Asymptotic Modelling of Carbon Nanotube (CNT) and CNT-Reinforced Composite Structures Using Strain Gradient Formulations

Abstract

Carbon nanotubes (CNTs) have garnered attention for their remarkable mechanical, thermal, and electrical properties, making them valuable in various applications. CNTs are particularly advantageous in aerospace structures as reinforcements in polymer matrix composites, enhancing structural performance while reducing weight. Furthermore, they offer the potential for multifunctionality, integrating structural, thermal, and electrical functionalities within components like wings. However, accurately modelling CNT behaviour poses challenges, especially considering their application in larger-scale aerospace structures. While accurate, molecular dynamics and molecular structural mechanics are computationally intensive and limited in length-scale. In this context, the present research proposes reduced-order continuum structural models using the Variational Asymptotic Method (VAM) to study CNT and its composite structures while incorporating length-scale effects using strain-gradient formulations. Using VAM, single-walled CNTs (SWCNTs) were first analysed by considering them as straight, hollow, circular tubes in a local continuum framework. This tube model accounts for the geometrically-nonlinear behaviour of standalone CNT when subjected to bending and buckling loads. Cross-sectional ovalisation leading to nonlinear bending and buckling behaviour has been studied. Combined loading cases of bending and compression; torsion and compression; & bending and torsion have been examined. The study aims to provide insights into the 3-D nonlinear deformation behaviour of SWCNTs, offering a more efficient approach for evaluating CNTs in aerospace composite applications. In the next step, recognising the significance of the structure's small size (such as used in MEMS, NEMS, and sensors), non-classical theories, such as the Modified Strain Gradient Theory, which account for the size effect in the material, have been employed to develop a pioneering beam and plate models tailored for CNT-reinforced composite structures. Emphasising the critical nature of size effects, characterised by length-scale parameters, this study delves into the nuances of the length-scale effects in nanoscale structures. To develop the asymptotically-correct strain-gradient beam model, a prismatic beam with a rectangular cross section has been considered to derive zeroth-order and subsequent higher-order models while capturing the strain-gradient effects. Notably, this work is the first application of non-classical theories in developing VAM-based beam models. Different orders for length-scale parameters have been considered, and the validity of each choice is scrutinised, followed by guidance on the appropriate choice of the length-scale parameters. Following the development of the strain-gradient beam model, a modified strain gradient theory-based plate model has also been developed using VAM, which is again a first-of-its-kind work in the context of VAM and reduced-order structural models. Using the variational methods, fourth-order differential equations were obtained for the non-classical case, and similarly, an additional set of boundary conditions (non-classical) were also derived. The warping solutions and the plate stiffnesses are obtained by solving this boundary value problem. It was noted that the material length-scale parameters appear only in the bending and twist stiffness terms. Further, the classical results can be derived by setting the material length-scale parameters to zero. Zeroth- and first-order approximations have been derived, followed by detailed validation of the results with literature for bending and buckling load cases. Parametric studies involving variations in thickness and plate width have been conducted to assess their influence on mechanical behaviour. The developed plate model is then applied to CNT-reinforced composites, and their bending and buckling studies have been carried out. The parametric studies have also considered evaluating all influencing parameters like CNT volume fraction, material length-scale parameter, plate thickness and width.