On the nature of constitutive boundary and interface conditions in stress-driven nonlocal integral model for nanobeams

In the stress-driven nonlocal theory (SDNT), the integral form (IF) can be transformed into an equivalent differential form (DF) with two constitutive boundary conditions (CBCs). In addition, two constitutive interface conditions (CICs) can be established for nanobeams subjected to discontinuous loads. The current literature indicates that CBCs and CICs are essential in DF, while their importance in IF remains uncertain. Furthermore, the critical features of CBCs and CICs in IF have yet to be fully understood. In this study, we reformulate the CBCs and CICs using space convolution integrals, revealing that they are indeed directly obtained from IF. CBCs and CICs are crucial in both IF and DF, they are explicitly represented in DF but are implicitly expressed in IF. This implicit representation reflects their true existence, which has not yet been documented. When addressing nanobeam problems using IF, CBCs and CICs are automatically satisfied, thus eliminating the requirement for their presence in the solutions. Moreover, the existence of CBCs and CICs is closely linked to IF and the kernel function. A series of representative nanobeam examples are presented to substantiate this assertion. To the authors' knowledge, this study is the first to theoretically clarify the real existence and essential features of CBCs and CICs in IF, thereby offering benchmark insights into the subject.