Introduction

Soils near the ground surface experience multiple drying and wetting cycles according to weather and climatic conditions. As soil dries, it experiences shrinkage and tensile cracks. These cracks affect the integrity of soil structures such as slopes, dams, and embankments in terms of permeability and strength. Cracks in the soil provide an easy pathway for rainwater infiltration and such phenomenon is commonly associated with failures in slopes, dams, and earth structures. Tensile cracks are highly influenced by the tensile strength of soils (Morris et al. 1992; Trabelsi et al. 2012; Vaniceek 2013; Shi et al. 2014; Li et al. 2019; Tang et al. 2019). Research shows that the tensile strength of unsaturated soils is mainly influenced by matric suction of the unsaturated soils (De Souza Villar et al. 2009; Yin and Vanapalli 2018).

Tensile strength tests for soils are generally grouped based on the method of load application: direct and indirect tensile tests. Direct tensile strength tests usually involve constraining one of the boundaries of the test specimen and applying uniaxial tensile force to the opposite boundary. The direct tensile test is regarded as the only method where a specimen is subjected to true uniaxial tension and failure occurs along its longitudinal axis (Peters and Leavell 1988; Win 2006). In a direct tensile test, it is assumed that uniform stresses are applied to the specimen, and torsional and bending stresses are absent. The tensile strength is computed as a ratio of the maximum load sustained by the specimen and the cross-sectional area on which the load acts. Direct tensile strength tests have been conducted on soils (Tschebotarioff et al. 1953; Hasegawa and Ikeuti 1966; Ajaz and Parry 1975; Peters and Leavell 1988; Tang and Graham 2000; Trabelsi et al. 2010; Li et al. 2019; Murray and Tarantino 2019).

Despite the advantages of direct tensile strength test, the validity of the tests has been questioned due to difficulties associated with the test such as difficulty to effectively clamp or hold the specimen at the ends, misalignment, stress concentration, and eccentric loading (Kennedy and Hudson 1968). Creep and volume changes as a result of sustained loading during the test have also been reported (Win 2006).

To address some of the difficulties in direct tensile tests, indirect tensile tests have been developed. Although the failure mode in indirect tensile test is a combination of compression and tension, indirect tensile tests have several advantages over direct tensile tests. It is relatively simple, failure is located in a region of uniform tensile strength, the specimens and equipment are the same as the compression tests, the surface conditions of the specimen do not affect the failure, and there is less variation of the test results (Kennedy and Hudson 1968). Many studies have been conducted on soils using indirect tensile tests (e.g., Uchida and Matsumoto 1961; Kennedy and Hudson 1968; Khrishnayya and Eisenstein 1974; Al-Hussaini 1981; Dexter and Kroesbergen 1985; Das et al. 1995; Li and Wong 2013; Akin and Likos 2017; Pittaro 2019). Among the indirect tensile tests, the Brazilian tensile strength (BTS) test is the most frequently used (Khrishnayya and Eisenstein 1974; Das et al. 1995; Vesga and Vallejo 2006; De Souza Villar et al. 2009; Beckett et al. 2015; Iravanian and Bilsel 2016; Akin and Likos 2017). Comparison between direct and indirect tensile tests shows no significant differences between the tensile strength (e.g., Vesga and Vallejo 2006; Fahimifar and Malekpour 2012; Kim et al. 2012).

Several tensile strength models for unsaturated soils considering the effects of matric suction have been proposed. However, the tensile strength model is either for cohesionless (coarse-grained) soils or clayey (fine-grained) soils. Generally, tensile strength models considering the effects of matric suction have been developed for coarse-grained soils rather than for fine-grained soils and hence tensile strength data together with soil–water characteristic curve data for fine-grained soils are scarce in the literature. In some literature, unsaturated soils were prepared by compacting a soil at various water contents to the same dry density, but the soils prepared in this manner have a different soil structure. Such data are not used in the present study.

The present study aims to develop a tensile strength model considering the effect of matric suction for both coarse and fine-grained soils. As tensile strength data for fine-grained soils with the same soil structure and soil–water characteristic curve data are lacking in the literature, the present study includes Brazilian tensile strength tests on fine-grained soils initially compacted wet of optimum and allowed to dry to various water contents. The matric suctions of the compacted soil specimens were measured using the contact filter paper method (ASTM D5298–16 2016) as well as estimated from its soil–water characteristic curve (ASTM D6836-16 2016).

