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

摘要

Raman elastic thermobarometry has recently been applied in manypetrological studies to recover the pressure and temperature (P –T ) conditions ofmineral inclusion entrapment. Existing modelling methods in petrology eitheradopt an assumption of a spherical, isotropic inclusion embedded in anisotropic, infinite host or use numerical techniques such as the finite-elementmethod to simulate the residual stress and strain state preserved in thenon-spherical anisotropic inclusions. Here, we use the Eshelby solution todevelop an analytical framework for calculating the residual stress andstrain state of an elastically anisotropic, ellipsoidal inclusion in aninfinite, isotropic host. The analytical solution is applicable to any classof inclusion symmetry and an arbitrary inclusion aspect ratio. Explicitexpressions 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 garnethost. Quartz is demonstrated to be the least affected, while rutile is themost affected. For prolate quartz inclusion (c ?axis longer than a ?axis), theeffect of varying the aspect ratio on Raman shift is demonstrated to beinsignificant. When c / a = 5 , only ca.?0.3?cm~(?1) wavenumber variation isinduced as compared to the spherical inclusion shape. For oblate quartzinclusions, 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 thereference spherical inclusion case. We also show that it is possible to fitan effective ellipsoid to obtain a proxy for the averaged residualstress or strain within a faceted inclusion. The difference between thevolumetrically averaged stress of a faceted inclusion and the analyticallycalculated stress from the best-fitted effective ellipsoid is calculated toobtain 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 randomcrystallographic orientation) faceted inclusions, we show that thevolumetrically averaged stress serves as an excellent stress measure and theassociated RMSD is less than 2?%, except for diamond, which has a systematicallyhigher RMSD (ca.?8?%). This expands the applicability of the analyticalsolution for any arbitrary inclusion shape in practical Raman measurements.