1. Introduction
Urban developments have raised challenges such as land shortages and traffic congestion which have become increasingly prominent with the growth of cities [
1]. The development and utilization of underground space can help alleviate issues caused by urbanization [
2]. However, in some complex and crowded environments, the excavation of foundation pits affects the stress state of soil in adjacent areas [
3,
4], which will lead to strength failure, deformation, and instability. At present, methods for excavation stability research mainly consist of on-site monitoring, theoretical research, numerical simulation, and model tests [
5,
6,
7,
8,
9]. For some intricate large-scale projects, on-site monitoring imposes difficulty with controlling variables and cannot explore the state of stress changes within the rock mass structure [
10,
11,
12]. Though theoretical research and numerical simulation have been widely utilized to describe the excavation of foundation pits, they still have limitations under complicated working conditions and face difficulties in recreating the real site’s mechanical state. As a result, model tests become an indispensable method. Geomechanical model tests using similar materials can effectively simulate the complete process of structural stress in complex environments in addition to more intuitively revealing earth pressure characteristics and deformation [
13]. Zhang [
14] studied the mechanical properties of a subway station retaining structure adjacent to the pile foundation by using laboratory the model testing method. Wang [
15] developed a centrifugal model test of a power plant excavation project in order to reveal the interaction mechanism between piles and soils during foundation pit excavation. On the other hand, however, these model tests only reduced the size without giving consideration to the scale change of material mechanical parameters. In combination with the similarity theory, it is known that test materials must maintain an appropriate scale ratio with the prototype medium. Therefore, the selection of an appropriate similar material is the guarantee for successful geomechanical model tests [
16,
17].
It is of great significance to study the influence of material components and proportions on the mechanical properties of a similar material [
18]. Scholars have researched similar materials for model tests in the past. In the 1960s, two institutes, ISMES in Italy and LNEC in Portugal, successively proposed model materials for geomechanics engineering model tests [
19,
20]. The MIB material developed by Han et al. [
21] is a similar material made by stirring and mixing iron ore powder, barite powder, rosin alcohol solution, and membraniferous iron powder. However, this material produces toxic substances during the filming of iron ore powder. Ma et al. [
22] developed a geomechanical model material (NIOS), which is characterized as non-toxic, heavy, low-price, and of stable performance. However, the model dries slowly and the curing time is long. Zhu et al. [
23,
24] combined the advantages of MIB and NIOS to develop a similar material which is more suitable for the geomechanics model tests of geotechnical engineering, and applied it to boundary loading model tests. Zhang et al. [
25] systematically analyzed the principle of similar material combinations and proposed a new similar material which can simulate different rock masses. At present, some achievements have been made in the relevant studies, but there is still a lack of systematic development of similar materials for foundation pit excavation model tests.
In summary, in order to accurately simulate the mechanical evolution process during the excavation of a foundation pit structure, it is urgent to develop a new type of similar material that satisfies the similarity conditions of model tests. In comparisons with existing research [
26,
27,
28], this study makes full use of the brittleness of rosin and the plasticity of liquid paraffin, and a new type of similar material with large adjustable characteristic is proposed for which barite powder, iron ore powder, and quartz sand are selected as aggregates, gypsum powder is used as a regulator, and non-water-soluble liquid paraffin and rosin are selected as cementing agents. Based on the orthogonal experimental method [
29,
30], mechanical parameter tests of different proportioning materials are performed. Subsequently, the effect of each raw material on the parameters of the similar material is studied by range analysis. Finally, in accordance with the above research, a further physical model test based on the excavation of metro stations is conducted using the new similar material, revealing that the earth pressure changes in limited spaces with soil masses of different aspect ratios. In conclusion, this research can provide a reference for the mechanical evolution law of practical metro station excavation and construction.
2. Similarity Principle
Based on the principle of similitude, the test model and the prototype must have the same physical quantities and be able to be expressed by the same relationship. Therefore, all physical parameters with the same dimension should maintain the same constant ratio [
31,
32]. The similarity scale C is defined as the ratio of physical quantities with the same dimension between the prototype and model [
33]. The geomechanical model test requires that the similarity ratios of dimensionless quantities are 1 and that the similarity ratio of the same dimensional physical quantities is equal. According to displacement boundary conditions, stress boundary conditions, and physical equations, the parameters of materials for model tests should satisfy the following similar relationships [
34,
35].
where
,
,
,
,
,
,
and
represent the similarity ratio of the strain, friction coefficient, internal friction angle, Poisson’s ratio, stress, elastic modulus, cohesive force, compressive strength, and tensile strength, respectively.
The similar relationship among stress similarity ratio, bulk density similarity ratio, and geometric similarity ratio is given as follows:
The similar relationship among displacement similarity ratio, strain similarity ratio, and geometric similarity ratio is given as follows:
The similar relationship among stress similarity ratio, strain similarity ratio, and elastic modulus similarity ratio is given as follows:
4. Range Analysis
By conducting nine sets of orthogonal tests with various proportions, the physical and mechanical parameters were measured for material specimens with different proportions, including density
(g/cm
3), compressive strength
, tensile strength
, cohesion
, elastic modulus
, and friction angle
.
