1 Introduction

There are significant loess deposits in the world, the most recognized and studied being those found, for example, in China, Russia, Argentina, United States, Germany, Bulgaria and New Zealand (Smalley and Smalley 1983; Li et al. 2019; Zhu et al. 2019). The main deposits of loess in South America are located in the Pampas Region of Argentina (Rinaldi and Francisca 1999; Zárate 2003). However, literature on the mechanical behavior of South American loess is limited (Rogers et al. 1994).

Argentine loess has an unstable structure, formed by silt and sand particles, jointed by clay bridges and precipitated salt crystal (Reginatto and Ferrero 1973; Rinaldi et al. 2007). This collapsible soil has an unstable mechanical behavior, which mainly depends on water content and mean stress. The collapse of soil microstructure produces a sudden volume reduction that has, for several decades, been analyzed by means of double oedometer tests (Jennings and Knight 1957). Francisca (2007) showed that maximum collapsibility can be associated with the blow count determined from Standard Penetration Tests (SPT), and that the stress and wetting history are also constraining factors controlling the macroscale behavior of loess. Large deformations usually associated with collapse can also significantly affect buried structures and foundations in loess (Francisca et al. 2002; Francisca and Redolfi 2003; Jin et al. 2019). Loess with similar collapsible behavior has been extensively studied over the past 60 years around the world (Terzariol 2009; Li et al. 2016; Jing et al. 2020).

Pereira and Fredlund (2000) define three distinct phases to explain metastable soil behavior. In the first phase known as pre-collapse, the soil has relativity high suction values, and only small volumetric strains occur in response to a decrease in matric suction. At this stage, there is neither collapse nor relative sliding of particles. A collapse phase occurs at intermediate suctions, with significant volumetric deformation and breakage of bonds connecting large particles when matric suction reduces. In the third or post-collapse phase, matric suctions are very low, and soil structure remains unaltered, with negligible volumetric changes as moisture content increases.

Unsaturated soil mechanics explains collapse phenomena as changes in the stress state variables, and this is a precise way to predict volume changes by suitable constitutive relations (Alonso et al. 1990; Li et al. 2016). In the last two decades, there have been numerous advances in the development of constitutive models in unsaturated soil mechanics and it remains an active area in current research (Gens et al. 2006). Several constitutive models were developed to characterize the stress–strain behavior of partially saturated soils (Alonso et al. 1990; Kohgo et al. 1993; Sun et al. 2007; Sheng et al. 2008; Ghorbani et al. 2016). Models such as the Basic Barcelona model satisfactorily describe the stress–strain behavior of unsaturated soils (Alonso et al. 1990; Patil et al. 2016, 2018). The Extended Basic Barcelona model (EBB) introduces modifications to the original model to simplify its computer implementation and also to allow simulation of the elastoplastic behavior during cycles of both mechanical and hydraulic loading (Pedroso and Farias 2011). This model response is the same as that of the so-called “modified” Cam Clay when the soil is fully saturated (Roscoe and Burland 1968; Wood 1992). Recently, great efforts have been made to define the mechanical behavior of collapsible soils. Li and Vanapalli (2018) present a simple method for predicting soil collapse due to wetting by using Soil Water Characteristic Curves (SWCC). In the same way, Jiang et al. (2014) investigate the effect of water content and void ratio on the compression and collapse behavior of loess by using discrete element models (DEM). Jiang et al. (2016) showed the influence of biaxial stress conditions on loess collapse due to wetting and Jiang et al. (2017) showed that the macroscopic behavior of unsaturated structural loess can be related to force-chain distribution and contact orientation between soil particles at the microscopic scale. Also, recently, the capacity of models such the Basic Barcelona model and Bounding Surface (BS) has been evaluated to analyze the strain hardening–softening response of silty sand specimens (Patil et al. 2016, 2018).

The purpose of this work is to evaluate the capacity of different elastoplastic models to predict the strain–stress behavior of collapsible loess. Experimental results from double-oedometer tests are used to calibrate the MCC model for saturated specimens and the EBB model for unsaturated and saturated specimens. The aim of these calibrations is to reproduce the stress–strain behavior of loess with a limited number of variables and without the need to perform suction-controlled oedometer tests. This new approach has the advantage of using data generated in most soil mechanics laboratories, both in industry and universities.

