Fundamental Topics in Continuum Mechanics: Grand Ideas, Errors & Horrors
Giovanni Romano, Raffaele Barretta
TL;DR
This work advocates a four-dimensional spacetime differential-geometric formulation of Continuum Mechanics, arguing that traditional 3D spatial treatments and referential equilibria obscure fundamental concepts and invite misstatements.It develops a rate-elasticity framework in which elastic-state variables evolve via the Lie derivative along the spacetime motion, linking elastic-stretching to stress through a nonlinear tangent compliance $oldsymbol{H}(oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{}}}}}}}}}}}}}}}}}$ and invoking elastic potentials $oldsymbol{ ext{Xi}}$, $oldsymbol{ ext{Xi}}^*$; mass conservation and metric compatibility anchor the theory.Material Frame Indifference (MFI) is recast as a natural covariance under frame changes, and the historical claim of Lie-derivative objectivity in 3D is shown to be flawed within a spacetime context.The work critiques referential equilibrium and chain (multiplicative) elastoplastic models for introducing nonphysical redundancy and proposes evolutive equilibrium with a Rate Virtual Power Principle (RVPP) to underpin robust computational methods, including variational boundary conditions and traveling control windows.
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.