Graphene sheets (GSs), which are a monolayer of bonded carbon atoms organized into a hexagonal structure, have gained increased attention because of their excellent mechanical, thermal, electrical, and optical properties. These properties include high strength, stiffness, aspect ratio, thermal conductivity, and electronic mobility as well as low density and light absorption. For example, the elasticity modulus of GSs is 10 times higher than that of steel. In addition, GSs possess thermal conductivity of (4.84 ± 0.44) x 10³ to (5.30 ± 0.48) x 10³ W/mK and electronic mobility of up to 2 x 10⁵ cm² V^-1s^-1. With these remarkable properties, single-layer GSs (SLGSs) exhibit potential in microelectromechanical and nanoelectromechanical systems, particularly in modern Industry 4.0 (fourth industrial revolution). The most important issues associated with Industry 4.0 are sensor use, communication, and computation, which can be addressed using GSs. Hence, the properties and performance of GSs must be elucidated through experimental, analytical, and theoretical investigations.

Experimental studies on the properties of GSs remain challenging because of difficulties in handling monolayer graphene with a monoatomic thickness and the required low level of forces required for imposing load. In this regard, non-experimental methods are required to investigate the properties of GSs. In the viewpoint of physicists, "structure," not "materials," exists. Hence, all macroscopic physical phenomena are dominated by microstructures. The behavior of a specified object can thus be predicted using ab initio calculations, rather than relying on experimental measurements. Ab initio calculations, also called first-principle quantum mechanics (QM) calculations, are based on density functional theory and belong to atomistic modeling. This method provides accurate description of the mechanical behavior of a material by calculating interactions involving electrons. However, QM calculations are computationally expensive. Although several assumptions can be employed to develop other atomistic modeling strategies (e.g., tight-binding method and molecular dynamics simulations) and reduce computational cost, satisfying the engineering requirements remains difficult.

In the perspective of dynamists, the behavior of a specified object can be predicted using the concept of "materials" instead of "structure." As such, the properties of an object should be studied at the phenomenal level, rather than at the microstructural level. Microstructures are averagely wiped into nonindividual body, that is, "materials," whose behavior can be described using general and specific equations. General equations are all-purpose to any "material" and include equations of mass, momentum, and energy conservation. Specific equations, also called constitutive equations, describe the characteristic of a specified "material." Dynamists mainly aim to establish constitutive equations for research "materials."

Computational cost is remarkably reduced when GSs are studied using the "materials" concept. However, the validity of using traditional constitutive equations to describe GSs remains questionable. For small-sized materials, lattice spacing between individual atoms is important and the discrete structure of the material cannot be homogenized into a continuum. Hence, material properties at small scale are size dependent, and traditional constitutive equations must be modified to consider the scale effect. Different have been introduced to consider the small-scale effect. In this work, nonlocal elasticity theory is employed to modify traditional constitutive equations.

An element-free computational framework and also known as element-free kernel particle (kp)-Ritz method is proposed to study the mechanical behavior of GSs. The construction of shape functions and their derivatives are described in detail based on the kp concept because the shape function is a critical factor in the development of an element-free method. Based on kp approximations for the field variables, Ritz method is employed to obtain discretized governing equations. The accuracy and efficiency of the method are validated using convergence and comparison studies.

This research is a systematic theoretical and numerical study on the mechanical behavior of GSs. A comprehensive study of the mechanical behavior of GSs, including SLGSs and bilayer GSs (BLGSs) with different aspect ratios and boundary conditions, was conducted to address static and dynamic issues. Bending stiffness is evaluated using a stabilized conforming nodal integration scheme. Several numerical techniques are also employed to eliminate shear locking of a thin plate. Computational results are then compared with published data to validate the proposed element-free kp-Ritz approach. A good agreement is obtained between computational and published data. The degrees of freedom of the system can be considerably decreased because of nodes are freely selected in the element-free computational framework, thereby saving computational resources. Therefore, typical mechanical behavior, including free vibration, buckling, large deformation, and nonlinear vibration, of GSs under various constraints are numerically simulated using the proposed element-free computational framework. Detailed case studies are also assessed to investigate the effects of geometric and nonlocal parameters on the mechanical responses of GSs. Finally, the application of the proposed approach is extended to study of the nonlinear vibration behavior of BLGSs with consideration of interlayer Lorentz magnetic force between adjacent layers.