Introduction

Expansive soils are widely found in many countries such as China, India, Canada, Israel, and Australia (Jones and Holtz 1973; Alonso et al. 1987). They are typically distributed close to the ground surface where they are exposed to various climate events such as evaporation, infiltration, and plant transpiration. Expansive soils are sensitive to moisture content variation, during which their bulk volume and microstructure change significantly. This results in complex evolution in their engineering properties such as permeability, shear strength, modulus, and volumetric behaviours (Tripathy et al. 2002; Vilar and Rodrigues 2011; Lu and Godt 2012; Ajdari et al. 2013; Zha et al. 2013; Zhang et al. 2015; Massat et al. 2016; Zou et al. 2018).

In seasonally frozen regions, in addition to the moisture fluctuation, expansive soils are also subjected to freezing and thawing processes in winter and spring. These environmental effects constitute periodical drying-wetting (DW) cycles, freeze–thaw (FT) cycles, and more realistically, combined drying-wetting-freeze–thaw (DWFT) cycles that act on surficial expansive soils. The thermal and moisture exchange associated with DW, FT, and DWFT cycles cause remarkable alterations in the bonding and arrangement of soil particles (Edwin and Anthony 1979; Xue et al. 2014; Liu et al. 2016), which lead to the deterioration in the engineering properties of expansive soils (Aubert and Gasc-Barbier 2012; Aldaood et al. 2014; Cui et al. 2014; Liu et al. 2016; Hu et al. 2019; Qu et al. 2020; Zhou et al. 2020). This poses challenges to the design of geotechnical structures in expansive soils in seasonally frozen regions.

Over the past few decades, the effects of DW cycles and FT cycles on various soils have been investigated (Alonso et al. 2005; Nowamooz and Masrouri 2008; Tripathy et al. 2009; Tang et al. 2016; Zhao et al. 2019; Othman and Benson 1993; Kraus et al. 1997; Hewitt and Daniel 1997; Yuen et al. 1998; Hohmann-Porebska 2002; Luo et al. 2018; Ye and Li 2019; Dalla et al. 2019). Generally, it was found that both DW and FT cycles introduce cracks to the soils’ structure. Cracks induced by DW cycles are mainly associated with the non-uniform development of the volumetric strain (Pires et al. 2008; Seiphoori et al. 2014; Monroy et al. 2010; Lin and Cerato 2014; Burton et al. 2015), while cracks induced by FT cycles are mainly the results of the internal pressure associated with the growth of ice lenses (Tian et al. 2019; Tang and Yan 2015).

DW- or FT-induced cracks weaken the integrity of soils’ structure and create large and unstable pore spaces (Pires et al. 2008; Seiphoori et al. 2014; Graham and Au 1985; Zhang et al. 2016) that lead to (i) an increase in hydraulic conductivity (Albrecht and Benson 2001; Malusis et al. 2011; Edwin and Anthony 1979; Konrad and Samson 2000), (ii) remarkable changes in the bulk volume (Al-Homoud et al. 1995; Peng et al. 2007; Liu et al. 2016; Kalkan 2011; Wang et al. 2017; Luo et al. 2018), and (iii) degradation in the stiffness and strength properties (Wang et al. 2015; Tang et al. 2018; Xu et al. 2018; Lu et al. 2019). However, the combined effects of DW and FT cycles (i.e. DWFT cycles) on the microstructure and mechanical behaviours of soils are less reported, especially for expansive soils.

This paper investigates the influence of DW, FT, and DWFT cycles on the mechanical behaviours and microstructure of a compacted expansive soil collected from Northeastern China. The investigated mechanical behaviours include elastic modulus and shear strength (i.e. cohesion and internal friction angle) that were obtained from consolidated undrained triaxial tests. Microstructural observations were performed through mercury intrusion porosimetry (MIP) and scanning electron microscope (SEM) tests to reveal the evolution of the soil structure. A constitutive model is proposed to describe the stress–strain relationships of the tested expansive soil considering the influence of DW, FT, and DWFT cycles.