1.1 Mechanical Behavior of Soils: Constitutive Models

This section presents the fundamentals of the MCC and the EBB models. The MMC model explains the mechanical behavior of saturated soil, while the EBB model has been used to describe the response of partially saturated soils.

The evolution of the void ratio with pressure increases in the Modified Cam Clay model is calculated as:

$$e = e_{ref} - (\lambda - \kappa )\log \frac{{p_{c} }}{{p_{c0} }} - \kappa \log \frac{p}{{p_{ref} }}$$
(1)

where, \({e}_{ref}\) is the reference void ratio or initial void ratio, \(p\) is the nonlinear pressure, \({p}_{c0}\) is the initial consolidation pressure, \({p}_{ref}\) is the reference pressure at the reference void ratio, \(\lambda\) is the compression index, and \(\kappa\) the swelling index. The nonlinear pressure is defined as:

$$p = p_{ref}^{{ - \left( {\frac{{1 + e_{ref} }}{\kappa }} \right)\varepsilon_{p}^{p} }}$$
(2)

where \({\varepsilon }_{p}^{p}\) is the volumetric plastic strain. Also, the evolution of the consolidation pressure is defined by the following equation:

$$\delta p_{c} = \frac{{1 + e_{ref} }}{\lambda - \kappa }p_{c} \delta \varepsilon_{p}^{p}$$
(3)

And, finally, the yield function for the MCC model is:

$$F_{y} = q^{2} + M^{2} (p - p_{c} )p = 0$$
(4)

where M is the slope of the critical state line and q the deviatoric stress. The shape of the yield function is mainly affected by the evolution of the consolidation pressure and the nonlinear pressure in the k0 condition.

The EBB model extends the Barcelona Basic model to the two-surface formulation, which guarantees a smooth transition between the elastic and elastoplastic behavior (Pedroso and Farias 2011). In this model, changes in void ratio are also represented by Eq. (1) but considering suction dependent compression index(es)\(.\) For the saturated condition, the EBB model collapses into the MCC model. In the unsaturated condition, the relation between the compression index at current suction (\(\lambda \left(\mathrm{s}\right))\) and the compression index at saturation (\(\lambda \left(0\right))\) is defined as follows (Alonso et al. 1990):

$$\lambda (s) = \lambda (0)\left[ {\left( {1 - w} \right)\exp \left( { - \frac{s}{m}} \right) + w} \right]$$
(5)

where \(w\) is the weighting parameter, \(m\) is a soil stiffness parameter and \(s\) is the suction. Also, for the Basic Barcelona model, the load-collapse curve is obtained by:

$$\left( {\frac{{p_{0} }}{{p_{ref} }}} \right) = \left( {\frac{{p_{0}^{*} }}{{p_{ref} }}} \right)^{{\frac{{\left[ {\lambda (0) - \kappa } \right]}}{{\left[ {\lambda (s) - \kappa } \right]}}}}$$
(6)

where \({p}_{0}\) is the consolidation pressure at current suction, \({p}_{\mathrm{ref}}\) is the reference pressure, \({p}_{0}^{*}\) is the yielding pressure at saturation, \(\lambda \left(0\right)\) is the compression index at saturation, \(\lambda \left(s\right)\) the compression index at the current suction, and \(\kappa\) is the swelling index.

The nonlinear pressure and the consolidation pressure are defined in the same way as in the MCC model, Eqs. 2 and 3, respectively.

2 Materials and Methods

Laboratory tests were carried out on 7 undisturbed loess soil samples (ML-CL according to the USCS) from Córdoba City. The laboratory program consisted in determining water content, Atterberg limits, grain size distribution and double-oedometer tests. The main physical properties of the tested loess are presented in Table 1. Samples were trimmed from soil blocks with the oedometer ring dimensions.

Table 1 Main physical properties of tested loess soils
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Double-oedometer tests were carried out by running confined compression tests under zero lateral deformation. Two specimens were tested for each soil sample and therefore, the overburden pressure, initial void ratio and natural moisture content were identical. The first specimen for each sample was tested at natural moisture content while the second one was tested flooded with water. The double oedometer test was performed by one-dimensional consolidation test according to ASTM 2435-03, using Method A, in samples at natural water content and saturated (Fig. 1). The load was constant for 24 h and the axial load increments started approximately at 0.1 kPa and stopped at 400 kPa. The time recorded for change in high samples was 0, 8, 15, 30 s, 1, 2, 3, 4, 8, 15, 30 min, and 1, 2, 4,8,16 and 24 h. The load was doubled for each increment. In all cases the primary consolidation was observed. Saturated samples were flooded with water for 24 h, before the first load step. All samples flooded with water achieved final water contents near saturation, although full saturation could not be guaranteed (Post-collapse stage according to Pereira and Fredlund 2000).

