Design of auxetic plates with only one degree of freedom
Introduction
Architected structures and metamaterials can be designed to achieve certain desired macroscopic properties which are induced by the particular geometry and connectivity of constituent elements, and which are not necessarily dependent on the properties of particular materials employed. In particular, morphing and reconfigurable structures, such as textured materials, bistable auxetic metamaterials, origami- and kirigami-inspired structures, self-foldable/deployable systems, just to mention a few examples, have attracted increasing attention in the past few years.
In [1] a profiled metallic sheet with discrete, self-locking modes of deformation is analyzed. In [2] a sufficient condition for equiauxetic behavior of a 2D material is derived. The design of auxetic structures with enhanced mechanical properties is addressed in [3]. In [4], metal and plastic auxetic metamaterials are modeled and tested, while in [5] the two-dimensional (2D) elastic problem for flexible metamaterial is mapped into that of nonlinear interaction of elastic charges. In [6], a class of switchable architected materials exhibiting simultaneous auxeticity and structural bistability are analyzed. The design, fabrication and testing of bistable lattices with tensegrity architecture and nanoscale features are investigated in [7]. In [8], recent origami and kirigami techniques for obtaining curved geometries from flat sheets are reviewed. Multi-stable metasurfaces with programmable non-Euclidean geometries are proposed in [9]. In [10], shape-memory composites are used to recreate fundamental folded patterns, derived from computational origami, that can be extrapolated to a wide range of geometries and mechanisms. In [11], shape morphing kirigami mechanical metamaterials are demonstrated. In [12], the analysis and testing of modular origami-like transformable metamaterials are presented. Interleaved kirigami assemblies are proposed in [13]. In [14] kirigami-inspired inflatables with programmable shapes are designed. Origami sunscreens with energy harvesting capabilities are presented in [15]. A geometrical method for designing multistable structures is provided in [16]. 4D printing of reconfigurable and tunable metamaterials is performed in [17]. In [18], it is demonstrated a strategy based on external light for programmed self-folding of polymer sheets. A magnetically responsive origami system is introduced in [19].
In this work, we present a simple novel architecture for thin plates having only one degree of freedom, in which the design of the microstructure translates into a continuum system with only one deformation mode. Such plates are realized as finite periodic tessellations of rigid hexagonal tiles, hinged to each other along the sides. The geometry of each tile is described by two parameters, identifying all possible hexagons with two symmetry axes.1 By exploiting both the discrete and the continuum descriptions of their kinematics, such structures are shown to feature a single admissible infinitesimal motion, leading to a synclastic (auxetic) deformation of the plate. Closed-form expressions of the principal curvatures are given in terms of the geometric parameters, making it possible to design plates deforming always into the same desired synclastic shape, independently of the activating loads.
Quite remarkably, under uniaxial bending these plates can achieve a ratio between transverse and longitudinal curvatures higher than one. The bending stiffness of a plate can be computed by attaching an angular spring to each hinge, and it is found to depend on the spring constants, the hexagon’s geometry, and the overall dimensions of the plate. Additive manufacturing techniques have been employed to realize some samples in order to verify experimentally the analytical findings. Different tessellated plates have been fabricated using polylactic acid filaments for the hexagons and polyurethane elastic filaments for the network of hinges, verifying the analytical predictions on the ratio between principal curvatures.
The paper is organized as follows. The kinematic analysis of the tessellated plate is performed in Section 2, at both the discrete level (Section 2.1) and the continuum level (Section 2.2), while energetic considerations are stated in Section 3. Section 4 is devoted to report the experimental results, and in Section 5 the implications of the present findings are discussed.
Section snippets
Kinematics
We consider a plate with a microstructure composed of non-regular rigid hexagonal tiles, connected between them by an elastic device, able to store energy whenever the angle between two adjacent hexagons changes (see Fig. 1). The relevant geometric parameters are: the length of four sides of the hexagon; the length () of two opposite sides, and the angle between two different sides; for and the hexagons are regular.
We introduce the orthogonal basis and the
Energetics
Let us now suppose that between the tiles there is a spring, storing energy whenever the angle between the tiles changes. The stiffness () of the springs is supposed to be different for each of the three edges sharing a nodal point (see Fig. 3, a).
In light of (9), the energy of the system can be written as
Tesselated plates
The additive manufacturing technique based on fused deposition modeling (FDM) is adopted to realize physical models of our tessellations. The structures are composed of non-regular hexagonal tiles provided with v-shaped re-entrant edges, nested within elastic nets, that materialize the hinges. The tiles are fabricated with polylactic acid (PLA) filaments, with 1.75 mm diameter, while the nets are manufactured with polyurethane elastic filaments, with the same diameter. An i3 MK3 3D Prusa
Discussion and concluding remarks
We presented a simple novel architecture for thin plates having a single deformation mode, independently of the applied loads and characterized by an unprecedented auxetic bending behavior. The honeycomb tessellation of the plate determines such deformation mode by means of the two geometric parameters identifying the hexagonal tile. Although the Poisson coefficient cannot be defined, the ratio between principal curvatures, always positive, provides a measure of the auxetic bending behavior of
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
A.M. acknowledges financial support from the Italian Ministry of Education, University, and Research (MIUR) under the PRIN 2017 National Grant “3D printing: a bridge to the future” (grant number 2017L7X3CS_004).
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