SCIENCE CHINA Earth Sciences

• RESEARCH PAPER • January 2010 Vol.53 No.1: 1–15 doi: 10.1007/s11430-010-0062-7

Influence of fault geometry and fault interaction on strain partitioning within western Sichuan and its adjacent region

WANG Hui1,2*, LIU Jie3, SHEN XuHui1, LIU Mian2, LI QingSong4, SHI YaoLin5& ZHANG GuoMin1

1 Institute of Earthquake Science, China Earthquake Administration, Beijing 100036, China; 2 Department of Geological Sciences, University of Missouri, Columbia, MO 65211, USA; 3 China Earthquake Network Center, China Earthquake Administration, Beijing 100045, China; 4 Lunar and Planetary Institute, Houston, TX 77058, USA; 5 Laboratory of Computational Geodynamics, Graduate University of Chinese Academy of Sciences, Beijing 100049, China;

Received March 31, 2009; accepted November 9, 2009

There are several major active fault zones in the western Sichuan and its vicinity. Slip rates and seismicity vary on different fault zones. For example, slip rates on the Xianshuihe fault zone are higher than 10 mm/a. Its seismicity is also intense. Slip

rates on the Longmenshan fault zone are low. However, Wenchuan Ms8.0 earthquake occurred on this fault zone in 2008. Here we study the impact of fault geometry on strain partitioning in the western Sichuan region using a three-dimensional viscoe- lastoplastic model. We conclude that the slip partitioning on the Xianshuihe-Xiaojiang fault presents as segmented, and it is related to fault geometry and fault structure. Slip rate is high on fault segment with simple geometry and structure, and vice versa. Strain rate outside the fault is localized around the fault segment with complex geometry and fault structure. Strain par- titioning on the central section of the Xianshuihe-Xiaojiang fault zone is influenced by the interaction between the An- ninghe-Zemuhe fault and the Daliangshan fault zone. Striking of the Longmenshan fault zone is nearly orthogonal to the direc- tion of eastward extrusion in the Tibetan Plateau. It leads to low slip rate on the fault zone.

Xianshuihe-Xiaojiang fault zone, Longmenshan fault zone, fault geometry, strain partitioning, 3D viscoelastoplastic model

Citation: Wang H, Liu J, Shen X H, et al. Influence of fault geometry and fault interaction on strain partitioning within western Sichuan and its adjacent region. Sci China Earth Sci, 2010, doi: 10.1007/s11430-010-0062-7

Located in the southeast borderland of the Tibetan Plateau, gence faults. The crustal deformation pattern in the region the western Sichuan region and its vicinity is a transition shows obvious deformation localization [1]. zone between the active Tibetan Plateau and the stable Two of the most important active fault zones in the South China. Tectonics in the region is active. There are western Sichuan area are the Xianshuihe-Xiaojiang fault many northwest, northeast and near north-south striking zone and the Longmenshan fault zone. These two faults faults cutting through the crust in the region. The north- form a ‘Y’-shaped fault system. They divide this region into west-west striking faults are mainly lateral strike-slip faults, three tectonic blocks: the Sichuan-Yunnan Block, the Bayan and the near north-south striking faults are mainly conver- Har Block, and the South China Block (Figure 1). The Xianshuihe-Xiaojiang fault zone is also one of the most active fault zones in Chinese mainland. Its northwestern *Corresponding author (email: [email protected]) section extends in northwest direction, while its southern

