ABSTRACT
An experimental program was designed to study the behavior of unsaturated soil at low degree of saturation and to verify theoretical models. Two types of test were performed on unsaturated soil. These tests include: (1) constant normal stress and many tests with different degrees of saturation, (2) constant degree of saturation and many tests with different normal stress. In both types of tests the air pressure was allowed to drain but water pressure was held in. Also many tests were conducted on unsaturated soil to show the effect of particle size. Tests carried out on fully saturated samples to determine the effective angle of friction.
It was found that as the degree of saturation increases from 2% up to 40% the shear strength increases and then it decreased as the saturation level increases. All the measured values of unsaturated shear strength have been compared with predicted values from the theoretical model that was suggested by Maaitah (2000). Predicted and measured values were in good agreement. Furthermore, it was found that there is no unique value for the ratio, w, and it is a function to soil type and to the degree of saturation.
It has been found that unsaturated critical state lines (CSL) are parallel to the saturated lines where the slope, M, is equal to 1.18 and independent of the saturation state. On contrast, the intercept, qo, of the CSL for unsaturated soil on the (q:p) plane as a function of saturation state.
Keywords: Degree of saturation, critical state, unsaturated soil, shear strength, predicted and measured strength
INTRODUCTION
Soils in Jordan are mostly in an unsaturated state since the rainfall is low in the majority of the country. Soil in such environment has apparent strength due to negative pore water pressure, which varies between summer and winter. In wet areas, the same situation exists as a result of the applied suction to the soil from vegetation. Environment plays a significant role in the soil conditions. Many problems involve the presence of partially saturated zones, the most common case being that of a capillary zone above the groundwater table. These zones are usually ignored in practice, and the soil is assumed to be either fully saturated or completely dry. This simplification is in many cases conservative since it does not take account of the increase in shear strength due to partial saturation. However, features of partially saturated soil behavior, which can have an impact on the global behavior, such as collapse upon wetting, cannot be reproduced.
From construction prospective, the problem in unsaturated soil is not associated with the shear strength at certain degrees of saturation, but arises from changing in shear strength due to change in the degree of saturation. To determine the bearing capacity of shallow foundations, slope stability, and many other geotechnical structures a full picture is needed to understand the behavior of unsaturated soil.
In addition, to the fact that compacted soils which is used in earth structures are in the unsaturated state during construction and during operating conditions. Their behavior is thus affected by the simultaneous presence of water and air in the pore spaces, which makes the pore fluid mixture compressible and influences the stress state through the water and air pressures
The key of valuable prediction into the behavior of unsaturated soil is to understand the real force and its direction that acts between the soil particles. The main aim of this paper is to verify the theoretical model that suggested by Maaitah (2002) to predict the shear strength of unsaturated soil using different soil types. On the second hand, this paper would examine the behavior of unsaturated soil against the critical state models that were suggested in recent years (Wheeler and Sivakumar 1995; Alonso et al. 1990).
THEORETICAL BASIS
The critical sate models are an attempt to expand the Cam Clay model that is suggested for saturated soil. The critical state frameworks for unsaturated soil mechanics was proposed and compared with those of saturated soil mechanics (Sun et al. 2000; Wheeler and Sivakumar 1995; Maatouk et al. 1995). The smooth transition from the stress state variables for an unsaturated soil, (s - ua) and (ua - uw) to the single stress variable (si - uw) for a saturated soil (Fredlund and Morgenstern 1977), forms the basis for this extension. The concepts of yielding, hardening, and critical state are the key elements comprising the critical state framework for saturated soils. These concepts can be extended using two independent stress state variables, (s - ua) and (ua - uw), to build a critical state framework for unsaturated soils. The critical state equation that suggested by Alonso et al. 1990 can be written as follows:
q = Mp + qo(s) | (1a) |
where
In contrast, Toll (1999) proposed that the critical state for unsaturated soil could be expressed in terms of q, p-ua , and ua-uw . The unsaturated critical state requires five parameters, namely, p, ua , uw , Ma , and Mb. The deviator stress was expressed as follows:
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(1b) |
Note that Ma and Mb is the slope of the CSL where the contributions due to unsaturated state have been labeled Mb and the contribution of friction is labeled Ma.
