Mechanical behavior of a new similar material for weathered limestone in karst area: An experimental investigation

: Considering the important e ﬀ ect of mineral composition, this article deduces the similarity criterion based on the dimensional method. Similar materials sui - table for tu ﬀ s with di ﬀ erent degrees of weathering in karst areas are made. By virtue of the orthogonal experi - ment, the mechanical behaviors of the similar materials with various mix proportions are systematically investi - gated. It provides an important reference and basis for the next proposed model tests on the stability of tu ﬀ strata in karst areas. The results indicate that a stable mechanical behavior can be achieved for the similar material made of quartz sand, cement, gypsum, lime - stone powder, diatomite, red clay, and water. Moreover, it is also found that the aggregate and the binder have an observable in ﬂ uence on the mechanical behavior of the similar material. With an increase in the amount of cement and gypsum, the homogeneity of the sample increases, failure mode in which the specimen shows lateral com - pression damage. However, with increasing the amount of quartz sand, there is a tendency that a weak structural plane will be generated within the sample. Thus, this similar material can be satisfactorily used for simulating limestone of di ﬀ erent weathering degrees in a scaled model test.


Introduction
A great deal of engineering practices proves that the bearing capacity and stability of the foundation in karst area are vastly controlled by the karst development features [1][2][3][4][5][6]. Within the range of the influence of the superstructure supporting the foundation in karst area, collapse failure of the karst cave can be easily induced by the upper additional load. This will lead to a sharp deformation of the ground, which jeopardizes severely the safety of engineering structures such as buildings, bridges, and highways during their construction and operation phases [7][8][9][10]. Figure 1 shows the typical disasters that occurred for the pile foundation in karst area.
To resolve these difficult problems, investigations based on theoretical derivations [11], numerical simulations [12][13][14][15], and scaled model tests [16] have been conducted by many scholars. Compared to theoretical derivations and numerical simulations, scaled model tests can better simulate the construction technique, loading mode, and time effect in complex environments [17]. In addition, by performing scaled model tests, the entire development of the stress field surrounding the karst cave from the elastic stage to the failure stage can be clearly revealed. In a scaled model test, selection of similar material and determination of the mix proportion of the selected similar material are extremely important for generating reasonable results [18].
It is well acknowledged that the similarity between the similar material and the prototype material is the primary prerequisite for the success of the scaled model test. In fact, the mechanical behavior of the similar material controls whether the physical and mechanical behavior of the prototype material can be accurately considered in the scaled model test. In general, the mechanical behavior of the similar material is primarily affected by the type of the basis material and the mix proportion [19]. At present, in the area of geomechanics, significant advances have been made in the preparation of similar materials used for scaled model tests [20][21][22][23][24]. Based on the similarity theory, Xu et al. [25] developed similar materials suitable for tunnel lining crack evolution. The similar materials used quartz sand as coarse aggregate, barite powder as fine aggregate, and oil, petroleum jelly, and paraffin as binder. Li et al. [26] developed a new similar material for solid-fluid coupling, in which the aggregate is composed of sand and steatite, while the paraffin is regarded as the binder. Chen and Bai [20] developed a new similar material of stable mechanical behavior for simulating rockburst in a scaled model test. This similar material is composed of aggregate (i.e., quartz sand), binder (i.e., cement and gypsum), and additive (i.e., glycerinum and gelatin).
Although significant achievements have been made in terms of similar material, most of the previous studies are primarily based on specific experiences. Moreover, as for the preparation of scaled model tests on bearing capacity of piles in karst area, there is no suitable material for a good simulation of limestone. In addition, previous studies have indicated that mineral composition has a prominent effect on the mechanical behavior of limestone [27]. Generally, a rock containing abundant clay minerals is of low strength and poor engineering properties. In the meantime, the existence of clay minerals is also an important factor affecting sandstone reservoir productivity. Furthermore, the type and amount of clay minerals exert a remarkable influence on the mechanical behavior of geotechnical materials [28,29].
In this study, considering the important effect of mineral composition on the mechanical behavior of weathered limestone, unconfined compression tests and direct shear tests are performed on similar material simulating weathered limestone. For similar materials, Quartz sand is used as aggregate, cement plus gypsum as a cementing agent, diatomite plus red clay and limestone powder as a mineral additive material conditioner. The orthogonal test method is used to develop the mechanical characteristics test scheme of similar materials with varied fitting ratios, the extreme difference analysis method is used to examine the sensitivity and influence law of each factor on physical and mechanical parameters, and the influence of the content of different components on the mechanical properties and damage modes of similar materials in tuff strata is investigated. Meanwhile, the test data of developing tuff similar materials can provide important references and basis for the next proposed stability model tests and numerical simulations of tuff strata in karst areas.

