Research Article  Open Access
Haicheng She, Zaiqiang Hu, Zhan Qu, Hongru Li, Hu Guo, Xiaoli Ma, "Structural Strength Deterioration Characteristics and a Model of Undisturbed Loess under the Action of Wetting and FreezeThaw Cycles", Mathematical Problems in Engineering, vol. 2019, Article ID 4790250, 23 pages, 2019. https://doi.org/10.1155/2019/4790250
Structural Strength Deterioration Characteristics and a Model of Undisturbed Loess under the Action of Wetting and FreezeThaw Cycles
Abstract
Undisturbed loess is affected by external environmental disturbances, such as wetting and freezethaw cycles, which cause microstructural changes that have an important impact on the structural strength of the loess. These changes in turn affect the stability of structures such as embankments, slopes, and guards. This article takes the Q_{3} undisturbed loess in Lintong District, Xi’an, as an example. The effects of wetting and freezethaw cycles on the loess expansion ratio and pore structure were studied by wetting tests, freezethaw cycle tests, and scanning electron microscopy (SEM). The changes in the compression index and compression modulus were studied by a confined compression test. The loess elgp compression curve was obtained according to the confined compression test, and the newly defined concepts of the loess structural strength, residual structural strength, and structural strength damage variable, in addition to the elgp compression curve, were combined with the experimental data to calculate the damage value generated by the disturbance during the sampling and preparation of loess. The deterioration of the structural strength and damage variable of loess was analyzed. Based on the microscopic statistical damage theory and Weibull distribution, the model used the volume expansion ratio as a variable to establish a statistical damage model under wetting and freezethaw cycles. Finally, on the basis of the test, the model parameters were determined. The models were verified by taking loess from a foundation pit in the northern suburbs of Xi’an and were in good agreement with the results of the test. Ultimately, the models have good practicability and can provide guidance for engineering design and construction.
1. Introduction
Loess is a porous, weakly cemented Quaternary sediment and is mostly distributed [1, 2] in arid and semiarid regions in Asia, Europe, North America, and South America. Among all the countries in the world, China has the widest loess distribution area, the thickest and most complete strata, and the most complex genetic types and typical characteristics [3, 4]. As shown in Figure 1, China’s loess is distributed mainly in Shanxi, Shaanxi, southeastern Gansu, and western Henan. In addition, there are sporadic distributions in Beijing, western Hebei, eastern Qinghai, Xinjiang, the Songliao Plain, Sichuan, Three Gorges, the Huaihe River Basin, and Nanjing. Loess forms a unique mineral composition with underconsolidated tiny structural units in a special sedimentary environment and complex and variable climatic conditions; consequently, loess has notable characteristics [5] such as a high water sensitivity, large pores, and lowpressure compaction, as shown in Figure 2.
The notable structural properties of loess [6] are the main reason why loess has unique physical and mechanical properties, such as its collapsibility, compressibility, and shear strength; these structural properties are also the underlying cause of serious disasters in loess areas. Due to the differences among the geological features of loess areas, such as their geographical distribution, thickness variation, geological sequence, and material composition (particles, minerals and salts, etc.), the properties of loess are also regionally variable [7]. An indepth understanding of the structural geological characteristics of regional loess is of great importance for solving the engineering geological problems in regional agriculture, industry, and urban construction. The core of geotechnical engineering research in loess regions is the analysis of the deformation, strength, and stability of loess in a complex external environment under mechanical disturbances. The most common loess disasters, as shown in Figure 3, primarily include loess landslides [8–11], collapses [12], loess collapsing [13, 14], ground collapse or subsidence [15, 16], ground fissures, and unstable slopes [17]. These disasters are of global concern because of their great cost to humans [8, 18–21]. The main factors inducing these loess disasters are natural causes such as earthquakes, rainfall, temperature changes, erosion of surface water, infiltration, and rivers as well as human activities such as excavation slopes, slope loading, blasting, reservoir water storage, and mining. Due to the disturbances of the external environment (such as the infiltration of underground and rainwater and temperature changes), the physical properties of loess microstructures, including their bonding and arrangement, are altered, and the structural strength deteriorates. The degree of deterioration of the loess structure, pore changes, and whether the soil structure is destroyed under external loading are closely related to wetting and freezethaw cycles and their interaction.
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The structural strength of loess is a complex problem that has attracted extensive attention from scholars worldwide. As early as the early 20^{th} century, people began to pay attention to the structure of the soil. A honeycomb structure [22], laminar structure [23], and disperse granular structure [24] have been proposed for soil; accordingly, the microstructure is important for evaluating soil strength and deformation. Soga [25] and Lafeber [26] presented the definition of soil structure based on Mitchell’s understanding of the soil microstructure, namely, the geometric arrangement of soil particles, aggregates, and pores usually defined as the microscopic configuration of soil. Lohnes and Demirel [27] studied the strength and structure of laterites and lateritic soils. Gao [28] scanned the microstructures of loess in various parts of northern China. Shen [29] asserted that the study of soil structure is the core of 21^{st} century soil research. Xie and Qi [30] constructed a quantitative parameter reflecting structural properties based on compression deformation, also known as the comprehensive structural potential. Subsequently, Luo et al. [31] and Chen et al. [32] performed extensive work based on the idea of the comprehensive structural potential. Reznik [33] proposed a method for the calculation of soil structural pressure values. Liu et al. [34], Ahad [35], and Xu et al. [36] studied clay soil residual strength and shear strength characteristics.
