Fracture mechanism of rock collapse in the freeze–thaw zone of the eastern Sichuan–Tibet Mountains under seasonal fluctuating combinations of water and heat

In the freeze–thaw zone of the eastern Sichuan–Tibet Mountains, the phases of water in cracks show strong seasonal variations, which significantly affect the stability of perilous rocks. However, few studies have clearly addressed the role of water/ice in crack development from a fracture mechanics viewpoint to explain the seasonality of rock collapse. In this study, we built physical models from a fracture mechanics viewpoint to calculate water-freezing stress, hydrostatic pressure, and their combinations induced by water/ice in cracks and show the crack propagation mechanism under temperature fluctuations in different seasons in mountainous regions. On the basis of these models, we calculate fracture conditions, simulate the crack process, and illustrate the rock collapse mechanism in different seasons using the extended finite element method. The results indicate that different phases of water, which induce stress under spatiotemporal fluctuations of temperature, determine the various propagation styles and influence what kind and when a collapse will occur. The collapse of fractured rocks in different seasons generally results from rock damage accumulation owing to the initiation, propagation, and connection of primary cracks under freezing stress or hydrostatic pressure or different combinations of these processes.


Introduction
Rock collapse widely occurs in the mountains of the eastern Tibetan Plateau owing to the geomorphic features of active crustal stress and large elevation differences. The occurrence of rock collapse and avalanche disasters has drastically increased over the past 50 years (Hore et al. 2018). These collapse events and their secondary disasters pose a serious danger to towns, roads, and other inhabited regions. Outstanding examples include the Yigong rock avalanche that blocked the Yigong River for 2 months and killed 94 people, the Baga collapse in Tibet that dammed the Chayu River, and the Xinmo collapse and rock slide in Sichuan that buried 64 houses and left dozens of people missing (Hu et al. 2018;Xu et al. 2012).
Rock collapse is a process in which a relatively intact mass on a well-defined rupture surface rapidly detaches and undergoes extensive internal disaggregation, which directly leads to a rockslide or avalanche disaster (Eisbacher 1979). Numerous studies have mainly focused on the dynamical mechanism of rockfalls or avalanches after collapse (Locat et al. 2006). However, relatively few studies have clearly addressed the rock collapse mechanism, although the character, distribution law, and induced factors of collapse have been well studied in a qualitative or semiquantitative manner (Ez Eldin et al. 2013). Nevertheless, these results cannot be used to explain how a collapse develops.
The perilous rocks situated on steep scarps or cliffs in the eastern Sichuan-Tibet Mountains are cut into crushes by various cracks (Chen et al. 2012), and a field survey indicates that numerous collapse events have occurred along the dominant cracks, which are mostly distributed in the freeze-thaw zone (Fig. 1a). As shown in Fig. 1b, these deep and controlled cracks are often filled by ice or water in different seasons and bear complicated loads with the seasonal fluctuations of combined water and heat. This effect causes large collapses in the late spring or early summer with increasing precipitation and temperature, and small but frequent collapses when the temperature decreases in late autumn.
Many studies have shown that cracks influence rock stability. Panek et al. (2009) noted that the gradual propagation of cut surfaces weakens the rock mass and leads to collapse. Alippi et al. (2010) suggested that rock collapse results from crack development in rocks. However, few studies have elucidated how the cracks in perilous rocks develop and cause collapse.
Other studies have addressed the effects of freeze-thaw action on rock-soil mass. Neaupane and Yamabe (2001) researched the coupled thermo-hydro-mechanical model of frozen rock according to continuum mechanics. Nicholson and Nicholson (2000) analyzed the freeze-thaw damage evolution of rocks. Unfortunately, these studies did not involve the mechanism of collapses with deep cracks in freeze-thaw mountainous settings (Fig. 1b). Fig. 1 Collapse bodies, perilous rock, and cracks in mountains Cracks and seasonal freeze-thaw action are two sides of the same issue regarding rock collapse in mountains. Primary cracks reduce the rock-soil mass strength, provide water channels, and facilitate weathering. The freezing and thawing of water/ice in different seasons, in turn, extend the crack and further reduce the rock stability. The effects of water and temperature on cracks in perilous rocks hence play a significant role in collapse processes within freeze-thaw zones. However, the environmental conditions and seasonal effects of temperature and moisture on rock collapse remain an open question in the eastern Sichuan-Tibet Mountains. Aside from preliminary studies regarding the collapse mechanism at different altitudes (Wu et al. 2020), few investigations have clearly addressed the role of seasonal water/ice in crack development from the viewpoint of fracture mechanics and explained the seasonality of rock collapse.
Hence, the major objective in our paper is to investigate the rock collapse mechanism in the freeze-thaw zone of the eastern Sichuan-Tibet Mountains from a fracture mechanics standpoint by considering the phase change of water under seasonal fluctuations of the surrounding temperature. We analyzed various water-heat combinations in different seasons and developed four models to characterize the driving forces of crack growth, including ice gravity, friction, water hydrostatic pressure, ice-induced stress, and their combinations. On the basis of these models, we calculated stress intensity factors (SIFs) and analyzed crack stability according to fracture mechanics. Our results provide a way to solve crack extension problems and quantitatively study the effects of crack characteristics on rock collapse patterns under different seasonal water-heat combinations. The entire crack propagation process was ultimately simulated using the extended finite element method (XFEM).
We first determined the research area and built models of the crack driving forces in perilous rocks in different water-heat environments based on a field survey. We calculated crack initiation and propagation using XFEM and studied different variations on the crack models to explain the rock collapse mechanism under the seasonal fluctuations of water and heat. The results are applied to practical examples to demonstrate the new theory of rock collapse.

