A simple approximation of the rheological constitutive equations for isotropic poro-viscoelastic material is proposed. The model is developed by modifying Biot's theory in the transient viscoelsatic regime. This is achieved by considering a liquid-filled spherical shell consisting of Maxwell viscoelastic material as the simplest representation of the poro-viscoelastic materials subjected to large volumetric strains. It is shown that in transition to viscous asymptotic, the stress-strain relations deviates from the canonic poroelastic form and Biot's constants become time-dependent. The proposed model is tested using the published experimental data on the deformation of partially melted rocks at elevated PT conditions. Experimental results are usually interpreted in terms of the power law viscous materials. However, in this work we consider the effect of strain damage on viscosity by treating the latter as a dynamic time-dependent parameter; with the variation rate proportional to the second invariant of strain rate. By taking healing into account, the dynamic power law viscosity has constant asymptotic at a given strain rate. The proposed rheological model is implemented in a 2D FEM code and used to study the formation of partial melt in biaxial tests. It is found that the numerically calculated stress-strain curves demonstrate maxima similar to those found in experiments. Also, the computed pattern of melt redistribution and strain localization at the contact with a stiff spacer is qualitatively similar to the experimental observations. The results also indicate that the matrix sensitivity to damage affects the scale of strain localization and liquid re-distribution. Additionally, the problem of the liquid migration in a folding of poro-viscoelastic layer was considered.
1. INTRODUCTION
There is increasing interest in using damage theory in the formulation of geomechanics problems [1-3]. This is in response to the need to model the onset and accumulation of micro-cracks in the process of mechanical rapture of elastic materials. Healing of micro-scale damage is also possible. This occurs in the form of recrystallization when a fluid phase is present or due to thermally activated processes of grain boundary migration and dislocation motion. This approach can potentially provide an alternative to the traditional method of tracking crack propagation using the stress intensity analysis or classic theory of the plastic deformations. The concept of evolution of distributed damage has been used by Lyakhovsky et al. (2001) [2] to describe the non-stationary nature of the effective elastic moduli of porous and nonporous elastic geomaterials undergoing deformation. Such models produce a very realistic portrait of spatial strain localization in the elastic crust underlain by a flowing viscoelastic mantle. Spatial dynamics of the rheological parameters of viscous and viscoelastic materials lead to strain localization and formation of localized rapture zones similar to the cracks in elastic solids. Description of the compliance (viscosity) tensor as a dynamic parameter exposed to strain-weakening and thermally activated healing was suggested by Sleep (2001) [4]. At a constant strain rate, the conventional power law strain-rate dependent viscosity becomes the asymptote of the dynamic power law (for isotropic deformation).
For geomechanical applications of the dynamic power law formalism, we use a simplified variant of the poro-viscoelasticity theory.
