Mathematical modelling of the elastic behavior of structured geophysical media

Abstract

Article is retracted From the viewpoint of modern concepts of nonlinear geophysics, the presence of the hierarchical block structure, anisotropy and heterogeneities is the defining property of the natural medium in particular of the Earth’s crust. A lot of achievements of modern self-organization theory in geophysics are based on the existence of basic structured media models. The structure plays a key role in Earth’s crust dynamic that is important for tectonic stress origin and localization as well as for next stress-relieved processes. As shown in [Starostenko et al., 2001], it is necessary to describe such a structured media at the micro-, meso- and macro-level as a sets of interacting blocks. Mathematical models created in such a way in particular numerical models of the dynamics of block-structured lithosphere al-low us to describe reliably such tectonic processes as the formation and evolution of the rift sedimentary basins [Starostenko et al., 2001], Earth’s crust compression in the subduction zones [Vengrovich, 2017], or faster tectonic processes of halo kinesis [Vengrovich, 2010]. The mechanism of local accumulation and emission of energy in the seismic waves form, which could be a way of creating a new model of earthquake source, was revealed during the investigation of rifting [Starostenko et al., 1996] and new subduction process modelling in the frame of block-structured lithosphere theory. However tectonic and seismic processes go far beyond the spatial and temporal scales. Used approaches need to be implemented in numerous models on the micro and mesolevel where it is extremely costly to calculate the dynamics of a huge number of interacting blocks. Therefore, we propose the mathematical model of the elastic behavior of the structured geophysical media allowing to obtain analytical dependencies between its elastic parameters and structure. We describe such environment in the first approximation as an elastic solid matrix with the inclusions of granules with excellent rheological properties. As usual rock formations keep irregular positions of particles different shapes and sizes in the space. These particles can be separated from the binder by fracturing. However, in this paper, we neglect nonregularity, fracturings, cap it all the granules will be considered as spherical. We combined the optical method of photo-elasticity studies [Malezhyk, 2001; Sirotin, Shaskolskaja, 1975; Sneddon, 1958; Christensen , 1979] and numerical calculations (FEM model) of stress fields dynamic in structured media using finite element analysis, overall, in such a way the proposed analytical model will be proofed. The numerical and analytical calculations of the stress fields evolution in the real earth with an internal structure are presented.