The bending behavior of graphene sheets under various multi-part asymmetric conditions using both continuum and molecular mechanics theories

Abstract

The primary objective of this study is to examine the bending behavior of graphene nanoplates under asymmetric multi-part conditions. Efforts focus on distinguishing the commonly used Modified Couple Stress Theory (MCST) and the newly proposed Nonlocal Modified Couple Stress Theory (NMCST) in capturing size dependency, including a detailed investigation of the influences of material length scale parameters (MLSPs). Constitutive equations for nonlinear bending analysis of orthotropic rectangular micro/nanoplates are derived using three-dimensional elasticity theory and NMCST. Unlike MCST, which uses a single MLSP and suits isotropic materials, NMCST incorporates three MLSPs, better suited to the non-isotropic nature of graphene nanoplates. This enables studying distinct effects of each MLSP, particularly under asymmetric conditions. Furthermore, NMCST offers deeper insight into the influence and material dependency of MLSP values. Employing 3D elasticity theory, the study avoids typical approximations in displacement fields defined by conventional plate theories. Consequently, governing equations based on 3D elasticity theory and NMCST are developed in Cartesian coordinates for the first time. These equations are numerically solved using a newly introduced Semi-Analytical Polynomial Method (SAPM), tailored to graphene’s orthotropic properties. The method addresses complex asymmetric conditions such as multi-part boundary scenarios, loading distributions, and elastic foundations—by segmenting the nanoplate and applying distinct conditions to each part. SAPM is presented as a robust technique for solving various partial differential equations under symmetric and asymmetric configurations. The numerical solution explores behavior and interaction of MLSPs across diverse conditions, emphasizing their dependency on the base material. Due to scarce data on bending behavior under multi-part conditions, validation is performed using Molecular Mechanics Method (MMM) simulations of graphene nanoplates. Results obtained via MM, FE, and SAPM approaches are compared and verified for reliability and consistency.