DOI: 10.5061/dryad.34tmpg4vd
title: Data from: Revisiting Volterra defects: Geometrical relation between edge dislocations and wedge disclinations

description: This study presents a comprehensive mathematical model for Volterra
 defects and explores their relations using differential geometry on
 Riemann–Cartan manifolds. Following the standard Volterra process, we
 derived the Cartan moving frame, a geometric representation of plastic
 fields, and the associated Riemannian metric using exterior algebra.
 Although the analysis naturally defines the geometry of three types of
 dislocations and the wedge disclination, it fails to classify twist
 disclinations owing to the persistent torsion component, suggesting the
 need for modifications to the Volterra process. By leveraging the
 interchangeability of the Weitzenböck and Levi-Civita connections and
 applying an analytical solution for plasticity derived from the
 Biot–Savart law, we provide a rigorous mathematical proof of the
 long-standing phenomenological relationship between edge dislocations and
 wedge disclinations. Additionally, we showcase the effectiveness of novel
 mathematical tools, including Riemannian holonomy for analysing the Frank
 vector and complex potentials that encapsulate the topological properties
 of wedge disclinations as jump discontinuities. Furthermore, we derive
 analytical expressions for the linearized stress fields of wedge
 disclinations and confirm their consistency with existing results. These
 findings demonstrate that the present geometrical framework extends and
 generalizes the classical theory of Volterra defects.
description: # Data from: Revisiting Volterra defects: Geometrical relation between
 edge dislocations and wedge disclinations ## Description of the data and
 file structure This dataset contains the source data in the VTK XML
 format. They can be used to plot the complex potential of the disclination
 dipole and monopole on the Riemann sphere and the complex plane. **Files
 and variables** **File: data.zip** **Description:**  The folder contains
 the raw source data in the VTK XML format for the complex potential of the
 disclination dipole and monopole plotted on Riemann sphere, and compelx
 plane as well. We used [ParaView](https://www.paraview.org/) to import and
 visualize the source data. The description of each files are as follows: *
 `fig3ab`: Raw data for plotting the complex potential of the disclination
 dipole in VTK XML format. * `riemann_sphere_left.vts` and
 `riemann_sphere_right.vts` are the Riemann sphere subdivided into two
 hemispheres. They include the real and imaginary part of the complex
 potential. * `complex_plane_left_array_potential.vts` and
 `complex_plane_right_array_potential.vts` are the complex plane subdivided
 into two. They also include the real and imaginary part of the complex
 potential. * `great_circle_1.vtu`, `great_circle_2.vtu` and
 `great_circle_3.vtu` are the circles on the Riemann sphere. They
 corresponds to the real and imaginary axis on the complex plane, and the
 equator of the Riemann sphere, respectively. * `fig3cd`: Raw data for
 plotting the complex potential of the disclination monopole in VTK XML
 format. * `riemann_sphere_monopole_left.vts` and
 `riemann_sphere_monopole_right.vts` are the Riemann sphere subdivided into
 two hemispheres. They include the real and imaginary part of the complex
 potential. * `complex_plane_left_monopole_potential.vts` and
 `complex_plane_right_monopole_potential.vts` are the complex plane
 subdivided into two. They also include the real and imaginary part of the
 complex potential. * `great_circle_1.vtu`, `great_circle_2.vtu` and
 `great_circle_3.vtu` are the circles on the Riemann sphere. They
 corresponds to the real and imaginary axis on the complex plane, and the
 equator of the Riemann sphere, respectively. **Code/software** The
 following source codes are used to generate the raw data contained in this
 repository. * `fig3ab.py`: Python script to generate raw data for figure
 3(a) and (b). * `fig3cd.py`: Python script to generate raw data for figure
 3(c) and (d). **Software** We used Python 3.11.10 and the following
 packages for the python scripts to generate raw data in the VTK XML
 format. * `pyevtk`: v1.6.0 * `numpy`: v1.26.4 We used ParaView 5.13.0 to
 import and visualize the files in the VTK XML format contained herein. No
 additional packages are required.