Existing unsaturated elasto-plastic stress-strain models tend not to provide information on the variation of degree of saturation, Sr , which is essential for coupled flow-deformation analyses (Gallipoli et. al., 2003). Numerical analyses performed to date with unsaturated elasto-plastic stress-strain models (e.g. Gens et al., 1995; Thomas and He, 1998) have generally overcome this by assuming either a unique water retention curve. The latter is relating degree of saturation solely to suction or a unique ‘state surface’ expression, relating degree of saturation to suction and mean net stress.
Gallipoli et. al. (2003) stated that “in practice, the relationship between degree of saturation and suction for a given soil will be non-unique for at least two reasons:
(a) In a deformable soil, variation of the void ratio produces changes in the dimensions of voids and of connecting passageways between voids, which cause corresponding changes in the water-retention curve.
(b) Even in a rigid soil, the occurrence of ‘hydraulic hysteresis’ during inflow and outflow of water to individual voids means that retention curves followed during wetting and drying are different.
Maaitah (2002) suggested a theoretical model to predict the increase in shear strength of unsaturated soil due to confining stress, suction and surface tension based on the pore and meniscus geometry. The model assumed the meniscus shape is parabolic. The determination of contact angle, b, is based on mathematical formula. The increase in strength was calculated on the basis of measuring particle size, void ratio, e, degree of saturation, Sr , and surface tension, T, and other meniscus properties. The model is able to estimate the increase in contact stress as the degree of saturation changes from zero to 30% for any void ratio. The shear strength was expressed as follows:
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(2a) |
Where the Ds is the contribution of surface tension and suction in shear strength which he stated as follows:
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(2b) |
However, in real soil, it is impossible to find mono size and spherical particles as he suggested. Therefore, equation 2b should be reproduced to introduce the effect of difference among the particle size in soil. This can be addressed as a correction factor for grading and particle shape. The correction factor for the shape needs more investigation for different soil. This factor, y, can be expressed as a ratio of the perimeter of greater particle size and the smaller size as follows:
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(2c) |
The correction factor y for the entire groups that tested in this paper is presented in Table 1. It is clear that as the variation between the particle size increases the factor y increases. This means that the number of contact increases and the component of contact force per unit volume increases. This explains why the apparent strength in unsaturated fine material is higher than the coarse one.
Table 1. Correction factor for different particle size and void ratio
according to suggested equation (2c).
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e | N | y | |
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Diameter vary (0.074-0.002) | Diameter vary (0.074-0.15 | ||
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0.68 | 7.84 | 29.73214286 | 1.628861004 |
0.75 | 7.28 | 32.01923077 | 1.754158004 |
0.8 | 6.88 | 33.88081395 | 1.856143935 |
0.9 | 6.08 | 38.33881579 | 2.1003734 |
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And the s/3 is the effect of net confining stress in increasing the shear strength which can be expressed as follows:
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(2d) |
The contact angle b can be calculated from the following formula
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(2e) |
The equation 2a can be re-written as follows
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(3) |
where
The behavior of unsaturated soil is in accordance with the above equation by which allows the understanding of how a water-air meniscus affects the “stress state” of two soil particle. The meniscus water at the particle contact induces a stress (Ds) normal to the plane passing through the contact point and orthogonal to the line connecting the particle centers. This stress is the only one arising from menisci water, even in the case of more complex particle shapes, and increases as suction increases (e.g. degree of saturation). Thus the effects of suction result in a greater normal stress holding the particle together and, if their contact performs frictionally, greater limiting slippage strength. It follows that unsaturated aggregates of spheres show a stiffer and more resistant load response with respect to that of dry contacts or submerged particle. This beneficial effect does not increase indefinitely, however, since Ds tends towards a limiting value due to the progressive reduction in the meniscus radius as suction increases.
MATERIAL AND TESTING
Direct shear tests have been conducted on unsaturated soil with specific soil particles. The specific gravity of soil particles is 2.69. The surface tension of water during testing is of 0.074 Nm-1. Two stages of the test in this work were made. In the first stage of the experimental program; three consolidated drained tests on saturated samples were conducted to determine the effective friction angle f/ to be used in the prediction of the shear strength of unsaturated soil. The second stage of the testing program can be classified into three groups according to the magnitude of applied normal stress, degree of saturation, particle size, and void ratio as illustrated in Table 2.
