Nonlinear Evolutionary Mechanisms of Instability of Plane-Shear Slope: Catastrophe, Bifurcation, Chaos and Physical Prediction

Transcription

1 Rock Mech. Rock Engng. (2006) 39 (1), DOI /s Nonlinear Evolutionary Mechaniss of Instability of Plane-Shear Slope: Catastrophe, Bifurcation, Chaos and Physical Prediction By S. Q. Qin 1, J. J. Jiao 2, and Z. G. Li 1 1 Engineering Geology and Applied Geophysics Departent, Institute of Geology and Geophysics, Chinese Acadey of Sciences, Beijing, P.R. China 2 Departent of Earth Sciences, The University of Hong Kong, Hong Kong, P.R. China Received July 13, 2004; accepted January 12, 2005 Published online March 31, 2005 # Springer-Verlag 2005 Suary A cusp catastrophe odel is presented and the necessary and sufficient conditions leading to landslides are discussed. The sliding surface is assued to be planar and is a cobination of two edia: ediu 1 is elastic-brittle or strain-hardening and ediu 2 is strain-softening. The shear stress-strain constitutive odel for the strain-softening ediu is described by the Weibull s distribution law. This paper is a generalization and extension of the paper by Qin et al. (2001b), in which the shear stress-strain constitutive odel for ediu 2 was described by a negative exponent distribution; a special case of the Weibull s distribution law. It is found that the instability of the slope relies ainly on both the stiffness ratio of the edia and the hoogeneity index and that a new role of water is to enlarge the aterial hoogeneity or brittleness and hence to reduce the stiffness ratio. A nonlinear dynaic odel (also called a physical forecasting odel), which is derived by considering the tie-dependent behavior of the strain-softening ediu, is used to study the tie prediction of landslides. An algorith of inversion on the nonlinear dynaic odel is suggested for seeking the precursory abnorality and abstracting echanical paraeters fro the observed series of a landslide. A case study of the Jiingsi landslide is analysed and its nonlinear dynaic odel is established fro the observation series of this landslide using the suggested odel and the algorith of inversion. It is found that the catastrophic characteristic index jdj shows a quick rise till reaching an extreely high peak value after the slope evolves into the tertiary creep, and subsequently approaches a zero value prior to instability, which can be regarded as an iportant precursory abnorality index. By taking into account the evolutionary characteristic of the slope being in the secondary creep, a siplified nonlinear dynaic odel is proposed for studying the properties of bifurcation and chaos. It is shown that the eergence of chaos depends on the echanical paraeters of the sliding-surface edia. Keywords: Cusp catastrophe, instability, stiffness ratio, hoogeneity index, chaos, physical prediction.

