Plate Tectonics: Theory, Mathematical Models, Computational Methods, Algorithms & Applications -- Volume 1

The book is intended to be self-contained and assessable to a large number of readers, and it is divided into two volumes. The book deals with the mathematical plate tectonic theory based on the authors ideas and scientific results. The book gives the uniform and consistent mathematical theory that facilitates theoretical study of Earths evolution during all of the geological periods (spanning about 4 billion years), and namely, during the relatively short time intervals from a geological point of view. The main objective of the book is to provide a compressive introduction to the main aspects of global tectonics and to introduce the corresponding mathematical plate tectonic theory. The presented mathematical plate tectonic theory is based, from a mathematical point of view, on the dynamic multi-body contact theory in three main rheologies the thermo-visco-plastic Bingham rheology and the special (non)-linear thermo-visco-elastic rheologies with short and long memories, respectively, that seem to be optimal for the present-day mathematical and computational studies of plate tectonic problems. The presented theory is consistent with the global model of the Earth. Volume 1 is divided into nine chapters. This volume looks at the fundamental topics and questions relating to the mechanism of evolution of the upper parts of the Earth the lithosphere and the mantle over several billion years. Following an introduction to the modeling of geodynamic processes in the lithosphere and the mantle, functional, convex and numerical analyses, the volume devotes individual chapters to introduce readers to: the structure of the Earth, global tectonics, geodynamic mantle processes in its lower and upper parts, including mantle convection, the origin of rift zones, and geodynamic processes in the lithosphere; the fundamentals of continuum mechanics, fracture rock mechanics and anisotropy of the Earths rocks, models of earthquakes and seismic wave propagations; the construction of models of the Earths mantle convection based on the temperature-depended Newtonian rheology, the infinite Prandtl number in the Boussinesq equations with temperature-depth-dependent coefficients and on the thermo-Bingham rheology. The last convection model couples the geodynamic and the phase change processes in the core-mantle boundary and the upper parts of the Earth; the variational formulations of the model problems and corresponding finite element methods for their discrete approximation. The author provides a unified presentation of new models for fracture mechanics, propagation of seismic waves through the Earth with a broken lithosphere into a great number of plates and blocks, based on the unilateral contact problems with (or without) friction and their discrete approximations. The author shows that throughout the low-velocity channels (e.g., the asthenosphere, etc.), the transverse channel waves propagate. Moreover, he also shows that the models of mantle convections based on the Boussinesqs approximations are special cases of models based on the (thermo-) Bingham rheology, when the threshold of plasticity (yield limit) is zero.