#### 3.1. Instrumentation Results

Figure 4 shows the suction and rainfall data for a period of about one year (15 May 2018–15 May 2019), along with indication of the periods in which crack depths were measured. It should be noted that only the suction data from MPS6-1 located at a depth of 0.5 m from top of embankment was used in the modelling of shrinkage crack depth as it was the nearest sensor in proximity to the crack.

From the middle of May to the end of October 2018, which was the rainy period, there was a significant amount of rainfall (about 750 mm) compared to the period between November and April 2019 with rain of only about 100 mm. The suction change agreed well with the rainfall pattern. In June 2018, the value of suction at the 0.5 m depth was the lowest (about 9 kPa) while in the following months as rainfall became scant, the suction continuously increased and reached the highest value of 935 kPa in the month of March 2019. The shrinkage crack started appearing in December 2018, when the value of suction rose to 318 kPa. The increase in suction (i.e., decrease in pore-water pressure) during the dry season was more considerable at the middle part of the slope (MPS6-2) than the top part (MPS6-1), while the suction at the base (MPS6-3) was the lowest. The higher pore-water pressure at the base of a slope was expected due to gravitational flow either as runoff or seepage and accumulation at the bottom. The underlying ground with a lower permeability could pose additional resistance to vertical flow, potentially resulting in ponding. The rise of pore-water pressure in response to rain infiltration appeared to be faster at the middle and lower parts than at the top. This is expected to be due to a lesser degree of compaction and the vegetation effect on the slope surface (i.e., at the middle and the base) which resulted in a more permeable ground than the well-compacted soil at the top surface of the dyke.

Figure 5 shows the response of volumetric water content from the TDR measurement. The response of the water content at a 0.1 m depth was expectedly faster than the suction at a 0.5 m depth. It can be seen that the sudden rise in volumetric water content took place whenever rainfall occurred, despite the relatively small amount of rain. Interestingly, by comparing TDR1 and MPS6-1 readings in

Figure 4 and

Figure 5, the decrease in water content during the drying period (October to April 2019) seemed to be much less than that of the pore-water pressure.

#### 3.2. Soil Water Retention Curve (SWRC)

Figure 6 shows the soil water retention curve (SWRC) of the undisturbed soil sample over the entire range of suction (0 to 1,000,000 kPa). The SWRC demonstrates the bimodality nature as indicated in

Figure 6a, which can be attributed to the macro- and micro-voids associated with the undisturbed sample obtained from the field. Shrinkage of the soil upon drying (

Figure 6b) can be clearly seen as a decrease in void ratio, which showed a linear trend against logarithmic soil suction for up to 10,000 kPa while it demonstrated a constant void ratio after 10,000 kPa.

For modelling of the SWRC, the measured data points were fitted with the equations proposed by [

34]. The model is comprised of the parameters that have clear physical meanings and the degree of saturation in the model is a function of ten parameters, as demonstrated in Equation (3),

The value of the parameters used to fit the measured data points in this study are given in

Table 3, where

${\mathsf{\psi}}_{\mathrm{b}1}$ is the air-entry suction,

${\mathsf{\psi}}_{\mathrm{r}\mathrm{e}\mathrm{s}}$ is the residual suction, and

${\mathrm{S}}_{\mathrm{r}\mathrm{e}\mathrm{s}}$ is the degree of saturation at the residual stage. The Subscripts 1 and 2 represent two levels of soil structures.

${\mathrm{S}}_{\mathrm{b}2}$ is the degree of saturation at the air entry of the second structure level. Furthermore, a plot of gravimetric water content (w) with suction is shown in

Figure 7.

This relationship was also used for correlating the FFR results with suction. The fitted gravimetric water content for the soil with specific gravity (Gs) 2.65 was obtained from Equation (4), in which the void ratio (e) is a function of suction.

