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# Stress of a spatially uniform dislocation density field

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(To appear in Journal of Elasticity).

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 curl of a sufficiently smooth two-dimensional rotation

field cannot be a non-vanishing constant on a domain. It is classically known in continuum

mechanics, stated first by the brothers Cosserat [Shi73], that if a second order tensor field on

a simply connected domain is at most a curl-free field of rotations, then the field is necessarily

constant on the domain. It is shown here that, at least in dimension 2, this classical result is in

fact a special case of a more general situation where the curl of the given rotation field is only

known to be at most a constant.

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## Comments

## An interesting corollary

It is classically known in continuum mechanics, stated first by the brothers Cosserat [Shield, 1973], that if a second order tensor field on a simply connected domain is at most a curl-free field of rotations, then the field is necessarily constant on the domain. A corollary of the work above is that, at least in dimension 2, this classical result is in fact a special case of a more general situation where the curl of the given rotation field is only known to be at most a constant.

The classical result can be directly read off from the Rigidity Estimate of Friesecke, James, and Muller (and of course the Generalized Rigidity estimate of Muller, Scardia, Zeppieri (MSZ)). Reading off the present corollary from the Generalized Rigidity Estimate of MSZ would seem to require a little work (Irene Fonseca has shown me such a proof provided the constant in the MSZ Generalized Rigidity Estimate can be shown not to depend on the domain).