2 60 S. Q. Qin et al. 1. Introduction The coonly used liit equilibriu approach for the stability analysis of slopes, which is incapable of considering the creep displaceent along the sliding surface, ay result in an unconvincing assessent of slope stability. Many nuerical ethods, such as finite-eleent, boundary-eleent, and discrete-eleent ethods, can do well in siulating the nonlinear behavior of slope deforation and failure, but ost of the are not suitable for odeling the nonlinear behavior of brittle rock (Tang, 1997). However, there is a tendency in this field that the theoretical fraework for nonlinear and discontinuu probles becoes ore and ore coplicated and soeties becoes less practicable. Hence, it is suggested that a convenient and practical approach to evaluate the stability of a slope should be developed which allows for the displaceent along the sliding surface. The coon ethods for predicting the failure tie of landslides are epirical and statistical. For exaple, Saito (1969) forecasted the tie of slope failure by tertiary creep. Voight (1989) presented a ulti-paraeter-forecasting forula according to the statistical function aong tie, displaceent and epirical constants. Qin and Wang (2000) suggested a hooorphic odel for identifying abrupt abnoralities of landslide forerunners. These statistical predictions, as pointed out by Rat (1988), are generally very unreliable and often very tricky due to lack of physical basis. The essential route for landslide prediction is to turn an epirical or statistical prediction into a physical one. The ethods and approaches based on nonlinear dynaic systes (NDS) theory have been recently applied to investigate various probles in rock echanics and geoechanics, including landslides (Saunders, 1980; Keilis-Borok, 1990; Qin et al., 1993; Phillips, 1995; and Qin et al., 2001). A brief review of these applications can be found in the papers by Qin et al., Qin et al. (2001a) developed a cusp catastrophe odel of instability of slip-buckling slope, and hence analyzed the echaniss of instability as well as the connection between the slope state and behavior in creep. Qin et al. (2001b) also studied the unstable echaniss of a plane-shear slope and the chaotic dynaic echaniss of its evolutionary process by NDS theory. They assued that the sliding surface of the landslides was planar and was a cobination of two edia: ediu 1 was elastic-brittle and ediu 2 was strain-softening. The shear stress-strain constitutive odel for ediu 2 was described by a negative exponent distribution. This paper is a generalization and extension of the paper by Qin et al. (2001b). The sliding surface of the landslide is also assued to be planar and is a cobination of two edia, but the ediu 1 can be elastic-brittle or strain-hardening. The shear stress-strain constitutive odel for the strain-softening ediu is described by the Weibull s distribution law, which is ore general and practical than a negative exponent distribution. The discriinating criteria leading to a fast-oving landslide will be suggested. A nonlinear dynaic odel derived by considering the tie-dependent behavior of the strain-softening ediu will be used to study the tie prediction of landslides. An algorith of inversion on the nonlinear dynaic odel will be suggested for seeking the precursory abnorality and abstracting echanical paraeters fro the observed series of a landslide. A case study of the Jiingsi landslide will be used to deonstrate the applicability of the odel.

3 Mechaniss of Instability of Plane-Shear Slope Cusp Catastrophe Model of a Plane-Shear Slope and its Instability Mechaniss 2.1 Mechanical Model Figure 1 shows the odel used in this study. The sliding surface with the dip angle is a nonunifor intercalation, the rock ass with the weight Wg (g is gravity acceleration) above is a rigid body. Under the action of the driving force caused by the weight of the rock ass, the creep displaceent is u along the intercalation. Due to different stress levels, aterial coposition, textures and structures at different segents of the intercalation, the interaction can coprise any kinds of edia with different echanical properties, such as elastic-brittle, elastic-ductile (strain-hardening) and strain-softening, etc. To siplify the analysis, we regard the interaction edia as a cobination of only two kinds of edia with different echanical properties, i.e., one (ediu 1) possesses a coplicated elastic-brittle (such as a strong interlocking rock block or so-called rock bridge) or strain-hardening (such as hard clay or incopact sandy soil) behavior and the other (ediu 2) has a strain-softening property. Whether ediu 1 is elastic-brittle or is strain-hardening, the sae conclusions will be drawn in this paper. For siplicity, we will consider the elastic-brittle property of ediu 1 in the analysis below. The shear stress-displaceent relation, for the strong rock block with pre-existing ulti-cracks or flaws, can generally be assued to be linear prior to failure. At failure, the rock block is bisected and a sudden stress drop will take place. After failure of the rock block the cobined cracks could reseble a stepped or sawtoothed, throughgoing discontinuity, resulting in a slow, soeties quick (e.g. Lajitai, 1967, and Gehle and Kutter, 2003) stress increent instead of a decrease with growing displaceent. Finally, the stepped or sawtoothed discontinuity is levelled out, which ay lead to a rapid stress decrease with growing displaceent. According to the above assuptions and direct shear tests on interittent rock joints perfored by Gehle and Kutter (2003), the constitutive odel for ediu 1 (Fig. 2) can be written as 8 u >< G e1 ¼ h ðu < u bþ u ð1þ >: G e2 h ðu b u u 2 Þ; where u b is the critical displaceent corresponding to the failure of the rock block, u 2 is the displaceent when the stepped or sawtoothed discontinuity inside the rock Fig. 1. Mechanical odel of a plane-shear slope instability

