应变能
每个微小体积 \( dV \) 中局部应变能的通用公式为 —
\begin{align} \label{eq:13301a} \textsf{Local strain energy} = \small\frac{1}{2} \normalsize \, \textsf{stress} \, \textsf{strain} \, dV \end{align}
则整个构件的总应变能为 —
\begin{align} \label{eq:13302a} \textsf{Total strain energy} = \sum \textsf{Local strain energy} \end{align}
其中 —
| \( \sum \) | = 对整个体积求和 |
拉压载荷
对于仅承受拉伸或压缩的构件（轴向谐振器的典型情况），局部应变能由下式给出 —
\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}
其中 —
| \( \sigma \) | = 轴向应力 |
| \( \epsilon \) | = 轴向应变 |
| \( E \) | = 弹性模量 |
且其中胡克定律为 —
\begin{align} \label{eq:13304a} \sigma = E \, \epsilon \end{align}
纯剪切载荷
对于仅承受纯剪切载荷的构件（扭转谐振器的典型情况），局部应变能由下式给出 —
\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}
其中 —
| \( \tau \) | = 剪切应力 |
| \( \gamma \) | = 剪切应变 |
| \( G \) | = 剪切模量 |
且其中胡克定律为 —
\begin{align} \label{eq:13306a} \tau = G \, \gamma \end{align}
另见 —
动能
储能
细长纵向谐振构件中储存的能量
参考文献 —
Juvinall，第 145 - 149 页
Shigley，第 66 - 71 页
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 |
Tension-compression loading
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}
Pure shear loading
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}
Also see —
Kinetic energy
Stored energy
Energy stored in a thin longitudinally resonant member
References —
Juvinall, pp. 145 - 149
Shigley, pp. 66 - 71