1. Introduction
Earthen sites are evidenced all over the world [
1]. Earthen-site soils as primary construction materials have been widespread since ancient times in many areas of Africa, Asia, and Central and South America. About 30% of the world’s population still lives in earthen buildings as per the United Nations [
2]. The outdoor immovable earthen sites are located in complex environmental conditions, exposed to the natural environment for a long time, with their surfaces severely weathered (
Figure 1). Consolidation may be imperative to promote and restore the strength of the earthen-site soils.
Research efforts on the protection of earthen sites have aroused wide interest since the 1960s. While remarkable advances have been made in the consolidation and repair of earthen sites, the conservation of earthen sites is still a great challenge, with surface weathering being one of the critical issues to be further solved.
The research on anti-weathering materials is increasingly prosperous. There are many kinds of anti-weathering materials, which can be divided into organic materials, inorganic materials, and organic–inorganic hybrid materials according to their chemical compositions. Organic materials have significant improvement in stone artefacts due to their favorable permeability and adhesiveness [
3,
4,
5]. Acrylic polymeric resins and silicon-based strengthening agents are used for the conservation of monumental buildings and other porous materials [
6,
7,
8,
9]. Inorganic materials have strong aging resistance, low cost, and suitable compatibility with ancient ruins [
10]. New calcium alkoxides reacted with atmosphere to generate calcium carbonate, which was used to consolidate carbonate rocks [
11,
12]. Alkaline solutions for the consolidation of earthen structures revealed a positive change in water resistance and mechanical strength [
13]. Barajas et al. [
14] compared the physicochemical effects of organic and inorganic consolidants on the Tlaltecuhtli monolith. Innovative phosphate-based consolidants were also compared with a commercial ethyl silicate in terms of effectiveness and compatibility [
15]. Zhang et al. [
16] found that the consolidation effect of combined materials on earthen-site soils is better than that of single materials in accordance with laboratory results.
The change in mechanical properties of soils after treatment is one of the core issues in evaluating improvement effects. The mechanical properties of soils can be divided into two categories, namely static properties and dynamic properties. The static properties of soils include stiffness, strength, constitutive relations, etc. Uniaxial compressive strength is the peak stress of the material resistance to axial compression when the load is slowly applied along the axis at a certain rate without confining pressure, which can directly indicate the strength of the material. The dynamic properties of soils include dynamic strength, dynamic constitutive relations, vibration characteristics, wave characteristics, etc. The nondestructive ultrasonic technique is prevalent in the field of cultural relic protection because of its small interference with the detected object. Previous studies have suggested the relationship between strength and acoustic parameters [
17,
18]. An exponential relation exists between unconfined compressive strength (UCS) and ultrasonic pulse velocity (UPV) for materials, such as foamed mixture lightweight soil [
19] and concrete [
20,
21,
22]. Tan et al. [
23] established mathematical models of coral aggregate concrete to explore the effects of sisal and age on the relationship between UCS and UPV. A linear correlation between UCS and UPV was found for foam-cemented paste backfill [
24] and carbonate rocks [
25]. The UPVs of most of consolidated earthen-site soils decrease to a certain extent compared to the untreated soils, while their UCSs increase [
16]. Reasons for the different changes in strength and acoustic properties are not well understood. Attempts thus far have been made to explore the relationship between them from statistics, without considering the microstructure of materials. Few studies were carried out to reveal the quantitative relationships between UCS, UPV, and pore characteristics; thus, it is difficult to predict the engineering properties of materials.
The change in physical and mechanical properties of materials is the result of the interaction of their composition [
26,
27], micromechanical properties, and microstructure [
28,
29]. The microstructure of materials can be observed by environmental scanning electron microscopy (ESEM) [
30] and mercury intrusion porosimetry (MIP) [
31]. The combination of ESEM and digital image correlation (DIC) technology can determine the deformation and failure mechanism of mudstone and consolidated earthen-site soil from the microscopic scale [
32], investigate the influence of humidity on the mechanical behavior of mudstone [
33], and perform full-field strain measurements for low-strain geomaterials [
34]. Zhao et al. [
35] analyzed the influence of microstructure on fracture mode by using image processing and analysis techniques. Liu et al. [
36] fitted the expression of correlation between pore size and UCS of concrete.