Experimental investigations

Material

The expansive soil investigated in this study was collected from a failed canal slope in the north of Qiqihar, Heilongjiang Province, China, which is a seasonally frozen region (see Fig. 1). The climate type of Qiqihar belongs to the temperate continental monsoon climate which is featured by hot and rainy summer and cold and dry winter. The average temperature ranges from 23.1 °C in summer to −18.6 °C in winter and the average precipitation ranges from 103.0 to 3.2 mm. The average frost depth and active zone depth in Qiqihar are approximately 2 m and 3 m, respectively.

Fig. 1
figure 1

Location of the tested expansive soil

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The canal (i.e. sampling site) was designed for water diversion from Nen River, mainly to meet the water demands of the petrochemical industry and agricultural irrigation in the vicinity. The average slope angle was between 12.2 and 19.9°. The local geology comprises weak and medium expansive soils and locally dispersive soils. At the end of 2011, more than a dozen large or small landslides were found on the left-side slope of the canal. The total volume of the landslides was 32,925 m3. A representative failed slope is shown in Fig. 2a. Due to the large-scale landslides, the concrete slabs on the canal bed were jacked up and the total damaged concrete is 717.6 m3 (see Fig. 2b).

Fig. 2
figure 2

a Representative failed canal slope and the b crushing failure of concrete slabs on the canal bed

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Field investigations and laboratory tests were conducted to investigate the causes of the slope failure. It was revealed that after excavation, the canal slope was directly exposed to the atmosphere with no protection. Besides, no slope drainage system was designed during the canal construction. Thus, cracks gradually develop on the slope surface due to the natural weathering effects (including cyclic drying-wetting-freeze–thaw effects), resulting in the acceleration of the infiltration of snowmelt and rainwater and the evaporation of soil moisture. During the weathering, the expansive soils experienced cyclic swelling and shrinkage, which facilitates the development of cracks. Such damages led to the decrease of soil strength and further caused the failure of the canal slope.

The soil samples were collected from the debris of the failed slope at a depth of approximately 1 m (i.e. within the frost and active zone depth) before the slope renovation. The natural moisture content (wn) and dry density (ρdn) were measured during the sampling, which are 26.3% and 1540 kg/m3, respectively. The basic index properties of this soil are summarized in Table 1.

Table 1 Basic index properties of the tested expansive soil
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X-ray fluorescence spectrometry (XRF) tests were performed to reveal the soil’s dominant chemical components, which are SiO2: 60.48%, Al2O3: 18.53%, Fe2O3: 6.63%, K2O: 3.05%, and CaO: 4% (by dry mass). The soil’s major mineral components determined by X-ray diffraction (XRD) tests are quartz, illite, albite, and calcite. The free swelling ratio (i.e. percentage of volumetric increment of dry and grounded soil powders when soaked in water) of the soil is 67%. According to the Chinese technical code for buildings in expansive soil regions GB 50,112–2013 (Ministry of Housing and Urban–Rural Development 2013), this soil is classified as an expansive soil with medium-swell potential. The soil has a plasticity index (IP) of 21. According to the guideline of Peck et al. (1974) which is based on the IP, this soil has medium to high swelling potential. Although from the mineralogical perspective (generally in agricultural and geological practice), the soil has a significant portion of silt and contains illite that is not as hydrophilic as the montmorillonite, the free swelling ratio, physical properties, and the associated form of slope failure all suggest that the soil demonstrates nonnegligible swelling potentials that pose threats to the geo-structures. Thus, from the geotechnical engineering perspective, the soil can be regarded as expansive soil.

Specimen preparation

The collected soil samples were air-dried, pulverized, and passed through a 2.0-mm-opening sieve to remove large particles. To simulate the field condition, the prepared dry soil was mixed with distilled water to reach the natural moisture content of 26.3%. The moist soil was statically compacted in three layers in a cylindrical stainless-steel mould with an inner diameter of 38 mm and a height of 76 mm. The specimens were compacted to the natural dry density of 1540 kg/m3. Compacted specimens were immediately extracted from the mould, wrapped by plastic films, and placed in sealed plastic bottles for 72 h at room temperature of 25 ± 1 °C to achieve moisture equilibrium. Thereafter, some as-compacted specimens were directly used for microstructure investigations and triaxial tests (referred to as untreated specimens hereafter). Other specimens were exposed to DW, FT, or DWFT cycles before subjecting to these tests.