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Figure 1 Simplified tectonic map of the western Sichuan and its adjacent region and geological fault slip rates (mm/a). section extends approximately from north to south. Its according to velocity profiles across faults. However, GPS length is more than 1000 km. Slip rates on the fault zone are results depend on seismic cycle on the fault because the higher than 10 mm/a [2, 3]. Seismicity in the fault zone is fault is locked when seismicity is quiet [8]. The geological also intense. In the 20th century, several earthquakes with suvey and GPS results are not enouth to explain the dynam- magnitude over M7.0 occurred on the fault zone. Distribu- ics of fault activity, because they only reflect the kinematic tion of major earthquakes indicates the segmentation of the characteristics. Xianshuihe-Xiaojiang fault zone [3]. The Longmenshan There are many factors influencing the distribution of fault zone strikes in northeast direction. The fault zone ac- fault slip rates, such as evolution history of fault system [9], comodates crust shortening between the Tibetan Plateau and fault mechanical parameters [10], fault geometry [11, 12], the South China [4]. Although present-day activity on the and interaction between faults [9, 13] etc. Crustal motion in Longmenshan fault zone is weak, the seismogenic fault the western Sichuan area presents a clockwise rotation sustained occurrence of Ms8.0 Wenchuan earthquake on around the Eastern Himalayan Syntax (EHS for short) [6, May 12th, 2008. 14–16]. The most significant feature of fault system in the Fault slip rates are the basis for describing regional strain study region is along-strike variation of the Xianshuihe- partitioning and seismicity. Both geology and GPS survey Xiaojiang fault zone. The northwest striking Xianshuihe provide slip rates on each segment of the Xianshuihe- fault is the western section of the Xianshuihe-Xiaojiang Xiaojiang and the Longmenshan fault zone [3, 5–7]. How- fault system, and the north-south striking Xiaojiang fault is ever, the observations are limited. Geological study pro- the southern section of the fault system. The trace of entire vides long-term motions of several given sites on fault zone. Xianshuihe-Xiaojiang fault system looks like an arc section Uncertainty of the dating of samples always introduces er- around the EHS. Fault geometry of the Xianshui-Xiaojiang rors for long-term motion. GPS survey establishes slip rates fault zone may play a significant role in strain partitioning WANG Hui, et al. Sci China Earth Sci January (2010) Vol.53 No.1 3 in the western Sichuan and its adjacent area. collision [6, 14–16]. The crust of the eastern Tibet moves In this paper, we construct a three-dimensional geody- about 13 mm/a toward east with respect to the stable South namic model to study the strain partitioning in the western China Block. Sichuan and its vicinity. We employ 3D viscoelastoplastic finite element model to model slip rates on the 1.2 Major fault systems and their kinematics in the Xianshuihe-Xiaojiang and the Longmenshan fault zone. western Sichuan area Based on our modeling results, the impact of fault geometry on slip rates and regional strain partitioning is discussed in The western Sichuan region is divided into the South China detail. block, the Sichuan-Yunnan block, and the Bayan Har block by the Xianshuihe-Xiaojiang and the Longmenshan fault systems. Different movement between two blocks is cen- 1 Tectonic background of the western Sichuan tralized mainly on these two fault systems. Kinematics and and its adjacent area seismicity on the two fault systems are different. The Xianshuihe-Xiaojiang fault zone formed in Cenozoic is ac- 1.1 Tectonics in the western Sichuan and its adjacent tive in Late Quaternary. Dozens of historical earthquakes area with magnitude over M7.0 were recorded on the fault zone Both shallow and deep structures are complex in the west- in the past 300 years [3]. The most significant feature of the ern Sichuan and its adjacent area. The upper crust is thin [17, fault system is along-strike variation [21]. The Longmen- 18] and the middle-lower crust is weak [19, 20]. The study shan fault system was formed in the Indosinian and in the region is divided into several tectonic units by major faults. Yanshanian. It is a convergence fault with striking in north- Tectonic motion of each unit is accounted to be a complex east direction. The Ms8.0 Wenchuan earthquake occurred on or superimposition of three basic types of motions: sliding, this fault on May 12th, 2008. rotation, and uplift [5]. The regional motion patterns lend The Xianshuihe-Xiaojiang fault system defines the support to a model with a mechanically weak lower crust northern and eastern boundaries of the Sichuan-Yunnan experiencing deformation beneath a stronger, highly frag- Block. The fault system consists of several active fault zones. mented upper crust [6]. Its northern section is the Xianshuihe fault zone, which is a Contemporary GPS velocity field with respect to the sta- narrow linear tectonic zone. The Xianshuihe fault zone con- ble South China presents a clockwise rotation around the sists of the Luhuo fault in the northwest and the Moxi fault EHS in the Sichuan-Yunnan region (Figure 2). The crust in the southeast, and conjoins the Anninghe-Zemuhe fault moves southward in the interior of Sichuan-Yunnan Block, and the Daliangshan fault near Shimian in Sichuan Province. and moves southwestward in the southwest Yunnan region. The central section of the Xianshuihe-Xiaojiang fault sys- This motion pattern might be the reflection of the eastward tem consists of the Anninghe-Zemuhe fault zone, the Dali- extrusion in the Tibetan Plateau during the Indo-Asian angshan fault zone, and Xiaoxiangling fragment between the two fault zones. The Anninghe-Zemuhe fault zone con- sists of the north-south striking Anninghe fault zone and the northwest striking Zemuhe fault. The Daliangshan fault zone consists of the Haitang-Yuexi fault, the Puduhe fault, the Butuo fault, and the Jiaoji fault etc. [22]. The An- ninghe-Zemuhe fault zone and the Daliangshan fault zone conjoin near Ningnan-Qiaojia in Yunnan province. The Xiaojiang fault cuts through the crosspoint. The southern section of the Xianshuihe-Xiaojiang fault system is the Xiaojiang fault zone. The Xiaojiang fault zone includes two parallel branch faults with a distance less than 20 km be- tween them. Motions of each fault segment on the Xianshuihe- Xiaojiang fault zone are different based on geological sur- vey. The left-lateral slip rate on the Luhuo fault is 15±5 mm/a [3]. Total left-lateral slip rate in Holocene on south- eastern section of the Xianshuihe fault is 9.6±1.7 mm/a [23]. Therefore, horizontal slip rate on the Xianshuihe fault zone striking NW45° is 14±2 mm/a. The left-lateral slip rate on the Anninghe fault is only 6.5±1 mm/a [5]. Slip rate on the Zemuhe fault zone is about 4.9±0.6 mm/a [24]. Mean Figure 2 GPS velocity field with respect to South China (after ref. [6]). left-lateral slip rate on the Daliangshan fault is about 3 4 WANG Hui, et al. Sci China Earth Sci January (2010) Vol.53 No.1 mm/a in Late Quaternary [5, 25, 26]. Left-lateral slip rate on to low slip rates on the fault section. the Xiaojiang fault is about 10±2 mm/a [27]. Previous observations provided basic kinematic patterns Comtemporary fault slip rate can be measured using GPS of the Xianshuihe-Xiaojiang fault zone and the Longmen- velocity profile across fault. The slip rate on the northwest- shan fault. Using a three-dimensional model, we further ern segment of the Xianshuihe fault is 10±2 mm/a. It is investigate the geodynamic factors contributing to the fault 10±2 mm/a on the central section, and is 11±2 mm/a on the slip rates. eastern section. Left-lateral slip rates on the Anninghe fault zone and on the Daliangshan fault zone are both 4±2 mm/a. 2 Three-dimensional viscoelastoplastic model Therefore, total slip rates on the central section of the Xianshuihe-Xiaojiang fault zone is about 8 mm/a. Left- in the western Sichuan and its adjacent region lateral slip rate on the Zemuhe fault zone is 7±2 mm/a, and 2.1 Introduction of viscoelastoplastic model the same value is true on the Xiaojiang fault zone [6]. The Longmenshan fault zone consists of the Wen- We employ a three-dimensional viscoelastoplastic model to chuan-Maowen thrust, the Yingxiu-Beichuan thrust, the simulate long-term motion and deformation of elastoplastic Penxian-Guanxian fault, and many buried piedmont thrust crust underlying viscous lower crust and upper mantle. De- faults [4]. Geological crust shortening on the Longmenshan tails of the viscoelastoplastic model are given in the appen- fault zone is about 4–6 mm/a [28]. Contemporary shorten- dix. ing derived from GPS survey is less than 3 mm/a. Right- Plastic deformation occurs when stress reaches the plas- lateral slip rate on the fault zone is indetectable [6, 29]. tic yield criterion (yield envelope), described here by the Slip rates on the Xianshuihe-Xiaojiang fault zone and the Drucker-Prager yield function [32]: Longmenshan fault zone given by previous studies are listed FI= α +− Jk, in Table 1. The data in Table 1 show difference between the 12 geological and the geodetic results. The geological slip rates where I1 is first invariant of the stress tensor and J2 is sec- include fault creeping and coseismic dislocation. They not ond invariant of the deviatoric stress tensor. The parameters only indicate the long-term motion of the fault, but also α and k are related to cohesion and inner effective frictional include information about fault evolution [9]. On the other coefficient, respectively. hand, the GPS results mainly present interseismic crustal A paralleled software package of three-dimensional vis- motion [8]. The difference between the two kinds of data is coelastoplastic finite element (FE) model was developed by reasonable [30, 31]. Li et al. [32]. It has been used to model long-term motion The geological slip rates reflect long-term fault activity and evolution of the San Andreas fault system, which is the and the geodetic slip rates show short-term fault activity. boundary between the Pacific Plate and the North America Dispite the different value, they have the same slip pattern. Plate [11, 13]. We use the software package in this study. Slip rate on the Xianshuihe-Xiaojiang fault system is higher The fault zone in our 3D FE model is assumed to be a vis- than that on the Longmenshan fault system. Seismicity on coelastoplastic layer with finite thickness. Calculated results the Xianshuihe-Xiaojiang fault has a higher frequency than include fault slip rate and effective strain rate outside the that on the Longmenshan fault. Slip rates on each fault fault. We take one point on fault and one point outside the segment are different. Structures of the Xianshuihe fault and fault to see the evolution process of slip rate and effective the Xiaojiang fault are both simple. It leads to high slip rates strain rate, respectively (Figure 3). Positions of these two on these two faults. The central section of the Xianshuihe- points are shown in Figure 4(b). Xiaojiang fault zone consists of the Anninghe-Zemuhe fault In a viscoelastoplastic model with constant loading, the and the Daliangshan fault. The complex fault structure leads effective strain rate is related to deformation time. If defor-

