Abstract
The elasticity of continuous media with topological defects is described naturally by differential geometry, since it relates metric to strain. We construct a geometrical field theory, identifying disclinations, dislocations and extra-matter defects with the curvature, torsion and nonmetricity tensors, respectively. Connection and metric are given explicitly in the presence of dislocations and extra-matter. The density of extra-matter is a scalar source of isotropic strain, described by a local length scale or gauge. The logarithm of the gauge is related to the density of extra-matter by a Poisson equation. The corresponding integral equation, similar to Gauss' law in electrostatics, measures the amount of extra-matter contained inside a contour.
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