11.2 Strain energy and complementary strain energy

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11.2.1 Strain energy

In an elastic medium, the energy is stored as recoverable internal energy.

\[\begin{equation*} U_0 = \int_0^{\varepsilon}\sigma d {\varepsilon} \end{equation*}\]

\[\begin{equation*} U = \int_V U_0 \; dV \end{equation*}\]

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For the uniaxial bar:

\[\begin{equation*} U = A l \, U_0 \end{equation*}\] - \(U_0\) – specific strain energy or strain energy density - strain energy per unit volume

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11.2.2 Linear elastic material

\[\begin{equation*} \sigma = E {\varepsilon} \end{equation*}\]

\[\begin{equation*} U_0 = \int_0^{\varepsilon}E {\varepsilon}d {\varepsilon} \end{equation*}\] \[\begin{equation*} U_0 = \frac{1}{2} E {\varepsilon}^2 \end{equation*}\] \[\begin{equation*} U_0 = \frac{1}{2} \frac{\sigma^2}{E} \end{equation*}\] \[\begin{equation*} U_0 = \frac{1}{2} \sigma {\varepsilon} \end{equation*}\]

• For elastic materials,
  • the work due to external load is stored as energy in the bar
  • \(U=W\)

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• Therefore:
  • \(U=\int_0^y P dy\)
  • \(P = {\frac{\partial U}{\partial y}}\)
  • This equation is commonly known as “Castigliano’s 1st theorem.” (1879)