Tensile strength models

Several theoretical models (Fisher 1926; Morris et al. 1992; Lu et al. 2009; Varsei et al. 2016) and empirical models (Kim and Hwang 2003; Munkholm and Kay 2014; Yin and Vanapalli 2018; Wang et al. 2020) have been developed to predict the tensile strength of soils. Tensile strength models based on the macromechanics approach compared to the micromechanical approach are more attractive for practical applications (Yin and Vanapalli 2018). A summary of the tensile strength models based on the macromechanics approach is presented in Table 1. In Table 1, the models of Morris et al. (1992), Trabelsi et al. (2012), Tang et al. (2015), and Varsei et al. (2016) were developed for clayey soils while the models of Lu et al. (2009), Yin and Vanapalli (2018), and Wang et al. (2020) were developed for cohesionless soils. In most of the models shown in Table 1, the tensile strength σt of unsaturated soils is usually obtained from the Mohr–Coulomb criterion with the y-axis intercept given by the apparent cohesion capp as shown in Fig. 1. The σt is obtained from either direct uniaxial tensile test or indirect tensile test. If the uniaxial tensile test is used, the Mohr circle will have the minor principal stress as − σt and major principal stress as 0. Based on Fig. 1, this case will give

$$ {\sigma}_{\mathrm{t}}=\frac{2\cos {\phi}_t}{1+\sin {\phi}_t}{c}_{\mathrm{app}}^{\ast };\kern0.5em \mathrm{or}\ \mathrm{alternatively}\kern0.5em {\sigma}_{\mathrm{t}}=2\tan \left(\frac{\pi }{4}-\frac{\phi_t}{2}\right){c}_{\mathrm{app}}^{\ast } $$
(1)
Table 1 Summary of tensile strength models with consideration of matric suction using the macroscopic approach
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Figure 1
figure 1

Representations of uniaxial and Brazilian tensile strength tests using Mohr–Coulomb failure criterion

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The parameters of ϕt and c*app are defined in Fig. 1.

If an indirect tensile test such as the Brazilian tensile test (ASTM D3967–16 2016) is used, the Mohr circle will have the minor principal stress as − σt and the major principal stress as about 3.1σt (Li and Wong 2013; Sivakugan et al. 2014; Consoli et al. 2015). In this case, σt is given by Eq. 2.

$$ {\sigma}_{\mathrm{t}}=\frac{\cos \phi \hbox{'}}{\left(2.05-1.05\sin \phi \hbox{'}\right)}{c}_{\mathrm{app}} $$
(2)

The parameters of ϕ′ and capp are defined in Fig. 1.

Except for Snyder and Miller (1985) in Table 1, the other tensile strength models can be expressed in the form of either Eq. 1 or 2 where capp can be expressed in the form of the total cohesion in the extended Mohr–Coulomb criterion for unsaturated soils (Fredlund et al. 1978) as shown in Eq. 3.

$$ {c}_{\mathrm{a}\mathrm{pp}}=c\hbox{'}+\left({u}_{\mathrm{a}}-{u}_{\mathrm{w}}\right)\tan {\phi}^b $$
(3)

where c′ is the effective cohesion, ua is pore air pressure, uw is pore water pressure, and ϕb is the angle indicating the rate of shear strength increase due to matric suction (ua–uw). Equation 3 can also be expressed more generally in term of a suction stress σs as shown in Eq. 4.

$$ {c}_{\mathrm{app}}=c\hbox{'}+{\sigma}^s\tan \phi \hbox{'} $$
(4)

In the tensile strength model of Morris et al. (1992), the coefficient of 0.5 based on Frydman (1967) and Baker (1981) and the coefficient of (0.5 to 0.7) cot ϕ′ based on Bagge (1985) shown in Table 1 can be examined in the context of Eqs. 1 and 2. Figure 2 shows a plot of coefficient \( \left(\frac{2\cos {\phi}_t}{1+\sin {\phi}_t}\right) \) from Eq. 1 and coefficient \( \left[\frac{\cos \phi \hbox{'}}{\left(2.05-1.05\sin \phi \hbox{'}\right)}\right] \) from Eq. 2 versus the friction angle ϕt or ϕ′ for the typical range from 15 to 50°. It can be seen that the coefficient \( \left(\frac{2\cos {\phi}_t}{1+\sin {\phi}_t}\right) \) is close to 0.7cot ϕ′ for the range of ϕ′ = ϕt from 28 to 40° and the coefficient \( \left[\frac{\cos \phi \hbox{'}}{\left(2.05-1.05\sin \phi \hbox{'}\right)}\right] \) is close to 0.5 for the full range of friction angle.

Fig. 2
figure 2

Comparison of coefficients in Eqs. 1 and 2 with Morris et al. (1992) model

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For the other tensile strength models in Table 1, Eq. 1 is used and only differs in the definition of σs and the presence or absence of a cohesion term. The tensile strength models of Tang et al. (2015), Yin and Vanapalli (2018), and Wang et al. (2020) originated from Lu et al. (2009). The differences among these models lie in the definition of σs which can be expressed in the general form of Eq. 5.

$$ {\sigma}^{\mathrm{s}}={S}_{\mathrm{e}}^k\left({u}_{\mathrm{a}}-{u}_{\mathrm{w}}\right)+f $$
(5)

where Se is the effective degree of saturation \( \left(=\frac{S-{S}_{\mathrm{r}}}{1-{S}_{\mathrm{r}}}\right) \), Sr is the residual degree of saturation, and exponent k is a parameter. The first term on the right-hand side of Eq. 5 is attributed to capillary suction while the second term (f) is attributed to surface tension of the “capillary bridges” (Yin and Vanapalli 2018; Wang et al. 2020). Only the exponent k in Yin and Vanapalli (2018) model is not unity. Yin and Vanapalli (2018) relate k to be a function of van Genuchten (1980) soil–water characteristic curve (SWCC) equation parameter n and capillary degree of saturation Sc which is dependent on the coefficient of uniformity Cu. The equations are shown in Table 1. The effective degree of saturation Se is defined differently in Varsei et al. (2016) who make use of a microscopic degree of saturation \( {S}_{\mathrm{r}}^{\mathrm{m}} \) following Alonso et al. (2010) instead of residual degree of saturation Sr as shown in Eq. 6.

