\documentclass{article}
\usepackage{fullpage}
\usepackage{amsmath}
\usepackage{enumitem}
\usepackage{graphicx}
\usepackage{pstricks}
\usepackage{pst-plot}
\usepackage{subfigure}

\newcommand{\bigo}{\ensuremath{\mathcal{O}}}
\newcommand{\XX}{\ensuremath{\mathcal{X}}}
\newcommand{\X}{\ensuremath{x}}

\begin{document}
\begin{center}
{\bf CME306 / CS205B Homework 5}
\end{center}
\section*{Mass-Spring stress-strain relationship}
Consider the mass-spring model discussed in class.  In many ways, this is just a specific constitutive model for solids -- as such, there should be a way to write the stress-strain
relationship explicitly, given an appropriate choice of measure of strain.  This would show that mass-spring models are a subset of the material properties that can be modeled via finite
elements.  For this problem, we'll consider a two-dimensional triangle and derive some of the finite element components for this mass-spring system.

%\begin{figure}[htb]
%\centering
%\begin{pspicture}(0,0)(3,3)
%\psline[linewidth=.04]{-}(1,1)(3,1)
%\psline[linewidth=.04]{-}(1,1)(2,2.7)
%\psline[linewidth=.04]{-}(2,2.7)(3,1)
%\uput[l](1,1){$\XX_{1_0}$}\uput[ur](2.5,1.85){$\ell_{1_0}$}
%\uput[u](2,2.7){$\XX_{2_0}$}\uput[d](2,1){$\ell_{2_0}$}
%\uput[r](3,1){$\XX_{3_0}$}\uput[ul](1.5,1.85){$\ell_{3_0}$}
%\end{pspicture}
%\caption{\label{fig:triangle}An equilateral triangle in object space, whose edge springs are at rest.  For this example, take $\XX_{1_0}$ as the origin in material space.}
%\end{figure}
\begin{enumerate}
\item The Cauchy strain is a measure of strain that is invariant to a number of operations... Briefly explain why this measure is an {\em inappropriate} choice of strain for the constituative model of our mass-spring system.   \\ \\ \\ \\ \\ \\
\item Sketch the triangle in world space (labeling the edge-lengths as $\ell_1,\ell_2,\ell_3$), then write down the force contributions ($f_1,f_2,f_3$) from the triangle acting on nodes $\X_1,\X_2$ and $\X_3$ respectively.  You should assume that the springs are {\em not damped}. \\ \\ \\ \\ \\ \\
\item These forces represent different linear combinations of the traction that appears on each face of the triangle.  Write down the equations for these forces in terms of the Cauchy
  stress tensor (which is a matrix of four unknowns) and the area-weighted normals.  \\ \\ \\ \\ \\ \\
\item Solve these equations to get $\sigma$.


\end{enumerate}

\end{document}
