Equiauxetic Hinged Archimedean Tilings

There is increasing interest in two-dimensional and quasi-two-dimensional materials and metamaterials for applications in chemistry, physics and engineering. Some of these applications are driven by the possible auxetic properties of such materials. Auxetic frameworks expand along one direction when subjected to a perpendicular stretching force. An equiauxetic framework has a unique mechanism of expansion (an equiauxetic mode) where the symmetry forces a Poisson’s ratio of −1. Hinged tilings offer opportunities for the design of auxetic and equiauxetic frameworks in 2D, and generic auxetic behaviour can often be detected using a symmetry extension of the scalar counting rule for mobility of periodic body-bar systems. Hinged frameworks based on Archimedean tilings of the plane are considered here. It is known that the regular hexagonal tiling, {63}, leads to an equiauxetic framework for both single-link and double-link connections between the tiles. For single-link connections, three Archimedean tilings considered as hinged body-bar frameworks are found here to be equiauxetic: these are {3.122}, {4.6.12}, and {4.82}. For double-link connections, three Archimedean tilings considered as hinged body-bar frameworks are found to be equiauxetic: these are {34.6}, {32.4.3.4}, and {3.6.3.6}.

An Auxetic System Based on Interconnected Y-Elements Inspired by Islamic Geometric Patterns

A 2D mechanical metamaterial exhibiting perfectly auxetic behavior, i.e., Poisson’s ratio of −1, is proposed in this paper drawing upon inspiration from an Islamic star formed by circumferential arrangement of eight squares, such as the one found at the exterior of the Ghiyathiyya Madrasa in Khargird, Iran (built 1438–1444 AD). Each unit of the metamaterial consists of eight pairs of pin-jointed Y-shaped rigid elements, whereby every pair of Y-elements is elastically restrained by a spiral spring. Upon intermediate stretching, each metamaterial unit resembles the north dome of Jameh Mosque, Iran (built 1087–1088 AD), until the attainment of the fully opened configuration, which resembles a structure in Agra, India, near the Taj Mahal. Both infinitesimal and finite deformation models of the effective Young’s modulus for the metamaterial structure were established using strain energy approach in terms of the spiral spring stiffness and geometrical parameters, with assumptions to preserve the eight-fold symmetricity of every metamaterial unit. Results indicate that the prescription of strain raises the effective Young’s modulus in an exponential manner until full extension is attained. This metamaterial is useful for applications where the overall shape of the structure must be conserved in spite of uniaxial application of load, and where deformation is permitted under limited range, which is quickly arrested as the deformation progresses.

Programming shape using kirigami tessellations

Kirigami tessellations, regular planar patterns formed by partially cutting flat, thin sheets, allow compact shapes to morph into open structures with rich geometries and unusual material properties. However, geometric and topological constraints make the design of such structures challenging. Here we pose and solve the inverse problem of determining the number, size and orientation of cuts that enables the deployment of a closed, compact regular kirigami tessellation to conform approximately to any prescribed target shape in two or three dimensions. We first identify the constraints on the lengths and angles of generalized kirigami tessellations that guarantee that their reconfigured face geometries can be contracted from a non-trivial deployed shape to a compact, non-overlapping planar cut pattern. We then encode these conditions into a flexible constrained optimization framework to obtain generalized kirigami patterns derived from various periodic tesselations of the plane that can be deployed into a wide variety of prescribed shapes. A simple mechanical analysis of the resulting structure allows us to determine and control the stability of the deployed state and control the deployment path. Finally, we fabricate physical models that deploy in two and three dimensions to validate this inverse design approach. Altogether, our approach, combining geometry, topology and optimization, highlights the potential for generalized kirigami tessellations as building blocks for shape-morphing mechanical metamaterials.

Simplified 2D Skin Lattice Models for Multi-Axial Camber Morphing Wing Aircraft

Conventional fixed wing aircraft require a selection of certain thickness of skin material that guarantees structural strength for aerodynamic loadings in various flight modes. However, skin structures of morphing wings are expected to be flexible as well as stiff to structural and coupled aerodynamic loadings from geometry change. Many works in the design of skin structures for morphing wings consider only geometric compliance. Among many morphing classifications, we consider camber rate change as airfoil morphing that changes its rate of the airfoil that induces warping, twisting, and bending in multi-axial directions, which makes compliant skin design for morphing a challenging task. It is desired to design a 3D skin structure for a morphing wing; however, it is a computationally challenging task in the design stage to optimize the design parameters. Therefore, it is of interest to establish the structure design process in rapid approaches. As a first step, the main theme of this study is to numerically validate and suggest simplified 2D plate models that fully represents multi-axial 3D camber morphing. In addition to that, the authors show the usage of lattice structures for the 2D plate models’ skin that will lead to on-demand design of advanced structure through the modification of selected structure.

