Theoretical Examination for the Consistency of Eringen’s Nonlocal Theories in Nanomaterial Modeling

In Eringen’s nonlocal theory and its refined version, referred to as Eringen’s two-phase local/nonlocal theory, it is well-received that the integral forms can be converted into differential forms with constitutive boundary conditions. While constitutive boundary conditions are deemed essential in a differential form, their actual existence and necessity in an integral form remain uncertain in Eringen’s nonlocal theory. This ambiguity has not yet been clarified in Eringen’s two-phase local/nonlocal theory, despite both integral and differential forms being well-posed. To address this issue, it is necessary to revisit the nature of constitutive boundary conditions and examine their existence and necessity. This work has established the following key conclusions. First, in both theories, constitutive boundary conditions can be directly derived from an integral form, and they are crucial for ensuring that both integral and differential forms are well-posed. The presence of constitutive boundary conditions is closely associated with the integral form and the kernel function, but it does not relate to the differential form. It appears that both integral and differential forms in Eringen’s nonlocal theory are generally ill-posed, while they only become well-posed under certain special conditions. Conversely, in Eringen’s two-phase local/nonlocal theory, two constitutive boundary conditions and two integral forms at both boundaries of nanobeams are equivalent in both integral and differential forms. It is thus concluded that Eringen’s two-phase local/nonlocal theory can satisfactorily and adequately replace Eringen’s nonlocal theory in structural analysis at nanoscale.