The suction modulus (H) defined in Equation (5) is the ratio of change in suction

$(\Delta {(\mathrm{u}}_{\mathrm{a}}{\text{}-\text{}\mathrm{u}}_{\mathrm{w}}\left)\right)$ to change in volumetric strain (

$\Delta {\mathsf{\u03f5}}_{\mathrm{vol}}$) and is calculated using the drying SWRC results. It is plotted against suction as shown in

Figure 8, which is seen to be increasing with an increase in suction, a typical trend observed in previous studies [

7,

12]. This function was then used in the modelling of crack depth, presented in

Section 3.6#### 3.3. FFR Results

The fundamental transverse frequency from the FFR tests was observed to be between 475 and 3350 Hz for various moisture contents. The shear wave velocity and the small strain shear modulus were calculated using Equations (1) and (2), respectively, the range of which were between 99.75 and 703.50 m/s and 20.96–896.19 MPa. The small strain shear modulus increased with decreasing moisture content and increasing suction, as shown in

Figure 9. This positive correlation between small-strain shear modulus with suction agrees well with findings of many previous studies [

24,

35,

36]. The overall trend of G

_{0} with moisture content and suction is bi-linear, with the turning point at a moisture content of about 10% and suction about 26 MPa.

The trend of a sharp increase in G

_{0} with decreasing moisture content (or increasing suction) at the initial stage of drying (suction from zero to 26 MPa) was expected and due to the greater influence of suction stress on the modulus [

37] in this stage, as well as the initial decrease in the void ratio on the initial drying that could further induce a rise in modulus (

Figure 6b and

Figure 9b). This process continued until the turning point (at a suction of around 26 MPa) was reached where shrinkage became minimal, after which the rate of change in G

_{0} became less. It is noteworthy that the moisture content at 26 MPa suction was 10.2%, which was lower than the soil’s shrinkage limit of 13.5% and smaller than the second air-entry point of the SWRC (

Figure 6 and

Figure 7). As drying continued, the sample showed a more gradual rate of increase in G

_{0} for suction higher than 26 MPa as shrinkage became progressively less with increasing suction.

#### 3.4. SASW Results

SASW tests were set up longitudinally at the top of the embankment and were conducted at different time intervals to determine the change in modulus (G

_{0}) with varying in-situ suctions. Shear wave velocity (Vs) profiles for the embankment at different periods are shown in

Figure 10. Only the Vs values obtained from zero to 0.5 m depth were used to calculate the G

_{0} of the upper layer of soil where the shrinkage crack was observed.

The measured suction at the depth of 0.5 m (MPS6-1) was plotted against G

_{0} as shown in

Figure 11. The G

_{0} values from May to November 2018 positively correlate with measured suctions, while from January to March 2019, after the shrinkage crack occurred in December, the G

_{0} appeared to decrease significantly and varied disproportionately with suction. In general, G

_{0} values from January to March should have been higher than those from May to November, since the suction values were higher for those periods, if they were to follow the trend of intact soil. A possible explanation is that the crack was responsible for the reduced values of G

_{0}.

#### 3.5. Suction–Small-Strain Shear Modulus Relationship

Small-strain shear modulus (G

_{0}) is dependent on net stress, suction history, void ratio, over-consolidation ratio (OCR), strain rate, and plasticity index [

24,

36,

38,

39,

40,

41]. Many empirical and semi-empirical expressions have been proposed for describing the relationship between suction and modulus. In this study, a modelling approach by [

42] was used to determine the relationship between G

_{0}, suction (

$\mathsf{\psi}$), and water content due to its viability and directness. Amongst various modeling options, the model that considers the influence of suction stress and net stress separately is depicted in Equation (6),

where

${\mathrm{S}}_{\mathrm{r}}$ is degree of saturation,

${\mathsf{\sigma}}_{\mathrm{o}}$ is total vertical stress,

${\mathrm{u}}_{\mathrm{a}}$ is pore air pressure,

$\mathsf{\psi}$ is soil suction (matric suction if <1500 kPa, total suction if >1500 kPa); f(e) = 1/(0.3 + 0.7e

^{2}) and is the void ratio function given by [

43] for sands and clays, A, C, and k are empirical parameters for obtaining the best fit between measured and predicted values.

