解孟雨， 史保平

中國(guó)科學(xué)院大學(xué)， 北京　100049

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數(shù)值模擬靜態(tài)應(yīng)力擾動(dòng)下的斷層失穩(wěn)：結(jié)果分析兼與Dieterich模型和Coulomb模型的對(duì)比

解孟雨， 史保平

中國(guó)科學(xué)院大學(xué)， 北京100049

摘要基于大量巖石力學(xué)實(shí)驗(yàn)，Dieterich和其他研究者(Dieterich, 1978; Ruina, 1983)首先提出了描述巖石摩擦過(guò)程的速率-狀態(tài)摩擦定律(R-S摩擦定律).如今R-S摩擦定律已成為研究地震成核等地震演化機(jī)制的有效手段.Dieterich (1992, 1994)最早提出了描述受靜態(tài)剪應(yīng)力擾動(dòng)后斷層失穩(wěn)時(shí)間提前或推后的余震觸發(fā)機(jī)制的解析模型.現(xiàn)在Dieterich模型已經(jīng)成為解釋余震隨時(shí)間衰減規(guī)律的Omori定律等地震觀測(cè)現(xiàn)象的有力工具.與之相對(duì)應(yīng)，廣泛使用的Coulomb應(yīng)力失穩(wěn)模型也可以給出斷層受到靜態(tài)剪應(yīng)力擾動(dòng)后，斷層失穩(wěn)時(shí)間的提前和推后量.Dieterich模型和Coulomb應(yīng)力失穩(wěn)模型基于不同的物理方法，所以在進(jìn)行地震危險(xiǎn)性評(píng)估時(shí)，二者均有各自的局限性.本文利用R-S摩擦定律控制的1-D彈簧-滑塊模型，模擬計(jì)算了理論地震循環(huán)以及在不同靜態(tài)剪切應(yīng)力擾動(dòng)下，失穩(wěn)時(shí)間的提前和推后量的變化情況，然后將計(jì)算得到的時(shí)間提前和推后量分別與Dieterich模型和Coulomb模型的相應(yīng)計(jì)算結(jié)果進(jìn)行了定量化對(duì)比和差異性分析，并給出了相應(yīng)的解釋.數(shù)值模擬結(jié)果顯示，對(duì)于R-S摩擦定律在參數(shù)不變的條件下，斷層模型失穩(wěn)時(shí)間的提前和推后量的大小強(qiáng)烈依賴(lài)于靜態(tài)剪應(yīng)力擾動(dòng)的大小和作用時(shí)間，而且絕對(duì)值相同的正、負(fù)向靜態(tài)剪應(yīng)力擾動(dòng)造成的失穩(wěn)時(shí)間的提前和推后量的變化情況并不完全一致.在震后松弛/滑移階段和閉鎖階段，時(shí)間提前和推后量是常數(shù)，且隨靜態(tài)剪應(yīng)力擾動(dòng)絕對(duì)值的增大而增大，兩者的比值接近于1.0，這與Dieterich模型和Coulomb模型的結(jié)果是一致的，相應(yīng)的差值小于兩模型結(jié)果的10%.而在自加速階段，模擬計(jì)算結(jié)果則存在與Dieterich模型和Coulomb模型結(jié)果不同的特征.首先，在自加速階段模擬計(jì)算結(jié)果均偏離Coulomb模型，而且時(shí)間提前和推后量的比值小于1.0，相異于Coulomb模型的論斷.不過(guò)當(dāng)受到正向靜態(tài)剪應(yīng)力擾動(dòng)后，Dieterich模型的結(jié)果和模擬計(jì)算結(jié)果是一致的，最大相差量不超過(guò)Dieterich模型結(jié)果的7%，可接近0.對(duì)于負(fù)向靜態(tài)剪應(yīng)力擾動(dòng)，當(dāng)其絕對(duì)值較小時(shí)，Dieterich模型的結(jié)果很接近模擬計(jì)算結(jié)果，相差量不超過(guò)該結(jié)果的14%.但對(duì)于絕對(duì)值較大的情況，模擬計(jì)算結(jié)果遠(yuǎn)大于Dieterich模型的結(jié)果，最大可達(dá)Dieterich模型結(jié)果的35倍，這是由于負(fù)向靜態(tài)剪應(yīng)力擾動(dòng)后使得θ/dc?1的條件不再成立，進(jìn)而使得Dieterich模型不再成立.總的來(lái)說(shuō)，與模擬計(jì)算相比Dieterich模型可以很好地描述1-D斷層受擾后失穩(wěn)時(shí)間提前和推后量的變化情況，并且可以體現(xiàn)出正、負(fù)靜態(tài)剪切應(yīng)力擾動(dòng)后失穩(wěn)時(shí)間提前量和推后量變化的差異性，而Coulomb模型則不能完整地給出受到靜態(tài)剪應(yīng)力擾動(dòng)后斷層失穩(wěn)時(shí)間提前或推后的估計(jì)值.

