The Cauchy Stress Tensor

Abstract

Cauchy’s development of the stress tensor is discussed along with Cauchy’s spatial equations of motion. The corresponding referential equations of motion are obtained in terms of the Piola/Kirchhoff stress tensor. Examples are given of several special stress fields possessing particular structures, and the stress power is introduced.

...[I]t might seem that Cauchy’s achievement in formulating and developing the general theory of stress was an easy one. It was not. Cauchy’s concept has the simplicity of genius. Its deep and thorough originality is fully outlined only against the background of the century of achievement by the brilliant geometers who preceded, treating special kinds and cases of deformable bodies by complicated and sometimes incorrect ways without ever hitting upon this basic idea, which immediately became and has remained the foundation of the mechanics of gross bodies.

...[T]his work of Cauchy’s marks one of the great turning points of mechanics and mathematical physics, ... a turning point that could well stand comparison with Huygens’ theory of the pendulum, Newton’s theory of the solar system, Euler’s theory of the perfect fluid, and Maxwell’s theories of the monatomic gas and the electromagnetic field.

Truesdell (1967)