# Effects of Barriers on Fault Rupture Process and Strong Ground Motion Based on Various Friction Laws

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## Abstract

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## Featured Application

**In this paper, we report an interesting simulated result in which spontaneous fault rupture with barrier significantly influenced the strong ground motion, and this provides important insights for seismic hazard assessment. This study is significant in earthquake source simulation and seismic hazard evaluation, and it will have a great impact on understanding the mechanisms of earthquakes.**

## Abstract

## 1. Introduction

_{r}on a fault exceeds the propagation velocity V

_{s}of the shear wave (S-wave) in the medium around the earthquake source area. Earthquake observations have shown that the rupture speed of the fault in most earthquakes is lower than the S-wave speed, but many strike-slip earthquakes produce a supershear rupture, such as the 1979 Imperial Valley earthquake in the United States (Mw = 6.4) [1], the 1999 Kocaeli (Izmit) earthquake in Turkey (Mw = 7.4) [2], the 1999 Duzce earthquake in Turkey (Mw = 7.2) [2], the 2001 West Kunlun earthquake in China (Mw = 7.8) [3], the 2002 Denali earthquake in Alaska of the United States (Mw = 7.9) [4], and the 2013 Craig earthquake in the United States (Mw = 7.5) [5]. In addition, it has been proven by theoretical analysis [6,7,8,9], numerical simulations [10,11,12,13,14,15,16,17], and laboratory experiments [18,19,20,21,22,23,24,25,26,27] that fault rupture speed can exceed the S-wave speed and produce a supershear rupture. When a fault ruptures at supershear wave velocity, the wavefronts that are formed by the new and old ruptures interfere with each other, and the superposition effect greatly increases the amplitude of ground motion and generate Mach waves, which greatly intensify the seismic hazard. Therefore, it is important to study the formation mechanism of supershear rupture earthquakes in order to understand earthquake sources and to evaluate seismic hazards.

_{s}, τ

_{0}, and τ

_{d}represent the fault static shear strength, initial shear stress, and dynamic shear stress, respectively), in the rupture process, by using two-dimensional numerical simulation and a slip-weakening model, a sub-Rayleigh wave velocity rupture occurs when S is large enough, a supershear rupture occurs when S is small enough, and the critical value of S ranges from 1.7 to 1.8. However, due to the existence of a barrier, in certain conditions, the rupture speed of faults changes from a subshear rupture speed to a supershear rupture speed.

## 2. Models and Methods

#### 2.1. Models

_{nz}) was 3 km; the thick blue line represents the barrier, the length of which (L

_{rz}) was 5 km, and the distance between the barrier and the nucleation zone was 70 km. Moreover, uniform compressive normal stresses -σ

_{0}(positive in tension) and shear stress τ

_{0}were applied in the whole model region as initial stress conditions. An additional effective normal stress Δσ

_{n}was applied in the barrier so that the initial normal stress in the barrier was higher than that in other areas of the fault. To prevent seismic waves from affecting the results by boundary reflection, an infinite element-based absorbing boundary (gray part in Figure 1) was set around the model.

#### 2.2. Finite Element Formulation

**v**is the velocity field vector, U is the internal energy in terms of unit mass,

**f**is the body force vector,

**T**

^{l}is the surface distributed load,

**T**

^{qb}is the solid infinite element radiation traction,

**T**

^{f}is the frictional traction, ${\dot{E}}_{W}$ is the rate of work done to the body by external forces, ${\dot{E}}_{QB}$ is the energy that is dissipated by the damping effect of solid medium infinite elements, and ${\dot{E}}_{F}$ is the rate of the energy that is dissipated by contact friction forces between the contact surfaces.

**F**

_{t}(t) and ${\mathit{F}}_{f}(D(t),\dot{D}(t))$ are the tectonic force and friction force vectors, respectively, D denotes the slip on the fault, and $\dot{D}$ denotes the sliding velocity on the fault. We used the dynamic finite element method for discontinuous contact mechanics to simulate the slide on the fault in the calculation.