Table 2. Forty-eight tests were conducted on unsaturated sample
with initial void ratio of 0.68 and the particle diameters vary 0.074 to 0.002 mm.
First Group
Thirty-six tests were conducted with initial degree of saturation (2, 5, 10, 15, 20, 25, 30, 40,50, 60, 70, and 80)% for each test. Each test was repeated for three different cell pressures 10, 50 and 100 kPa. In all these tests the void ratio was 0.68 and the effective particle radius was 0.001 mm (the diameter of particles varies between 0.074 mm and 0.002 mm).
Second Group
Sixteen tests were conducted with initial void ratio 0.68, 0.75, 0.8, and 0.9 and the initial degree of saturation (2, 5, 10, 15, 20, 25, 30, 40, 50, 60, 70, and 80 percent) for each test. Each test was repeated for two different cell pressures, 10 and 50 kPa. In all these tests the effective particle radius is 0.001 mm (the diameter of particles varies between 0.074 mm and 0.002 mm).
Third Group
Seven tests were conducted with initial degree of saturation for each test (2, 5, 10, 15, 20, and 25) percent. Each test was conducted at constant cell pressure equal to 10 kPa. In all these tests the void ratio is 0.68 and the effective particle radius is 0.001 mm (the diameter of particles varies between 0.15 mm and 0.002 mm).
In the second stage of the experimental work, the soil particles that were selected for the laboratory-testing program were silty sand with significant amount of clay as demonstrated in table 2. Specimens were prepared by compacting the soil using direct shear box. Air-dried soil passing through a sieve No.200 was mixed with distilled, deaired water to make the sample homogenous.
Choosing the low stress level was due to considerable amount of which is carried out at relatively shallow depths where soil is likely to be unsaturated and subjected to low stress levels. One particular example is the installation or replacement of underground of pipes, and pavement.
The experiments in this work were conducted under drained conditions to maintain the pore air pressure is atmospheric. In addition, the tests performed under low saturation, which means that the water is held around the contact point and will not drain due to gravity (i.e. capillary forces is greater than gravity forces). This is the main reason for choosing simple direct shear test. For low degrees of saturation and normal stress, the rise in air pressure during shearing is very low and can be ignored.
During shearing no water was added to the top of the specimen to ensure that degree of saturation was maintained. At low saturation, the water cannot be drained because the capillary force is greater than gravity force. The unsaturated soil will suck water from the porous stone disc, which will change the degree of saturation. In order to control the degree of saturation the sample was sealed by a glass cap at the bottom of the box rather than a porous disc.
Mixing the material with distilled water until the sample reached the desired degree of saturation. A high level of care was taken in the preparation of the sample in order to determine, as accurately as possible, the initial void ratio and the exact initial degree of saturation. In order to achieve the accuracy in the initial void ratio, the sample weight and volume were measured before and after the sample preparation. The target and actual void ratio values were recorded in most of the tests. The maximum difference between the target and actual initial void ratio was found to be between 1 and 1.5%. The differences between the target and the measured initial degree of saturation, was less than 0.4%. The target void ratio of 0.91 is equivalent to open packing and the number of contact per unit area is equal to 6, but the measured values vary between 0.89 and 0.92. The target void ratio of 0.65 is equivalent to close packing, but the measured values vary between 0.63 and 0.67. This little variation on the initial void ratio has no significant effect on the contact number per unit area or on the co-ordination number.
Specimens prepared in this way should have had a relatively simple structure and stress history. The definition of critical state used herein is a condition of constant volume, and constant shear strength that a soil reaches. The deviator stress increases with strain and reaches a maximum or stable value at an axial strain range between 10 and 15%, indicating a critical state.
A decrease in soil saturation increased the shear strength. However, decreases in soil saturation did not affect the general shape of the stress-strain relationship. The shapes of the stress-strain curves for the unsaturated specimens are similar to those for saturated specimens. Similar to the saturated soil specimens, the shear stress of the unsaturated specimens stabilizes at an axial strain between 10 and 15%.