Application of DW, FT, and DWFT cycles

Relevant studies suggested that the mechanical behaviour and microstructure of clayey soils generally come into equilibrium after 5 to 7 cycles of DW and FT actions (Liu et al. 2016; Lu et al. 2019; Hu et al. 2019). Therefore, to reveal the evolution and final status of the soil’s structure and mechanical properties, specimens in this study were subjected to 1, 4, 6, and 10 DW, FT, or DWFT cycles.

During DW cycles, the moisture content of specimens varied between the saturated water content (wsat) and shrinkage limit (ws). In one DW cycle, specimens were firstly dried from wn to ws and then wetted to wsat. They were afterward dried to the wn. During drying, specimens were entirely wrapped in filter papers and placed in a temperature and humidity control box where they were dried under a temperature of 25 °C and relative humidity (RH) of 60%. It was found that under an RH = 60%, the specimens dried slowly, and only very few desiccation cracks were found on the surface of the specimens. It typically took about 5 to 6 days to dry the specimens from wsat to ws. During wetting, specimens were entirely covered in filter papers and were periodically (every 24 h) sprayed with distilled water. After each spray, specimens along with the moist filter papers were wrapped in plastic film for moisture equilibrium of 24 h. This procedure was repeated until wsat was achieved. After wetting, the specimens were cut into 10 to 15 small blocks to measure their moisture contents. The differences between the moisture content of soil blocks near the surface and the soil blocks at the core of the specimens were always within 0.2%, confirming that the moisture distribution was uniform within the specimens. Such slow drying and wetting processes help to reduce the nonuniform volumetric strain and the associated development of cracks (Han and Vanapalli 2016).

Closed-system FT cycles were utilized to apply FT histories to the specimens considering the low permeability of expansive soils (especially under unsaturated and frozen conditions) which does not allow significant water migration during freezing (Chamberlain 1973). Specimens wrapped in plastic film and sealed in bottles were directly placed in the temperature and humidity control box. In one FT cycle, specimens were firstly frozen at −20 °C for 12 h and then allowed to thaw at 25 °C for 12 h. The temperature range during FT cycles was determined based on the local annual temperature range in Qiqihar (i.e. −18.6 to 23.1 °C).

In one DWFT cycle, specimens were firstly subjected to one DW cycle and then subjected to one FT cycle immediately afterward. This procedure repeats until the designated number of DWFT cycles was applied.

Measurement of volumetric change

To trace the volumetric behaviour during DW, FT, and DWFT cycles, volume measurements were conducted on specimens after each drying, wetting, freezing, and thawing process using an electronic Vernier calliper with an accuracy of 0.005 mm. Diameter (i.e. d1, d2, and d3) and height (i.e. h1, h2, and h3) measurements were taken at three respective cross-sections that are evenly distributed on the surface of the specimen. The measurements were taken very carefully and gently in order not to disturb the specimens, especially the weak and wet specimens after thawing. The average values of diameter and height measurements were used to calculate the volume of the specimens and further, the volumetric strain (ɛv) during DW, FT, and DWFT cycles.

MIP and SEM tests

MIP tests were conducted on untreated specimens and specimens subjected to 1, 4, and 10 FT, DW, and DWFT cycles. SEM tests were performed on untreated specimens and specimens subjected to 10 FT, DW, and DWFT cycles. Samples used for MIP and SEM tests were 1–2 g in weight and were sampled from compacted specimens following the procedures introduced by Lin and Cerato (2014). The freeze-drying method using nitrogen was employed to dehydrate the samples before the MIP and SEM tests. This method was considered capable of preserving the soil structure during drying (Delage and Lefebvre 1984; Penumadu and Dean 2000; Sasanian and Newson 2013). A PoreMaster 33 porosimeter was used to perform the MIP tests, and A FEI Quanta 200 machine was used for the SEM tests.

