Flexural wave propagation in small-scaled functionally graded beams via a nonlocal strain gradient theory

Abstract An analytic model of small-scaled functionally graded (FG) beams for the flexural wave propagation analysis is developed based on the nonlocal strain gradient theory, in which the stress accounts for not only the nonlocal elastic stress field but also the strain gradients stress field. By using the analytic model, the acoustical and optical dispersion relations between phase velocity and wave number are explicitly derived. It is found that an asymptotic phase velocity of both the acoustical and optical branches can be observed. The asymptotic phase velocity can be increased by decreasing the nonlocal parameter or increasing the material characteristic parameter. Furthermore, the power-law index has a significant effect on the acoustical and optical dispersion relations of nano-scaled FG beams. The effects of nonlocal parameter and material characteristic parameter on the acoustical and optical dispersion relation are significant at high wave numbers, however, may be ignored at low wave numbers. The acoustical and optical phase velocities can generally increase with the increasing material length scale parameter or the decreasing nonlocal parameter.