The recent threads of applying computed tomography (CT) and scanning electron microscopy (SEM) have promoted the development of real microstructures, which has also laid the foundation for the study of the relationship between the microstructure and mechanical behavior of consolidated earthen-site soils. While previous studies focus on explaining the macroscopic physical and mechanical properties of materials from the perspective of the microstructure, the macroscopic mechanical analysis based on microscopic images fails to consider the influence of the microstructure parameters, thus failing to clarify the essential causes of deformation and failure. With the rapid development of computer technology, the study of mechanical behavior extends from laboratory and field tests to numerical simulation. Particle agglomerates modeled by particle flow codes were used to investigate the effects of size [
37], porosity, and friction coefficient [
38] on strength properties of soils. The two-dimensional particle assembly model can study the influence of contact laws [
39], particle shape [
40], and particle breakage [
41,
42,
43] on mechanical parameters, and simulate various laboratory tests to capture the macroscopic behavior of granular media [
44,
45,
46]. Nguyen et al. [
47] undertook simulations of concrete with an X-ray CT image-based model to study the failure mechanism during in situ compression. Zhang et al. [
48] visualized the pore distribution of consolidated earthen-site soils and simulated the UCS test by the finite element method, but did not consider the influence of pore characteristic parameters. In addition, stress wave propagation can be simulated by the finite element model [
49]. Numerical simulation of ultrasonic pulse effectively detected the early-stage corrosion of steel rebar [
50], revealed the physical mechanisms of wave modulus of elasticity as being higher than the static modulus of elasticity through sinusoidal wave and compression testing techniques [
51], and correlated the ultrasonic parameters with the elastic modulus, UCS, and density of the materials [
19].
Due to complex structure of the earthen-site soils, the quantitative relationships between pore parameters and macroscopic properties are not well understood. Traditional testing methods lack quantitative investigation of the relationship between micro-pore parameters and strength, stiffness, and acoustic characteristics of soils; hence, it is difficult to directly and comprehensively analyze the reasons for the changes in the mechanical properties of the consolidated earthen-site soils. After the introduction of numerical simulation, the above-mentioned problems are expected to be well solved. The numerical model can strictly control the variables and conform to the principle of reducing intervention in the cultural heritage conservation criteria. Micromechanics-based finite element simulation may provide a new route for exploring the causes of macroscopic phenomena.
In the current study, we aimed to develop a physics-based numerical model for quantifying the relationship between micro-pore structure characteristics, strength, and acoustic parameters of earthen-site soils. A microstructure-based finite element model was established to control the change in pore parameters. The model reveals how pore equivalent diameter, pore sphericity, and porosity affect the macroscopic mechanical properties of soils under linear axial load and harmonic pulse load. This study analyzes the reasons for the variations in unconfined compression strength (UCS) and ultrasonic pulse velocity (UPV), as well as deepens the understanding of soil improvement. The porosity has a great influence on the macroscopic mechanical behaviors of soil. The compression wave velocity of the non-porous medium is more than twice that of the medium with 40% porosity. The quantitative investigations between the macroscopic performance and microscopic characterization relationships will be helpful to guide the design and construction of cultural relic conservation projects.
2. Materials and Methods
2.1. Image Processing and Characterization of Pores
The purpose of quantifying the microstructure of soils is to acquire the sphericity, size, quantity, distribution location, fractal dimension, and other indexes of particles or pores. In order to determine these parameters, it is necessary to observe the morphology of soils qualitatively by using microstructure testing equipment to obtain microstructure images, followed by extracting microstructure parameters through a series of image processing and analysis procedures.