_{n}is the normal contact stress, τ is the shear stress, and μ is the coefficient of friction.

^{−4}s.

#### 2.3. Friction Laws

_{s}is the static friction coefficient, μ

_{d}is the dynamic friction coefficient, |s| is the slip of the fault, and D

_{c}refers to the characteristic slip distance.

_{s}). Then, the modified friction constitutive relation is used to calculate and analyze the effect of fault barriers on the rupture process. The mathematical expression of the modified slip-weakening friction law is as follows [43,44]:

_{s}is the static friction coefficient, μ

_{d}is the dynamic friction coefficient, |s| is the slip of the fault, $\left|\dot{s}\right|$ is the slip rate, and D

_{c}is the characteristic slip distance.

^{-7}- 1 mm/sec), which is inconsistent with the actual slip rate of large earthquakes, its mathematical expression is more complex. A linear rate-state friction law, which combines the advantages of laboratory results and the slip-weakening friction law, is proposed. Its mathematical expression is simple and can describe the friction behavior of faults under high-speed slip conditions. The specific expression is as follows [45,46,47,48,49,50]:

_{s}is the static friction coefficient, V is the slip rate, V

_{c}is the characteristic velocity, α

_{f}and β

_{f}are the constants of the evolution effect, θ is the state parameter, t

_{c}is the characteristic time scale, and the characteristic slip distance D

_{c}is approximately equal to the product of the characteristic velocity V

_{c}and the characteristic time scale t

_{c}.

## 3. Results

#### 3.1. Comparing the Effect of Barrier with Various Friction Laws

#### 3.1.1. Barrier with Classical Slip-Weakening Friction Law

_{s}= 0.63, the dynamic friction coefficient u

_{d}= 0.525, the characteristic slip distance D

_{c}= 0.4m, and the barrier strength Δσ

_{n}/σ

_{n}= 16%.

#### 3.1.2. Barrier with Rate-State Friction Law

_{c}in the rate-state friction law is approximately equal to the product of the characteristic velocity V

_{c}and the characteristic time scale t

_{c}. When the characteristic slip distance D

_{c}= 0.4m, a spontaneous rupture cannot be produced by the fault when the parameters of the other parametric models are consistent with those of Section 3.1.1, which indicates that there is a great difference when using the two different friction constitutive relations to simulate the spontaneous rupture process of the fault. For this section, V

_{c}= 0.1m/s, t

_{c}= 0.1 s were set up to enable the fault to spontaneously rupture and to study the effect of the barrier on the rupture process.

_{n}+ Δσ

_{n})·u

_{d}), but the shear stress decreased at the corresponding position in the model without a barrier (τ = σ

_{n}·u

_{d}). This phenomenon caused the stress state in front of the barrier to reach the critical state faster than the model without a barrier, as shown in Figure 5b.

#### 3.1.3. Barrier with Modified Slip-Weakening Friction Law

#### 3.2. Effect of the Barrier Strength on the Fault Rupture Process

_{n}(Δσ

_{n}/σ

_{n}= 1%–15%) in the barrier and simulated more than ten different models with the classical slip-weakening friction law for a comparative analysis. Figure 7 shows the simulated velocity curves of a rupture front on a fault under different strengths of a barrier. We can see from the figure that when there was a barrier on the fault surface, the rupture process of the fault changed; when the rupture propagated to the barrier, the rupture speed of the fault obviously reduced; and when the rupture front of the fault left the barrier zone for a certain distance, the rupture speed obviously accelerated. As the barrier strength increased, the curve of rupture speed on the fault fluctuated more and more; when Δσ

_{n}/σ

_{n}exceeded 3%, a supershear rupture zone appeared on the fault; and with the increase of Δσ

_{n}/σ

_{n}, the propagation distance of the rupture with supershear wave velocity correspondingly increased. However, after analyzing the simulation results, it was found that the overall rupture duration of the fault was basically unchanged and slightly increased with the increase of the barrier strength. It was concluded that the barrier could not shorten the overall rupture duration of the fault, but it could regulate the distribution of the rupture speed on the fault surface and promote the fault to produce supershear rupture.

