Introduction

The emergence of unsaturated soil mechanics in the last three decades attests to the amount of geotechnical applications in the partially saturated zone. Contrast to saturated soils, distinct volume change and shear strength variation are exhibited when soils are partially saturated. This has been the major challenge to researchers in developing a constitutive model consistent with both saturated and unsaturated soils. Therefore, the understanding of volume change and shear strength behaviour of unsaturated soils is still being developed.

Shear strength characteristics of unsaturated soils are highly dependent on suction and the confining stress, and they vary with the variation of suction within the soil mass and the stress applied. Generally, the stiffness and the effective cohesion of unsaturated soils increase with increasing suction. Under the influence of decreasing suction, compacted unsaturated soils may suffer loss of strength and large deformations, leading to catastrophic failures of engineering structures.

Wheeler and Sivakumar [27], Rahardjo et al. [19], Oh et al. [18], Kim et al.[10] and Zhai et al.[28] among many others, experimentally verified that the shear strength of unsaturated soils increases with increasing suction. They also found that the shear strength of unsaturated soils increases with increasing confinement as a result of hardening due to isotropic compression to higher mean net stresses. As per the literature, the change of volume during shearing does not show any specific trend with varying moisture contents or varying confinements. Sivakumar et al. [22] studied the effect of stress-induced anisotropy on the shear behaviour of compacted unsaturated soils, and their results showed that the reduction in specific volume of isotropically compacted soil during shearing increased with confining pressure. However, test specimens prepared by one-dimensional compaction did not show any pattern for the reduction in specific volume with increasing confining pressure. By contrast, the results reported by Zhang and Li [29] and Liu and Peng [14] from their constant moisture content triaxial test programs on locally available silts, showed an increasing trend for the change of specific volume during shearing with increasing moisture content. Rahardjo et al. [19] reported that the volume change of a compacted unsaturated specimen during shearing is more complicated than its saturated counterpart.

With the identification of these various behavioural patterns under the influence of deviatoric stress, significant attention was gained for the development of constitutive models that can satisfactorily predict the deviatoric behaviour of unsaturated soils. Similar to the volume change behaviour, at first the shear strength of unsaturated soils was defined by combining the two constitutive variables. Later, with the introduction of elasto-plastic constitutive modelling, shear strength behaviour was attributed to constitutive models by translating them to more general elasto-plastic critical state frameworks. Barcelona Basic Model (BBM) proposed by Alonso et al. [2] was the first constitutive model to link volume change and shear strength behaviour. This model considers the volume change due to suction and volume change due to stress separately. Some of the other constitutive models developed after BBM followed the same approach [4, 6], while some researchers used the effective stress concept [3, 16, 17] and isotropic stress states [20].

Mostly constant suction testing had been used to verify these phenomenological behavioural patterns as most of the models utilise suction as a state variable. However, suction-based testing is time consuming and requires specific experimental set-ups and expertise. More recently number of researchers conducted constant water content (CW) tests with suction measurements to investigate shear strength and pore-water pressure characteristics [19, 23, 25]. They showed that suction and net stress are influential in controlling the shear strength, pore-water pressure and volume change in unsaturated soils. These experimental works were followed by other researchers to utilise alternative state variables in modelling unsaturated soil behaviour in order to overcome the problems associated with suction. One such model is the Monash-Peradeniya-Kodikara (MPK) framework for volumetric constitutive behaviour of compacted unsaturated soils introduced by Kodikara [11]. The model has shown promising results in the volumetric space where the model was validated for 1-D stress conditions by Islam and Kodikara [9]; Kodikara et al. [12] and isotropic stress conditions by Abeyrathne et al. [1]. The model utilises specific water volume and mean set stress as the state variables. The use of specific water volume has made the model simple and easily applicable in engineering practice. Furthermore, recently, Kodikara et al. [13] postulated the MPK generalised model incorporating mechanical behaviour through environmentally stabilised line and the hydraulic behaviour through constant plastic volumetric strain dependent soil water retention curve in the (\(v, {v}_{\mathrm{w}}, p\)) space.

In this context, the current experimental study evaluates the deviatoric behaviour of compacted unsaturated soils in (\(q, {v}_{\mathrm{w}}, p\)) space, where \(q\) is the deviator stress, \({v}_{\mathrm{w}}\) is the specific water volume and \(p\) is the mean net stress. A series of constant moisture content triaxial tests on compacted unsaturated kaolin was carried out under different initial moisture contents and at different confining pressures and the results verified the phenomenological observations of compacted unsaturated soils under shear loading.

