Graphene nanoplates bending under multipart asymmetric conditions based on 3D elasticity and new modified couple stress theories: comparison with the molecular mechanics method

Abstract

The main purpose of the present study is to examine the bending behavior of graphene nanoplates under asymmetric multipart conditions. Efforts have been made to discuss the differences between employing the commonly used modified couple stress theory (MCST) and the new modified couple stress theory (NMCST) in capturing size dependency, as well as to conduct a detailed investigation into the influences of material length scale parameters (MLSPs). For this purpose, the constitutive equations based on the 3D elasticity theory and the NMCST for the nonlinear bending analysis of orthotropic annular micro/nanoplates are initially derived. The rationale for suggesting the use of NMCST, which incorporates three MLSPs, instead of MCST, which uses only one MLSP (suitable for isotropic materials), lies in the non-isotropic nature of graphene nanoplates. This approach allows for the investigation of the potential differential effects of each MLSP under various conditions, particularly asymmetric conditions. Furthermore, employing NMCST provides a better understanding of the influence and dependency of MLSP values on the base material. The 3D elasticity theory is utilized to avoid the common approximations in displacement fields typically introduced by other plate theories. Consequently, the governing equations based on the 3D elasticity theory and NMCST in polar coordinates have been developed for the first time. These equations have been numerically solved using a new semi-analytical polynomial method (SAPM), taking into account the orthotropic properties of graphene. This method is designed to explore numerical solutions under various asymmetric conditions, such as multipart boundary conditions, loading, and elastic foundations. Multipart conditions refer to dividing the nanoplates into different sections and assigning distinct conditions to each section. The fundamentals of this new method are introduced in detail as a powerful semi-analytical approach for solving various partial differential equations under both symmetric and asymmetric conditions. The numerical solution of the derived equations aims to explore the effects of different MLSPs and their influences under various symmetric and asymmetric conditions, thereby providing insights into the nature and behavior of each MLSP. Additionally, the uniqueness or dependency of MLSP values on the base material is investigated. To validate the results, graphene nanoplates have been simulated using the molecular mechanics method (MMM), as there is a lack of information on bending under multipart conditions in the existing literature. All results are compared to ensure accuracy and reliability.