Background
Various hazards including landslides and debris flow are widely distributed in the eastern Sichuan-Tibet Mountains owing to the complicated geology, neotectonic movement, geomorphology, and climate. Among them, rock collapse poses a serious threat to local inhabitants and usually results in rock avalanches/debris flows, which can block rivers and induce floods.

Characteristics of the natural environment
The study area, Maojiagou gully, is located approximately 11.2 km southwest of Kangding town in Sichuan, China. As shown in Fig. 2, many important transport lines, such as the G318 national road and Sichuan-Tibet railway (under construction), pass near this area. The mountain slopes in the study area are very steep, and thus, numerous potential rock detachment zones have developed on both sides of the valley. Maojiagou gully has a subtropical mountain monsoon climate. The local meteorological data show that the maximum and minimum average monthly air temperature values at the gully mouth are 8.3 °C and − 14.1 °C, respectively (3220 m altitude). The annual precipitation is approximately 1 3 810 mm, and nearly 80% occurs in summer from May to September. The Zheduohe River flows through the front of the Maojiagou gully from northwest to southeast.

Geological and slope structure
Two active geologic tectonic structures are present in the study area: the SW-NE-oriented Longmenshan thrust fault and the SE-NW-oriented Xianshuihe strike-slip fault belts, which intersect nearby. Because of vigorous tectonic activity, the rock strata are heavily fragmented and fractures have extensively developed. The exposed lithology within the study area mainly includes weathered granite, metamorphic granite, and plagiogranite with intense disintegration, which enhances the development of high-risk rock zones (Fig. 3a, b). The gully bed is thus vulnerable to valley rock avalanches caused by several large collapses at different altitude positions of the mountain on both valley slopes. At the study point P (29°58′00" N, 101°50′30" E), the elevations of the highest collapse and mountain feet are approximately 4600 and 3580 m, respectively, with an average slope angle of approximately 34.2°. A variety of loose Quaternary accumulation is distributed on the inclinations or foot of the slope.

Character of collapse and accumulation
Through field surveys, we obtained the basic parameters of the collapse: position; slope angle; accumulation volume; and particle size. According to the survey data, the region in which the collapses occur is mainly between the snowline and timberline. In this region, the temperature frequently varies around zero in different seasons, and hence, the freeze-thaw action of the soil-rock body is intensive. Most fractured rocks are easily broken under hydraulic splitting or ice wedging forces with temperature fluctuations.
Rock collapse can occur in this region during any season but is more common in early spring, a rainy summer, and late autumn. Collapses are widely distributed in the freeze-thaw alternation area or near the glacial retreat area caused by global warming. Their accumulations are not thick or far from the source. The particles vary in size, shape, and sorting owing to different formation mechanisms, but most particles are small on the mm to cm scale, aside from some meter-scale boulders.
To determine the characteristics of the collapse particles, we collected accumulation source material in the Maojiagou gully and measured the grain size distribution. As shown in Fig. 3c, the radial dimensions of 91.6% of previous fall particles from in situ collapse accumulations observed on the slope or gully are smaller than 0.25 m, whereas the material with the largest radius (0.6 m) was collected from the most dangerous rock collapse.

