Figure 1 shows the particle size distribution curve of the expansive soil tested in this study. The soil has been characterized as a highly swelling soil using a twofold rationale: (1) a rigorous experimental campaign comprising multiple odometer tests to quantify swelling in terms of both swell potential and pressure and (2) historical perspective, whereby published studies reported that the Nandipur clays exhibit very high swelling potential (Akbar and Farooq, 2002; Israr et al., 2014; Mumtaz et al., 2020). First, the soil has been classified based on visual and physical observations at the sampling site in Nandipur, Pakistan. The natural expansive soil was collected from 1 m depth with the help of a shovel, preserved in sealed plastic bags, and conveyed to the laboratory. Table 1 presents the physical properties of the soil specimen used in this study. It is to be noted that the aforementioned rationale could sufficiently establish that the currently tested soil is a high-swelling clay and that the observed cracking has been the result of shrinkage due to the loss of moisture. The double punch test method was performed to determine the tensile strength of the soil (Fang and Chen, 1971). The tensile strength (σ_t) value was obtained from Eq. 1. σ t = P π 1.0 b H − a 2 , (1) where tensile strength (σ_t) is in psi, P is the applied load in lbs, a is the radius of the punch in inches, b is the radius of the specimen in inches, and H is the height of the specimen in inches. The values of a, b, and H are 12.7 mm (0.5 in), 101.6 mm (4 in), and 116.84 mm (4.6 in), respectively.

The relationship between tensile strength at different moisture contents is plotted in Figure 2. The maximum value of tensile strength was noted as 65.086 kPa at 24% moisture content with 55 blows using the modified compaction test procedure, and the same value was carried out for further analysis.

A shrinkage strain test was performed on the soil samples, and an image-based technique called the DCI method was used to study the shrinkage behavior of both the soil and the cracking area. The samples were compacted at different moisture contents of 16%, 19%, 21%, and 24%. For brevity, the compacted samples were placed between two filter papers and porous stones for wet and dry cycles in the modified odometer assembly. In this study, a wet–dry cycle consisted of submerging the soil sample in water until saturation and air-drying to the desired moisture content, after which no further shrinkage occurred (Ekrem Kalkan, 2011). The saturation was ensured when the Skempton’s B-value exceeded 0.95 (i.e., the ratio between monitored pore water pressure and the applied total stress). The cyclic swell–shrink test was performed on the prepared sample under 12 kPa pressure. The sample was allowed to fully swell in the wet cycle, and a constant temperature of 45 ± 5°C was maintained during dry cycles. After 24 h, the moisture content dropped to 4% and no further shrinkage was observed. Cyclic swell–shrink behavior was studied by subjecting the compacted soil specimens to two alternate wet–dry cycles. At the end of each drying cycle, the surface camera images were captured using a digital camera. The images taken on each day were analyzed on ‘ImageJ’ software to compute the results. The water content of the sample was measured at the end or during the dry cycles.

The 20-MP digital cameras were used in this test to capture the images of the soil specimens. The binary image processing technique using ‘ImageJ’ software is suitable for various sample shapes. The selected total surface area of the soil specimen, including the crack area (A_T), and then, the whole area of the image (A₁) were measured using the ‘Analyze–Measure’ command; the white area (A₂) which includes the area of cracks and the surrounding white area other than the soil surface, were also measured. The next step involved the separation of the crack area from the total surface area. The area of the shrunken soil A _s and the area of the cracks A _c were measured by Eqs. 2, 3, respectively. A s = A 1 − A 2 , (2) A c = A T − A s . (3)

The area of cracks obtained at the end of the second wet–dry cycle was 0.511, 0.845, 2.533, 2.748, and 5.787 cm² at 16.2, 19.04, 21, 24%, and 28.2% moisture contents, respectively. From the areas A _s and A _c , the volume of the crack and soil was measured through Eqs. 4, 5, and then compared with the Vernier caliper readings. V o l u m e   o f   t h e   c r a c k s = A c t + A c b 2   h e i g h t   o f   t h e   s o i l   s p e c i m e n (4) where A _ct is the crack area obtained from top surface images and A _cb is the crack area obtained from bottom surface images. V o l u m e   o f   t h e   s o i l = A s × H e i g h t   o f   t h e   s o i l   s p e c i m e n (5)

The volume of cracks obtained at the end of the second wet–dry cycle was 1.2264, 1.859, 5.5726, 6.3204, and 12.7314 cm³ at 16.2, 19.04, 21, 24%, and 28.2% moisture, respectively.

