Nonlinear elastic response of a topological mechanical metamaterial

We consider the static response of a finite 1D lattice of pivoting rigid bars connected end-to-end by harmonic springs of zero equilibrium length. The linearized model is equivalent to the Kane-Lubensky model of rigid rotors and is topologically polarized when the pivot point of each bar is off center. We fix the angular displacement of the leftmost bar and solve the nonlinear torque balance equations for the equilibrium configuration. For the unpolarized case, we find an algebraic decay of the rotation angle θ_n due to nonlinear effects associated with the bulk zero mode. For one sign of the polarization, θ_n decays exponentially and the elastic energy is nearly zero, consistent with the excitation of the zero mode at the left edge. For the other sign, θ_n also decays exponentially with the same decay length as the zero mode, but with a finite energy, and there is a turning point beyond which θ_n grows exponentially toward the free boundary. Numerical solutions are explained in detail by an analysis that necessarily includes terms of up to fifth order in θ_n. The results provide insight into numerical results for the directional response of a 2D mechanical graphene model to an externally applied local strain.