Data from: Biot-Savart law in the geometrical theory of dislocations
Abstract
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.
README: 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, whileedge_circuit.vtu
andscrew_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
andsurface_real_Psi.vts
are for visualizing corresponding functions as a surface.contour_imaginary_Psi.vtu
andcontour_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
andscrew_stress_bz.vtr
contains the data of the stress components of correspodning dislocations.edge_circuit.vtu
andscrew_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
: v1.6.0numpy
: 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.