4 62 S. Q. Qin et al. Fig. 2. Constitutive curves of two edia along the sliding plane block is levelled out, G e1 and G e2 are the shear oduli for u < u b and u b u u 2, respectively, and h is the layer thickness of the intercalation. A siplified constitutive odel of ediu 2 can generally be expressed as a nonlinear function between the shear stress and the creep displaceent u. Qin et al. (2001b) used a negative exponent distribution of strength to describe the strain-softening property. However, the ost coonly used distribution is the Weibull s distribution which describes very well the experiental data (Hudson and Fairhurst, 1969), so the Weibull s distribution law is adopted for the strain-softening edia, i.e. u ¼ G s exp h u u 0 ; ð2þ where G s is the initial shear odulus, u 0 is a easureent of average strength and is the shape paraeter (Fig. 3). One of the attractive aspects of the Weibull s distribution Fig. 3. Weibull s distribution constitutive curves of ediu 2 for different values of (G s ¼ 10 MPa, u 0 ¼ 0.1, and h ¼ 0.1 )

5 Mechaniss of Instability of Plane-Shear Slope 63 is the presence of the shape paraeter which allows this function to take a wide variety of shapes (Fig. 3). For instance, for ¼ 1, this distribution is exponential; at about ¼ 2, it very closely approxiates a noral distribution. Since is a easureent of the local strength variability, it can be considered as a hoogeneity index (Tang, 1993). The larger the index, the ore hoogeneous is the rock. When trends to infinity, the variance trends to zero and a perfect rock is obtained. A aterial with such a property is so-called ideal brittle aterial, such as glass. Thus, the shape paraeter can also be referred to as the brittleness index. 2.2 Cusp Catastrophe Model The overall potential energy for the syste illustrated in Fig. 1 is equal to the su of the strain energy and driving potential energy, i.e. 8 ð u G s u u l s exp du þ 1 G e1 l e 0 h u 0 2 h u2 Wgu sin ðu < u b Þ >< V ¼ ð u G s u u l s exp du þ 1 G e2 l e 0 h u 0 2 h u2 þ 1 l e u 2 b 2 h ðg ð3þ e1 G e2 Þ >: Wgu sin ðu b u u 2 Þ; where l e and l s are the length of the sliding surface for edia 1 and 2, respectively, and l s þ l e ¼ H=sin ; u can be regarded as the state variable in the cusp catastrophe analysis. It is assued here that l e is far less than l s, because a slope probably is very stable when l e > l s. Let V 0 ¼ 0 and the equilibriu surface equation (Fig. 4) is expressed as V 0 ¼ G sl s u exp h u u 0 þ G el e u Wg sin ¼ 0; ð4þ h Fig. 4. Cusp catastrophe odel

6 64 S. Q. Qin et al. where G e ¼ G e1 ðu < u b Þ or G e ¼ G e2 ðu b u u 2 Þ. Equation (4) is the equilibriu condition of forces. By the soothness property of the equilibriu surface, the cusp can be solved using the condition V 000 ¼ 0, i.e. u ¼ u 1 ¼ þ 1 1 u0 ð5þ Equation (5) shows that the displaceent value at cusp is exactly the displaceent value at the turning point of the constitutive curve of ediu 2. By Taylor series expansion with respect to u 1 for Eq. (4), discarding all the ters but the first three because the third order ter is the iniu one away fro zero, and substituting Eq. (5) into Eq. (4), one has: 1 G s l s u 1 exp þ1 ð þ 1Þ 2 6 h u u 3 1 þ 6 u 1 6 þ ð þ 1Þ 2 G e l e expð þ1 Þ! 3 G s l s ð þ 1Þ 2 1 u u 1 ð þ 1Þ 2 u 1 1 þ G el e expð þ1 Þ Wgh sin expðþ1 Þ 7 5 ¼ 0 G s l s G s l s u 1 In order to transfor Eq. (6) into a standard for of cusp catastrophe, let x ¼ u u 1 ; ð7þ u 1 6 a ¼ ðk 1Þ; 2 ð þ 1Þ ð8þ ð6þ 6 b ¼ ð1 þ k Þ; 2 ð þ 1Þ ð9þ k ¼ G el e expð þ1 Þ ; ð10þ G s l s Wgh sin expðþ1 ¼ Þ ; ð11þ G s l s u 1 where k is the ratio of the stiffness of ediu 1 (k e ¼ G e l e =h) to the stiffness at the turning point of the constitutive curve of ediu 2 (k s ¼j G s l s exp½ ð þ 1=ÞŠ=hj) (for siplicity, it is called hereafter the stiffness ratio); is relative to the weight of rock ass, geoetric size of the syste, and echanical paraeters of edia (referred to as the geoetric-echanical paraeter). Substituting Eqs. (7) (11) into Eq. (6) leads to x 3 þ ax þ b ¼ 0: ð12þ Equation (12) is the standard cusp catastrophe odel of the equilibriu surface, with a and b as its control paraeters and x as its state variable. The cusp catastrophe described by the equilibriu surface containing fold or pleat is illustrated in Fig. 4 where axes of three-diensional space are the control paraeters a, b (horizontal) and response paraeter x (vertical). As pointed out by Henley