Table 1 Kinematics of the Xianshuihe-Xiaojiang fault zone and the Longmenshan fault zone

Active fault zone Striking Motion mode Geological slip rate (mm/a) GPS slip rate (mm/a) Xianshuihe NW40° Left-lateral strike slip Western Xianshuihe NW40° Left-lateral strike- slip 15±5 10±2 Eastern Xianshuihe NW20° Left-lateral strike slip with thrust 9.6±1.7 11±2 Anninghe NS Left-lateral strike slip 6.5±1 ~4 Zemuhe NW25° Left-lateral strike slip with normal 4.9±0.6 7±2 Daliangshan NS Left-lateral strike slip ~3 ~4 Xiaojiang NS Left-lateral strike slip 10±2 7±2 Shortening rate <3 with little right-lateral Longmenshan NE45° Reverse with right-lateral strike slip Shortening rate, 4–6 strike slip WANG Hui, et al. Sci China Earth Sci January (2010) Vol.53 No.1 5

Figure 3 Sketch map for evolution of the predicted fault slip rate and effective strain rate outside the fault. (a) Evolution of slip rate on point A shown in Figure 4(b); (b) grey line shows evolution of effective strain rate on point A shown in Figure 4(b). Dark line shows evolution of effective strain rate on point B shown in Figure 4(b).

Figure 4 Numerical mesh for the finite element model. The entire Xianshuihe-Xiaojiang fault and the Longmenshan fault (white line) are explicitly in- cluded in the model. (a) Span of the finite element model and boundary constraints; (b) three-dimensional numerical mesh for the finite element model. mation time is much shorter than relaxation time, the effec- Plateau. We construct a three-dimensional finite element tive strain rate is mainly elastic strain rate when stress does model for it. The crust thickness shows large lateral varia- not reach the plastic yield criterion, and is mainly plastic tion in the study area. It reaches 60–70 km in the eastern strain rate when stress reaches the plastic yield criterion. If Tibet [33, 34]. The focal depths indicate a shallow elastic deformation time is long enough, the effective strain rate is upper crust in the western Sichuan region. According to the mainly viscous strain rate when stress does not reach the results of earthquake relocation, the focal depths in the plastic yield criterion, and is mainly plastic strain rate when western Sichuan plateau are centralized mainly in 0–15 km stress reaches the plastic yield criterion. If deformation time [18, 35]. To study the long-term motion and deformation of is much longer than relaxation time, then stress in rock will the upper crust in the western Sichuan area, we assume the reach litho-static pressure without shear stress. The effective mean thickness of regional upper crust to be 15 km and ig- strain rate is mainly viscous strain rate (no shear deforma- nore the influence of up and down of the Moho discontinu- tion) at that time. The crustal effective strain rate mentioned ity. Although lateral and vertical variations of the crust have in the paper consists of plastic and viscous strain rate. effects on the deformation in the western Sichuan area [36], they are also ignored to simplify the model. Such processing 2.2 Three-dimensional finite element model of the highlights the influence of fault geometry on regional strain western Sichuan area partitioning. The viscosities for the finite element model are 1×1025 Pa·s for the upper crust and 1×1020 Pa·s for the lower The study area spans from 98°E to 106°E and from 24°N to crust and upper mantle. Mean Young’s modulus is estab- 32°N. It is located in the southeast borderland of the Tibetan lished using three-dimensional velocity structure in the 6 WANG Hui, et al. Sci China Earth Sci January (2010) Vol.53 No.1 western Sichuan area [36]. ternary tectonics [7, 37–39]. It is reasonable to use the GPS We focus on the Xianshuihe-Xiaojiang fault zone and the results as the boundary constraints for our models. Longmenshan fault zone. These two fault zones are simpli- In order to study how the strain distributes along the fied as connective fault zones though they are composed of Xianshuihe-Xiaojiang fault zone, we constrain boundary many disconnected faults. They are assumed to cut through conditions using the GPS results respect to the stable South the whole model according to their tectonic background. All China Block (Figure 4(a)). The eastern boundary of our faults are modeled as viscoelastoplatic layers with 400 m in model is located in the interior of South China. It is as- width in the three-dimensional finite element model. The sumed to be fixed since the South China Block is stable and Longmenshan fault is defined as a thrust fault zone trending deforms little. The western boundary is located in the east- northwest. Its dip angle is about 30°. The Xianshuihe- ern Tibet and the southeast Yunnan block. Its boundary Xiaojiang fault zone is simplified as a vertical fault. Previ- constraints are the interpolation of the observed GPS veloc- ous studies show that the effective friction coefficient on the ity nearby. The southern and northern boundaries are 24°N Xianshuihe-Xiaojiang fault zone is less than 0.1–0.08 [10], and 32°N respectively. The southern and northern bounda- we assume 0 as the fault effective friction coefficient. Rock ries not only cut across the channel for the lateral extrusion cohesions are based on previous results [11, 13, 32]. Mate- of the Tibetan Plateau [40, 41], but also cut across the rial parameters for the three-dimensional model are listed in Xianshuihe-Xiaojiang fault zone and the Longmenshan fault Table 2. Figure 4 shows the scale and the mesh of the FE zone respectively. To reduce the influence of boundary con- model. The total area is about 6.99×105 km2. The three- straints on the results, we assume that the northern and dimensional finite element model is divided into 2591 ele- southern boundaries are free. The surface of our model is ments in each layer. Mean size for the elements is about also set free. The bottom (at 60 km depth) is fixed in verti- 16.4 km. cal direction, and free in horizontal direction. We construct four models with the same boundary con- The three-dimensional boundary constraints are assumed straints to investigate the influence of the Xianshuihe- to be invariant with depth, for the study region is small Xiaojiang fault zone and the Longmenshan fault zone on the enough to be simplified as a planar thin shell model. The regional strain partitioning. The simple descriptions for the GPS survey results in the study area represent crustal mo- four models are shown in Table 3. The details of different tion and deformation well. Although previous studies dis- models are introduced in section 3. cussed mechanical coupling between the crust and the upper mantle using the observations of the directions of principal stress [42, 43], the constraints on deep motion are still 2.3 Boundary constraints scarce. Since our model is a simplified physics model to Crustal motion within the western Sichuan area is controlled discuss the influence of fault geometry on long-term strain by the eastward extrusion of the Tibetan Plateau, material partitioning, using the GPS results as the approximate heterogeneity, and deep process etc. The first factor might boundary constraints for our model is reasonable. be the most important [1, 37]. Based on GPS results, the crust of the eastern Tibet moves eastward respect to the sta- 3 Impact of fault geometry on the long-term ble South China, while the crust of the southwest Yunnan moves southwestward [6]. Previous studies show that the strain partitioning present-day crustal motion pattern derived from GPS survey is consistent with that deduced by studying on the late Qua- We use a paralleled finite element package to simulate the

