Sensitivity and performance of transverse wave transmission in third-gradient one-dimensional continua

We investigate the propagation of transverse waves and the associated energy transmission in one-dimensional third-gradient continua. Starting from a variational formulation for small transverse displacements, we derived the corresponding dispersion relation. Owing to the contribution of the third-gradient kinematics, such a relation takes the form of a sixth-order polynomial in the wavenumber. Furthermore, we obtained the expression of the energy flux averaged over the period, which enabled the definition of a transmission coefficient for third-gradient continua. In the present study, we examine in detail the dependence of the admissible wavenumbers and of the transmission coefficient on specific third-gradient parameters: the characteristic length of the third-gradient (defined as the ratio between the third-gradient and the second-gradient parameters) and the inertial–stiffness ratio (given by the ratio between the mass density and the second-gradient parameter). In addition, for the transmission coefficient, we analyze its sensitivity to variations of the third-gradient’s characteristic length and its dependence on other constitutive parameters, with particular attention to the role played by the third-gradient contributions compared to second-gradient configurations. This analysis highlights the influence of higher-order microstructural effects on both wave propagation and energy transport, thereby providing further insight into the mechanical response of generalized continua.