Experimental work

Experimental work included a series of constant moisture content shear tests conducted on compacted unsaturated kaolin with different moisture contents to analyse the deviatoric behaviour of compacted unsaturated soils. The effect of moisture content and confining pressure on the shear strength behaviour of compacted unsaturated soils was evaluated at four different moisture contents, 20, 27, 30 and 37%, and three different confining pressures, 200 kPa, 400 kPa and 600 kPa as listed in Table 1. Test specimens initially subjected to loading and unloading state paths were also sheared at constant moisture content to investigate the effect of over-consolidated stress history on the subsequent deviatoric behaviour of compacted unsaturated soils.

Table 1 Summary of Constant Moisture Content Shear Tests

Sample preparation

Commercially available powdered kaolin, known by the trade name Eckalite, a white-coloured lightly reactive clay was used as the testing material. The kaolin used in the present experimental program has a clay content of 75%, liquid and plastic limits of 60.5 and 27.9%, respectively. The specific gravity of the soil is 2.65 and optimum moisture content is 30.2% under standard proctor compaction. A complete set of material and geotechnical properties of Eckalite can be found in Islam [8].

The required amounts of dry powder of kaolin and de-aired water measured by weight were mixed in a mixing bowl with the use of an electric mixer while adding water slowly to the mixture. The kaolin and water mixture was mixed thoroughly for 10 min before transferring it to an air-tight plastic bag to avoid loss of moisture. The sealed plastic bag was stored in a temperature-controlled room for at least 24 h to allow moisture equilibrium throughout the soil mix prior to further testing. A moisture content verification test was done before each test.

Then, the soil mixture was statically compacted in 5 layers using a compression frame at a fixed axial displacement rate of 1.5 mm per min into a cylindrical compaction mould to prepare a test specimen of 65 mm in diameter and 130 mm in height. A rubber membrane was placed around the inner surface of the mould with the help of O-rings before placing the first layer of soil to minimise the wall friction. For each layer the required amount of soil mixture was put into the mould and a steel plug was placed on top of the soil mixture inside the mould, and then the fixed rate of displacement was applied until a 50 kPa axial load was reached. Once the final load has been reached, the steel plug was removed from the mould and the top surface of the compacted layer was scarified before adding the next layer of soil mixture. Once the test specimen is prepared at the targeted moisture content, it was carefully installed in the tri-axial cell.

Experimental Procedure

After setting up the test specimens in the GCTS unsaturated tri-axial cell, the specimens were initially isotropically compressed at constant moisture content condition to the targeted confining pressure by ramping the cell pressure at a rate of 10 kPa per min. The test specimens were then sheared to critical state at constant moisture content under strain-controlled mode by ramping the axial load at an axial displacement rate of 0.001 mm per min while maintaining the cell pressure constant. The strain rate was selected to be slow enough to avoid the building up of excess pore water pressure. The air phase was kept drained by opening the air drainage line throughout the shearing stage. Shearing was terminated once the change in mean net stress (p), deviator stress (q) and sample volume (v) almost ceased. The final moisture content of the test specimen was measured to ensure that the test was conducted under the constant moisture content condition.

For test specimens subjected to loading and unloading state paths in isotropic stress space before shearing, first the specimens were initially isotropically compressed at constant moisture content condition to the targeted confining pressure by ramping the cell pressure at a rate of 10 kPa per min and then unloaded to a lower confining pressure at the same moisture content by reducing the cell pressure at the same rate of 10 kPa per min.

Results and discussion

Figures 1, 2 and 3 shows the results of the constant moisture content triaxial tests carried out on unsaturated kaolin samples prepared with different moisture contents and then subjected to isotropic compression up to three different stress levels, specifically 200 kPa, 400 kPa and 600 kPa. Figure 1 presents the results for the confining pressure of 200 kPa, while Figs. 2 and 3 present the results for 400 kPa and 600 kPa confinements, respectively. Each figure contains five sub-plots; (a) the stress–strain behaviour, (b) the volume change behaviour during shearing, (c) the variation of degree of saturation during shearing, (d) the stress paths in (\(q, p\)) plane, and (e) the volume change behaviour for the complete test including the initial isotropic compression stage during which the test specimens were brought to the targeted confinement in (\(v, p\)) plane. In plot (e) a dashed vertical line shows the beginning of the shearing stage.