Qualitative analysis of collapse in the mountainous freeze-thaw zone during different seasons
The temperature in the mountains (Fig. 4) drops by 0.6 °C with an elevation rise of 100 m (Du et al. 2017), and rainfall increases with increasing altitude over a certain range; therefore, the average monthly air temperature at different altitudes in study area can thus be estimated (Table 1) using the surface temperature data mentioned in Sect. 2.1.  As geologic bodies, perilous rocks contain numerous primary fractures (e.g., cracks, joints, faults), many of which are filled with snow or ice in the high mountains of the study area, as shown in Fig. 5a and b. Because ice mechanics is sensitive to temperature, the stability of ice-filled cracks is influenced by the variation of water-heat combinations at different altitudes and/or in different seasons. The field survey revealed that the fractures mainly distribute on the surface of perilous rocks and penetrate to substantially smaller depths than the climate-influenced depths. Hence, the temperature of the filled cracks is directly affected by the atmosphere and can be assumed to be the same as the local air temperature (Table 1).
Moreover, the temperature of cracks in the study area changes and varies seasonally. At altitudes lower than 3800 m, the temperature typically decreases below zero in cold seasons and rises above zero in warm seasons, which induces serious freezing stress in autumn or winter and high hydrostatic pressure in spring or summer. When the altitude is 3800 m and higher, the temperature is negative year-round and freezing stress dominates the development of perilous rocks.
Data from the field investigation also indicate that seasonal freezing and hydraulic splitting under different water-heat combinations contribute to the collapses in study area. Hence, to illustrate the entire propagation process of cracks in different seasons, the influence of water phase changes at different temperatures on the collapse mechanism in mountains must be considered (Fig. 5c).

Models of crack stress caused by filled water/ice in perilous rock with season variation
For simplicity, we idealize perilous rock as a 2D model in Fig. 6 with a major crack that is l long, w wide, and with an inclination angle θ. The crack mouth is taken as the origin, the x-axis is built along the cracks, and the following assumptions are made.
(a) The water in the crack is incompressible, but the ice is elastic. The constraint provided by the crack is rigid prior to initiation and propagation. (b) The ice in the crack is a rigid elastic solid, which has a high cohesion (0.5-1 MPa) and can form dry calving cliffs or deep cracks with tens of meters in depth in a real iceberg (Bassis and Walker 2011). Hence, the lateral pressure on the crack induced by ice gravity does not to be considered. (c) Because the temperature of filled cracks is observed to be the same as the local air temperature (Sect. 2.4), the water-ice phase change can be assumed to instantaneously occur without considering the hysteresis process of heat conduction. (d) The crack surface is rough, and the friction with the ice induced by ice gravity must be considered, although its influence on cracking is typically less than that of the transverse stress induced by ice expansion. (e) The width w of the crack is substantially smaller than its length l. Hence, ice expansion along the x-axis can be neglected and no shear strain nor friction occurs owing to ice expansion.

Hydrostatic pressure on the surface of a water-filled crack in hot summer
Previous studies revealed that the rockfall frequency significantly increases in mountainous permafrost areas during warm months, which has been attributed to the loss of ice-bonded forces when filling ice melts, and the disappearance of the "adhesion" of the ice to rock has been suggested to cause the subsequent failure (Dramis et al. 1995). However, these studies do not explain how cracks develop and damage accumulates for a rock slope that is stable prior to being filled with ice. Here, we propose that the meltwater of ice during warm weather or abundant rainfall in rainy seasons fills the cracks, which generates hydrostatic pressure and causes hydraulic fracturing. As shown in Fig. 7a, the triangular distribution of hydrostatic pressure acting on a crack surface is: where γ w is the unit weight of water and the maximum value (γ w lcosθ) occurs at the crack tip. We point out that the length of the immersion surface differs on either side of an inclined crack; therefore, real hydrostatic stress on both surfaces is only expected to be similar but not identical. However, in our study, the crack length is substantially longer than its width, and hence, a reasonable approximate value is given as Eq. (1).

Stress on surface of an ice-filled crack in cold winter
In cold winter, water in cracks completely freezes. There are two stresses on the crack surface (Fig. 7b). The first is the gravity component perpendicular to the crack: where p g is the normal pressure on the low surface caused by ice gravity and γ i is the unit weight of ice. The second part is friction on the rough surface induced by the ice gravity component along the x-direction, which can be expressed on the basis of the above assumption as: where f g is the surface friction and μ is the friction factor between ice and rock, which depends on temperature and is < 0.1 for the lubrication of melting ice at high negative temperature (> − 10 °C) (McCarthy et al. 2017). The x-component of ice gravity γ i lcosθ can thus not be balanced by only friction in Eq. (3) and imposes a residual pressure on the crack tip as follows: where p tip is the pressure on the crack tip caused by ice gravity.

Stress on the surface of an ice-water-filled crack upon increasing temperature in warm spring
Upon the arrival of spring, the ice within cracks begins to melt with temperature rise. Meltwater in the upper section of a crack with length l w poses a hydrostatic pressure (Eq. 1) on the crack surface, as shown in Fig. 7c. For the lower crack segment (l i ), the ice gravity results in normal pressure p g and friction f g on the lower surface, as given in Eqs. (2) and (3). The friction f g caused by the normal component of ice gravity in Eq.
(3) cannot usually balance the ice gravity component γ i lcosθ and let alone the maximum hydrostatic pressure imposed on top of the ice column. Hence, residual pressure is generated on the crack tip as: where p tip is the pressure induced by ice gravity on the crack tip. (1)

