Bond-based nonlocal models by nonlocal operator method in symmetric support domain

The present study focuses on the applications of energy decomposition in diverse nonlocal models, such as elasticity, thin plates, and gradient elasticity, with the aim of establishing bond-based nonlocal models in which the bond force is solely dependent on the deformation of a single bond. Through the adoption of an appropriate bond force form and the application of energy equivalence between local and nonlocal models, several kinds of highly succinct bond -based models are obtained. The present study involves a reexamination of nonlocal operator methods, with a particular focus on the simplified version within a symmetric support domain. A three-point bent-bond model has been proposed to characterize the curvature and bending moment. A crack criterion for normal strain of the bond based on Griffith theories is proposed. This approach is analogous to the phase field model and allows for individual application to each bond, resulting in strain localization. By implementing this rule, the path of the crack can be predicted in an automated manner through the act of cutting the bond, yielding outcomes that are akin to those obtained via the phase field method. Simultaneously, a crack rule for critical shear strains in shear fractures is presented. Moreover, an incremental version of the plasticity model associated with bond force has been formulated. The nonlocal bond-based models are further validated through several numerical examples.