2-D multistable structures under shear: equilibrium configurations, transition patterns, and boundary effects

Multistable structures have a promising potential in a wide range of engineering and scientific applications, such as shock absorption, soft robotics, superelastic structures, vibration mitigation, foldable structures, configurable structures, programmable materials, and tunable shape-memory structures. In addition, they are directly relevant to the study of materials undergoing martensitic phase transformations, macromolecular networks, and the development of new metamaterials. In this paper, we study the quasistatic behavior of 2-D bistable lattices subjected to shear, with emphasis on the multitude of equilibrium configurations, overall stress-strain relation, sequence of phase transition, and statistics of stress jumps. In particular, the influence of material (properties of the individual bistable interaction) and microstructure geometry (architecture of the lattice) on the above mentioned characteristics of the overall behavior is investigated. To this end, we perform extensive numerical simulations with four different periodic lattice geometries. We find that, for the same loading conditions, different lattice geometries or different material (bistable) properties of the building block may result in fundamentally different overall (macro) behaviors. This is manifested both in the overall stress-strain relation and also in the evolution of the phase-transition patterns. Also, hysteresis, which is a macroscopic manifestation of the energy dissipated during change of configuration, is significantly affected by the lattice architecture. Similar effects of geometrical incompatibility, but at the level of the atomic lattice, have been observed in shape-memory alloys. Our results also reproduce stress peaks, associated with nucleation of a new phase. The magnitude of these nucleation peaks, their location, and number is dictated by the geometry of the lattice and boundary effects that lead to stress concentrations.