图  1   地震复发模型示意图(引自Shimazaki,Nakata,1980)

(a)周期模型；(b)时间可预测模型；(c)滑动可预测模型τ₁和τ₂分别为地震发生后和地震发生前的应力,均为常数(虚线),S_c为累积同震位移

Figure  1.   Schematic diagram of earthquake recurrence models(after Shimazaki,Nakata,1980)

(a)Strictly periodic model；(b)Time-predictable model;(c)Slip-predictable model τ₁ and τ₂ are stresses after and before an earthquake,respectively,and they are constants denoted by dashed lines. S_c represents the cumulative coseismic slip

图  2   库仑模型断层失稳示意图

(a)静态应力扰动引起的断层失稳(修改自仲秋,史保平,2012)；(b)断层摩擦强度降低引起的断层失稳虚线为演化状态变化后的断层演化图,Δt为断层演化状态变化后第一次失稳发生时间的提前量

Figure  2.   Schematic diagram of the fault instability in Coulomb failure model

(a)Fault instability caused by applying static stress perturbation(modified from Zhong,Shi,2012);(b)Fault instability caused by lowering the fault frictional strength. Dashed lines represent the fault evolution after changing state. Δt represents the time advance when the first instability occurs after changing the evolution state of fault

图  3   一维弹簧-滑块模型

Figure  3.   1D spring-slide block model

τ and σ represent shear and normal stress on the fault plane,respectively; δ is the displacement of the block along fault plane; v_pl is the remote loading rate; δ_pl is the remote loading displacement; k is the effective stiffness coefficient of spring

图  4   基于一维断层模型的滑移速率周期演化图(a)和相平面内的闭合轨迹图(b)

T_r为断层模型的完整演化周期,黑色三角形为各个阶段的分界标志,下同

Figure  4.   Periodic evolution of slip rate based on 1D fault model(a)and the closed trajectories in phase plane(b)

T_r represents a complete earthquake recurrence time. The colour represents the time percentage of the cycle period at each point,the black triangles are the boundary between different stages,the same below

图  5   一个地震循环周期内的一维断层演化图

(a)断层滑移速率演化图；(b)断层模型中Ω=vθ/D_c随时间的变化

Figure  5.   1D fault evolution during an earthquake cycle

(a)Slip rate evolution in a seismic cycle;(b)Variation of Ω=vθ/D_c versus the time in a seismic cycle

图  6   滑移速率v(a)和状态变量θ(b)的近似解与数值解对比图

Figure  6.   Comparison of analytic approximate solution with numerical solution for slip rate v (a)and state variable θ(b)

图  7   不同a/b值情况下断层滑移速率和剪切应力的相图

Figure  7.   Phase plane solutions of slip rate and shear stress for different a/b values

图  8   施加和未施加正向静态剪切应力扰动时的滑移速率演化示意图

实线为未扰动时的演化,虚线为存在扰动时的演化,t₀为施加静态剪切应力扰动的时刻,T和T_v分别为未扰动时和扰动后从t₀时刻至断层模型失稳所需的时间

Figure  8.   Schematic diagram of the time evolution of slip rate with and without positive static stress perturbation

The solid and dashed lines represent the unperturbed and perturbed evolution,respectively. t₀ is the onset time of the static shear stress perturbation. T and T_v represent the time from t₀ to unperturbed failure time and the time from t₀ to perturbed failure time,respectively

图  9   正向(左)和负向(右)静态剪切应力加载产生的非地震滑移

(a)滑移速率v演化图；(b)状态变量θ演化图；(c)断层面上剪切应力τ和滑移速率v的相图黑线和红线分别为未受应力扰动和受应力扰动时一维弹簧-滑块模型的演化情况,τ_ss和v_ss分别为稳定状态下断层面上的剪切应力和滑移速率

Figure  9.   Aseismic slip produced by a positive(left)and negative(right)static shear stress loading

(a)Evolution of slip rate v;(b)Evolution of state variable θ;(c)The plots of the closed trajectory(cycle)of shear stress τ and slip rate v in the phase plane. The black and red lines represent the evolution of 1D spring-slide block system on unperturbed and perturbed stress loading conditions,respectively. τ_ss and v_ss represent the shear stress and slip rate,respectively,when the system is at steady state