Modified Eshelby's theory for equivalent continuum modelling of carbon nanotube-based composites

The present work is a first step towards a refined modeling and simulation of the dissipative phenomena characterizing carbon nanotube composites. In the past decade, nano-structured materials have gained significant importance from a technological point of view for the wide range of engineering applications that require high levels of structural performance and multifunctionality. Therefore, the interest has been growing continuously towards carbon nanotubes that, due to their mechanical and electrical properties, have the potential to become the building blocks of a new generation of multifunctional composite materials. We propose a numerical investigation of the vibratory behavior of a suitable equivalent continuum model within a linearized elastodynamic context. The continuum model is based on the homogenization procedure of Mori and Tanaka which derives the elastic properties of the composite material by means of the Eshelby theory of elastic inclusions embedded into an elastic hosting material. Within this framework, we first investigate the properties of the equivalent continuum in terms of vibration modes, which highlight the enhanced elastic properties due to the presence of CNT inclusions. We also pause to elaborate on the homogenization procedure itself since several ways can be explored to obtain the elastic tensor of the equivalent continuum. While such approaches are completely equivalent in providing the linear elastic response, they may suggest different ways of setting up a nonlinear procedure in the presence of internal dissipation.