$$ {S}_{\mathrm{e}}^{\mathrm{m}}=\frac{S-{S}_{\mathrm{r}}^{\mathrm{m}}}{1-{S}_{\mathrm{r}}^{\mathrm{m}}}={(S)}^{\alpha \ast } $$
(6)

Instead of determining \( {S}_{\mathrm{r}}^{\mathrm{m}} \), Varsei et al. (2016) proposes Eq. 7 where α* is a material parameter and is always equal to or greater than 1.

$$ {S}_{\mathrm{e}}^{\mathrm{m}}={(S)}^{\alpha \ast } $$
(7)

The term f is zero in Lu et al. (2009), Trabelsi et al. (2012), Tang et al. (2015), and Varsei et al. (2016). Wang et al. (2020) followed Yin and Vanapalli’s (2018) formulation of f to be a product of surface tension of water Ts and the specific air–water interfacial area per pore volume aaw, i.e., f = Ts·aaw. The equations are given in Table 1.

Lu et al. (2009), Tang et al. (2015), Yin and Vanapalli (2018), and Wang et al. (2020) make use of the van Genuchten (1980) SWCC equation in their tensile strength model. However, Wang et al. (2020) propose equations to determine the van Genuchten (1980) SWCC equation parameters α and n, as shown in Table 1.

As the models of Lu et al. (1992), Yin and Vanapalli (2018), and Wang et al. (2020) were developed for cohesionless soils, the cohesion term is zero. However, three out of ten soils that Yin and Vanapalli (2018) used in the development of their model are fine-grained soils. In contrast, the models of Trabelsi et al. (2012), Tang et al. (2015), and Varsei et al. (2016) were developed for clayey soils and hence contain a cohesion term. Only Varsei et al. (2016) kept the cohesion term as the effective cohesion. In Trabelsi et al. (2012) model, the cohesion term is a function of porosity while in the Tang et al. (2012) model, the cohesion is given by the residual tensile strength at full saturation. However, Tang et al.’s (2015) model was developed for soils compacted at various water contents to the same dry density. Thus, Tang et al.’s (2015) model assumes that the influence of the soil structure on σt, α, and n is negligible.

In summary, the macroscopic approach tensile strength models showed variations only in the definition of the suction stress σs and the presence of a cohesion term c for clayey soils only. The general form of the macroscopic approach tensile strength model is expressed in Eq. 8.

$$ {\sigma}_{\mathrm{t}}=2\tan \left(\frac{\pi }{4}-\frac{\phi }{2}\right)\left({\sigma}_{\mathrm{s}}\tan \phi +c\right) $$
(8)

The suction stress σs consists of either one term (Lu et al. 2009; Tang et al. 2015; Varsei et al. 2016) or two terms (Yin and Vanapalli 2018; Wang et al. 2020).

In the development of the tensile strength model for unsaturated soils, the grain size distribution curve, SWCC, and effective shear strength parameters for the saturated soil, c′ and ϕ′, are required. The tensile strength tests should also be performed for a soil either undergoing drying or wetting process and not on soils prepared at various water contents to avoid effects of soil structure on the test results. Tensile strength test data with SWCC are more commonly found for coarse-grained (cohesionless) soils than for fine-grained (clayey) soils in the literature, hence tensile strength tests and SWCC tests are performed for fine-grained soils in the present study.

Materials and methods

Soil materials

Two residual soils, one from the Bukit Timah Granite (BT) and the other from the Jurong Formation (JF) in Singapore, were used as the fine-grained soils in the present study. The Bukit Timah Granite is mainly an acidic igneous rock system while the Jurong Formation is predominantly a folded sedimentary rock system (Leong et al. 2002b). The basic properties of the two residual soils are summarized in Table 2.

Table 2 Basic properties of the soils used in study
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The standard (SP) and modified Proctor (MP) compaction curves of the residual soils according to ASTM D698 - 12e2 (2015) and ASTM D1557-12e1 (2015), respectively, are shown in Fig. 3. The maximum dry densities for the standard Proctor effort for BT and JF are 1.69 and 1.76 Mg/m3, respectively, corresponding to an optimum water content of 16% for both soils. The maximum dry densities for the modified Proctor effort for BT and JF are 1.86 and 1.94 Mg/m3, corresponding to optimum water contents of 13.3 and 13.5%, respectively. The compaction properties of BT and JF residual soils are also included in Table 2. The preparation of the samples is described below.

