Unified analysis of stiffening and weakening size effects in nanobeams on Winkler–Pasternak foundations via local/nonlocal stress gradient elasticity

The free transverse vibration of Rayleigh nanobeams resting on a Winkler-Pasternak two-parameter elastic foundation is investigated within the local/nonlocal stress-driven gradient (LNStressG) elasticity framework. The constitutive model incorporates three length-scale parameters—the nonlocal characteristic length LC , stress-gradient length scale Lₗ , and mixture parameter α —to capture both stiffness-stiffening and stiffness-weakening size-dependent responses within a thermodynamically consistent formulation. The model is validated against three independent benchmark solutions, achieving excellent agreement with the reference results. Four model hierarchies are systematically examined for clamped–clamped and pinned–pinned boundary conditions: the classical Rayleigh beam, purely stress-driven nonlocal model, two-phase local/nonlocal stress-driven model, and full LNStressG model. The Winkler and Pasternak parameters consistently increase stiffness, with the strongest influence observed in the fundamental mode of pinned–pinned beams and diminishing effects at higher modes as nonlocal constitutive stiffness dominates. The purely stress-driven nonlocal model exhibits exclusively stiffening behavior, whereas the full LNStressG model reveals competing effects: increasing LC stiffens the response, while increasing Lₗ softens it. Their relative influence depends on mode number, boundary condition, and α . The results provide insight into substrate–size-effect interactions in nanobeam resonators, mass sensors, and energy harvesters.