A nonlinear unit cell for adaptive lattice structures with shape-memory-like behaviour

We present and classify a fundamental building block for constructing macro-scale lattice structures with shape-memory-like behaviour. Shape memory alloys (SMAs) are materials that undergo a reversible phase transition at the atomic scale from a high-symmetry crystal structure above a transition temperature to a lower-symmetry crystal structure below the transition temperature, thereby imbuing these alloys with superelastic properties and the ability to recover a previous shape when heated. We present a unit cell for creating latticed metamaterials at the macro- rather than the micro-scale that shows similar smart, adaptive behaviour in two and three spatial dimensions. Specifically, we study a square unit cell with rigid outer edges and two nonlinear springs on the diagonals that undergoes instabilities into planar, rhombic and/or non-planar, folded states. We identify two non-dimensional parameters that govern the planar multistable behaviour, and derive boundaries that split the domain into monostable (square cell), bistable (rhombic cell) and tristable (square and rhombic cells) regimes. Transitions between the square and rhombic configurations can be smooth/soft or sudden/hard. We illustrate how changes in the springs’ rest lengths and stiffness properties define the possible types of phase transitions, irrespective of the physical mechanisms (such as temperature, electromagnetic fields or swelling) driving those changes. In addition to planar multi-stability, we define the parameter regime wherein the initially planar unit cell can deform out of the plane.