Introduction

Measurement of displacements on slopes has been often adopted to evaluate the instability of the slope or inspect the possibility of an onset of a landslide in practice. It has been recently adopted on the small-scale slopes on which the collapse of surface layer of the slopes might have been expected to occur (Askarinejad et al. 2018; Ochiai et al. 2004, 2021; Crawford et al. 2019). Displacement gauge, extensometer and laser displacement gauge have been mainly adopted on small-scale slopes (Askarinejad et al. 2018; Loew et al. 2017; Ochiai et al. 2004, 2021; Crawford et al. 2019).

It was often reported that the displacements indicated an accelerative increase just prior to a failure of a slope (Askarinejad et al. 2018; Hao et al. 2017; Kroner et al. 2015; Loew et al. 2017; Ochiai et al. 2004, 2021; Sasahara 2021). The accelerative displacement before the failure has been modelled by many researchers, and prediction methods of failure time have been developed based on the model (Fukuzono 1985; Hao et al. 2016; Saito 1965; Saito and Yamada 1973; Varns 1982; Voight 1988, 1989; Xiao et al. 2009). Fukuzono (1985) found the linear relationship on logarithmic scale between the velocity and the acceleration derived from displacement measured on a model slope subjected to sprinkling water as below.

$$\frac{\mathrm{d}v}{\mathrm{d}t}=a\cdot {v}^{\alpha }$$
(1)

where v and t are the velocity and time, respectively. a and α are experimental constants. It can be recognized that a and α represent an intercept on the vertical axis and a gradient of a line for the relationship, respectively. The constants a and α can be derived by the regression analysis of the relation on logarithmic scale. Fukuzono (1985) proposed the procedure for predicting failure time of the slope based on the relationship. Integrating Eq. (1) produces Eq. (2) for the relationship between the time (t) and the velocity (v) as follows:

$$v=\left\{a\left(1-\alpha\right)\right\}^\frac1{1-\alpha}\left(t_r-t\right)^\frac1{1-\alpha}\;\;\;\; \left(\alpha>1\right)$$
(2)

where tr indicates the failure time of a slope. Rearranging Eq. (2) produces an equation for deriving the failure time (tr) of the slope as shown in Eq. (3). Failure time (tr) can be calculated from the constants a and α derived by the procedure as mentioned previously, and present time (t).

$${t}_{{r}}=\frac{{v}^{1-\alpha }}{a\left(\alpha -1\right)}+t$$
(3)

Voight (1988, 1989) examined the relationship not only for soils but also for other materials such as steel and concrete. He also applied the prediction method for failure time not only to landslide but also to other types of failure of geomaterials such as the eruption of volcano. Hao et al. (2016, 2017) proposed a generalized form of the relationship proposed by Fukuzono (1985) and Voight (1988, 1989), and developed the equation from time derivatives to stress derivatives which produces stress–strain relationships. This can predict the influence of applied stress on the failure of the material.

Methods proposed for predicting a failure time of the slope based on the relationship between the velocity and the acceleration adopted the assumption that the displacement proceeded with monotonic increasing velocity as in the tertiary creep stage. While the displacement does not always increase monotonically with accelerating velocity, it shows a series of steep and gentle increase of the displacement with time (Fig. 1) in the slope under a series of rise and lowering of water level in a reservoir (Yao et al. 2015; Yang et al. 2019; Zhou et al. 2018; Li et al. 2018; Miao et al. 2018) and repeated rainfall (Sasahara and Sakai 2017). The displacement can terminate its increase, and the slope does not fail in the case with decelerating velocity, while it increases up to the failure of the slope in the case with accelerating velocity at the final stage of the displacement as in Fig. 1. Predicting failure time can succeed in the case of accelerating velocity of the displacement based on proposed models as above, while it fails in the case of decelerating velocity of the displacement by the models. It is necessary to judge whether the displacement increased significantly up to failure or not to predict a failure time of the slope only based on the measurement of the displacement without shear test for strength parameter. It cannot be judged only from measured displacement at current time because no method has been already established for the judgement only from the displacement data.

