Bending analysis of nano-sized beam-elastic substrate systems: Unification of the stress-driven nonlocal model and Winkler-Pasternak foundation model

Nano- and micro-scale beam–substrate assemblies are key components of sensors, actuators, and flexible electronics, yet their mechanical responses deviate from classical elasticity because intrinsic material length scales govern their behavior. This study develops a unified, paradox-free bending framework for Euler–Bernoulli nano-beams on elastic media by combining a stress-driven nonlocal constitutive law—expressed in an equivalent differential form with constitutive boundary conditions—with a Winkler–Pasternak foundation that ensures shear continuity. The equilibrium equation and natural boundary conditions (NBs) are derived via the principle of minimum potential energy, and the overall formulation is summarized through a Tonti diagram. An equivalent end spring compactly represents semi-infinite substrate continuation. Three key findings highlight the contribution of this work: (i) the stress-driven nonlocal formulation eliminates the pure-bending paradox by restoring curvature consistency and predicts size-induced stiffening; (ii) the Winkler–Pasternak substrate produces physically coherent displacement, moment, and curvature fields, while the Winkler foundation yields non-rational rigid-body settlements; and (iii) parametric studies reveal that normalized contact stiffness exceeds unity, confirming nanoscale stiffening that diminishes as the substrate stiffness increases. The proposed framework offers closed-form insight and a reproducible analytical tool for the design and performance evaluation of nano-devices and nanocomposites interacting with elastic substrates.