\documentclass[11pt,reqno]{article}
\usepackage{graphicx}
\usepackage{xspace}
\usepackage{doublespace}
\usepackage{calc,ifthen}
\usepackage{version}
\usepackage{array}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{lscape}
\usepackage[raggedright,figtopcap,TABTOPCAP,tight]{subfigure}
\usepackage{fancyhdr}
\usepackage{lastpage}
\usepackage{amsmath_jsb}
\usepackage[notcite,saferef,notref]{saferef_jsb}
\usepackage{url_jsb}
\usepackage{float_jsb}
\usepackage{variables_jsb2}
\usepackage{jsb}
\headheight 0.in %added 1/25/90
\headsep 0.in %added 1/25/90
\textheight 9.in %from 8.in before 1/25/90
\textwidth 6.46in
\oddsidemargin 0.02in
\evensidemargin 0.02in
\parskip 0.6ex


\setstretch{1.2}
\renewcommand{\arraystretch}{1.2}
\begin{document}

%From SRD final report
\include{JSB_definitions.1999-09-02}

\excludeversion{test}
\excludeversion{comment}

\setlength{\topmargin}{-0.5in}
\setlength{\headsep}{0.5in}
\pagestyle{fancy}
\lhead{\textsc{PeGn624a}}
\chead{Compositional Modeling - Application to Enhanced Oil Recovery }
\rhead{Spring 2009}
\lfoot{}
\cfoot{\thepage\ of \pageref{LastPage}}
\rfoot{}

%========  v TitlePage v =======================
{\large\textbf{\textsc{PeGn624} Homework \# 7}}
\begin{itemize}
\item Assigned: 2009-01-29
\item Due: 2009-02-09
\end{itemize}


%========  v Definitions v =======================

\refname{figure}{Figure}{Figures}
\refname[Fig]{figure}{Fig.}{Figs.}
\refname[Figure]{figure}{Figure}{Figures}

\refname{table}{Table}{Tables}
\refname[Tab]{table}{Table}{Tables}
\refname[Table]{table}{Table}{Tables}

\refname{section}{Section}{Sections}
\refname[Sec]{section}{Sec.}{Secs.}
\refname[Section]{section}{Section}{Sections}

\refname{equation}{Eq.}{Eqs.}
\refname[Eq]{equation}{Eq.}{Eqs.}
\refname[Equation]{equation}{Equation}{Equations}

\newcommand{\XX}{\textsf{X}}
\newcommand{\fug}{\mathfrak{f}}
\newcommand{\ff}{\textsf{f}}
\newcommand{\liq}{\textsc{l}}
\newcommand{\vap}{\textsc{v}}

\newcolumntype{R}{>{$}r<{$}}
\newcolumntype{C}{>{$}c<{$}}

%========  v Main Paper v =======================

Please derive the effective horizontal stress equation \Eqref{effstress}.
\begin{equation}\label{effstress}
  \sigma'_{h} = \frac{\nu}{1-\nu} \left( \sigma_{v} - \alpha_P P - 3 \beta_T K (T-T_{R}) \right)
\end{equation}
For both derivations, use the following assumptions, \Eqrefrange{ass1,ass3}.
\begin{equation} \label{ass1}
  \sigma_h = \sigma_{xx} = \sigma_{yy}
\end{equation}
\begin{equation} \label{ass2}
  \sigma_{v} = \sigma_{zz}
\end{equation}
\begin{equation} \label{ass3}
  \epsilon_{xx} = \epsilon_{yy} = 0
\end{equation}
The following relationships may be useful, \Eqrefrange{rel1,rel3}.
\begin{equation} \label{rel1}
  G = \frac{E}{2 (1+\nu)}
\end{equation}
\begin{equation} \label{rel2}
  \lambda = \frac{\nu E}{(1+\nu)(1-2 \nu)}
\end{equation}
\begin{equation} \label{rel3}
  K = \frac{E}{3 (1-2 \nu)}
\end{equation}

\begin{enumerate}
\item   Start with Hooke's Law, \Eqrefrange{simple1,simple2}.
\begin{equation} \label{simple1}
  \epsilon_{xx} = \frac{1}{E} \left( \sigma_x - \nu (\sigma_y + \sigma_z) \right) 
\end{equation}
\begin{equation} \label{simple2}
  \epsilon_{yy} = \frac{1}{E} \left( \sigma_y - \nu (\sigma_x + \sigma_z) \right)
\end{equation}
\item  Start with the general stress-strain relationship, \Eqref{general}.
\begin{equation}\label{general}
  \sigma_{ij} = 2 G \epsilon_{ij} + \lambda \epsilon_b \delta_{ij} + \alpha_{P} P \delta_{ij} + 3 \beta_{Tb} K_{b} (T-T_{R}) \delta_{ij}
\end{equation}
\end{enumerate}


\end{document}