In the FFR test, since the test was done in a free boundary condition state, the sample was devoid of any net confining pressure. However, in the SASW test, the interested depth involving the observed shrinkage crack was within 0.3 m, so the net confining pressure involved can be considered marginal but not zero. Hence, the term,

$\mathrm{A}*\mathrm{f}\left(\mathrm{e}\right){\left({\mathsf{\sigma}}_{\mathrm{o}}{\text{}-\text{}\mathrm{u}}_{\mathrm{a}}\right)}^{\mathrm{n}}$ in Equation (6), which was assumed to be constant, D, would be higher in the SASW test than in the FFR test. Based on [

37], suction stress, defined as suction multiplied by effective wetted area present at the interface, can be introduced as

${\mathsf{\sigma}}_{\mathrm{s}}={{\mathrm{S}}_{\mathrm{r}}}^{\mathrm{k}}\mathsf{\psi}$, (k = 1/2), reducing Equation 6 to the form of,

The values of degree of saturation used in the calculation of

${\mathsf{\sigma}}_{\mathrm{s}}$ were estimated using the SWRC and the measured suction. A reasonable goodness of fit (R

^{2} ranging between 0.77 and 0.98) was observed for the relationships G

_{0}—

$\mathsf{\psi}$ (

Figure 12a) and G

_{0}—

${\mathsf{\sigma}}_{\mathrm{s}}$ (

Figure 12b) for both the SASW and FFR results. In the FFR tests, the soil followed a bilinear relationship for both G

_{0}—

$\mathsf{\psi}$ and G

_{0}—

${\mathsf{\sigma}}_{\mathrm{s}}$ as explained earlier. Regarding the SASW, the G

_{0}—

$\mathsf{\psi}$ relationship holds well (R

^{2} = 0.905) until the initiation of the crack after which, despite the increase of suction, G

_{0} decreased. It is interesting to note that after the crack, the values of G

_{0}, in comparison to the expected G

_{0}—

$\mathsf{\psi}$ and G

_{0}—

${\mathsf{\sigma}}_{\mathrm{s}}$ relationships, reduced and became similar to the FFR test. This may be because the effect of the crack was to reduce the net confining stress effect and thus to reduce the constant D in Equation (7). The values of D and C obtained by curve fitting (

Figure 12b) for different test conditions are listed in

Table 4.

#### 3.6. Modelling of Crack Depth

An attempt was made to model the measured crack depth by using a semi-empirical approach based on the isotropic elasticity principle (Equation (8)) given by [

12]

where

${\mathsf{\epsilon}}_{\mathrm{h}}$ is a normal strain in the horizontal direction,

$\mathsf{\mu}$ is Poisson’s ratio,

${\mathsf{\sigma}}_{\mathrm{v}}$ and

${\mathsf{\sigma}}_{\mathrm{h}}$ are the total normal stress in the vertical and horizontal directions, respectively,

${\mathrm{u}}_{\mathrm{a}}$ and

${\mathrm{u}}_{\mathrm{w}}$ are pore-air and pore-water pressures, respectively, E and H are elastic moduli with respect to changes in (

${\mathsf{\sigma}\text{}-\text{}\mathrm{u}}_{\mathrm{a}}$) and

${(\mathrm{u}}_{\mathrm{a}}{\text{}-\text{}\mathrm{u}}_{\mathrm{w}}),$ respectively. According to [

5], the cracking mechanism mainly involves decreasing horizontal stress and, upon unloading, the soil tends to behave elastically. In addition, the soil closer to the surface behaves more isotropically than the deeper soil [

5]. Thus, the isotropic elastic model was considered acceptable as a first approximation. For the at rest condition,

${\Delta \mathsf{\epsilon}}_{\mathrm{h}}$ can be set to zero. As

${\Delta \mathsf{\sigma}}_{\mathrm{v}},{\Delta \mathrm{u}}_{\mathrm{a}},$ and

${\mathrm{u}}_{\mathrm{a}}$ equal zero, and given the matric suction,

${\mathsf{\psi}\text{}=\text{}\mathrm{u}}_{\mathrm{a}}-{\mathrm{u}}_{\mathrm{w}}$, Equation (8) reduces to the following,

Provided that the initial ground condition is saturated and suction equals zero, the initial horizontal stress can be calculated using the expression

${\mathrm{k}}_{0}{\mathsf{\gamma}\mathrm{Z}}_{\mathrm{c}}$, where

${\mathrm{k}}_{0}$ is the coefficient of earth pressure at rest,

$\mathsf{\gamma}$ is the unit weight of the soil, and

${\mathrm{Z}}_{\mathrm{c}}$ is the depth of the soil considered.