A published study [
16] indicated that cubic samples consolidated by Bio Line
® Ethyl silicate+Bio Line
® Micron lime have obvious advantages over other materials in terms of physical and mechanical properties. Therefore, scanning electronic microscopy (SEM) images with 100× magnification of consolidated and untreated samples were selected for microstructure analysis in this study (
Figure 2). The untreated and consolidated earthen-site soils are regarded as a two-phase system consisting of a solid (matrix) and a gas (pore). The quantization process of the images’ microstructure parameters can be divided into four steps. The first step is to format the images to BMP files and import them into image processing software for segmentation. The second step is to preprocess the images before segmentation. The images were filtered to reduce the noise, as it would interfere with the observation of information. Thereafter, the artifacts were removed, and the features of interest were enhanced. The third step is to identify the interface between different substances based on pixel intensity. The pre-processed images were separated into different regions—matrix and pores—by the threshold segmentation method [
35,
52]. Then, isolated pixels or pixel blocks were removed and the nodes that had little influence on the particle morphology were reduced to improve the computational efficiency. The fourth step is parameter extraction. The porosity of samples was calculated according to the area ratio of each region. Parameters such as pore equivalent diameter and sphericity were obtained by image analysis module.
2.2. Selected Area and Threshold Settings
Threshold has a great influence on modeling. Different threshold values were set for images in the range of 70–125 with 5 as threshold interval to determine the optimal threshold value. Porosities of soils were calculated as per their area proportion, and the relationship between porosity and threshold was ascertained.
In addition, the study area should be representative due to the heterogeneity of microstructure of soils. The image should contain enough particles and pores. The pixel size of the SEM image with 100× magnification is 960 × 1280. Eight selection areas, i.e., 120 × 160, 240 × 320, 360 × 480, 480 × 640, 600 × 800, 720 × 960, 840 × 1120, and 960 × 1280, were set to establish the relationship between threshold and porosity so as to determine the optimal selection area.
2.3. Simulations of Linear Axial Load Transfer in Soils
Simulations were carried out on Marc Mentat 2020.0.0 developed by MSC software Corporation. The soil is an inhomogeneous medium with complex and diverse particle composition and pore structure. In continuum mechanics, constitutive relations are normally applied to study the responses of uniform media to loads. Therefore, heterogeneous soil can be equivalent to uniform material, and the representative volume element (RVE) should be selected.
The samples and loads in the process of uniaxial compression were symmetric. Only half of the model was selected, and the left boundary was taken as the symmetric boundary. Therefore, the horizontal displacement of the left boundary was constrained to zero. The geometric properties of the elements were structural plane stress. The vertical displacement was loaded on all nodes of the upper boundary at a loading rate of 0.05 mm/s. The nodes of the lower boundary were fixed, and those of the right boundary were free. The static structural analysis in Marc software was conducted to simulate the mechanical behavior of the soils under linear axial load. As the yield and failure behaviors of soils were involved, large strain was employed for calculation.
The isotropic elastoplastic model was applied to analyze mechanical response of microstructure under axial compression, and the Young’s modulus and Poisson’s ratio of the medium had to be given. The Mohr–Coulomb yield criterion was adopted to describe the plastic properties of soils, and the yield stress of the material had to be defined. The Mohr–Coulomb criterion is a combination of Hooke’s law of isotropic linear elasticity and Coulomb’s law of perfect plasticity. In finite element analysis, the response of force and displacement was determined by the path graph. In post-processing, the vertical stress of a cross-section could be calculated as per the force in the vertical direction as,
where
σ is the axial stress,
RF2 is the reaction force in the vertical direction, and
A is the area of the cross-section.
The reaction force in the vertical direction can be calculated by multiplying the stress at each node with the area of element. The formula of stress can be rewritten as
where
is the average vertical stress,
is the vertical stress of the node,
is the area of the element, and n is the number of nodes at the cross-section.