_{n}/σ

_{n}is small), the peak ground acceleration near the fault did not obviously change. With the increase of Δσ

_{n}/σ

_{n}, the barrier started to cause the fault to produce a supershear rupture, the propagation distance of supershear wave velocity was longer and longer, and the peak ground acceleration near the fault was also increasingly larger. This shows that the existence of a barrier may lead to the enhancement of seismic hazards.

## 4. Discussion

_{c}led to a supershear rupture of the fault when the stress ratio S remained constant.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Model geometry, the initial conditions, and boundary conditions. The model domain is 300 by 300 km. The black line in the figure represents the location of the fault, and the red star marks the nucleation patch. The blue line denotes the barrier, and the gray region around the model is the absorbing boundary.

**Figure 2.**(

**a**) Distribution diagram of the rupture speed on the fault surface of Weng’s model (SW for short). (

**b**) Distribution diagram of the slip along the interface over time.

**Figure 3.**The slip and slip rate of the center point of the barrier on the SW model fault change with time.

**Figure 4.**(

**a**) Distribution diagram of the rupture speed of the fault surface of the RS model (a model that uses the rate-state friction law to simulate the effect of a barrier on rupture propagation). (

**b**) Distribution diagram of the slip along the interface over time.

**Figure 5.**(

**a**) Shear stress at the receiver P1 on the barrier patch (corresponding position in the model without a barrier) and (

**b**) shear stress at the receiver P2, 5 km in front of the barrier patch.

**Figure 6.**(

**a**) Distribution diagram of the rupture speed of the fault surface. (

**b**) Distribution diagram of the slip along the interface over time.

**Figure 7.**The distribution of rupture speed on the fault varies with barrier strength. The figure shows that near the barrier, the rupture speed of the fault first decreased and then rose. As the barrier strength increased, the curve of rupture speed on the fault fluctuated more and more.

**Figure 8.**Distribution diagram of peak ground acceleration near faults with barriers of different strengths.

**Figure 9.**(

**a**) Distribution diagram of the slip along the interface over time of Model 1. (

**b**) Distribution diagram of the slip along the interface over time of Model 2.

Description | Parameter | Value |
---|---|---|

Shear wave speed | V_{s} (m/s) | 3333 |

Dilatational wave speed | V_{p} (m/s) | 5774 |

Initial normal stress | σ_{0} (MPa) | 50 |

Initial shear stress | τ_{0} (MPa) | 28 |

Density | ρ (kg/m^{3}) | 2700 |

Poisson’s ratio | v | 0.25 |

Shear stress within nucleation zone | τ_{nucl} (MPa) | 31.7 |

Barrier strength | Δσ_{n}/σ_{n} | 1%~16% |

Nucleation size | L_{nz} (km) | 3 |

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**MDPI and ACS Style**

Yuan, J.; Wang, J.; Zhu, S.
Effects of Barriers on Fault Rupture Process and Strong Ground Motion Based on Various Friction Laws. *Appl. Sci.* **2020**, *10*, 1687.
https://doi.org/10.3390/app10051687

**AMA Style**

Yuan J, Wang J, Zhu S.
Effects of Barriers on Fault Rupture Process and Strong Ground Motion Based on Various Friction Laws. *Applied Sciences*. 2020; 10(5):1687.
https://doi.org/10.3390/app10051687

**Chicago/Turabian Style**

Yuan, Jie, Jinting Wang, and Shoubiao Zhu.
2020. "Effects of Barriers on Fault Rupture Process and Strong Ground Motion Based on Various Friction Laws" *Applied Sciences* 10, no. 5: 1687.
https://doi.org/10.3390/app10051687