Fig. 1
figure 1

Schematic diagram of the time variation of the displacement with accelerating and decelerating velocity

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It was reported that the vertical displacement approached to constant with the shear displacement increase to failure in sandy model slope due to the rise in groundwater level (Sasahara et al. 2014). This was due to the dilatancy of sandy soil under direct shear and the variation in normal displacement converged to zero at steady state, which means to be just prior to failure. It suggests that the variation of normal displacement can be an indicator for a failure of a soil. Measurement of normal displacement can directly suggest an occurrence of a failure without deriving strength constant by indoor shear test before the measurement in actual slope. The condition of the shear and normal displacement development until failure was examined based on 2-dimensional measured displacement data in model slopes under repeated rainfall to examine the possibility of normal displacement against shear displacement as an indicator for a failure of a slope in this paper.

Methodology

Measured displacement data of 2 kinds of model slopes under repeated rainfall, namely H21NIED (Sasahara and Sakai 2014, 2017) and H23NIED model slopes (Sasahara 2022), were examined in this paper. The models were constructed, and the experiments were conducted in a large-scale rainfall simulator at National Research Institute for Earth Science and Disaster Resilience, Japan. They had same dimensions: 600 cm long, 150 cm wide and 50 cm thick in slope section and 300 cm long, 150 cm wide and 50 cm thick in horizontal section as in Figs. 2 and 3. The vertical depth was 57.7 cm in slope section. The slope section had an inclination of 30° from the horizon. Dimensions of the model were almost of the same scale with actual slope failure in Japan (Osanai 2009). The model slopes were constructed in a steel flume with side walls of glass. Vertical blades with the height of 1 cm were placed across the base of the flume at every 50 cm in longitudinal direction to prevent the slippage of the model. The soil was put into the flume and consolidated by human stamping at every 50 cm thickness. The model was made of the granite soil with physical and mechanical properties shown in Table 1. The arrangement of the measurement instruments in the H23NIED model slope is shown in Fig. 2. Extensometer, displacement gauge, water level gauge, tilt meter, soil moisture sensor and tensiometer were installed to measure the surface displacement, normal displacement, groundwater level (GWL), shear strain, volumetric water content (VWC) and suction in the H23NIED model slope, respectively. The extensometer consisted of a rotating sensor fixed at the upper boundary of the flume and invar wire with targets to measure the displacement between the sensor and the target as the surface displacement (Fig. 4). Normal displacement was measured as the distance between the steel plate on the surface and the displacement gauge fixed on the steel bar above the slope (Fig. 4). Measured normal displacement should be corrected to exclude the component which was due to the surface displacement. The correction method is referred to Sasahara and Sakai (2017). The surface displacement (SD), normal displacement (ND) and GWL were analysed. The resolution of the extensometer (CPP-60, Midori Precisions) and the displacement gauge (SDP-100R, Tokyo Sokki, Inc.) was 0.1 mm while that of the water level gauge (TD4310, Toyota Koki, Inc.) was 1 cm. Although soil moisture sensors, tensiometers and tilt meters were also installed, the specification of these devices are just the same with that of H21NIED (Sasahara and Sakai 2014). The time interval for the measurement and record was 10 s. The geometry of the model slope was the same with the same soil, and the arrangement of the measurement devices was a little bit different but the same devices were installed in the H21NIED model slope as explained by Sasahara and Sakai (2014, 2017).

Fig. 2
figure 2

Experimental apparatus and arrangement of measuring instruments in the H23NIED model slope

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Fig. 3
figure 3

H23NIED model slope in a large-scale rainfall simulator at National Research Institute for Earth Sciences and Disaster Resilience, Japan

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Table 1 Physical and mechanical properties of a soil in the H21NIED and H23NIED model slope
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Fig. 4
figure 4

Measurement of surface and normal displacements

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The experimental condition for the H23NIED model slope is shown in Table 2 (Sasahara 2022). Four pre-rainfall events were given to simulate the antecedent rainfalls for natural slopes. Rainfall intensities in pre-rainfall events were 15–25 mm/h, and their duration was around 2 to 4.5 h which was the duration in which the VWC increased and reached to maximum at the deepest soil moisture gauge. The duration between the pre-rainfall events was 3 days to 1 week. The final rainfall event, namely rain 5, with a rainfall intensity of 50 mm/h, continued until the failure of the model slope. These rainfall conditions were decided based on the rainfall conditions of the event which, usually, caused landslides in Japan. The experimental condition for the H21NIED model slope is shown in Sasahara and Sakai (2014, 2017). The series of rainfall events consisted of 3 pre-rainfall events and final rainfall which caused the failure of the model slope for the H21NIED model slope.