DISCUSSION OF THE TEST RESULTS
AND THEORETICAL VERIFICATION
The meniscus shape and size is the key of understanding the behavior of unsaturated soil. Therefore, it is important to discuss the model parameters that suggested by Maaitah (2002) in the light of result that is found in this paper. One of the most important parameters is the ratio, w, which plays an important role in determining the shape of the meniscus as the degree of saturation and particle arrangement changes. The ratio, w, varies between 0.5 and 1. If the ratio, w, is greater than 0.9, the meniscus will be nearly flat which means that the saturation state has been reached. On the other hand, if the ratio, w, is smaller than 0.5 the meniscus is too sharp which means that there is a contradiction with well-known facts “the water will not take the smallest size due to surface tension” (e.g. the drop of water is spherical). Then, it is not realistic in both cases (w > 0.9 and w < 0.5) according to the microscopic observation and to the well-known facts. Therefore, the ratio, w, should be greater than 0.5 and smaller than 0.9.
It is found that the more reliable ratio, w, for silty sand is 0.7 whereas for soil with significant amount of clay is nearly 0.6 as shown in Figure 1a. Figure 1b shows the relation between the contact angle, and the ratio w, which this also is illustrated in Equation 2e. Figure 1c presents the influence of degree of saturation on the ratio, w, which already affect the meniscus shape and then the shear strength. To conclude, it is difficult to find a unique value for the ratio, w, which means that it is function of soils kind and to the degree of saturation.
Figure 1a. Effect of the ratio, w, on predicting the unsaturated shear strength
Figure 1b. Relationship between the ratio, w, and the contact angle
Figure 1c. Relationship between the ratio, w, and the degree of saturation
The increase of the soil strength as stated in Equation 2b can be predicted for a given value of water angle, q, and appropriate value of the ratio, w. Thus the volume of water and water angle can be derived over a range of values of void ratio. Predicting Ds from Sr requires a trial and error method to determine the value of Ds to give the correct value of Sr , which can be performed by simple worksheet (Excel). Table 3 shows an example of calculation (as a sequence steps). Comparison between the predicted and measured unsaturated shear strength is presented in Figure 2 (bold line is using to present predicted value). The test that used in the comparison was group 1 that stated on table 2. Also, Figure 2 shows that as confining stress and the degree of saturation increase the shear strength increase. It is obvious that there is a good agreement between the measured and predicted values. Figures 3 and 4 present the effect of void ratio on predicting the unsaturated shear strength. The test that used in the comparison was group 2. It is clear that as the void ratio decreases and the confining stress increases the shear strength increases. This can be explained as the void ratio decreases the contact number increases and the number of capillary forces increases per unit area and then the contact stress and the shear strength increases.
Table 3. Example of theoretical calculation based on Maaitah (2002) model
Figure 2. Comparison between the predicted and measured unsaturated shear strength
At low degrees of saturation the shape of water at points of contact between the soil particles is assumed to be the same everywhere in the sample if the soil sample consists of uniform radius spheres arranged in systematic way. The pore radius is then similar throughout the soil sample. In reality this is not the case, and large pores will loose water faster than small pores. It is well-known fact that the greater effect on behavior of unsaturated media (i.e. capillary) is caused by smaller pores; hence, in unsaturated soil the finer particles dominate the behavior due to the capillarity effect. Soil consists of different particle shapes and sizes; hence, the pore throat has a different cross section in the soil profile. Consequently, the degrees of saturation and suction have a different value for each pore. This is why the smallest particles radius has been used to predict the unsaturated shear strength of soil containing different sizes in this paper.
Figure 3. Effect of void ratio on predicting the unsaturated shear strength
Figure 4. Effect of void ratio on predicting the unsaturated shear strength
Figure 5. Effect of particle size on predicting the unsaturated shear strength
Figure 5 stated the influence of particle size on predicting the unsaturated shear strength. The test that used in the comparison was groups 1, 2 and 3. It is obvious from Equation 2-b the effect of the particle radius on the predicted shear strength. Particle size has greater influence on the behavior of unsaturated soil than any other factor. This is because the particle size determine the meniscus radius and then the capillary force and number of contact per unit area. A result of this paper has shown that the force induced by non-saturation increases as the particle size, and hence void size, of soil decreases. Also, for low stress level (low normal stress) will usually be less than 100 kPa, and relatively low degree of saturation are likely to have an great effect on the soil behavior. In the light of Figures 5, it can be concluded that there is a good agreement between measured and predicted shear strength.
