A New Insight into the Paradoxical Integral and Differential Constitutive Relations of Eringen Nonlocal Theory

Using the Eringen nonlocal theory (ENT), the integral constitutive relation (ICR) can be transformed into a corresponding differential constitutive relation (DCR). The inequivalence and equivalence between ICR and DCR have been documented in the current literature, indicating a paradoxical relationship between ICR and DCR. Despite some provided explanations, the actual mathematical connection between ICR and DCR remains an open question. Moreover, there has been limited focus on infinite-length nanostructures. In this study, using a vigorous mathematical approach, we determine the relationship between solutions of ICR and DCR for both finite- and infinite-length nanobeams. ICR is only a particular solution of DCR for finite-length nanostructures. The general solution of DCR differs from ICR (the particular solution), which reveals that for finite-length nanostructures, DCR is not equivalent to ICR for general cases and they are equivalent only for some special cases. This reasoning directly explains the paradoxical relationship between ICR and DCR mathematically. Additionally, it is evident that the static analysis for infinite-length nanostructures on foundations is also significant. Some typical examples are chosen to validate the above conclusions. This study is the first to reveal the genuine mathematical relationship between ICR and DCR, explaining their differences and similarities, while also offering a new perspective on the issue.