Macroscopic description of microscopically strongly inhomogenous systems: A mathematical basis for the synthesis of higher gradients metamaterials

TL;DR

This paper develops a mathematical framework for approximating complex inhomogeneous systems with higher gradient metamaterials using microscopic mechanical models, ensuring convergence as the scale diminishes.

Contribution

It introduces a rigorous method to approximate inhomogeneous continua with microscopic systems and proves their convergence, providing a foundation for designing advanced metamaterials.

Findings

Established a convergence theorem for microscopic approximations

Validated the approximation method mathematically

Provided a basis for synthesizing higher gradient metamaterials

Abstract

We consider the time evolution of a one dimensional $n$-gradient continuum. Our aim is to construct and analyze discrete approximations in terms of physically realizable mechanical systems, called microscopic because they are living on a smaller space scale. We validate our construction by proving a convergence theorem of the microscopic system to the given continuum, as the scale parameter goes to zero.