Fig. 1
figure 1

Schematic layout of the double oedometer test system

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The compression index \(\lambda\) and the swelling index \(\kappa\) were theoretically calculated from the experimental results. These can be obtained from Eqs. (7) and (8) as follows (Roscoe and Burland 1968):

$$\lambda = \frac{{C_{c} }}{\ln (10)} = \frac{{C_{c} }}{2.3} = 0.434C_{c}$$
(7)
$$\kappa = \frac{{C_{s} }}{\ln (10)} = \frac{{C_{s} }}{2.3} = 0.434C_{s}$$
(8)

where \({C}_{c}\) is the compression coefficient and \({C}_{s}\) is the load/reload coefficient obtained of compressibility curves, which relate the effective vertical stress (log scale) and void ratio.

The results were used to develop numerical models trying to reproduce the observed stress–strain behavior and relative collapse. Two different elastoplastic constitutive models were implemented: the MCC model to reproduce the saturated behavior, and the EBB model for both saturated and unsaturated behavior. The calibration parameters for the MCC model were: compression index \({\lambda }_{0}^{*}\), expansion index \({\kappa }_{0}^{*}\), and preconsolidation pressure \({p}_{0c}^{*}\). For the EBB model, the calibration parameters were: \({\lambda }_{0}^{*}\), \({\kappa }_{0}^{*}\), \({p}_{0c}^{*}\), the weight parameter (\(w\)), the soil stiffness parameter (\(m\)), the dimensionless smoothing parameter (\(b\)), the reference pressure (\({p}_{ref}\)), the suction at saturation (\({S}_{0}\)) and the suction at natural water content (\({S}_{n}\)). All numerical simulations were performed by using Comsol Multiphysics 5.4 software.

The geometry adopted in all models represented an oedometer ring with the same dimensions as those used for testing. A two-dimensional geometry with an axisymmetric axis was used. Boundary conditions allowed vertical displacement (y-direction) at the soil-cell interface, the bottom edge had both directions fixed, and the top edge had a prefixed vertical displacement for the simulation of loading. A structured mesh was adopted, with quadratic serendipity elements. The maximum element size was 1 mm. No influence of mesh configuration (structured or unstructured) or element shape (triangular or rectangular) was observed on the trends obtained. The strain tensor was determined as a large deformation by Green–Lagrange and the stress tensor was calculated through the second Piola–Kirchhoff tensor.

3 Results and Analysis

Figure 2 presents the evolution of void ratio versus vertical stress for the seven loess samples tested under confined compression with zero lateral displacement. It shows the experimental results obtained for the specimens tested at natural moisture content and saturated, and also the MCC model response that best fits the experimental data for saturated specimens. The result shows that specimens saturated with water suffered higher deformations than those tested unsaturated, as expected for collapsible soils. Compression indexes λ, expansion indexes κ, and preconsolidation pressures pc, determined from experimental results and as fitting parameters in the MCC model, are presented in Table 2.

Fig. 2
figure 2

Confined compression stress–strain behavior under the zero lateral displacement condition of tested loess. Symbols represent experimental results and dashed lines are EBB model results

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Table 2 Compression index λ, expansion index κ, and preconsolidation pressure pc determined from experimental results and as fitting parameters in the MCC model
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Model parameters λ and κ resulted in values close to those computed from Eqs. (1) and (2) identified as experimental values in Table 2. Predictions of λ, κ, and pc were within a 95% confidence interval, with \({R}^{2}\) = 0.963, 0.983 and 0.973, respectively.

The mechanical behavior of loess is highly dependent on water content and, for this reason, saturated specimens showed large deformations, significant increases in compression indexes and a reduction in preconsolidation pressures. These trends can be attributed to the collapse of the loess microstructure.

The expansion of the yield function is guaranteed as the elastoplastic strain variables converge. Nonconvergence occurs when there is no smooth transition between the compression index and the swelling index for a given void ratio, yield pressure and reference pressure. For ratios between the compression index and the swelling index lower than 20, non-convexity problems have not been found. All parameters can be estimated from compressibility curves from one consolidation test.