Triaxial consolidated undrained test

After the application of FT, DW, and DWFT cycles, specimens were vacuum saturated for 24 h. They were thereafter transferred to the triaxial cell where they were further back-pressure saturated. An initial cell pressure (σc) of 30 kPa and an initial back pressure (σb) of 20 kPa were applied. They were then increased simultaneously at an interval of 20 kPa until the pore-water pressure parameter; B exceeds 0.90 (Kamruzzaman et al. 2009). After back-pressure saturation, the consolidation stage followed during which the σb kept constant, while the σc increases to consolidate the specimens at effective confining pressure σ′c (σ′c = σc −σb) levels of 50, 100, 200, and 300 kPa. The consolidation process was considered finished when the volume of discharged water approached constant and the excess pore water pressure had dissipated (Thu et al. 2006). Finally, the specimens were sheared under undrained conditions at an axial displacement rate of 0.066%/min until the axial strain reached 20%.

Test results and discussions

Volumetric behaviour

The volumetric strain (ɛv) of specimens during DW, FT, or DWFT cycles is defined by Eq. 1.

$${\varepsilon }_{v}=\left({V}_{N}-{V}_{0}\right)/{V}_{0}\times 100\%$$
(1)

where V0 is the initial volume of the untreated specimen, VN is the volume of the specimen after N cycles of DW, FT, or DWFT processes. Positive ɛv indicates swelling while negative ɛv refers to shrinkage.

The evolution of ɛv during DW, FT, and DWFT cycles is shown in Fig. 3 where letters D, W, F, and T indicate drying, wetting, freezing, and thawing processes, respectively. Eventually, DW cycles result in shrinkage (ɛv = −3% after 10 DW cycles), while FT cycles lead to slight swelling (ɛv = 1.2% after 10 FT cycles) in the specimens’ volume. The scale of the variation in the ɛv during DW cycles is higher than that during FT cycles. The two opposite volumetric behaviours offset during DWFT cycles and result in shrinkage of the specimens (ɛv = −2.2% after 10 DWFT cycles). The ɛv during the initial several DW, FT, and DWFT cycles are significant, but its scale reduces with an increasing number of DW, FT, and DWFT cycles (hereafter denoted as NDW, NFT, and NDWFT).

Fig. 3
figure 3

Evolution of volumetric strains during the a DW, b FT, and c DWFT cycles

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The volumetric behaviour becomes elastic after approximately 4 DW, FT, or DWFT cycles. In other words, the accumulation of non-recoverable ɛv becomes almost negligible when NDW, NFT, or NDWFT exceed 4. These observations are consistent with similar studies (Viklander 1998; Tripathy et al. 2002; Wang and Wei 2014; Estabragh et al. 2015; Zeng et al. 2018) and indicate that specimens’ microstructure has come into an equilibrium status (Sridharan and Allam 1982; Dif and Blumel 1991; Basma et al. 1996; Zhang et al. 2006).

Evolution of soils’ microstructure

Based on the volumetric behaviours shown in Fig. 3, it is reasonable to assume that the soil’s microstructure has reached equilibrium after 10 DW, FT, and DWFT cycles. Figure 4 shows MIP results of untreated specimens and specimens subjected to 10 DW, FT, and DWFT cycles in the form of cumulative intrusion (CI) curves (Fig. 4a) and pore size distribution (PSD) curves (Fig. 4b). The CI curves are defined as the relationship between the void ratio intruded by mercury eMIP (eMIP = Vm/Vs where Vm is the volume of intruded mercury and Vs is the volume of the soil solids) and the corresponding pore diameter d. The derivatives of CI curves (i.e. −δeMIP/δlogd versus logd relationships) are the PSD curves. The corresponding SEM images are shown in Fig. 5.

