diff --git a/doc/neohooke_der.tex b/doc/neohooke_der.tex new file mode 100644 index 0000000000000000000000000000000000000000..a1079900aa9e30281fcdb181bb9c302770d77c80 --- /dev/null +++ b/doc/neohooke_der.tex @@ -0,0 +1,127 @@ +\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} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: t +%%% End: