Analytical solution for residual stress and strain preserved in anisotropic inclusion entrapped in an isotropic host
<p>Raman elastic thermobarometry has recently been applied in many petrological studies to recover the pressure and temperature (<span class="inline-formula"><i>P</i></span>–<span class="inline-formula"><i>T</i></span>) condit...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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Copernicus Publications
2021-04-01
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| Series: | Solid Earth |
| Online Access: | https://se.copernicus.org/articles/12/817/2021/se-12-817-2021.pdf |
| _version_ | 1818657663501729792 |
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| author | X. Zhong X. Zhong M. Dabrowski M. Dabrowski B. Jamtveit |
| author_facet | X. Zhong X. Zhong M. Dabrowski M. Dabrowski B. Jamtveit |
| author_sort | X. Zhong |
| collection | DOAJ |
| description | <p>Raman elastic thermobarometry has recently been applied in many
petrological studies to recover the pressure and temperature (<span class="inline-formula"><i>P</i></span>–<span class="inline-formula"><i>T</i></span>) conditions of
mineral inclusion entrapment. Existing modelling methods in petrology either
adopt an assumption of a spherical, isotropic inclusion embedded in an
isotropic, infinite host or use numerical techniques such as the finite-element
method to simulate the residual stress and strain state preserved in the
non-spherical anisotropic inclusions. Here, we use the Eshelby solution to
develop an analytical framework for calculating the residual stress and
strain state of an elastically anisotropic, ellipsoidal inclusion in an
infinite, isotropic host. The analytical solution is applicable to any class
of inclusion symmetry and an arbitrary inclusion aspect ratio. Explicit
expressions are derived for some symmetry classes, including tetragonal,
hexagonal, and trigonal.</p>
<p>The effect of changing the aspect ratio on residual stress is investigated,
including quartz, zircon, rutile, apatite, and diamond inclusions in garnet
host. Quartz is demonstrated to be the least affected, while rutile is the
most affected. For prolate quartz inclusion (<span class="inline-formula"><i>c</i></span> axis longer than <span class="inline-formula"><i>a</i></span> axis), the
effect of varying the aspect ratio on Raman shift is demonstrated to be
insignificant. When <span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M5" display="inline" overflow="scroll" dspmath="mathml"><mrow><mi>c</mi><mo>/</mo><mi>a</mi><mo>=</mo><mn mathvariant="normal">5</mn></mrow></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="40pt" height="14pt" class="svg-formula" dspmath="mathimg" md5hash="f5d3c7503b34dfd63ac2429dac7e9fda"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="se-12-817-2021-ie00001.svg" width="40pt" height="14pt" src="se-12-817-2021-ie00001.png"/></svg:svg></span></span>, only ca. 0.3 cm<span class="inline-formula"><sup>−1</sup></span> wavenumber variation is
induced as compared to the spherical inclusion shape. For oblate quartz
inclusions, the effect is more significant, when <span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M7" display="inline" overflow="scroll" dspmath="mathml"><mrow><mi>c</mi><mo>/</mo><mi>a</mi><mo>=</mo><mn mathvariant="normal">0.5</mn></mrow></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="49pt" height="14pt" class="svg-formula" dspmath="mathimg" md5hash="3a103ffbd4517c04cefd7e3f3fe2326e"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="se-12-817-2021-ie00002.svg" width="49pt" height="14pt" src="se-12-817-2021-ie00002.png"/></svg:svg></span></span>, ca. 0.8 cm<span class="inline-formula"><sup>−1</sup></span>
wavenumber variation for the 464 cm<span class="inline-formula"><sup>−1</sup></span> band is induced compared to the
reference spherical inclusion case. We also show that it is possible to fit
an effective ellipsoid to obtain a proxy for the averaged residual
stress or strain within a faceted inclusion. The difference between the
volumetrically averaged stress of a faceted inclusion and the analytically
calculated stress from the best-fitted effective ellipsoid is calculated to
obtain the root-mean-square deviation (RMSD) for quartz, zircon, rutile,
apatite, and diamond inclusions in garnet host. Based on the results of 500 randomly generated (a wide range of aspect ratio and random
crystallographic orientation) faceted inclusions, we show that the
volumetrically averaged stress serves as an excellent stress measure and the
associated RMSD is less than 2 %, except for diamond, which has a systematically
higher RMSD (ca. 8 %). This expands the applicability of the analytical
solution for any arbitrary inclusion shape in practical Raman measurements.</p> |
| first_indexed | 2024-12-17T03:45:03Z |
| format | Article |
| id | doaj.art-188860a5ddc0488da490efa25469986d |
| institution | Directory Open Access Journal |
| issn | 1869-9510 1869-9529 |
| language | English |
| last_indexed | 2024-12-17T03:45:03Z |
| publishDate | 2021-04-01 |
| publisher | Copernicus Publications |
| record_format | Article |
| series | Solid Earth |
