  Helical structures are ubiquitous in nature and engineering, ranging from DNA
molecules to plant tendrils, from sea snail shells to nanoribbons. While the
helical shapes in natural and engineered systems often exhibit nearly uniform
radius and pitch, helical shell structures with changing radius and pitch, such
as seashells and some plant tendrils, adds to the variety of this family of
aesthetic beauty. Here we develop a comprehensive theoretical framework for
tunable helical morphologies, and report the first biomimetic seashell-like
structure resulting from mechanics of geometric frustration. In previous
studies, the total potential energy is everywhere minimized when the system
achieves equilibrium. In this work, however, the local energy minimization
cannot be realized because of the geometric incompatibility, and hence the
whole system deforms into a shape with a global energy minimum whereby the
energy in each segment may not necessarily be locally optimized. This novel
approach can be applied to develop materials and devices of tunable geometries
with a range of applications in nano/biotechnology.
