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The giant perilous rock at Wangxia (named Wangxia perilous rock) is representative in Three Gorges of the Yangtze River, China, has threatened badly the navigation of the Yangtze River channel for a long period. The giant perilous rock is composed of siliceous limestone and argillaceous limestone, and includes two elements marking by W1 and W2, respectively. The W1 is an isolated pillar while the W2 is in clintheriform. The linking segment of dominant fissure in the W2 is composed by moniliform solution funnels at its back, and the locked segment of the dominant fissure at the base of the W2 is composed by two parts. For the locked part of the dominant fissure of the W2, the upper segment shows the same lithology with the perilous rock and it can be simplified as an elastic medium, for the lower segment composed by argillaceous shale and mudstone can be simplified as a strain-softening medium. Introducing the water-softened function, the constitutive curves with two kinds of medium materials for the locked segment in the dominant fissure of the W2 have been proposed. Based on energy principle, the cusp catastrophe model for perilous rock rupture is built and formulas for the transient elastic and impulsive acceleration and the elastic-impulsive velocity of perilous rock catastrophe rupture have been established. By the calculation, the elastic-impulsive acceleration for the catastrophe rupture of the W2 is 531.4 m/s2, while the average elastic-impulsive velocity is 2.608 m/s. Further, it is deduced that the elastic-impulsive velocity at the base of the W2 is about 5.2 m/s. For the transient ruture of the W2, there is a greater speed difference between the top and the base of the W2, which impels the giant perilous rock to be retroverted sliding rupture, coinciding with the fact. Undoubtedly, studies in this paper must play an important role to analyze the catastrophe rupture mechanism of giant perilous rocks at both banks in Three Gorges of the Yangtze River, China.

As one of global mountain hazards, collapse stemes from perilous rock and possesses some fundamental characteristics, such as outburst, celerity and disaster [

The height for the top of Wangxia perilous rock varies from 1220 m a.s.l. to 1230 m a.s.l and for the base of the perilous rock is from 1137 m a.s.l to 1147 m a.s.l., with about 70 m elevation difference. Length of the perilous rock along the cliff is about 120 m, the average thickness is 30 - 35 m, and the volume is about 40 × 104 m^{3}. Based on exist position, Wangxia perilous rock can be divided into two parts/blocks, marking with W1 and W2, respectively (^{3} in volume. For W2, whose height is about 70 m, length is about 80 m, thickness is 10 - 15 m, volume is about 7 × 104 m^{3} and the dominant fissure at its tail is transfixed and composed by moniliform solution funnels. The section of the perilous rock is locked patch and it can be divided into upper segment and lower segment. The upper segment is composed by the siliceous and argillaceous limestone layer with different thickness from thin to medium thick and the chert layer, where, its lithology is flinty and it can be simplified as elastic segment. As for the lower segment, it is composed by argillaceous shale and mudstone, possessing strain-softening characteristic, can be simplified as strain-softening segment.

Wangxia perilous rock appeared deformation between Jul. and Aug. in 1999 for the first time, then, it has been actualized professional monitoring by 2003 and it didn’t appear distinct deformation by Jul. 2010. However, on Aug. 21, 2010, due to heavy rainfall in Wu Gorge area, W2 of Wangxia perilous rock appeared aggravation of deformation at the east segment of the perilous rock, the surface layer of the perilous rock fell its blocks frequently with 8 to 40 m ^{3} in volume for single block. At 7: 40 a .m. on Oct. 21, 2010, W2 proceeded catastrophe rupture, whose base slid forth for 10 - 15 m, the whole perilous rock mass dislocated down for about 10 m, it impulsed the soil body at the base of cliff to slid down to the downside of highway, and then, volumes of soil body and rock blocks collapsed down along the slope (

The geological section of W2 of Wangxia perilous rock is shown as in

respectively, and the components along y-axis are

where, m is the weight of perilous rock (kg); α is the dip angle for the un-linking patch of dominant fissure (˚); g is the acceleration of gravity (9.8 m/s^{2}).

The constitutive models for elastic segment and strain-softening segment of dominant fissure in Wangxia perilous rock are respectively described by using Equation (3) and Equation (4), and its representative constitutive curves are expressed in

where, G_{1} is the shear modulus of elastic segment (MPa); _{2} is the shear modulus of strain-softening segment (MPa); u_{0} is the corresponding displacement for the peak point of shearing strength on perilous rock mass (m).

The permeability of water for the linking patch of dominant fissure at the tail of Wangxia perilous rock is larger, the rock mass in locked patch ruptures and under the effect of groundwater, its intensity is easily to appear weakening-water phenomenon. According to the achievement that intensity of quartzose, shale will change with different moisture content from Vutukuri et al. [_{r}), as follows.

where, S_{r} is the saturation of rock and soil medium in dominant fissure; g(S_{r}) is the decreasing function, when the rock and soil sediment in dominant fissure is at the dry state, S_{r} = 0, g(S_{r}) = 0, and when it is at the saturation state, S_{r} = 1, g(S_{r}) = β; β is the shear strength parameter as the rock and soil sediment in dominant fissure is at the saturation state, the smaller numerical value, the more obvious degradation. β for the elastic segment and strain-sof- tening segment of dominant fissure in Wangxia perilous rock are respectively 0.8 and 0.2.

