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

Metamaterials of Auxetic Geometry for Seismic Energy Absorption

The propagation of earthquake energy occurs primarily through elastic waves. If the seismic force input to a structure can be directly reduced from the source, then the structure can be protected from seismic wave energy. Seismic metamaterials, regarded as periodic structures with properties different from conventional materials, use wave propagation characteristics and bandgaps to dissipate seismic wave energy. When the seismic wave is located in the bandgap, the transmission of seismic wave energy is effectively reduced, which protects the structure from the damage caused by seismic disturbance. In practical application, locating seismic frequencies below ten Hz is a challenge for seismic metamaterials. In the commonly used method, high-mass materials are employed to induce the effect of local resonance, which is not economically feasible. In this study, a lightweight design using auxetic geometry is proposed to facilitate the practical feasibility of seismic metamaterials. The benefits of this design are proven by comparing conventional seismic metamaterials with metamaterials of auxetic geometry. Different geometric parameters are defined using auxetic geometry to determine the structure with the best bandgap performance. Finite element simulations are conducted to evaluate the vibration reduction benefits of auxetic seismic metamaterials in time and frequency domains. Additionally, the relationship between the mass and stiffness of the unit structure is derived from the analytical solution of one-dimensional periodic structures, and modal analysis results of auxetic metamaterials are verified. This study provides seismic metamaterials that are lightweight, small in volume, and possess low-frequency bandgaps for practical applications.

Metamaterial with Tunable Positive and Negative Hygrothermal Expansion Inspired by a Four-Fold Symmetrical Islamic Motif

A metamaterial with controllable positive and negative thermal and hygroscopic expansions is investigated herein by inspiration from a range of Islamic geometric patterns. Constructing from eight pairs of pin-jointed Y-elements, each unit cell manifests eight rhombi that are arranged circumferentially, thereby manifesting four axes of symmetry. By attachment of bimaterial spiral springs of contrasting expansion coefficients to the far arms of the paired Y-elements, a change in the environment’s thermal or hygroscopic condition alters the offset angle of the paired Y-elements such that the unit cell of the metamaterial ranges from the eight-pointed star to the regular octagon. The effective coefficient of thermal expansion (CTE) and the coefficient of moisture expansion (CME) of this metamaterial were developed for small and large changes in environmental fluctuations using infinitesimal and finite models, respectively. Generated data indicates that the sign and magnitude of the effective thermal and hygroscopic expansion coefficients can be controlled by geometrical descriptors of the bimaterial spiral spring—such as its coil number and the ratio of its mean radius to its thickness—as well as the properties of the bimaterial’s layers such as their expansion coefficients, Young’s moduli and, in the case of effective hygroscopic expansion, their relative absorptivity. The obtained results suggest that the proposed metamaterial can be designed to perform as highly sensitive thermal and/or moisture sensors, as well as other functional materials or devices that take advantage of environmental changes as stimuli.

Metamaterials and Symmetry

How are metamaterials related to symmetry [...]

A Generalized Strain Energy-Based Homogenization Method for 2-D and 3-D Cellular Materials with and without Periodicity Constraints

A generalized strain energy-based homogenization method for 2-D and 3-D cellular materials with and without periodicity constraints is proposed using Hill’s Lemma and the matrix method for spatial frames. In this new approach, the equilibrium equations are enforced at all boundary and interior nodes and each interior node is allowed to translate and rotate freely, which differ from existing methods where the equilibrium conditions are imposed only at the boundary nodes. The newly formulated homogenization method can be applied to cellular materials with or without symmetry. To illustrate the new method, four examples are studied: two for a 2-D cellular material and two for a 3-D pentamode metamaterial, with and without periodic constraints in each group. For the 2-D cellular material, an asymmetric microstructure with or without periodicity constraints is analyzed, and closed-form expressions of the effective stiffness components are obtained in both cases. For the 3-D pentamode metamaterial, a primitive diamond-shaped unit cell with or without periodicity constraints is considered. In each of these 3-D cases, two different representative cells in two orientations are examined. The homogenization analysis reveals that the pentamode metamaterial exhibits the cubic symmetry based on one representative cell, with the effective Poisson’s ratio v¯ being nearly 0.5. Moreover, it is revealed that the pentamode metamaterial with the cubic symmetry can be tailored to be a rubber-like material (with v¯ ≅0.5) or an auxetic material (with v¯ < 0).

Infinitesimal Periodic Deformations and Quadrics

We describe a correspondence between the infinitesimal deformations of a periodic bar-and-joint framework and periodic arrangements of quadrics. This intrinsic correlation provides useful geometric characteristics. A direct consequence is a method for detecting auxetic deformations, identified by a pattern consisting of homothetic ellipsoids. Examples include frameworks with higher crystallographic symmetry.

Extreme Poisson’s Ratios of Honeycomb, Re-Entrant, and Zig-Zag Crystals of Binary Hard Discs

Two-dimensional (2D) crystalline structures based on a honeycomb geometry are analyzed by computer simulations using the Monte Carlo method in the isobaric-isothermal ensemble. The considered crystals are formed by hard discs (HD) of two different diameters which are very close to each other. In contrast to equidiameter HD, which crystallize into a homogeneous solid which is elastically isotropic due to its six-fold symmetry axis, the systems studied in this work contain artificial patterns and can be either isotropic or anisotropic. It turns out that the symmetry of the patterns obtained by the appropriate arrangement of two types of discs strongly influences their elastic properties. The Poisson’s ratio (PR) of each of the considered structures was studied in two aspects: (a) its dependence on the external isotropic pressure and (b) in the function of the direction angle, in which the deformation of the system takes place, since some of the structures are anisotropic. In order to accomplish the latter, the general analytic formula for the orientational dependence of PR in 2D systems was used. The PR analysis at extremely high pressures has shown that for the vast majority of the considered structures it is approximately direction independent (isotropic) and tends to the upper limit for isotropic 2D systems, which is equal to +1. This is in contrast to systems of equidiameter discs for which it tends to 0.13, i.e., a value almost eight times smaller.