On the topological nature of Volterra's theorem

TL;DR

This paper explores the topological aspects of Volterra's theorem, clarifying the relationship between strain compatibility, homotopy, and holonomy in manifolds with dislocations and disclinations.

Contribution

It clarifies the topological interpretation of Volterra's theorem, linking strain compatibility to holonomy and topological properties of manifolds with non-trivial connectivity.

Findings

Homotopy plays a key role in strain compatibility.

Flat connections can have non-trivial holonomy in multiply connected manifolds.

The original Volterra construction relates to topological properties of dislocations.

Abstract

It is first observed that the original formulation of the Volterra construction for dislocations and disclinations was related to the role that homotopy plays in strain compatibility, whereas the modern discussions are chiefly concerned with how it relates to the holonomy groups of connections that have non-vanishing torsion and curvature. However, the Saint Venant conditions that follow from assuming infinitesimal strain compatibility imply that both torsion and curvature must vanish. The resolution of the confusion is in the fact that when a manifold is multiply connected a flat connection might still have non-trivial discrete holonomy.