To date, some studies have examined the influence of freezethaw cycles on the structural strength of loess. Edwin and Anthony [37] and Qi et al. [38] determined that the soil structure is affected by freezethaw cycles, which mainly change the microparticle cementation and arrangement of the soil and affect the changes in shear strength parameters. Johnson et al. [39] studied the effects of freezethaw cycles on the resilient properties of finegrained soils. Similarly, Lu et al. [40] studied the effects of freezethaw cycles on the cracking of cohesive soils. Zhou et al. [41] and Xu et al. [42, 43] studied the effects of freezethaw cycles on mesostructural characteristics and mechanical behaviors using scanning electron microscopy (SEM) and soil mechanics tests.
Based on the above research, the present paper quantitatively studied the wetting and freezethaw cycles and the coupling of these two cycles. First, through laboratory experiments, the volume and structural pore variation of undisturbed loess samples and the deterioration of loess structural strength under the conditions of wetting and freezethaw cycles were studied. Second, the variables related to the structural strength were newly defined according to the undisturbed loess elgp compression curve, and the basic equation of damage was established. Finally, based on microscopic damage mechanics theory and mathematical statistics, the statistical damage variables under wetting and freezethaw cycles and a statistical damage model of loess structural strength were constructed, the model parameters were determined, and the statistical damage model was verified.
2. Loess Deterioration Test
The soil samples were taken from the Q_{3} loess of a foundation ditch in Lintong District, Xi’an. The soil has strong structural properties, and its buried depth is approximately 16 m. The essential properties and mineral composition of loess were analyzed by routine experiment and Xray diffraction, as shown in Table 1.

The waterfilm transfer method (graded titration water injection method) was used to prepare loess samples with water contents of 9%, 15%, 20%, 25%, 30%, and saturated, and these samples were placed in a closed moisturizing tank for more than 3 days to ensure uniform water transport in the sample. The soil samples with the original water content and different water contents were subjected to 1, 2, 3, 6, 9, and 15 freezethaw cycles. Then, the confined lateral compression test was performed. Water migration loss occurred during the freezethaw cycles, and the test water was quantitatively supplemented according to the experimental statistical rules.
2.1. Test Schemes and Methods
The experiment simulated the changes in the volume and pore size of loess with an increase in the water content before and after the freezethaw cycles. The test schemes are shown in Table 2.

The samples were taken from free expansion deformation, in which the lateral direction is restricted, and to improve the measurement accuracy, the test measured the mass of the expanded portion. It can be seen from Table 1 that the loess samples represented expansive loess, and the amount of expansion was considered to depend upon water swelling and frost heaving. When undisturbed loess was used in the process of preparing samples with different water contents, a certain amount of expansion occurred in the sample during the wetting process, and a convex shape was formed on the upper and lower sides of the ring cutter. This convex part was considered to be caused by water swelling. With different water contents, after each freezethaw cycle, the loess sample also produced a certain amount of expansion, and a convex shape was formed on the upper and lower sides of the ring cutter. This convex part was considered to be caused by frost heaving. Specific research ideas and test procedures are shown in Figure 4.
2.1.1. Test Equipment
The main equipment used were as follows: ring cutter, ripper, plastic wrap, sealed bag, moisturizing jar, burette, electronic balance, freezer, and confined compression instrument.
2.1.2. Test Steps
(a)The prepared undisturbed loess samples were labeled according to the nomenclature in Table 2. The samples were moistened by the fractional titration water injection method, and 6 loess samples were prepared with water contents of 9%, 15%, 20%, 25%, 30%, and saturated.(b)The prepared samples were wrapped with plastic wrap and placed in sealed bags and moisturizing jars and then allowed to stand for 72 hours to allow uniform water diffusion and sufficient water swelling. The convex parts from the upper and lower sides of the ring cutter were trimmed with a ripper, and the trimmed part of the loess was weighed and recorded as .(c)Loess samples 10, 20, 30, 40, 50, and 60 were subjected to a confined compression test (samples without freeze and thaw), and the remaining samples were placed in a freezer for 24 hours.(d)After freezing for 24 hours, the samples with different water contents were observed for changes in volume; the convex parts from the upper and lower sides of the ring cutter were trimmed with a ripper, and the trimmed part of the loess was weighed and recorded as Δm_{i}, where i has a value of 1 for each freezethaw cycle.(e)Loess samples 11, 21, 31, 41, 51, and 61 were subjected to a confined compression test, and the remaining samples were placed in a freezer for 24 hours.(f)Test steps “d” and “e” were repeated, and finally, the cumulative frost heaving amount Δm_{n} after n freezethaw cycles was obtained, and the value was obtained by superposing Δm_{i} of each freezethaw cycle.
