A universal measure of nonlocality in peridynamics

In peridynamics, the influence function determines the “nonlocality” of the model, which depends on the support of the influence function (the “horizon size”) and its specific shape. In this paper, we introduce a universal scalar parameter - the nonlocality constant - to quantify the strength of nonlocal interactions in bond-based peridynamic models. The nonlocality constant derives rigorously from the nonlocal factor, a key component in analytical solutions of peridynamic equations, and establishes a one-to-one correspondence with the difference between peridynamic and classical solutions. By analyzing eight distinct influence functions, we demonstrate that the nonlocality constant universally governs the deviation of peridynamic responses from their classical counterparts. Using the analytical solution for elastic membrane deflection derived via eigenfunction expansion, we validate that the proposed measure accurately ranks influence functions by their nonlocality strength. This work provides a systematic framework for selecting influence functions in multiscale modeling of materials with microstructural heterogeneities.