Fig. 4
figure 4

The a cumulative intrusion (CI) curves and b pore size distribution (PSD) curves of specimens untreated and after 10 DW, FT, and DWFT cycles

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Fig. 5
figure 5

SEM images of a untreated specimens and specimens after b 10 FT cycles, c 10 DW cycles, and d 10 DWFT cycles

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The PSD curve of the untreated specimen is bimodal with two peaks at about 1.3 μm and 30 μm, respectively. Such a bimodal PSD curve is typical for compacted soils (Burton et al. 2015). Macropores about 30 μm in dimension can be observed from the SEM image (Fig. 5a). To evaluate the influence of DW, FT, and DWFT cycles on the microstructure, the CI and PSD curves were divided into A, B, and C zones by two boundaries defined at 0.1 μm and 5 μm (see Fig. 4). A total of 0.1 μm is a demarcation value at the left side of which there is nearly no effect of cyclic treatments on PSD curves. Referencing the criterion suggested by Burton et al. (2015), 5 μm is the general turning point for all PSD curves and thus is used as the delimiting boundary which separates the macropores and micropores. Void ratio in the A, B, and C zones (denoted as eMIP,A, eMIP,B, eMIP,C) can be computed from eMIP,A = eMIP,0.01 −eMIP,0.1, eMIP,B = eMIP,0.1 −eMIP,5, eMIP,C = eMIP,5 where eMIP,N is the eMIP value at Nμm. Their values are summarized in Table 2.

Table 2 Evolution of the eMIP after 1, 4, 10 FT, DW, and DWFT cycles
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The following can be observed in Fig. 5 and Table 2:

  1. (i)

    In Zone A, DW, FT, and DWFT cycles impose trivial influence on the shape of the CI and PSD curves as well as the value of eMIP,A.

  2. (ii)

    In Zone B, the eMIP,B decreases during DW, FT, and DWFT cycles. The decrease is most significant during DWFT cycles but is slight during DW cycles. The peak of the micropores shifts to the right, indicating a slight increase in the diameter of the dominant micropores. The frequency (i.e. −δeMIP/δlogd) corresponding to the peak, however, increases after 10 DW cycles but decreases after 10 FT or DWFT cycles.

  3. (iii)

    In Zone C, the peak of macropores in the PSD curve completely disappears after 10 DW cycles, suggesting that the macropores are unstable upon moisture changes. This is accompanied by a significant decrease in the eMIP,C. The collapse of the macropores is also reflected in the SEM image in Fig. 5c where the soil structure becomes uniform and the compaction-induced macropores (see in Fig. 5a) are erased.

  4. (iv)

    In Zone C, dominant macropores with diameters of 30 μm also collapse but FT-induced cracks emerge and develop after 10 FT cycles (see Fig. 5b). The dimension of these cracks is random and varies between 5 and 20 μm, constituting a plateau in the PSD curve. Similar behaviours are also observed for specimens subjected to 10 DWFT cycles. However, due to the offset between the effects of FT cycles and DW cycles, the dimension and amount of the induced cracks are smaller than the FT-induced cracks (compare cracks in Fig. 5b and d) and concentrates at about 7 μm.

It should be noted that the total intruded void ratio (i.e. eMIP0.01) of the specimen after 10 FT cycles is lower than that of the untreated specimen (see Fig. 4a), while the volumetric expansion was observed from FT specimens (see Fig. 3b). This is due to the existence of FT-induced cracks with dimensions that are larger than the maximum pore diameter a MIP test can detect (about 250 μm). These cracks are visible to naked eyes, but their volume is not reflected by the CI curves.

The volumetric behaviour of the soil is closely related to the evolution of its microstructure. During DW cycles, the collapse of the macropores and the remarkable reduction in the eMIP,C lead to the final shrinkage of the global volume. On the other hand, the development of the cracks during FT cycles creates extra pore space and increases the eMIP,C as well as in global volume. The collapse of the macropores and development of cracks offset during DWFT cycles and lead to an intermediate volumetric behaviour.

Figure 3 shows that the scale of the ɛv decreases with increasing NDW, NFT, or NDWFT in the first few cycles before reaching equilibrium. This is generally due to two reasons: (i) the initial ɛv contains plastic and elastic strain, while the plastic strain quickly decreases during the first few cycles and (ii) cracks induced by DW and FT cycles reduce the integrity of the soil structure and segregate the soil matrix. The deformation during DW and FT cycles was partly offset by the widening of these cracks.

