A novel porosity-dependent homogenization procedure for wave dispersion in nonlocal strain gradient inhomogeneous nanobeams

.Present research aims to consider the reciprocal impacts of Young moduli and mass density while investigating the wave propagation behaviors of functionally graded (FG) nanobeams via a nonlocal strain gradient based shear deformable beam theory for the first time. The porosity influences are regarded within the framework of a newly developed method. The motion equations are obtained by extending the dynamic form of the principle of virtual work for a higher-order beam model. Thereafter, the constitutive relations are modified based on the nonlocal strain gradient theory in order to account for the effects of small scale. At the end, the dispersion curves are achieved by solving the problem with an analytical method. The results show that there is a remarkable difference between the results of this homogenization method and those of former simple methods that were utilized to capture porosity effects. Also, it is proven that the presented methodology is efficient enough to estimate the dynamic behaviors of FG nanobeams.