2.1.3. Test Considerations
(a)The test operation required a closed room with humidity of 40 to 60% and a temperature of 3°C to 6°C, but the temperature of the moisturizing jar was controlled at approximately 20°C(b)In addition to trimming the expanded part, the plastic wrap and sealed bag were removed, but the remaining test samples needed to be resealed at any other time period(c)If the amount of moisture lost from the samples by evaporation was too great, samples were replenished according to the constant mass of the test process
2.2. Volume and Pore Changing Test
The soil samples show axial free expansion deformation, which is confined laterally. To improve the measurement accuracy, the expansion ratio is calculated by measuring the mass of expansion, and the expansion is considered to be composed of water swelling and frost heave. Water swelling and frost heave change the microstructure, which in turn changes the density of loess. The expressions for density at different water contents () and density after each freezethaw cycle (ρ_{n}) are as follows:where ρ is the wet density of the original loess, g/cm^{3}, and and are density coefficients of change (test determination).
After each freezethaw cycle, the frost heave portion of the samples is trimmed. Then, the water swelling and each frost heave are cumulatively calculated. The free expansion rate (η) is calculated as follows:where is the expansion volume after water swelling, mm^{3}; is the cumulative volume after frost heave, mm^{3}; is the expansion quality after water swelling, ; is the cumulative mass after frost heave, ; and is the mass increase of frost heave per freezethaw cycle, .
After the volume variable in the calculation formula of the expansion ratio is changed to the mass variable, the water swelling rate () and the frost heave rate (η_{n}) can be expressed as follows:where is the water swelling rate, η_{n} is the frost heave rate, η is the free expansion rate, and m is the ring cutter mass, .
The calculation formula for calculating the void ratio of the soil sample according to the volume expansion ratio is as follows:where is the void ratio after water swelling and is the void ratio after frost heave.
According to the experimental measurements of the pore ratio and volume change of samples and the use of equations (4) and (5) to calculate the volume and pore change of the loess after water swelling and frost heave, the results are as shown in Figures 5–7.
Figures 5–7 show the following. (1) As the water content increases, the initial volume changes slowly, and after the water content exceeds 16%, the volume change rate increases significantly, indicating that the loess is an expansive loess. (2) The volume change rate is faster during the first 3 freezethaw cycles and then slows down. After 9 cycles, the change is gradual, indicating that the loess structure is broken at the initial stage of the freezethaw cycles, and a new structure is formed, and as the number of freezethaw cycles increases, the new structure will tend to be stable. (3) Due to the correlation between the void ratio and the expansion ratio, the variation in the void ratio with the water content and the freezethaw cycles is consistent with the change in the expansion ratio.
In short, the volume expansion of loess samples with wetting and freezethaw cycles is worthy of our attention; the reasons for these changes can be summarized as follows. (1) When the water content of the loess sample is low, the volume expansion produced is small. On the one hand, the reason is that the matrix suction of loess makes the water difficult to diffuse or spread unevenly, the local structural strength in the sample is weakened, and the overall structural strength is disturbed a little. On the other hand, due to the insufficient hydration of the loess sample, it produces a small hydration expansion force, which is difficult to overcome the structural strength of the loess. When the water content of the loess sample is high, the volume expansion ratio of the sample is significantly accelerated. The reason is that the water diffusion and hydration of the loess sample can be fully achieved. (2) In the previous freezethaw cycles, the rate of frost heave increases rapidly, and as the water content increases, the growth rate is more significant. The reason is that the water inside the loess sample is frost heave, which changes the internal structure of the loess and also increases voids and pores that contain moisture. After freezethaw cycles reach a certain value, the rate of frost heave becomes slower. The reason is that the loess sample is not subjected to any load and there is no thawing collapse after frost heaving; that is, the loess structure change is basically not recovered. The expansion volume is mainly caused by local frost heaving after moisture migration, so the expansion rate is slowed down.
2.3. Loess Microstructural Test
A JSM6700F scanning electron microscope (SEM) was a field emission scanning electron microscope made in Japan, which was mainly composed of an electrooptical system, a scanning system, a signal detection amplification system, an image display and recording system, a power supply, and a vacuum system (see Figure 8). The SEM was used to scan the undisturbed sample, the sample with a water content of 20%, and samples exposed to 1, 2, 6, and 9 freezethaw cycles with a water content of 20%. The changes in the microscopic pore structure were studied.
2.3.1. SEM Image Processing and Analysis
The SEM images were processed by MATLAB for grayscale adjustment, enhancement, iterative denoising, etc., and the binarization images were obtained as shown in Figure 9.
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A large number of fine pores and fewer large pores are found in the undisturbed loess, and its microstructure exhibits a dense overall structure (Figure 9(a)). During the wetting process, the rapid flow of water along the large pores will increase the throat diameter of the original pores. Then, the water infiltrates into the surrounding fine pores along the large pore wall, and large granules and clots appear due to the wetting expansion and suction (Figure 9(b)). As the number of freezethaw cycles increases, the fine pores decrease, the large pores increase significantly, the maximum pore throat diameter can reach 35 μm, and the shape and distribution of the pore structure also change (see Figures 9(c)–9(f)).