Fig. 3
figure 3

Compaction curves and sample sets

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Sample preparation and test procedures

Compaction

The air-dried residual soils were passed through sieve no. 4 and mixed with distilled water to the target moisture contents. The soils were then sealed in Ziploc bags and stored in a humidity chamber for at least 5 days to allow for moisture equalization.

To obtain test specimens of about 50 mm in diameter for the BTS test, a two-part split PVC cylindrical mold (internal diameter of 52 mm and height of 102 mm) was fabricated. A compaction apparatus was fabricated to ensure that the compaction procedure is consistent for the preparation of the soil samples. The compaction apparatus, shown in Fig. 4, was equipped with a pulley system to lift a cylindrical drop mass of 4.5 kg and dimensions of 75 mm diameter and 150 mm height with a center hole. The drop mass center hole slides along a PVC tube guide which is slotted onto a dowel on top of a solid steel cylinder, diameter 51.6 mm and height of about 100 mm, which acts as the anvil. The anvil is placed on top of the soil in the PVC two-part split mold. A PVC collar sits on top of the PVC two-part split mold to constrain the anvil. The PVC two-part split mold is held together with hose clips and rests on the standard compaction mold base plate. The PVC two-part split mold is held upright at mid-height by an acrylic square plate with a center hole that is held in place using the screws of the standard compaction mold base plate and nuts. The height of fall of the drop mass was calibrated to deliver standard and modified Proctor energies to the soil. For the standard Proctor energy, the drop mass falls through a height of 200 mm and the soil is compacted in three layers while for the modified Proctor energy, the drop mass falls through a height of 450 mm and the soil is compacted in five layers. For both compaction energies, each layer in the PVC mold was subjected to six blows. The compacted sample produced has a height of about 100 mm.

Fig. 4
figure 4

Compaction set-up to prepare soil specimens in PVC mold

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Before each compaction, the internal walls of the PVC split mold were lightly oiled. During compaction, each layer was scarified before placing another layer to allow for bonding between the layers. The soils were compacted wet of optimum on both the standard and modified Proctor curves, indicated as A to D for BT soils and E to G for JF soils in Fig. 3a and b, respectively. As each of these points are compacted at different water contents wet of optimum, the soil structure is different and thus each point can be said to be a “different” soil. For each point, A to G, multiple samples were prepared and collectively, these samples are denoted as sample sets A to G and the average soil properties are summarized in Table 3.

Table 3 Properties of sample sets A to G
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Brazilian tensile strength (BTS) test

Tensile strength of soil is commonly determined in the laboratory by direct and indirect tensile test methods. Although the indirect tensile test has been criticized as being less reliable, research has shown that the indirect tensile tests have advantages over the direct tensile test such as indirect tensile test is relatively simpler to perform; the specimens and equipment can be the same as for compression tests; the failure is located in a region of uniform tensile strength; the surface condition of the specimen does not affect the failure and there is less variability in the test results (Al-Hussaini and Townsend 1973; Vaniceek 2013; He et al. 2018).

The Brazilian tensile strength (BTS) test is an indirect tensile test originally developed for rock (ASTM D3967–16 2016) but has been used to obtain the tensile strength of soils (e.g., Khrishnayya and Eisenstein 1974; Das et al. 1995; Vesga and Vallejo 2006; De Souza Villar et al. 2009; Beckett et al. 2015; Iravanian and Bilsel 2016; Akin and Likos 2017). A review on the development of the BTS test can be found in Li and Wong (2013). In the BTS test, the applied load is compressive but the failure mode is tensile if specific constraints on specimen geometry (thickness to diameter, t/d, and ratio) are met. In ASTM D3967–16 2016, the t/d ratio is recommended to be within the range of 0.2 to 0.75. Akin and Likos (2017) found that the tensile strength is essentially constant when t/d ratio is greater than 0.42 for kaolin. In the present study, the t/d ratio of the test specimens was maintained as close to 0.6 as possible. Hence, each compacted sample of 100 mm length was sawn using Buehler IsoMet™ 4000 precision saw into three cylindrical disk specimens of about 30 mm thick and 52 mm in diameter. Although the precision saw can minimize sample deformation and have low kerf loss, the compacted sample was wrapped in two layers of cling film followed by a single layer of masking tape along the sample height to prevent moisture loss and to limit surface cracking during sawing. The disk specimens were then dried to various water contents in airtight desiccators with saturated sodium chloride (NaCl) solution at the base of the desiccators. The saturated NaCl solution provides a constant relative humidity of about 76% (Young 1967) to dry the test specimens. Weight and volume of the disk specimens were measured regularly to provide the shrinkage curve. Test specimens that attained the targeted weight (and hence moisture content) were wrapped in cling film and thereafter kept in a temperature-controlled humidity chamber to allow for moisture equalization for about 10 days as recommended by Mendes (2011) before testing.

