Large-strain Elastic and Elasto-Plastic Formulations for Host-Inclusion Systems and Their Applications in Thermobarometry and Geodynamics

Mineral inclusions are trapped in a variety of geological environments and physical conditions. If brought to conditions different than their entrapment, mineral inclusions will generally experience different stress conditions than their hosts due to differences in their thermo-elastic properties and the associated deformation. These stress differences develop both in prograde and retrograde metamorphic conditions. The currently available analytical solutions consider isotropic materials and employ either fully linear-elastic behavior or they account for the non-linear-elastic volumetric deformation of minerals. Here we show that, by taking into account the finite volumetric deformation, we are able to explain the systematic differences amongst the available linear and non-linear elastic solutions. Furthermore, we employ a newly derived analytical solution for fully non-linear elastic materials (generalized Varga materials) to the host-inclusion problem. This solution considers both the geometric non-linearity and the material non-linearity by employing a Murnaghan equation of state. Our results show that the complete non-linear, hyperelastic behavior is not needed to explain the pressure differences that develop in common, unreacting, host-inclusion systems. The effects of plastic yielding are also investigated for the case of large finite deformations that can be relevant for the cases of phase transitions and mineral reactions that induce significant volume changes. Our results show that in the case of very large volumetric deformations the incorporation of finite strain effects may become important. Moreover, depending on the yield stress of the materials, the effects of plasticity may be dominant. In the latter case, significant pressure gradients will be developed as a consequence of stress balance. These results are general and they can also be used for elastic-barometry/volcanology applications and for benchmarking compressible Navier-Stokes geodynamic models. Accurate stress predictions in mechanical problems with large volumetric deformation can be significant in modeling the effects of mineral reactions that are generally non-isochoric.