Shell design from planar pre-stressed structures*

The interplay between mechanics and geometry is used to construct a theoretical framework able to describe the class of three-dimensional objects that can be fabricated from suitable planar designs by using relaxation of pre-strains/stress in ultra-thin films. Small deformations and large rotations are used here to model the elastic relaxation into various three-dimensional shapes. Over the kinematics associated with the designed mid-surface, a small perturbation of Love-Kirchhoff type is considered in order to deduce the design plate-to-shell equations for orthotropic materials with an important pre-stress/strain heterogeneity. The resulting equations for the efforts average and efforts moments provide the supplementary equations to compute the in-plane pre-strain/stress. In particular, for materials with a weak material transversal heterogeneity the moments equations involve only the thickness, the curvature tensor, and the pre-strain/stress moments. Special attention is devoted to materials that can be obtained by layer-by-layer crystal growth (molecular beam epitaxy), which posses an in-plane isotropic pre-strain. We have found that a rectangular plate could relax both into a cylindrical surface or on a part of a sphere in which case it should have a small diameter with respect to the sphere radius. In both cases, the theoretical estimates have been compared with the experimental realizations and finite-element numerical computations and we found very good agreement among all of them. In addition, for the cone geometry we found that the design is not possible from an isotropic pre-stress with an in-plane homogeneity. However, the 3D finite-element computation of the relaxed surface with a (necessary) non-isotropic pre-stress obtained from the theoretical estimates matches remarkably well the designed conical surface.