The test specimens for the BTS in the present study were loaded using two specially fabricated curved bearing blocks as recommended in ASTM D3967–16 2016 to reduce contact stress. The radius of the contact arc and width of contact with the test specimen were 10° and 30 mm, respectively. During testing, the bearing blocks were attached to the loading platens of an unconfined compression test machine using magnets embedded into the base of the bearing block (Fig. 5). The BTS test was conducted at a loading rate of 0.5 mm/min and all test specimens failed within 1 to 10 min as recommended in ASTM D3967–16 2016. The tensile strength σt was computed from the BTS test results using Eq. 9 (ASTM D3967–16 2016).

$$ {\sigma}_t=1.272\frac{P}{\pi tD} $$
(9)

where P is the maximum applied load in the BTS test, t is the thickness of the specimen, and D is the diameter of the specimen. The number of tensile tests conducted for each sample set ranges from 35 to 66 giving a total of 343 tensile test results. Observations during the BTS tests show that crack initiated at the center of the disk and the stress–displacement curves exhibited brittle failure in almost all the tests indicating the general applicability of the BTS test for the compacted soils in the present study (Frydman 1964, 1967).

Fig. 5
figure 5

Brazilian test set-up and curved bearing block

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Suction measurements

Suction measurements were conducted on the compacted soil samples using the contact filter paper method following Leong et al. (2002a). Whatman No. 42 filter paper of diameter 42.5 mm was used. The matric suction calibration equations for Whatman No. 42 filter paper used were those suggested by Leong et al. (2002a) and reproduced in Eq. 10.

$$ \log \psi =\left\{\begin{array}{c}2.909-0.0229{w}_{\mathrm{f}},\kern0.5em {w}_{\mathrm{f}}\ge 47,\\ {}\begin{array}{cc}4.945-0.0673{w}_{\mathrm{f}},& {w}_{\mathrm{f}}<47\end{array}\end{array}\right. $$
(10)

where ψ is the matric suction and wf is the water content of the filter paper in percent.

For the suction measurement, the compacted soil sample was sawn into four cylindrical disk specimens of about 25 mm height each. A piece of filter paper was sandwiched between two disk specimens and then wrapped in cling film and taped to minimize moisture loss as suggested by Bulut et al. (2001). Each of the taped specimens was then placed in an airtight container and kept in a constant temperature chamber for 28 days to allow for suction equilibration. At the end of 28 days, procedures recommended by Bulut et al. (2001) were followed to determine the water content wf of the filter paper. The matric suction of the compacted soil was then computed using Eq. 10.

Soil–water characteristic curves

The SWCCs of the compacted soils were established using both the pressure plate apparatus (method C) and the chilled-mirror dewpoint technique (method D) following ASTM D6836-16 (2016). For suctions up to and including 1000 kPa, a pressure plate apparatus with a 15-bar ceramic plate cell was used. For suctions higher than 1000 kPa, the WP4C dewpoint potentiometer (Leong et al. 2003) was used.

Before commencement of the pressure plate test, the compacted samples were saturated following the procedure suggested by Agus et al. (2001) and then disk specimens of 50 mm diameter and 30 mm height were used. The suction level in the pressure plate extractor was increased incrementally from 10 to 1000 kPa as recommended in ASTM D6836-16 (2016). At equilibrium, the weight and volume of the specimens were measured to provide the shrinkage curve.

For the WP4C dewpoint potential meter, a cylindrical disk specimen of diameter 35 mm and thickness 5 mm was cookie-cut from the saturated compacted sample using a PVC ring with a sharpened edge and placed into the WP4C sample cups. These specimens in the sample cups were dried in airtight desiccators containing saturated NaCl solution. The weights of the sample cups with the soils were monitored periodically as the soil dries and concurrently, the suction was measured using the WP4C dewpoint potentiometer. The drying continued until no further change in weight could be observed (less than 0.001 g). The sample cups with the soils were then left outside the desiccator (laboratory environment at 22 ± 4 °C and relative humidity at ~ 60%) to allow for further drying and measuring the suction using the WP4C dewpoint potentiometer until no further weight change could be detected (less than 0.001 g).

Results and discussion

Tensile strength tests

The test results for sample sets A to D and E to F are summarized in Fig. 6 a and b, respectively. For all the sample sets, drying of the soil specimens from the wet of optimum to dry of optimum showed an increase in σt. The σt values increase up to a certain limit and then generally becomes constant despite further drying or increase in soil suction. Some data show a slight drop in σt at extremely low water contents. The degree of saturation at which σt ceases to increase significantly varies between 95 and 97% for all the sample sets. Locating these degrees of saturation on the shrinkage curves indicates that they correspond approximately to the curvature of the shrinkage curve, which begins to depart from the 100% degree of saturation line and approaches the minimum void ratio. As the soil dries, its volume decreases and suction increases. These factors collectively influence σt. However, the contribution of suction to σt becomes increasingly minimal past the air-entry value (AEV) of the soil since the water phase becomes increasingly discontinuous in the soil. Hence, σt ceases to increase and becomes almost constant when the shrinkage limit is reached. Any deviation from the constant value is attributed to micro-fissures in the specimen and can be classified as experimental errors.

Fig. 6
figure 6figure 6

a Summary of test results for sample sets A to D. b Summary of test results for sample sets E to G

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Evaluation of tensile strength models

Review of the tensile strength models showed a general form and two variations of suction stress which can be represented by the models of Lu et al. (2009) and Yin and Vanapalli (2018).