7 Mechaniss of Instability of Plane-Shear Slope 65 (1976), path B B 0 and path A A 0 denote a stable evolutionary process and an unstable evolutionary process, respectively. The bifurcation set (cusp) (Tho, 1972) can be expressed as D ¼ 4a 3 þ 27b 2 ¼ 0 ð13þ Substituting Eqs. (8) and (9) into Eq. (13) leads to 2 D ¼ 4 3 ðk 1Þ 3 þ 27 ð1 þ k Þ 2 ¼ 0 ð14þ where ¼ 6=ð þ 1Þ 2. The bifurcation set (Fig. 4) defines the thresholds where sudden changes can take place. As long as the state of the syste reains outside the bifurcation set (D > 0), the behavior varies soothly and continuously as a function of the control paraeters. Even on entering the bifurcation set (D < 0) no abrupt change is observed. When the control point passes all the way through the bifurcation set (D ¼ 0), however, a catastrophe is inevitable. Thus, Eqs. (14) or (13) is the sufficient and necessary echanical criteria for the plane-shear slope instability (fast-oving landslide). In the following analysis D is referred to as the catastrophic characteristic index. Obviously, only when k 1, the condition of Eq. (14) ay be satisfied. Thus, the necessary condition of instability is k ¼ G el e expð þ1 Þ 1 ð15þ G s l s Equation (15) shows that the saller the stiffness of ediu 1 is, the larger the post-peak stiffness (the absolute value of post-peak stress constitutive curve slope) of ediu 2, and the ore possible it is for the slope syste to lead to catastrophe. It is known fro Eq. (15) that k decreases with an increase of for the fixed values of G e l e =G s l s, deonstrating ore prone to landsliding for a ore hoogenous rock. It is generally believed that the instability of a slope has a close connection with the action of water, which reduces the anti-shear strength and increases the pore-water or uplift pressure in the intercalation. Figure 5 shows that water also has a new role of enlarging the hoogeneity or brittleness index and hence reducing the stiffness ratio, ore readily leading to landsliding. 2.3 Sybol b and the Regions with Creep Behaviour Substituting Eqs. (10) and (11) into Eq. (9) leads to þ1 6 exp b ¼ ð þ 1Þ 2 G e l e u 1 þ G s l s u 1 exp þ 1 Wgh sin ; ð16þ G s l s u 1 b depends on the relative agnitude of resisting and driving forces at the turning point of the strain-stress curve of ediu 2. It is clear fro Eq. (16) that b > 0, b ¼ 0 and b < 0 correspond to the sliding acceleration of the slope ass: <0 (priary creep), ¼ 0 (secondary creep) and >0 (tertiary creep), respectively. Figure 6 illustrates the corresponding relation between b and the regions with creep behaviour.