Table 2 Experimental condition for the H23NIED model slope (Sasahara 2022)
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Results

Time variation of the displacements in model slopes

Figure 5 shows the time variation of the SD, ND and GWL at 150 cm and 300 cm from the toe of the H21NIED model slope. The SD and ND increased during every rainfall event largely, and these increased slightly after rains 1, 2 and 3. Their increase was almost 0.5–1.0 cm during rains 1 to 3 while it was much larger during rain 4. They increased significantly to failure during rain 4. Their increase was only 0.2–0.4 cm after rains 1, 2 and 3. The displacements increased without the generation of the GWL during rains 1 and 2 while they increased with the rise in the GWL during rains 3 and 4. The GWL was zero before every rainfall event at 150 cm and 300 cm in the model slope.

Fig. 5
figure 5

Time variation of the surface displacement (SD), normal displacement (ND) and groundwater level (GWL) at 150 cm and 300 cm from the toe of the H21NIED model slope

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Figure 6 shows the time variation of the SD, ND and the GWL at 200 cm from the toe of the H23NIED model slope (Sasahara 2022). The SD and ND increased during rains 1, 2 and 5 while they remained almost constant during rains 3 and 4. The increase of the SD during rain 1 was slight, and it increased up to 0.3 cm after rain 1 and was around 1.0 cm during rain 2. The increase of the ND during rain 1 was so slight, and it increased until 0.2 cm after rain 1 and was around 0.6 cm during rain 2. The SD and ND increased significantly until failure during rain 5. The GWL generated during rains 2 to 5. The SD and ND increased under an unsaturated condition during and after rain 1. The increase of the GWL during rains 3 and 4 was almost the same with that during rain 2. It might have been the reason of no progress of the displacements during rains 3 and 4. Larger GWL in rainfall event might have been necessary for the development of the displacements.

Fig. 6
figure 6

Time variation of the surface displacement (SD), normal displacement (ND) and groundwater level (GWL) at 200 cm from the toe of the H23NIED model slope

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Relationship between the SD and the ND

Variation of the displacements could not be clearly shown in graphs for the relationship between the time and the displacements because the SD and ND developed remarkably only within a short time during rainfall event in the model slopes. The development of the SD and ND during final rainfall events was recognized only as a vertical line in Figs. 5 and 6 while the SD versus time was nonlinear as shown in Sasahara (2022). The relationship between the SD and the ND is shown in this section to express the variation of displacements clearly.

Figure 7 shows the relationship between the SD and the ND, the GWL and the velocity of surface displacement at 150 cm and 300 cm from the toe of the H21NIED model slope. The GWL did not rise during rains 1 and 2, and it rose and lowered during rain 3, which could not be recognized well in Fig. 5. The GWL rose significantly at first from 2 cm of SD and then showed a gentler rise with the increase of the SD at both places during rain 4. The GWL seemed not to rise until an ultimate value at 150 cm while it reached to maximum and then lowered a little bit from 31 to 28 cm at 300 cm. The lowering in the GWL might be negligible; thus, the GWL after 5.5 cm in the SD could be regarded as constant at 300 cm. The ND monotonically increased until the failure at 150 cm while it increased and then stayed almost constant at around 7.5 cm after the SD of 22 cm at 300 cm. The velocity of the surface displacement stayed at almost zero until 2 cm of SD at both locations at first, and then significantly increased to around 0.07 cm/s at 4 cm and 0.08 cm/s at 5 cm of the SD with the rise in the GWL at 150 cm and 300 cm, respectively. It showed a series of small increase and decrease from those to 16 cm and 22 cm of the SD at 150 cm and 300 cm, respectively, and finally significantly increased up to failure. This fluctuation might have been recognized to occur during failure because the increase of the GWL to the SD lowered at 150 cm and the GWL reached to maximum before this stage at 300 cm. Thus, 4 cm and 5 cm in the SD (arrows in Fig. 7) could have been recognized as the start of the failure at 150 cm and 300 cm, respectively. The movement of failed mass in the H21NIED model slope might have been restricted by the steep slip surface at the foot of the slope (Sasahara 2022); thus, the failed mass could not have been displaced acceleratively. The final increase in the velocity at both locations was due to the second failure with smaller scale by visual observation. Thus, the ND showed a monotonic increase at both locations before the failure. Figure 8 shows the enlarged section until 3 cm of the relationship between the ND, the GWL and the SD at 300 cm from the toe of the H21NIED model slope, to recognize the behaviour well. It was recognized that the GWL rose up to 12 cm with the increase in the SD from 1.5 to 1.7 cm following rain 3, and then it lowered also with the increase in the SD from 1.7 to 2.0 cm until the start of rain 4. Negative GWL might have been an error due to the disappearance of water around the water level gauge. It was recognized that the ND monotonically increased with the increase of the SD during and after the rise in the GWL following rain 3 and rain 4 before the failure as shown in Figs. 7 and 8.

