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author: Sheng Fan, Department of Geology, University of Otago, Dunedin, New Zealand
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</font><div><!--introduction--><!--/introduction--><h2id="1">The sphericity as measure for grain boundary irregularity</h2><p>Deformed polycrystalline materials such as ice often contain large grains interlocking with smaller grains, with many irregular grain boundaries. Boundary irregularity is hard to judge by visual inspection and it is better to use quantitative measures of boundary irregularity to infer processes across different deformation conditions. Here, we quantify the irregularity of each grainÿs boundary by introducing a sphericity parameter \(\Psi\) which is calculated in 2-D from grain area <codeclass="language-plaintext highlighter-rouge">A</code>, grain boundary perimeter <codeclass="language-plaintext highlighter-rouge">P</code>, and area equivalent grain radius <codeclass="language-plaintext highlighter-rouge">R</code> by the formula</p><p>\[\Psi = \frac{A}{P \cdot R}\]</p><p>The grain sphericity \(\Psi\) is a useful indicator for grain boundary irregularity because it measures how closely a grainÿs boundary resembles the circumference of a perfect circle. It decreases from \(\Psi = 0.5\), where the grain has a perfect circular shape, to \(\Psi = 0\) where the grain boundary is infinitely irregular. The statistics of grain boundary sphericity can be used to segregate recrystallised grains from remnant original grains (please refer to the paper for more details).</p><h2id="3">Data</h2><p>The EBSD data set used in this demonstration (PIL185.ctf) is available from <ahref="https://doi.org/10.6084/m9.figshare.13456550">https://doi.org/10.6084/m9.figshare.13456550</a>. The EBSD data were collected with a step size of 5 µm and representeds an ice sample deformed at -20°C to 12 percent axial strain. Let's import the data and reconstruct some grains.</p>
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</font><div><!--introduction--><!--/introduction--><h2id="1">The sphericity as measure for grain boundary irregularity</h2><p>Deformed polycrystalline materials such as ice often contain large grains interlocking with smaller grains, with many irregular grain boundaries. Boundary irregularity is hard to judge by visual inspection and it is better to use quantitative measures of boundary irregularity to infer processes across different deformation conditions. Here, we quantify the irregularity of each grain’s boundary by introducing a sphericity parameter \(\Psi\) which is calculated in 2-D from grain area <codeclass="language-plaintext highlighter-rouge">A</code>, grain boundary perimeter <codeclass="language-plaintext highlighter-rouge">P</code>, and area equivalent grain radius <codeclass="language-plaintext highlighter-rouge">R</code> by the formula</p><p>\[\Psi = \frac{A}{P \cdot R}\]</p><p>The grain sphericity \(\Psi\) is a useful indicator for grain boundary irregularity because it measures how closely a grain’s boundary resembles the circumference of a perfect circle. It decreases from \(\Psi = 0.5\), where the grain has a perfect circular shape, to \(\Psi = 0\) where the grain boundary is infinitely irregular. The statistics of grain boundary sphericity can be used to segregate recrystallised grains from remnant original grains (please refer to the paper for more details).</p><h2id="3">Data</h2><p>The EBSD data set used in this demonstration (PIL185.ctf) is available from <ahref="https://doi.org/10.6084/m9.figshare.13456550">https://doi.org/10.6084/m9.figshare.13456550</a>. The EBSD data were collected with a step size of 5 µm and representeds an ice sample deformed at -20°C to 12 percent axial strain. Let's import the data and reconstruct some grains.</p>
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{% highlight matlab %}
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% plotting convention
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<center>
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{% include inline_image.html file="IceSphericity_02.png" %}
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</center><h2id="5">Influence of EBSD step size on sphericity parameter</h2><p>Next we investigate how step size influences grain boundary irregularity measurements. To do this, we can artificially increase the step size of the EBSD data to from 10 up to 100 ÿm. Then, we choose one representative grain (one with a large number of pixels) and see how the sphericity parameter changes as the EBSD step size increases.</p>
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</center><h2id="5">Influence of EBSD step size on sphericity parameter</h2><p>Next we investigate how step size influences grain boundary irregularity measurements. To do this, we can artificially increase the step size of the EBSD data to from 10 up to 100 μm. Then, we choose one representative grain (one with a large number of pixels) and see how the sphericity parameter changes as the EBSD step size increases.</p>
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Copy file name to clipboardExpand all lines: pages/examples_matlab/SlipSystemAlumina.html
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title: Slip System Analysis in \(\alpha\)-Alumina
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last_updated: 05-Oct-2022
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last_updated: 01-Mar-2024
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sidebar: examples_sidebar
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permalink: SlipSystemAlumina.html
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folder: examples
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<!--
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This HTML was auto-generated from MATLAB code.
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--><title>Slip System Analysis in \(\alpha\)-Alumina</title><metaname="generator" content="MATLAB 9.13"><linkrel="schema.DC" href="http://purl.org/dc/elements/1.1/"><metaname="DC.date" content="2022-10-05"><metaname="DC.source" content="script_SlipSystemAlumina.m"></head><body><fontsize="2"><ahref="https://github.com/mtex-toolbox/examples/blob/master/PlasticityExamples/SlipSystemAlumina.m">
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--><title>Slip System Analysis in \(\alpha\)-Alumina</title><metaname="generator" content="MATLAB 9.14"><linkrel="schema.DC" href="http://purl.org/dc/elements/1.1/"><metaname="DC.date" content="2024-03-01"><metaname="DC.source" content="script_SlipSystemAlumina.m"></head><body><fontsize="2"><ahref="https://github.com/mtex-toolbox/examples/blob/master/PlasticityExamples/SlipSystemAlumina.m">
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edit page</a>
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author: Ruben Wagner, Robert Lehnert, TU Bergakademie Freiberg, Institute
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</font><div><!--introduction--><p>of Materials Engineering, Germany</p><!--/introduction--><h2id="1">Data Import</h2><p>The following EBSD maps has been measured by Ruben Wagner TUBAF, Institute of Materials Engineering, 2022 within the project SFB 920. It shows an alumina inclusions in 42CrMo4 steel after nanoindentation.</p>
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<h2id="11">References</h2><div><ul><li>Ruben Wagner, Robert Lehnert, Enrico Storti, Lisa Ditscherlein, Christina Schröder, Steffen Dudczig, Urs A. Peuker, Olena Volkova, Christos G. Aneziris, Horst Biermann, Anja Weidner, <ahref="https://www.sciencedirect.com/science/article/abs/pii/S1044580322005393"><i>Nanoindentation of alumina and multiphase inclusions in 42CrMo4 steel</i></a>, 2022.</li></ul></div></div></body></html>
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