8 66 S. Q. Qin et al. Fig. 5. The effect of pore-pressure on the brittle-ductile transition in liestone at a confining pressure of 6.9 MPa. Nubers on the curves are values of pore pressure in MPa (Fig in Jaeger and Cook, 1979) Fig. 6. Regions of behavior in creep. Priary, secondary and tertiary creep correspond to decreasing rate, steady rate and increasing rate with tie, respectively Equation (12) (b < 0) can be used to deterine the critical displaceent value at the catastrophic points corresponding to failure of ediu 1 as follows: pffiffiffi 2 u b ¼ u 1 1 ð1 kþ1=2 ðk 1Þ ð17þ þ 1

9 Mechaniss of Instability of Plane-Shear Slope Nonlinear Dynaic Model of Evolutionary Process of Slope In the above analysis, we considered the quasi-static otion of the slope, but not the dynaic process of instability. To study the chaotic echanis of the evolutionary process of a slope and to predict the landslide instability, a nonlinear dynaic odel is needed. 3.1 Nonlinear Dynaic Model If the viscosity or creep (i.e. tie-dependent) property of the strain-softening ediu is considered (Fig. 7), then its shear strength can be expressed as u u ¼ s þ v ¼ G s exp þ du h u 0 h dt ; ð18þ where is the viscosity coefficient. Substituting Eq. (18) into the equilibriu surface Eq. (4) and using Eqs. (5) (11), one obtains dx dt ¼ 1 G s expð þ1 Þð þ 1Þ2 ðx 3 þ ax þ bþ: ð19þ 6 Equation (19) is a nonlinear dynaic or a physical forecasting odel with a definite physical eaning of each paraeter. As long as we know the paraeters of the odel by eans of laboratory tests and field investigations, the prediction of the landslide behavior can be ade. It is known fro the above-entioned analysis that the paraeters a and b denote the possible instability and the creep phase of a landslide, respectively. Obviously, the diensionless displaceent rate grows as the paraeter b (b < 0) decreases when a < 0. Now, we study the character of the equilibriu state as given by Eq. (19). It is known by letting dx=dt ¼ 0 that Eq. (19) also is a cusp catastrophe and its instability condition is the sae as Eq. (13). It is realized fro Eq. (19) that the diensionless displaceent x is wholly doinated by the echanical and geoetric paraeter of the slope body itself. Thus, it is deduced that the variations of the echanical paraeters can be reflected in the observed displaceent-tie series of a landslide when the geoetric paraeters are fixed, which deonstrates that we can estiate the echanical paraeters fro the observation series using an algorith of inversion. 3.2 Algorith of Inversion on the Nonlinear Dynaic Model The only inforation available at present is the data derived fro observation and description of deforation and failure phenoena for the coplex evolution process Fig. 7. A shear stress odel considering the viscosity or creep property, siilar to Kelvin or Voigt odel

10 68 S. Q. Qin et al. of landslides. That is to say, we know a series of specific solutions of the dynaic odel. If we regard such solutions as a series of discrete values of the dynaic odel, the quasi-ideal nonlinear dynaic odel for the evolution of the landslide can, thus, be obtained through an algorith of inversion. The following gives the analytical procedures and steps. (1) Substituting Eqs. (7) (11) into Eq. (19) in order to transforing x-t series into u-t series, one has where du dt ¼ c 1u 3 þ c 2 u 2 þ c 3 u þ c 4 ; c ¼ G s exp þ 1 ð þ 1Þ 2 =6; ð20þ ð21þ c 1 ¼ c=u 2 1 ; c 2 ¼ 3c=u 1 ; c 3 ¼ð3 þ aþc; ð22þ ð23þ ð24þ c 4 ¼½b ðaþ1þšcu 1 : ð25þ (2) The paraeters c 1, c 2, c 3 and c 4 can be solved by best fitting the original observation data with Eq. (20) by a fitting algorith. It should be noted (Bakus and Gilbert, 1970) that, in ost cases, the solutions of Eq. (20) usually are unstable when the least squares ethod is adopted. Therefore, it is suggested that an iproved iterative algorith of inversion presented by Qin et al. (2002) should be applied to the solution of Eq. (20). The predicted values can be calculated using the Runge-Kutta integration ethod. 3.3 Case Study The Jiingsi landslide at Zigui county in Hubei province, PR China, with a volue of ca , occurred on 29 June 1991 (Fig. 8). The landslide ass consists of hard liestone containing soft and thin arlite intercalations. The landslide was induced by ining liestone and is a typical plane-shear instability with a slope angle of 35. The upper elevation of the stored ore at the site is 250. The liestone below elevation was exploited, foring a steep cutting surface with a height of and width of 153. Fractures at the two sides of the landslide ass were present and fored the side boundaries. The upper part of the landslide ass was cut off by a fault. This slope with a soft arlite sliding surface was characterised by a set of boundary conditions leading to landslide instability. Under such conditions, due to beast vibrations and rainfalls, the slope body began to defor. The annual average rainfall in this area is The total rainfall was fro 1 May to 15 June in 1991.

