Content area
This paper aims to present an understanding and prediction of the drying shrinkage behavior of clayey soils through a combined approach involving experimental investigations and numerical simulations. Initially, desiccation experiments were conducted on fully saturated slurry as well as pre-consolidated clayey soils. The experimental results indicate that the deformation observed in the fully saturated slurry during the drying shrinkage process is related to its moisture loss. Therefore, this research directly employed the Fredlund’s function to estimate the shrinkage volumetric strain. Subsequently, a constitutive model for the hydraulic mechanical properties of clay was developed based on a stress state variable framework, and simulations were performed for the desiccation experiments involving fully saturated slurry. The drying experiment conducted on pre-consolidated clayey soils revealed that these soils does not deform during the drying process, and external loading emerges as a critical factor influencing their deformation. As moisture is lost from pre-consolidated clay, an internal suction gradient develops, thus, suction distribution was derived by integrating Darcy’s law with SWCC. Ultimately, this research provides a theoretical explanation for the deformation processes associated with clay drying through numerical modeling while highlighting differences in clay drying behavior between varying initial states.
Highlights
A constitutive model describing the hydraulic and mechanical behavior of clay under controlled environment was established through the stress state variables framework.
The experimental relationship between the physical variables in desiccation of material was established.
For pre-consolidation specimens, the desaturation process was relatively slow in clayey soil due to low permeability, the high-water pressure gradient was generated with a significantly heterogeneous suction distribution in the clayey soils during the drying process.
Introduction
Clayey soil is widely distributed on the Earth’s surface, and due to its affordability and accessibility, it has been extensively utilized as a building material throughout history [1]. Additionally, clay can serve not only as a natural barrier in the dam cores but also as a matrix for the storage of radioactive wastes due to its retention properties [2]. The mechanical behavior of clay materials is complex. One of the challenges lies in its sensitivity to water, clayey soil becomes hard and brittle after desiccation because of its fine particle size and high water content [3, 4]. Desiccation shrinkage of clayey soil is widespread in nature and significantly affects the mechanical behavior of the soil, including disintegration, permeability, residual shear, tensile strength, etc [5, 6], which in practice can compromise the sustainability of geotechnical structures as mentioned in different studies reported in the literature [7, 8–9]. Owing to the significant concern of this question, the mechanism of desiccation of clayey soils has gained the attraction of many researchers during the last decades, in both experimental investigations and numerical modeling.
The desiccation of clayey soils has been the subject of several works from an experimental perspective [10, 11, 12–13]. Lakshmikantha [10] conducted drying tests on clayey soil slurry in an environment-controlled laboratory, aiming to determine whether changes in environmental conditions such as temperature and humidity during soil drying can cause changes in soil material properties. The results showed that material properties of the soil, such as tensile strength or fracture toughness, vary considerably as a function of the soil’s water content. Tang [11] observed the desiccation experiments of clayey soils and obtained experimental results using computer image processing techniques. These tests indicate that soil dries faster when the humidity is lower, and thicker layers of soil generally exhibit a slower desiccation rate. Péron [14] conducted desiccation experiments of clayey soils, and the soils was applied the boundary conditions during the drying process, the experimentally relationships between the physical variables such as water content, suction, degree of saturation and strain in the desiccation process were established at the macroscale, the relationship between desiccation shrinkage of clayey soils and cracking was studied. Desiccation cracks in clayey soils are generally considered to be the consequence of constrained shrinkage, during which the internal induced tensile stress exceeds the corresponding resistance strength [12, 15, 16]. Therefore, it is essential to first conduct a thorough study of the drying process of clayey soils in order to establish a foundation for investigating the mechanism of cracking.
From a numerical perspective, numerical modeling has been employed to approach the desiccation in clayey soils, and more generally in porous medium. In the theory of poro-mechanics, Coussy [17] established general constitutive framework, general poroelasticity theory can be seen as an extension of Biot’s theory of poroelasticity. Coussy’s constitutive framework is presented as being the consequence of an unbiased and systematic application of thermodynamics of open continua. Bishop proposed the principle of effective stress suitable for unsaturated soils, that is, the generalized effective stress principle [18, 19]. In the stress frameworks for modelling of unsaturated soils, the skeleton deformations are fully governed by a single effective stress of the Bishop type, to which a second stress variable is added [20], often to complete the description of the hydraulic behaviour. This constitutive modelling can achieve the straightforward transition between saturated and unsaturated states of clay and directly accounts for the hydraulic hysteresis effect on the mechanical response. Subsequently, various authors [21, 22–23] further developed the generalised effective stress approach to construct a more sophisticated model. Fredlund [24] proposed that the constitutive analysis of soil behaviour can depend on the stress state variables. In the constitutive framework of the stress state variables, the use of two independent stress state variables will enable the decomposition of total strain into drying shrinkage induced part due to water mass removal, and a mechanical part due to the total stress variation [4, 25]. By using this method, the drying shrinkage of clay can be simulated as an independent component, the total strain is divided into two parts and described separately, making it more intuitive and easier to operate.