Fig. 1
figure 1figure 1

Constant moisture content triaxial tests at 200 kPa confining pressure

Fig. 2
figure 2figure 2

Constant moisture content triaxial tests at 400 kPa confining pressure

Fig. 3
figure 3figure 3

Constant moisture content triaxial tests at 600 kPa confining pressure

Stress–strain behaviour

Figure 1a, 2a and 3a present the stress–strain behaviour of compacted unsaturated soils with different moisture contents but at constant confining pressures. As the figures show, with the increasing moisture content the gain of shear strength reduces, with the soil reaching the critical states at lower values of deviator stresses and axial strains. This behavioural pattern is consistent with the fact that unsaturated soils exhibit an increase in shear strength with increasing suction [18, 19, 27], because the lower the moisture content, the higher the suction, and suction adds a confinement to the soil particles, leading to higher strength.

Change in specific volume

The change in specific volume during shearing does not show a specific trend. Figure 2b shows a decrease in the change of specific volume with increasing moisture content, with the driest sample having the highest reduction in specific volume. However, in Fig. 1b and 3b the test specimen with the highest moisture content shows the largest reduction in specific volume. This indicates that the specific water volume and the confining mean net stress may affect volume change during shearing. A similar variation in specific volume, without a certain pattern, was observed by Sivakumar et al. [22] and as reported by them, it may be due to the intricate nature of the soil fabric of compacted unsaturated soils. Houston et al. [7] predicted that the shear-induced volume change of unsaturated soils is more likely to be related to the soil water characteristic curve of the soil, as the volume change behaviour is dependent on both net normal stress and soil suction.

If it is assumed that, similar to saturated soils, the critical state lines for unsaturated soils with different moisture contents are parallel to the corresponding isotropic compression lines in (\(v, p\)) plane, the isotropic compression line and the critical state line for a certain moisture content can be schematically represented as shown in Fig. 4. AB represents the shearing path and the volumes at A and B can be written as:

Fig. 4
figure 4

Schematic diagram of the constant moisture content shearing test path for unsaturated normally consolidated soil in (\(v, p\)) plane

$${v}_{i}={v}_{0}-{\lambda }_{w}ln\left(\frac{{p}_{i}}{{p}_{0}}\right)$$
(1)
$${v}_{c}=\Gamma -{\lambda }_{w}ln\left(\frac{{p}_{c}}{{p}_{0}}\right)$$
(2)

where, \({v}_{i}\) is the specific volume at A and \({v}_{c}\) is the specific volume at B or at critical state. Since the tests were conducted under air-phase drained conditions, \({p}_{i}\) and \({p}_{c}\) are the mean net stresses at A and B, respectively, \({v}_{0}\) and \(\Gamma\) are the specific volumes at reference stress \({p}_{0}\) corresponding to the isotropic compression line and the critical state line, respectively. \({\lambda }_{w}\) is the slope of the isotropic compression line, which is assumed to be equal to the slope of the critical state line for that moisture content, and \({v}_{0},\Gamma\) and \({\lambda }_{w}\) are assumed to be functions of specific water volume,\({v}_{w}\).

From Eqs. 1 and 2, the volume change during shearing can be obtained as follows:

$${v}_{i}-{v}_{c}=\left({v}_{0}-\Gamma \right)+{\lambda }_{w}ln\left(\frac{{p}_{c}}{{p}_{i}}\right)$$
(3)

If the same behaviour is schematically represented in (\(q, p\)) plane as given by Fig. 5, a relationship between the \({p}_{i}\) and \({p}_{c}\) can be obtained as described below.

Fig. 5
figure 5

Schematic diagram of the constant moisture content drained shearing test path in (\(q, p\)) plane

Here it is assumed that the critical state relationship for unsaturated soils with a certain moisture content has the same format as proposed by Wheeler and Sivakumar [27] for a certain suction and can be given by Eq. 4. The slope of the critical state line, \(M,\) and the intercept, \(\mu ,\) are assumed to be functions of specific water volume.

$$q=M\left({v}_{w}\right)p+\mu \left({v}_{w}\right)$$
(4)

Since the drained shearing test path features a gradient of 1:3, the deviatoric stress, at critical state can be written as \(3\left({p}_{c}-{p}_{i}\right)\). Substituting this into Eq. 4, the relationship in Eq. 5 can be obtained:

$$3\left({p}_{c}-{p}_{i}\right)=M\left({v}_{w}\right){p}_{c}+\mu \left({v}_{w}\right)$$
$${p}_{c}=\frac{3{p}_{i}+\mu \left({v}_{w}\right)}{3-M\left({v}_{w}\right)}$$
(5)