For evaluation of the tensile strength models, ϕ′ is needed. The ϕ′ values for sample sets A to G were obtained from consolidated undrained triaxial tests for saturated soil specimens of each sample set conducted following ASTM D4767–11 (2011) (details are reported in Kizza 2019). The effective stress shear strength parameters (c′ and ϕ′) and the van Genuchten (1980) SWCC equation parameters (α and n) for sample sets A to G are tabulated in Table 4.

Table 4 Shear strength and SWCC equations’ parameters for sample sets A–G
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Lu et al. (2009) suggest that ϕt be obtained using a non-linear Mohr–Coulomb envelope in the low-stress zone and can be greater than the effective friction angle ϕ′ (see Fig. 1). For the presentation of results, two cases of ϕt values are shown, i.e.

  1. Case 1.

    All parameters were as evaluated from experimental data, i.e., α and n of the van Genuchten (1980) SWCC equation from curve fitting the SWCC data while ϕt = ϕ′ as given in Table 4.

  2. Case 2.

    Values of α and n of the van Genuchten (1980) SWCC equation as for case 1 while ϕt was adjusted to obtain the best match with the experimental data.

The performance of Lu et al. (2009) tensile strength model is shown in Fig. 7 a. In Fig. 7 a, the SWCC is also plotted. To evaluate the performance of the tensile strength model against the experimental data, the root mean square error (RMSE) was computed. The RMSE is defined in Eq. 11.

$$ RMSE=\sqrt{\frac{\sum_{i=1}^n{\left({X}_{\mathrm{obs},i}-{X}_{\mathrm{model},i}\right)}^2}{n}} $$
(11)
Fig. 7
figure 7figure 7

a Evaluation of Lu et al. (2009) tensile strength model (sample sets A-D). b Evaluation of Lu et al. (2009) tensile strength model (sample sets E–G)

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where Xobs, i = observed/measured value, Xmodel, i = model value as predicted at the same conditions as observed value, and n = number of data points. A smaller RMSE indicates a better agreement.

As expected, case 2 gives the better estimate as compared to case 1 for all sample sets A to G. The values of α and n of the van Genuchten (1980) SWCC equation seem too low for Lu et al. (2009) tensile strength model to estimate the test data. In case 2, the value of ϕt needs to be reduced from ϕ′ to get a better match with the test data, but there is no justification for the reduction in ϕt. Table 5 summarizes the ϕt and RMSE values for all the sample sets A to G. If a cohesion term was included in Lu et al.’s (2009) tensile strength model, the value of ϕt would need to be further reduced to achieve a good match with the test data.

Table 5 Summary of ϕt and RMSE for Lu et al. (2009) tensile strength model
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Similar to Lu et al.’s (2009) tensile strength model, Yin and Vanapalli’s (2018) tensile strength model could not estimate the test data well. For the presentation of results, two cases that gave the best estimate of the test data are presented:

  1. Case 1.

    The parameters k and ηs were computed using the equations in Table 1while ϕt = ϕ′.

  2. Case 2.

    The parameters k and ηs were as for case 1, while ϕt was adjusted to give the best match with the experimental data.

The void ratio e in Eq. 8 is taken to be emin from the shrinkage curve (Leong and Wijaya (2015). For the computation of k, the ratio S0/Sc was set at 0.9 and 0.6 for BT and JF soils, respectively, within the range of values given by Yin and Vanapalli (2018).

Figure 8a shows the two cases where σt is plotted against S for all the sample sets A to G together with their SWCCs. Table 6 summarizes the parameters and the RMSE values for all the sample sets A to G. As expected, case 2 gives the lower RMSE for all the sample sets. Similar to Lu et al.’s (2009) tensile strength model, ϕt in Yin and Vanapalli (2018) model needs to be reduced from ϕ′ to obtain a better match with the test data and a further reduction is needed if a cohesion term is included in Yin and Vanapalli (2018) model. The model requires the use of either larger values of k and ηs or a smaller ϕt value to get a better match.

Fig. 8
figure 8figure 8

a Evaluation of Yin and Vanapalli (2018) tensile strength model (sample sets A–D). b Evaluation of Yin and Vanapalli (2018) tensile strength model (sample sets E–G)

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Table 6 Summary of parameters and RMSE for Yin and Vanapalli et al.’s (2018) tensile strength model
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In both Lu et al.’s (2009) and Yin and Vanapalli’s (2018) tensile strength models, a better estimate of σt can only be obtained by using ϕt < ϕ′ as shown in Tables 5 and 6. This is undesirable as ϕt becomes a fitting parameter and the model cannot be used in the predictive sense. Hence, a more suitable tensile strength model needs to be developed keeping ϕt = ϕ′ to get a better estimate of σt.