Stress on the surface of an ice-water-filled crack when the temperature drops in cool autumn
When the crack water begins to freeze with the temperature drop, it expands by a factor of 0.09 upon freezing and can linearly expand to a factor of nearly 0.135 at − 22 °C; therefore, an outward force is generated on crack, which wedges the rock apart (Hallet 2006). Assuming that the crack provides a rigid constraint prior to cracking and ice strain only occurs perpendicular to the crack surface, the freezing stress applied on the crack can be expressed as: where E(t) is the ice elastic modulus (MPa) and expressed as 6600(1-0.012t) (Godbout et al. 2000), ε(t) is the strain, and t is the temperature (°C). Considering that ice exhibits viscous-plastic behavior under high pressure, although it is brittle under most natural conditions, the freezing stress may be overestimated by Eq. (6) and need to be modified with a correction factor (0.185) (Wu et al. 2020). Natural water in cracks usually gradually freezes from the top (crack mouth) to the bottom (crack tip) with decreasing temperature. In most cases (e.g., Fig. 8a), when the weather becomes cool in autumn, a water-filled crack in cold mountains will fill with ice in the upper segment (l i ) and water in the lower segment (l w ).

Stress on the upper segment of the crack surface
Because the normal component of the ice gravity is given as the normal vector to the surface in Eq. (2), the pressure on each side of the crack surface in Fig. 8(b) differs as: Friction on the rough surface is mainly induced by the tangential component (γ i wcosθ) of ice gravity along the x-direction. In most cases, the friction is sufficient to support the ice pillar and can be expressed as: However, when the bearing capacity provided by the surface along the x-direction is less than the gravity component γ i wcosθ, the friction on the surface and excess hydrostatic pressure generated on the water in the lower crack segment are: where p iw is the excess hydrostatic pressure generated by water that, together with friction on the surface, upholds the upper ice.

Stress on the lower segment of the crack surface
On the lower crack surface, not only does hydrostatic pressure occur owing to water gravity (Eq. (1)) but also excess hydrostatic pressure p iw (Eq. (9)). The surface pressure in Fig. 8c is thus: Particularly at the crack tip, the pressure reaches its maximum value of γ w wsinθ + p iw . The temperature fluctuates during not only the transition between seasons but also between day and night. Particularly in warm spring or cool autumn, the temperature can fluctuate daily above or below freezing. The above model includes the process of crack cooling/heating and can be used to illustrate crack propagation with daily temperature changes in the same way.

Mechanism of cracking and its simulation
To study the fracture mechanism of rock collapse in the freeze-thaw zone of the eastern Sichuan-Tibet Mountains using the above models, we review the criteria of crack initiation, study the crack extension, and use XFEM to simulate cracking according to fracture mechanics.

Criteria of crack initiation
In fracture mechanics (Erdogan and Sih 1963), the mode-I and mode-II SIFs, K I and K II , induced by ice gravity, ice expansion forces, friction, and hydrostatic pressure given in Sect. 3 must satisfy the following formulations to maintain crack stability: where K 0 = cos (0.5f ) K I cos 2 (0.5f ) − 1.5K II sin (f ) is a compound value of SIF that considers the contributions of mode-I and mode-II fracturing, K C is the rock fracture toughness, and φ is the fracture angle, which is measured anticlockwise from the original crack direction (Fig. 9).
By solving these formulas, the fracture angle and critical stress that determine the initiation of crack failure can be obtained.

Crack extension (length) after initiation and propagation
Most deep cracks of perilous rocks continue to extend after initiation to propagate and lead to collapse because of increasing water depth and hydrostatic pressure during abundant rainfall, unless the propagation direction or extension path is not suitable for failure.
However, for cracks that initiate to propagate under water-freezing stress, their extension would be easier to terminate with crack opening and ice stress release. Hence, a certain extension length must exist during a continuous cooling process.
In this case, the freezing stress on a crack would continue to decline from the maximum value (Eq. 6) with ongoing propagation. When the stress becomes lower than its critical value, the extension eventually terminates. Thus, the releasing ice strain ice during this process (Fig. 10) can be given as: where σ c1 is the critical stress that determines whether a crack (l 0 ) with a propagation length (Δl 0 ) initiates and propagates, p i and ε 0 are the freezing stress on the crack and free strain of ice after crack propagation, respectively, when the temperature drops from zero to t 0 , and Δε is the release of ice strain during crack opening. Note that the freezing stress in Eq. (6) is a maximum value on the assumption that the crack can provide a rigid constraint prior to initiation. However, when the freezing stress exceeds the critical stress (σ c0 ), the crack begins to propagate and the stress no longer increases. Hence, the work of freezing stress on cracks with unit width (along the horizontal extension direction of the perilous slope) can be calculated as: where σ c0 is the critical stress that causes the initiation of a crack with an original length (l 0 ), and W is the work of the freezing stress.
According to Griffith fracturing theory (Griffith 1921), the extension length of a crack is: where Δl 0 is the propagation length and G C is the energy release rate.

