Discrete Mathematical Model of Earthquake Focus: An Introduction
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Pure and Applied Geophysics
Discrete Mathematical Model of Earthquake Focus: An Introduction SERGEY A. ARSEN’YEV,1 LEV V. EPPELBAUM,2 and TATIANA B. MEIROVA2 Abstract—The process of earthquake appearance in its focus is analyzed on the basis of the oscillation theory. The earthquake focus consisting for simplicity of two blocks (granitic and basaltic) is studied mathematically and physically. The block sizes, density and Young’s modulus of the rocks composing these blocks are considered to be known. We assume that the blocks are located on an edge of a regional tectonic fault. The tectonic plate or subplate, moving with a given speed u of shearing, is on the other edge of the fault. The mechanical interaction of the fault edges is due to the friction, which depends on the relative velocity V = u - dx/dt, where x is a coordinate of the concrete block. Physical-mathematical equations of block motion are solved using analytical methods. As a result, we find complete information about seismic vibrations in the focus and their characteristics. The evolutions of kinetic, potential and total energy as well as the function of dissipation in the focus and magnitude of the earthquake are calculated. The computations were carried out at different speeds of movement u. This allowed us to study the dependence of the earthquake magnitude on the velocity u of the main plate. The constructed original model of the earthquake focus unmasks the mechanism of seismic oscillations and their properties. Keywords: Earthquake mechanics, mathematical modelling, earthquake sources, faults, seismic oscillations.
List of Symbols A Amplitude of the Berlage’s impulse in Eq. (28) a Side of a block B Berlage’s impulse in Eq. (28) b Dimensionless empirical constant in Eq. (8) C Dimensionless constant of integration in Eqs. (34), (40), (41) c Dimensionless connectedness of blocks D Dimensionless value in Eq. (19) d Normalization factor for function Dis
1 Department of Earth and Planetary Physics of Schmidt’s Institute of the Earth’s Physics, Russian Academy of Sciences, 10 Bolshaya Gruzinskaya, Moscow 123995, Russia. E-mail: [email protected] 2 Department of Geosciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, 6997801 Tel Aviv, Israel. E-mail: [email protected]
Dis E F Fmax G g h In J KE Kj k L M m N n O Pmax p Q q r S T t U u uc V W X? x Y
Dissipation function in Eq. (1) Total energy (Lagrange’s function) in Eq. (1) Function of friction in Eqs. (4)–(7) Maximal force of an earthquake Block loading in Eq. (7) Gravity acceleration Coefficient for lateral friction in Eq. (4) Interval, i.e. distance between blocks Young’s modulus Energy class of an earthquake Constant of integration Coefficient of elastic stiffness in Eq. (3) Operator of differentiation by time in Eqs. (2), (4) Earthquake magnitude in Eq. (55) Mass of block Dimensionless load Normalization factor for interval In Dimensionless operator of the time differentiation in Eqs. (13), (14) Maximal pressure of an earthquake Dimensionless empiri
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