The primary objective of this article is to comprehend the desiccation phenomena in clayey soils through combined numerical simulation and laboratory experiments research. The relationships between the physical variables of clayey soils during the drying process are established at a macroscale through experiments. A constitutive model for clayey soils can be developed in the simulation, the desiccation of clayey soils can be reproduced. This research serves as a basis for further study on the desiccation cracking process of clayey soils. For this purpose, the desiccation experiments on fully saturated slurry and pre-consolidation clayey soils are analyzed and simulated. The material properties of the test material are first introduced. Then, details about the desiccation experiments are provided along with an analysis of results and identification of numerical model parameters. Subsequently, a constitutive model for clayey soils is established and the numerical simulation for the desiccation is realized using finite element method. This approach allows for estimation of deformations, suction, and stresses developed in the sample from the fully saturated state to the completely dry state. Finally, the proposed numerical approach is validated by reproducing the desiccation experiments. The numerical results demonstrate good agreement with experimental observations, providing a deeper understanding of the formation mechanism of desiccation in clayey soils through a combination of numerical simulation and experiments. A preprint has previously been published [26].
Material properties and experimental methods
The experiments were performed on an industrial clayey soil called Kaolin. In this section, the fundamental properties of Kaolin and the employed experimental methods are presented. The experiments can be divided into two groups, the first group includes desiccation tests of the slurry specimens, while the second group involves desiccation tests of the pre-consolidation specimens.
Material properties and specimen preparations
Mineralogy analysis
X-ray diffraction analysis conducted in laboratory was utilized to analyze the composition of Kaolin. The results, depicted in Fig. 1, indicate that a large proportion of kaolinite, larger than 90%, contained in this clay (peak 1), with less than 10% of quartz alpha grains (peak 3), traces of illite (peak 2) and some other minor constituents (peak 4). Kaolinite is a common clay mineral with a chemical formula of Al2Si2O5(OH)4. It is a layered silicate mineral, comprising a tetrahedral sheet of silica (SiO4) linked through oxygen atoms to an octahedral sheet of alumina (AlO6) octahedra. It should be noted that the kaolinite typically exhibits a low shrink-swell and a low cation-exchange capacity. Mineral composition of Kaolin of the technical document provided by the manufacturer is given in Table 1.
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Fig. 1
X-ray diffraction analysis of Kaolin
Table 1. Chemical type composition of Kaolin
Chemical type | |||||
|---|---|---|---|---|---|
Content (%) | 60 | 28 | 1.0 | 0.6 | 0.5 |
Grain size distribution
The Scanning Electron Microscopy (SEM) observation on the Kaolin powder reveals that kaolinite particles are mainly assembled in face-to-face form and most of the clay aggregates have a size between 5 μm and 10 μm, as shown in Fig. 2. The results show that kaolinite particle is plate-like and the size of an individual kaolinite particle is generally less than 2 μm.
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Fig. 2
SEM images of Kaolin powder
Figure 3 presents the “particle size distribution” of the raw material identified using the laser diffraction method. It should be noted that the “particle size” given in Fig. 3 for Kaolin might be in reality the “aggregate size”. Approximately 60% of the clay aggregates are smaller than 9 μm, and nearly 20% of the clay aggregates have a size larger than 20 μm with a maximum size of 80 μm.
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Fig. 3
Particle size distribution of Kaolin
Physical properties
The physical properties of Kaolin are as follows: the specific gravity of the solids is approximately 2.63, the geotechnical tests indicate that the liquid and plastic limits are about 40% and 21%, the plastic index is about 19%. The physical properties were determined according to the Chinese Standard GB/T50123-2019.
Specimen preparations
The soil selected for the study was an industrial clay with a stable source. The sample preparation was based on the requirements of this study and previous research [5, 33]. Moreover, researchers have conducted extensive studies on this and achieved some results [13]. Therefore, the preparation and testing were considered repeatable.
Slurry specimen preparation
Specimens used for desiccation tests were prepared from a saturated homogenized slurry state. Initially, Kaolin powder was mixed with deaerated water to achieve an initial water content of 1.2 times the liquid limit (w = 1.2wL). The slurry was then mechanically stirred at a velocity of 280 revolutions per minute for 15 min. Afterwards, the slurry was reserved in a recipient carefully sealed by film paper and aluminum foil, rested at temperature room for 24 h. This method allows to ensure a homogeneous mixture, and the initial water content of Kaolin slurry close to a value as 48%.