Numerical simulation according to XFEM
Crack loading conditions are complicated in nature, which makes it difficult to analyze the fracture mechanism of rock collapses using an analytical solution. Numerical methods provide a simple way to analyze and visualize the crack propagation process and its impacts on perilous rock stability. For example, FEM is a common way to evaluate SIFs by coupling with J-integral methods. However, the mesh near the crack tip must be refined to conform to the crack geometry, which can generate inaccurate fracture mechanics solutions and high computational costs.
In this study, we use XFEM using high-order terms (Wang and Waisman 2018) to solve our fracture problems, which alleviates the cost of re-meshing with crack propagation and offers a more accurate solution near the crack tip. The Heaviside function L(x) is used to describe the displacement jump on the crack face, and the XFEM displacement approximation u h (x) is applied as described in our previous study (Wu et al. 2020): where u O denoted by N O (x) is the standard finite element shape functions related to the DOF (standard degree of freedom), x represents the spatial coordinate, S L is the node set, which can be entirely split by the crack, S T is node set located at the crack tip to support their basis functions, S is the domain node sets, f a (x) are crack-tip branch functions expressed by (r, θ) terms in polar coordinates taking the tip as the origin, a O and b aO are DOFs of nodes corresponding to functions L and f a , and n is the number of enrichment functions using the near-tip asymptote.

Calculation and analysis
According to the field survey in the study area, we take perilous rock of the most common dimensions as an example to illustrate the fracture mechanism of rock collapse in a freeze-thaw zone. As shown in the 2D model in Fig. 11a, the width (not including the crack width) and height of the perilous rock are 3.7 and 3.5 m, respectively. A crack with a length of 1.3 m and vertical inclination of 10° developed in the upper part of the rock within 2.0 m to the far-right end (point D) of the rock-free face. In the model, all DOFs on the bottom BC and horizontal DOF on boundary AB are fixed.
To quantitatively carry out a parametric study on the SIFs and fracture angles, we model the free face of the perilous rocks as a double broken line ODC in Fig. 11b, where the horizontal inclination of OD is 20° and the vertical inclination of DC is a. The crack length and vertical inclination are expressed as l 0 and θ 0 , respectively. Note that all these parameters are sensitive to the specific calculation purposes described in Sect. 5.4. The stress on the crack induced by hydrostatic or ice expansion pressure can be given according to Eqs. (1)-(10).

Material properties
The samples are medium-fine-grained biotite granite collected from the collapse accumulation site, which is grayish-white and has a blocky structure. As an igneous rock, the samples mainly consist of 41.2% microcline, 22.4% quartz, and 27.3% plagioclase. Experiments such as density tests and MTS (mechanical testing and simulation) tests (Fig. 12) were performed to obtain the rock properties ( Table 2). The standard geometrical dimensions of the specimens used to test the fracture mechanics properties are given in Fig. 12b and c.
According to the Cracked Chevron-Notched Brazilian Disk (CCNBD) tests (Fowell et al. 1995) and the Punch-Through Shear with Confining Pressure (PTS/CP) experiments (Backers and Stephansson 2012), the fracture toughness is given as: where P max is the yield load (kN), P c is the confining pressure (MPa), F max is the peak load (kN), and K IC and K IIC represent the mode-I and mode-II fracture toughness (MPa·m 0.5 ), (16) K IC = 0.10224P max K IIC = 7.74 × 10 −2 F max − 1.80 × 10 −3 P c Fig. 11 Computational analysis model of perilous rocks 1 3 respectively. The crack energy release rate G c is given as K 2 C 1 − v 2 /E, and K C can be obtained in Eq. (11).