關(guān)鍵詞速率-狀態(tài)摩擦定律； 靜態(tài)剪應(yīng)力擾動(dòng)； Coulomb應(yīng)力失穩(wěn)模型； Dieterich模型； 失穩(wěn)時(shí)間提前量； 失穩(wěn)時(shí)間推后量

We first use the 1-D spring-slide block model governed by the R-S law to simulate earthquake cycle and triggered seismic activities caused by different static shear stress changes. Then we compare the simulation results separately with the solutions given by the Dieterich model and the Coulomb failure model, and the differences between our numerical simulation and the Dieterich and Coulomb models are discussed.

The numerical solutions show that without parameters changing, the advanced or delayed time strongly depends on both the onset time when static shear stress is applied and the amplitude of the applied shear stress. Moreover, the variation of the advanced or delayed time for positive or negative static shear stress which absolute values are equal isn′t same. Compared with our numerical simulation, the Dieterich model gives a more proper description of the time advance or time delay than that resulted from the Coulomb failure model which is a very popular model used in the earthquake forecasting model development. And the Dieterich model can show the different variation of the advanced or delayed time for positive or negative static shear stress.

1引言

圖1　Coulomb模型中破裂時(shí)間的提前或推后. 當(dāng)給定時(shí)，Δt只同ΔCFF的大小相關(guān)，與加載時(shí)刻無(wú)關(guān).當(dāng)|ΔCFF|給定時(shí)，時(shí)間的提前或延后量是相等的Fig.1　Schematic illustration of the time advance or delay for the Coulomb failure model. When , the remote loading of shear stress rate, is given, Δt, the time advance or delay depends only on the amount of ΔCFF. The onset time while the static shear stress is applied doesn′t affect Δt. In this model, the advanced or delayed times are the same if |ΔCFF| is given

在本文中，利用R-S摩擦定律，從1-D彈簧-滑塊模型出發(fā)，數(shù)值模擬在不同加載時(shí)刻，不同大小靜態(tài)應(yīng)力擾動(dòng)(本文中暫不考慮正應(yīng)力變化)下，對(duì)應(yīng)的發(fā)震時(shí)刻(斷層失穩(wěn)的時(shí)間)提前或推后的變化情況.通過(guò)模擬計(jì)算可以發(fā)現(xiàn)在R-S摩擦定律控制下，失穩(wěn)時(shí)間提前或推后量的大小依賴(lài)于靜態(tài)應(yīng)力擾動(dòng)的大小和加載時(shí)刻，而且正、負(fù)向靜態(tài)應(yīng)力擾動(dòng)后對(duì)應(yīng)了不同失穩(wěn)時(shí)間提前或推后的變化情況；另一方面，通過(guò)將上述結(jié)果分別與Coulomb模型和Dieterich模型的計(jì)算值分別進(jìn)行比較和討論后，可以證明Coulomb模型中提出的斷層失穩(wěn)時(shí)間的提前或推后的解析表達(dá)式僅可以描述震后松弛/滑移階段(Postseismic/Afterslip Stage)和閉鎖階段(Interseismic Stage)的情況，而Dieterich模型則能完整地給出斷層演化不同階段受不同靜態(tài)應(yīng)力擾動(dòng)后斷層失穩(wěn)時(shí)間的提前或推后量的變化情況，并能解釋正、負(fù)向靜態(tài)應(yīng)力擾動(dòng)的差異性.