Pre-consolidation specimen preparation
The Kaolin slurry prepared above was poured into the consolidometer with full of water for an one-dimensional compression. The consolidometer is in rigid PVC and in cylindrical form with the diameter of 95 mm as shown in Fig. 4, and the clay core is continuously saturated. “upper filter” is a saturated filter paper for water exchange, and “upper drains disc” is a circular tray with a honeycomb-like hole structure for pressure transmission. For ensuring good drainage during the consolidation, the compression load increases step by step. The final designed axial stress was σ’v0 = 120 kPa, the value of 120 kPa is not immutable, there are also researchers who take 100 kPa as the final design compressive stress, 120 kPa can be used as an experience value. Finally, the material was fully saturated and allowed to consolidate under σ’v0 for at least 3 weeks.
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Fig. 4
Schematic diagram of consolidometer
Experimental methods
Soil - water characteristic curve
Soil - Water Characteristic Curve (SWCC) is crucial for analyzing the desiccation shrinkage process in clayey soils, as it represents the relationship between water content (w), degree of saturation (Sr), void ratio (e) and matric suction (s).
Slurry specimens were utilized in SWCC. The SWCC of the Kaolin was determined through experimental methods, which involved measuring the volume of samples (50 mm*50 mm dimensions and thicknesses of 10 mm, previously subjected to a specific suction level) by hydrostatic weighing of the sample while immersed in the liquid. Before managing with liquid the air substitution, the small samples were submitted first to suction controlled by salt solutions in a desiccator, suction values ranging from approximately 0.27 MPa up to about 363.7 MPa. Details on these experiments can be found in Cheng’s paper [13].
Desiccation test procedure of slurry specimens
The homogenized Kaolin slurry was poured into a square frame (a square mold of 180 mm x 180 mm in dimensions and 5 mm in thickness), placed on a base plate. Subsequently, the square frame was removed and the sample remained on the base plate. The drying took place on a Teflon support treated with hydrophobic substances containing silicon and ethanol, desiccation tests were conducted to avoid, any mechanical constraints at the boundary caused by the bonding friction effect during the shrinkage process as much as possible.
The tests were carried out in a controlled chamber where relative humidity was regulated using salt solutions, the temperature of the controlled test room was constant at T = 20℃, an electronic hygrothermograph was used for monitoring, and the electronic hygrothermograph can continuously record. For measurements, the sample with its support was positioned on a balance permitting to measure mass loss during drying, the water content as consequence changes during the test. Vernier calipers were used to measure the displacement and deformation of the sample.
The digital camera located above the sample captured images (each 2 min) of the sample evolving during the experiment, these images were automatically saved and analyzed. Thus, the evolution of water content, displacement and strain of the sample were deduced. Further details regarding these calculations can be found in Cheng’s paper [13].
Desiccation test procedure of pre-consolidation specimens
The pre-consolidation specimens for desiccation tests were cut from the obtained consolidated clay cores. They were cylindrical with dimensions of 75 mm height and 50 mm diameter. Two specimens labeled A and B are presented in this paper, their initial state parameters are listed in Table 2.
Table 2. Initial state of two specimens
Specimen | Initial wet mass m0 | Initial water content w0 | Initial void ratio 0 |
|---|---|---|---|
A | 269.46 | 36.69 | 0.987 |
B | 280.50 | 36.50 | 0.960 |
The specimens were placed in sealed desiccators, on the top of the container, while the lower part of the container was saturated saline solution, as shown in Fig. 5. Two different saturated saline solutions were used to control the humidity in the sealed container. For specimen A, the relative humidity was controlled by saturated Na2SO4 solution corresponding to 13 MPa of suction; for specimen B, it was controlled by saturated NaCl solution corresponding to 38 MPa of suction. The two containers were placed in a constant ambient environment at a temperature of 20 °C. Initially, before desiccation, the specimens were in fully saturated state. The technique imposed suction on the material by transferring water from the soil to the external environment of the desiccator and absorbing it with saline solution. During the experiments, the mass of each specimen was measured at different desiccation times, and the water content was then calculated at each step.
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Fig. 5
Schematic diagram of desiccator with specimen
Experimental results and discussion
Soil – water characteristic curve determination
Experimental results
Figure 6 presents results showing the relationship between void ratio, suction, and water content. During the drying process, when the sample was in the saturated state, the void ratio decreased as the water content decreased, following the straight saturation line with the slope of , as shown in Fig. 6a. Before air entry, the degree of saturation remained close to 100%, air entry corresponded onset air invasion within the sample. In this way, the air-enter suction value was determined. Then, when the sample entered an unsaturated state, the rate decrease void ratio sharp slowed down until reaching the minimum void ratio , this point could be seen as a shrinkage limit. Kaolin slurry, initially prepared as remoulded saturated slurry, air-entry value and shrinkage limit were close.
