Neighborhood Watch in Mechanics: Nonlocal Models and Convolution

Abstract

SIAM Review, Volume 67, Issue 1, Page 176-193, March 2025.
Abstract.This paper is intended to serve as a low-hurdle introduction to nonlocality for graduate students and researchers with an engineering mechanics or physics background who did not have a formal introduction to the underlying mathematical basis. We depart from simple examples motivated by structural mechanics to form a physical intuition and demonstrate nonlocality using concepts familiar to most engineers. We then show how concepts of nonlocality are at the core of one of the most active current research fields in applied mechanics, namely, in phase-field modeling of fracture. From a mathematical perspective, these developments rest on the concept of convolution in both its discrete and its continuous forms. The previous mechanical examples may thus serve as an intuitive explanation of what convolution implies from a physical perspective. In the supplementary material we highlight a broader range of applications of the concepts of nonlocality and convolution in other branches of science and engineering by generalizing from the examples explained in detail in the main body of the article.