Modelling of flexoelectric graphene-based structures: beam, plate, wire and shell

Abstract

Owing to its unique multifunctional and scale-dependent physical properties, a graphene is emerged as a promising reinforcement to enhance the overall response of its nanotailored composite materials. Most recently, the piezoelectricity phenomenon in graphene sheets was found through interplay between different non-centrosymmetric pores, curvature and flexoelectricity concept. This has added new functionality to the existing non-piezoelectric graphene. An overview of the literature revealed that the graphene-reinforced polymer matrix nanocomposite-based structures find numerous nanoelectromechanical systems (NEMS) and allow researchers to tailor their mechanical, thermal and electrical properties as per requirements. Such a piezoelectric graphene reinforced in the polymer matrix may be called as “graphene-reinforced nanocomposite (GRNC)”. Surprisingly, the application of piezoelectric graphene for modelling of graphene-based structures is not explored yet and this has provided the motivation for this Thesis. Therefore, the purpose of present research is to model the GRNC-based beams, plates, wires and shells. The prediction of effective elastic, piezoelectric as well as dielectric properties of GRNC are required priori. Therefore, the effective properties of GRNC were determined first as the open literature do not provide the same accounting the piezoelectric, flexoelectric and surface effects. In the first, the elastic properties of pristine and defective graphene sheets were determined via molecular dynamics simulations and the obtained results are found to be in good agreement with the existing experimental and numerical results. In the second, the micromechanical models based on the mechanics of materials (MOM), strength of materials (SOM) and finite element (FE) were developed to predict the effective elastic, piezoelectric and dielectric properties of GRNC. The developed models predict that the piezoelectric coefficients of GRNC account for the actuation capability of a graphene layer in the transverse direction due to the applied electric field in the plane. The predictions by analytical and numerical models are found in good agreement. Finally, the obtained effective properties of GRNC were used to study the electromechanical behaviour of GRNC-based beams, plates, wires and shells. An analytical model based on the linear piezoelectricity and Euler-Bernoulli theory was developed to investigate the electromechanical response of GRNC cantilever beam under both electrical and mechanical loads accounting the flexoelectric effect. In another attempt, the electromechanical behavior of GRNC beams with flexoelectric and surface effects were investigated using size-dependent Euler-Bernoulli theory and Galerkin’s weighted residual method. Analytical and FE models were developed to study the static response of flexoelectric GRNC beams under point load with various boundary conditions: cantilever, simply-supported and clamped-clamped. The cantilever nanobeam shows a softer elastic behavior compared to that of simply-supported and clampedclamped nanobeams for positive surface stress and the reverse is true for negative surface stress. On the contrary, simply-supported and clamped-clamped nanobeams show stiffer elastic behavior due to positive surface stress effect and vice versa. The results predicted by both analytical and FE models are found to be in better agreement. Outcomes reveal that the flexoelectric and surface effects on the static response of GRNC beams are significant and should be taken into account. The electromechanical behavior of GRNC plates with flexoelectric effect was studied by deriving an analytical model based on Kirchhoff’s plate theory and Navier’s solution. The static and dynamic responses of simply-supported flexoelectric GRNC plates under different loadings such as uniformly distributed, non-uniformly distributed, inline and point loads were investigated. Our results reveal that the flexoelectric effect on the static and dynamic responses of GRNC plate is substantial and cannot be neglected. Analytical and FE models were developed to study the electromechanical responses such as electric potential and deflection of cylindrical GRNC cantilevered nanowire with flexoelectric effect. Results show that the piezoelectric potential in the GRNC nanowire depends on the transverse force but it is not a function of the force acting along its axial direction. The electric potential in the tensile and compressive sections of nanowire is antisymmetric along its cross-section, making it as a “parallel plate capacitor” for nanopiezotronics applications. The predictions of potential distributions across the GRNC nanowire show better agreement with FE predictions. Finally, the analytical and FE models were developed for the elastic cylindrical shell laminated with flexoelectric GRNC layer based on Kirchhoff–Love theory considering both piezoelectric and flexoelectric effects to investigate the electric potential distributions in it. Developed models envisage the results for the distribution of electric potentials in GRNC shell and results predicted by analytical model with piezoelectric effect are found to be in better agreement with FE predictions. It is found that the electromechanical behavior of laminated shell is significantly improved due to the incorporation of flexoelectric effect. To summarize, this Thesis reports the enhancement in electromechanical response of GRNC structures due to the incorporation of flexoelectric effect. The electromechanical response of GRNC structures such as beam, plate, wire and shell can be engineered to achieve the desired electromechanical characteristics using different boundary and loading conditions as well as different parameters such as aspect ratio, thickness, diameter, length and volume fraction of graphene. Our study highlights the possibility of developing light-weight and high-performance piezoelectric graphenebased NEMS such as sensors, actuators, nanogenerators and distributors as the existing piezoelectric material such as Lead Zirconate Titanate (PZT) is heavy, brittle and toxic (Ibn-Mohammed et al., 2017). Keywords: Graphene; Piezoelectricity; Flexoelectricity; Surface effect; Micromechanics; Mechanics of materials; Strength of materials; Finite element method; Nanocomposite structures; NEMS.