11 Mechaniss of Instability of Plane-Shear Slope 69 Fig. 8. Geological section of Jiingsi landslide On 5 March 1990, a few cracks at the top of the landslide were observed. Since 24 April 1990, an increase in the crack width was easured with an autoatic onitoring syste. The onitored crack width at C4 and C7 observation points (Lu, 1994) are illustrated in Fig. 9. The displaceent (crack width) curves versus tie are very siilar to the typical creep curves shown in Fig. 6. At C4 observation point, the nonlinear dynaic odel obtained using the aboveentioned algorith of inversion is as follows: dx dt ¼ 0:5161ðx3 1:03701x 0:4282Þ: ð26þ Figure 9 shows that the predicted values are in good agreeent with the observed values. By choosing the data between the beginning of the observation and a certain tie, we can get a series of values for a, b and D. It is also observed fro Fig. 9 that the catastrophic characteristic index jdj behaves as a relatively steady change during the secondary creep phase, then, it begins to have a rapid increase and reaches an extreely high peak value after entering into the tertiary creep, and eventually it quickly falls to approxiately zero prior to the landslide instability as expected. Thus, the index D can be regarded as an iportant index indicating the precursory abnorality of the landslide. Taking the observation series at C4 observation point as an exaple, we can estiate the possible paraeter scales using the conditions a ¼ and b ¼ obtained fro Eq. (26). It is seen fro Fig. 10 that the paraeters k, and are in the range 0 k , and 0 < 1.4, respectively. Considering the worst scenario, i.e., l e ¼ 0 (then, k ¼ 0, ¼ 1.4 and ¼ 1.582), and applying the known conditions of l s ¼ 237.6, u 1 ¼ 0.928, Wg ¼ kn, h ¼ 0.2 and ¼ 35, we obtain G s ¼ 1.17 MPa and ¼ Pa.s, and realize that the shear odulus is very low, close to landslide instability.

12 70 S. Q. Qin et al. Fig. 9. Curves of the onitored values, forecasting ones and the catastrophic characteristic index jdj versus tie at C4 (a) and C7 (b) observation points Fig. 10. Variations of k and versus