2理論模型

2.1速率-狀態(tài)摩擦定律(R-S摩擦定律)

R-S摩擦定律為我們提供了一個(gè)基本的物理途徑用以定量化描述斷層內(nèi)部復(fù)雜的摩擦情況.該摩擦關(guān)系是由巖石力學(xué)實(shí)驗(yàn)結(jié)果所總結(jié)出的定律，它已成為研究地震成因和斷層演化力學(xué)機(jī)制的重要理論基礎(chǔ)，用以描述幾乎所有觀察到的同震滑移和震間斷層響應(yīng)，并被廣泛地用于描述地震行為規(guī)律性的變化 (Marone, 1998).目前使用最廣泛的R-S摩擦定律為Dieterich定律 (Segall, 2010)，即

(1)

(2)

2.21-D彈簧-滑塊模型

除了R-S摩擦定律，為了模擬斷層摩擦應(yīng)力隨時(shí)間變化的特征，我們還需要一個(gè)描述摩擦面和其周?chē)橘|(zhì)之間彈性作用的斷層力學(xué)模型.為了定性地說(shuō)明問(wèn)題，通常采用1-D彈簧-滑塊模型，其斷層上的加載應(yīng)力τ可以表示為

(3)

(4)

(5)

其中，η=G/2β，G為介質(zhì)的剪切模量，β為S波的速度.

2.3靜態(tài)應(yīng)力擾動(dòng)下的速度突變

(6)

2.4靜態(tài)應(yīng)力擾動(dòng)下的Dieterich解析表達(dá)式

(7)

而在初始時(shí)刻受靜態(tài)應(yīng)力擾動(dòng)后的失穩(wěn)時(shí)間f為

(8)

圖2　正向靜態(tài)應(yīng)力擾動(dòng)示意圖. 其中實(shí)線表示未擾動(dòng)時(shí)的演化，而虛線表示存在擾動(dòng)時(shí)的演化.其中t0表示施加靜態(tài)應(yīng)力擾動(dòng)的時(shí)刻，T表示未擾動(dòng)時(shí)從t0時(shí)刻到斷層模型失穩(wěn)所需的時(shí)間，f表示擾動(dòng)后從t0時(shí)刻到斷層模型失穩(wěn)所需的時(shí)間Fig.2　Schematic illustration of the slip velocity evolution with and without positive static stress perturbation. The solid and dashed lines show the unperturbed and perturbed response, respectively, and t0 is the onset time of the shear stress perturbation. T and f denote the time from t0 to the unperturbed failure time and the time from t0 to perturbed failure time, respectively

3數(shù)值結(jié)果分析

圖3　未受擾動(dòng)時(shí)1-D斷層模型的演化(a) 斷層速率的演化圖，其中Tr為斷層模型的完整演化周期； (b) 斷層所受摩擦應(yīng)力的演化圖； (c) 斷層滑移距離的演化圖； (d) 斷層模型中θ/dc在一個(gè)周期內(nèi)的變化情況，其中曲線的色度表示了曲線上某一點(diǎn)所處的時(shí)刻占整個(gè)演化周期的比例； (e) 斷層演化在速率-應(yīng)力相平面上的相圖，其中曲線的色度表示了曲線上某一點(diǎn)所處的時(shí)刻占整個(gè)演化周期的比例.Fig.3　Frictional responses of the 1-D fault model(a) Slip velocity evolution during a multi-seismic cycle. Tr represents a typical earthquake recurrence period. (b) Frictional shear stress variations on the fault during a multi-seismic cycle. (c) The stick-slip motion related time history of slip displacement. (d) The variation of in a seismic cycle, and the color represent the percentage of the cycle period. (e) Plot of the closed trajectory (cycle) in the phase plane, and the color represent the percentage of the cycle period.

前人對(duì)于靜態(tài)應(yīng)力擾動(dòng)的研究主要集中在正向靜態(tài)應(yīng)力擾動(dòng)(Δτ>0)對(duì)斷層演化的影響(Gomberg et al., 1998; Gomberg et al., 2000; Gomberg et al., 2005a; Kaneko and Lapusta, 2008)，而對(duì)負(fù)向靜態(tài)應(yīng)力擾動(dòng)(Δτ<0)情形下時(shí)間推后量Δtd同時(shí)間提前量Δta之間差異性的討論相對(duì)較少.2003年，Perfettini等討論了2D斷層模型受靜態(tài)應(yīng)力擾動(dòng)后時(shí)間提前和推后量之間的關(guān)系，他們的結(jié)果表明兩者之間的比值為一震蕩的曲線.因此，我們有必要從數(shù)值模擬入手，探討其差異性出現(xiàn)的緣由以及同Coulomb模型和Dieterich模型作對(duì)比.表1給出了本文在計(jì)算模擬中所使用的模型參數(shù)值(主要取自Kame et al., 2013b).

