Stress of a spatially uniform dislocation density field

It can be shown that the stress produced by a spatially uniform dislocation density field in a body comprising a linear elastic material under no loads vanishes. We prove that the same result does not hold in general in the geometrically nonlinear case. This problem of mechanics establishes the purely geometrical result that the $\text{curl}$ of a sufficiently smooth two-dimensional rotation field cannot be a non-vanishing constant on a domain.