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Fig. 6
The results of void ratio and suction versus water content of Kaolin
Fitting results
Figure 6a shows the experimental results of theoretical shrinkage curve of Kaolin in (w-e) plane. Empirical law had been proposed to fit the theoretical shrinkage curve among them Braudeau, Fredlund and Leong [27, 28–29]. Fredlund function [28] was used to fit our experimental measurement, its expression being:
1
where .
is the minimum void ratio; is a parameter determining the slope of the tangency line; is the curvature of the shrinkage curve; is the initial degree of saturation; is the specific gravity of the solids. The water content is expressed as a percentage. This equation can well fit the relationship between the moisture content and the void ratio.
The fitting curve allowed to identify Fredlund parameters presented in Table 3, where: , and . Sr0 being equal to 100%. The result of fitting curve is shown in Fig. 6a.
During the drying process, the volumetric strain induced could be related to the void ratio through:
2
In which, is the initial void ratio of the sample, while is the actual void ratio, for ease of presentation, the experimental results take the compressed strain as positive. The equation also implies the small deformation hypotheses.
The initial void ratio of the Kaolin sample was measured at 1.26 (w0 = 48%), the variation of the volumetric strain of the Kaolin in the drying path was obtained by Eq. 2 and experimental evolution of void ratio e. The variation of the volumetric strain of Kaolin as shown in Fig. 7. At the same time, combining the above Eq. 1 and Eq. 2, the theoretical shrinkage volumetric strain of the soil sample generated in the drying path can be related to its water content through the Fredlund function, as in Eq. 3. Therefore, the variation of the theoretical shrinkage volumetric strain of Kaolin during the drying process was obtained as shown in Fig. 7.
3
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Fig. 7
The evolution of volumetric strain of sample versus average water content in the drying path
The experimental relation of volumetric water content versus suction was fitted to the Van Genuchten equation [30], expressed by Eq. 4.
4
is the volumetric water content; is the suction; is the residual volumetric water content, 0.001; is the saturated volumetric water content, 0.56; n, m and are parameters related to unsaturated state of the soil.
corresponds to the reciprocal of the entry air value. n and m are the parameters governing the curvature of the curve. The fitting experimental data gives the parameters values, where: = 1000 kPa, n = 1.05 and m = 0.87. The result of fitting curve is shown in Fig. 6b.
Desiccation test of slurry specimens
Variation of water content
Firstly, the variation of water content of the slurry specimen in the desiccation test was analyzed. The initial water content of the slurry specimen was about . The desiccation test was carried out at a temperature of 20 °C and a relative humidity of 7% (controlled by the KOH saturated salt solution, corresponding to a suction of 360 MPa). Figure 8 shows the evolution of the average water content of the sample, indicating that the water content decreased following the variation curve with a rate of 0.83, then, the water content stabilized at a residual water content of 1.02%.
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Fig. 8
The variation of water content of the specimen versus time in desiccation test
Evolution of strain
The evolution of strain components of the slurry sample in desiccation test have been showed in Fig. 9 (compression is positive). Due to its square shape, the experimental results of the horizontal strain and longitudinal strain of the sample during the drying process were basically the same. In the early stage of drying, the deformation increased with the loss of moisture. When the shrinkage limit was reached, the deformation of the sample stops, and at this moment, the and reache 5.8%.
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Fig. 9
The strain components of the slurry sample in desiccation test
The evolution of volumetric strain of the slurry specimen in desiccation test was obtained by adding the strains of all directions of the sample, + + . Then, the results of the volumetric strain of the slurry sample in desiccation test with the results of theoretical shrinkage strain obtained were compared, it is possible to obtain that the volumetric strain of the samples obtained by the two methods is almost identical, as shown in Fig. 7. This means that the desiccation shrinkage of the slurry sample is not limited in desiccation test, the volumetric strain and strain component of the sample is the theoretical desiccation shrinkage strain, the experimental results of the sample in desiccation test can be used to describe and analyze the desiccation shrinkage process of clayey soils.
Desiccation test of pre-consolidation specimens
The volume of the pre-consolidation specimens did not change in the desiccation test, so the change of the water content of the samples were calculated based on mass variations and initial volume measurements. As shown in Fig. 10, the water content of pre-consolidation specimens changes with time under suction 13 MPa and 38 MPa. Obviously, the greater the suction force applied, the faster the rate of water loss of the samples. And the residual water content of the samples under the greater suction is lower than that of the sample under the low suction. During the drying process, the suction applied to the surface of the sample does not reach the expected value immediately, it needs a process. Under the influence of suction 38 MPa, the sample has an inflection point at 65 days, and the water loss rate of the sample changes. At this time, the suction applied on the surface of the sample has reached 38 MPa. At 200 days, the entire sample have reached 38 MPa, and the water content of the sample no longer changes. Under the influence of suction 13 MPa, this inflection point is at 120 days.
