# Strain energy

A form of potential_energy in which the energy is stored in the form of deformation or strain.

The general equation for the the local strain energy in each small volume $$dV$$is —

\begin{align} \label{eq:13301a} \textsf{Local strain energy} = \small\frac{1}{2} \normalsize \, \textsf{stress} \, \textsf{strain} \, dV \end{align}

Then the total strain energy for the entire member is —

\begin{align} \label{eq:13302a} \textsf{Total strain energy} = \sum \textsf{Local strain energy} \end{align}

where —

 $$\sum$$ = summation over entire volume

For a member that sees only in tension or compression (typical of axial resonators), the local strain energy is given by —

\begin{align} \label{eq:13303a} \textsf{Local strain energy} &= \small\frac{1}{2} \normalsize \, \sigma \, \epsilon \, dV \\[0.7em]%eqn_interline_spacing &= \small\frac{1}{2} \normalsize \, E \, \epsilon^2 \, dV \nonumber \\[0.7em]%eqn_interline_spacing &= \small\frac{1}{2} \normalsize \, \frac{\sigma^2}{E} \, \, dV \nonumber \end{align}

where —

 $$\sigma$$ = axial stress $$\epsilon$$ = axial strain $$E$$ = modulus of elasticity

and where Hooke's law is —

\begin{align} \label{eq:13304a} \sigma = E \, \epsilon \end{align}

For a member that is loaded only in pure shear (typical of torsional resonators) the local strain energy is given by —

\begin{align} \label{eq:13305a} \textsf{Local strain energy} &= \small\frac{1}{2} \normalsize \, \tau \, \gamma \, dV \\[0.7em]%eqn_interline_spacing &= \small\frac{1}{2} \normalsize \, G \, \gamma^2 \, dV \nonumber \\[0.7em]%eqn_interline_spacing &= \small\frac{1}{2} \normalsize \, \frac{\tau^2}{G} \, \, dV \nonumber \end{align}

where —

 $$\tau$$ = shear stress $$\gamma$$ = shear strain $$G$$ = shear modulus

and where Hooke's law is —

\begin{align} \label{eq:13306a} \tau = G \, \gamma \end{align}

References —
Juvinall, pp. 145 - 149
Shigley, pp. 66 - 71