The mechanical behavior of 2-D lattices with bi-stable springs

Abstract

Multi-stable structures offer a unique set of mechanical properties, such as the ability to undergo large reversible deformations, the ability to provide mechanical protection and efficient shock absorption, and the ability to retain variety of geometrical configurations after loads have been removed. In addition, the study of lattice-based multi-stable structures is of relevance to a wide range of engineering and physical phenomena, such as atomic models of shape memory materials, mechanics of protein networks, foldable structures for engineering applications, and the development of new meta materials that display extraordinary behaviors. In this paper, we study theoretically and numerically the quasi-static behavior of 2-D multi-stable lattices. Special emphasis is placed on equilibrium configurations, evolution of transition patterns, stability, force-displacement relations, and hysteresis. In addition, the influence of the lattice geometry and the properties of the bi-stable springs on the abovementioned features are investigated. We show that approximating the non-monotonous force-strain relation of the bi-stable springs by a tri-linear relation enables closed-form formulation and analytical insights. These, together with extensive numerical simulations, demonstrate the wealth of equilibrium configurations and that the overall response as well the transition sequence may exhibit an ordered pattern, a disordered pattern, or a combination of both, depending on the lattice parameters. The closed-form analytical formulation has also been found advantageous in analyzing the stability of equilibrium configurations. We show that necessary conditions for stability can be neatly expressed in terms of the Lucas sequence and the corresponding metallic mean, which for some class of lattice parameters, can be replaced by the Fibonacci sequence and the Golden ratio.