Proposed tensile strength model

Development

As mentioned earlier, Eq. 1 is generic to the tensile strength models of Lu et al. (2009), Trabelsi et al. (2012), Yin and Vanapalli (2018), Tang et al. (2015), Varsei et al. (2016), and Wang et al. (2020). However, the friction angle is not ϕ′ for the models of Lu et al. (2009), Tang et al. (2015), and Yin and Vanapalli (2018). By assuming that the extended Mohr–Coulomb criterion is tangent to the BTS test Mohr circle, Eq. 2 was adopted instead. The biggest difference in the tensile strength models lies in the definition of σs. The σs can be made more general by adopting the form given in Eq. 12.

$$ {\upsigma}^{\mathrm{s}}=A\cdot {S}^{k_1}\cdot \left({u}_{\mathrm{a}}-{u}_{\mathrm{w}}\right) $$
(12)

In Eq. 12, A is a constant and is function of basic soil properties; degree of saturation S is used which implies that Sr = 0; exponent k1 is used instead of k to indicate that it is a different exponent from that used in Yin and Vanapalli (2018). Similar non-linear suction stress–matric suction relationship as given in Eq. 12 has been suggested by others, e.g., Kim et al. (2010). The proposed model is thus given by Eq. 13.

$$ {\sigma}_{\mathrm{t}}=\frac{\cos \phi \hbox{'}}{\left(2.05-1.05\sin \phi \hbox{'}\right)}\left[{\sigma}^{\mathrm{s}}\tan \phi \hbox{'}+c\hbox{'}\right] $$
(13)

According to Yin and Vanapalli (2018), the exponent k is heavily related to the SWCC through van Genuchten (1980) SWCC equation parameter n and the capillary degree of saturation corresponding to the AEV (Sc) and somewhat related to the grain size gradation parameter Cu. Hence, it is reasonable to suggest that k1 is a function of parameters associated with void ratio e, grain size distribution, and SWCC. It is highly likely that both A and k1 have different expressions for coarse and fine-grained soils. Hence, expressions for A and k1 were obtained separately for coarse and fine-grained soils. The soil properties used to obtain the proposed model parameters are α and n of van Genuchten (1980) SWCC equation (Sr = 0), void ratio e, gradation parameters, d50, Cu and Cc, and percentage passing no. 200 sieve, P200. A summary of the soil parameters used is tabulated in Table 7. For the coarse-grained soils, Ottawa sand from Goulding (2006); Perth sand, Silty sand, and Fine sand from Lu et al. (2007); and Medium and Esperance sand from Lu et al. (2009) were used to develop the relationships for the proposed model parameters, A and k1. For the fine-grained soils, the data reported in the present study together with the mine tailings from Narvaez et al. (2015) were used to develop the relationships for the proposed model parameters, A and k1.

Table 7 Parameters used in the development of the proposed tensile strength model
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To determine the relationships of parameters, A and k1, with basic soil properties, an iterative optimization process was adopted. In the first iteration, the best-fit values of A and k1 to each dataset in the calibration dataset were determined. A correlation matrix between A and k1 with the basic soil properties and combinations (α, n, e, d50, Cc, Cu, P200, α·Cu, α·Cc, n·Cu, n·Cc, e·P200, e·n) was determined. A regression analysis (using common functions of linear, exponential, power, and quadratic polynomial) was then performed on the parameter (A or k1) with the basic soil property or combination having the highest value in the correlation matrix. When more than one basic soil property or combination have the highest correlation values within 0.01, regression analysis was performed on each soil property or combination and the regression equation with the highest coefficient of determination R2 was selected. Once the regression equation was obtained for the parameter (A or k1), the value of the parameter was given by the regression equation while the other parameter (k1 or A) was optimized again for each dataset of the calibration dataset. Subsequently, a regression analysis (using common functions of linear, exponential, power, and quadratic polynomial) was then performed on the other parameter. Finally, the coefficients of the regression equations obtained for parameters A and k1 are optimized with the calibration dataset to obtain the final regression equations.

The correlation matrices for the coarse-grained and fine-grained soils are shown in Tables 8 and 9, respectively. For coarse-grained soils, parameter A has the highest correlation with Cc (0.9247) while parameter k1 has the highest correlation with n·Cc (− 0.7230). As the correlation between parameter A and Cc is higher than the correlation between k1 and n·Cc, a regression analysis was performed for parameter A with Cc as the independent variable to give Eq. 14.

$$ A=9.973{\left({C}_{\mathrm{c}}\right)}^{6.2}\kern9em \left({R}^2=0.9010\right) $$
(14)
Table 8 Correlation matrix for coarse-grained soils
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Table 9 Correlation matrix for fine-grained soils
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Using Eq. 14 to give the value of parameter A for each dataset in the calibration dataset, values of k1 were then optimized again to give k1_2 and a new correlation matrix was formed for k1_2 as shown in the last row of Table 8. The values of k1_2 have the highest correlation with n·Cc (− 0.7890). This correlation value is higher than the previous correlation between k1 and n·Cc (− 0.7230). Hence, a regression equation was obtained for k1_2 with n·Cc as the independent variable to give Eq. 15.

$$ {k}_{1\_2}=2.835\exp \left(-0.341\cdot n\cdot {C}_c\right)\kern3em \left({R}^2=0.7031\right) $$
(15)

