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\begin{document}

\section{Finite Elements}

\begin{tabular}{|c|c|cc|l|}
\hline
Symbol & Definition & Dimensions & Units & Meaning \\
\hline
$\o$ & $\o = \m{1 & \cdots & 1}^T$ & $d$ & $1$ & All ones vector \\
$\S$ & $\S = \m{\I & -\o}$ & $d \times (d+1)$ & $1$ & Scatter matrix \\
\hline
$\X_m$ & $\m{X_1 & X_2 & X_3 & X_4}_m$ & $d \times (d+1)$ & $m$ & Material space element node positions \\
$\X$ & $\m{X_1 & X_2 & X_3 & X_4}$ & $d \times (d+1)$ & $m$ & World space element node positions \\
$\Dm$ & $\Dm = \X_m \S^T$ & $d \times d$ & $m$ & Material space relative node positions \\
$\Ds$ & $\Ds = \X \S^T$ & $d \times d$ & $m$ & World space relative node positions \\
$\F$ & $\F = \Ds \Dm^{-1}$ & $d \times d$ & $1$ & Deformation gradient \\
\hline
$J$ & $J = \det(\F)$ & scalar & $1$ & Jacobian \\
$I_1$ & $I_1 = \tr(\F \F^T)$ & scalar & $1$ & First invariant \\
\hline
$V$ & $V = \frac{1}{d!} \det(\Dm)$ & scalar & $m^d$ & Material space element volume \\
$\N$ & $\N = V \Dm^{-T}$ & $d \times d$ & $m^{d-1}$ & Material space area weighted normals \\
\hline
$\psi$ & - & scalar & $kg \, m^{2-d} s^{-1}$ & Energy density \\
$\phi$ & $\phi = V \psi$ & scalar & $kg \, m^2 s^{-1}$ & Potential energy of element \\
$\f$ & $\f = -\pp{\phi}{\X}$ & $d \times (d+1)$ & $kg \, m s^{-1}$ & Force on element nodes \\
$\P$ & $\f = -\P \N \S^T$ & $d \times d$ & $kg \, m^{2-d} s^{-1}$ & First Piola-Kirchoff stress \\
\hline
\end{tabular}
\vspace{.2in}

Consider that a small position change $\delta \X$ is made.  Then
\begin{eqnarray*}
0 & = & \f : \delta \X + \delta \phi \\
& = & \tr(\f \delta \X^T) + \delta \phi \\
& = & \tr(-\P \N \S^T \delta \X^T) + V \delta \psi \\
& = & -V \tr(\P \Dm^{-T} \S^T \delta \X^T) + V \delta \psi \\
\delta \psi & = & \tr(\P \Dm^{-T} \S^T \delta \X^T) \\
& = & \tr(\P \delta (\Dm^{-T} \S^T \X^T)) \\
& = & \tr(\P \delta (\Dm^{-T} \Ds^T)) \\
& = & \tr(\P \delta \F^T) \\
& = & \P : \delta \F \\
\P & = & \pp{\psi}{\F}
\end{eqnarray*}

\section{Potential Energy for Neo Hookean}

Let $\psi = \frac{\mu}{2} (I_1 - d) - \mu \ln J + \frac{\lambda}{2} \ln^2 J$.

\begin{eqnarray*}
\pp{J}{\F} & = & \pp{}{\F} \det(\F) = \det(\F) \F^{-1} = J \F^{-1} \\
\pp{I_1}{\F} & = & \pp{}{\F} \tr(\F \F^T) = 2 \F \\
\pp{}{\F} & = & \pp{}{\F} \ln J = \F^{-1} \\
\P = \pp{\psi}{\X} & = & \pp{}{\X} \left( \frac{\mu}{2} (I_1 - d) - \mu \ln J + \frac{\lambda}{2} \ln^2 J \right) \\
& = & \frac{\mu}{2} \pp{I_1}{\X} - \mu \pp{}{\F} \ln J + \lambda \ln J \pp{}{\F} \ln J \\
& = & \mu (\F - \F^{-1}) + \lambda \ln J \F^{-1}
\end{eqnarray*}


\end{document}
