Closed-Form Solution of the Bending Two-Phase Integral Model of Euler-Bernoulli Nanobeams

Recent developments have shown that the widely used simplified differential model of Eringen’s nonlocal elasticity in nanobeam analysis is not equivalent to corresponding and initially proposed integral models, pure two-phase model, all cases loading boundary conditions. This has resolved a paradox with solutions are line expected softening effect theory appears other cases. In addition, it revived interest which were due their complexity solving relevant integro-differential equations, respectively. this article, we use direct operator method for value problems nth order linear Volterra–Fredholm equations convolution type construct closed-form Euler–Bernoulli nanobeams bending under transverse distributed load various types