Now, substituting Eq. 5 into Eq. 3, a relationship between the specific water volume, confining mean net stress and the shear induced volume change can be obtained as follows:

$${v}_{i}-{v}_{c}=\left({v}_{0}-\Gamma \right)+{\lambda }_{w}ln\left(\frac{3+\mu \left({v}_{w}\right)/{p}_{i}}{\left(3-M\left({v}_{w}\right)\right)}\right)$$
(6)

Equation 6 defines that the volume change during shearing is a function of both specific water volume,\({v}_{w}\), and the confining mean net stress (confining cell pressure)\({p}_{i}\). Once the relationships of \({v}_{0}\), \(\Gamma ,\) \({\lambda }_{w}\),\(\mu \left({v}_{w}\right)\) and \(M\left({v}_{w}\right)\) with specific water volume are defined, volume change during shearing under constant confinement can be defined.

Degree of saturation

Plot (c) of Figs. 1, 2 and 3 indicates the variation in degree of saturation during shearing. The soil samples isotropically compressed to a higher mean net stress show the highest degree of saturation at the beginning of shearing, representing the effect of mean net stress (confining pressure), and accordingly, they are at a higher degree of saturation at the critical state. Furthermore, in the experiments it was noticed that a small amount of water drained out through the air drainage line for specimens with a higher starting degree of saturation towards the end of the shearing stage as the soil was approaching full saturation. The stress path presented in plot (d) of each figure is drained for all the unsaturated tests, as the air phase was held drained throughout the testing and therefore shows a gradient of 1:3. However, the saturated triaxial test shown in Fig. 1d was undrained. Figures 1e, 2e and 3e demonstrate the volume change behaviour of the complete test, with both the isotropic compression stage and the shearing stage. Considering the volume change behaviours for all the tests, a change in slope of the volume change curves after the initiation of the shearing is clearly visible in all the figures. This indicates that similar to saturated soils where the critical state line is located below the Normal Compression Line (NCL) with the same slope, for unsaturated soils there may also exist a set of critical state lines in parallel with the isotropic compression lines at different moisture contents. In fact, there may be a critical state surface generated by the critical state lines for different moisture contents. Lloret‐Cabot [15] analysed the constant suction shearing experimental data by Sivakumar [21], where the critical state surface lied below the yield surface. Hence, it can be concluded that a critical state surface can be obtained parallel to the LWSBS of the MPK framework irrespective of the type of test utilised.

Figure 6 shows the same set of results for the moisture content of 36.7% analysed with respect to the confining pressure. Similar to Figs. 1, 2 and 3, Fig. 6 also has five sub-plots. It is apparent that with increasing confining pressure, the gain of shear strength increases for the same moisture content (Fig. 6a). This is because the strength of a soil isotropically compressed to a higher level of mean net stress is much greater than that of an identical soil sample compressed to a lower level of mean net stress as a result of hardening due to loading. On the other hand, the soil sheared at 200 kPa confinement shows a higher volume change compared to the soil sheared at 400 kPa (Fig. 6b). As expected for drained shearing, all the stress paths are parallel in (\(q, p\)) plane with a gradient of 1:3 as shown by Fig. 6d. The isotropic compression lines shown in plot (e) have approximately identical slopes which change during shearing.

Fig. 6
figure 6figure 6

Constant moisture content triaxial tests at w = 36.7%

Loading–unloading test

Figure 7 shows the results of a compacted unsaturated soil specimen at 26.6% moisture content subjected to isotropic compression (loading) up to 400 kPa, then unloaded to 200 kPa and then sheared to critical state at the same moisture content. The results are presented along with the results of an identical specimen sheared at 400 kPa confinement without unloading. The unloaded specimen with overconsolidated soil reached the critical state at a much lower value of deviator stress and at a lower value of axial strain. As can be seen from Fig. 7b and e, the change in volume during shearing is much less in the unloaded specimen compared to the other specimen. This is because, as a result of unloading, the soil state moves into the elastic loading space, and until it yields again, the soil behaves elastically, showing a minor variation of the specific volume during subsequent shear loading. An interesting feature to note is that if a critical state line is drawn through the end points of the curves shown in Fig. 7e as shown by the dashed line, it appears to be parallel to the isotropic compression line for that particular moisture content, indicating the possibility of having a set of critical state hyperlines in (\(v, p\)) plane parallel to the corresponding isotropic compression lines.