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Fig. 10
Water content of pre-consolidation specimens changes with time
Theory and numerical approach
In current research, the combined numerical simulation and laboratory experiments research were conducted for investigating the drying process of a remolded clayey soil. Therefore, after the experimental results of the samples in the desiccation experiments were analyzed, the simulation methods for describing the drying process of clayey soils were proposed. According to the above test results, there are differences in the deformation of the two samples during the drying process: the deformation of slurry in the test is mainly caused by drying shrinkage; the pre-consolidation sample does not deform during the drying process, the external load is the key factor leading to its deformation. Therefore, the theoretical methods and constitutive establishment of the two type of samples also need to be analyzed separately.
Slurry specimens
Governing equations
The total strain of slurry specimens was divided into two parts (Eq. 5), the desiccation shrinkage strain part and the mechanical strains part that are generated by the total stress variation due to the boundary conditions imposed on the sample. The first consideration was how to describe the desiccation shrinkage strain . In fact, the desiccation shrinkage strain of the slurry specimens during the drying process can be divided into two stages one related to saturated state and the other to the unsaturated state. In these two stages, the sample will change its internal pore water pressure due to the evolution of water content. Many researchers tried to use constitutive models to construct the relationship between pore water pressure and desiccation shrinkage strain. However, for the remolded clayey mud, the stress and the stiffness of the slurry are quite difficult to identify especially in the saturation stage. The stress being almost negligible it is thus difficult to use constitutive approaches to accurately describe the desiccation shrinkage strain. Therefore, in this research one directly uses the Fredlund’s function in Eq. 3 to estimate the desiccation volumetric shrinkage strain related to its water content generated by the drying process. Because the water loss rate of the sample can be obtained through experiments, it is easy to construct the water content field of the sample, and then the desiccation volumetric shrinkage strain .
5
The experimental observations show that, the desiccation shrinkage deformation of the slurry specimens during the drying process is not always isotropic, and there may also be anisotropy. Therefore, in order to make the simulation more realistic, the strain components of the sample also need to be simulated. It is necessary to convert the desiccation volumetric shrinkage strain into the desiccation shrinkage strain component, and then the desiccation shrinkage strain component is used in the constitutive. The strain component of the sample in the three principal directions ( can be deduced as from the volumetric strain in supposing of an isotropic desiccation shrinkage strain, as follows:
6
where are the transverse, the longitudinal and the vertical desiccation shrinkage strain of the sample, respectively.
The experimental observations of desiccation tests show that, the deformation which due to the loss water of slurry specimens during the desiccation process was far away from isotropic. In the case of anisotropic desiccation shrinkage strain, which is closer to the experimental results, the coefficient of shrinkage ratio was used. The coefficient of shrinkage ratio used in this research was based on the desiccation experiments.
7
So far, the desiccation volumetric shrinkage strain and desiccation shrinkage strain components of soils during the drying process can be described. Then the coupling of the desiccation shrinkage strain and the mechanical strain is considered. The latter which relate the mechanical and the hydraulic problems may be performed through sequential resolution of the two problems and the interactions between them.
The proposed constitutive model was formulated in terms of mechanical strain, the total stress in soils during the drying process was generated by mechanical strain. The mechanical strain was equal to the difference between total strain and desiccation shrinkage strain, under the form: , such the constitutive relationship could be expressed by:
8
In this article one uses simple linear elastic matrix to construct constitutive and to describe the relationship between mechanical strain and total stress. In the Eq. 8, is the elastic matrix, notice that total stress and strain are negative in compression. In fact, the experimental results show that no total stress is involved in the desiccation test of slurry specimens, therefore, , and the total strain corresponds to the desiccation shrinkage strain .
Implementation of desiccation shrinkage algorithm
The 3D formulation of the model was implemented in the ABAQUS via the subroutine of UMAT.
To achieve numerical simulation of desiccation shrinkage process of the sample, a water content field in the model should be created first. The water content of each element in the model is obtained from the duration and the rate of water loss.
9
w is the water content at a given stage of drying, which is represented through the drying duration t.
w0 is the initial water content, which is considered as constant in all the sample at t = 0.
is the water loss rate parameter. For uniform distribution for instance, is constant.
If one considers a no-uniform water content distribution, can be set as more or less complex function depending on the space variables and the geometrical form of the sample.
and were set in the material parameters of ABAQUS, the time was set in the step of ABAQUS (Corresponds to time (2) in UMAT). After obtaining the actual water content of each element, the void ratio and the desiccation shrinkage strain of each element were obtained by using the Fredlund equation (the Fredlund equation was implemented in UMAT, the parameters of the formula were set in the user material parameters of ABAQUS). The desiccation shrinkage strain in each direction was obtained by using the coefficient of shrinkage ratio .
