Infinitesimal isometries on developable surfaces and asymptotic theories for thin developable shells

We perform a detailed analysis of first order Sobolev-regular infinitesimal isometries on developable surfaces without affine regions. We prove that given enough regularity of the surface, any first order infinitesimal isometry can be matched to an infinitesimal isometry of an arbitrarily high order. We discuss the implications of this result for the elasticity of thin developable shells

Disclination-mediated thermo-optical response in nematic glass sheets

Nematic solids respond strongly to changes in ambient heat or light, significantly differently parallel and perpendicular to the director. This phenomenon is well characterized for uniform director fields, but not for defect textures. We analyze the elastic ground states of a nematic glass in the membrane approximation as a function of temperature for some disclination defects with an eye towards reversibly inducing three-dimensional shapes from flat sheets of material, at the nano-scale all the way to macroscopic objects, including non-developable surfaces. The latter offers a new paradigm to actuation via switchable stretch in thin systems.Comment: Specific results for spiral defects now added. References to Witten, Mahadevan and Ben Amar now added

Equilibrium Shapes with Stress Localisation for Inextensible Elastic Mobius and Other Strips

We formulate the problem of finding equilibrium shapes of a thin inextensible elastic strip, developing further our previous work on the Möbius strip. By using the isometric nature of the deformation we reduce the variational problem to a second-order one-dimensional problem posed on the centreline of the strip. We derive Euler–Lagrange equations for this problem in Euler–Poincaré form and formulate boundary-value problems for closed symmetric one- and two-sided strips. Numerical solutions for the Möbius strip show a singular point of stress localisation on the edge of the strip, a generic response of inextensible elastic sheets under torsional strain. By cutting and pasting operations on the Möbius strip solution, followed by parameter continuation, we construct equilibrium solutions for strips with different linking numbers and with multiple points of stress localisation. Solutions reveal how strips fold into planar or self-contacting shapes as the length-to-width ratio of the strip is decreased. Our results may be relevant for curvature effects on physical properties of extremely thin two-dimensional structures as for instance produced in nanostructured origami

Free-form, form finding and anisotropic grid shell

p. 966-876The new geometrical developments open new perspectives for free-form design, making it possible to escape from planar triangular or quadrilateral discretizations. Recent advances in theory algorithms allow for the discretization of any surface using only single curvature panels thus allowing the realisation of smooth double curvature glazed envelops of any form. Grid shell structures usually present a nearly in plane uniform behaviour, but previous realisations have shown that grid shells can be designed also according to an anisotropic inplane arrangement. The control of principal direction and the fine tuning of the stiffness of the different structural elements (arcs, cables etc.) is a tool for adjusting the form-finding thus controlling the resulting geometry. Moreover, the form-finding can also be performed without researching a constant stress (self weight); in this case an even wider range of forms become possible. These new geometrical and structural approaches have been coupled together and tested in re-designing, as a case study, the glazed roof of the Neumunster Abbey in Luxembourg. Such approach allowed for the conception of an efficient structure supporting a smooth double curvature glass skin, made out of only single curvature panels, perfectly coherent with the perimeter of the courtyard i.e. matching all the edges without any gaps.Baldassini, N.; Raynaud, J. (2010). Free-form, form finding and anisotropic grid shell. Editorial Universitat Politècnica de València. http://hdl.handle.net/10251/696