Parametric study on fracture characters
Under seasonal fluctuations of water-heat combinations, rock collapse is determined by whether a crack in the perilous rock can fracture, along which direction, and how far it propagates. All of these can be estimated by some fracture factors (e.g., SIFs, fracture angles, propagation length), which are usually affected by the geometry of the crack or perilous rock and ambient temperature variation. Considering the seasonal change of temperature and rainfall, extensive parametric studies including the perilous rock geometry characteristics and ambient temperature (cases 1-7, Table 3) were conducted as follows. a) In the rainy season (late spring or summer), the hydrostatic pressure calculated by Eq. (4) is applied to the water-filled cracks with different lengths l 0 (case A) and vertical inclinations θ 0 (case B), which follows a triangular distribution, and its maximum value is given in Table 4 at the crack tip. b) In the cool season (autumn or early winter), water-freezing stress owing to frost heave is applied on the surfaces. With a temperature drop of 5 °C during a single night from 0 °C, its value given by Eq. (6) is 129.6 MPa regardless of how large the vertical inclinations     In case E, the freezing stress at different length cracks changes with different temperature drops (Δ t) during the seasonal alternations, as shown in Table 5.
Note that the friction along the crack face induced by ice gravity begins to occur in these three cases. Its value on both crack surfaces is nearly the same (38.289 Pa) even in different cases according to Eq. (10) because the freezing stress is substantially higher than the ice gravity in the crack.
On the basis of case E, we also study crack extension (length) according to Eq. (17). In the calculation, the critical stresses σ c0 and σ c1 , which determine whether propagation continues, are obtained by trial. We first assume a propagation length (Δl 0 ) and calculate the critical stress and then obtain the calculated extension length using Eq. (17) and compare it with our assumed value. When the extension length given by the formula is consistent with the trial length, it is taken as the solution.
(c) When the temperature drops 20 °C from 0 °C during autumn, the water gradually freezes and different lengths of ice (l i ) form in the crack of case F. In this case, the freezing stress remains at 198.2 MPa and the water pressure changes according to Eqs. (6) and (7), as listed in Table 6.
In case G, the ice melts gradually in spring. Except for the hydrostatic pressure, stress induced by gravity of the unmelted ice (l i ) calculated by Eqs. (1)-(5) is also generated, as listed in Table 6.
The average static friction coefficient in our calculation model between rock and ice is given as 0.45 according to laboratory tests and literature values (Mamot et al. 2018), without considering its change with temperature or pressure.

Simulation of the dynamic evolution of crack extension and investigation of the collapse mechanism during different seasons
In the freeze-thaw zone of the eastern Sichuan-Tibet Mountains, the crack temperature is directly determined by the surrounding air temperature and fluctuates dramatically with the seasons, resulting in the phase change of water in cracks and breakage of fractured rocks. Hence, the seasonal fluctuation of temperature must be considered in collapse simulations. On the basis of the local meteorological data given in Table 1, a temperature change series (Table 7) is adopted to calculate crack propagation in a realistic perilous rock at an altitude of 3800 m in Fig. 11c, which shows the collapse mechanism in the freeze-thaw zone.

Fracture mechanism of rock collapse induced by the hydrostatic pressure of crack water in rainy seasons
In late spring or hot summer, the ice in cracks of perilous rocks completely melts and the abundant seasonal rainfall leads the hydrostatic pressure to dominate the initiation and propagation of such deep cracks, which leads to collapse events on mountains. In this subsection, we systematically study the influence of crack length (case A) and vertical inclination (case B) on the fracture characteristics of perilous rocks under gravity and hydrostatic pressure (Table 4). Figure 13a depicts the variation of SIFs with crack length. It is clear that longer cracks counteract crack stability. There exists a critical crack length value l c that induces the onset of propagation, which is 5.75 m. Hence, the scale of collapse induced by rainfall in summer can be several meters. Because our model ignores the influence of the mechanical property deterioration of perilous rock during rainfall, the calculated result is slightly larger but nearly agrees with our field study, as discussed in subsect. 2.3.
The effect of hydrostatic pressure is increasingly significant with increasing crack length compared with the contribution of gravity terms to the SIF values. This can be expected because the gravity does not change with crack length, but hydrostatic pressure loading does, which can also be verified by the change of fracture angle in Fig. 13b.
We can also observe how a crack propagates with a mixed mode considering both hydrostatic pressure and gravity loading in Fig. 13b. The crack propagation angle, which ranges from − 34.2° to 22.3°, is initially similar to the gravity contribution and then gradually tends to that of gravity with increasing crack length. More importantly, as shown in Fig. 13c, compared with a shorter water-filled crack, the propagation of a longer crack is more toward the outside of the slope and facilitates collapse. Apart from crack length, crack inclination also strongly affects the fracture character. For a vertical crack (θ 0 = 0) in Fig. 14a, K I achieves its highest value, whereas K II is nearly zero, which indicates the fracture follows a purely tensile mode. However, K II increases as the crack tends inward (θ 0 < 0) or outward (θ 0 > 0), which implies that the role of hydrostatic pressure dominates at low crack angles but is gradually replaced by rock gravity as the angle increases.
The highest compound SIF K 0 , which indicates the most unstable scenario, appears at a fracture angle of 20°. A. negative related fracture angle in Fig. 14b results in larger rock mass collapse, as illustrated in Fig. 14c. However, in this case, the propagation requires more energy owing to its inward direction (i.e., toward the inner slope).
In late spring or summer, fractured rocks with longer and modest outward-inclined cracks are more prone to serious collapse during rainfall. The critical crack length is typically no less than several meters, which indicates a large volume collapse of perilous rocks unless collapse cannot occur under abundant rainfall.

