Abstract
In this work, we examine the dynamical behaviour of the “single mass-springs” model for earthquake subjected to the strength due to the up flow of magma for the period of volcanism, considering the fractional viscous damping force, the fractional weakening friction and fractional power law of elastic force. The numerical simulation method used in this paper is that of Grünwald-Letnikov based on the generalization of the classical derivative, and the approximately analytical solution obtained by the harmonic balance method. The results have shown that the fractional-order derivative can affect the dynamical properties of fault rock, which is characterized by the equivalent damping coefficient and the equivalent stiffness coefficient. Moreover, the amplitude-frequency equation for the steady-state solution was established. It appears that the resonant amplitude and resonant frequency are strongly dependent on the fractional-order damping r, fractional-order friction q, the fractional deflection 𝛼, the nonlinear stiffness coefficient and the fractional viscous coefficient. We have also shown that, the recurrence time of an event, the duration time of an event and the slip size of an earthquake can be controlled by the fractional-order derivative, the fractional-order deflection and the magnitude of the magma strength. The model allowed us to better interpret the earthquake as a stick-slip motion.
Highlights
An earthquake is a ground shock resulting from the sudden release of energy accumulated by the stresses on the rocks [Kanamori, 2001]
The dynamics of an improved one-dimension nonlinear spring-block model for earthquake with fractional viscous damping subjected to the strengths due to the motion of the tectonic plates and the up flow of magma is studied
The resonance amplitude is only affected by the linear and nonlinear damping coefficients, and the resonance frequency is only affected by the linear and nonlinear stiffness, contrarily, in this work the resonance amplitude and the resonance frequency are affected by fractional-order derivative and the fractional power
Summary
An earthquake is a ground shock resulting from the sudden release of energy accumulated by the stresses on the rocks [Kanamori, 2001]. Volcanic eruptions require a flow of magma and/or aqueous fluids through rock, and there is potential for long-period seismic signals to provide valuable information on changes in the location, velocity, and types of fluids (e.g., gas, magma, bubbly magma) under volcanoes Such analysis requires understanding potential source mechanisms of the ground oscillations and the characteristics of the resulting signals [Julian, 1994]. There are numerous potential sources of the pressure disturbance required to trigger resonance including an earthquake, Fractional earthquakes under magma flow a new crack network connection, shock waves from “choked” flow [Morrissey and Chouet, 2001], or bubble coalescence leading to a rising gas slug [Cruz and Chouet, 1997].
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