Shell design from planar pre-stressed structures

Abstract

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.

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

“…The main geometric assumption in Danescu and Ionescu (2020) is that there exists a transformation x : R 0 → S 0 of the Lagrangian mid-surface R 0 into the designed Eulerian one S 0 such that the associated deformation of the geometric transformation is small, i.e.,…”

“…However, it is well-known that the class of isometries between planar and three-dimensional surfaces, extensively studied in Fosdick and Fried (2016), is too narrow to cover simple non-developable surfaces occurring in prestressed relaxation design problems. To circumvent this theoretical drawback, in a recent paper Danescu and Ionescu (2020) we developed a shell design model built on a non-isometric perturbation assumption of Love-Kirchhoff type superposed on a plate-to-shell theory. Extending shell models in Steigmann (2013), Ciarlet and Mardare (2018), Steigmann (2007b), Steigmann (2007a), Steigmann and Ogden (2014)) the geometric description involves a single small parameter δ 1, the product between the thickness of the shell and its curvature.…”

“…The main difficulty in applying the shell-design model in Danescu and Ionescu (2020) is of a geometric nature. Indeed, for several common mid-surfaces the smallstrain assumption drastically reduces the surface width.…”

“…To encompass this limitation, in Danescu and Ionescu (2021) another type of shell, (called a strip-shell) is constructed, for which this assumption can be fulfilled by an appropriate choice of an additional geometric parameter, namely the strip width. In this restricted framework, if the product between the strip-shell width and its curvature is of order δ 1/2 the assumptions of plate-to-shell theory Danescu and Ionescu (2020) are fulfilled so that, for any strip of a given shell we provide a simple model able to design the corresponding plate-strip (i.e., to compute the shape and pre-stress momentum of the plate). The next step analysed here is to cover the given surface (shell) with one or several strips, situation in which we can provide an explicit design of the corresponding planar (plate) strips.…”

“…The paper is organized as follows : the first two sections recall the geometric and mechanical assumptions of the plate to shell model for design proposed in Danescu and Ionescu (2020). We relate the geometric aspects to the pre-stress via constitutive relations and field equations in finite strains through the assumption of weak-transversal heterogeneity, assumption fulfilled here by the weak variation of the composition in our crystal growth process.…”