Extraordinary stiffness tunability through thermal expansion of nonlinear defect modes

Incremental stiffness characterizes the variation of a material's force response to a small deformation change. Typically materials have an incremental stiffness that is fixed and positive, but recent technologies, such as super-lenses, low frequency band gap materials and acoustic cloaks, are based on materials with zero, negative or extremely high incremental stiffness. So far, demonstrations of this behavior have been limited either to a narrow range of frequencies, temperatures, stiffness or to specific deformations. Here we demonstrate a mechanism to tune the static incremental stiffness that overcomes those limitations. This tunability is achieved by driving a nonlinear defect mode in a lattice. As in thermal expansion, the defect's vibration amplitude affects the force at the boundary, hence the lattice's stiffness. By using the high sensitivities of nonlinear systems near bifurcation points, we tune the magnitude of the incremental stiffness over a wide range: from positive, to zero, to arbitrarily negative values. The particular deformation where the incremental stiffness is modified can be arbitrarily selected varying the defect's driving frequency. We demonstrate this experimentally in a compressed array of spheres and propose a general theoretical model.