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

Abstract

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.