Table 2 Material parameters for the three-dimensional viscoelastoplastic model

Young’s modulus Viscosity Effective friction coef- Cohesion Poisson’s rate (MPa) (Pa·s) ficient (MPa) Upper crust 7.85×104 0.25 1×1025 0.4 50 Lower crust and upper mantle 1.03×105 0.25 1×1020 0.4 50 Fault in upper crust 7.85×104 0.25 1×1025 0 10 Fault in lower crust and upper mantle 1.03×105 0.25 1×1020 0 10

Table 3 Simple descriptions for the four finite element models in the western Sichuan and its adjacent region

Model No. Fault zones included in the FE model and their cohesions 1 The Xianshuihe-Anninghe-Xiaojiang fault zone (10 MPa) 2 The Xianshuihe-Anninghe-Xiaojiang fault zone (10 MPa), the Daliangshan fault zone (10 MPa, 20 MPa) 3 The Xianshuihe-Daliangshan-Xiaojiang fault zone (10 MPa) 4 The Xianshuihe-Anninghe-Xiaojiang fault zone (10 MPa), the Daliangshan fault zone (20 MPa), the Longmenshan fault zone (10 MPa)

WANG Hui, et al. Sci China Earth Sci January (2010) Vol.53 No.1 7 lithospheric deformation including faults. We adopt 5 years that of the South China Block. The entire Bayan Har Block as one time step, and calculate 50000 steps. Boundary dis- moves northeastward. The crustal velocity of the South placement constraints increase linearly with time step. If China Block is nearly zero. On the west side of the calculating were longer enough, both fault slip rates and Xianshuihe-Xiaojiang fault zone, the Sichuan-Yunnan effective strain rate outside the faults would reach the Block rotates clockwise around the EHS. The crust motion steady state (Figure 3). The steady state represents the patterns of the northwestern Sichuan sub-block, the middle long-term crustal motion. The constant slip rates represent Yunnan sub-block, and the southwest Yunnan sub-block are long-term fault slip rate. different from each other. The northwestern Sichuan sub-block moves southeastward. The middle Yunnan 3.1 Impact of geometry of the Xianshuihe-Anninghe- sub-block moves nearly southward. The southwest Yunnan Xiaojiang fault sub-block moves southwestward. The predicted slip rates show that the left-lateral In the first model, only the Xianshuihe-Anninghe- strike-slip dominates the entire Xianshuihe-Xiaojiang fault Zemuhe-Xiaojiang fault zone is assumed active. We com- zone (Figure 5(b)). However, slip rates on each fault seg- pare the predicted crustal velocity and the GPS velocity ments are different. Slip rates on the northwestern segment field in the western Sichuan and its adjacent region (Figure of the Xianshuihe fault are the highest. The mean value is 5(a)). Although two velocity fields differ obviously, they 12.4 mm/a with extension component. Mean slip rate on the show the same pattern and clockwise rotation around the southeastern segments of the Xianshuihe fault is 8.4 mm/a. EHS. The differences are small at the points in the vicinity Slip rate on the Anninghe-Zemuhe fault is the smallest, 4.4 of the faults. mm/a on the Anninghe fault and 4.7 mm/a on the Zemuhe The crustal motion patterns in the study area are different fault, respectively. Slip rate on the Xiaojiang fault varies at two sides of major faults. The Xianshuihe-Anninghe- from 8.0 to 8.7 mm/a. Predicted fault slip rates are consis- Zemuhe-Xiaojiang fault zone consumes most of the differ- tent with geological observations. Slip rates are high on the ent motion between the Sichuan-Yunnan Block and the northern and the southern sections of the Xianshuihe- South China Block. The displacement across the fault sys- Xiaojiang fault zone since they are straight. The Xian- tem shows gradient variation. On the east side of the shuihe-Xiaojiang fault zone changes its orientation sharply Xianshuihe-Xiaojiang fault zone are two blocks: the Bayan on its central section. It leads to low slip rates on this sec- Har block and the South China Block. The Sichuan-Yunnan tion. Block and the southwest Yunnan Block are located at its Effective strain rates outside the faults are localized in west side. The blocks move faster on the west side than on the regions where the fault striking changes sharply, such as the east side of the Xianshuihe-Xiaojiang fault zone. The the region near Moxi, the region to the east of the An- crustal motion pattern of the Bayan Har Block differs from ninghe-Zemuhe fault, and the region near Dongchuan etc.

