Rate-of-strain tensor

The rate-of-strain tensor is the symmetric part of the velocity gradient,

The shear rate is the scalar rate at which adjacent material layers acquire shear strain. In simple shear $u_x=u_x(y)$, its signed value is $\dot{\gamma }=d{u}_x/dy$.

A simple shear flow has velocity $\mathbf{u}=(\dot{\gamma }y,0,0)$ in Cartesian coordinates. Its velocity gradient has one nonzero off-diagonal entry.

The spin tensor is the antisymmetric part of the velocity gradient, $\mathbf{\Omega }=(\nabla \mathbf{u}-(\nabla \mathbf{u}{)}^T)/2$. It describes local rigid-body rotation.