The microstructure quantitative analysis of all the images in Figure 9 employs the image analysis software ImagePro Plus 6.0 (IPP 6.0), which is widely used in medical, biological, industrial, and other fields. IPP 6.0 is image processing software that can capture the area, diameter, angle, perimeter, and roundness of the image. A calibration space scale is first applied on an image of a certain magnification to establish a unit of measurement associated with the image. The spatial measurement scale associated with the magnified image is constructed later. In general, IPP 6.0 will measure all the space in pixels; therefore, the spatial scale is changed so that IPP performs the measurement in microns. The specific processing steps are as follows. (a) The pixel ratio is set because the actual image is calculated in pixels, and the spatial scale is defined by the length; therefore, the length needs to be defined according to the scale of the SEM image (Figures 10 and 11). (b) The unknown parameters are selected and calculated. The parameters of the microstructure of the loess are mainly the void ratio and the particle size (Figure 12). (c) According to statistical analysis of the target parameters of the image, the statistical results of particle size and void change with wetting and freezethaw cycles are obtained, and the microstructural changes of the loess are quantitatively analyzed.
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Each selected microarea was analyzed using PPI software as shown in Figure 13, and the maximum and minimum pore and particle size were obtained. The average value of the maximum value and the minimum value is taken as the parameter value of the minute area. Finally, the values of each regional parameter are cumulatively calculated, and the results of quantitative analysis of pores and particles are obtained as follows.
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2.3.2. Quantitative Analysis of Particles
The statistical results were used to analyze the variation in the average particle size of the loess particles with the undisturbed loess and 20% water content loess following 0, 1, 3, 6, and 9 freezethaw cycles.
Figure 14 shows that particle sizes (d < 2 μm) gradually decreased with the freezethaw cycles, and particles (2 < d < 5 μm, 5 < d < 20 μm, and d > 20 μm) increased in fluctuation, indicating that freezethaw cycles reduce cohesiveness and increase both particle size and agglomeration.
2.3.3. Quantitative Analysis of Pore Size
According to the average pore size, the pores in the soil were divided into extralarge pores (r > 50 μm), large pores (20 < r < 50 μm), medium pores (5 < r < 20 μm), small pores (2 < r < 5 μm), and micropores (r < 2 μm). The statistical results were used to analyze the variation in the pore diameter of loess with the undisturbed loess and 20% water content loess following 0, 1, 3, 6, and 9 freezethaw cycles.
Figure 15 shows that the loess sample pores were mainly mesopores, small pores, and micropores, with relatively few extralarge pores and large pores. Microporosity decreased with the distribution of freezethaw cycles. The sizes of small pores and small and medium pores increased with the number of freezethaw cycles. The sizes of extralarge pores and large pores increased slightly.
The areas occupied by the pores or soil particles in the crosssectional area were calculated using IPP 6.0. The plane porosity ratio can be approximated as follows at a 20% water content or after different freezethaw cycles:where is the porosity ratio at different water contents; e_{n} is the porosity ratio after different freezethaw cycles; S_{aw} and S_{an} are the areas occupied by the pores at different water contents or after different numbers of freezethaw cycles, respectively; and S_{sw} and S_{sn} are the areas occupied by soil particles at different water contents or after different numbers of freezethaw cycles, respectively.
The expansion ratios and η_{n} at different water contents or after different numbers of freezethaw cycles can be obtained by equations (4) and (5):
By comparing the geotechnical test method and the digital analysis method of loess microstructure images, the conclusions are as follows.
Figure 16 shows that the loess expansion rate determined according to the digital image analysis method is slightly higher than the loess expansion rate determined by the geotechnical test method, but as the number of freezethaw cycles increases, the results of the two methods exhibit the opposite trend, although the difference is small. Therefore, the use of digital image analysis has unique advantages in studying the pore structure and variation of undisturbed loess.
2.4. Confined Compression Test
The strength test was applied to different samples after the simulated wetting and freezethaw cycles. The void ratio of the samples was not the initial void ratio; instead, the void ratio after the freezethaw cycles was used as the void ratio for each test. In this paper, the confined compression test was used to measure the deterioration of the structural strength of the loess, as shown in Figures 16 and 17.
(1) Figure 17 shows that as the water content increases, the compression index C_{c} increases and the compression modulus E_{s} decreases, which indicates that the cementation strength and the friction strength between the soil microstructures of the loess decrease with the increase in the water content, reducing the strength of the loess structure. (2) Figures 18(a) and 18(b) show that when ≤ , the compression index C_{c} and the compression modulus E_{s} change little as the number of freezethaw cycles increases, which indicates that the volume increase due to the water phase change has little effect on the structural strength of the loess. (3) Figures 18(c)–18(f) show that when < ≤ , the compression index C_{c} increases greatly with the number of freezethaw cycles, and the compression modulus E_{s} changes less, which indicates that the volume increase due to the water phase change has a greater influence on the spatial arrangement of the microstructure, and thus, the structural strength is reduced at a faster rate. When > , the compression index C_{c} and the compression modulus E_{s} both change slowly with the number of freezethaw cycles, which indicates that the loess has rheological properties and that the microstructure of the loess is selfhealing due to the rheological action after the solid water melts; thus, the rate of structural strength reduction is slowed down.
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3. Deterioration of the Loess Structural Strength
Loess structural properties include the composition, shape and arrangement of the skeleton particles, internal pore characteristics, types of cementing material, and degree of cementation. Under external environmental disturbances such as wetting and freezethaw cycles, the volume of loess changes, which causes the particle distribution and pore structure to change. Soil and water action can reduce the bonding force of the cementing material between the loess microparticles or the microparticle aggregates, and the lubricating property of water significantly reduces the friction between the microparticles or the microparticle aggregates, reducing the structural strength of the loess. Because water in the micropores produces uneven migration and distribution and phase change occurs during the freeze and thaw process, the volume of water is changed; then, the soil samples produce frost heave, which causes the loess internal structure to be further damaged.
