Physics-informed Neural Network for Bending Analysis of Two-Dimensional Functionally Graded Nano-Beams based on Nonlocal Strain Gradient Theory

This paper presents the bending analysis of two-dimensionally functionally graded (2D FG) nano-beams using a physics-informed neural network (PINN) approach. The material properties of the nanobeams vary along their length and thickness directions, governed by a power-law function. Hamilton's principle, combined with the nonlocal strain gradient theory (NSGT) and Euler-Bernoulli beam theory, is employed to derive the governing equation for the bending analysis of 2D FG nanobeams. Due to the incorporation of size dependency and the variation of material properties in two dimensions, the governing equation becomes a high-order variable-coefficient differential equation, which is challenging, if not impossible, to solve analytically. In this study, the applicability of PINN for solving such high-order complex differential equations is investigated, with potential applications in nanomechanical engineering. In the PINN approach, a deep feedforward neural network is utilized to predict the mechanical response of the beam. Spatial coordinates serve as inputs, and a loss function is formulated based on the governing equation and boundary conditions of the problem. This loss function is minimized through the training process of the neural network. The accuracy of the PINN results is validated by comparing them with available reference solutions. Additionally, the effects of material distribution, power-law index (in both length and thickness directions), nonlocal strain gradient parameters, and material length scale parameters are investigated. This study demonstrates the versatility of the PINN approach as a robust tool for solving high-order differential equations in structural mechanics.