A couple of books on elasticity, e.g. Gould and Soutas-Little, in the section on 2d elasticity in cylindrical coordinates, mention the case of 'quasi-axisymmetric' problems. These are problems where the stress and strain retain and axial symmetry (i.e., no dependence on the angular coordinate $\theta$), whereas the displacements $u_r$ and $u_\theta$ may carry $\theta$ dependence.
In particular, on writing the expression for $u_\theta$: $$ u_\theta = A*r\theta + B*\cos\theta + C*\sin\theta + D*r $$ with A,B,C,D constants, both mention that the last three terms rigid body motions, and hence do not contribute to any strain.
This statement per se I understand. Rigid body motions cannot give rise to strains or stresses. However, while I can imagine that the $\cos\theta$ and $\sin\theta$ represent rotations, I do not see how the $r$ term cannot give rise to strains! If $u_\theta$ is (say) zero at the inner radius and finite at the outer radius, it must surely give rise to strains! What is going on? What am I misunderstanding?
Addition to answer below --
Please see my comment to the answer for an explanation of the $r\theta$ term as well.