3.1. Loess Structural Strength and Damage Variables
Loess structural strength is a kind of cementing strength formed during the formation of the soil structure. Structural strength is generated with the formation of soil structure and disappears with the destruction of soil structure. Its value is determined by the cementing force and friction between the microparticles or the microparticle aggregates.
The cementing force is a shortrange force, including the chemical bonding force, molecular gravity, and suction between the microparticles or the microparticle aggregates, which form the cementing structural strength of loess. Friction is a longrange force, including the friction and interlock force between soil particles or the particle aggregates that form the friction structural strength of loess. Friction is the force that prevents the deformation of loess when the connecting force between the microparticles or the microparticle aggregates is overcome by mutual dislocation, displacement, and rearrangement movements.
Dang and Li [44] analyzed the unsaturated loess composition to determine the strength and sources of structural strength, developed a method for determining the structural strength, and studied its variation. Hu et al. [45] used the confinement compression test to determine undisturbed loess structural strength and studied its variation. Tian et al. [46, 47] quantitatively described the loess structural parameters, that is, the variation in the cementing structural strength and friction structural strength of loess. Yin et al. [48] analyzed transport of liquid water, vapor, stress, and heat in unsaturated freezing soil. Lu et al. [40, 49] and Tang et al. [50] analyzed cracking characteristics and established constitutive relations and mechanical degradation of expansive soil under freezethaw cycles. Liu et al. [51] researched mechanical behavior and the constitutive model of tailing soils subjected to freezethaw cycles. Liu et al. [52] established an elastoplastic model for saturated freezing soils based on thermoporomechanics. Zhang et al. [53] used a new novel constitutive model to simulate the rock failure under freezethaw cycles.
The present authors believe that the elgp compression curve can be analyzed from the flat linear phase of the elastic deformation to the local microstructure damage, and the curvature of the line segment suddenly increases in plastic deformation. The pressure corresponding to the maximum point of curvature is defined as the disturbance to the original structural strength of the original loess caused by the sampling process. To investigate the influences of sampling and the process of preparing samples on the structural strength of undisturbed loess, the pressure difference between the point corresponding to the preconsolidation pressure and the pore ratio at the point corresponding to the saturated remolded loess indicates that the structural strength of the undisturbed loess is greater reasonable. The structural strength of the undisturbed loess is shown in Figure 19.
The structural strength of the undisturbed loess can be expressed as follows (see Figure 19):
The preconsolidation pressure is most often determined by the Casagrande [24] drawing method, which requires strict testing operations and data reading to ensure accuracy.
For the structural strength of the loess affected by the external environment (wetting, temperature change, loading, etc.), the elgp compression curve has a significant turning point, which is also considered to be the starting point of the failure of the loess microstructure and the corresponding void ratio. The external environment disturbs the maximum pressure difference between the original loess and the saturated remolded loess. This value is defined as the residual structural strength of loess, as shown in Figure 20.
Based on the principle of damage mechanics, the damage variable is defined as follows:
In general, D is between 0 and 1. When D = 0, it indicates that there is no disturbance damage, that is, the state of the original stratum. When D = 1, it indicates that the state of the original stratum is fully disturbed, that is, the state of saturation remolding.
replaces to indicate the residual structural strength of undisturbed loess after sampling and preparation of samples; then, equation (10) changes to the following:where D_{d} is the sampling and sample preparation disturbance damage variable. Its numerical size reflects the different degrees of disturbance caused by the release of pressure on the formation and sampling and preparation of samples after excavation. The higher the value, the larger the disturbance.
D_{d} can be obtained according to the elgp compression curve. First, the pressure values at the turning point of the fitting curve (the maximum curvature point) and the preconsolidation pressure are determined.
3.2. Pressure at the Turning Point
According to the test curve (Figure 19), a univariate function model with a higher order is satisfied. Since the abscissa is a function, let x = lg(p/Pa); then, set the function as follows:where n is a natural number. Its size depends on the shape of the curve. The general value is 35, and the value of the saturated remolded loess is 1. a_{i} is the coefficient of fitting the mathematical expression, i = 1, 2, 3, …, n.
For convenience of discussion, take n = 3 and substitute e(x) into the curvature calculation formula to obtain the curvature value of the curve:
To perform the maximum curvature method, the absolute elimination in equation (13) is changed to the function extremum problem, all the maxima and the limit values are compared to the absolute value, and the maximum value is the maximum curvature of the test curve. The point corresponding to the maximum curvature is the turning point and the point of maximum curvature. The point corresponding to the void ratio and pressure is the starting point of the failure of the loess microstructure. Therefore, let
Equation (14) can be simplified to obtain a onedimensional quadratic equation:where , , , , and .
