Multiscale nonlocal beam theory: An application of distributed-order fractional operators

TL;DR

This paper develops a multiscale nonlocal beam theory using distributed-order fractional operators, enabling accurate and efficient simulation of anisotropic heterogeneous beams across multiple scales with reduced computational cost.

Contribution

It introduces a novel multiscale nonlocal beam model based on distributed-order fractional operators, providing a more efficient and accurate approach for heterogeneous material simulations.

Findings

Distributed-order operators effectively capture microstructure effects.

The model reduces computational costs compared to fully-resolved models.

Simulations align well with ground truth models.

Abstract

This study presents a comprehensive theoretical framework to simulate the response of multiscale nonlocal elastic beams. By employing distributed-order (DO) fractional operators with a fourth-order tensor as the strength-function, the framework can accurately capture anisotropic behavior of 2D heterogeneous beams with nonlocal effects localized across multiple scales. Building upon this general continuum theory and on the multiscale character of DO operators, a one-dimensional (1D) multiscale nonlocal Timoshenko model is also presented. This approach enables a significant model-order reduction without compromising the heterogeneous nonlocal description of the material, hence leading to an efficient and accurate multiscale nonlocal modeling approach. Both 1D and 2D approaches are applied to simulate the mechanical responses of nonlocal beams. The direct comparison of numerical simulations produced by either the DO or an integer-order fully-resolved model (used as ground truth) clearly illustrates the ability of the DO formulation to capture the effect of the microstructure on the macroscopic response. The assessment of the computational cost also indicates the superior efficiency of the proposed approach.