To substitute Equation (5) into Equation (3), the constitutive model of elastic segment in the locked patch of dominant fissure under the consideration of weakening-water effect is expressed as,

where, g_{1}(S_{r}) is the weakening-water function in the elastic segment of dominant fissure;

To substitute Equation (5) into Equation (4), the constitutive model of strain-softening segment in the locked patch of dominant fissure under the consideration of weakening-water effect is expressed as,

where, g_{2}(S_{r}) is the weakening-water function in the strain-softening segment of dominant fissure;

To perilous rock with unit width, when the locked patch of dominant fissure produces creep displacement u and before it reaches the peak of shear resistance, namely,

where, S_{1} is the length for the elastic segment in the locked patch of dominant fissure (m); other variables are the same as above.

When the creep displacement u producing by the locked patch of dominant fissure is

The variables in formula (9) are the same as above.

For the strain-softening segment in the locked patch of dominant fissure, the formula to estimate the elastic deformation energy of perilous rock is established as follows.

where, S_{2} is the length for the strain-softening segment in the locked patch of dominant fissure (m); other variables are the same as above.

The formula to estimate gravitational potential energy of perilous rock mass is as followings.

The variables in formula (11) are same as above.

When the medium layer of dominant fissure produces creep displacement u, locked patch accumulates elastic deformation energy and the gravitational potential energy of perilous rock mass is in release state, the potential function of perilous rock mass can be divided into two cases to discuss, respectively.

Case 1: combining Equation (8) with Equations (10) and (11), the formula to express potential function W_{1} of perilous rock is established, as follows.

Case 2: combining Equation (9) with Equations (10) and (11), the formula to express potential function W_{2} of perilous rock is established, as follows.

Taking partial derivative on Equation (12) and Equation (13), respectively. Then

According to the catastrophe theory, dW/du = 0 is the catastrophe manifold. Based on the quality of smooth manifold, the third-order partial derivative for potential function of perilous rock is gained as

where,

In the same way, to expand Equation (15) at ut by using Taylor integration method and take the value to third-order terms as,

As u < u_{b}, Equation (16) is coordinated as followings,

Let

equation of cusp catastrophe model:

To substitute parameters_{1}, and combine Equation (6) with Equation (7), another expression for parameter b_{1} is established.

In catastrophe theory, parameter b_{1} can be used to distinguish the creep state of perilous rock, discussed as follows from three aspects:

The first, b_{1} > 0 expresses that the creep acceleration of dominant fissure is less than zero and the deformation of perilous rock is at state of decelerated creep.

The second, b_{1} = 0 expresses that the creep acceleration of dominant fissure is equal to zero and the deformation of perilous rock is at state of constant creep.

And the third, b_{1} < 0 expresses that the creep acceleration of dominant fissure is more than zero and the deformation of perilous rock is at state of accelerated creep.

Obviously, u < u_{b} means that perilous rock is at creep state and catastrophe failure does not appear, however, u ≥ u_{b} means some other valuable conclusions, discussing in detail as follows.

when u ≥ u_{b}, Equation (17) is coordinated as Equation (21).

Let

To substitute parameters k_{2} and _{2}, and combine Equation (6) with Equation (7), the another expression for parameter b_{2} is established.

The physical meaning of parameter b_{2} is the same as parameter b_{1}.

In addition, in the state u ≥ u_{b}, perilous rock is likely to produce catastrophe failure and the governing equation that catastrophe point to obey is established, as follows.

To substitute parameters a_{2} and b_{2} into Equation (24), then

The smooth catastrophe manifold figure M shown in _{2}-b_{2} plane is called bifurcation set expressed as

B_{1} and B_{2}, respectively. Each point in equilibrium surface, namely, the phase point, denotes a certain status in the evolution process of perilous rock. Further, equilibrium surface can be divided into three parts, i.e., superior leaf, middle leaf and inferior leaf.

The state variable x of cusp catastrophe is instable at the critical points set of vertical tangent in equilibrium surface from inferior leaf to superior leaf. The state variable in critical points set needs to fit in with the Equation as follows,

Instability needs to span the left branch B1 of bifurcation set, and then, three roots of Equation (22) are solved, as follows.

Combining

tial point u_{j} and at end point us needed by the catastrophe instability of perilous rock are achieved followings, respectively.

According to the integral on x in Equation (19), the potential function for catastrophe model is expressed as,

where, c_{2} is a constant and the other variables are the same as above.

From the differential

Based on Equations (27) to (32), then

It indicates that the total potential energy after instability of perilous rock is lower than the one before its instability, thus, Equation (15) is coordinated, as follows.

