26 Oct 2020

26 Oct 2020

Review status: this preprint is currently under review for the journal SE.

Analytical solution for residual stress and strain preserved in anisotropic inclusion entrapped in isotropic host

Xin Zhong1,2, Marcin Dabrowski1,3, and Bjørn Jamtveit2 Xin Zhong et al.
  • 1Institut für Geologische Wissenschaften, Freie Universität Berlin, Malteserstrasse 74–100, 12449 Berlin, Germany
  • 2Physics of Geological Processes, The Njord Center, University of Oslo, Norway
  • 3Computational Geology Laboratory, Polish Geological Institute - NRI, Wrocław, Poland

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.

Xin Zhong et al.

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Xin Zhong et al.


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Short summary
Elastic thermobarometry is an useful tool to recover the paleo-pressure and temperature. Here, we provide an analytical model based on the Eshelby solution to calculate the residual stress/strain preserved in a mineral inclusion exhumed from depth. The method applies to ellipsoidal, anisotropic inclusion in infinite isotropic host. Finite element method is also used for facet effect. It is found that the volumetrically avereged stress is a good proxy for the overall heterogeneous stress stage.