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Helical ribbons arise in many biological and engineered systems, often drivenby anisotropic surface stress, residual strain, and geometric or elasticmismatch between layers of a laminated composite. A full mathematical analysisis developed to analytically predict the equilibrium deformed helical shape ofan initially flat, straight ribbon, with prescribed magnitudes and orientationsof the principal curvatures when subjected to arbitrary surface stress and/orinternal residual strain distribution. The helix angle, radius, axis andchirality of the deformed helical ribbons are predicted with a comprehensive,three-dimensional model that incorporates elasticity, differential geometry,and variational principles. In general, the mechanical anisotropy (e.g., insurface/external stress, residual strain or elastic modulus) will lead tospontaneous, three-dimensional helical deformations. Ring shapes are formedwhen the principle axes of deformation coincide with the geometric axes of theribbon. The transition from cylindrical helical ribbons to purely twistedribbons, or tubular structures is controlled by tuning relevant geometricparameters. Analytic, closed-form predictions of the ribbon shapes arevalidated with simple, table-top experiments. This theoretical approachrepresent a tool to inform the design of materials and systems in order toachieve desired helical geometries on demand.

Continuum Elasticity Theory Approach for Spontaneous Bending and Twisting of Ribbons Induced by Mechanical Anisotropy