A solution to the equation according to the Ferrari method can be obtained as follows:where
Considering the boundary conditions, the invalid solution of equation (16) is omitted, and the point R_{min} (lgp_{c max}, e_{c max}) at which the maximum value of K is obtained is the maximum curvature point, that is, the minimum radius of curvature at this point. Then,
When n equals 1 or 3, the result is an irrational number, which is an invalid solution. When n equals 2 or 4, the results are opposite to each other and take a positive value.
3.3. Preconsolidation Pressure
The preconsolidation pressure is determined by the Casagrande method. First, the line equations of I, II, and IV (Figure 19) need to be determined separately.
The line I equation is obtained by linear regression analysis, and the test data (e_{i}, lgp_{i}), i = 1, 2, 3, …, m are taken at the end of the compression curve.
Then, the linear regression equation is set towhere and are estimated values of β_{0} and k_{1}, respectively, and and must be estimated using the least squares method.where and .
The line II equation is expressed by a point oblique type:where
The line IV equation is also represented by a point oblique type:where
Simultaneously solving equations (19) and (23) obtains the intersections E(lgp_{c}, e_{c}) of line I and line IV:
3.4. Loess Damage Value of Sampling and Sample Preparation Disturbance
The saturated remolded loess compression curve is approximately straight:where A_{s} and B_{s} are the linear fitting coefficients of the saturated remolded loess.
Then, and (equation (10)) change as follows:
Substituting equations (27) and (28) into equation (10) obtains the damage variable equation:where measured without the action of wetting and freezethaw cycles is . Additionally, substituting equations (27) and (28) into equation (10) obtains the sampling and sample preparation disturbance damage variable equation:
According to the test data, the compression curve of sampling and sample preparation disturbance of the loess can be fitted as follows:
The regression line of the sampling and sample preparation disturbance of the loess at the end compression curve is as follows:
The compression curve of saturated remolded loess can be fitted as follows:
Based on the above deduction process, the calculation parameters need to be determined as follows (Table 3).

Substituting the calculation parameters into equation (30) obtains D_{d} = 0.177.
3.5. Structural Strength Damage of Loess under Wetting and FreezeThaw Cycles
Using the same method, the calculated parameters under different water contents and freezethaw cycles can be obtained and substituted into equations (28) and (29) to obtain the residual structural strength and damage variables of the loess under wetting and freezethaw cycles. The changing rules are shown in Figures 21–24.
With an increase in water content, the loess residual structural strength (Figure 21) decreases according to the following pattern. When ≤ , the decrease rate of is slower, which indicates that the cementing structural strength and the friction structural strength are slightly reduced in the loess structure. When ≤ ≤ , the decrease rate of is faster, which indicates that cementing structural strength and friction structural strength are greatly reduced in the loess structure. When > , the decrease rate of slows down, which indicates that the friction structural strength is basically reduced to zero, and the cementing structural strength reaches a limit in the loess structure.
The residual structural strength of the loess (Figure 22) will continue to decrease with an increase in the number of freezethaw cycles on the basis of moisturerelated degradation, and the change trend is as follows. When the water content is very low, the residual structural strength does not change, or there is a slight decrease with the increase in the number of freezethaw cycles, which indicates that the cementing structural strength and friction structural strength do not change or change slightly. When ≤ ≤ , the residual structure strength of the loess decreases the fastest before 6 freeze and thaw cycles, which indicates that the frozenphase transformation causes the loess structure to produce many large pores, the cementing structural strength is reduced greatly, and the friction structural strength is reduced slightly. After 6 freeze and thaw cycles, the decrease rate of the loess residual structural strength becomes slower, which indicates that secondary structure is produced slowly. When > , the residual structure strength of loess also decreases faster before 6 freeze and thaw cycles, which indicates that the larger pore skeleton structure has been produced and increases in quantity. After 6 freeze and thaw cycles, the rate of decrease slows, which indicates that the secondary structure is basically formed.
Figures 23 and 24 show the following. (1) With an increase in water content, the damage variable of the loess increases, which shows that when < , the damage variable changes slowly; when < < , the damage variable changes faster; and when > , the damage variable becomes slower again. (2) The degradation behavior of the loess structural strength during the freezethaw cycles with different water contents continues to increase based on wetting damage. (3) When the water content is low, the structural strength of the undisturbed loess is not affected or is only slightly affected by the freezethaw cycle. (4) Under different water content conditions, as the freezethaw cycles increase, the degree of deterioration of the loess structural strength increases first and then slows down. The concrete performance is as follows. When < < , the intensity degradation curve of loess structure is steeper between 1 and 6 cycles. After 6 cycles, the curve becomes slower and the secondary substructure is slowly formed; when > , the structural deterioration curve of loess structure is somewhat steep between 1 and 6 cycles. After 6 cycles, the curve is basically gentle and the structure of the lower secondary structure is gradually formed.
