Fundamental conical defects: The d-cone, its e-cone, and its p-cone.

We consider well-known surface disclinations by cutting, joining, and folding pieces of paper card. The resulting shapes have a discrete, folded vertex whose geometry is described easily by Gauss's mapping, in particular, we can relate the degree of angular excess, or deficit, to the size of fold line rotations by the area enclosed by the vector diagram of these rotations. This is well known for the case of a so-called "d-cone" of zero angular deficit, and we formulate the same for a general disclination. This method allows us to observe kinematic properties in a meaningful way without needing to consider equilibrium. Importantly, the simple vector nature of our analysis shows that some disclinations are primitive; and that other types, such as d-cones, are amalgamations of them

k-cones and kirigami metamaterials.

We are inspired by the tensile buckling of a thin sheet with a slit to create a foldable planar metamaterial. The buckled shape comprises two pairs of identical e-cones connected to the slit, which we refer to as a k-cone. We approximate this shape as discrete vertices that can be folded out of plane as the slit is pulled apart. We determine their kinematics and we calculate generic shape properties using a simple elastic model of the folded shape. We then show how the folded sheet may be tessellated as a unit cell within a larger sheet, which may be constructed a priori by cutting and folding the latter in a regular way, in order to form a planar kirigami structure with a single degree of freedom

Nonlinear Growing Caps

Engineers capture growth strains in two ways, reflecting the inherent bending-stretching nature of shells: by a strain gradient through the thickness or by an average in-plane value. We analyse their interaction by assuming a uniform displacement and growth-strain field in shells with elastic spring supports and a radial force applied to their outer boundary. The increased degree of statical indeterminancy enriches the variety of existing solutions and we distinguish two in-plane actuation modes which can induce Gaussian curvature via radially varying quadratic expansions in either the circumferential or radial direction. Using a Rayleigh-Ritz approach, we find closed-form solutions of the Föppl-von Kármán shell equations for the buckling thresholds, bistable limits and the post-buckled shape, which show good agreement with finite element reference solutions and available results from the literature. Moreover, we show that ‘natural’ growth modes, which evoke a change of shape without incurring elastic strain energy, can be achieved by employing quadratic radial expansions only. Additionally, we study unsupported shells subjected to higher-order actuation distributions, which give rise to natural growth modes with varying wavenumbers. Our approach dramatically simplifies an otherwise non-trivial general solution, and may be applied in novel generations of smart materials with actively tunable material properties.Friedrich-Ebert-Foundatio

Spherical images and inextensible curved folding.

In their study, Duncan and Duncan [Proc. R. Soc. London A 383, 191 (1982)1364-502110.1098/rspa.1982.0126] calculate the shape of an inextensible surface folded in two about a general curve. They find the analytical relationships between pairs of generators linked across the fold curve, the shape of the original path, and the fold angle variation along it. They present two special cases of generator layouts for which the fold angle is uniform or the folded curve remains planar, for simplifying practical folding in sheet-metal processes. We verify their special cases by a graphical treatment according to a method of Gauss. We replace the fold curve by a piecewise linear path, which connects vertices of intersecting pairs of hinge lines. Inspired by the d-cone analysis by Farmer and Calladine [Int. J. Mech. Sci. 47, 509 (2005)IMSCAW0020-740310.1016/j.ijmecsci.2005.02.013], we construct the spherical images for developable folding of successive vertices: the operating conditions of the special cases in Duncan and Duncan are then revealed straightforwardly by the geometric relationships between the images. Our approach may be used to synthesize folding patterns for novel deployable and shape-changing surfaces without need of complex calculation

Efficient optimisation of structures using tabu search

This paper presents a novel approach to the optimisation of structures using a Tabu search (TS) method. TS is a metaheuristic which is used to guide local search methods towards a globally optimal solution by using flexible memory cycles of differing time spans. Results are presented for the well established ten bar truss problem and compared to results published in the literature. In the first example a truss is optimised to minimise mass and the results compared to results obtained using an alternative TS implementation. In the second example, the problem has multiple objectives that are compounded into a single objective function value using game theory. In general the results demonstrate that the TS method is capable of solving structural optimisation problems at least as efficiently as other numerical optimisation approaches

Numerical analysis of morphing corrugated plates

Abstract In this paper a numerical model for investigating the moment-rotation response of corrugated plates is presented. In particular, the effect of the geometry of the plate on the bending response is considered. Results are compared with a simplified theoretical model recently appeared in the literature. Combining geometrical effects and prestress, corrugated plates can become multistable forming the basis of new morphing structures

On the shape of bistable creased strips

We investigate the bistable behaviour of folded thin strips bent along their central crease. Making use of a simple Gauss mapping, we describe the kinematics of a hinge and facet model, which forms a discrete version of the bistable creased strip. The Gauss mapping technique is then generalised for an arbitrary number of hinge lines, which become the generators of a developable surface as the number becomes large. Predictions made for both the discrete model and the creased strip match experimental results well. This study will contribute to the understanding of shell damage mechanisms; bistable creased strips may also be used in novel multistable systems

Buckling of Naturally Curved Elastic Strips: The Ribbon Model Makes a Difference

International audienceWe analyze the stability of naturally curved, inextensible elastic ribbons. In experiments, we first show that a loop formed using a metallic strip can become unstable if its radius is smaller than its natural radius of curvature (undercurved case): the loop then folds onto itself into a smaller, multiply-covered loop. Conversely, a multi-covered, overcurved metallic strip can unfold dynamically into a circular configuration having a lower covering index. We analyze these instabilities using a one-dimensional mechanical model for an elastic ribbon introduced recently (Dias Audoly, 2014), which extends Sadowsky's developable elastic ribbon model in the presence of natural curvature. Combining linear stability analyses and numerical computations of the post-buckled configurations, we classify the equilibria of the ribbon as a function of the ratio of its natural curvature to its actual curvature. Our ribbon model is formulated in close analogy with classical rod models; this allows us to adapt classical stability methods for rods to the case of a ribbon. The stability of a ribbon is found to differ significantly from that of an anisotropic rod: we attribute this difference to the fact that the tangent twisting modulus of a ribbon can be negative, in contrast to what is possible in the well-studied case of linearly elastic rods. The specific stability properties predicted by the curved ribbon model are confirmed by a finite element analysis of cylindrical shells having a small height-to-radius ratio

In-situ multiscale shear failure of a bistable composite tape-spring

A bistable composite tape-spring (CTS) is stable in both the extended and coiled configurations, with fibres oriented at ±45°. It is light weight and multifunctional, and has attracted growing interest in shape-adaptive and energy harvesting systems in defence-, civil- and, especially aerospace engineering. The factors governing its bistability have been well-understood, but there is limited research concerning the mechanics of structural failure: here, we investigate the shear failure mechanisms in particular. We perform in-situ neutron diffraction on composite specimens using the ENGIN-X neutron diffractometer at Rutherford Appleton Laboratory (STFC, UK), and shear failure is characterised at both macroscopic and microscopic scales. Elastic and viscoelastic strain evolutions at different strain levels reveal the fundamentals of micromechanical shear failure, and their temperature dependency. Multiscale shear failure mechanisms are then proposed, which will benefit the optimisation of structural design to maintain structural integrity of CTS in aerospace applications