Figure 5 Results produced by the first model. (a) Predicted velocity field (red arrows show observed GPS velocity, black arrows show predicted velocity); (b) predicted fault slip rates and effective strain rate outside the fault. 8 WANG Hui, et al. Sci China Earth Sci January (2010) Vol.53 No.1

(Figure 5(b)). As a transfer segment connecting the The first model reveals that the geometry of the Xianshuihe fault zone and the Anninghe fault zone, the Xianshuihe-Anninghe-Zemuhe-Xiaojiang fault zone influ- Moxi fault segment rotates clockwise about 15° than the ences the strain partitioning in the western Sichuan area. Luhuo fault segment. Strike altering of fault restrains the The region with high effective strain rate is located on the activity on the Moxi fault segment. Slip rate is lower on the east side of the Anninghe-Zemuhe fault. It extends to the Moxi fault segment than on the Luhuo fault segment. Strike neighborhood of the Daliangshan fault zone. High effective of the Anninghe-Zemuhe fault changes from north-south strain rate in the region is in favor of initiation of the Dali- direction to northwest direction. Since the Anninghe fault angshan fault zone. To investigate the impact of the Dali- strikes in north-south and prevents the eastward motion, angshan fault zone on regional strain partitioning, we mod- crustal motion changes from southeastward in the west Si- ify the first model and construct the second model. In the chuan plateau to southward in the central Yunnan region. second model, the Daliangshan fault zone is assumed active. The Anninghe-Zemuhe fault zone has the smallest slip rates Assuming the cohesion of the Daliangshan fault zone to on the whole fault system. Effective strain rates around the be the same as that of the Anninghe-Zemuhe fault zone, we Anninghe-Zemuhe fault zone are located at the Xiaoxian- calculate slip rates on each of the fault segments and effec- gling fragment, which is located at the east side of the An- tive strain rate outside the fault (Figure 6(a)). The initiation ninghe-Zemuhe fault. It leads to a large area with high of the Daliangshan fault makes the whole Xianshuihe- strain rate in the region. Toward south, fault strike changes Xiaojiang fault smooth. It makes the striking of the fault from northwest to near north-south near Dongchuan, where system more consistent with the crustal motion pattern. the Zemuhe fault conjoins the Xiaojiang fault. It results in Since the Daliangshan fault initiates, the whole Xianshuihe- high effective strain rate near Dongchuan. High effective Xiaojiang fault system slips more easily. In the second strain around the western boundary of the model may be model, slip rates on the northwestern segment of the related to the boundary effect. Xianshuihe fault increase to 12.9 mm/a. Slip rates on the High effective strain rate outside the fault zone correlates Xiaojiang fault zone increase to 8.4–10.0 mm/a. Slip rates to large crustal deformation around the Xianshuihe- on the central section of the Xianshuihe-Xiaojiang fault Xiaojiang fault zone. The highest mountain in the western zone are distributed on the Annhehe-Zemuhe and the Dali- Sichuan region, the Gongga Mountain, is located on the angshan faults. Because the geometry of the Daliangshan west side of the Moxi fault. The Gongga region was uplifted fault is relatively simple and is consistent with trending of in Late Cenozoic. Vertical movement in the region absorbs regional crustal motion, slip rates on the Daliangshan fault part of strain off faults. The active Daliangshan and the are higher than that on the Anninghe-Zemuhe fault. Slip Mabian fault zones are located on the east side of the An- rates on the Daliangshan fault are about 6.9 mm/a, and only ninghe-Zemuhe fault zone. The three fault zones are nearly 2.4–2.8 mm/a on the Anninghe-Zemuhe fault. The complex parallel. Field investigations reveal that the newly generated fault geometry may contribute to lower slip rate on the An- Daliangshan fault zone consists of several discontinuous ninghe-Zemuhe fault. The sum of slip rates on the An- fault zones. Slip rate on the fault zone is low [25], so is the ninghe-Zemuhe fault zone and the Daliangshan fault zones seismicity [44]. Initiation of the Daliangshan fault and the is nearly 10 mm/a. Mabian fault is later than that of the Anninghe-Zemuhe High effective strain rates outside the Xianshuihe- fault zone [26]. Distribution of predicted effective strain Xiaojiang fault system are concentrated in the region near rates indicates that the initiation of the Daliangshan faul the Moxi fault, the Xiaoxiangling fragment, and the region zone and the Mabian fault zone might be related to the near Dongchuan, respectively. High effective strain rate that strain localization on the east side of the Anninghe-Zemuhe centralized near the Moxi fault and near Dongchuan is al- fault zone. most the same as that derived from the first model. How- ever, the initiation of the Daliangshan fault zone changes 3.2 Impact of the Daliangshan fault zone the strain partitioning around the central section of the Xianshuihe-Xiaojiang fault zone. It makes strain rate local- The Xianshuihe fault zone is separated into two branches ized on the Anninghe-Zemuhe fault zone and the Daliang- near Shimian town. The west branch is the Anninghe- shan fault zone. It also makes strain rate decline in the Zemuhe fault, and the east branch the newly generated Xiaoxiangling fragment. Daliangshan fault zone. The Daliangshan fault zone consists Results produced by the second model indicate that the of four fault segments, and its striking is about 330°–360° Daliangshan fault zone plays a significant role in strain par- [5]. The Anninghe-Zemuhe fault zone and the Daliangshan titioning in the western Sichuan area. However, there is a fault zone conjoin with the Xianshuihe fault near Shimian big discrepancy between the predicted and observed slip and with the Xiaojiang fault zone near Qiaojia in Yunnan. rates on the Daliangshan and the Anninghe-Zemuhhe fault The Daliangshan fault is a newly generated fault zone. Its zones. As the Daliangshan fault zone is a newly generated maturity is lower than the Anninghe-Zemuhe fault zone fault zone, the cohesion on the Anninghe-Zemuhe fault [26]. zones and the Daliangshanfault zone should be different. WANG Hui, et al. Sci China Earth Sci January (2010) Vol.53 No.1 9

Figure 6 Predicted slip rates and effective strain rate outside the fault produced by the second model. (a) Predicted results when cohesion of the Daliang- shan fault zone is the same as that of the Anninghe-Zemuhe fault zone; (b) predicted results when cohesion of the Daliangshan fault zone is twice of that of the Anninghe-Zemuhe fault zone.

