Geometry and continuum mechanics: An essay

Abstract

Geometry, analysis, and numerics all apply to a good modelling of continua. Mathematics is the natural language of physics (Galileo Galilei), geometry is the natural language of continuum mechanics. This essay emphasizes the more or less elementary notions of differential geometry that are hidden in the bases of classical continuum mechanics (Killing’s theorem, covariance, Riemannian curvature). Then it examines the intervening of more modern and sophisticated notions that have been introduced for pedagogical purpose in harmony with present day mathematics and others of which the need appeared in the twentieth century development of this science: connections, torsion, Cartan’s forms and spaces. The influence of Einstein’s theory of gravitation on this increased geometrization and the role played by the formulation of a geometric theory of evolving structural rearrangements and defects such as dislocations and material inhomogeneities is of prime importance. The main actors in this historical perspective appear to be Pfaff, Lie, Riemann, Killing, Cartan, Kondo, Kröner, Bilby, and Noll.