基于速度-状态摩擦本构定律的三维DDA方法

原有三维非连续变形分析(DDA)方法采用常摩擦系数的Mohr-Coulomb定律作为切向破坏准则,然而当描述更大尺度构造块体的运动与变形时,常摩擦系数不再适用.速度-状态摩擦本构定律能够定量描述地震周期各阶段断层面剪应力变化,解释发震断层行为.本文将速度-状态摩擦定律与三维DDA方法相结合,首先推导了计算摩擦系数的实用公式,随后通过滑动-保持-滑动实验与速度步进实验算例对改进的三维DDA方法进行了验证.结果表明,应用速度-状态摩擦本构定律的三维DDA方法能够比较准确地模拟静摩擦的时间依赖性与动摩擦的速度依赖性,解决了将三维DDA方法在地学中应用的基本问题.

Development of the 3D DDA method with rate- and state-friction laws

The elastic medium in lithosphere should be considered as discontinuities since the complex tectonic background has produced many active faults as separation. The Mohr-Coulomb joint failure criterion with a constant friction coefficient, adopted in the original 3D discontinuous deformation analysis (DDA) method, cannot meet the requirement of highly accurate calculations for motion and deformation of tectonic block systems.The rate- and state-friction laws, which are capable of reproducing virtually the entire range of observed fault behaviors, are combined into the 3D DDA method. Firstly, the formula of computing coefficient of friction on the interface with rate- and state friction laws is derived. In order to calculate the value of state variable θ, slip velocity V and friction coefficient μ in every time step, a first-order differential equation about V is deduced. The increment of V is calculated by the second-order Taylor series expansion in our scheme. Secondly, the evolution law is determined by the Runge-Kutta scheme with adaptive step-size control. The friction submatrix, which consists of discrete forms of V and θ, is rewritten and then combined into the 3D DDA method. Finally, on the basis of reasonable geometry and mechanical properties of the numerical model, slide-hold-slide tests and velocity stepping tests are designed to examine the accuracy of the modified method.Suits of numerical slide-hold-slide tests are performed using hold time from 1 to 10000 seconds and then we take the numerical test of the 10-second hold time as an example to make a brief illustration. A rigid block moves along the base in uniform linear motion at the first five seconds, because of the equality between friction and point loading. At the fifth second, the point loading is set to zero, and the friction strength drops immediately. After ten seconds, the block is reset as the previous loading. Strength increases, reaches a peak value and returns to its previous steady-state value subsequently. The numerical results are consistent with the laboratory data measured by Beeler et al. (1994) with a 0.999-goodness of fits. Even though both data sets are mixed together, there is also a 0.997-goodness of fits. In addition, the friction healing rate β can be evaluated by the friction parameter b when the slip rate approximates to 0. The slope of these numerical data comes close to the b value of assumption with a 5% relative error. Besides, velocity stepping tests are also modeled using the improved 3D DDA. A slider keeps a steady sliding under a loading rate of 10 μm/s with a distance of 25 μm, and then decreases to 1 μm/s during a characteristic distance. The results show a strong velocity dependence of friction which is consistent with laboratory data.Comparison between numerical results and laboratory data shows that the 3D DDA method in combination with rata- and state-dependent friction laws is capable of simulating velocity dependence of sliding friction and time dependence of static friction, which resolves a basic problem when using 3D DDA in geodynamics research. The improved 3D DDA method still has the limitation such as unbalanced embedment due to the uncertainty of stiffness on the discontinuous interface, which leads to inexact results of contact force determination. In the future, this modification can be used in quantitative simulation of regional crustal deformation in combination with observations, such as GPS data.