DOI: 10.5061/dryad.pk0p2ngz4
title: Data from: Biot-Savart law in the geometrical theory of dislocations

description: Universal mechanical principles may exist behind seemingly unrelated
 physical phenomena, providing novel insights into these phenomena. This
 study sheds light on the geometrical theory of dislocations through an
 analogy with electromagnetics. In this theory, solving Cartan's first
 structure equation is essential for connecting the dislocation density to
 the plastic deformation field of the dislocations. The additional
 constraint of a divergence-free condition, derived from the Helmholtz
 decomposition, forms the governing equations that mirror Ampère's and
 Gauss' law in electromagnetics. This allows for the analytical
 integration of the equations using the Biot-Savart law. The plastic
 deformation fields of screw and edge dislocations obtained through this
 process form both a vortex and an orthogonal coordinate system on the
 cross-section perpendicular to the dislocation line. This orthogonality is
 rooted in the conformal property of the corresponding complex function
 that satisfies the Cauchy-Riemann equations, leading to the complex
 potential of plastic deformation. We validate the results through a
 comparison with the classical dislocation theory. The incompatibility
 tensor is crucial in the generation of the mechanical field. These
 findings reveal a profound unification of dislocation theories,
 electromagnetics, and complex functions through their underlying
 mathematical parallels.
description: # Data from: Biot-Savart law in the geometrical theory of dislocations ##
 Description of the data and file structure This dataset contains the
 plastic and elastic field formed by dislocations in the VTK XML format,
 which can usually be imported by ParaView. ### Files and variables ####
 File: data.zip **Description:**  The folder contains the raw source data
 in the VTK XML format to plot the plastic deformation field, complex
 potential, and stress fields of an edge and screw dislocations. We used
 ParaView to import and visualize the source data. The description of each
 files are as follows: * fig1/edge, fig1/screw: raw data for plotting the
 plastic deformation field of a straight edge and screw dislocations.
 `edge.vtr`, `screw.vtr` contains the plastic deformation field, while
 `edge_circuit.vtu` and `screw_circuit.vtu` includes the data to
 schematically illustrate the Burgers circuits. * fig2/imaginarypart,
 fig2/realpart: raw data for plotting plastic deformation potential of
 dislocations. `surface_imaginary_Psi.vts` and `surface_real_Psi.vts` are
 for visualizing corresponding functions as a surface.
 `contour_imaginary_Psi.vtu` and `contour_real_Psi.vtu` are prepared to
 show the contour curves of these functions. `floor_grid.vtu`, which is
 contained in both folders is used to schematically illustrate the
 coordinate grid. * fig3/edge, fig3/screw: raw data for plotting the stress
 components of a straigh edge and screw dislocations. `edge_stress_bx.vtr`
 and `screw_stress_bz.vtr` contains the data of the stress components of
 correspodning dislocations. `edge_circuit.vtu` and `screw_circuit.vtu` are
 used to schematically illustrate the Burgers circtuis. ## Code/software
 ### Code The following source codes are included in `code.zip`. They were
 used to process mathematical equations, and to generate the raw data
 contained in this repository. *
 `BiotSavartLawGeometricalTheoryDislocations.nb`: Mathematica Notebook to
 check the validity of the mathematical equations in the paper. *
 `fig1.py`: Python script to generate raw data contained in fig1/edge and
 fig1/screw. * `fig2a.py`, `fig2b.py`: Python script to generate raw data
 contained in fig2/imaginarypart and fig2/realpart. * `fig3a.py`,
 `fig3bc.py`: Python script to generate raw data contained in fig3/edge and
 fig3/screw. ### Software We used Mathematica (Wolfram 14.0.0.0) to execute
 `BiotSavartLawGeometricalTheoryDislocations.nb`. We also used Python
 3.11.7 and the following packages for the python scripts to generate raw
 data in the VTK XML format. * `pyevtk`: v1.6.0 * `numpy`: v1.26.3 We used
 ParaView 5.13.0 for macOS with Silicon arm64 processors to import and
 visualize the files in the VTK XML format contained herein. No additional
 packages are required. For the visualization in ParaView, we used the
 filters such as Slice, Clip, Glyph, StreamTracer,
 StreamTracerWithCustomSource, Transform, and Contour to plot the sectional
 distribution, vector fields, integral curves, and contour lines. We also
 used the sources such as Cone, Cylinder, and Plane to schematically plot
 the coordinate axes, sectional planes, and the Burgers vector and tangent
 vector of dislocation lines.