表1　模擬計(jì)算時(shí)使用的模型參數(shù)值

3.1正向靜態(tài)應(yīng)力擾動(dòng)

在模擬計(jì)算中，正向靜態(tài)應(yīng)力擾動(dòng)的取值分為Δτ= 1, 2, 5, 10和15 MPa，分別在整個(gè)演化周期內(nèi)的不同時(shí)刻t0加載，這樣就可以計(jì)算出不同時(shí)刻t0對(duì)應(yīng)的未擾動(dòng)時(shí)失穩(wěn)時(shí)間T和擾動(dòng)后失穩(wěn)時(shí)間f，然后就能得到失穩(wěn)時(shí)間的提前量Δt=T-f，接著可以分別繪出模擬計(jì)算結(jié)果與Coulomb模型和Dieterich模型的對(duì)比圖和差異性分析圖(圖4).首先是模擬結(jié)果和Coulomb模型的對(duì)比，如圖4a 和圖4c，不難看出，對(duì)于正向加載(Δτ>0)，在斷層模型演化早期(震后松弛/滑移階段和閉鎖階段)加載，Coulomb模型給出的時(shí)間提前量(虛線)比模擬計(jì)算的(實(shí)線)結(jié)果小，而且兩者的差值隨Δτ的增大而增大，不過(guò)其差值不超過(guò)Coulomb模型的10%(圖4c)，說(shuō)明兩者在斷層模型演化早期階段基本保持一致.而在演化后期的應(yīng)力加載，模擬結(jié)果得到的時(shí)間提前量隨加載時(shí)刻的推后而不斷減少趨于零，即當(dāng)t0→Tr，f→ 0，明顯不同于Coulomb模型給出的時(shí)間提前量值.而且兩者相近的時(shí)間段也隨著應(yīng)力擾動(dòng)量的增大，由80%的演化周期Tr減小至50%的演化周期Tr.

圖4　不同正向靜態(tài)應(yīng)力擾動(dòng)(Δτ>0)對(duì)應(yīng)的加載時(shí)刻和失穩(wěn)時(shí)間提前量Δt關(guān)系及差異分析其中t0表示加載時(shí)刻，Tr表示模型的演化周期. (a) 模擬計(jì)算結(jié)果(實(shí)線)與Coulomb模型(虛線)的比較； (b) 模擬計(jì)算結(jié)果(實(shí)線)同Dieterich模型(虛線)的比較； (c) Coulomb模型與模擬計(jì)算的差值，Difference=(ΔtCoulomb-ΔtSimulation)/ΔtCoulomb； (d) Dieterich模型與模擬計(jì)算的差值，Difference=(ΔtDieterich-ΔtSimulation)/ΔtDieterich.Fig.4　The advanced times after applying different positive static stress perturbations (Δτ>0) as a function of the onset time of the stress perturbationst0 is the onset time of the application of the stress perturbation. Tr is the cycle period. (a) A comparison between the simulations (solid lines) and the results derived from Coulomb failure model (dashed lines); (b) A comparison between the simulations (solid lines) and results derived from Dieterich model (dashed lines); (c) The differences between the Coulomb failure model and simulation with given different stress perturbations, and the Difference=(ΔtCoulomb-ΔtSimulation)/ΔtCoulomb; (d) The differences between the Dieterich model and simulation, and the Difference=(ΔtDieterich-ΔtSimulation)/ΔtDieterich.

