Design of auxetic plates with only one degree of freedom

doi:10.1016/j.eml.2020.101091

Extreme Mechanics Letters

Volume 42, January 2021, 101091

https://doi.org/10.1016/j.eml.2020.101091 Get rights and content

Abstract

A continuum elastic plate has infinite degrees of freedom: according to the applied loads, it assumes the shape that the minimization of the total energy prescribes, in dependence of the material it is made of. The possibility to control the shape of a morphing structure is receiving an increasing attention in several fields; in this connection, if a classical elastic plate is considered, it is not possible, in general, to tune the elastic properties of the material in order to select, between the infinite deformation modes the plate may have, the one desired. We present a prototypical case that tries to satisfy this desideratum, opening the way to the systematic design of microstructure’s geometries to fully control a system’s shape independently of the applied loads. A novel yet simple architecture for thin plates having only one degree of freedom is proposed. The plate is realized as a tessellation composed by rigid equal hexagonal tiles hinged to each other along the sides, and it can deform in just one way, that is, into a predetermined synclastic surface, whatever loads are applied to it. Such tessellated plate also has a remarkable auxetic behavior in bending, with the ratio between transverse and longitudinal curvatures in uniaxial bending reaching values larger than one. Modeling assumptions and analysis results, at both the discrete and the continuum level, are verified by tests carried out onto additively manufactured bi-material specimens, showing that it is possible to design the deformed configuration by controlling the hexagons’ geometry. The proposed architecture for realizing auxetic plates with only one degree of freedom is highly scalable and easily manufacturable, and it can find applications for auxetic scaffolds, prosthetic stress shields, energy harvesters, and wearable devices.

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 $ℓ$ of four sides of the hexagon; the length $β ℓ$ ( $β > 0$ ) of two opposite sides, and the angle $ϑ$ between two different sides; for $β = 1$ and $ϑ = π ∕ 6$ the hexagons are regular.

We introduce the orthogonal basis ${e_{1}, e_{2}}$ 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 $k_{i}$ ( $i = 1, 2, 3$ ) of the springs is supposed to be different for each of the three edges sharing a nodal point $x^{ℓ} \in N (ℓ) ≔ {x \in R^{2} : x = n_{1} ℓ d^{1} + n_{2} ℓ d^{2} with (n_{1}, n_{2}) \in Z^{2}}$ (see Fig. 3, a).

In light of (9), the energy of the system can be written as $U_{ℓ} (φ) = \sum_{x^{ℓ} \in N (ℓ)} \frac{1}{2} k_{1} {(φ^{1} ℓ)}^{2} + \frac{1}{2} k_{2} {(φ^{2} ℓ)}^{2} + \frac{1}{2} k_{3} {(φ^{3} ℓ)}^{2} = \sum_{x^{ℓ} \in N (ℓ)} \frac{1}{2} k_{1} {(φ ℓ)}^{2} + \frac{1}{2} k_{2} {(φ ℓ)}^{2} + \frac{1}{2} k_{3} {(2 φ ℓ sin ϑ)}^{2} = \sum_{x^{ℓ} \in N (ℓ)} \frac{1}{2} \frac{k_{1} + k_{2} + 4 {sin}^{2} ϑ k_{3}}{2 (sin ϑ + β) cos ϑ} | E^{ℓ} (x^{ℓ}) | (φ)$

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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View full text

Design of auxetic plates with only one degree of freedom

Abstract

Introduction

Section snippets

Kinematics

Energetics

Tesselated plates

Discussion and concluding remarks

Declaration of Competing Interest

Acknowledgments

Scr. Mater.

Extreme Mech. Lett.

Extreme Mech. Lett.

Mater. Today

Mech. Res. Commun.

Compos. Sci. Technol.

Mater. Des.

Sensors Actuators A

Symmetry detection of auxetic behaviour in 2D frameworks

Europhys. Lett.

Geometric charges and nonlinear elasticity of two-dimensional elastic metamaterials

Proc. Natl. Acad. Sci. USA

Bistable auxetic mechanical metamaterials inspired by ancient geometric motifs

Extreme Mech. Lett.

Invariant and smooth limit of discrete geometry folded from bistable origami leading to multistable metasurfaces

Nature Commun.

A method for building self-folding machines

Science

Shape morphing kirigami mechanical metamaterials

Sci. Rep.

A three-dimensional actuated origami-inspired transformable metamaterial with multiple degrees of freedom

Nature Commun.

Keeping it together: Interleaved kirigami extension assembly

Phys. Rev. X

Kirigami-inspired inflatables with programmable shapes

Adv. Mater.

Geometrically controlled snapping transitions in shells with curved creases

Proc. Natl. Acad. Sci. USA