Covering a surface with pre-stressed ribbons : from theory to nano-structures fabrication

The paper deals with the fabrication of nano-shells from pre-stressed nano-plates release. Due to geometrical and technological restrictions we have to cover a given surface with three-dimensional thin ribbons. We discuss the key role of the geodesic curvature in the design of such shell-ribbons. We show that including small-strains but large rotations we are able to control the metric tensor of both Lagrangian and Eulerian ribbons by an appropriate choice of the width and thickness of the ribbons. Moreover, the Green-Lagrange strain tensor is controlled by the distance between the curvature of the planar ribbon and the geodesic curvature of the supporting curve of the shell-ribbon. Under suitable constitutive assumptions, we deduce the field equations, the boundary conditions and the design equations. The former relate the pre-stress in the planar layer to the final geometry of the desired shell-ribbon. A fine tuning of the composition, geometry and of the pre-stress of the plate-ribon is necessary to design and fabricate the shell-ribbon. We design and fabricate a partial cover of the sphere with constant latitude ribbons starting from planar multi-layer semiconductor materials grown by molecular beam epitaxy. The details of fabrication method and its limitations are discussed.