Fracture mechanism of rock collapse induced by freezing stress on a fully ice-filled crack during cooling seasons
In autumn or early winter, freezing stress is generated on a crack face when the water begins to freeze with decreasing temperature, which becomes a new dominant factor that determines whether a crack will propagate and if a collapse will occur. In this subsection, the effects of vertical inclination of the rock-free face (case C), crack width (case D), and different temperature drops (case E) on the fracture characteristics are systematically studied by considering the stresses driving crack evolution such as freezing stresses (Sect. 5.2b, Table 5) and gravity. In Fig. 15a, we illustrate how rock-free surface inclination α influences the fracture character of a frozen crack. The compound SIF K 0 rapidly declines with a reduction of a from positive to negative, indicating that the slope with an inward-inclined face (a > 0) is more prone to fail than one with an outward face (a < 0). The fracture angle (φ) remains positive and is beneficial to collapse during this entire process.
The crack width also influences the fracture character. As shown in Fig. 15b, a wider crack with a smaller slenderness ratio has smaller K 0 and φ values and is comparatively stable.
The SIFs in the ice-filled crack with the temperature drop in autumn are approximately two orders of magnitude larger than those for the water-filled crack in rainy seasons. The critical length calculated for a crack under freezing stress in case E is only 0.016 m, which is substantially smaller than that given in a water-filled crack in Fig. 13a, which indicates that collapses caused by freezing stress would be very minor most of the time compared with rainfall-induced collapses in summer. Figure 16a shows the effect of a temperature drop (Δt) on SIFs when the crack is frozen in autumn. It can be easily concluded that larger Δt from 0 °C are associated with higher SIFs at the tip, regardless of the ice-filled crack length. Hence, a continuous, sharp, widerange cooling process could lead to a large scale but minor collapse of saturated-fractured rocks, such as freeze-thaw weathering in mountains, which explains the character of collapse accumulation and agrees well with our field study discussed in subsection 3.3.
We also study the crack propagation length with different Δt values according to Eqs. (12)-(14). Figure 16b clearly shows that the extension increases for all length cracks in case E with larger Δt, which further confirms that minor collapses (freeze-thaw weathering) are more likely to occur during rapidly cooling weather, such as cold waves.
An important detail to mention is the temperature drop, and freezing stress may more strongly affect the propagation length for longer cracks. As shown Fig. 16b, the extension increases by no less than five orders of magnitude, while the original length of crack varies from 0.01 to 1.3 m, which indicates that propagation during continuous several freeze-thaw Fig. 15 Variation of SIFs and φ with a rock-free surface inclination a (case C) and b crack width w (case D) 1 3 cycles is an accelerated process and that longer cracks become longer. This well explains why fractured rocks containing deep cracks can develop comparatively quickly and are prone to collapse.
In autumn or early winter, longer-thinner ice-filled cracks can easily crack under abrupt temperature drops and lead to fracturing, especially for rocks with an inward-inclined face collapse. The freezing stress is so high that it leads to a very small critical length, which means that nearly all of the visible cracks can be frozen, propagate, and easily cause small collapses, such as erosion.

Fracture characteristics of cracks filled with different ice lengths in autumn or meltwater in spring
The natural variation of temperature is not instantaneous, and the freezing/melting of water in cracks is a progressive process. In this subsection, the effects of ice length (case F) and meltwater length (case G) on the fracture characteristics are systematically studied considering the different stresses during the freezing or melting process (Table 6, Sect. 5.2c) and gravity. Figure 17a shows that the compound SIF K 0 and its individual modes (K I and K II ) on the tip increase with the ice length increment during crack freezing in autumn. This indicates that more complete freezing occurs during a rapid cooling process, which implies that longer ice in cracks can more easily cause collapse.
Note that the compound SIF K 0 is very similar to its individual mode K I but not K II , which means that the freezing stress dominates the cracking. It can be inferred that a purely tensile fracturing mode may nearly occur when the ice in a crack is sufficiently long. For example, the fracture angle sharply declines when the ice is longer than 0.1 m (Fig. 17a).
As shown in Fig. 17b, longer meltwater in a crack is associated with smaller SIFs. Without considering the softening effect of saturated rock, the stress induced by ice gravity rather than hydrostatic pressure is easily sufficient to split a crack. Moreover, the absolute value of a fracture angle increases when the crack ice melts, indicating that the mixed mode of the crack changes gradually from the purely tensile fracturing mode to the purely shear fracturing mode.
In summary, as the temperature continually drops in autumn, cracks extend more easily with the ice growing within them. When the ice melts in warm spring, the cracks tend to be stable if the rainfall supply and rock softening are not considered.