4. Loess Structural Strength Deterioration Model
After the sampling and sample preparation processes are disturbed, the loess structural strength can be referred to as the loess residual structural strength after sampling and sample preparation disturbances and expressed as q_{0r}. After wetting, the loess structural strength can be called the wettingdamaged loess residual structural strength, expressed by q_{wr}. After freezethaw cycles, the loess structural strength can be referred to as frost heavedamaged loess residual structural strength and expressed as q_{nr}. The damage variables corresponding to the loess residual structural strength are as follows: the sampling and sample preparation disturbance damage variable, expressed as D_{d}; the wetting damage variable, expressed by ; and the frost heave damage variable, expressed by D_{d}. The damage variables are all microscopic statistical damage variables, so they do not affect each other. Therefore, the basic equations of damage to loess structural strength can be established as follows:
4.1. Loess Microscopic Statistical Damage Variable
The loess microstructure is considered to be a comprehensive representation of loess microparticles or microparticle aggregates and their geometric arrangement and cementation. The loess microstructure has a certain structural strength, which is the ability of loess to maintain the original basic unit structure without being destroyed. Under external environmental disturbances and loads, the loess microstructure is weakened, and the soil skeleton particles are deformed and destroyed. The mechanical behavior of these microstructures shows that the macroscopic physical factors follow the Weibull distribution. In accordance with the Weibull distribution, the probability of the values of these random variables can be calculated and inferred, and the related problems of engineering technology can be solved.
Krajcinovic and Silva [54] proposed a statistical damage model based on the theory of continuous damage mechanics and strength statistics. Subsequently, the method was adopted by other scholars. In recent years, Wang et al. [55] and Li et al. [56] used the statistical method to describe rock evolution (initiation, growth, and coalescence) from damage to fracture with a continuous process and proposed a damagesoftening statistical constitutive model for rock according to the Weibull distribution of mesoscopic element strength. Xie et al. [57] also established a loessial soil damage constitutive model based on the strength of loessial soil microunits with a Weibull distribution. Lai et al. [58] investigated the strength distribution patterns for warmed frozen clay and warmed icerich frozen clay based on the characteristics of the random distribution of defects in both clays. Jian and Gu [59] used the theory of continuous damage mechanics together with statistical mesoscopic strength theory based on a maximum entropy distribution to present a statistical damage constitutive model for rocks exhibiting strain softening behavior. Wen et al. [60] proposed the acoustic emission (AE) evolution model of rock failure according to the view that rock damage and AE were consistent. Zhang et al. [61] established a constitutive model for freezethaw damage to rock under threedimensional stresses considering the random characteristics of the defects of rock materials. Zhu et al. [62] and Zhang et al. [63] also conducted indepth research on the frozen soil damage model.
The loess microstructure is assumed to be composed of microunits such as different soil particles or particle clusters. The following statistical damage model was proposed by Krajcinovic and Silva [54]:where E is the elastic modulus; ε is the strain; N_{i} is the number of microcells that fail under the action of the i^{th}; and N is the total number of microunits.
In this paper, the volume expansion ratio is taken as the test variable, and the probability density function P(η) of the Weibull distribution can be expressed as follows:where β is the shape parameter; F_{0} is the scale parameter; and η is the volume expansion rate.
By analyzing equations (34)–(36) and equation (37) separately, the random damage variable can be defined as the ratio of N_{i} to N. The singlefactor damage variables in this paper are D_{d}, , and D_{n}. We further assume that when each volume expansion rate reaches a certain level η_{d}, η_{w}, and η_{n}, the number of damaged units is , , and , respectively.
At a certain depth, the soil is in a state of stress balance. During the sampling process, the complex formation pressure of the loess is released, which causes a certain volume expansion. It is impossible to determine the volume change of the loess with a direct test, so the statistical damage model cannot be used to determine D_{d}. However, the D_{d} size has been obtained from the above experimental data fitting curve, i.e., D_{d} = 0.177.
According to Figure 19, the relationship between the volume of the preconsolidation pressure and the volume after the release of the sample pressure is as follows:
Figure 19 and Tables 1 and 3 show that = 0.92, p_{c} = 1035.18 kPa, and = 0.834. Substituting these data into equation (42) yields ΔV_{d} = 5.375 cm^{3}. Therefore, the sample of a ring knife can be reduced to the volume under the prior pressure as follows:
Therefore, the sampling process disturbance expansion rate is as follows:
Similarly, the test expansion rate of the loess under the wetting and freezethaw cycles can be reduced to the prior pressure as follows:
Substituting , , and into equations (39)–(41) provides the following equations:
Next, substituting equation (38) into equations (46)–(48), D_{d}, , and D_{n} can be obtained as follows:
Based on the irreversible and superposition principles of damage mechanics, the damage variable of the strength of the undisturbed loess structure and the combined effects of wetting and freezethaw cycles can be defined as follows:
4.2. Microscopic Statistical Damage Model of Loess
In this paper, the volume expansion ratio is taken as the test variable. Figure 25 shows that the volume expansion ratio of the loess increases with the water content. Initially, the growth is slow, and then the growth is accelerated. To improve the accuracy, the two straight lines are shown in Table 4.

The function of the relationship between the volume expansion ratio and the water content of loess is and divided into two straight lines, that is,where a_{1}, a_{2}, b_{1}, and b_{2} are fitting factors for the experiment and can be calculated from the test data, where .
Figure 26 shows that the undisturbed loess sample (water content is 2.5%) does not change the volume of the loess with an increase in the number of freezethaw cycles, which indicates that the freezethaw cycle does no damage to the structural strength of the loess. At other water contents, the number of freezethaw cycles increases the volume expansion ratio of loess in the first period, the growth is faster, and then the growth slows down. To improve the accuracy, the two straight lines are shown in Table 5.

