Analysis of Ultrasonic Wave Dispersion in Presence of Attenuation and Second-Gradient Contributions

In this study, we aim to analyze the dispersion of ultrasonic waves due to second-gradient contributions and attenuation within the framework of continuum mechanics. To investigate dispersive behavior and attenuation effects, we consider the influence of both higher-order gradient terms (second gradients) and Rayleigh-type viscoelastic contributions. To this end, we employ the extended Rayleigh–Hamilton principle to derive the governing equations of the problem. Using a wave-form solution, we establish the relationship between the phase velocity and the material’s constitutive parameters, including those related to the stiffness of both standard (first-gradient) and second-gradient types, as well as viscosity. To validate the model, we use data available in the literature to identify all the material parameters. Based on this identification, we observe that our model provides a good approximation of the experimentally measured trends of both phase velocity and attenuation versus frequency. In conclusion, this result not only confirms that our model can accurately describe both wave dispersion and attenuation in a material, as observed experimentally, but also highlights the necessity of simultaneously considering both second-gradient and viscosity parameters for a proper mechanical characterization of materials.