I am slightly confused about what the material derivative of displacement is. $$\frac{D}{Dt}=\frac{\partial}{\partial t}+ v\frac{ \partial}{\partial x}$$ which means that for the displacement we should have, $$\frac{Du}{Dt}=\frac{\partial u}{\partial t}+ v\frac{ \partial u}{\partial x}$$ but we also have that, $$v=\frac{Du}{Dt}=\frac{\partial u}{\partial t}$$
Does it mean $\frac{\partial u}{\partial x}=0$?
I noticed someone edited and changed my notations. Please do not do that without explaining why. I need an answer to the question, editing just creates more questions in my head plus I don't see anything wrong with my notations so I have got them back to their original form. Thanks.
The answer to your question depends upon whether you are taking an Eulerian or Lagrangian point of view. A discussion of this can be found here
http://en.wikipedia.org/wiki/Continuum_mechanics
In particular, section 7 which discusses Kinematics.