The function of the relationship between the volume expansion ratio of loess and the freezethaw cycles in cases of different water contents is and is divided into two straight lines, that is,where are the test fitting coefficients, which can be calculated from the test data, and .
Substituting the functions and into equations (38) and (50) yields a damage probability density function for wetting and freezethaw cycles and a damage variable for wetting and freezethaw cycles.
The damage probability density function in the process of wetting is expressed as follows:where i = 1 or 2, as determined by equation (51).
The damage probability density function with increasing number of freezethaw cycles in different water contents is expressed as follows:where i = 1 or 2, which is determined by equation (52).
The damage variable for the wetting and freezethaw cycles is expressed as follows:
The first derivative of equation (54) can be used to obtain the wetting damage evolution function and the freezethaw cycle damage evolution function :
Substituting equation (55) into equations (34)–(36), the basic model of statistical damage of loess structure strength wetting and freezethaw cycles is obtained:
4.3. Damage Model Parameter Determination
After a simple transformation of equations (58) and (59), the logarithm of the two sides of the equation can be obtained as follows:
The model parameters , β_{n}, , and F_{n0} can be calculated by substituting the test fitting coefficients in Tables 4 and 5 into equations (60) and (61), respectively.
By substituting the model parameters of Table 6 into equation (55), the model of waterabsorbing wetting and hydration damage to the loess structure can be obtained.

By substituting the model parameters of Table 6 into equation (55), the damage model coupling strength, wetting and freezethaw cycles of the loess structure can be obtained.
When ≤ ,
When > ,
4.4. Model Verification
Under the coupling of different water contents and freezethaw cycles, through a side compression test of Q_{3} loess taken from a foundation pit in the northern suburbs of Xi’an, the damage degradation value was obtained by using the calculation method described in this paper, and the damage model was verified.
As shown in equation (62) and Figure 27, when < , the water absorption and wetting hydration model curve predicts the damage value to be consistent with the experimental value. When > , the model predicts the damage value to be lower than the experimental value. Therefore, the model needs to be revised for specific projects.
Equations (63a) and (63b) and Figure 28 are used to produce the wetting and freezethaw cycles coupling model curve. To improve the accuracy of the model, the segmentation conditions are refined according to the water content and the number of freezethaw cycles. When < , the predicted damage value of the model is basically consistent with the experimental value with the increase in the number of freezethaw cycles. The predicted damage value of the model is lower than the experimental value at individual water contents. When > , when the number of freezethaw cycles is n < 3, the model predicts the damage value. Compared with the experimental values, when the number of freezethaw cycles n > 3, the predicted damage value of the model is higher than the experimental value; thus, the engineering design is conservative and safe.
From Equations (62), (63a), and (63b), the basic model of damage to the loess structure through the watersoil interaction and the coupling of wetting and freezethaw cycles can be applied to collapsible loess, which decreases in volume during wetting. During the subsequent process, the volume increases, the wetting volume decreases again, and the refreeze volume increases. Such repeated changes need to be corrected and discussed in equations (51) and (52), which is beyond the scope of this article.
5. Conclusions
(1)Laboratory experiments were conducted to study the volume and pore structure of loess samples with the changes caused by wetting and freezethaw cycles, and the change trends for the expansion ratio and void ratio were obtained. Second, microscopic testing technology (SEM) and digital image analysis technology (IPP 6.0) were used to investigate the variations in the microscopic particle size and pore ratio of loess with the number of freezethaw cycles at = 20%. The results are very close to those of the geotechnical test. This method represents a new approach for the analysis of the microscopic pore structure of loess. Finally, the variations in the compressive index C_{c} and compressive modulus E_{s} of loess with wetting and freezethaw cycles were investigated according to the confined compression test.(2)According to the elgp compression curve, the principle of yield failure of loess structure was analyzed, and the loess structural strength, residual structural strength, and damage variable were newly defined. Additionally, combined with confined compression test data and mathematical statistics, the damage value generated by the disturbances during the sampling and preparation of loess was determined, and the deterioration of the structural strength and damage variable of loess was analyzed.(3)Based on microscopic statistical damage theory and the Weibull distribution, statistical damage variables considering the disturbances in sampling, sample preparation processes, wetting and freezethaw cycles were established. Then, the volume expansion ratio of loess was used as the variable for the Weibull distribution. According to the experimental data, the expansion rate function under wetting and freezethaw cycles was fitted, and the statistical damage model parameters and the damage models of wetting and freezethaw cycles were solved by using the coefficient of expansion function and the basic equation of damage. Finally, the damage model was verified using data of loess taken from a foundation pit in the northern suburbs of Xi’an. The modelpredicted damage value was consistent with the experimental value. The statistical damage model has good practicability and can for provide guidance for practical engineering design and construction.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (nos. 51674200, 51874320, 51604285, and 51868075), the Science and Technology Innovation Project of Key Laboratory of Shaanxi Province China (no. 2014SZS15–Z02), the Yan’an University Doctor Scientific Research Initialization Foundation (no. YDBK201852), and the Open Foundation of Key Laboratory of Failure Mechanism and Safety Control Techniques of EarthRock Dam of the Ministry of Water Resources (YK916006), which are gratefully acknowledged.
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Copyright © 2019 Haicheng She et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.