On longitudinal dynamics of nanorods

The longitudinal dynamic problem of a size-dependent elasticity rod is formulated by utilizing an integral form of nonlocal strain gradient theory. The nonlocal strain gradient model accounts for the energies diffused from surrounding particles in a reference domain by utilizing the convolution integral over nonlocal kernel functions, and can account for micro/nano-structures with internal displacement field via gradient forms. Unlike the size-dependent differential models, the developed integral model is both self-consistent and well-posed. The governing equations and boundary conditions for the longitudinal dynamics of the rod are deduced by employing the Hamilton principle. In addition to the well-known classical boundary conditions, the developed integral rod model also contains non-classical boundary conditions. By reducing the complicated integro-differential equations to a sixth order differential equation with mixed boundary conditions, the asymptotic solutions for predicting the natural frequencies of the rods are derived for the nonlocal strain gradient rod under various boundary conditions. It is shown explicitly that the integral rod model can exert stiffness-softening and stiffness-hardening effects by considering various values of the size-dependent parameters. By studying the size-dependent effects on the longitudinal dynamics of monolayer graphene, the dispersion relation calculated by using the nonlocal strain gradient model can show good agreement with the experimental data obtained by inelastic X-ray scattering. The size-dependent effect can make monolayer graphenes possess softening frequencies.