  Thick rods are employed in Nanotechnology to build modern electro mechanical
systems. Design and optimization of such structures can be carried out by
nonlocal continuum mechanics which is computationally convenient when compared
to atomistic strategies. Bishop's kinematics is able to describe small-scale
thick rods if a proper mathematical model of nonlocal elasticity is formulated
to capture size effects. In all papers on the matter, nonlocal contributions
are evaluated by replacing Eringen's integral convolution with the consequent
(but not equivalent) differential equation governed by Helmholtz's differential
operator. As notorious in integral equation theory, this replacement is
possible for convolutions, defined in unbounded domains, governed by averaging
kernels which are Green's functions of differential operators. Indeed, Eringen
himself, in order to study nonlocal problems defined in unbounded domains, such
as screw dislocations and wave propagations, suggested to replace
integro-differential equations with differential conditions. A different
scenario appears in Bishop rod mechanics where nonlocal integral convolutions
are defined in bounded structural domains, so that Eringen's nonlocal
differential equation has to be supplemented with additional boundary
conditions. The objective is achieved by formulating the governing nonlocal
equations by a proper variational statement. The new methodology provides an
amendment of previous contributions in literature and is illustrated by
investigating the elastostatic behavior of simple structural schemes. Exact
solutions of Bishop rods are evaluated in terms of nonlocal parameter and
cross-section gyration radius. Both hardening and softening structural
responses are predictable with a suitable tuning of the parameters.