The numerical implementation of the MCC model considered the expansion of the yielding surface, showing a good capacity of the model to adjust to the experimental data. However, given that the MCC model was developed for saturated soil mechanics, it can reproduce neither the stress–strain behavior for the partially saturated state nor for loess collapse.

For this reason, the EBB model was adopted to calibrate the stress–strain relation in loess soil for the unsaturated condition. The response of the EBB model is identical to that of the MCC for saturated conditions. For the EBB model, it is necessary to define the load collapse (LC) curve form Eq. 6 (Fig. 3). The LC curves show a similar tendency according to the initial void ratio (T1-T4 compared with T5-T7). The experimental results show that in all cases the compression index at saturation is greater than when unsaturated, so the weight parameter is less than 1 (w < 1), as was formulated in the original model. The \({p}_{\mathrm{ref}}\) parameter for LC curves is obtained based on the transition between \({p}_{0}\) and \({p}_{0}^{*}\), considering the intersection of the normal lines in the compressibility curves at natural water content and saturated. In all cases, \({p}_{\mathrm{ref}}\) was greater than the saturated yielding pressure. One of the most important problems associated with the Basic Barcelona model prediction is the constraint imposed by the selection of the reference pressure (\({p}_{\mathrm{ref}}\)), at which the yield curve becomes a straight vertical line in the s:p plane (Wheeler et al., 2002). However, as reported in Fig. 3, the load collapse curve became straight for values greater than the yielding suction. Even if the \({p}_{\mathrm{ref}}\) was selected based on two normal lines, the yield curve does not present non-convexity problems. The EBB model parameter selected made it possible to recreate the stress–strain relation of loessial soil (Fig. 4).

Fig. 3
figure 3

Load collapse curve for tested loess

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Fig. 4
figure 4

Capacity of EBB model to simultaneously reproduce the stress–strain behavior under the zero lateral displacement condition of loess at natural moisture content and water-flooded

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Figure 4 shows a comparison between the experimental results and the EBB model for the seven loess samples at natural moisture content and flooded with water. Specimens with low initial void ratio correspond to tests T1 to T4, while those with higher initial void ratio are T5 to T7.

The EBB model assumes that the swelling index does not change with suction in the elastic field (Alonso et al. 1990). Hence, unique \(\kappa\) were adopted for each sample by considering an average value that better predicted the experimental results obtained for specimens tested at natural water content and saturated. Adjusted EBB model parameters are presented in Table 3. Fairly good adjustment of the EBB model was obtained for all samples with lower initial void ratio (T1 to T4), while the model predictions were less accurate for the samples with high initial void ratio (T5 to T7). This trend may be attributed to the significant relative displacement between particles that is expected to occur in highly collapsible soils having major open and loose structures (Francisca 2007).

Table 3 Extended Basic Barcelona Model parameters when assuming a suction independent κ in the elastic region
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The loess at natural water content presents a rigid skeleton, with small vertical deformations. For this reason, the weighting parameter (w), which relates the compression index at high suction values and the compression index at saturation, may be defined as the relation between the compression index at natural water content and the compression index at saturation. The adopted \({\lambda }_{n}\) must be lower than 0.1. Even though the EBB model assumes a unique swelling index, different values were allowed for the specimens having higher initial void ratio, to represent the stress–strain behavior at natural water content and water-flooded (T5 to T7). In these cases, the compressibility curves show differences in the slope within the elastic range (suction dependent \(\kappa\)). Table 4 shows the parameters obtained when this modification of the swelling index is allowed. In this case, more accurate predictions were obtained with the EBB model, and also the resulting swelling indexes were identical to those defined in the MCC model (Table 2). This can be attributed to the EBB model reducing to the MCC model upon soil saturation (Alonso et al., 1990). The modification of the swelling index gave slightly different suction values, which more adequately explain the mechanical behavior of the soil in a natural condition (\({S}_{n}\)). The results obtained suggest that different swelling indexes should be adopted when significant changes in the micro-structure of the soil are expected due to collapse. This is the case of the loess tested, which has larger initial void ratio and develops a very important collapse due to the expansion of clay bridges and the dissolution of precipitated salt crystal by water-flooding (Francisca 2007). However, negligible changes were obtained for the samples with lower void ratio. Therefore, the hypothesis that soil behavior does not depend on suction in the elastic region does not apply for loess with high initial void ratio. For these cases, matric suction is not enough to explain the increase in compressibility.