Computational Analysis of 3D Lattice Structures for Skin in Real-Scale Camber Morphing Aircraft

Conventional or fixed wings require a certain thickness of skin material selection that guarantees structurally reliable strength under expected aerodynamic loadings. However, skin structures of morphing wings need to be flexible as well as stiff enough to deal with multi-axial structural stresses from changed geometry and the coupled aerodynamic loadings. Many works in the design of skin structures for morphing wings take the approach either of only geometric compliance or a simplified model that does not fully represent 3D real-scale wing models. Thus, the main theme of this study is (1) to numerically identify the multi-axial stress, strain, and deformation of skin in a camber morphing wing aircraft under both structure and aerodynamic loadings, and then (2) to show the effectiveness of a direct approach that uses 3D lattice structures for skin. Various lattice structures and their direct 3D wing models have been numerically analyzed for advanced skin design.

An Anisotropic Auxetic 2D Metamaterial Based on Sliding Microstructural Mechanism

A new 2D microstructure is proposed herein in the form of rigid unit cells, each taking the form of a cross with two opposing crossbars forming slots and the other two opposing crossbars forming sliders. The unit cells in the microstructure are arranged in a rectangular array in which the nearest four neighboring cells are rotated by 90° such that a slider in each unit cell is connected to a slot from its nearest neighbor. Using a kinematics approach, the Poisson’s ratio along the axes of symmetry can be obtained, while the off-axis Poisson’s ratio is obtained using Mohr’s circle. In the special case of a square array, the results show that the Poisson’s ratio varies between 0 (for loading parallel to the axes) and −1 (for loading at 45° from the axes). For a rectangular array, the Poisson’s ratio varies from 0 (for loading along the axes) to a value more negative than −1. The obtained results suggest the proposed microstructure is useful for designing materials that permit rapid change in Poisson’s ratio for angular change.

Auxetic deformations and elliptic curves

In materials science and engineering, auxetic behavior refers to deformations of flexible structures where stretching in some direction involves lateral widening, rather than lateral shrinking. We address the problem of detecting auxetic behavior for flexible periodic bar-and-joint frameworks. Currently, the only known algorithmic solution is based on the rather heavy machinery of fixed-dimension semi-definite programming. In this paper we present a new, simpler algorithmic approach which is applicable to a natural family of three-dimensional periodic bar-and-joint frameworks with three degrees of freedom. This class includes most zeolite structures, which are important for applications in computational materials science. We show that the existence of auxetic deformations is related to properties of an associated elliptic curve. A fast algorithm for recognizing auxetic capabilities is obtained via the classical Aronhold invariants of the cubic form defining the curve. A related alternative is also considered.

Analytic analysis of auxetic metamaterials through analogy with rigid link systems

In recent years, many structural motifs have been designed with the aim of creating auxetic metamaterials. One area of particular interest in this subject is the creation of auxetic material properties through elastic instability. Such metamaterials switch from conventional behaviour to an auxetic response for loads greater than some threshold value. This paper develops a novel methodology in the analysis of auxetic metamaterials which exhibit elastic instability through analogy with rigid link lattice systems. The results of our analytic approach are confirmed by finite-element simulations for both the onset of elastic instability and post-buckling behaviour including Poisson's ratio. The method gives insight into the relationships between mechanisms within lattices and their mechanical behaviour; as such, it has the potential to allow existing knowledge of rigid link lattices with auxetic paths to be used in the design of future buckling-induced auxetic metamaterials.

New principles for auxetic periodic design

We show that, for any given dimension d ≥ 2, the range of distinct possible designs for periodic frameworks with auxetic capabilities is infinite. We rely on a purely geometric approach to auxetic trajectories developed within our general theory of deformations of periodic frameworks.

Geometric auxetics

We formulate a mathematical theory of auxetic behaviour based on one-parameter deformations of periodic frameworks. Our approach is purely geome- tric, relies on the evolution of the periodicity lattice and works in any dimension. We demonstrate its usefulness by predicting or recognizing, without experiment, computer simulations or numerical approximations, the auxetic capabilities of several well-known structures available in the literature. We propose new principles of auxetic design and rely on the stronger notion of expansive behaviour to provide an infinite supply of planar auxetic mechanisms and several new three-dimensional structures.

Liftings and stresses for planar periodic frameworks

We formulate and prove a periodic analog of Maxwell's theorem relating stressed planar frameworks and their liftings to polyhedral surfaces with spherical topology. We use our lifting theorem to prove deformation and rigidity-theoretic properties for planar periodic pseudo-triangulations, generalizing features known for their finite counterparts. These properties are then applied to questions originating in mathematical crystallography and materials science, concerning planar periodic auxetic structures and ultrarigid periodic frameworks.

Bi-stiffness property of motion structures transformed into square cells

Cellular solids with internal microstructures enable the reduction in some environmental loads because of their lightweight bodies, and deliver unique elastic, electromagnetic and thermal properties. In particular, their large deformability without topological change is one of their most interesting solid properties. In this study, we propose a bar-and-joint framework assembled with a basic unit of motion structure, which has eightfold rotational symmetry (MS-8). The MS-8 is made of tetragons, arranged in a concentric fashion, which are transformed into either one of two different aligned patterns of square cells according to the coordinated rotations of the inside squares. Square cells are extremely anisotropic, which is why the stiffness of the MS-8 changes dramatically in the transformation process. Thus, the MS-8 exhibits bi-stiffness according to the two different motions. Taking advantage of the bi-stiffness property, the possibilities of deformation behaviours for repetitive structures of MS-8s are discussed.