  We identify the restricted class of attainable effective deformations in a
model of reinforced composites with parallel, long, and fully rigid fibers
embedded in an elastic body. In mathematical terms, we characterize the weak
limits of sequences of Sobolev maps whose gradients on the fibers lie in the
set of rotations. These limits are determined by an anisotropic constraint in
the sense that they locally preserve length in the fiber direction. Our proof
of the necessity emerges as a natural generalization and modification of the
recently established asymptotic rigidity analysis for composites with layered
reinforcements. However, the construction of approximating sequences is more
delicate here due to the higher flexibility and connectedness of the soft
material component. We overcome these technical challenges by a careful
approximation of the identity that is constant on the rigid components,
combined with a lifting in fiber bundles for Sobolev functions. The results are
illustrated with several examples of attainable effective deformations. If an
additional second-order regularization is introduced into the material model,
only rigid body motions can occur macroscopically.