Figure 6 shows the relationship between degree of saturation and shear strength. It is clear that as the degree of saturation increases from 2% up to 40% the shear strength increases and then it decreased as the saturation increases. This is due to the reduction in the capillary forces. It is undisputable that the capillary demolish when the fully saturation state reach. The gas occurs as discrete bubbles if the degree of saturation between 40% to 65. The strength may rise at this range of saturation as a result of gas bubble effect.
Figure 6. Relationship between degree of saturation and shear strength
The shear strength of unsaturated soil, teq , values measured at a constant mean net stress show an increase with saturation for both optimum and wet of optimum compacted materials (Figure 6). It can be explained by considering that two types of water can be present in the pores of an unsaturated soil: bulk water, completely filling some voids, and menisci water. Thus the teq-Sr in Figure 6 can be subdivided into three different zones:
In zone 1, for low saturation (e.g. suction values high enough) to allow menisci water to have a dominant influence, suction-change effects conform to those expected on the basis of the Equations (2b and 3). The water phase in this zone is discontinuous which exist at the point of contact only whereas the air phase is continuous. Thus, the initial shear strength increases with suction at an initially fast rate and then tends toward a threshold value. This complies with the trend of compressibility from the Alonso et al. (1990) model and with the results already shown in the large strain range. Molding water content seems to cause some significant differences in soil response. This variable affects the ratio between the saturated value and the unsaturated threshold values of shear strength, and the suction values characterizing the transition between bulk-water regulated behavior and menisci-water regulated behavior. This latter point can be explained on the basis of intuitive physical considerations. In fact, changes in soil fabric due to an increase in molding water content induce smaller pore sizes in the wet material with respect to the optimum soil, even if the overall porosity follows the opposite trend. This is also suggested by the characteristic curves of compacted soils from Vanapalli et al. (1996), who observed that optimum compaction leads to a lower air-entry value than wet compaction.
In zone 2, i.e., intermediate saturation or suction values, the amount of air present in the pore voids becomes ever more significant as suction increases. The air and water phases in this zone are nearly continuous. The shear strength is approximately constant. The water at adjacent point of contact will be touching each other and the air must come into contact with the soil particles, and will affect the shear strength. In reality, this situation is not sustainable, as the air will diffuse and collect in a zone of soil with a low degree of saturation. Of more practical significance is the state where water exists only at particle contacts. A progressive shift of the soil response occurs from bulk-water regulated behavior to menisci-water regulated behavior.
The behavior of soils containing discrete bubbles can vary considerably, depending on the relative sizes of the bubbles and the soil particles. As the bubbles being in contact between any two or more particle the surface tension of these bubbles will pull the soil particle together and then causes of higher strength. It is difficult to model such phenomena mathematically due to the complexity of the pore geometry and the shape and size of the gas bubble. On the other hand, the gas bubble is affected by the applied stress. It is well known that at high pressure these bubble goes into solution and the fully saturation state is reached.
Zone 3 finished up in saturated conditions at null suction and is restricted to low suction values (Sr > 80%). Herein, bulk-water effects dominate the soil behavior, since the amount of air present in the specimens is negligible. In this zone, variations in Sr are practically equivalent to changes in mean effective stress. The gradient of the teq-Sr relationship is then expected to be the same as that of the saturated material, calculated at a mean effective.
In addition, Figure 6 shows the effect of confining stress on shear strength. However, the question is what is the real pore pressure within the voids. The assumption that the pore air pressure is equal to atmospheric pressure may be considered true for drained tests under the condition that the pore air is continuous (i.e. Sr < 40%) and is in contact with atmospheric air pressure. The pore air is discontinuous when the saturation exceeds 40%, which means that the pores have no contact with atmospheric air, so this the above assumption becomes invalid. Also, in the undrained condition the pore air has a higher pressure than atmospheric air.
