Nonlocal inflected nano-beams: A stress-driven approach of bi-Helmholtz type

Abstract

Size-dependent bending behavior of Bernoulli-Euler nano-beams is investigated by elasticity integral theory. Eringen’s strain-driven nonlocal integral formulation, providing the bending moment interaction field as output of the convolution between elastic curvature and bi-Helmholtz averaging kernel, is examined. Corresponding higher-order constitutive boundary conditions are established for the first time, showing inapplicability of strain-driven nonlocal integral law of bi-Helmholtz type to continuous nano-structures defined on bounded domains. As a remedy, an innovative stress-driven nonlocal integral elastic model of bi-Helmholtz type is presented for inflected Bernoulli-Euler nano-beams, by swapping input and output of Eringen’s nonlocal integral elastic law. The proposed stress-driven nonlocal integral constitutive law is proven to be equivalent to a fourth-order linear differential equation in the elastic curvature, fulfilling suitable homogeneous constitutive boundary conditions. The corresponding elastic equilibrium problem of an inflected nano-beam is well-posed and numerically solved for statics schemes of nano-technological interest, under standard loading and kinematic boundary conditions. New benchmarks for nonlocal computational mechanics are also detected.