Leading-Order Nonlocal Kinetic Energy in Peridynamics for Consistent Energetics and Wave Dispersion

This paper first considers the approximation of peridynamics by strain-gradient models in the linear, one-dimensional setting. Strain-gradient expansions that approximate the peridynamic dispersion relation using Taylor series are compared to strain-gradient models that approximate the peridynamic elastic energy. The dynamic and energetic expansions differ from each other, and neither captures an important feature of peridynamics that mimics atomic-scale dynamics, namely that the frequency of short waves is bounded and non-zero. The paper next examines peridynamics as the limit model along a sequence of strain-gradient models that consistently approximate both the energetics and the dispersion properties of peridynamics. Formally examining the limit suggests that the inertial term in the dynamical equation of peridynamics – or equivalently, the peridynamic kinetic energy – is necessarily nonlocal in space to balance the spatial nonlocality in the elastic energy. The nonlocality in the kinetic energy is of leading-order in the following sense: classical elasticity is the zeroth-order theory in both the kinetically nonlocal peridynamics and the classical peridynamics, but once nonlocality in the elastic energy is introduced, it must be balanced by nonlocality in the kinetic energy at the same order. In that sense, the kinetic nonlocality is not a higher-order correction; rather, the kinetic nonlocality is essential for consistent energetics and dynamics even in the simplest setting. The paper then examines the implications of kinetically nonlocal peridynamics in the context of stationary and propagating discontinuities of the kinematic fields.