Figure 7a. p-q diagram for unsaturated sample (R = 0.001 mm, e = 0.68)
Figure 7b. p-q diagram for unsaturated sample (R = 0.001 mm, e = 0.68)
The second part of this research is to analysis the result of the experiment in the light of the knowledge of critical state model as stated in Equation 1a and 1b. Figures 7a and 7b present the implementation of critical state model on unsaturated tests that have been conducted as a part of this paper. These test results seem to support that the net mean stress, p, and deviator stress, q, can be used as critical state variables for unsaturated soils. The variables p, and q, has been suggested as the critical state variables for unsaturated soils by several researchers (Wheeler and Sivakumar 1995; Maatouk et al. 1995; Rampino et al. 1998; and Adams and Wulfsohn 1997). Figures 7a, 7b and 8 show the CSLs on the (q:p) plane. The unsaturated critical state lines are parallel to the saturated lines where the slope is equal to 1.18. However, laboratory tests to determine the critical state behaviors of unsaturated soils are still limited in number due to the complex and time-consuming nature associated with testing of such soils.
Figures 7a, 7b and 8 consolidate the Equation 1a and 1b. The slope of the CSL on the (q:p) plane, M, is independent of degree of saturation (or matric suction) and can be determined from the saturated critical state line. This term belongs to the friction between the soil particles and is considered as material characteristics. Also, this term may be function of the same parameters that affects the friction angle. The intercepts qo(s) varies with degree of saturation as shown on Figure 8. The critical-state line in unsaturated case can be proposed as follows:
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(4) |
Where y is a function of pore geometry, meniscus properties, surface tension, and particle size distribution, number of contact per unit area and contact angle.
Figure 8. Variation of the intercept with degree of saturation
(R = 0.001 mm, e = 0.68)
Equations 1a and 1b has simpler forms because this equation is proposed based on the testing of specimens with a relatively simple soil fabric and stress history. M or Ma may be independent on degree of saturation (or matric suction) in other cases of more complex soil failure.
The findings in this paper confirm that the effective friction angle f/ is considered as material properties which means that it is not affected by saturation state. This can be clarified because of the slope of CSL related to the friction angle f/. This leads to be used in the prediction for unsaturated soil strength models. Fredlund and Rahardjo (1993), Wheeler and Sivakumar (1995); Alonso et al. (1990) and many other researchers have used the effective friction angle in their models to predict the unsaturated shear strength. The angle of friction, f/, then can be utilized in the unsaturated test, if just the void ratio was identical with saturated samples.
Finally, the mechanical behavior of a granular material is governed by its structure and the effective stress applied to it. Structure is considered to be function of the packing of particles, and giving overall density. Factor that affect structure are the particle size and shape distributions and the arrangement of grain contact. To discuss particle shape, however, the concepts of angularity, roundness and roughness are vague and it is evident that there is some difference in the use of these terms on a day-to-day basis. This has generated the search for a more quantitative method for measurement particle shape.
However, Maaitah model can provide more understanding of the behavior of unsaturated soil than the critical state model. In the light of the findings result that showed in the figures can prove the validity of model for different void ratios and degrees of saturation. The diversity in the test results could be explained due to differences in the soil fabric, size and shape of the soil particle, and the pore throat. The soil fabric resulting from compaction adds a complexity to isolating the fundamental shear strength component related to matric suction (Mancuso et al. 2000). The complex soil fabric also makes it difficult to determine whether a true critical state has been reached or not (Wheeler and Sivakumar 1995; Toll 1990). As a result, there remains a need to carry on additional laboratory research studies on unsaturated soils, particularly to obtain a fundamental understanding of the shear strength and critical state of unsaturated soils by testing soil specimens with a simple structure and stress history.
CONCLUSIONS
According to laboratory actual volume the following can be concluded in this research:
Finally, the behavior of unsaturated soil can be divided into three stages. The first one is at low saturation, which is governed by the meniscus properties and the effect of surface tension. The second one is dominated by the effect of gas bubble. The third stage is at high saturation, which can be understood by classical soil mechanics.
Rainfall induced instabilities in partially saturated slopes have been reported widely in the literature (e.g. Henkel and Skempton 1955; Brand et al 1984).
REFERENCES
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