In UMAT, the program was divided into two parts. The first part was to create the water content field in the model as described above, and then used the water content of each element to calculate the variables, e.g. void ratio, desiccation shrinkage strain, these values were retained and output as state variables (Corresponds to SDV in UMAT). The other part was used to calculate the total stress and strain of the model. The obtained desiccation shrinkage strain components were substituted into the linear elasticity isotropic constitutive of Eq. 8, and the total stress was obtained. Then the stage of finite element simulation can be performed.
Modeling and friction boundary establishment
In this section, the model for simulating desiccation experiments was established. The size of sample model was a square with 180 mm length and 5 mm in thickness, the finite element mesh contained 40 elements in the horizontal direction and 2 elements in the vertical direction. Then, the size of support model was a square with 200 mm length and 5 mm in thickness, it was only divided into one element. The model as shown in Fig. 11, the 8-node element (C3D8) was used.
In the case of desiccation test of slurry specimens, the boundary conditions used in the simulation phase are illustrated in Fig. 11, in theory, there is no need to add any boundary conditions to the sample model to achieve the purpose of simulating drying. However, for the sample model the displacement in both X direction and Y direction are blocked at the bottom of the centre. The purpose is to prevent the model from rotating during the simulation process. The base of the plate is considered rigid, so the displacement of the base plate model in all directions is restricted. Two contact surfaces (lower surface of sample and upper surface of base plate) need to be defined in ABAQUS, the base plate model can limit the displacement of the bottom of the sample model in the Z direction.
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Fig. 11
Geometry and boundary conditions for model
The principal model parameters were listed in Table 3. The initial water content and rate of water loss were used to create water content field in the model, the water content field was uniform. The material parameters applied in the Fredlund equation to calculate the desiccation shrinkage volumetric strain . The constitutive can be calculated by the elasticity modulus E and the Poisson ratio . The coefficient of shrinkage ratio is used to calculate the anisotropic strain.
Table 3. List of model parameters
Notation | Name of the model parameter, function | Value |
|---|---|---|
Parameters of model | ||
Initial water content | 48 | |
Initial void ratio | 1.26 | |
Coefficient of shrinkage ratio | 2.8 | |
the rate of water loss | 0.83%/h | |
E | Elasticity modulus | 1*105Pa |
Poisson ratio | 0.38 | |
Parameters of Fredlund equation | ||
0.63 | ||
50 |
Pre-consolidation specimens
Governing equations
Due to low permeability of the pre-consolidated specimen, the desaturation process is relatively slow in clayey soil. Therefore, a high-water pressure gradient is generated with a significantly heterogeneous suction distribution in the clayey soil during the drying process. This non-uniform distribution of moisture will induce the tensile stresses. This is one of the main reasons for the initiation and propagation of microcracks in clayey soil. Therefore, it is very important to understand the changes in capillary pressure and water content within the clayey soil during the drying process.
The clayey soil is composed of solid skeleton and porous and can be studied in the framework of porous media. It is supposed that the materials are partially saturated as its pores are occupied by a liquid water (lq) and a gas mixture (gm). The capillary pressure is then depends on the gas mixture pressure in the pore and the pore water pressure , as:
10
The variation of gas pressure is generally very small with respect to that of liquid pressure. For the reason of simplification, it is assume that the gas mixture pressure is constant with the same value of the atmosphere pressure, as = = 0.1MPa. Therefore, the capillary pressure is directly depends on the pore water pressure in soil.
It also supposes that the transfer of pore water in pre-consolidation specimens in the desiccator could be described by the generalized Darcy law, as given as:
11
In which, is the vector of fluid flow of pore water, µ is the viscosity of fluid. ρ is the density volumetric of fluid. k is the intrinsic permeability of pre-consolidation specimen and defines the relative permeability related to fluid, which is function of fluid saturation. Generally, the intrinsic permeability depends only on the pore and its distribution in the soil, the intrinsic permeability vital factor which control the desiccation process.
Mass conservation of water:
12
where is the mass of fluid.
The water saturation is related to the capillary pressure by van Genuchten equation:
13
where , N and M are material parameters. The relative permeability to liquid depends on the water saturation, and the following relation is used in this study:
14
Applying the Darcy’s law into the equation of mass conservation, the generalized diffusion equations for unsaturated porous media is obtained. By applying the variational method to the field equations and using the implicit time stepping, a set of non-linear equations is established. The non-linearity must be solved, principally related to the non-linear poroelastic diffusion. An iterative procedure is then needed for each time step.