13 Mechaniss of Instability of Plane-Shear Slope Siplified Nonlinear Dynaic Model When the slope syste evolves into the secondary creep phase, then b ¼ 0 and therefore Eq. (19) can be siplified as dx dt ¼ð 1x 2 x 3 Þ¼ 2 ðx x 3 Þ; ð27þ where 1 ¼ ca, 2 ¼ c > 0 and ¼ 1 = 2 ¼ a. Equation (27) represents a siplified nonlinear dynaic odel. 1 < 0(<0) and 1 0(0) denote that the slope is in the stable and unstable states, respectively. 4.1 Bifurcation The paraeter in Eq. (27) is a control paraeter. According to the Lyapunov s linear stability analysis theory (Wolf et al., 1985), it is easily shown that two groups of static solutions of Eq. (27) are as follows: 8 < <0; x 0 ¼ 0;!¼ 2 <0 : x 0 is asyptotically stable x >0; 0 ¼ 0;!¼ p 2 >0 : x 0 is unstable x ¼ ffiffiffi ð28þ : ;!¼ 22 <0 : x is asyptotically stable; where! is the eigenvalue of Eq. (27). It is found fro Eq. (28) that for the sae static solution x 0 ¼ 0, the syste becoes unstable at ¼ 0 fro the previously asyptotic stability, and hence p ffiffiffi produces two new asyptotically stable sub-solutions with an increent rate of, when changes fro a negative value to a positive one. This shows that the syste behaves as a bifurcation behavior and ¼ 0(k¼1) is the bifurcation point (Fig. 11). Also, this deonstrates that the conclusion drawn by the linear stability analysis is consistent with that by the catastrophe analysis, because <0, ¼ 0 and >0 correspond to k > 1, k ¼ 1 and k < 1, respectively. 4.2 Chaos If the values of x are easured at a fixed tie interval t, then, the previously continuous variables x(t) and t becoe discrete ones (x 0 ; x 1 ; x 2 ;...), where x n denotes Fig. 11. The stability and bifurcation of the syste (the thick real line and thick broken line denote asyptotic stability and unstability, respectively)

14 72 S. Q. Qin et al. the value x easured at the nth tie (n ¼ 0; 1; 2;...). Accordingly the continuous differential Eq. (27) becoes the following discrete difference Eq. (29), i.e. x nþ1 ¼ð1þ 2 Þx n þ 2 x n 2 : ð29þ Let ¼ 1 þ 2 ¼ 1 þ ca ð30þ y n ¼½ 2 =ð1 þ 2 ÞŠ 1=2 x n ð31þ and Eq. (29) is rewritten as y nþ1 ¼ y n ð1 y 2 n Þ: ð32þ By applying an iterative coputation for Eq. (32), it is found fro Fig. 12 that the syste has the solution with the period of 1 when 2. As increases, the solutions with the periods of 2, 4, 8 and 16 can be obtained, respectively, corresponding to ¼ 2.01, 2.24, and When >2.3, there occurs a solution with the period of 2 1. In other words, the period becoes infinite, i.e. the syste does not hold Fig. 12. Curves of y versus n as increases when solutions are stable

15 Mechaniss of Instability of Plane-Shear Slope 73 periodicity any ore and chaos appears. This analysis also shows that the evolution route leading to chaos of the slope syste is obtained by the bifurcation of perioddoublings. Substituting Eqs. (8) and (21) into Eq. (30) leads to 1 ¼ ca ¼ G s ð1 kþ exp þ 1 > 1:3 ð33þ We can judge whether the slope oveent is chaotic or not by Eq. (33). It is noted that the unit of (generally, Pa.s) should be consistent with the tie unit of t while using Eq. (33). It is known fro Eq. (33) that whether or not the chaotic phenoenon appears depends on the echanical paraeters G s, and of the strain-softening ediu as well as the stiffness ratio k. Obviously, the larger is and the saller k is, the ore prone to the occurrence of chaos the syste is. The physical iplication of Eq. (33) is that when a < 0(k < 1), i.e. a slope has the possibility of instability, the slope syste ay evolve into the chaotic state; whereas a slope is extraordinarily stable, no chaotic phenoenon appears. This eans that chaos also is an indicator of the possibility of instability of the slope. The reasons can be explained as follows: for extreely stable slopes, the environental factors, such as four-season cliate, teperature, rainfall, earthquake and vibration, have little ipact on their evolutionary states; contrarily, for possibly unstable slopes, especially those being in the critical state, the environental factors have a great influence on their evolutionary state, which indicates that the evolution of these slopes is sensitive to the change of the environental factors and therefore they could exhibit non-periodic otion (chaos). If a slope syste has no chaotic characteristics, a deterinistic prediction can be ade; otherwise, the predictable tie scale (Qin et al., 2001c) ust be considered and the accuracy of prediction during the tie scale should be studied. 4.3 Siplified Physical Forecasting Model Substituting Eq. (7) into Eq. (27) in order to transfor the variable x into the variable u, one has 3 du u dt ¼ u1 u u1 1 2 : ð34þ u 1 u 1 Equation (34) is siilar to the Logistic equation and is a siplified physical forecasting odel. Conducting the tie prediction of landsliding in ters of the observed data of a landslide, one can solve the paraeters 1 and 2 using the least squares ethod. The axiu displaceent value which can be easily derived fro Eq. (34) can be expressed as rffiffiffiffiffiffiffi 1 u ax ¼ u 1 1 þ : ð35þ 3 2 The occurrence tie of landsliding can be estiated according to the tie corresponding to u ax. For the Jiingsi landslide, for exaple, we obtain u ax ¼ 72.2 c and accordingly the unstable tie is on 26 April 1991, using the observation series in the

