Nonlocal strain gradient elastic beam models with two-step differential approach and decoupling of standard and extra boundary conditions - Part I

Within the framework of Eringen's nonlocal elasticity, a theory of coherent nonlocal strain gradient (NSG) beam models, i.e. admitting equivalent integral and differential approaches, is reported. Making reference to shear undeformable Euler-Bernoulli (EB) beams in bending, it is shown that the coherence requisite requires that the constitutive equations incorporate a pair of two-phase local/nonlocal models of which one is driven by strain, the other by strain gradient. The governing integro-differential equation is found to be of sixth differential order, accompanied by four standard plus two gradient BCs, whereas the conjugate differential equation is of eighth order, that is, two order units more, two being the number of the nonlocality BCs, herein also determined. A two-step differential approach is presented in which ODEs of fourth order are addressed with decoupling of the BCs.