Plate Tectonics: Theory, Mathematical Models, Computational Methods, Algorithms & Applications -- Volume 2

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 2 includes mathematical models that facilitate the study of plate tectonic and geodynamic processes in the upper parts of the Earth as well as analyses of tectonic and geodynamic processes in some subduction and collision zones. This volume is divided to three chapters. This volume includes: the fundamentals of the mathematical plate tectonic theory, based on the dynamic and quasi-static multi-body linear and non-linear contact-thermo-mechanics; several mathematical models of subduction and collision zones in different rheologies, such that they are consistent with the global model of the Earth and its mathematical description; unified presentations of new dynamic and quasi-static plate tectonic models in the linear and non-linear thermo-visco-elastic and thermo-visco-plastic of Bingham type rheologies (with or without friction (Coulomb or Tresca), acting on the contact boundaries between plates and blocks and formulated in displacements or velocities, respectively, as well as of the coupled contact-two-phase-change in thermo-elasticity; a systematic development of variational formulations, and a presentation of the possible existence and uniqueness of corresponding weak solutions; discrete approximation schemes and algorithms, which use the finite element method to discretise the spatial domain and finite differences for the time derivatives, and the corresponding error estimates; analyses of seismotectonics in some places on Earth, such as in the Aleutian and Andaman Arc Systems, the Andean subduction sone and the Himalayan collision zone; very useful methods for applications, i.e., an optimal shape design method for the construction of structures in high mountain massifs and a method for studying problems with uncertain input data. The last method will also be very useful for the extension mathematical plate tectonic theory under the assumption that all used physical data are uncertain. The book is suitable for graduate and postgraduate students and researches in geophysics, geology, computational geomechanics and seismology as well as in applied mathematics.