Cauchy’s stress theorem stated and proved on atomistic grounds

Abstract

This paper offers a novel proof of a hitherto neglected theorem by Cauchy, in which he established the existence of the stress tensor for a system of interacting point-particles, 5 years after his celebrated 1823 theorem, in which he had obtained the same result for a continuous medium. In the intervening 200 years from then until now, both continuum mechanics and atomic and molecular physics have undergone enormous progress—even though along separate and diverging lines. As a consequence, nowadays it is both possible and necessary to depart substantially from the original formulation of the atomistic stress theorem given by Cauchy and from the arguments he devised to prove it. Here, I consider general (i.e., non-necessarily binary) short-ranged interparticle interactions, their effective range setting the microscopic length scale. Under the crucial short-rangedness assumption, I show that the traction field over the boundary of suitably defined space cells—labeled as “mesoscopic”—may be characterized by the property of being equipowerful to the out-of-cell interaction forces applied to the particles in the cell, with respect to all affine velocity field. More precisely, the defining property of the traction field requires equipowerfulness “to within a mesoscopically negligible error”—another concept intrinsic to atomistic-to-continuum coarse-graining. On this basis, the standard continuum notions of contact force and traction are proved to be well-defined, and the Cauchy postulate, fundamental lemma, and stress theorem are proved to hold, to within a mesoscopically negligible error.