We present an analytical model to investigate the mechanics of 2-dimensional lattices composed of elastic beams of non-uniform cross-section.Our approach is based on reducing a lattice to a single beam subject to the action of a set of linear and torsional springs, thus allowing the problem to be solved through a transfer matrix method. We show a non-trivial region of design space that yields materials with short wavelength modes (closely associated with auxetic behaviour) for strains greater than that required to trigger elastic instability. The critical loading required to make this transition from long to short wavelength buckling modes is calculated. Furthermore, we present lattice parameters that provide direction-dependent deformation modes offering great tailorability of the mechanical properties of finite size lattices. Not only is our analytical formulation in good agreement with the finite element simulation results, but it provides an insight into the role of the interplay between structure and elastic instability, and gives an efficient methodology to pursue questions of rational design in the field of mechanical metamaterials.
The equations for the equilibrium of a thin elastic ribbon are derived by adapting the classical theory of thin elastic rods. Previously established ribbon models are extended to handle geodesic curvature, natural out-of-plane curvature, and a variable width. Both the case of a finite width (Wunderlich’s model) and the limit of small width (Sadowksky’s model) are recovered. The ribbon is assumed to remain developable as it deforms, and the direction of the generatrices is used as an internal variable. Internal constraints expressing inextensibility are identified. The equilibrium of the ribbon is found to be governed by an equation of equilibrium for the internal variable involving its second-gradient, by the classical Kirchhoff equations for thin rods, and by specific, thin-rod-like constitutive laws; this extends the results of Starostin and van der Heijden (2007) to a general ribbon model. Our equations are applicable in particular to ribbons having geodesic curvature, such as an annulus cut out in a piece of paper. Other examples of application are discussed. By making use of a material frame rather than the Frénet-Serret’s frame, the present work unifies the description of thin ribbons and thin rods.
We consider the equilibrium shapes of a thin, annular strip cut out in an elastic sheet. When a central fold is formed by creasing beyond the elastic limit, the strip has been observed to buckle out-of-plane. Starting from the theory of elastic plates, we derive a Kirchhoff rod model for the folded strip. A non-linear effective constitutive law incorporating the underlying geometrical constraints is derived, in which the angle the ridge appears as an internal degree of freedom. By contrast with traditional thin- walled beam models, this constitutive law captures large, non-rigid deformations of the cross-sections, including finite variations of the dihedral angle at the ridge. Using this effective rod theory, we identify a buckling instability that produces the out-of-plane configurations of the folded strip, and show that the strip behaves as an elastic ring having one frozen mode of curvature. In addition, we point out two novel buckling patterns: one where the centerline remains planar and the ridge angle is modulated; another one where the bending deformation is localized. These patterns are observed experimentally, explained based on stability analyses, and reproduced in simulations of the post-buckled configurations.
We develop recursion equations to describe the three-dimensional shape of a sheet upon which a series of concentric curved folds have been inscribed.In the case of no stretching outside the fold, the three-dimensional shape of a single fold prescribes the shape of the entire origami structure. To better explore these structures, we derive continuum equations, valid in the limit of vanishing spacing between folds, to describe the smooth surface intersecting all the mountain folds. We find that this surface has negative Gaussian curvature with magnitude equal to the square of the fold’s torsion. A series of open folds with constant fold angle generate a helicoid.
Folding a sheet of paper along a curve can lead to structures seen in decorative art and utilitarian packing boxes. Here we present a theory for the simplest such structure: an annular circular strip that is folded along a central circular curve to form a three-dimensional buckled structure driven by geometrical frustration. We quantify this shape in terms of the radius of the circle, the dihedral angle of the fold, and the mechanical properties of the sheet of paper and the fold itself. When the sheet is isometrically deformed everywhere except along the fold itself, stiff folds result in creases with constant curvature and oscillatory torsion. However, relatively softer folds inherit the broken symmetry of the buckled shape with oscillatory curvature and torsion. Our asymptotic analysis of the isometrically deformed state is corroborated by numerical simulations that allow us to generalize our analysis to study structures with multiple curved creases.
Recent experiments have imposed controlled swelling patterns on thin polymer films, which subsequently buckle into three-dimensional shapes. We develop a solution to the design problem suggested by such systems, namely, if and how one can generate particular three-dimensional shapes from thin elastic sheets by mere imposition of a two-dimensional pattern of locally isotropic growth. Not every shape is possible. Several types of obstruction can arise, some of which depend on the sheet thickness. We provide some examples using the axisymmetric form of the problem, which is analytically tractable.