Therefore, we adjust the cohesion on the Daliangshan fault zone replaces the Anninghe-Zemuhe fault zone. It indicates to fit the observed slip rates. The effective strain rate out- that the whole Xianshuihe-Xiaojiang fault zone consists of side the fault changes with the adjustment of cohesion on the Xianshuihe fault zone, the Daliangshan fault zone, and the Daliangshan fault. If the cohesion on the Daliangshan the Xiaojiang fault zone. The whole fault system is much fault zone doubles and reaches 20 MPa, the mean slip rate smoother than it is in the first model. Striking altering of the on the Daliangshan fault is about 3.3 mm/a. At the same fault system is more consistent with the crustal motion pat- time, slip rate on the parallel Aninghe-Zemuhe fault is tern within the study area. 3.8–3.7 mm/a. Slip rates on the Xianshuihe fault and the Results of the third model further show that the signifi- Xiaojiang fault are still higher than those predicted from the cant influence of the Daliangshan fault on strain partitioning first model (Figure 6(b)). in the western Sichuan area. The geometry of the As a newly-generated fault zone, initiation of the Dali- Xianshuihe-Xiaojiang fault influences the regional strain angshan fault zone not only makes the whole Xianshuihe- partitioning. The motion between the South China Block Xiaojiang fault system slip easier, but also adjusts strain and the Sichuan-Yunnan Block is mainly absorbed by the partitioning around the central section of the whole fault Xianshuihe-Daliangshan-Xiaojiang fault zone (Figure 7). system. The interaction between the parallel Daliangshan Long-term slip rates predicted by the third model are much fault zone and the Anninghe-Zemuhe fault zone also affects smoother than those produced by the first and the second strain partitioning around the central section of the whole models. Slip rates on the fault system are attenuated toward Xianshuihe-Xiaojiang fault system. With its development, the south in the third model. Predicted slip rates on the Da- the Daliangshan fault might replace the role performed by liangshan section in the third model are higher than those on the Anninghe-Zemuhe fault zone in the whole Xianshuihe- the Anninghe-Zemuhe segment from the first model. They Xiaojiang fault [26]. In the future, the Xianshuihe-Xiaojiang are also higher than the sums of slip rates on the Daliang- fault zone may finally develop to a smooth arc section shan fault and the Anninghe-Zemuhe fault in the second around the EHS. The strain partitioning processes produced model. At the same time, there is little effective strain rate by the evolution of parallel fault zones are also observed in located on the east of the Daliangshan fault zone. other area [9, 13]. Previous studies mainly focused on the Anninghe- To further investigate the effect of the Daliangshan fault Zemuhe fault zone and ignored the activity of the parallel zone on strain partitioning in the study region, we construct Daliangshan fault zone that also absorbs much crust short- the third model. In the third model, the Daliangshan fault ening. Previous geological studies illustrate the imbalance 10 WANG Hui, et al. Sci China Earth Sci January (2010) Vol.53 No.1

titioning. Our models demonstrate the impact of the Dali- angshan fault zone on the strain partitioning in the western Sichuan region. The results are also useful to explain the occurrence of historical devastating earthquakes in the re- gion extending from the east of the Anninghe fault zone to the Mabian fault zone.

4 Influence of the Longmenshan fault zone

The Longmenshan fault strikes in northeast direction. It is also an important block boundary in the western Sichuan area. It connects the Xianshuihe-Xiaojiang fault zone in the study region. The two fault zones divided the western Si- chuan region into three blocks. To discuss the impact of the Longmenshan fault, we construct the fourth model on the basis of our second model. We get the regional crustal motion pattern and the dis- tribution of crustal strain in the fourth model (Figure 8). The activity of the Longmenshan fault makes the crust move like block motion in the western Sichuan region. The Bayan Figure 7 Predicted slip rates and effective strain rate outside the fault produced by the third model. Har block moves northeastward. The Longmenshan fault zone absorbs the movement between the Bayan Har block and the South China Block. It is a dextral strike-slip fault between the total displacements and slip rates on each fault with thrust component. segment of the Xianshuihe-Anninghe-Xiaojiang fault [27, Results obtained in the fourth model show that the im- 45]. It is found that there are large displacement deficits on pact of the active Longmenshan fault on the regional strain the Anninghe-Zemuhe fault zone. Most of previous studies partitioning is less than that of the Xianshuihe-Xiaojiang presumed qualitatively that the southeastward slip rates on fault. The slip rate on the Longmenshan fault is about 1.3 the Xianshuihe fault may be distributed on the tectonics at mm/a, and the thrust component is less than 1 mm/a (Figure east of the Anninghe fault zone [22, 46]. They ignored the 8(b)). The active Longmenshan fault reduces the effective impact of the Daliangshan fault zone on regional strain par- strain rate around the Xianshuihe-Xiaojiang fault zone. In