At the transient of catastrophe instability for perilous rock, the elastic energy can be transferred into the kinetic energy by the equation as follows,

For the massive perilous rock, its failure velocity is fit to take the average velocity. However, for the columnar and platy perilous rock like W2 of Wangxia perilous rock, it possesses qualities such as: height is much larger than thickness. At the transient failure, it is difficult to reach the displacement triggering rupture of the whole perilous rock, actually, the end area of dominant fissure of perilous rock firstly obtains the released energy to reach the elastic-impulsive velocity as the velocity for top of perilous rock is zero. The elastic-impulsive velocity of failure and acceleration for the bottom of perilous rock are deduced as followings, respectively.

where, v_{b} is the departure violent-slide velocity for the bottom of perilous rock (m/s); a_{b} is the elastic-impulsive (or transient) acceleration for the bottom failure of perilous rock (m/s).

For the top of Wangxia perilous rock, the natural bulk density of the chert limestone is 26.3 kN/m^{3}, the saturation bulk density is 27.0 kN/m^{3}, the cohesion is 700 kPa and the internal friction angle is 45.38˚. For its base, the natural bulk density of carbonaceous shale is 22.5 kN/m^{3}, the saturation bulk density is 23.0 kN/m^{3}, the cohesion is 200 kPa and the internal friction angle is 29.9˚. For the locked patch of dominant fissure in W2 perilous rock, the dip angle α is 42˚, the thickness h is 0.3 m, the weight mg unit width perilous rock is 23198 kN. For the elastic segment of dominant fissure, the cohesion is 60 kPa, the internal friction angle is 19.38˚, shear modulus G_{2} is 18.71 Gpa, parameter β of water retaining strength is 0.79, and parameter S_{1} is 8.5 m. For the strain-softening segment, the cohesion is 30 kPa, the internal friction angle is 13.2˚, shear modulus G_{2} is 4.23 Gpa, parameter β of water retaining strength is 0.29, parameter S_{1} is 4.1 m, and u_{0} is 0.15 × 10^{−2} m. Moreover, the saturation S_{r} for elastic segment is 86% and about 84% for strain-softening segment.

Based on the related parameters above presentation, some intermediate variable are achived as follows.

It indicates that the dominant fissure of perilous rock is at the accelerated creep state. The energy release quantity ΔE at this moment is 7920 kJ, hereby, the transient acceleration a for the failure of the perilous rock is 531.4 m /s^{2} and the average elastic and impulsive velocity v is 2.608 m/s. According to that, it can be deduced that the elastic-impulsive velocity at the base of the W2 of Wangxia perilous rock is about 5.2 m/s. Therefore, for the fracture moment of the W2 of Wangxia perilous rock, characteristics with rotation and translation are visualized obviously due to the great speed difference between the top and the base of the perilous rock block. However, in virtue of the anti-slide at the base of the W2 of Wangxia perilous rock and the friction effect produced by the top of perilous rock back against matrix surface, the W2 stops on the matrix ultimately (

Firstly, the Wangxia perilous rock in Wu Gorge of the Yangtze River, China, is composed of siliceous limestone and argillaceous limestone, and includes two block marking in W1 and W2. W1 is an isolatedpillar and W2 is in clintheriform. The linking patch of dominant fissure in W2 is composed by moniliform solution funnels at its back, and the locked patch of dominant fissure at the base of perilous rock is composed by two parts. Where the upper segment shows the same lithology with perilous rock and it can be simplified as elastic medium, for the lower segment, it is composed by argillaceous shale and mudstone and can be simplified as strain-softening medium. The failure mode of Wangxia perilous rock manifests as retroverted sliding failure.

Secondly, according to the geological model of W2 of Wangxia perilous rock, its mechanical model is established. By using the weakening-water function, the constitutive curves for the locked patch of dominant fissure with two kinds of medium materials has been proposed.

Thirdly, based on energy principle, the cusp catastrophe model for the failure of perilous rock is built and formulas for the transient elastic and impulsive acceleration and the elastic-impulsive velocity of perilous rock catastrophe failure have been established.

Fourthly, by the calculation, the elastic-impulsive acceleration for the catastrophe failure of W2 of Wangxia perilous rock is 531.4 m /s^{2}, the average elastic and impulsive velocity is 2.608 m/s. Further, it deduces that the elastic-impulsive velocity at the base of perilous rock is about 5.2 m/s. For the transient failure of perilous rock, there is a great speed difference between the top and the base of W2 of Wangxia perilous rock, which impels perilous rock to be retroverted sliding failure and it coincides with the fact.

This study in the paper was funded by The National Natural Science Foundation of China (No. 11272185, 51678097 and 51378521), and special fund of the innovation team for Geological Disasters Mitigation along Highways & Shipping Lanes in Western China (CXTDG201602012). The authors are grateful for Prof. Xian Xue-fu, academician of Chinese academy of engineering, and Prof. Li Ji-jun, academician of Chinese Academy of Sciences.

Chen, H.-K. and Wang, S.-J. (2017) Study on Sinking-Sli- ding Failure Mechanism of Perilous Rock at Wangxia in Three Gorges of the Yangtze River, China. Journal of Geoscience and Environment Protection, 5, 30-43. https://doi.org/10.4236/gep.2017.52003