| spelling | doaj.art-188860a5ddc0488da490efa25469986d2022-12-21T22:04:54ZengCopernicus PublicationsSolid Earth1869-95101869-95292021-04-011281783310.5194/se-12-817-2021Analytical solution for residual stress and strain preserved in anisotropic inclusion entrapped in an isotropic hostX. Zhong0X. Zhong1M. Dabrowski2M. Dabrowski3B. Jamtveit4Institut für Geologische Wissenschaften, Freie Universität Berlin, Malteserstrasse 74–100, 12449 Berlin, GermanyPhysics of Geological Processes, The Njord Center, University of Oslo, Oslo, NorwayPhysics of Geological Processes, The Njord Center, University of Oslo, Oslo, NorwayComputational Geology Laboratory, Polish Geological Institute – NRI, Wrocław, PolandPhysics of Geological Processes, The Njord Center, University of Oslo, Oslo, Norway<p>Raman elastic thermobarometry has recently been applied in many petrological studies to recover the pressure and temperature (<span class="inline-formula"><i>P</i></span>–<span class="inline-formula"><i>T</i></span>) conditions of mineral inclusion entrapment. Existing modelling methods in petrology either adopt an assumption of a spherical, isotropic inclusion embedded in an isotropic, infinite host or use numerical techniques such as the finite-element method to simulate the residual stress and strain state preserved in the non-spherical anisotropic inclusions. Here, we use the Eshelby solution to develop an analytical framework for calculating the residual stress and strain state of an elastically anisotropic, ellipsoidal inclusion in an infinite, isotropic host. The analytical solution is applicable to any class of inclusion symmetry and an arbitrary inclusion aspect ratio. Explicit expressions are derived for some symmetry classes, including tetragonal, hexagonal, and trigonal.</p> <p>The effect of changing the aspect ratio on residual stress is investigated, including quartz, zircon, rutile, apatite, and diamond inclusions in garnet host. Quartz is demonstrated to be the least affected, while rutile is the most affected. For prolate quartz inclusion (<span class="inline-formula"><i>c</i></span> axis longer than <span class="inline-formula"><i>a</i></span> axis), the effect of varying the aspect ratio on Raman shift is demonstrated to be insignificant. When <span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M5" display="inline" overflow="scroll" dspmath="mathml"><mrow><mi>c</mi><mo>/</mo><mi>a</mi><mo>=</mo><mn mathvariant="normal">5</mn></mrow></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="40pt" height="14pt" class="svg-formula" dspmath="mathimg" md5hash="f5d3c7503b34dfd63ac2429dac7e9fda"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="se-12-817-2021-ie00001.svg" width="40pt" height="14pt" src="se-12-817-2021-ie00001.png"/></svg:svg></span></span>, only ca. 0.3 cm<span class="inline-formula"><sup>−1</sup></span> wavenumber variation is induced as compared to the spherical inclusion shape. For oblate quartz inclusions, the effect is more significant, when <span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M7" display="inline" overflow="scroll" dspmath="mathml"><mrow><mi>c</mi><mo>/</mo><mi>a</mi><mo>=</mo><mn mathvariant="normal">0.5</mn></mrow></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="49pt" height="14pt" class="svg-formula" dspmath="mathimg" md5hash="3a103ffbd4517c04cefd7e3f3fe2326e"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="se-12-817-2021-ie00002.svg" width="49pt" height="14pt" src="se-12-817-2021-ie00002.png"/></svg:svg></span></span>, ca. 0.8 cm<span class="inline-formula"><sup>−1</sup></span> wavenumber variation for the 464 cm<span class="inline-formula"><sup>−1</sup></span> band is induced compared to the reference spherical inclusion case. We also show that it is possible to fit an effective ellipsoid to obtain a proxy for the averaged residual stress or strain within a faceted inclusion. The difference between the volumetrically averaged stress of a faceted inclusion and the analytically calculated stress from the best-fitted effective ellipsoid is calculated to obtain the root-mean-square deviation (RMSD) for quartz, zircon, rutile, apatite, and diamond inclusions in garnet host. Based on the results of 500 randomly generated (a wide range of aspect ratio and random crystallographic orientation) faceted inclusions, we show that the volumetrically averaged stress serves as an excellent stress measure and the associated RMSD is less than 2 %, except for diamond, which has a systematically higher RMSD (ca. 8 %). This expands the applicability of the analytical solution for any arbitrary inclusion shape in practical Raman measurements.</p>https://se.copernicus.org/articles/12/817/2021/se-12-817-2021.pdf |
| spellingShingle | X. Zhong X. Zhong M. Dabrowski M. Dabrowski B. Jamtveit Analytical solution for residual stress and strain preserved in anisotropic inclusion entrapped in an isotropic host Solid Earth |
| title | Analytical solution for residual stress and strain preserved in anisotropic inclusion entrapped in an isotropic host |
| title_full | Analytical solution for residual stress and strain preserved in anisotropic inclusion entrapped in an isotropic host |
| title_fullStr | Analytical solution for residual stress and strain preserved in anisotropic inclusion entrapped in an isotropic host |
| title_full_unstemmed | Analytical solution for residual stress and strain preserved in anisotropic inclusion entrapped in an isotropic host |
| title_short | Analytical solution for residual stress and strain preserved in anisotropic inclusion entrapped in an isotropic host |
| title_sort | analytical solution for residual stress and strain preserved in anisotropic inclusion entrapped in an isotropic host |
| url | https://se.copernicus.org/articles/12/817/2021/se-12-817-2021.pdf |
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