Shear strength and elastic modulus during consolidated undrained shearing

Figures 6, 7, and 8 summarize the stress–strain relationships (i.e. deviator stress q versus axial strain ɛ1; q = σ1 −σ3, σ1 is the major principal stress and σ3 is the minor principal stress, σ3 = σc) during consolidated undrained shearing for specimens subjected to DW, FT, and DWFT cycles. Seating errors were corrected in these relationships following the approach suggested by Atkinson and Evans (1985). It can be observed that the stress–strain curves of all specimens (regardless of the type and number of treatments) belong to the strain-stabilization type under low σc (i.e. 50, 100 kPa). For higher σc (i.e. 200, 300 kPa), the stress–strain curves start to show strain-hardening characteristics.

Fig. 6
figure 6

Stress–strain relationships after DW cycles

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Fig. 7
figure 7

Stress–strain relationships after FT cycles

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Fig. 8
figure 8

Stress–strain relationships after DWFT cycles

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To determine the soil’s undrained elastic modulus and shear strength, the stress–strain curves were fitted using the hyperbolic model (Eq. 1) proposed by Kondner (1963).

$$q=\frac{{\varepsilon }_{1}}{a+b{\varepsilon }_{1}}$$
(2)

where ɑ and b are model parameters. According to Gutierrez et al. (2008), Moniz (2009), and Ladd et al. (1977), 1/ɑ is the slope of the initial curve and equals the undrained elastic modulus, Eu. 1/b is the ultimate deviator stress and indicates the undrained shear strength, qu.

Mohr–Coulomb model (Eq. 3) was used to interpret the undrained shear test results and determine the soil’s cohesion (c, kPa) and internal friction angle (ϕ, °) in terms of total stress.

$${\sigma }_{1}=\frac{2c\ \mathrm{cos}\phi }{1-\mathrm{sin}\phi }+\frac{1+\mathrm{sin}\phi }{1-\mathrm{sin}\phi }{\sigma }_{3}$$
(3)

Variation of the determined qu, Eu, c, and ϕ with N is summarized in Figs. 9, 10, and 11, respectively. A power function (Eq. 4) is adopted to describe the reduction of qu, Eu, c, and ϕ with N.

$$\frac{{\Omega }_{N}}{{\Omega }_{0}}=\frac{1+\alpha {e}^{-\beta N}}{1+\alpha }$$
(4)

where Ω0 collectively represents the qu, Eu c, or ϕ of untreated specimens; ΩN is the qu, Eu c, or ϕ after N cycles of treatment; α and β are model parameters; and e is the base of the natural logarithm and equals 2.7182. The fitting curves are shown in Figs. 9, 10, and 11, and the fitting parameters (i.e. α, and β) are summarized in Table 3. The following can be observed:

Fig. 9
figure 9

Variation of shear strengths after a DW, b FT, and c DWFT cyclic treatments

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Fig. 10
figure 10

Variation of elastic modulus of specimens after a DW, b FT, and c DWFT cyclic treatments

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Fig. 11
figure 11

Evolution of (a) total cohesion and (b) total friction angle with the increase of cycle numbers

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Table 3 Values of parameters in Eq. 4
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  1. (i)

    The Eu, c, and ϕ decrease with DW, FT, and DWFT cycles, and their behaviours are consistent. The decrease is comparatively more significant during DW cycles than FT cycles. The most remarkable reduction takes place during DWFT cycles due to the combined damages of FT and DW cycles. They reach an equilibrium after approximately 6 cycles. These observations are consistent with similar studies in the literature (Rosa et al. 2017; Zeng et al. 2018; Wang et al. 2019; Tian et al. 2019; Tang et al. 2018; Han et al. 2018; Hossain et al. 2016).

  2. (ii)

    The σc appears to have less impact on the Eu-N relationships, and the same pair of α and β values can suitably describe the Eu-N relationships under different σc.

  3. (iii)

    For qu specifically, the DW, FT, and DWFT cycles have similar effects on the qu-N relationships because a unique pair of α and β values well describes the qu-N relationships under all types of treatment under different σc.

