C_{1}   =  (7/16)G$,  $C_{2}  =  (1/16)G$.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{Uniaxial tension.}


The    case  of  uniaxial  tension  (or compression, as in soil mechanics tests)
has two equal principal stretches, $\lambda_{1} > \lambda_{2} =
\lambda_{3} = \lambda$.
Since the material is compressible ($\mu = 0.4$) and the constraint is uniaxial,
$\lambda_{1} >1$, $\lambda <1$, the only nonzero stress component is $\sigma_{11}$.

Figure  (\ref{puf}) shows  that  the  tensile  tractions (undeformed
reference) increases with the increase of tensile stretch 
for the Neo--Hookean, Logarithmic, Mooney--Rivlin--Simo and Ogden--Simo model.
%
It is interesting to note that the logarithmic model response exhibits softening
in response  (tractions    vs.  stretch) caused by the violation of
Legendre--Hadamard condition (eg. Bruhns et al. \cite{Bruhns2001}).
%This  is similar to the behavior earlier
%observed for pure shear deformation (eg. Jeremi{\'c} et al. \cite{Jeremic98g}).
%
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=12cm]{puForce.eps}
\caption{\label{puf}   Uniaxial   tension:   tensile  traction  $t$  (undeformed
reference) vs. tensile stretch $\lambda_1$}
\end{center}
\end{figure}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{Volumetric deformation.}

Volumetric  deformation  has  three  equal  principal  stretches, $\lambda_{1} =
\lambda_{2} = \lambda_{3} = \lambda$.
Only volumetric stress $p_{m} = \sigma_{11} = \sigma_{22} = \sigma_{33}$ is nonzero.
The   volume   ratio  $J=\lambda_{1}\lambda_{2}\lambda_{3}$  is used to
represent  volumetric deformation.
The absence of volumetric deformation is implied by $J = 1$.
Volumetric  dilation  applies when  $J  >  1$,  while  the  volumetric
contraction holds  when $ 0 < J < 1$.

Figure  (\ref{pvf}) 
% and  (\ref{pvs}) 
 shows the volumetric traction (undeformed
reference) versus volume ratio, respectively
for the Neo--Hookean, Logarithmic, and Simo--Pister models. These curves have the same slope when
$J = 1$, which is expected as  the deformations approach the small deformation case.
%
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=12cm]{pvForce.eps}
\caption{\label{pvf}    Volumetric   deformation:   hydrostatic   pressure   $p$
(undeformed reference) vs. volume ratio $J$}
\end{center}
\end{figure}
%   %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{CONCLUSIONS}

A  common deformation state with two  or  three equal principal stretches in
computational hyperelasticity was investigated.
%
A  closed    form    solutions  for deformation measures, stress measures and stiffness
tensor were analytically presented in 
Lagrangian format.
%
These  solutions  are  particularly  important in  view  of  possible  problems 
with   perturbation   techniques   that   are  used  to  overcome  computational
difficulties.
%
An  example  of illustrating problems with perturbation
technique  shows  that  even  when a solution is available, it might be the wrong
one.
%
In  addition  to that, a set of numerical examples, employing presented analytical
solutions  was  used  as  an  illustration  for  cases  with  two or three equal
principal stretches.
%

Of particular importance is a decisions on when
to consider principal stretches equal. This decisions has to be made within
computational implementation of large deformation hyperelasticity. 
%
A simple  suggestion is that if the difference between
two  or  three  principal  stretches   is within  a  value  of  $\chi^{\star}  =
10^{2} \sqrt{M}$, they can be
treated    as   numerically  equal  and  the analytical formulae for two or three
equal principal stretches (developed in this paper) should be used.
%

We  also  note  that  the described theory is implemented into the public
domain  finite  element  platform  OpenSees  (eg.  \cite{OpenSEES2000}) with the
source code and above examples available on-line.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\section*{Acknowledgment}
This work was supported in part by a number of grants listed below:
%
Earthquake    Engineering   Research  Centers  Program  of  the National Science
Foundation  under  Award  Number EEC-9701568 (cognizant program director Dr. Joy
Pauschke);
%
Civil  and Mechanical System program, Directorate of Engineering of the National
Science  Foundation, under Award NSF-CMS-0337811 (cognizant program director Dr.
Steve  McCabe);
%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage
\clearpage
\renewcommand{\baselinestretch}{1.2}
\small  % trick from Kopka and Daly p47.
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\begin{appendix}

\section{Isotropic Hyperelastic Models}
\label{Models}

Derivatives needed to obtain stiffness tensors and stress measures for a set of
hyperelastic models are given in Table (\ref{models}).
Detail descriptions for these models can be found in Ogden \cite{Ogden84}, Miehe
\cite{Miehe94c} and Simo and Marsden \cite{Simo84b}.
%
\begin{table}[!htbp]
\caption{\label{models} Derivatives for stiffness tensors and stress measures of
various hyperelastic material models.}
\begin{center}
\begin{small}
{%\footnotesize
\begin{tabular}{|c||c|c|c|c|c|} \hline
Model       &$\frac{\partial{}^{iso}\!W }{\partial \tilde{\lambda}_{A}}$
            &$\frac{\partial^2 \left( {}^{iso}\!W  \right)}{\partial \tilde{\lambda}_{A}^{2}}$
            &$\frac{d {}^{vol} \! W(J)}{dJ}$
            &$\frac{d^2 {}^{vol} \! W(J)}{dJ^2}$        \\ \hline\hline
Ogden       &$\sum_{r=1}^{N}c_{r} \tilde{\lambda}_{A}^{\mu_{r}-1}$
            &$\sum_{r=1}^{N}c_{r}(\mu_{r}-1) \tilde{\lambda}_{A}^{\mu_{r}-2}$
            &$$
            &$$  \\ \hline
%
Neo-Hookean  &$G\tilde{\lambda}_{A}$
            &$G$
            &$K_b(J-1)$
            &$K_b$  \\ \hline
%
Mooney--Rivlin          &$2C_{1}\tilde{\lambda}_{A}-2C_{2}\tilde{\lambda}_{A}^{-3}$
            &$2C_{1}+6C_{2}\tilde{\lambda}_{A}^{-4}$
            &$$
            &$$  \\ \hline
%
Logarithmic         &$2G\tilde{\lambda}_{A}^{-1}\ln{\tilde{\lambda}_{A}}$
            &$2G\tilde{\lambda}_{A}^{-2}(1-\ln{\tilde{\lambda}_{A}})$
            &$K_b J^{-1} \ln{J}$
            &$K_b J^{-2}(1- \ln{J})$  \\ \hline
%
Simo--Pister          &$$
            &$$
            &$(J-\frac{1}{J}) \frac{K_b}{2}$
            &$(1 +\frac{1}{J^2} ) \frac{K_b}{2}$  \\ \hline
\end{tabular}
}
\end{small}
\end{center}
\end{table}




\end{appendix}

\end{document}
\bye



