Journal of Earthquakes

Volume 2015, Article ID 434156, 9 pages

http://dx.doi.org/10.1155/2015/434156

## Dynamics of an Earthquake under Magma Thrust Strength

Laboratoire de Mécanique et de Modélisation des Systèmes Physiques (L2MSP), Faculté des Sciences, Université de Dschang, BP 69, Dschang, Cameroon

Received 22 November 2014; Accepted 9 February 2015

Academic Editor: Alejandro Ramírez-Rojas

Copyright © 2015 L. Y. Kagho et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper deals with the study of the behaviour of a one spring-block model subjected to the strengths due to the motion of the tectonic plates and the upflow of magma during volcanism. Using the direct integration method, we show that the sound velocity decreases (or increases) with the amplitude of the block’s oscillation when the external frequency is zero (or not). It appears that this amplitude grows with the amplitude of the magma thrust strength. For the resonance case, where this external frequency equals the pseudofrequency of the block’s oscillation, we establish that the slip which occurs during the periodic movement of a block is a stick-slip motion instead of a creep motion as usually known; therefore, the transition does not occur. We also obtain that the event probability decreases with the amplitude of the magma thrust strength while the power of the earthquake increases with it.

#### 1. Introduction

Many research works related to earthquake dynamics were focused on the study of a single spring-block model [1–3]. This single block model is deduced from the simplification of models of several blocks [4–8]. In these models, the motion of the fault is only stimulated by the strength due to the motion of the tectonic plates. As far as the one spring-block model is concerned, few studies have shown the appearance of chaotic behavior within the system when its motion is unidirectional [3] or bidirectional [2]. Moreover, Vasconcelos has investigated the phase transition in this single-block model and demonstrated that when the friction characteristic velocity is greater than 0.5, the block displacement vanishes after a slip event and the transition occurs [1]. In this study, the only force that provokes the rupture of the fault is induced by the movement of tectonic plates. One feature of our work is to examine this transition in the presence of a supplementary action referred to as the magma thrust strength.

In this paper, we consider a modified 1D spring-block model subjected to the strength due to the movement of the tectonic plates during subduction movement and to the magma thrust strength that appears during a volcanic activity. Therefore, we plan to present the origin and investigate the effects of this new force on the system and thereby contribute to the better understanding of earthquake dynamics.

This paper is outlined as follows. In Section 2, we present the earthquake model under consideration and determine the equation of motion. Section 3 is devoted to the analytical analysis of the motion equation exploiting the direct integration method. These investigations are done for two cases which deal with the closeness (or not) between the frequency of the magma and the fundamental frequency of the spring block. In Section 4, the magnitude and the event probability of an earthquake are examined. The last section is devoted to discussions and concluding remarks.

#### 2. Earthquake Model and Equation of Motion

The mechanical model under consideration (Figure 1) consists of a block of mass attached to a spring of stiffness that moves the upper line with a constant velocity which represents the speed of the tectonic plate. This block rests on a rough surface and is connected by a harmonic spring of stiffness which stands for the elasticity of rocks separating the block of mass and the loading point of the strength that represents the magma thrust.