Elastic geobarometry for anisotropic inclusions in cubic hosts

Mineral inclusions entrapped in other minerals may record the local stresses at the moment of their entrapment in the deep Earth. When rocks are exhumed to the surface of the Earth, residual stresses and strains may still be preserved in the inclusion. If measured and interpreted correctly through elastic geobarometry, they give us invaluable information on the pressures (P) and temperatures (T) of metamorphism. Current estimates of P and T of entrapment rely on simplified models that assumes that the inclusion is spherical and embedded in an infinite host, and that their elastic properties are isotropic. We report a new method for elastic geobarometry for anisotropic inclusions in quasi-isotropic hosts. The change of strain in the inclusion is modelled with the axial equations of state of the host and the inclusion. Their elastic interaction is accounted for by introducing a 4th rank tensor, the relaxation tensor, that can be evaluated numerically for any symmetry of the host and the inclusion and for any geometry of the system. This approach can be used to predict the residual strain/stress state developed in an inclusion after exhumation from known entrapment conditions, or to estimate the entrapment conditions from the residual strain measured in real inclusions. In general, anisotropic strain and stress states are developed in non-cubic mineral inclusions such as quartz and zircon, with deviatoric stresses typically limited to few kbars. For garnet hosts, the effect of the mutual crystallographic orientation between the host and the inclusion on the residual strain and stress is negligible when the inclusion is spherical and isolated. Assuming external hydrostatic conditions, our results suggest that the isotropic and the new anisotropic models give estimations of entrapment conditions within 2%.