A characteristic length scale causes anomalous size effects and boundary programmability in mechanical metamaterials

The architecture of mechanical metamaterials is designed to harness geometry^1,2,3,4,5,6, nonlinearity^7,8,9,10,11 and topology^{11,12,13,14,15} to obtain advanced functionalities such as shape morphing^{7,9,16,17,18,19,20,21,} programmability^18,22,23 and one-way propagation^11,13,14. Although a purely geometric framework successfully captures the physics of small systems under idealized conditions, large systems or heterogeneous driving conditions remain essentially unexplored. Here we uncover strong anomalies in the mechanics of a broad class of metamaterials, such as auxetics^2,5,24, shape changers^{16,17,18,19,20,21} or topological insulators^11,12,13,15; a non-monotonic variation of their stiffness with system size, and the ability of textured boundaries to completely alter their properties. These striking features stem from the competition between rotation-based deformations—relevant for small systems—and ordinary elasticity, and are controlled by a characteristic length scale which is entirely tunable by the architectural details. Our study provides new vistas for designing, controlling and programming the mechanics of metamaterials.