事實(shí)上，根據(jù)Rubin和Ampuero (2005)的結(jié)果，可知在模型的演化早期，滿足Ω遠(yuǎn)遠(yuǎn)小于1，從而有下式：

(9a)

(9b)

當(dāng)加載正向應(yīng)力擾動(dòng)后，若依然滿足Ω遠(yuǎn)遠(yuǎn)小于1，那么根據(jù)(6)式可以得到下式：

t>t0，

(10a)

(10b)

圖5　在震后松弛/滑移階段和閉鎖階段加載靜態(tài)應(yīng)力擾動(dòng)時(shí)模型的演化情況(點(diǎn)線)與初始條件改變時(shí)演化情況(實(shí)線)的對(duì)比. 其中虛線表示未加載擾動(dòng)時(shí)模型的演化情況.加載時(shí)刻為t0=0.5Tr，靜態(tài)應(yīng)力擾動(dòng)為Δτ=5 MPaFig.5　A comparison of the temporal slip velocity evolution (dotted line) while the static stress perturbation is applied at postseismic/afterslip stage or interseismic stage and the temporal slip velocity evolution (solid line) with a changing of the initial conditions. The dashed line represent the temporal slip velocity evolution without static stress perturbation. The triggering time t0 satisfies t0=0.5Tr, and the static stress perturbation is Δτ=5 MPa

由公式 (7) 和 (8)，可知在Dieterich模型中，應(yīng)力擾動(dòng)后失穩(wěn)時(shí)間f和未擾動(dòng)時(shí)失穩(wěn)時(shí)間T滿足以下關(guān)系(Kaneko and Lapusta 2008)：

(11)

(12)

3.2負(fù)向靜態(tài)應(yīng)力擾動(dòng)

和正向靜態(tài)應(yīng)力擾動(dòng)相對(duì)應(yīng)，我們選擇的擾動(dòng)值為Δτ=-1,-2,-5,-10和-15 MPa，相應(yīng)的加載時(shí)刻t0和失穩(wěn)時(shí)間變化量Δt的關(guān)系及對(duì)比如圖6所示.和正向靜態(tài)應(yīng)力擾動(dòng)一樣，在模型演化早期施加負(fù)向靜態(tài)應(yīng)力擾動(dòng)后，計(jì)算得到時(shí)間推后量(虛線)比Coulomb模型給出的時(shí)間推后量(實(shí)線)大，不過(guò)兩者的差值隨Δτ的增大而減小，而且其差值不超過(guò)Coulomb模型的7%，也就是說(shuō)兩者在斷層模型演化早期階段基本一致.進(jìn)一步而言，隨靜態(tài)應(yīng)力擾動(dòng)絕對(duì)值的增大，模擬值和Coulomb模型給出值相符的時(shí)間段會(huì)有所加長(zhǎng)，由85%的演化周期增至95%的演化周期.從圖6a 中可以看出當(dāng)擾動(dòng)的絕對(duì)值較大時(shí)，在95%到98%的演化周期內(nèi)，模擬計(jì)算得到的時(shí)間推后量隨加載時(shí)刻的推后而減少，之后從98%的演化周期到失穩(wěn)，時(shí)間推后量隨加載時(shí)刻的推后而增加，甚至在Δτ=-5 MPa時(shí)存在突跳，明顯也不同于Coulomb模型給出的時(shí)間推后量值.

圖6　不同負(fù)向靜態(tài)應(yīng)力擾動(dòng)(Δτ<0)對(duì)應(yīng)的加載時(shí)刻和失穩(wěn)時(shí)間推后量Δt關(guān)系及差異分析 其中t0表示加載時(shí)刻，Tr表示模型的演化周期.(a) 模擬計(jì)算結(jié)果(實(shí)線)與Coulomb模型(虛線)的比較；(b) 模擬計(jì)算結(jié)果(實(shí)線)同Dieterich模型(虛線)的比較； (c) Coulomb模型與模擬計(jì)算結(jié)果的差值， Difference=(ΔtCoulomb-ΔtSimulation)/ΔtCoulomb ；(d) Dieterich模型與模擬計(jì)算的差值，Difference=(ΔtDieterich-ΔtSimulation)/ΔtDieterich. Fig.6　The delayed times after applying different negative static stress perturbation (Δτ<0) as a function of the onset time of the stress perturbationt0 is the time of the application of the stress perturbation. Tr is the cycle period. (a) A comparison between the simulations (solid lines) and results derived from the Coulomb failure model (dashed lines); (b) A comparison between the simulations (solid lines) and results derived from the Dieterich model (dashed lines); (c) The differences between the Coulomb failure model and simulations, and Difference=(ΔtCoulomb-ΔtSimulation)/ΔtCoulomb ; (d) The differences between the Dieterich model and simulations, and Difference=(ΔtDieterich-ΔtSimulation)/ΔtDieterich.