Modelling stress–strain behaviours

A simple normalized model is developed to describe all stress–strain curves of the tested specimens after different DW, FT, and DWFT cycles, as shown in Figs. 6, 7, and 8. Reordering Eq. 2, substituting 1/a = Eu and 1/b = qu, and multiplying both sides by a normalization factor X yield (Eq. 5).

$$X\frac{{\varepsilon }_{1}}{q}=\frac{X}{{E}_{u}}+\frac{X}{{q}_{u}}{\varepsilon }_{1}$$
(5)

To normalize Eq. 5, factors X/Eu and X/qu must be constants such that the right side of Eq. 5 is independent of σc. The σc, qu, and Eu are typical candidates for factor X when normalizing stress–strain curves (Xu et al. 2005). qu is adopted in this study as the X due to two reasons: (i) qu is one of the simplest and most widely used soil properties in geotechnical practice and (ii) the qu-N relationships during DW, FT, and DWFT cycles are similar and can be described by a unique equation. Substituting X = qu into Eq. 5 yields Eq. 6.

$${q}_{u}\frac{{\varepsilon }_{1}}{q}=\frac{{q}_{u}}{{E}_{u}}+{\varepsilon }_{1}$$
(6)

The Eu-qu relationships for each type of treatment are consistent regardless of the differences in the σc and the number of treatments, as shown in Fig. 12. They can be well fitted by linear equations, which show that qu/Eu equals 0.00967 for FT cycles, 0.0134 for DW cycles, and 0.0129 for DWFT cycles.

Fig. 12
figure 12

Relationship of the elastic modulus and shear strength for specimens of DWFT cycle

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Based on Eq. 4, the qu-N relationship can be expressed as Eq. 7.

$${q}_{uN}={q}_{u0}\left(\frac{1+\alpha {e}^{-\beta N}}{1+\alpha }\right)$$
(7)

where quN and qu0 are the qu of specimens after N cycles of DW, FT, or DWFT treatment and before treatment, respectively. Substituting Eq. 7 into Eq. 6 yields the normalized expression of q-ε1 relationship.

$$q={q}_{u0}\left(\frac{1+\alpha {e}^{-\beta N}}{1+\alpha }\right)\frac{{\varepsilon }_{1}}{{\varepsilon }_{1}+\frac{{q}_{u}}{{E}_{u}}}$$
(8)

It is noted that α and β are constants regardless of σc and the number and type of treatments (α = 0.398, β = 0.395; see Table 3). The stress–strain curves predicted by Eq. 8 are shown in Figs. 6, 7, and 8. It can be observed that good agreements are achieved between the measurements and the predictions of Eq. 8.

Conclusions

In this paper, the influence of multiple DW, FT, and DWFT cycles on the microstructure and mechanical behaviours (including volumetric behaviour, stress–strain behaviour during consolidated undrained shearing, undrained shear strength parameters, and undrained elastic modulus) of an expansive soil was investigated. The following observations and conclusions were obtained from the experimental studies:

  1. (i)

    Compaction-induced macropores in the range of 5 to 250 μm collapse during DW cycles. In the same dimension range, FT cycles introduce cracks into the soil structure. These two types of opposite influences offset during the DWFT cycles and result in an intermediate soil structure. DW, FT, and DWFT cycles appear to have a minor influence on the micropores with diameters less than 5 μm.

  2. (ii)

    The tested soil swelled during FT cycles but shrank during DW cycles. The interplay of the two opposite volumetric behaviours results in slight shrinkage of the soil during DWFT cycles. The volumetric behaviours become elastic after approximately 4 cycles during all three types of treatments.

  3. (iii)

    The qu, Eu, c, and ϕ reduce during DW, FT, and DWFT cycles, and the reduction is most significant during DWFT cycles due to the combined damage of FT and DW processes. The reduction comes to equilibrium after approximately 6 cycles.

A simple normalized model is developed to predict the stress–strain curves of specimens undergoing 0, 1, 4, 6, and 10 DW, FT, and DWFT cycles. The performance of this model is verified by the triaxial compression test results. The model presented in this paper could be useful for the estimation of the mechanical behaviours of expansive soils in seasonally frozen regions.