Fig. 7
figure 7

The relationship between the surface displacement (SD) and the normal displacement (ND), the groundwater level (GWL) and the velocity of surface displacement (v) at 150 cm and 300 cm from the toe of the H21NIED model slope

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Fig. 8
figure 8

The relationship between the surface displacement (SD) and the normal displacement (ND), the groundwater level (GWL) at 300 cm from the toe of the H21NIED model slope enlarged for the surface displacement from 0 to 3 cm

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Figure 9 shows the relationship between the SD and the ND, the GWL and the velocity of the surface displacement at 200 cm from the toe of the H23NIED model slope. The SD developed without the rise in the GWL during rain 1 and with the rise in the GWL during rain 2, and then the GWL lowered without the significant increase in the SD after that. The GWL rose and then lowered with a slight increase in the SD during rains 3 and 4. The GWL rose once at the start of rain 5 with a slight increase in the SD (1st duration in Fig. 9), and then the SD developed under almost constant GWL until the value of 3.4 cm. Finally, the GWL rose significantly and then showed a gentler rise with the SD until failure with 4.3 cm of the SD (2nd duration in Fig. 9). The velocity increased with a slight increase in the SD at duration 1 and showed an accelerative increase at duration 2 just prior to failure. The ND monotonically increased at the start (duration 1) and final stage (duration 2) of rain 5 just prior to failure with the significant rise in the GWL.

Fig. 9
figure 9

The relationship between the surface displacement (SD) and the normal displacement (ND), the groundwater level (GWL) and the velocity of surface displacement (v) at 200 cm from the toe of the H23NIED model slope

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Discussion: condition for the increase of the displacement to failure

Basic concept

Experimental results shown in Figs. 7 and 9 indicated that the ND is not constant just prior to failure on the model slopes against the hypothesis that the ND converged to constant just prior to failure which was the steady state under direct shear condition in the slope. The reason for no-constant ND against the SD is examined in this section based on the simple assumption that sliding mass moves as a rigid body on the slip surface as in Fig. 10. Measured displacements on the slope surface are the SD and ND, respectively. Synthesizing both displacement vectors on the surface results in the synthetic displacement vector (RD vector). The angle between the direction of the slope surface and the RD vector, shown as α, is derived as follows:

Fig. 10
figure 10

Displacements on the slope surface and slip surface when the angle between the horizon and the slope surface is different from that between the horizon and the slip surface.

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Displacements parallel and normal to the slip surface are supposed to be the shear displacement (dS) and normal displacement (dN) on the slip surface. Synthesizing both displacement vectors on the slip surface produces the synthetic displacement (dR) vector on the slip surface.

If the shear deformation on the slip surface is assumed to be a direct shear, the direction of the synthetic displacement (dR) finally coincides with that of the shear displacement (dS) on the slip surface at steady state just prior to failure because the shear displacement (dS) increases significantly while the normal displacement (dN) remains small and constant on the slip surface at steady state just prior to failure. The direction of the synthetic displacement RD on the slope surface coincides with that of dR because the sliding mass moves as a rigid body on the slip surface, and it brings the condition that the angle (θ + α) between the horizon and the synthetic displacement SD is equal to the inclination of the slip surface β as in Fig. 10. This suggests that the angle α converges to (β − θ) with the increase in the SD at steady state as in Fig. 11a. And, the angle of ND against SD is equal to the angle α at steady state according to Eq. (1) as in Fig. 11b. This indicates the normal displacement does not stay constant at steady state but increases linearly with the surface displacement when the direction of slip surface is not parallel to that of the slope surface. The angle α converges to null with the increase in the SD if the direction of the slip surface is equal to that of the slope surface because the inclination of the slip surface β is equal to the angle of the slope surface to the horizon θ. The angle between the direction of the ND and that of the SD is also equal to null in this case.

Fig. 11
figure 11

Relationships between the surface displacement (SD) and the angle α between the slope surface and the synthetic displacement R and between the SD and the normal displacement (ND) when the angle between the horizon and the slope surface is different from that between the horizon and the slip surface. a SD and α. b SD and ND

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The angle between the direction of the slope surface and the synthetic displacement RD

The angle between the direction of the slope surface and the synthetic displacement RD, namely the angle α, was derived by the analysis of the measured displacements following the procedure as in previous section, and the results of the analysis are shown in this section.