Finally, the coefficients of the regression equations for parameters A and k1_2 (Eqs. 14 and 15, respectively) were optimized with the calibration dataset to give the final equations, Eqs. 16 and 17, respectively. Note the slight changes in coefficients between Eqs. 14 and 16 for parameter A, and between Eqs. 15 and 17 for parameter k1.

$$ A=9.869{\left({C}_{\mathrm{c}}\right)}^{6.405} $$
(16)
$$ {k}_1=2.643\exp \left(-0.326\cdot n\cdot {C}_{\mathrm{c}}\right) $$
(17)

For fine-grained soils, parameter A has the highest correlations with Cc (0.9859) and n·Cc (0.9815) while parameter k1 has the highest correlation with e·n (− 0.9289) (Table 9). As the correlations between parameters A and Cc and between parameters A and n·Cc are higher than between k1 and e·n, a regression analysis was performed for parameter A with Cc and n·Cc as an independent variable in turn. The regression equation for parameter A adopted was the one with the highest coefficient of determination, R2, shown in Eq. 18.

$$ A=0.0255{\left({C}_{\mathrm{c}}\right)}^2+0.971{C}_{\mathrm{c}}+0.249\kern3em \left({R}^2=0.9722\right) $$
(18)

Using Eq. 18 to give the value of parameter A for each dataset of the calibration dataset, the values of k1 were then optimized for the calibration dataset again to give k1_2 and a new correlation matrix was formed for k1_2 as shown in the last row of Table 9. The values of k1_2 have the highest correlation with n (− 0.9198) and e·n (− 0.9294). Hence, a regression equation was obtained for k1_2 with n and e·n as the independent variable in turn. The regression equation for k1_2 adopted was the one with the highest coefficient of determination, R2, shown in Eq. 19.

$$ {k}_{1\_2}=11.293\exp \left(-1.121n\right)\kern6em \left({R}^2=0.9279\right) $$
(19)

Finally, the coefficients of the regression equations for parameters A and k1_2 (Eqs. 18 and 19, respectively) were optimized with the calibration dataset to give the final equations, Eqs. 20 and 21, respectively. Note the slight changes in coefficients between Eqs. 18 and 20 for parameter A, and between Eqs. 19 and 21 for parameter k1.

$$ A=-0.395{\left({C}_{\mathrm{c}}\right)}^2+1.784{C}_{\mathrm{c}}+0.041 $$
(20)
$$ {k}_1=14.115\exp \left(-1.241n\right) $$
(21)

Validation

For validation of the proposed tensile strength model, three coarse-grained soils from Win (2006), Kim and Sture (2008), and Jindal et al. (2016) and three fine-grained soils from Zeh and Witt (2005), Wong et al. (2017), and Murray and Tarantino (2019) were used. These data were not used in the derivation of the proposed tensile strength model. Most important of all, these data are for soils of the same soil structure and contain the SWCC information. The parameters of the validation dataset used in the proposed model are summarized in Table 10. The performances of the modified tensile strength model, and Lu et al. (2009) and Yin and Vanapalli (2018) models are shown in Figs. 9 and 10 for coarse and fine-grained soils, respectively. The proposed model showed good agreement for the coarse and the fine-grained soils in Figs. 9 and 10 except for Fig. 10a at high degree of saturation (> 80%). The data in Fig. 10a was for a medium plasticity clay which shrinks on drying (Zeh and Witt 2005). For such a soil, it is expected that the soil remains fully saturated as the soil shrinks on drying and only becomes less than fully saturated near the air-entry value. Hence, the data at high degree of saturation were affected by error in the volume measurement. The RMSE for each set of data are shown in Table 11. Figures 9 and 10, and Table 11 show that the proposed tensile strength model outperforms Lu et al.’s (2009) and Yin and Vanapalli’s (2018) tensile strength models.

Table 10 Parameters of validation data used in the proposed tensile strength model
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Fig. 9
figure 9

Validation of proposed tensile strength model for coarse-grained soils

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Fig. 10
figure 10

Validation of proposed tensile strength model for fine-grained soils

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Table 11 Comparison of RMSE of the validation data for proposed, Lu et al. (2009), and Yin and Vanapalli (2018) tensile strength models
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Conclusion

The present study reviews the macroscopic approach tensile strength models proposed for unsaturated soils. The tensile strength models were proposed for either cohesionless (coarse-grained) or clayey (fine-grained) soils. The differences and discrepancies of the models were highlighted. Brazilian tensile tests were performed on fine-grained soils compacted on the wet side of optimum and subjected to drying at various water contents. Existing tensile strength models for unsaturated soils were not able to model the tensile strength for a wide range of soil types. Using the test data together with data collated from the literature, a new tensile strength model was developed for both unsaturated coarse and fine-grained soils. The proposed model uses the Brazilian tensile test Mohr circle and the extended Mohr–Coulomb criterion. Parameters of the suction stress in the proposed model were related to the grain size distribution and soil–water characteristic curve. Using a validation dataset of three coarse-grained soils and three fine-grained soils, the proposed tensile strength model was shown to perform better than existing tensile strength models.