Modeling and friction boundary establishment
In this section, finite element method is used for modeling. Owing to the symmetry of the problem to be solved, only a quarter of the sample is considered and meshed. The boundary conditions in the desaturation phase are illustrated in Fig. 12. The two boundaries (AB and AD) are impermeable while the horizontal displacement is blocked on AD and the vertical displacement is blocked on AB. The capillary pressure corresponding to the applied relative humidity is imposed on the boundary exterior BC and CD. The value of the applied suction which as a function of time can be changed arbitrarily over time in the simulation. However, it was shown that the value of the applied suction increases with time during the drying test, and it is assumed that the applied suction varies linearly with time to simplify the process in the simulation. The finite elements mesh is shown in Fig. 12, in the length direction, there are totally 49 elements, while in the vertical direction there are 76 elements. The 4-node element is used. The moisture of the sample diffuses from the surface into the air during the drying process, the exchange of the moisture of the sample occurs frequently on the surface. The water content of the surface decreases rapidly, in comparison, the water content of the inside of the sample decreased relatively gently. In order to better study the variation of water content in the sample, the division of elements which near the surface of the sample model becomes dense, while the division of elements which inside the sample is loose in this recherche.
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Fig. 12
Geometry, boundary conditions and mesh for clayey soil
The principal parameters used for the simulation are listed in Table 4. , N and M are parameters of the van Genuchten equation, their values are obtained by fitting experimental data [31, 32–33], as shown in Fig. 13, the water saturation of specimen is related to the applied capillary pressure. M can also be used to calculate relative permeability of the specimens. K is the initial permeability, it is only related to the nature of the solid skeleton. Since the specimens A and B are both Kaolin pre-consolidated specimens at 120 kPa, the parameters α, N, M and K of specimens should be approximately the same.
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Fig. 13
The variation of water saturation with capillary pressure
Table 4. Typical values of parameters
Parameter | Value |
|---|---|
2.63 | |
49.10 or 48.98 | |
1*106Pa | |
N | 2.15 |
M | 0.535 |
K | 1*10− 10m/s |
ρlq | 1000 kg/m3 |
µlq | 0.001 Pa*s |
Specific gravity and initial porosity have been introduced in the above, volumetric mass of liquid ρlq and viscosity of liquid µlq, which are useful in the simulation process, they are regular parameters, there is not much explanation here
Numerical simulation results and discussion
Slurry specimens
Evolution of water content
The water loss rate of 0.83%/h, obtained from experiments conducted under environmental conditions of 20 °C temperature and 7% relative humidity, was utilized in the simulation to ensure consistency with the evolution of water content in the dried soils. The slurry sample involved in this research was thin, the water loss on the surface of the sample accounts for a significant proportion of the total volume, thus there was little difference between the water content inside and on the surface of the sample, the water loss rate of the sample was approximately considered uniform in the simulation. The simulation results depicting the evolution and distribution of water content in the model over time are presented in Fig. 14. The simulation results well demonstrated that during the drying process, water was uniformly distributed within the clay slurry samples and continuously decreased at a water loss rate of 0.83%/h.
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Fig. 14
Simulation results illustrating the evolution and distribution of water content over time
Evolution of strain
The simulation results are compared and analyzed against experimental data from desiccation experiments. The simulation results of the evolution and distribution of the strain components of slurry sample model as a function of the water content are illustrated in Figs. 9 and 15.
In accordance with theoretical expectations, when no constraints were applied to the sample model, the deformation of the sample model was solely attributed to desiccation shrinkage strain without generation of mechanical strains, consequently, the total stress is 0, and the sample is in a state of contraction. Moreover, because the water content distribution was considered uniform in the simulation, the total strain of each element is the same in the simulation result of the desiccation experiment, as shown in Fig. 15. The figure shows the strain results of the sample model when the simulation reaches the shrinkage limit, the and reache 5.8%. Moreover, because anisotropic drying shrinkage was taken into account in the simulation, the obtained simulation results are more in line with the experimental results, and the vertical strain is greater than the horizontal strain.
The simulation considered anisotropic shrinkage expressed by Eq. 7, the strains along the three directions exhibiting the same values range on x- and y-axis and larger strains along the vertical z direction. These more realistic results are confirmed by the strain variations of the sample exhibited in Fig. 9 that highlights the experimental anisotropy phenomenon well reproduced by the simulations.
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Fig. 15
The uniform distribution of strain components (w = 20%)
Pre-consolidation specimens
In this section, the simulation results of the drying process of the pre-consolidation specimens are compared and analyzed with the experimental results, as shown in Fig. 16. The difference between the two groups of specimens in simulations lies in the boundary conditions for applying suction to the sample. For the first 65 days, the boundary condition of applying suction is increased from 0 to 38 MPa, and after 65 days, it remains continuously at 38 MPa. Under a suction influence of 13 MPa, for the first 120 days, the boundary condition of applying suction is increased from 0 to 13 MPa, and after that period it remains continuously at 13 MPa.
