# What is the difference between the rate of rotation tensor and spin tensor?

In continuum mechanics the velocity gradient l is additively splitted into a symmetric d and skew-symmetric w part, where d is the rate of deformation tensor and w is called spin tensor.

From the polar decomposition of the deformation gradient F we have F = RU, where R is a proper othogonal tensor and U is the right stretch tensor.

In many textbooks one can find the following connection between these tensor and their time rates, respectively:

$$w = \dot{R}R^T + \frac{1}{2} R(\dot{U}U^{-1} - U^{-1}\dot{U})R^T$$

My question is:

1) What is the physical interpretation of w? Many textbooks say, it describes the rotation of a material element, but this is only true for a rigid body motion where U is zero and the formula reduces to the well known form of the angular velocity tensor of rigid body: $$w = \dot{R}R^T$$ Otherwise w contains also stretch/strain rate. Can someone tell please, how to interpret w and its terms. And whats the difference compared to the time rate of rotation $$\dot{R}$$ or $$\dot{R}R^T$$ which should describe the rotation rate rather than w in my opinion.

Thank you!

The term involving $\mathbf{U}$ represents the spin of the stretch eigendirections. It is a purely mathematical quantity, moreover it cannot be measured as $\mathbf{U}$ is in the reference placement. This is probably why it is so hard to interpret.