Evaluating constitutive interface conditions in Eringen’s two-phase local/nonlocal nanobeam model under discontinuous loads

Abstract

In Eringen’s two-phase local/nonlocal theory (ETLT), the integral form (IF) can be transformed into a corresponding differential form (DF). Discontinuous loads give rise to non-smooth fields, resulting in the formation of constitutive interface conditions (CICs). Although CICs are essential in DF, their manifestation in IF remains unclear. In this study, CICs are derived by using IF, and their intrinsic features are also examined. Through in-depth theoretical derivation and analysis, it can be found that CICs can be directly obtained from IF and are necessary in both IF and DF, appearing explicitly in DF but implicitly in IF. The presence of CICs is closely related to IF and the kernel function. The two CICs are associated with the two IFs at the location where the load changes abruptly, which can be verified through nanobeam examples subjected to discontinuous loads. More importantly, the results demonstrate a correlation between the local phase volume fraction and the nonlocal length-scale parameter, allowing a natural transition between Eringen’s two-phase local/nonlocal and purely local models. This work evaluates the intrinsic features of CICs in ETLT for non-smooth fields, providing new insights into their role.