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We study elastic ribbons subject to large, tensile pre-stress confined to acentral region within the cross-section. These ribbons can buckle spontaneouslyto form helical shapes, featuring regions of alternating chirality (phases)that are separated by so-called perversions (phase boundaries). Thisinstability cannot be described by classical rod theory, which incorporatespre-stress through effective natural curvature and twist; these are both zerodue to the mirror symmetry of the pre-stress. Using dimension reduction, wederive a one-dimensional (1D) 'rod-like' model from a plate theory, whichaccounts for inhomogeneous pre-stress as well as finite rotations. The 1D modelsuccessfully captures the qualitative features of torsional buckling under aprescribed end-to-end displacement and rotation, including the co-existence ofbuckled phases possessing opposite twist, and is in good quantitative agreementwith the results of numerical (finite-element) simulations and modelexperiments on elastomeric samples. Our model system provides a macroscopicanalog of phase separation and pressure-volume-temperature state diagrams, asdescribed by the classical thermodynamic theory of phase transitions.

Twisting instabilities in elastic ribbons with inhomogeneous pre-stress: a macroscopic analog of thermodynamic phase transition