Table 2 shows the original images of the soil sample prepared for the DCI method before and after cracking. The relationship between the vertical shrinkage strain and moisture content was determined by shrink testing using Briaud’s method (Briaud et al., 2003). A total of four samples were prepared in a cylinder of 70 x 100 mm size at 16.2, 19.04, 21, and 24% moisture contents. Volumetric deformation was computed with the help of Eq. 6. Notably, Figure 3 shows the volumetric deformation of soil at different moisture contents. V o l u m e t r i c   d e f o r m a t i o n   % = Δ V V × 100 , (6)

where ΔV is the change in volume and V is the actual volume of the soil sample.

The values obtained from the shrink test were computed to obtain the swell–shrink modulus (E _w ) and shrinkage ratio parameters by substituting the values in Eqs. 7, 8. E w = Δ w Δ V V o , (7) f = Δ H H o Δ V V o , (8) where Δw=w _o-w represents change in the water content, w _o is initial water content corresponding to zero volume change, and w is the water content corresponding to relative volume change ( ∆ V V o ) and the relative height change ( ∆ H H o ).

These parameters were then used to obtain the vertical shrinkage strain at different moisture contents by substituting the value in Eq. 9 as shown as follows: Δ H = Δ w E w H f , (9) where H is the height of the sample and Δw is the change in moisture content as a function of depth. It is noteworthy that surface soil was analyzed in this study and no depth was involved, so only the initial moisture content was considered. Another shrinkage test was performed on the same soil samples using Dhowian’s method (1990). Δ H = C w . d r y H w i − w f , (10) where C w . d r y is the moisture index for drying, H is the height of the soil layer, and w _i and w _f are the initial and final gravimetric water content values, respectively. C w . d r y = α G s 1 + e o . (11)

In this study, the volume compressibility factor α for clay materials was selected as 0.33 (Dhowian, 1990; Bratton, 1996; Briaud et al., 2003), while specific gravity of 2.645 was obtained from the specific gravity test. The initial void ratio (e _o ) was obtained from relation between specific gravity and gravimetric water content. The moisture index obtained from Eq. 11 and other parameters were substituted in Eq. 10 to obtain the vertical shrinkage strain.

The filter paper test was performed to determine the matric suction values of the soil samples (Bicalho et al., 2007). The suction values were determined using Eqs. 12, 13 in tandem (Marinho and Oliveira, 2006): ω f   <   33   Ψ = 4.83   –   0.0839   w , (12) ω f   >   33   Ψ = 2.57   –   0.0154   w , (13) where ω_f is the filter paper water content break point, w is the moisture content in %, and Ψ is the log₁₀ suction in kPa. SWCC, as presented, was prepared by calibrating Eq. 14 (Fredlund and Xing, 1994). ω ψ = 1 − ln 1 + ψ ψ r ln 1 + 10 6 ψ r ψ s ln e + ψ a n m , (14) where ω is the gravimetric moisture content and Ψ is the suction in kPa. The values of the parameters a, n, e and m are 135 kPa, 0.6015, 2.71828 and 0.5718, respectively. The unknown parameters of SWCC were obtained by substituting the moisture content in Eq. 15: ω ψ = 1 − ln 1 + ψ 11234.89 ln 1 + 10 6 11234.89 0.36 ln 2.72 + ψ 138.09 1.4209 0.44 . (15)

Figure 4 shows both the theoretical and experimental SWCC, which agree closely with the previously developed mathematical models (Javid et al., 2020).

The value of the diffusivity coefficient is obtained through the following Eq. 16: α = 0.005824 + 0.00000713 T M I + 0.0495134 Δ h Δ w − 0.0034581 γ h . (16)

The α value obtained from Eq. 16 was 1.1 x 10⁻² cm²/s. The value of the error correction factor (erf) was taken as 0.0556.

Eq. 17 was used to obtain the suction compression index (Mckeen, 1992): γ h = − 0.02673 × Δ h Δ w − 0.38704 , (17) where ∆ h / ∆ w is the slope of the soil water characteristic curve (SWCC). Here, the diffusivity coefficient is obtained by the Thornthwaite Moisture Index (TMI) using Eq. 18: T M I = R − E T o E T o , (18) where R is the annual rainfall and ET _o is the annual potential evapotranspiration. The data on average annual rainfall were obtained from the National Center for Environmental Information (NCEI), as shown in Figure 5.

