# Topological defects and metric anomalies as sources of incompatibility for piecewise smooth strain fields

## TL;DR

This paper develops a comprehensive framework to analyze how topological defects and metric anomalies cause strain incompatibilities in piecewise smooth materials, enabling better prediction of internal stresses.

## Contribution

It introduces unified compatibility equations incorporating defect and anomaly densities for piecewise smooth strain fields with bonded or imperfect interfaces.

## Key findings

- Derived general compatibility equations for piecewise smooth strain fields.
- Unified framework for defects and metric anomalies including bulk and surface densities.
- Relations enabling internal stress prediction from defect and anomaly distributions.

## Abstract

The incompatibility of linearized piecewise smooth strain field, arising out of volumetric and surface densities of topological defects and metric anomalies, is investigated. First, general forms of compatibility equations are derived for a piecewise smooth strain field, defined over a simply connected domain, with either a perfectly bonded or an imperfectly bonded interface. Several special cases are considered and discussed in the context of existing results in the literature. Next, defects, representing dislocations and disclinations, and metric anomalies, representing extra matter, interstitials, thermal, and growth strains, etc., are introduced in a unified framework which allows for incorporation of their bulk and surface densities, as well as for surface densities of defect dipoles. Finally, strain incompatibility relations are derived both on the singular interface, and away from it, with sources in terms of defect and metric anomaly densities. With appropriate choice of constitutive equations, the incompatibility relations can be used to determine the state of internal stress within a body in response to the given prescription of defects and metric anomalies.