The determination of parameters can refer to the range of empirical data due to the limitations of current testing technology. Many simulations and specifications have given the range of Poisson’s ratios for soils. The specific gravity and plasticity index of the earthen-site soils are 2.72 and 10.16, respectively, thus being silty clay according to the
Code for Investigation of Geotechnical Engineering [
53]. Clays are generally assumed to be undrained. In an undrained analysis of cohesive soils such as silty clay, the Mohr–Coulomb criterion is reduced to the well-known Tresca yield criterion. For saturated clay, Poisson’s ratio is in the range of 0.4 to 0.5 and the friction angle is close to 0 [
54,
55]. Poisson’s ratio was 0.44 in this study, and the Young’s modulus and yield stress of the matrix were 161 MPa and 3.32 MPa, respectively [
48]. It is also noted that in the consolidation of heritage/earthen-site soils, the unconfined compressive strength is usually used to analyze the change in the strength of the heritage after consolidation, so as to evaluate the consolidation effect of the consolidants in the literature [
10,
56]. We studied the effect of different microstructural parameters on strength; therefore, the unconfined compressive strength test was used instead of the triaxial test.
2.4. Simulations of Ultrasonic Propagation in Soils
The propagation characteristics of ultrasonic waves are closely related to the property and structure of soils. The ultrasonic detector with two transducers can detect the physical and mechanical properties of soils. One sensor served as a pulse transmitter and the other was used for receiving the pulse. The time of flight of the compression wave (
) and shear wave (
) is measured by the input and output signals. According to the distance between the transmitting position and receiving point (
l) and the time of flight of compression wave (
) and shear wave (
), ultrasonic wave velocities can be calculated by
where
is the propagation velocity of the compression wave, and
is the velocity of the shear wave. Furthermore, the dynamic shear modulus of the material (
) can be assessed according to the calculated velocity of the shear wave:
where
ρ is the density of the material.
Based on the assumption that the medium is homogeneous, isotropic, and elastic, the dynamic elastic parameters of the material can be computed in accordance with the ultrasonic wave velocity:
where
is the wave modulus of elasticity, and
is the dynamic Poisson ratio.
Three-node triangle elements were used in the meshing. The dynamic transient structural analysis in Marc software was conducted to simulate the mechanical behavior of the soils under simple harmonic pulse. Only small elastic deformation occurs in soil because of the low energy of ultrasonic waves. The soil can return to its original state after removing the load; thus, it belongs to small strain.
A half-sine pulse with a frequency of 50 kHz was applied at one boundary of the RVE and transmitted signals were received at its opposite boundary. Four positions were selected for transmitting and receiving signals. The ultrasonic wave was excited from the top edge of RVE and received from the bottom edge, marked as TB. The signal was excited from the bottom edge of RVE and received from the top edge, named BT. The half-sine excitation was loaded to the nodes on the left boundary and output from the right boundary, marked as LR. The pulse was loaded to the nodes on the right boundary and output from the left boundary, named RL. Time of flight was estimated by subtracting a quarter period of input signal from the time corresponding to the first peak of transmitted signal, and then determining ultrasonic wave velocity according to Equations (3) and (4). The wave modulus of elasticity of the matrix was 161 MPa, the dynamic Poisson ratio was 0.44, and the mass density was 1.605 g/cm3.
2.5. The Establishment of Models
Finite element models of isotropic homogeneous medium with pores were developed, in which the pores were randomly distributed and of regular shape, and the size and shape of pores in the same medium were completely consistent. The micromechanics-based model including simple pores could control single variables easily, and explore the effect of structural characteristics on strength, stiffness, and acoustic parameters of soils. The pore equivalent diameter, pore sphericity, and porosity of medium were set to a series of gradient values to observe their influence on mechanical behavior. The static behavior of the medium was achieved by applying linear axial displacement to the model, while the dynamics of the medium were obtained by loading sinusoidal acceleration pulses.
4. Discussion
The pore equivalent diameter has little influence on the strength and stiffness characteristics of the medium, and the macroscopic stress–strain curves of the medium almost coincide (
Figure 8). The static parameters of the medium fluctuate in a small range with the increase in pore diameter, indicating that the static behavior of the medium is nearly unaffected by the pore equivalent diameter, and the subtle difference in the macroscopic static behavior is also related to the grid generation of the medium and the distribution of pores (
Figure 16). Pore equivalent diameter can influence the macroscopic dynamic behavior of medium. The propagation velocity of the compression wave decreases first and then increases with the rise in pore diameter. The maximum velocity is 1.05 times that of the minimum velocity within the pore equivalent diameter studied in this paper, and the variation is small.
