On a bound for deformations in terms of Strains in an elastic cylinder

Consider a long thin isotropic elastic cylinder with a self-equilibrated loading on each end face, but which is stress-free on the sides and which has no internal body forces. It is shown that if the displacement gradient is pointwise sufficiently small, then, in any subcylinder of length 1/4 a, it is possible to add a rigid body motion such that the L₂ norm of the resulting displacement gradient can be bounded by a constant times the L₂ norm of the strain in a subcyclinder of length 2 a centered at the same point. The parameter a depends upon the pointwise bound for the displacement gradient (the smaller the bound, the larger a can be) and the constant is independent of the length thickness ratio of the subcylinder.