Dynamic evolution of crack extension and collapse mechanism
Realistic fracture models (Fig. 11c) were studied to determine the collapse mechanism of fractured rocks during different seasons in the eastern Sichuan-Tibet Mountains. The load conditions are given by the models in Sect. 3 with the temperature data in Table 7.
As shown in Fig. 18, the original crack is 0.5-m long and filled with water. In autumn of the first year, only the upper 0.2 m of water is frozen. The calculation reveals that when the temperature drops 17.6 °C below zero, the freezing stress and friction on the crack face reach 186 MPa and 217.15 Pa, respectively, which directly causes the crack to propagate. Figure 18a-d shows that stress concentrates at the crack tip and causes 0.995 m of propagation under freezing stress during the first autumn. Note that although the crack grows to 1.495 m, it does not meet the critical length and continues to extend in the following winter, spring, and summer.
Propagation occurs again when autumn returns in the second year. As shown in Fig. 18d, the new crack is almost filled with water except for 0.5 m in the upper section.  When the temperature drops from zero to − 17.6 °C, about 0.3 m of ice forms. As shown in Fig. 18e-i, the crack propagates along nearly a straight line and extends 1.705 m with mode I, which differs from the counterclockwise propagation at the beginning of the first-year autumn. This indicates the overwhelming influence of freezing stress when the crack grows longer. When propagation terminates at the end of the temperature drop in the second-year winter, the final crack length is 3.2 m (Fig. 18i).
The rock damage is aggravated owing to crack growth over the two-year period, and the influence of gravity is enhanced. The calculation shows that the critical crack length decreases to 3.05 m under hydrostatic stress. Hence, in the third-year spring and autumn when the ice melts and precipitation is abundant, the crack fills with water and begins to propagate again. As shown in Fig. 19, the crack initially propagates clockwise with a fracture angle of − 62.2°, which differs substantially from the propagation direction under freezing stress in the previous two years. However, the trend reverses with later crack growth mainly because hydrostatic pressure dominates the fracture. During the final stage, the deformation sharply increases from Fig. 19f to 19g when the crack extends to 5.8 m, and collapse eventually occurs.
In summary, the collapse of fractured rocks in different seasons is the result of the initiation, propagation, and connection of a primary crack under the fluctuation of water-heat combinations. During this process, freezing stress in the cooling seasons is the major driving force that weathers, breaks, and ultimately destroys a fractured rock of any length crack, which is as a precondition and first step to collapse. Hydrostatic pressure in the rainy season exerts a strong promoting factor on long cracks with a sufficiently small critical length and can directly lead to collapse. Rock damage accumulation owing to crack growth can rapidly reduce stability and lead to collapse.

Conclusions
A theoretical model of different stresses that drive fracture propagation corresponding to different seasons is proposed on the basis of a field investigation. These models are used to calculate SIFs, fracture angles, and extension by fracture mechanics, and the effects of crack characteristics under the seasonal combination of water and heat on crack stability and rock collapse patterns are analyzed. Crack propagation is simulated using XFEM to explain the rock collapse mechanism by applying the new theory to practical examples. The conclusions are summarized as follows.
(1) Owing to the fluctuation of water-heat combinations during different seasons, freezing stress and hydrostatic pressure are generated on internal crack surfaces and become two dominant loads that drive crack evolution in cooling and rainy seasons, respectively. (2) Freezing stress dominates, whereas hydrostatic stress is relatively small. The critical length of a frozen crack is thus substantially smaller than that of a water-filled crack.
(3) Almost any longer crack in a natural perilous rock can easily fracture under freezing stress, which mostly results in more frequent, widespread, and small-scale collapses (i.e., rock disintegration) during cooling seasons, although some large collapses can also occur. Conversely, few sufficiently long cracks can propagate under hydrostatic pressure owing to the large critical length, which causes the collapse events in rainy seasons to become volumetrically larger. (4) The extension length of a frozen crack is relatively stable during one continuous cooling process, whereas crack extension during abundant rainfall continues until collapse. (5) Freezing stress in the cooling season is the major driving force that weathers, breaks, and destroys fractured rock with cracks of any length, which is a precondition and first step to collapse. Hydrostatic pressure in the rainy season plays more of a role in promoting long cracks with a critical length and can directly lead to collapse. (6) Fractured rocks with longer and modest outward-inclined cracks are generally more prone to serious collapse during rainfall in late spring or summer. In autumn or early winter, longer and thinner ice-filled cracks more easily crack under the sharp drop of temperature and cause fractured rocks, especially those with inward-inclined faces, to undergo serious collapse. (7) As the temperature continually decreases in autumn, cracks become easier to extend with ice growth. When the ice melts in warm spring, the meltwater is more unlikely to cleave the cracks if rainfall supply and rock softening are not considered. (8) The collapse of fractured rocks in different seasons is the result of the initiation, propagation, and connection of the primary crack under freezing stress or hydrostatic pressure. Collapse is essentially a process of rock damage accumulation owing to crack growth under fluctuating combinations of water and heat.
1 3 relationships that could have appeared to influence the work reported in this paper.