# Rate of deformation tensor

## Main Question or Discussion Point

Hello, I have difficulty interpreting the following fact (I'm reading Cotinuum Mechanics by Spencer). The relative velocity between two nearby points P and Q in the current configuarion is given by: $dv_i=D_{ik}dx_k + W_{ik}dx_k$
where $D_{ik}=\frac{d}{dt}e_{ik}$ is the rate of deformation tensor being equal to the material derivative of the linear eulerian deformation tensor and the part involving the vorticity tensor W_ik can be interpreted as a pure rotation about an axes through the point P. This expression is "exact" despite the appearance of the linear eulerian tensor. Now comes the paradox: if D_ik=0 then all points near P simply "rotate rigidly" and this strongly suggest that there is no change of the (non linear) eulerian tensor E_ik which is and exact measure of deformation around P however only the linear part of the tensor vanishes so it seems that despite the pure rotation about P there is a change in the eulerian deformation tensor E_ik,ie, this tensor changes but the deformation does not change.
Can someone point out my mistake?

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