Fundamental Topics in Continuum Mechanics: Grand Ideas, Errors & Horrors

TL;DR

This paper critically examines the foundational concepts and formulations in continuum mechanics, highlighting errors and misconceptions in traditional literature, and advocates for a geometric 4D spacetime approach for clearer understanding.

Contribution

It offers a rigorous critique of classical continuum mechanics formulations and introduces a geometric 4D spacetime framework for improved conceptual clarity.

Findings

Identifies errors in traditional continuum mechanics formulations

Proposes a 4D Euclidean spacetime geometric approach

Enhances clarity of fundamental notions and methods

Abstract

Shortly after the middle of the past century, a comprehensive presentation of Continuum Mechanics was written under supervision of Clifford Ambrose Truesdell III in two volumes of Siegfried Fluegge's Handbuch der Physik, a first in 1960 with Richard Toupin on The Classical Field Theories (the monster), including an Appendix on Tensor Analysis by Jerald LaVerne Ericksen, and a second volume in 1965 with Walter Noll on The Non-Linear Field Theories of Mechanics (the monsterino). Both nicknames are due to Truesdell. These contributions were gradually taken as turning points by the Mechanics Community worldwide, due to completeness of analysis and profoundness of documentation. Vastness of treatment acted however as a shield to careful reasoning on delicate but basilar notions which, in the wake of some scholars of the XIX century, were taken to be worthy of belief and incorporated in the presentation with a valuable historical background. Lack of engineering perspective didn't favour the necessary caution to be taken in facing a number of issues. Scholars in Continuum Mechanics, fascinated by the monumental work conceived and carried out by Truesdell and associates, did not dare any accurate revision. The analysis is here centred on unsatisfactory formulations presently disseminated in literature by followers of such authoritative treatment as Truesdell's opus magnum. The geometric approach in 4D Euclid spacetime adopted here is self-proposing even in classical context and provides clarity of notions, methods and results not achievable by the more familiar but less powerful and prone to confusing 3D treatment.