  We study elastic ribbons subject to large, tensile pre-stress confined to a
central region within the cross-section. These ribbons can buckle spontaneously
to form helical shapes, featuring regions of alternating chirality (phases)
that are separated by so-called perversions (phase boundaries). This
instability cannot be described by classical rod theory, which incorporates
pre-stress through effective natural curvature and twist; these are both zero
due to the mirror symmetry of the pre-stress. Using dimension reduction, we
derive a one-dimensional (1D) 'rod-like' model from a plate theory, which
accounts for inhomogeneous pre-stress as well as finite rotations. The 1D model
successfully captures the qualitative features of torsional buckling under a
prescribed end-to-end displacement and rotation, including the co-existence of
buckled phases possessing opposite twist, and is in good quantitative agreement
with the results of numerical (finite-element) simulations and model
experiments on elastomeric samples. Our model system provides a macroscopic
analog of phase separation and pressure-volume-temperature state diagrams, as
described by the classical thermodynamic theory of phase transitions.
