Unified strain-gradient frameworks for micro/nano-beams: Bridging the gap between strain-gradient and stress-driven nonlocal theories

Although both stress-driven nonlocal theory (SDNT) and strain-gradient theory (SGT) can capture the stiffness hardening effect of micro/nano-scale beams and share identical governing equations, it remains uncertain whether these two theories can be unified within a single theoretical framework. In this work, we introduce virtual higher-order springs into the conventional SGTs, enabling unified strain-gradient frameworks that represent the differential forms of SDNT and SGT for Euler and Timoshenko micro/nano-scale beams. Furthermore, the constitutive boundary conditions in SDNT and the higher-order boundary conditions in SGT can be unified as higher-order elastic boundary conditions through virtual higher-order springs. Corresponding integral models of these unified strain-gradient frameworks are also developed to provide integral formulations for SDNT and SGT. The similarities and differences between them are systematically analyzed. More importantly, this study is the first to propose unified strain-gradient Euler and Timoshenko beam frameworks for both SDNT and SGT, which can elucidate their relationship and fill the knowledge gap in nonlocal modeling.