E T o was obtained by Eq. 19 that was proposed by the Blaney-Criddle method (1950). The value obtained for the suction compression index from Eq. 20 was 0.01386. E T o = 0.254 p 32 + 1.8 T a , (19) where T a is the average monthly temperature ℃ calculated from Eq. 20: T a = T max + T min 2 . (20)

T _max and T _min are monthly maximum and minimum temperatures shown in Figure 6. p is the mean daily percentage of annual daytime hours, which is as shown in Figure 7. T _a and p were used in Eq. 21 to find out the value of E T o . The values of E T o and R are used in the previous Eq. 18 to find out the values of TMI. Afterward, the relationship between tensile stresses and depth with varying time was calibrated by using Eq. 21

Figure 9 shows the variation of volumetric deformation obtained from the Vernier scale and DCI methods at 16.2, 19.04, 21, 24%, and 28% moisture content under two wet–dry cycles. At the end of the second wet–dry cycle, values of volumetric shrinkage deformation by the Vernier scale method were −21.868, −23.14, −23.66, −24.54, and-25.132 %at 16.2, 19, 21, 24, and 28% moisture content, respectively. However, the DCI method showed the volumetric shrinkage deformation to be −26.84, −28.661, −29.46, −29.846, and −32.92 %at 16.2, 19, 21, 24, and 28% moisture content, respectively. The circular gap along the crack increased during the successive drying cycles; therefore, it becomes evident that the Vernier scale method underestimates the volumetric deformation. Shrunken soil shows predominant cracks appearing at the circumference of the soil sample. These cracks extend from the top surface to the bottom surface of the soil sample, which supports the assessment of the cracking pattern using the DCI method. The vertical shrinkage strain could be computed through Eq. 23: ∂ ϵ = ∆ h ∂ log Ψ h ∆ log Ψ × 100 . (23)

Figure 10 shows the comparison of predictions from the current model, Briaud’s method, and Dhowian’s method against the experimentally observed vertical shrinkage strain values. The results showed that the currently proposed suction-based method of this study slightly overestimates the vertical shrinkage, whereas the moisture-based methods of both Briaud and Dhowian overestimate the vertical shrinkage deformation. This difference can be attributed to the different indices utilized in both types of methods. For instance, the moisture-based methods consider the change in volumetric water content at the middle of the soil layer, whereas the current suction-based methods more realistically consider the suction change at the surface of the soil. In essence, the comparison shows that the values predicted by the current model are in good agreement with those observed during the experiments. Similarly, the two existing models overestimate the vertical shrinkage strain values more than the current model, which yields conservative values that agree closely with the observed values, thus validating the current model.

The model developed based on Eq. 21 predicts the tensile stress profile in relation to soil depth. However, none of the published studies have reported the relation between crack depth and drying period. This study uses climate data such as average drying temperature and annual rainfall for the study area. The time was estimated based on the hit and trial method for 1, 2, and 3 months. For instance, time t was kept constant, and the crack depth z was estimated against different values (0–85 kPa) of tensile stress σ_x during the period of a month.

Figure 11 shows that the crack will reach 0.7, 1.3, and 1.8 m depth after 1, 2, and 3 months, respectively. Moreover, the dotted line on the figure against a tensile stress of 65 kPa shows the actual depth of the crack for the soil specimen tested in the present study. It is to be noted that the crack remains within the dry cycle and achieves a depth of 1 m during the second month.

Figure 12 graphically depicts the correlation between tensile stress and crack depth using previous Eq. 21, which was used to estimate the drying time for the same soil against varying tensile stresses. For brevity, curves at three different crack depths, i.e., 0.8 m, 1 m, and 1.48 m, have been plotted, and the tensile stress magnitude corresponding to 1 m deep crack has been determined (i.e., 65 kPa). It can be observed that the crack will reach a depth of 1 m at 40th day of drying time. In essence, the results obtained from both Figures 11, 12 could have considerable direct implications for designing safer pavements and shallow foundations on expansive soils. Although laboratory experiments still remain the most reliable approach to ascertaining the mechanical response of expansive soils, the current propositions may be used with caution for prompt and preliminary assessments only.