Table 4 Modification of Extended Basic Barcelona Model parameters when assuming a suction-dependent κ in the elastic region
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Calibrated numerical models of this work show similar behavior to that observed by Pereira and Fredlund (2000). At natural water content, the soil samples present small volumetric strains and no collapse of the micro structure is observed. This can be explained by the relatively high values of matric suction. When soil samples are flooded with water, matric suction decreases, large deformations occur, and the soil structure collapses. When the soil is saturated, the soil structure remains unaltered after bond breakage. Partial wetting produces partial collapse. With 50% soil saturation, 85% collapse deformation occurs, and for 65%–70% saturation, there is full collapse (El-Ehwany and Houston 1991). For this reason, while the oedometer test without suction control cannot guarantee 100% saturation, it allows values very close to it to be inferred and therefore deformation values near to those developed for 100% collapse. The EBB model enables the stress–strain behavior of loess at different moisture contents and therefore matric suction to be reproduced. If double-oedometer tests are available, the EBB model can reproduce experimental results if matric suction is considered as a fitting parameter to achieve the resulting relative collapse (change in void ratio at a given pressure due to water-flooding). The matric suctions (Sn) that explain the experimental results of this work are summarized in Tables 3 and 4.

Figure 5 shows the influence of the volumetric content of water on matric suction for undisturbed loess according to data compiled from different authors and the UNSODA database. The data compiled in Fig. 5 include results obtained for loess from Argentina (Zeballos et al. 1999), China (Huang et al. 2010; Ng et al. 2016, 2020 and France (Muñoz-Castelblanco et al. 2012). The matric suctions (Sn) computed from the EBB model are also shown in Fig. 5. The results obtained show good agreement with the experimental data reported in literature.

Fig. 5
figure 5

Influence of volumetric water content on matric suction for loess

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In geotechnical practice, double-oedometer tests are performed without controlling suction. Results obtained by Munõz-Castelblanco et al. (2011) showed that the constant water content compression test occurs at a fairly constant suction, even at yielding. They consider that the compression-induced collapse mainly affects the largest pores with little effect on the smaller pore sizes that control suction changes. Thus, the water retention curve is governed by capillarity in the large pores between coarse grains and by water absorption in the clayey fraction with the smallest pores, where the microstructure is sensitive to changes in water content (Muñoz-Castelblanco et al. 2012). For this reason, even though suction changes during loading were neglected in this work, the EBB was still able to satisfactorily represent the experimental result by adopting the matric suctions shown in Tables 3 and 4 and Fig. 5.

The capacity of the EBB model to represent the zero lateral displacement condition of loess soils enables it to reproduce the collapse phenomena accurately. Under infinite embankment or shallow flexible foundations, the soil behaves like in an oedometer condition. Also, as in the axis of flexible structures, such as fuel or water tanks, the stress path occurs near the k0 line. The double oedometer test in collapsible soils is widely used in professional practice to analyze the soil response, and the correct calibration of the parameters used in constitutive models is necessary to explain the mechanical behavior of loessial soils.

4 Conclusions

This work analyzed the capacity of the Modified Cam Clay (MCC) model and the Extended Basic Barcelona (EBB) model to simulate the mechanical behavior of undisturbed loess samples in a zero lateral displacement condition. The following conclusions were obtained:

  • The MCC model adequately simulates the mechanical behavior of water-flooded loess. However, given that this model was created for saturated soils, it cannot capture the collapse phenomena.

  • The EBB model is able to reproduce the stress–strain behavior of loess at natural water content and flooded with water simultaneously. Thus, it successfully recreates the soil collapse expected for unsaturated loess due to water-flooding.

  • The loess tested was characterized by a rigid skeleton with high matric suction at natural moisture content. Therefore, changes in soil suction during the oedometer tests induced negligible effects on its mechanical behavior. In this case, the EBB model was able to reproduce the experimental results obtained in the oedometer tests without controlling suction. The numerical results obtained show good correlation with the experimental measurements, and the suctions computed from the adjusted EBB model matched data reported in the literature. These results demonstrate the capacity of the model to predict soil collapse.

  • Oedeometer tests without controlling suction are widely used in geotechnical practice to evaluate the collapse potential of metastable soils. The EBB model has difficulties in its implementation because of the large number of soil parameters needed for calibration. The results obtained in this work show the potential of the EBB model to predict loess collapse with a limited number of experiments and without the need of performing suction-controlled oedometer tests.