Geobarometry for host-inclusion systems: the role of elastic relaxation

Conventional thermo-barometric methods can be challenged in UHPM terraines as the temperatures of deep subduction often exceed the closure temperature of geothermobarometers, and they are also afflicted by the effects of further reactions and re-equilibration on exhumation. The elastic behavior of mineral inclusions trapped in host mineral phases contained in UHPM rocks provides an alternative method that is independent of chemistry and chemical equilibria. Minerals trapped as inclusions within other host minerals will develop residual stresses on exhumation as a result of the differences between the thermo-elastic properties of the host and inclusion phases. Measurement of the residual stress in the inclusions in combination with the equations of state (EoS) of the two phases, can be used to infer the pressures of entrapment. However, until now, even the simplest elastic system of a single inclusion embedded in an isotropic host has not been properly addressed for geological systems. Previous analyses (i.e. Zhang, 1998) have relied on the assumption of linear elasticity and invariant elastic properties of the minerals with P and T, or assume that the host material is completely rigid. These assumptions are not physically correct. We will present a solution to the single-inclusion problem that incorporates non-linear elasticity and can be applied to determine the stress distribution in the host and inclusion that arises from any change in P and T. Our solution shows that the previous calculations of residual inclusion pressures are incorrect in the relaxation term. The relaxation arises from the difference in stress at the host/inclusion interface that will force the interface outwards thus increasing the radial stress in the host adjacent to the inclusion, and decreasing the P inside the inclusion. The errors from linear elasticity theory are greater with softer hosts, and when the final conditions are not at ambient P and T. The general form of our solution relies on the concept of the isomeke, a line in P-T space along which the fractional volume changes of the host and inclusion are the same. This allows our solution to be used in combination with any form of equation of state and/or thermal expansion, and is not restricted to linear elasticity or just invertible EoS. Calculations can be performed with Eosfit7c (Angel et al. 2014).