Neighbourhood watch in mechanics: non-local models and convolution

TL;DR

This paper introduces non-locality and convolution concepts in mechanics, providing an accessible explanation for engineers and demonstrating their importance in phase-field fracture modeling and other scientific fields.

Contribution

It offers an intuitive, example-based introduction to non-locality and convolution, connecting physical intuition with advanced modeling techniques in mechanics.

Findings

Non-locality is fundamental in phase-field fracture modeling.

Convolution provides a mathematical basis for understanding non-local interactions.

The paper illustrates applications across various science and engineering fields.

Abstract

This paper is intended to serve as a low-hurdle introduction to non-locality 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 non-locality using concepts familiar to most engineers. We then show how concepts of non-locality are at the core of one of the moste active current research fields in applied mechanics, namely in phase-field modelling of fracture. From a mathematical perspective, these developments rest on the concept of convolution both in its discrete and in its continuous form. 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 non-locality and convolution in other branches of science and engineering by generalizing from the examples explained in detail in the main body of the article.