Numerical and analytical solutions for the large-strain elastoplastic Lame problem and its geological applications

Minerals and multiphase rocks in general may have non-trivial material models (constitutive relations) with respect to their volume change as a response of changing pressure and temperature (P-T) conditions. However, natural minerals within rocks do not freely expand/contract. When mineral phases are enclosed by phases that have different thermoelastic properties, a difference in volumetric strain develops upon the loading/unloading of the host-inclusion system. The difference of the volumetric strain between the two phases can lead to the significant stress build up in the vicinity of the host-inclusion interface. This behavior is in fact expected in geological scenarios where mineral reactions and phase transitions are responsible for significant volumetric changes. One of the most classical problems in elasticity theory is the Lame problem of an internally and externally pressurized thick cylinder. When adapted for spherical symmetry, this problem has been extensively used in geological applications in order to evaluate the stress distribution around a pressurized rock or mineral. Using linear elasticity theory and standard mineral properties it can be shown that the level of stresses that can develop around pressurized inclusions may be in the order of ~ 1 GPa. Such stress predictions are well beyond typical values of the yield stress of rocks which leads to large plastic deformations. Therefore, the incorporation of plasticity and finite strains is crucial in such models.

Here we present new analytical and numerical solutions for the classic host-inclusion problem assuming hyperelastic-plastic materials that follow a Drucker-Prager (non-associative) plasticity model under finite strains. Our analytical solution is based on the recently published solution of Levin and others (2021) that reduces to the Murnaghan model for purely hydrostatic loading. Our solutions have been developed to consider the effects of physical and geometrical non-linearities in deforming geomaterials. For stiff mineral hosts that can support GPa-level differential stresses, non-linear formulations provide accurate stress predictions even if the effects of geometrical non-linearities are ignored. For systems that reach the plastic yield, a plastic zone develops that can lead to the reduction of the pressure difference between the host and the inclusion phase. Nevertheless, the development of a plastic zone is occurring simultaneously to the development of pressure variations at the mineral hosts. Therefore, the development of pressure gradients in host-inclusion systems from the mineral to the outcrop scale are to be expected when the host material reaches the yield conditions.

Acknowledgments:

E.M. would like to acknowledge the Johannes Gutenberg University of Mainz for financial support. Y.P., K.Z., A.V. and V.L. were financially supported by Russian Science Foundation (project No. 19-77-10062) in the part related to the geomechanical problem statement and its analysis, and by Ministry of Education and Science of Russian Federation (grant №075-15-2019-1890) in the part related to the development of analytical and numerical algorithms for problem solving.