How Do I Include Aligned Equations In a Table?

The primary problem with including aligned equations in a tabular environment is that LaTeX uses the same alignment character '&' for indicating columns in the tabular environment and for indicating the alignment location in the aligned equation environment. So the LaTeX parser has trouble telling which & characters go with which environment. One solution to this is to typeset all the aligned equations into a saved box, and then place that box in the tabular environment. This prevents the parser from seeing extraneous & characters.

\documentclass{article}
\usepackage{amsmath}
\begin{document}

\newsavebox\unistrain
\begin{lrbox}{\unistrain}
  \begin{minipage}{0.3\textwidth}
    \begin{align*}
      \epsilon_1 &= \frac{\sigma_1}{E} \\
      \epsilon_2 &= -\nu \epsilon_1 \\
      \epsilon_3 &= -\nu \epsilon_1
    \end{align*} 
  \end{minipage}
\end{lrbox}

\newsavebox\unistress
\begin{lrbox}{\unistress}
  \begin{minipage}{0.3\textwidth}
    \begin{align*}
      \sigma_1 &= E \epsilon_1 \\
      \sigma_2 &= 0 \\
      \sigma_3 &= 0
    \end{align*} 
  \end{minipage}
\end{lrbox}

\begin{table}
  \label{tab:stress_strain_relations}
  \caption{Elastic Stress-Strain Relations}
  \begin{center}
    \begin{tabular}{lcc}
      Type of Stress & Principal Strains & Principal Stresses \\ \hline \hline
      Uniaxial & \usebox{\unistrain} & \usebox{\unistress} \\
    \end{tabular}
  \end{center}
\end{table}

\end{document}

Result: