Iterative Nonlocal Residual Elasticity

TL;DR

This paper introduces an iterative nonlocal residual elasticity model that simplifies solving nonlocal elasticity problems by iteratively correcting classical solutions, supported by convergence analysis.

Contribution

It develops a new nonlocal elasticity formulation based on residual stress fields and an iterative solution procedure, addressing complexities of existing models.

Findings

The iterative method converges to the nonlocal solution.

The model is supported by lattice and continuum mechanics perspectives.

Boundary value problems are effectively solved using the proposed approach.

Abstract

Motivated by the existing complications of finding solutions of Eringen nonlocal model, an alternative model is developed here. The new formulation of the nonlocal elasticity is centered upon expressing the dynamic equilibrium requirements based on a nonlocal residual stress field. This new nonlocal elasticity is explained from the lattice mechanics and continuum mechanics points of view. Boundary value problems obtained based on the new nonlocal elasticity are solved following a proposed iterative procedure. This iterative procedure is centered upon correcting the solution of the classical field problem for the nonlocal residual field of the elastic domain. Convergence analyses are presented to show the convergence of the iterative procedure to the solution of the nonlocal field problem. The iterative procedure is an integrated part of the proposed nonlocal elasticity. Therefore, the newly developed nonlocal elasticity is give the name iterative nonlocal residual elasticity.