Abstract
Abstract. Raman elastic thermobarometry has recently been applied in many petrological studies to recover the pressure and 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 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. 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 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.
Highlights
Raman elastic thermobarometry has been extensively used to recover the pressure and temperature (P –T ) conditions of mineral inclusion entrapment, e.g. the mostly studied quartzin-garnet inclusion–host pair (Ashley et al, 2014; Bayet et al, 2018; Enami et al, 2007; Gonzalez et al, 2019; Kouketsu et al, 2014; Taguchi et al, 2016, 2019; Zhong et al, 2019)
Quartz-in-garnet elastic barometry has been calibrated with experiments by synthesizing almandine garnets and quartz inclusions at high P –T conditions and comparing the entrapment pressure recovered based on the residual pressure measured in quartz with the pressure applied in experiments (Bonazzi et al, 2019; Thomas and Spear, 2018)
We use the classical Eshelby solution combined with the equivalent eigenstrain method to calculate the residual strain and stress in an anisotropic, ellipsoidal mineral inclusion embedded in an elastically isotropic host
Summary
Raman elastic thermobarometry has been extensively used to recover the pressure and temperature (P –T ) conditions of mineral inclusion entrapment, e.g. the mostly studied quartzin-garnet inclusion–host pair (Ashley et al, 2014; Bayet et al, 2018; Enami et al, 2007; Gonzalez et al, 2019; Kouketsu et al, 2014; Taguchi et al, 2016, 2019; Zhong et al, 2019). Elastically anisotropic inclusion entrapped in an infinite isotropic host, an exact closed-form analytical solution has been available for a long time (Eshelby, 1957; Mura, 1987) This solution has been widely applied in Earth science for many problems, such as viscous creep around inclusions (Freeman, 1987; Jiang, 2016), flanking structures (Exner and Dabrowski, 2010); elastic stress of inclusions at various scales (Meng and Pollard, 2014), microcracking and faulting (Healy et al, 2006), and magma-chamber-induced deformations (Bonaccorso and Davis, 1999). This may potentially provide a useful guide for future applications of elastic thermobarometry for any natural faceted mineral inclusions
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