The range of pore equivalent diameter in the actual microstructure of soil is wider; hence, the equivalent diameter greatly changes the propagation velocity of a simple harmonic wave. The average pore equivalent diameter for the untreated soil is 0.17 mm, while that for consolidated soil is slightly reduced to 0.12 mm. The static behavior of the soil does not change visibly after consolidation, and the propagation velocity of the half-sine pulse in the soil decreases in terms of the pore equivalent diameter. Therefore, ultrasonic pulse velocity may decrease when the strength of the soil remains unchanged, which may be one of the reasons why the compression wave velocity of the consolidated soil presents a downward trend in the ultrasound-based test.
The pore sphericity has an obvious effect on the static behavior of the medium (
Figure 11). The stress–strain curve of the medium changes distinctly with pore sphericity, and the pore sphericity has a considerable influence on the strength and deformation characteristics of the medium. In order to compare the influence of pore sphericity on the macroscopic static and dynamic parameters of the medium more intuitively, the macroscopic mechanical parameters of the medium were extracted to explore their relationship with the pore sphericity of the medium (
Figure 17). The unconfined compression strength, secant modulus, and compression wave velocity are positively correlated with the pore sphericity, while the strain at the peak stress is negatively correlated with the pore sphericity. The strength and stiffness of the medium with circular pores are the largest in a two-dimensional structure, and the propagation velocity of simple harmonics is also the fastest. The decline in pore sphericity tends to reduce the strength, stiffness, and propagation velocity of simple harmonics. In the pore sphericity range of 0.600–1, the top strength of the medium is 2.57 times that of the lowest value, the maximum secant modulus of the medium is 2.79 times that of the minimum value, and the fastest propagation velocity of the compression wave in the medium is 1.23 times that of the slowest value. The average pore sphericity of soil after consolidation is 0.6525, and that of the untreated soil is 0.6880. Therefore, the average pore sphericity of soil drops after consolidation, leading to a decrease in the ultrasonic pulse velocity.
For the consolidated soil, the average pore equivalent diameter becomes smaller, the propagation velocity of simple harmonic wave decreases slightly, and the strength and deformation characteristics of soil hardly change. However, its average sphericity decreases, the ultrasonic pulse velocity increases greatly, and the stiffness and strength of soil also increase obviously. Pore sphericity has a more significant effect on the macroscopic static and dynamic parameters of soil than pore equivalent diameter (
Figure 16 and
Figure 17). Therefore, the macroscopic mechanical properties of the medium are more sensitive to pore sphericity than to pore equivalent diameter within the range of pore parameters studied in this paper.
The macroscopic static behavior of the medium is affected enormously by porosity (
Figure 14). The peak stress and secant modulus decrease by roughly the same amplitude with the increase in porosity. As the porosity of the medium rises, the ultrasonic pulse velocity decreases, and the velocity of the medium with 10% porosity decreases significantly compared with that of the non-porous medium. In the range of 10–40% porosity, the variation in the propagation velocity of elastic wave is roughly consistent with the strength and secant modulus (
Figure 18). The strain at the peak stress also tends to decrease with the increase in porosity. In the porosity range studied in this paper, the top strength of the medium is 4.80 times that of the lowest value, the maximum secant modulus of the medium is 2.99 times that of the minimum value, the maximum strain at peak stress is 1.43 times that of the lowest value, and the fastest ultrasonic pulse velocity in the medium is 2.11 times that of the lowest value. This demonstrates that the static behavior of the medium is more sensitive to porosity than the acoustic characteristics; hence, the strength and ultrasonic pulse velocity of the soil can be improved by the decrease in porosity after consolidation, and the strength increases more than the velocity.
In terms of the equivalent diameter, sphericity, and porosity, porosity has the greatest influence on the macroscopic mechanical behavior of the medium, followed by sphericity, and the mechanical behavior of the medium is the least sensitive to the pore equivalent diameter. Therefore, the change in porosity and pore sphericity of soil should be paid more attention when studying the macroscopic mechanical properties of soil.