3.3時(shí)間提前量和推后量的比值

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圖7　絕對(duì)值相同的應(yīng)力擾動(dòng)對(duì)應(yīng)的失穩(wěn)時(shí)間提前和推后量比值變化圖其中Δta為失穩(wěn)時(shí)間的提前量，Δtd為失穩(wěn)時(shí)間的推后量，t0表示加載時(shí)刻，Tr表示模型的演化周期.實(shí)線表示模擬計(jì)算的結(jié)果，虛線是Dieterich模型給出的結(jié)果.Fig.7　Ratio of the time advance and time delay for the same absolute value of static stress perturbation as a function of the triggering timeΔta and Δtd represent the time advance and time delay, respectively. The solid and dashed lines show the simulation and calculation derived from the Dieterich model.

4討論

在文章中我們使用了1-D彈簧-滑塊模型，在這里我們對(duì)這個(gè)模型做一簡(jiǎn)單的討論.首先1-D彈簧-滑塊模型等價(jià)于空間上滑動(dòng)和應(yīng)力均勻分布的斷層模型(Rice, 1983)，它可以用來(lái)研究不同情況下斷層模型的一階近似情況(Dieterich, 1981)，能夠突出基本物理特征，進(jìn)而為我們進(jìn)一步研究震源機(jī)制提供線索.例如，研究者們通過(guò)對(duì)1-D彈簧-滑塊模型的分析認(rèn)識(shí)到存在臨界剛度kc，只有當(dāng)彈簧的剛度系數(shù)滿足k

5結(jié)論

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(本文編輯胡素芳)

基金項(xiàng)目國(guó)家基金委(41574040)，科技部《中國(guó)—東南亞毗鄰區(qū)大震活動(dòng)地球動(dòng)力學(xué)研究》(2015DFA21260)和中國(guó)科學(xué)院創(chuàng)新團(tuán)隊(duì)項(xiàng)目(KZZD-EW-TZ-19)共同資助.

作者簡(jiǎn)介解孟雨，男，博士研究生，主要從地震觸發(fā)機(jī)制和地震演化數(shù)值模擬等方面的研究. E-mail: jiemengyu13@mails.ucas.ac.cn

doi:10.6038/cjg20160217 中圖分類(lèi)號(hào)P541

收稿日期2015-05-06，2015-11-30收修定稿

Numerical simulation of fault instability due to an arbitrary static stress perturbation:A comparison with the Dieterich model and Coulomb failure model

XIE Meng-Yu, SHI Bao-Ping

UniversityofChineseAcademyofSciences,Beijing100049,China

AbstractBased on the laboratory′s experiment related to the rock friction, Dieterich and other researchers (Dieterich, 1978; Ruina, 1983) first introduce rate-and-state friction law (R-S Friction Law). Currently, R-S law has emerged as an effective tool for the study of earthquake faulting, such as earthquake nucleation. Later, Dieterich (1992, 1994) first proposed an earthquake triggering model used to describe the time advance or time delay of an earthquake due to the static shear stress change. The Dieterich model has become a powerful tool in explaining many seismic observations, including the Omori law related to the aftershock decays with time. In other hand, the widely-used Coulomb failure model can also be used to estimate the time advance and time delay of an earthquake after applying a static shear stress change. Due to the different physical approaches between the Dieterich model and the Coulomb failure model, there exists a certain limitation for both models used in the seismic hazard analysis. In this study, we try to find out what decides the time advance or time delay of an earthquake due to the static shear stress change, the difference of variation of the time advance or time delay and figure out the limitations for Dieterich model and the Coulomb failure model.

KeywordsRate-and-state friction law；Static shear stress change；Coulomb failure model；Dieterich model；Time advance；Time delay

解孟雨， 史保平. 2016. 數(shù)值模擬靜態(tài)應(yīng)力擾動(dòng)下的斷層失穩(wěn)：結(jié)果分析兼與Dieterich模型和Coulomb模型的對(duì)比.地球物理學(xué)報(bào)，59(2):593-605,doi:10.6038/cjg20160217.

Xie M Y, Shi B P. 2016. Numerical simulation of fault instability due to an arbitrary static stress perturbation: A comparison with the Dieterich model and Coulomb failure model.ChineseJ.Geophys. (in Chinese),59(2):593-605,doi:10.6038/cjg20160217.