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Discrete Mathematical Model of Earthquake Focus: An Introduction

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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.

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Abbreviations

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

Dis:

Dissipation function in Eq. (1)

E :

Total energy (Lagrange's function) in Eq. (1)

F :

Function of friction in Eqs. (4)–(7)

F max :

Maximal force of an earthquake

G :

Block loading in Eq. (7)

g :

Gravity acceleration

h :

Coefficient for lateral friction in Eq. (4)

In :

Interval, i.e. distance between blocks

J :

Young's modulus

K E :

Energy class of an earthquake

K j :

Constant of integration

k :

Coefficient of elastic stiffness in Eq. (3)

L :

Operator of differentiation by time in Eqs. (2), (4)

M :

Earthquake magnitude in Eq. (55)

m :

Mass of block

N :

Dimensionless load

n :

Normalization factor for interval In

O :

Dimensionless operator of the time differentiation in Eqs. (13), (14)

P max :

Maximal pressure of an earthquake

p :

Dimensionless empirical constant in Eq. (8)

Q :

Dimensionless value in Eq. (20)

q :

Empirical dimension coefficient in Eq. (28)

r :

Dimensionless empirical constant in Eqs. (11), (12)

S :

Speed of the block motion

T :

Kinetic energy of an earthquake

t :

Time

U :

Potential energy of an earthquake

u :

Movement velocity of the main plate

u c :

Optimal velocity of movement

V :

Relative velocity in Eq. (4)

W:

Dimensionless relative velocity

X ∞ :

Residual displacement of blocks in Eqs. (36), (37)

x :

Displacement of the block

Y :

Acceleration of the block motion

α :

Coefficient of the block connection

β :

Empirical dimensionless constant in Eq. (8)

γ :

Dimensionless real part of a root in the formula for λ

δ :

Dimensionless coefficient h in Eqs. (13) and (14)

ε :

Empirical dimensional constant in Eq. (28)

ϑ :

Numerical constant in Eq. (39)

λ :

Root of Eq. (27)

μ :

Dimensionless coefficient of friction in law (7)

ν :

Partial frequency in Eq. (9)

ξ :

Dimensionless coordinate in Eqs. (13) and (14)

ρ :

Density of a block

σ :

Dimensional empirical coefficient in Eq. (28)

τ :

Dimensional time

Φ :

Function in Eq. (26)

φ :

Dimensionless imaginary part of a root in the formula for λ

χ :

Order of roots in Eq. (27)

Ψ :

Auxiliary function in Eqs. (21) and (22)

ω :

Base frequency of the seismic oscillations in Eq. (29)

ω 0 :

Typical frequency of the seismic oscillations in Eq. (30)

i :

Index for block numeration

j :

Summation index in the formula for Φ

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Acknowledgements

The authors thank two anonymous reviewers, who thoroughly reviewed the manuscript; their critical comments and valuable suggestions were very helpful in preparing this paper.

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Authors and Affiliations

  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

    Sergey A. Arsen’yev

  2. Department of Geosciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, 6997801, Tel Aviv, Israel

    Lev V. Eppelbaum & Tatiana B. Meirova

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  1. Sergey A. Arsen’yev
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Correspondence to Lev V. Eppelbaum.

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Arsen’yev, S.A., Eppelbaum, L.V. & Meirova, T.B. Discrete Mathematical Model of Earthquake Focus: An Introduction. Pure Appl. Geophys. 177, 4097–4118 (2020). https://doi.org/10.1007/s00024-020-02485-1

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