Exact and "exact" formulae in the theory of composites

TL;DR

This paper reviews methods for determining the effective properties of composites, discusses the ambiguous use of the term 'exact solution' in applied mathematics, and highlights the importance of precise terminology.

Contribution

It clarifies the definitions and usage of 'exact' and related terms in the context of composite material theory, emphasizing rigorous mathematical standards.

Findings

Review of classical and recent methods for composite properties

Analysis of terminology and its misuse in applied mathematics

Examples illustrating the ambiguity of 'exact solution'

Abstract

The effective properties of composites and review literature on the methods of Rayleigh, Natanzon--Filshtinsky, functional equations and asymptotic approaches are outlined. In connection with the above methods and new recent publications devoted to composites, we discuss the terms "analytical formula, approximate solution, closed form solution, asymptotic formula" etc... frequently used in applied mathematics and engineering in various contexts. Though mathematicians give rigorous definitions of exact form solution the term "exact solution" continues to be used too loosely and its attributes are lost. In the present paper, we give examples of misleading usage of such a term.