Two-phase local/nonlocal gradient mechanics of elastic torsion

The higher order two-phase local/nonlocal elasticity model and the higher order strain gradient theory are unified via an abstract variational scheme. The higher order constitutive integral convolution is established in a consistent variational framework governed by ad hoc functional space of test fields. Equivalent differential constitutive law equipped with nonclassical boundary conditions of constitutive type is determined. The proposed higher order elasticity theory provides as special cases a range of well-known size-dependent elasticity models such as nonlocal, two-phase local/nonlocal, strain gradient, modified nonlocal strain gradient, and nonlocal strain-driven gradient models. Evidences of well-posedness of the introduced higher order two-phase local/nonlocal gradient problems are elucidated by rigorous examination of the elastostatic torsional response of structural schemes of applicative interest in nano-mechanics. The exact analytical solution of the torsion problem of elastic nano-beams is derived, graphically demonstrated, and compared with analogous outcomes in the literature. The conceived higher order elasticity theory can efficiently characterize advanced nano-materials and structural elements of modern nano-systems.