${\mathrm{k}}_{0}$ is calculated using Jaky’s formula,

${\mathrm{k}}_{0}=1-\mathrm{s}\mathrm{i}\mathrm{n}{\mathsf{\varphi}}^{\prime}$, where

${\mathsf{\varphi}}^{\prime}$ is a friction angle obtained from the direct shear test. As suction increases, thereby causing a reduction in horizontal stress to the point when it becomes negative and equal to the tensile strength of the soil,

${\mathsf{\sigma}}_{\mathrm{t}}$, thus initiating a crack from the surface down to depth,

${\mathrm{Z}}_{\mathrm{c}}$. So, the required change in horizontal stress becomes

${\mathrm{k}}_{0}{\mathsf{\gamma}\mathrm{Z}}_{\mathrm{c}}+{\mathsf{\sigma}}_{\mathrm{t}}$. Then, Equation (9) can be arranged to give the value for depth of crack as [

44],

Based on field observations, the crack started appearing when the suction value reached 318 kPa in December 2018. Using this value and back-calculating the tensile strength from the material properties shown in

Table 5, the obtained value of

${\mathsf{\sigma}}_{\mathrm{t}}$ was 32.7 kPa at this suction level. The value of

${\mathsf{\sigma}}_{\mathrm{t}}$ can be assumed as either a constant [

44] (called Approach A in this study) or a function of suction according to [

5] (called Approach B). For Approach A, differentiating Equation (10) gives,

For Approach B,

${\mathsf{\sigma}}_{\mathrm{t}}$ is assumed as a function of unsaturated shear strength, which is dependent on suction as follows,

where

${\mathrm{c}}^{\mathrm{s}}=\mathsf{\psi}\xb7{\mathrm{S}}_{\mathrm{r}}\xb7\mathrm{tan}{\mathsf{\varphi}}^{\prime}$. The value of

${\mathsf{\alpha}}_{\mathrm{t}}$ corresponding to a suction level of 318 kPa (at the initiation of the crack) is equal to 0.13357 based on back-calculation of

${\mathsf{\sigma}}_{\mathrm{t}}$ using Equation (12). As a first approximation,

${\mathsf{\alpha}}_{\mathrm{t}}$ is assumed to be independent of suction. Equation (10) can again be differentiated as

After December 2018, the tensile strength of the soil was exceeded by the suction-induced horizontal stress change and a crack started propagating. The modelling of the crack depth (

Figure 1) was done by using an incremental form, as in Equations (11) and (13). The incremental cracking depth,

${\mathrm{dZ}}_{\mathrm{c}},$ was obtained with an incremental change in suction

$\mathrm{d}\mathsf{\psi}=\mathrm{d}({\mathrm{u}}_{\mathrm{a}}-{\mathrm{u}}_{\mathrm{w}})$. The variation of the H modulus with suction in

Figure 8 was then used to find the E/H variation as shown in

Figure 13, which was inputted in Equation (13) to calculate

${\mathrm{dZ}}_{\mathrm{c}}$ for each increment. The modelled crack depth,

${\mathrm{Z}}_{\mathrm{c}}$, non-linearly increased at a reduced rate as the suction value increased. The range of the E/H ratio for this study was obtained by varying the E value until attaining the best fit, and was shown to vary between 0.005 and 0.002 for the suction value ranging from 318 kPa to 1000 kPa. It should be noted that the E modulus in

Table 3 corresponds to a large-strain condition as opposed to the small-strain modulus shown in

Figure 12. Approach B (varying tensile strength) was also used to fit the variation of

${\mathrm{Z}}_{\mathrm{c}}$ as shown in

Figure 13. Both approaches (A and B) appeared to yield similar results, which indicated that the

${\mathsf{\sigma}}_{\mathrm{t}}$ can be reasonably assumed as a constant, at least up to the investigated suction level.