Figure 8 Results produced by the fourth model. (a) Predicted velocity field (red arrows show observed GPS velocity, black arrows show predicted veloc- ity); (b) predicted fault slip rates and effective strain rate outside the fault. WANG Hui, et al. Sci China Earth Sci January (2010) Vol.53 No.1 11 the fourth model, slip rate on each fault segment of the Xianshuihe-Xiaojiang fault zone decreases about 0.5 mm/a than that produced by the first and the second models. However, the distribution of effective strain shows that the Xianshuihe-Xiaojiang fault is more active than the Long- menshan fault. The insignificant slip on the Longmenshan fault is consistent with the results from historical seismicity and geodetic survey [6, 29, 47]. Difference between the activity of the Longmenshan fault and the Xianshuihe-Xiaojiang fault is caused by the geometry differences of these two fault systems. During the uplift of the Tibetan Plateau, huge gravity potential energy makes extension in the east-west direction in the plateau [48]. The eastward extrusion in the Bayan Har region is blocked by the steady Sichuan basin. Crust shortening oc- curred in the Longmenshan region. The Longmenshan thrust fault zone was formed during the shortening process. Its strike is orthogonal to the direction of regional crustal motion. At the same time, no old craton like the Sichuan basin exists in the southwest Yunnan region to block the crust motion. The lateral extrusion of the Tibetan Plateau Figure 9 Predicted crustal uplift produced by the fourth model. causes the initiation of the sinistral Xianshuihe-Xiaojiang fault system. The different geodynamic environments and different fault strikes lead to the different activities on the significant for understanding regional strain partitioning Xianshuihe-Xiaojiang fault system and the Longmenshan process. fault system. Slip rates on the Longmenshan fault zone are much lower 5.1 The effect of material parameters than that on the Xianshuihe-Xiaojiang fault system. Thus, the recurrence interval of major earthquakes on the Long- Many material parameters are used in our models. The im- menshan fault zone is longer than that on the Xianshuihe- portant parameters are viscosity, effective friction coeffi- Xiaojiang fault system [49]. Historical earthquake records cient, and cohesion etc. Li et al. [32] discussed the influence and paleoseismicty reveal millennium quiescent seismicity of different material parameters on predicted results in the on the Longmenshan fault [50]. The accumulated strain in three-dimensional viscoelastoplastic model in detail. Based past millenniums should be enough to sustain the Wenchuan on Li et al’s results [32], we briefly introduce the influence of viscosity and effective friction coefficient on the model- Ms8.0 earthquake on May 12th, 2008 on the Longmenshan fault zone. ing results. 25 The vertical movement shows difference in southern and We take 10 Pa·s as the viscosity of upper crust in our northern parts of the western Sichuan area (Figure 9). Crust three-dimensional model. Deformation time of material in uplift dominates the deformation in northwestern Sichuan upper crust is shorter than its relaxation time. The effective region. The crust uplift rate around the Gongga Mountain is strain rate in upper crust is mainly elastic strain rate and larger than 1.5 mm/a. The mean elevation is about plastic strain rate. The viscosity of the lower crust and upper 4500–5000 m in the northwestern Sichuan region, whereas mantle beneath the study area is about 1020 Pa·s [20]. Pre- it is only 2500m in the central Yunnan [41]. Geological vious study [11] shows that the less the viscosity difference uplift rate of the Gongga Mountain is about 3.2 mm/a [5]. is, the stronger the coupling between the upper and lower The predicted vertical movement pattern is consistent with layer is. the pattern obtained from the geological results. The effective friction coefficient of fault also affects the distribution of the effective strain rate outside the faults. A low friction coefficient was assumed on fault when simu- 5 Discussions lating the long-term fault slip in many papers. In that case, the long-term fault activity is simplified as steady creeping. Three-dimensional finite element model in the western Si- For example, He et al. [10] thought that the effective fric- chuan region in the paper is in fact a simplified mechanical tion coefficient is lower than 0.1–0.08 on the Xianshuihe- model. The model describes the first order characteristics of Xiaojiang fault zone. Zhang et al. [51] adopted 0.0001 as the real geological unit. Although limited by the computa- the effective friction coefficient on the Bangongcuo-Nuji- tional complexity and computing capability, the model is ang-Red River fault and the Zemuhe fault zone when simu- 12 WANG Hui, et al. Sci China Earth Sci January (2010) Vol.53 No.1 lating the recent crust movement and the fault activities in west of the Jinpingshan-Yulongxueshan thrust zone. The the Tibetan Plateau. The effective friction coefficient in our Jinpingshan-Yulongxueshan thrust zone is the largest to- model is assumed as zero. The assumed low value is not the pography transform zone in the southwestern China true rock inner friction coefficient. It is used to simulate the mainland, and is the sharp gradient zone of crustal depth long-term slip rate on major fault zone, which may be between the central Yunnan region and the Western Sichuan equivalent to the mean dislocation of historical earthquakes Plateau [27]. The eastward extrusion of the Tibetan Plateau in a given time span. Numerical experiments show that a extends to the east of the Jinpingshan-Yulongxueshan thrust small shift (0–0.1) of the effective friction is insignificant zone in Quaternary. The Lijiang-Xiaojinhe fault zone ab- for modeling results, especially for predicted fault slip rates sorbs part of southeastward motion of the northwestern Si- and distribution of effective strain rate outside the fault [11]. chuan region. Geological left-lateral slip rate with thrust The constitutive model in the paper is nonlinear. The component is 3.8±0.7 mm/a [5]. numerical solution therefore depends strongly on the initial Although the Lijiang-Xiaojinhe fault is regarded as a stress field. The stress field produced by gravity is part of sub-block boundary zone in the Sichuan-Yunnan Block the initial stress field. Gravity does not affect neotectonic based on the studies of surface active tectonics [53]. The displacement field, such as fault slip rate, but it affects the earthquake relocation results show the connection of the plastic yield of material. It is necessary to study the impact Litang fault zone and western segment of the Lijiang- of gravity when we study the distribution of the plastic Xiaojinhe fault [18]. Therefore, the northern segments of strain rate. Different initial stress would produce different the Lijiang-Xiaojinhe fault system (north of Muli) may play distribution of plastic strain. Gravity is ignored since we an insignificant role in strain partitioning around the eastern mainly focus on fault slip rate. Stress produced by the re- boundaries of the Sichuan-Yunnan Block. gional elevation difference and undulation of the Moho The Jinshajiang fault zone and the Lijiang-Xiaojinhe discontinuity may also be insignificant. fault zone are ignored in our model for their tectonic com- Numerical experiments show that although material pa- plexity. Numerical experiments show that slip rates on these rameter variations would produce different fault slip rate two fault zones are less than 0.5 mm/a. The influences of these and effective strain rate outside the fault, the spatial distri- two fault zones on strain partitioning on the Xianshuihe- bution of strain rate changes little in the case of small de- Xiaojiang fault zone are also insignificant. formation. On a fault zone, slip rates are high on the seg- The impact of the Red River fault is also ignored because ment with simple structure, and vice versa. Regional strain we focus on strain partitioning between the Sichuan- is localized around the fault segment with complex orienta- Yunnan block and the South China Block. The influences of tion and structure. Such deformation distribution makes the Jinshajiang fault zone, the Lijiang-Xiaojinhe fault, and regional crustal move more harmoniously. the Red River fault need further study.

5.2 The effect of other major faults in the western Si- 6 Conclusions chuan area

There are still other fault zones affecting regional strain The western Sichuan and its vicinity is an important region partitioning in the western Sichuan area, including the Jin- to study the evolution and geodynamics of the Tibetan Pla- shajiang fault zone, Lijiang-Xiaojinhe fault zone, and the teau. The crustal motion presents altering from lateral shear Red River fault zone etc. on faults striking northwest-west to convergence with shear The structures of the Jinshajiang fault zone and the Liji- components on faults striking near north-south. The same ang-Xiaojinhe fault are complex. Activities on these two crustal motion pattern is also found in other areas in the fault zones may be controlled by the motion of the deep eastern borderland of the Tibetan Plateau, such as in the strucutres. The Jinshajiang fault zone is a thrust fault system Haiyuan-Liupanshan fault zone and in the east Kunlun- with dozen kilometers width. It consists of several arc-shape Minshan fault zone etc. We provide a case study for the thrust faults protruding eastward [52]. The fault system not special deformation pattern in the eastern borderland of the only changes the eastward extrusion of the Tibetan Plateau Tibetan Plateau. The research is helpful to understand the into crustal shortening and uplift in the western Sichuan geodynamics of the eastern Asia, especially the evolution of region, but also transfers the remaining horizontal motion to the Tibetan Plateau. Major conclusions of this study are: the northwest Sichuan block. The geological right-lateral (1) Fault geometry obviously influences regional strain slip rate with reverse component on the Jinshajiang fault partitioning. Slip rates on each segment of the Xianshuihe- zone is about 5 mm/a [5]. The Lijiang-Xiaojinhe fault is a Xiaojiang fault are influenced by the fault geometry and northeast striking fault that was formed on the Jinping- fault structure. Slip rates are high on fault segment with shan-Yulongxueshan thrust zone. The eastward extrusion of simple geometry and structure, and vice versa. Regional the Tibetan Plateau has been confined mainly in the west strain is concentrated around the fault segment with com- Sichuan area before Quaternary, which is located in the plex geometry and structure. WANG Hui, et al. Sci China Earth Sci January (2010) Vol.53 No.1 13

