# Limits of distributed dislocations in geometric and constitutive paradigms

## TL;DR

This paper explores the relationship between geometric and constitutive paradigms in modeling dislocations in solids, demonstrating their equivalence under certain symmetries and analyzing homogenization limits of dislocation distributions.

## Contribution

It establishes the equivalence of geometric and constitutive models for dislocations with discrete symmetries and compares their homogenization limits, unifying two foundational paradigms in material defect modeling.

## Key findings

- Geometric and constitutive paradigms are equivalent with discrete symmetries.
- Homogenization of dislocations can be viewed as manifold convergence or energy functional $ ext{Γ}$-convergence.
- The two homogenization theories are consistent and identical under symmetry conditions.

## Abstract

The 1950's foundational literature on rational mechanics exhibits two somewhat distinct paradigms to the representation of continuous distributions of defects in solids. In one paradigm, the fundamental objects are geometric structures on the body manifold, e.g., an affine connection and a Riemannian metric, which represent its internal microstructure. In the other paradigm, the fundamental object is the constitutive relation; if the constitutive relations satisfy a property of material uniformity, then it induces certain geometric structures on the manifold. In this paper, we first review these paradigms, and show that they are equivalent if the constitutive model has a discrete symmetry group (otherwise, they are still consistent, however the geometric paradigm contains more information). We then consider bodies with continuously-distributed edge dislocations, and show, in both paradigms, how they can be obtained as homogenization limits of bodies with finitely-many dislocations as the number of dislocations tends to infinity. Homogenization in the geometric paradigm amounts to a convergence of manifolds; in the constitutive paradigm it amounts to a $\Gamma$-convergence of energy functionals. We show that these two homogenization theories are consistent, and even identical in the case of constitutive relations having discrete symmetries.