Table 3 shows the test results, and the density of the similar material is 2.23–2.65 g/cm
3. The similar material has a higher bulk density, which is consistent with the actual cohesionless soil bulk density on site. Therefore, it can simplify the conversion of physical parameters between the model and the prototype. On the other hand, the tensile–compression ratio of the similar material is 1/8.5–1/11.7, indicating that the similar material can better simulate the tensile properties of the prototype. Furthermore, the compressive strength ranges from 0.37 to 5.37 MPa, the elastic modulus ranges from 42.0 to 279.0 MPa, the cohesion ranges from 42.7 to 57.3 kPa, and the friction angle ranges from 28.37° to 37.04°. The above experiment results demonstrate that these mechanical parameters have a large variation range which can meet the requirements of the various mechanical parameters of the similar material in different testing conditions.
This paper adopts range analysis to study the results. Generally,
is defined as the value of the
i-th level of factor
j (
i = 1, 2,..., n,
j = A, B, C, D). Experiments under
are conducted in order to obtain the result index
of the
i-th level of factor
j, where
are random variables that follow the normal distribution. Conducting
P1 tests under
can produce
test results (
= 1, 2,…,
). In Formula (7),
is the statistical parameter of the
j-th factor at the
i-th level, while
is the index value of the
k-th test result of the
j-th factor at the
i-th level.
The range refers to the difference between the extreme values of the data, which reflects the degree of dispersion of the data. The larger the range, the more sensitive the factor [
36]. The factors with the largest range would be the most significant. Generally, the five main parameters of density, compressive strength, elastic modulus, cohesion, and friction angle can effectively control the physical and mechanical characteristics of materials. Therefore, these five parameters are utilized to carry out range analysis and determine the order of effect of the factors. The range analysis results of influencing factors are shown in
Table 4.
In order to more intuitively investigate the effect of each factor on the five parameters, a diagram of the relationships among the levels of each factor and parameter was produced, and this is shown in
Figure 5. The results show that the four factors have certain effects on the mechanical properties of similar materials.
In terms of the densities of specimens, the range of D is greater than those of A, B, and C. The range of factor D is the largest, slightly larger than factor B. Factor D mainly controls the density of the similar materials, and factor B also has a significant effect. As show in
Figure 5a, the density decreases as factors C and D increase, but it increases as factor B increases. In terms of the compressive strengths of specimens, the range of values of factor B is the largest, indicating that B is the most important factor influencing compressive strength. As show in
Figure 5b, the compressive strength first increases and then decreases under the influence of factors A and B, and it increases with the increase of factor C. The influence of each factor on the compressive strength ranged from large to small for B, C, A, and D, respectively. In terms of the elastic modulus of specimens, the range of factor B is much larger than other factors and is the main factor affecting the elastic modulus. As is shown in
Figure 5c, the elastic modulus decreases significantly as factors A and B increase. The range of factors C and D is small and has little effect on the elastic modulus. The influence of each factor on the elastic modulus ranged from large to small for B, A, C, and D, respectively. In terms of the cohesion of specimens, the range of factor D is the largest, and it is slightly larger than that of factor B. Factor D is an important factor influencing cohesion, and factor B also has a certain effect on cohesion. As is shown in
Figure 5d, the cohesion increases as factor D increases. The cohesion first decreases and then increases under the influence of factor B. The range of factor A is small, so the data have good concentration. The influence of each factor on the cohesion ranged from large to small for D, B, C, and A, respectively. In terms of the friction angle of specimens, the range of values of factor C is the largest, and it is slightly larger than that of factor B. Factor C mainly controls the friction angle of the similar materials, and factor B also has an obvious influence. As shown in
Figure 5e, the friction angle decreases as factors A, B, and C increase. The range of factor D is the smallest, and its influence is not significant. The influence of each factor on the friction angle ranged from large to small for C, B, A, and D, respectively.
By analyzing the five main parameters, the ranges of compressive strength and elastic modulus corresponding to factor B are 2.55 and 177.67, respectively, which are much larger than the ranges corresponding to the other three factors. On the other hand, the ranges of density, cohesion, and friction angle are also relatively large. Therefore, factor B is the dominating factor affecting the mechanical characteristics of similar materials.
The elastic modulus and compressive strength are the key parameters for analyzing the mechanical properties of materials, which determine deformation law and the ultimate strength of model materials, respectively. The orthogonal test and range analysis proved that cementing agent content has the greatest effect on these two mechanical parameters. Through adjusting the cementing agent content in order to investigate the changing rules of similar materials’ compressive strength and elastic modulus under the promise of keeping other factors unchanged.
Figure 6 shows the relationship between the cementing agent content and compressive strength and elastic modulus. With the increase of cementing agent content, the elastic modulus of the material decreases gradually, while the compressive strength first increases and then decreases. The change trend is highly consistent with the results of the range analysis. The use of paraffin as cement has a significant impact on similar materials. Notably, controlling the content of cement within 9% can effectively guarantee the molding of the similar materials.