Figure 12 shows the relationship between the SD and the GWL and the angle α at 150 cm and 300 cm from the toe of the H21NIED model slope. A very slight increase in the SD was recognized during and after rains 1 and 2 without the generation of the GWL and with the rise in the GWL after rain 3, while the SD significantly increased during rain 4 to failure with a significant rise in the GWL. The SD slightly increased with the significant rise in the GWL at first, and then it became larger with the increase in the GWL during rain 4. The SDs at failure were around 4 cm and 5 cm for 150 cm and 300 cm, respectively. Thus, the SDs before these values are the scope of discussion. Figure 7 indicates that the NDs at 150 cm and 300 cm still increased at failure, which indicates that the slip surface might have inclined against the slope surface as shown in Fig. 11b. The angle α showed a significant increase and decrease during and after rains 1 to 3 until 3 cm of the SD while it showed a slight fluctuation during rain 4 until failure at both locations as in Fig. 12. It slightly increased just prior to failure at both locations, and this slight variation just prior to failure might be able to be recognized as quasi-constant. The angle α between the slip surface and the slope surface was recognized as 15° and 20–25° at 150 cm and 300 cm, respectively.

Fig. 12
figure 12

The relationship between the surface displacement (SD) and the groundwater level (GWL) and the angle α between the direction of the synthesis displacement and the slope surface at 150 cm and 300 cm from the toe of the H21NIED model slope

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Figure 13 shows the relationship between the SD and the GWL and the angle α between the direction of the synthesis displacement and the slope surface during rain 5 at 200 cm from the toe of the H23NIED model slope. The GWL significantly increased at the start of rain 5 (duration 1) with a slight increase in the SD and before failure (duration 2) with a significant increase in the SD. The angle α monotonically decreased from the start of rain 5 including duration 1, and it stayed constant at duration 2 just prior to failure as in Fig. 13. The angle α between the slip surface and the slope surface was recognized as around 12°.

Fig. 13
figure 13

The relationship between the surface displacement (SD) and the groundwater level (GWL) and the angle α between the direction of the synthesis displacement and the slope surface during rain 5 at 200 cm from the toe of the H23NIED model slope

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These facts indicate the normal displacement almost converged constant at steady state (just prior to failure) on the slip surface in the H21NIED and H23NIED model slope. The slip surface was estimated to be inclined against the slope surface in both model slopes.

Conclusion

Measurement of the displacement parallel and normal to the model slope surface was implemented on the model slopes under repeated rainfall, and the measured displacement was analysed to examine the mechanical condition which causes accelerating displacement to the failure of the slope based only on the measured displacements on the slope. The following results were derived from the analysis of measured data:

  1. 1.

    The measured normal displacement approached to constant at a point but did not converge to constant at 2 points on the model slope under repeated rainfall just prior to failure. These results were against the idea that the normal displacement approached to constant at steady state in direct shear condition on the slip surface parallel to the slope surface.

  2. 2.

    The reason of the variation in the normal displacement just prior to failure was examined. It was supposed to be due to the difference in the direction of the slip surface to the slope surface. Some consideration with simple assumption showed that normal displacement monotonically increased on slope surface just prior to failure if the slip surface inclines against the slope surface. The angle α between the direction of slope surface and that of synthetic displacement on the surface was introduced instead of the normal displacement to explain the variation of normal displacement with the increase of shear displacement on the slip surface in direct shear condition just prior to failure. No variation of the angle with the surface displacement increase just prior to failure indicates constant normal displacement on the slip surface just prior to failure which occurred in steady state.

  3. 3.

    The angle α indicated a very slight variation with the surface displacement increase just prior to failure in the H21NIED model slope and constant with the surface displacement increase just prior to failure in the H23NIED model slope. The angle was supposed to be quasi-constant just prior to failure in the H21NIED model slope. These indicated the normal displacement on the slip surface converged to almost constant with the surface displacement increase just prior to failure under artificial rainfall.

  4. 4.

    The result indicated the angle α converged to constant just prior to failure in the model slope. It suggested that the normal displacement on the slip surface converged to constant just prior to failure, which meant the steady state. The angle α converged to constant just prior to failure and can be an indicator to judge whether the surface displacement increased significantly until failure or not.