Fig. 7
figure 7figure 7

w = 26.6% loading, unloading and shearing test

Presenting unsaturated behaviour in (\({\varvec{q}},{\varvec{p}}\)) and (\({\varvec{v}},{\varvec{p}}\)) planes

Figures 8 and 9 present the critical state data on the constant moisture content triaxial tests presented thus far in (\(q, p\)) and (\(v, p\)) stress planes, respectively. Even with the limited amount of data available, it is evident that for a certain moisture content the critical state line is unique in both (\(q, p\)) and (\(v, p\)) planes. As evident from the above experiments and analysis of previous studies [15], it can be concluded that there exists a set of constant moisture content critical state hyperlines in both stress planes for compacted unsaturated soils. However, the variations of the slopes of the critical state lines in both (\(q, p\)) and (\(v, p\)) stress planes are uncertain based on the available data. The constant moisture content triaxial tests conducted by Thu et al. [24] on coarse kaolin soil and by Liu and Peng [14] on a railway embankment material also resulted in a set of critical state hyperlines for different moisture contents, and interestingly, both groups of researchers reported that critical state lines in (\(q, p\)) plane are parallel with their slopes independent of moisture content. On the other hand, the dependency of the slopes of the critical state lines on suction has been reported by several researchers [5, 22, 26, 27].

Fig. 8
figure 8

Critical state hyperlines in (\(q, p\)) plane

Fig. 9
figure 9

Critical state hyperlines in (\(v, p\)) plane

If the critical state points for unsaturated soils and the saturated critical state points are plotted in one figure as shown in Fig. 10, they appear to fall on a single unique line. This means that the critical state behaviour in (\(v, p\)) plane does not depend on the moisture content. Therefore, based on the available data, it can be argued that for the kaolin soil in the present study, critical state behaviour of compacted unsaturated soils in (\(v, p\)) is approximately equal to the critical state behaviour of saturated kaolin. Further testing and more data are necessary to substantiate this argument.

Fig. 10
figure 10

Critical state points in (\(v, p\)) plane approximated to single line

With the existence of a set of constant moisture content critical state hyperlines in (\(v, p\)) plane, a critical state surface can be constructed in (\(v, {v}_{w}, p\)) space as shown in Fig. 11. It should be noted that only a part of the critical state surface can be generated with the available data. From Fig. 11a, it appears that the constant mean net stress contours of the critical state surface are almost horizontal Therefore, the critical state surface looks relatively flat and aligned to the horizontal with a smaller angle in (\(v, {v}_{w}, p\)) space, as shown by Fig. 11b. Figure 12 shows the comparison of the developed critical state surface with the isotropic Loading Wetting State Boundary Surface (LWSBS) generated for the same kaolin soil under isotropic compression as reported in [1]. It is evident that the critical state surface is located below the LWSBS, and appears to be parallel to the LWSBS within the limited mean net stress range within which the critical state surface was generated. However, with the proposed single line to represent the critical state behaviour of compacted unsaturated soils in (\(v, p\)) plane, the critical state surface will not be in parallel with the isotropic LWSBS over the full range of the mean net stress.

Fig. 11
figure 11

Critical state surface in (\(v, {v}_{w}, p\)) space

Fig. 12
figure 12

Comparison of isotropic LWSBS and critical state surface for kaolin soil

Conclusions

Deviatoric behaviour of compacted unsaturated kaolin was evaluated in (\(q, {v}_{\mathrm{w}}, p\)) space in this study. It was found that the shear strength of compacted unsaturated soils would increase with the decreasing moisture content, while the same would increase with the increasing confinement. The critical state lines constructed by joining the critical state points for different specific water volume values in both (\(q, p\)) and (\(v, p\)) planes resulted in a set of critical state hyperlines at different specific water volumes. This is evidence for the possible existence of a unique critical state relationships for compacted unsaturated soils at different specific water volumes. With the available data, it was found that these critical state lines in (\(v, p\)) plane can be approximated to a single line that defines a critical state relationship independent from specific water volume. This signified that the effect of moisture content on the volume change behaviour at critical state is minimal. This hypothesis of a unique critical state relationship needs to be validated with more experiments and data.

One of the key findings of this study is the existence of a critical state surface in (\(v, {v}_{w}, p\)) space. This critical state surface was positioned below the corresponding LWSBS and was relatively flat as the effect of moisture content was very low as described in the text.