Hyperelasticity as a Γ-limit of Peridynamics When the Horizon Goes to Zero

Peridynamics is a nonlocal model in Continuum Mechanics, and in particular Elasticity, introduced by Silling (2000). The nonlocality is reflected in the fact that points at a finite distance exert a force upon each other. If, however, those points are more distant than a characteristic length called horizon, it is customary to assume that they do not interact. We work in the variational approach of time-independent deformations, according to which, their energy is expressed as a double integral that does not involve gradients. We prove that the Γ-limit of this model, as the horizon tends to zero, is the classical model of hyperelasticity. We pay special attention to how the passage from the density of the non-local model to its local counterpart takes place.