16 74 S. Q. Qin et al. secondary creep phase at C4 observation point. The coputed value of 1 is < 1.3, indicating no chaotic phenoenon of this landslide and that a deterinistic prediction can be ade. 5. Conclusions This paper presents a cusp catastrophe odel based on the catastrophe theory and discusses the necessary and sufficient conditions leading to landslide instability. It is assued that the sliding surface of the landslide is planar and is a cobination of two edia: one is elastic-brittle or strain-hardening and the other is strain-softening. The shear stress-strain constitutive odel for the strain-softening ediu is described by the Weibull s distribution law. The conditions leading to a fast-oving landslide are derived. The following findings can be obtained fro this paper: 1. The instability of a slope relies ainly on the ratio of the stiffness of ediu 1 to the post-peak stiffness of ediu 2, and the hoogeneity index of ediu A newly-found role of water is to enhance the aterial hoogeneity or brittleness and hence to reduce the stiffness ratio of the syste. 3. A nonlinear dynaic odel of the evolutionary process of a slope is presented by taking into account the viscosity or creep property of ediu 2. If the echanical and geoetric paraeters for a slope are deterined, a physical prediction can be ade. 4. The paraeters of the nonlinear dynaic odel can be obtained fro the observation data of a landslide by an algorith of inversion. The observation series contains plenty of inforation associated with the echanical paraeters that can be estiated using the odel and the algorith presented in this paper. 5. The nonlinear dynaic analysis on the Jiingsi landslide shows that the catastrophic characteristic index jdj has a relatively steady variation during the secondary creep phase. Then, this index shows a quick rise till reaching an extreely high peak value after entering into the tertiary creep, and in succession closes to zero prior to instability. We can judge the occurrence of landslide fro this feature and regard the index D as a paraeter reflecting the precursory abnorality of a landslide. 6. A siplified nonlinear dynaic odel is derived by considering the evolutionary characteristics of the slope being in the secondary creep. The properties of bifurcation and chaos of a landslide are studied. It is found that the bifurcation point, k ¼ 1, is the turning point of the slope syste fro stability to potential instability, and that chaos possibly arising fro a potentially unstable slope can be generated in a anner of the bifurcation of period-doublings. The discriinant leading to chaos presented in this paper shows that the chaotic phenoenon is related to the echanical paraeters of the edia along the sliding surface. Acknowledgeents The work was funded by China national 973 prograe (No. 2002CB412702).

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18 76 S. Q. Qin et al.: Mechaniss of Instability of Plane-Shear Slope Tho, R. (1972): Stabilite structurelle et orphogenese. Benjain, New York. Voight, B. (1989): Materials science law applies to tie forecasts of slope failure. Landslide News (3), Wolf, A., Swift, J. B., Swinney, H. L., Vastano, J. A. (1985): Detering Lyapunov exponents fro a tie series. Physica 16D, Authors address: Dr. Siqing Qin, Chinese Acadey of Sciences, Engineering Geology and Applied Geophysics Departent, Beijing, P.R. China; e-ail: qsqhope@sohu.co