The crooked road to effective stress

Abstract

Two Lagrangian functions are said to be variationally equivalent if they differ by a null Lagrangian (a Lagrangian whose associated Euler-Lagrange equations are identically satisfied). Kibble [1] noted in his seminal paper of 1961 that variationally equivalent Lagrangians lead to inequivalent gauge field theories, after which this important observation was actively ignored. There is an understandable reason for this situation; variationally equivalent Lagrangian functions are distinguished only by their distinct natural Neumann data, while elementary particle physics rarely if ever considers problems with imposed Neumann data. On the other hand, problems with imposed Neumann data demand inclusion of appropriate null Lagrangians in order that the imposed data be made variationally natural. It is thus clear that gauge theories with imposed Neumann data must make due allowances for the gauge-theoretic inequivalence of variationally equivalent Lagrangians. A specific case in point is the gauge theory of materials with defects that are subjected to imposed tractions on their boundaries.

This body of ideas is the subject of this paper. We first analyze the breaking of variational equivalence of Lagrangians that is induced by the minimal replacement operator of gauge theory in a general context. The results of this study are then particularized to the gauge theory of materials with defects. This allows us to show that the dislocation fields are driven by effective stress rather than the total stress, as has long been known, and that disclination fields are driven by the effective couple stress and the orbital dislocation couples. An interesting by product of the analysis is the emergence of the Cosserat continuum as the underlying reversible modality.