11.3 Complementary energy

• We also define the term complementary energy as: \[\begin{equation*} U^c_0 = \int_0^\sigma {\varepsilon}d \sigma \end{equation*}\] \[\begin{equation*} U^c = \int_V U^c_0 \; dV \end{equation*}\]
• It can be shown that (for our uniaxial bar) \[\begin{equation*} y = {\frac{\partial U^c}{\partial P}} \end{equation*}\]
• This equation is commonly known as “Castigliano’s 2nd theorem.”

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11.3.1 Note: for Linear elastic materials

\[\begin{equation*} U = U^c \end{equation*}\]