(2) Strain partitioning around the central section of the strain component is related with stress. Elastic strain com- Xianshuihe-Xiaojiang fault zone is controlled by interaction ponent is related with stress increment: between the parallel Anninghe-Zemuhe fault zone and the {dQεσvt }= []{−1 }, dt Daliangshan fault zone. The initiation of the Daliangshan (A3) fault makes the whole Xianshuihe-Xiaojiang fault zone slip {}[]{},dDdεσe = −1 much easier. t (3) Striking of the Longmenshan fault zone is nearly or- where, {σ } stress tensor at t moment, dt is time increment, thogonal to the direction of lateral extrusion in the Tibetan {dσ} is increment of stress tensor, [Q] is matrix for viscous Plateau. It leads to low slip rates on the thrust fault. The material parameter, [D] is matrix for elastic material pa- activity of the Longmenshan fault decreases the strain rate rameter. on and around the Xianshuihe-Xiaojiang fault zone. ⎛⎞111 (4) The impact of fault geometry on strain partitioning is ⎜⎟−−000 significant in the western Sichuan and its adjacent region. ⎜⎟366 ⎜⎟11 1 Eastward motion about 13 mm/a in the eastern Tibet is ab- −−000 sorbed mainly by the deformation in the Xianshuihe- ⎜⎟63 6 −1 1 ⎜⎟ Xiaojiang fault system and the Longmenshan fault zone. []Q = ⎜⎟111 , (A4) q −− 000 Slip rates on the Longmenshan fault zone are about 1 mm/a. ⎜⎟663 The remnant motion is absorbed by slip on the Xianshuihe- ⎜⎟ ⎜⎟0 0 0 100 Xiaojiang fault zone and strain outside the fault. The whole ⎜⎟0 0 0 010 Xianshuihe-Xiaojiang fault can be divided into three sec- ⎜⎟ tions based on their different motion pattern. The north sec- ⎝⎠0 0 0 001 tion consists of the Xianshuihe fault zone. The left-lateral E slip rate is about 12 mm/a. The uplifting of the Gonggashan []D = region near the Xianshuihe fault zone absorbs part of crustal (1+−νν )(1 2 ) strain. The central section of the Xianshuihe-Xiaojiang fault ⎛⎞1−νν ν 000 ⎜⎟ system consists of the parallel Anninghe-Zemuhe fault and ννν1− 000 the Daliangshan fault. Slip rates on these two fault zones ⎜⎟ ⎜⎟νν1− ν 000 change with the change of fault cohesions. The southern ⋅ ⎜⎟ , section of the fault system is the Xiaojiang fault. Slip rate ⎜⎟0000.50−ν 0 on the fault is about 8.0 mm/a. ⎜⎟000 00.50−ν ⎜⎟ ⎝⎠000 0 00.5−ν (A5) Appendix: control equation for three-dimen- sional viscoelastoplastic model where q is viscosity, E is Young’s module and v is Pois- son’s rate. {σt} is stress at t moment: Three-dimensional viscoelastoplastic model used in the pa- ttdt− per is developed based on classical elastoplastic model. {}{σσσ=+d }{ }. (A6) Viscosity term is added to classical elastoplastic model. According to eqs. (A2), (A3), and (A6), we get Physical equations that control the model include static p equilibrium equation: {}[]({}{dDddσ =−+% εε }){} d σ% , (A7) ∂σ ij +=f 0 , (A1) where ∂x i j []DD% =+ ([]−−−111 [] Qdt ), (A8) −−1 tdt where σij is stress tensor, fi is body force. {}dDQdtσσ% =− [][]% { }. Stress increment consists of viscous component, elastic component, and plastic component: Plastic deformation occurs when stress reaches the plas- tic yield criterion (yield envelope), described here by the {}{ddε =++εεεvep }{ d }{ d }, (A2) Drucker-Prager yield function

v e p where ε , ε and ε are viscous stress increment, elastic FI= α 12+− Jk, (A9) stress increment, and plastic stress increment, respectively. {} represents tensor form of stress increment. When stress where I1 is first invariant of the stress tensor and J2 is sec- is beyond the yield, relationship between stress and strain ond invariant of the deviatoric stress tensor. The parameters follows the rule that controls the Maxwell material. Viscous α and k are related to cohesion and effective frictional coef- 14 WANG Hui, et al. Sci China Earth Sci January (2010) Vol.53 No.1 ficient, respectively. Plastic strain direction (the flow rule) 5 Xu X W, Wen X Z, Zheng R Z, et al. Pattern of latest tectonic motion is specified with a plastic potential function G : and its dynamics for active blocks in Sichuan-Yunnan region, China. Sci China Ser D-Earth Sci, 2003, 46(Suppl): 210–226 6 Shen Z, Lv J, Wang M, et al. Contemporary crustal deformation GJ= 2 . (A10) around the southeast borderland of the Tibetan Plateau. J Geophys Res, 2005, 110: B11409, doi: 10.1029/2004JB003421 Plastic strain increments are given by 7 Zhang P Z, Wang M, Gan W J, et al. Slip rates along major active faults from GPS measurements and constraints on contemporary con- p ⎧⎫∂G tinental tectonics. Earth Sci Front, 2003, 10(Suppl): 81–92 {}ddε = ⎨⎬λ , (A11) ⎩⎭∂σ 8 Meade B J, Hager B H. Block models of crustal motion in southern California constrained by GPS measurements. J Geophys Res, 2005, where dλ is a ‘plastic multiplier’. With the condition that 110: B03403, doi: 10.1029/2004JB003209 9 Bennett R A, Friedrich A M, Furlong K P. Codependent histories of the stress stays on the yield envelope during plastic yielding, the San Andreas and San Jacinto fault zones from inversion of fault dF=0, we obtain from eqs. (A7), (A9), and (A11) that displacement rates. Geology, 2004, 32: 961–964, doi: 10.1130/ G20806.1 T ⎧⎫∂F 10 He J K, Lu S J. 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