Analytical and meshless numerical approaches to unified gradient elasticity theory. (1st September 2021) - Kamil Żur, Krzysztof; Faghidian, S. Ali
Highlights: Examination of unified gradient theory with application to nano-mechanics of torsion. Introducing an efficient numerical approach based on Reissner variational functional. Proposing independent series solutions of the kinematic and kinetic field variables. Realizing excellent agreement between meshless method and exact analytical solution. Fast convergence rate and admissible convergence region of proposed meshless method. Abstract: The unified gradient elasticity theory with applications to nano-mechanics of torsion is examined. The Reissner stationary variational principle is invoked to detect the differential and boundary conditions of equilibrium along with the consistent form of the constitutive laws. An efficient meshless numerical approach is established by making recourse to the Reissner variational functional wherein independent series solution of the kinematic and kinetic field variables are proposed. Suitable forms of the coordinate functions, in terms of the Chebyshev polynomials, are introduced to fulfill a set of kinematic and higher-order boundary conditions in the elastic torsion of nano-bars with practical kinematic constraints. Torsional behavior of the unified gradient elastic bar is studied for structural schemes of applicative interest. An excellent agreement between the torsional responses of the nano-bar detected based on the established meshless method and obtained exact analytical solution is realized. The proposed meshless numerical approach is confirmed to have a fast convergence rate and an admissible convergence region in determination of the torsional rotation field with high accuracy. The introduced meshless method is demonstrated to be highly efficacious in characterizing both the softening and stiffening structural behaviors at nano-scale. The presented numerical approach therefore paves the way ahead in mechanics of nano-structures.