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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 477486, 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{NSFPEER ; NSF}{EEC9701568; CMS0337811}
\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}
%
% \begin{center}
% {\Large {\sc Version: \today{}, \hhmm{} }}
% \end{center}
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% \newpage
% \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 hyperelasticplastic approach and most of them stem from pioneering work by
Simo et al. \cite{local12, Simo88a, Simo88b, Simo92}.
%
%
A consistent hyperelastic computational formulation forms a basis of
all the developments for large deformation hyperelastoplasticity.
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
PiolaKirchhoff 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
nonequal 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 selfsufficient 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 (nonequal 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.
%
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\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 PiolaKirchhoff 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}).
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\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 PiolaKirchhoff stress is known. For example, the First
PiolaKirchhoff 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
$\tau_{ij}=F_{iA}(F_{jB})^{t}S_{AB}$.
%
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\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}
%
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\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 SimoSerrin 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}
%
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\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}
%
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\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 $G$.
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\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 7541985, 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}{rrrr} \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 MooneyRivlin 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 NeoHookean, Logarithmic, MooneyRivlinSimo and OgdenSimo model.
%
It is interesting to note that the logarithmic model response exhibits softening
in response (tractions vs. stretch) caused by the violation of
LegendreHadamard 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 NeoHookean, Logarithmic, and SimoPister 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 online.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\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 EEC9701568 (cognizant program director Dr. Joy
Pauschke);
%
Civil and Mechanical System program, Directorate of Engineering of the National
Science Foundation, under Award NSFCMS0337811 (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}{cccccc} \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
%
NeoHookean &$G\tilde{\lambda}_{A}$
&$G$
&$K_b(J1)$
&$K_b$ \\ \hline
%
MooneyRivlin &$2C_{1}\tilde{\lambda}_{A}2C_{2}\tilde{\lambda}_{A}^{3}$
&$2C_{1}+6C_{2}\tilde{\lambda}_{A}^{4}$
&$$
&$$ \\ \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
%
SimoPister &$$
&$$
&$(J\frac{1}{J}) \frac{K_b}{2}$
&$(1 +\frac{1}{J^2} ) \frac{K_b}{2}$ \\ \hline
\end{tabular}
}
\end{small}
\end{center}
\end{table}
\end{appendix}
\end{document}
\bye