Actually, due to symmetry in solving this problem, the simulation result of the residual water weight (equivalent to water content) is half of the sample. Therefore, it is necessary to multiply by two times and compare with the experimental results. In the end, the simulation results are consistent with the experimental results, the changes in capillary pressure and water content in the consolidated clayey soil during the drying process are obtained.
As shown in Fig. 16, the greater the suction force applied, the faster the rate of water loss of the samples. And the residual water content of the samples under the greater suction is lower than that of the sample under the low suction. This is consistent with the experimental results. The difference between the current experiment and the simulation was mainly due to the use of the Van Genuchten equation in the simulation to describe the relationship between water content and suction. This equation was a continuous and smooth curve. Therefore, when this curve was used to describe the sharp turning point of the change in moisture content of clay due to the influence of suction, there was a slight difference. The maximum of the difference was w = 2% under the same suction, and it mainly occurred after the sample’s shrinkage limit (w = 24%). In fact, after the shrinkage limit, the sensitivity of clay moisture content changes to suction decreases. Therefore, this difference was still acceptable in the simulation results. This aspect will be improved and refined in future research.
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Fig. 16
Comparison of experimental results and simulation results of changes in residual water content of samples
In this research, the software GID (developed by the International Center for Numerical Methods in Engineering) is used for post-processing, purposes which helps obtain the changes in capillary pressure within the consolidated clayey soil during its drying process.
As shown in Fig. 17, the distribution of capillary pressure within the samples when their surfaces reach preset capillary pressures. When the capillary pressure on the surface of the sample A reaches 13 MPa, the maximum difference between inside and outside capillary pressure is only 10 MPa. When the capillary pressure of the surface of sample B reaches 38 MPa, the maximum capillary pressure difference between inside and outside reaches 35 MPa. This explains why the greater the environmental suction, the easier the clay sample will crack. The capillary pressure in most areas of sample A is close to 13 MPa, while in most areas of sample B it is far away from 38 MPa. This difference caused sample A to lose water uniformly as a whole, whereas the surface of sample B dried first and the interior still had a high degree of saturation. This also explains why the initial water loss rate of sample B was faster than that of sample A, but both samples reached similar final water content over time. This research serves as a basis for further study on the cracking process of consolidated clayey soil due to drying.
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Fig. 17
The distribution of capillary pressure within the samples when the surface of the two samples reaches a preset capillary pressure
Discussions
The simulation results are compered with the experimental results to verify the validity of the finite element simulation. For slurry specimens, the evolution of strains were not constrained in the drying process and the mechanical strains were equal to zero, the total strains of the sample corresponded to the desiccation shrinkage strains. As a consequence, in this study, no stress was generated by the calculations except for the uniformly distributed suction within Kaolin, and then, no crack initiation could possibly occur in this case. Through this relatively simple modelling approach using finite elements and appropriate unsaturated laws for clayey soils, we showed that desiccation shrinkage development within the clayey soils could be well reproduced. For pre-consolidation specimens, the desaturation process was relatively slow in clayey soil due to low permeability, the high-water pressure gradient was generated with a significantly heterogeneous suction distribution during the drying process. The greater environmental suction led to the greater maximum capillary pressure difference between the interior and exterior, making it easier for clay samples to crack, the research was the basis for the further study on the cracking process of consolidated clayey soil due to drying.
Conclusions
In this paper, the desiccation experiments and simulations were conducted on the slurry and pre-consolidation clayey soils. The SWCC of Kaolin was derived from the experimental tests. Fredlund and van Genuchten curves had been selected as appropriate laws fitting well the desiccation shrinkage and retention properties of the material. The deformation that occurred during the drying shrinkage process of fully saturated slurry is related to its own loss of moisture. Therefore, this research directly employed the Fredlund’s function to estimate the shrinkage volumetric strain. A constitutive model for the hydraulic mechanical properties of clay had been established based on a stress state variable framework. For pre-consolidation specimens, the desaturation process was relatively slow in clayey soil due to low permeability, the high-water pressure gradient was generated with a significantly heterogeneous suction distribution in the clayey soils during the drying process. Through the combination of numerical simulation and laboratory experiments, the changes in capillary pressure and water content within the consolidated clayey soils during the drying process were obtained. This research is the basis for the further study on the cracking process of clayey soils due to drying.
Author contributions
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Wenqing Cheng，Hanbing Bian，Qianfeng Gao，Liang Wang and Fangfang Xun reviewed the manuscript.
Funding
This research is funded by Natural Science Research Project of Anhui Educational Committee Refs: 2024AH050379.
Data availability
Available from the first author upon request.
Declarations
Ethical approval
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Consent to participate
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Competing interests
The authors declare no competing interests.
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