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Commit 0015ce36 authored by Oliver Sander's avatar Oliver Sander Committed by sander
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The derivatives of the functional for a Neo-Hookean material

[[Imported from SVN: r9275]]
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\documentclass{article}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{graphicx}
\usepackage{natbib}
\usepackage{psfrag}
\graphicspath{{gfx/}}
\providecommand{\parder}[2]{\frac{\partial #1}{\partial #2}}
\providecommand{\secparder}[3]{\frac{\partial {#1}^2}{\partial #2 \partial #3}}
\newcommand{\tr}{\operatorname{tr}}
\begin{document}
\section{The Derivatives of the Energy Functional of a Neo-Hookean Material}
\subsection{Preliminaries: The Derivative of the Determinant of the Deformation Gradient}
Let
\begin{equation*}
J = \det F = \det \nabla (X + u) = \det (I + \nabla u)
= \det (I + \frac{\partial u_i}{\partial X^j}).
\end{equation*}
The deformation $u$ is a finite element function $u = \sum_{i,j} u_i^j \phi_i^j$.
\subsubsection{In Two Space Dimensions}
\begin{eqnarray*}
\parder{J}{u_i^j}
& = & \parder{}{u_i^j}\det (I + \frac{\partial u_i}{\partial X^j}) \\
& = & \parder{}{u_i^j} \det
\begin{bmatrix}
1 + \sum_k u^0_k \parder{\phi^0_k}{X^0} & \sum_k u^1_k \parder{\phi^1_k}{X^0} \\
\sum_k u^0_k \parder{\phi^0_k}{X^1} & 1 + \sum_k u^1_k \parder{\phi^1_k}{X^1}
\end{bmatrix} \\
& = & \parder{}{u_i^j}
[ ( 1 + \sum_k u^0_k \parder{\phi^0_k}{X^0} ) (1 + \sum_k u^1_k \parder{\phi^1_k}{X^1})
- (\sum_k u^1_k \parder{\phi^1_k}{X^0}) (\sum_k u^0_k \parder{\phi^0_k}{X^1}) ]
\end{eqnarray*}
Thus,
\begin{eqnarray*}
\parder{J}{u_i^0}
& = &
\parder{\phi_i^0}{X^0} ( 1 + \sum_k u^1_k \parder{\phi^1_k}{X^1})
-
(\sum_k u^1_k \parder{\phi^1_k}{X^0}) \parder{\phi_i^0}{X^1} \\
%
\parder{J}{u_i^1}
& = &
\parder{\phi_i^1}{X^1} ( 1 + \sum_k u^0_k \parder{\phi^0_k}{X^0})
-
(\sum_k u^0_k \parder{\phi^0_k}{X^1}) \parder{\phi_i^1}{X^0}
\end{eqnarray*}
The four second derivatives are
\begin{eqnarray*}
\secparder{J}{u_i^0}{u_j^0}
& = & 0 \\
%\delta_{ij} \parder{\phi^0_i}{X^0} (1 + \sum_k u^1_k \parder{\phi^1_k}{X^1})
%- \delta_{ij} \parder{\phi^0_i}{X^1} (\sum_k u^1_k \parder{\phi^1_k}{X^0})\\
%
\secparder{J}{u_i^0}{u_j^1}
& = &
\parder{\phi_i^0}{X^0} \parder{\phi^1_j}{X^1}
- \parder{\phi^1_j}{X^0} \parder{\phi^0_i}{X^1}\\
%
\secparder{J}{u_i^1}{u_j^0}
& = &
\parder{\phi^1_j}{X^0} \parder{\phi^0_i}{X^1}
- \parder{\phi_i^0}{X^0} \parder{\phi^1_j}{X^1} \\
%
\secparder{J}{u_i^1}{u_j^1}
& = & 0
%\delta_{ij} \parder{\phi^1_i}{X^1} (1 + \sum_k u^0_k \parder{\phi^0_k}{X^0})
%- \delta_{ij} \parder{\phi^1_i}{X^0} (\sum_k u^0_k \parder{\phi^0_k}{X^1})
\end{eqnarray*}
\subsection{The Derivatives of $\tr E$}
\begin{eqnarray*}
\tr E = \frac 12 \tr (\nabla u + \nabla^T u + \nabla^T u \nabla u)
= \tr \nabla u + \frac 12 \tr \nabla^T u \nabla u
\end{eqnarray*}
\subsection{First Derivatives of $W$}
\begin{equation*}
W(u) = \frac{\lambda}{4} ( J^2 -1 ) - (\frac \lambda 2 + \mu) \ln J + \mu \tr E
\end{equation*}
Thus,
\begin{eqnarray*}
\parder{W}{u_i^j}
& = &
\frac{\lambda}{4} \parder{}{u_i^j} J^2
- (\frac \lambda 2 + \mu) \parder{}{u_i^j} \ln J
+ \mu \parder{}{u_i^j} \tr E \\
%
& = &
\frac{\lambda J}{2} \parder{J}{u_i^j}
- (\frac \lambda 2 + \mu) J^{-1} \parder{J}{u_i^j}
+ \mu \tr \parder{}{u_i^j} E
\end{eqnarray*}
\subsection{Second Derivatives of $W$}
\begin{eqnarray*}
\secparder{W}{u_i^j}{u_k^l}
& = &
\parder{}{u_k^l} \Big[ \frac{\lambda J}{2} \parder{J}{u_i^j}
- (\frac \lambda 2 + \mu) J^{-1} \parder{J}{u_i^j}
+ \mu \tr \parder{}{u_i^j} E \Big] \\
%
& = &
\frac{\lambda}{2} \Bigg[ \parder{J}{u_i^j} \parder{J}{u_k^l}
+ J \secparder{J}{u_i^j}{u_k^l} \Bigg] \\
& &
- (\frac \lambda 2 + \mu) J^{-2}
\Bigg[ \secparder{J}{u_i^j}{u_k^l} J - \parder{J}{u_i^j} \parder{J}{u_k^l} \Bigg]
+ \mu \tr \secparder{}{u_i^j}{u_k^l} E
\end{eqnarray*}
\end{document}
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