Unified higher-order theory of two-phase nonlocal gradient elasticity

The unified higher-order theory of two-phase nonlocal gradient elasticity is conceived via consistently introducing the higher-order two-phase nonlocality to the higher-order gradient theory of elasticity. The unified integro-differential constitutive law is established in an abstract variational framework equipped with ad hoc functional space of test fields. Equivalence between the higher-order integral convolutions of the constitutive law and the nonlocal gradient differential formulation is confirmed by prescribing the non-classical boundary conditions. The strain-driven and stress-driven nonlocal approaches are exploited to simulate the long-range interactions at nano-scale. A range of generalized continuum models are restored under special ad hoc assumptions. The established unified higher-order elasticity theory is invoked to analytically examine the wave dispersion phenomenon. In contrast to the first-order size-dependent elasticity model, the higher-order two-phase nonlocal gradient theory can efficiently capture the wave dispersion characteristics observed in experimental measurements. The precise description of nano-scale wave phenomena noticeably underlines the importance of applying the proposed higher-order size-dependent elasticity theory. A viable approach to tackle peculiar dynamic phenomena at nano-scale is introduced.