Abstract. Raman elastic thermobarometry has recently been applied in many petrological studies to recover the pressure-temperature (P-T) 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 finite element method to simulate the residual stress and strain state preserved in the non-spherical anisotropic inclusion. 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 e.g. tetragonal, hexagonal and trigonal. 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 (c-axis longer than a-axis), the effect of varying the aspect ratio on Raman shift is demonstrated to be insignificant. When c/a = 5, only ca. 0.3 cm⁻¹ wavenumber variation is induced as compared to the spherical inclusion shape. For oblate quartz inclusions, the effect is more significant, when c/a = 0.5 ca. 0.8 cm⁻¹ wavenumber variation for the 464 cm⁻¹ 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/strain within 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 inclusion, we show that the volumetrically averaged stress serves as an excellent stress measure and the associated RMSD is less than 2 %, except for diamond with a systematically higher RMSD (ca. 8 %). This expands the applicability of the analytical solution for any arbitrary inclusion shape in practical Raman measurements.

中文翻译：

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各向同性基质包裹在各向异性夹杂物中的残余应力和应变的解析解

摘要。拉曼弹性热压法最近已在许多岩石学研究中用于恢复压力-温度（P - T）夹杂物夹带的条件。岩石学中的现有建模方法要么采用在各向同性无限主体中嵌入球形各向同性夹杂物的假设，要么使用诸如有限元法之类的数值技术来模拟非球形各向异性夹杂物中保留的残余应力和应变状态。在这里，我们使用Eshelby解决方案来开发一个分析框架，用于计算无限大的各向同性主体中的弹性各向异性椭圆形夹杂物的残余应力和应变状态。该解析解适用于任何种类的夹杂物对称性和任意的夹杂物纵横比。对于某些对称类别（包括例如四边形，六边形和三边形），可以导出显式表达式。研究了改变长径比对残余应力的影响，包括石榴石主体中的石英，锆石，金红石，磷灰石和金刚石夹杂物。事实证明，石英受影响最小，而金红石受影响最大。对于扁长石英包裹体（c轴比a轴长），事实证明改变纵横比对拉曼位移的影响不明显。当c / a  = 5时，只有ca。与球形夹杂物形状相比，引起0.3cm ^-1波数变化。对于扁石英夹杂物，当c / a  = 0.5 ca时，效果更明显。464 cm ^-1的0.8 cm ^-1波数变化与参考球形夹杂物的情况相比，感应带被诱导。我们还表明，有可能拟合有效的椭球体以获得多面夹杂物内平均残余应力/应变的代理。计算多面夹杂物的体积平均应力与最适合的有效椭球体的分析计算应力之间的差，可得出石榴石基质中石英，锆石，金红石，磷灰石和钻石夹杂物的均方根偏差（RMSD）。根据500个随机生成（宽高比和随机晶体学方向的范围广泛）的多面夹杂物的结果，我们表明，体积平均应力可以作为一种出色的应力度量，并且相关的RMSD小于2％，系统地提高了RMSD（约8％）。