Dynamic behaviour of state-based peridynamic media through analysis of potential fields

Abstract

Unlike classical theories, which rely on local interactions and differential equations, peridynamic theory employs integro-differential equations to describe the mechanics of materials and structures. This distinctive approach allows peridynamics to naturally incorporate long-range forces and discontinuities, such as cracks, which are challenging to handle using classical partial differential equations. In this study, a novel approach for the implementation of nonlocal Helmholtz-Hodge decomposition is used to decompose the displacement field into components that are divergence-free and curl-free. State-based peridynamics, which was introduced to overcome the limitations of bond-based peridynamics, has been considered in this work. As a consequence, two governing equations involving integrals are derived, which are associated with potential fields. An analysis of the propagation of waves has been performed to determine the dispersion relation for both longitudinal and transverse waves. The general solutions for initial-value problems are derived, and the closed-form expression for Green’s function in terms of potential fields is obtained. This study considers Gaussian, exponential, and constant functions as nonlocality functions. Graphs are demonstrated to illustrate the variations in frequency, phase velocity with changes in the size of the horizon (which represents nonlocal length) and nonlocality functions. The validation is achieved both analytically and numerically by ensuring classical correspondence in the limit of the nonlocality parameters approach zero. This work implements nonlocal Helmholtz decomposition, a robust framework that simplifies the study and provides comprehensive insight while analysing vector fields.