A unifying perspective on rigidity in under-constrained materials

We present a novel approach to understand rigidity in under-constrained materials, including sub-isostatic spring networks as well as 2D and 3D vertex models for dense biological tissues. We show that the onset of rigidity is determined by a purely geometric criterion. This allows us to analytically predict the elastic material properties close to the transition, which depend only on few geometric coefficients. We obtain exact expressions for the magnitudes of bulk modulus and shear modulus discontinuities at the rigidity transition, several scaling relations of the shear modulus, and the magnitude of the anomalous Poynting effect. Moreover, we show that the ratio of the excess shear modulus to the shear stress is inversely proportional to the critical shear strain with a prefactor of three, which we expect to be a general hallmark of rigidity in under-constrained materials induced by geometric incompatibility.