- #1
stanley.st
- 30
- 0
Hello!
I read somewhere about intro to continuum mechanics. There was a vector \vec{\mu} and displacement vector \delta\vec{\mu}. As vector \vec{\mu} move, it will get new position
\vec{\mu}'=\vec{\mu}+\delta\vec{\mu}
\vec{\mu}'=\vec{\mu}+\frac{\partial\vec{\mu}}{\partial x_i}\delta x_i=\vec{\mu}+\left(\frac{\partial\vec{\mu}}{\partial x_i}+\frac{1}{2}\frac{\partial\vec{\mu}}{\partial x_j}-\frac{1}{2}\frac{\partial\vec{\mu}}{\partial x_j}\right)\delta x_i=\vec{\mu}+\left[\frac{1}{2}\left(\frac{\partial\vec{\mu}}{\partial x_i}+\frac{\partial\vec{\mu}}{\partial x_j}\right)+\frac{1}{2}\left(\frac{\partial\vec{\mu}}{\partial x_i}-\frac{\partial\vec{\mu}}{\partial x_j}\right)\right]\delta x_i
Last component
\frac{1}{2}\left(\frac{\partial\vec{\mu}}{\partial x_i}-\frac{\partial\vec{\mu}}{\partial x_j}\right)
represent rotation. Can you explain me that? I don't understand this rotation.
I read somewhere about intro to continuum mechanics. There was a vector \vec{\mu} and displacement vector \delta\vec{\mu}. As vector \vec{\mu} move, it will get new position
\vec{\mu}'=\vec{\mu}+\delta\vec{\mu}
\vec{\mu}'=\vec{\mu}+\frac{\partial\vec{\mu}}{\partial x_i}\delta x_i=\vec{\mu}+\left(\frac{\partial\vec{\mu}}{\partial x_i}+\frac{1}{2}\frac{\partial\vec{\mu}}{\partial x_j}-\frac{1}{2}\frac{\partial\vec{\mu}}{\partial x_j}\right)\delta x_i=\vec{\mu}+\left[\frac{1}{2}\left(\frac{\partial\vec{\mu}}{\partial x_i}+\frac{\partial\vec{\mu}}{\partial x_j}\right)+\frac{1}{2}\left(\frac{\partial\vec{\mu}}{\partial x_i}-\frac{\partial\vec{\mu}}{\partial x_j}\right)\right]\delta x_i
Last component
\frac{1}{2}\left(\frac{\partial\vec{\mu}}{\partial x_i}-\frac{\partial\vec{\mu}}{\partial x_j}\right)
represent rotation. Can you explain me that? I don't understand this rotation.