Nonlinear Constitutive Model for Axisymmetric Bending of Annular Graphene-Like Nanoplate with Gradient Elasticity Enhancement Effects: Fid-Bau Portal

Nonlinear Constitutive Model for Axisymmetric Bending of Annular Graphene-Like Nanoplate with Gradient Elasticity Enhancement Effects

Yu, Y. M. / Lim, C. W.

Abstract

Abstract

Based on Eringen’s nonlocal elasticity theory, a new nonlinear, nonlocal constitutive relation in polar coordinates is presented for axisymmetric bending of annular graphene-like nanoplate. Instead of the common strain gradient theory, an iterative procedure is developed to solve the coupled nonlinear constitutive relations and to express the nonlocal stresses asymptotically in stress gradients with increasing orders. Subsequently, the nonlocal strain energy is formulated similarly to that of gradient elasticity, and the potential energy of external forces is obtained. Because analytical solutions are not available to date, a computational approach is developed to compute the minimum total energy and to establish the bending equilibrium condition of the nonlocal nanoplate. In a mathematically asymptotic manner, the nonlocal bending deflection function is approximated by finite polynomials that satisfy the admissible geometric boundary conditions. A numerical algorithm based on a minimum energy approach is subsequently developed to solve the coefficients of the nonlocal deflection function. To demonstrate the accuracy and computational efficiency, four practical examples with different boundary conditions and external loadings are presented. After verifying numerical convergence with increasing orders of polynomials, the numerical solutions show that the dimensionless maximum deflection decreases with increasing nonlocal effect. The analytical and numerical solutions presented here will assist in behavioral analyses for graphene and graphene-like structures and their performances.