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.

多稳定结构提供一组独特的机械性能，例如承受大的可逆变形的能力、提供机械保护和有效减震的能力，以及在去除负载后保持各种几何结构的能力。此外，基于晶格的多稳态结构的研究与广泛的工程和物理现象相关，例如形状记忆材料的原子模型、蛋白质网络的力学、工程应用的可折叠结构以及开发表现出非凡行为的新元材料。在本文中，我们从理论上和数值上研究了二维多稳态晶格的准静态行为。特别强调平衡配置、过渡模式的演变、稳定性、力-位移关系和滞后。此外，研究了晶格几何形状和双稳态弹簧的特性对上述特征的影响。我们表明，通过三线性关系来近似双稳态弹簧的非单调力-应变关系能够实现封闭形式的公式和分析见解。这些与广泛的数值模拟一起证明了平衡配置的丰富性，并且根据晶格参数，整体响应以及过渡序列可能表现出有序模式、无序模式或两者的组合。还发现封闭形式的分析公式有利于分析平衡构型的稳定性。