Similarity theory
The design of a scaled model test is based on the similarity theory and dimensional analysis. A reasonable scaled model test requires that the model is similar to the prototype in terms of model size, similar material, loading mode, load distribution, etc. In general, three principles of similitude should be obeyed when designing a scaled model test. These principles of similitude are called the positive theorem of similarity, π theorem, and third similarity theorem.
The first thing during designing a scaled model test is to determine the ratio of similitude. As the dimensional method applies to various cases, this method is adopted herein to deduce the ratio of similitude for the scaled model test on the bearing behavior of pile in karst area. In this method, two fundamental dimensional systems are generally used: the first is the fundamental dimensional system consisting of MASS (M), LENGTH (L), and TIME (T); the second is the fundamental dimensional system consisting of FORCE (F), LENGTH (L), and TIME (T).
Considering the factors affecting the bearing behavior of piles in karst areas such as physical and mechanical properties of the ground, ground size, pile size, pile material, and load applied on the pile, the first fundamental dimensional system mentioned earlier is selected for the design of the scaled model test. The main physical quantities involved in the scaled model test on the bearing behavior of the pile in karst area have been summarized in Table 1.
According to the π theorem, the physical quantities listed in Table 1 can be expressed in the form of For convenience, one can choose P, g, and D in equation (1) as the independent parameter. In this case, the other nine parameters can be expressed by a function of P, g, and D. For example, when deducing the ratio of similitude of the elasticity modulus E, E can be expressed as The dimensional expression has the form of According to the homogeneity principle of the equation, we have By manipulating equation (6), we get a = 1, b = 0, and c = 0. Accordingly, the π relationship is expressed as Similarly, the similarity criteria for the other parameters can be obtained as EI π EA π π Convert equation (7) into the expression describing the relationship among the nine functions: Physical and mechanical properties of limestone Modulus of elasticity, According to the positive theorem of similarity, the π relationships for the prototype and the model should be identical. Therefore, we have (9) Thus, the ratios of similitude for these parameters are finally obtained as In the scaled model test on the pile in karst area, the acceleration of gravity is consistent with that in the prototype. In this case, C g = 1. When preparing the similar material simulating the limestone in karst area, the ratio of similitude for material density is set as C ρ = 1.5. Moreover, the ratio of similitude for height, C D , also needs to be determined. In this study, C D has been set to be C D = 1:λ. Based on this, the ratios of similitude for other parameters can be finally obtained by using equation (10). A summary of the ratios of similitude used for the scaled model test on the bearing behavior of piles in karst area is presented in Table 2.

Selection and testing on similar material
Based on the ratios of similitude obtained earlier, several basic materials are selected for preparing similar materials simulating the limestone in an actual pile foundation project. Unconfined compression and direct shear tests are then performed to investigate the mechanical behavior of the similar material.

Geological conditions of the pile foundation project
The G318 Chizhou Section Karst Area Pile Foundation Project is a part of the construction of the Qiupuhe Bridge. This Bridge is located in Chizhou, Anhui, China, as depicted in Figure 2. At this site, the ground is mainly composed of two layers: one is the strongly weathered limestone layer, and the other is the moderate-weathered limestone layer. By conducting in-situ geological exploration, the mechanical parameters of the prototype limestone material and pile are obtained as summarized in Table 3. When the ratios of similitude for the height and the density are set as, respectively, α D = 30, and α ρ = 1.5, then the ratios of similitude for φ, ε, σ, c are obtained as, respectively, α φ = 1, α ε = 1, α σ = 45, and α c = 45. Based on these ratios of similitude, the targeted parameters of the similar materials for limestone and pile are finally obtained as summarized in Table 4.

Material selection
When designing a scaled model test in geomechanics, the selection of similar material should conform to the following six principles. First, the strength and deformation behavior of the selected material should meet the principle of similitude. Second, the selected material is homogeneous, and its physical and mechanical properties should be stable. Third, Poisson's ratio of the similar material equals that of the prototype material. Fourth, creep does not occur. Fifth, the failure characteristics of similar material are analogous to that of prototype material. Finally, the material is easy to get. According to the similarity theory and the six principles mentioned earlier, the similar material for weathered limestone is characterized by large unit weight, and low strength and elasticity modulus. Therefore, quartz sand is selected as the aggregate. Cement and gypsum are selected as the cementing agent. The mineral additive ingredients consist of red clay, diatomite, and limestone powder. In order to optimize the grading and adjust the mechanical properties of the similar material, quartz sand of 1.0-1.5 mm in particle diameter is used for uniform gradation. Gypsum has a specification of 1,200 mesh, a flexural strength of 6-6.5 MPa, and a density of 0.9 g/cm 3 , and it as a gas-hardening mineral binder hardens through hydration reaction. The water-cement ratio, defined as the ratio of the water's weight to the gypsum's weight, ranges between 0.8 and 3.5. The unit weight, Poisson's ratio, and elasticity modulus ranges, respectively, between 3.5 and 10.0 kN/m 3 , between 0.17 and 0.20, and between 0.3 and 5.5 GPa. Pure gypsum is suitable for the simulation of a linear elastic model, which cannot simulate the residual strength and deformation. When the water-cement ratio is large, its elastic modulus changes less and cannot simulate the deformation of similar materials, which is easy to cause dissociation. Therefore, the mechanical properties of similar materials are improved by adding P.O42.5 silicate cement, whose similar materials can better simulate the deformation and damage characteristics of the prototype materials, and diatomite has a particle size of about 0.15-0.3 mm and a density of 0.44 g/cm 3 ; red clay powder has a specification of 1,300 mesh and a density of 0.6 g/cm 3 ; limestone powder has a particle size of 0.05-0.3 mm and a density of 2.56 g/cm 3 .

Sample preparation
A steel double-open mold, 50 mm in internal diameter and 100 mm in height, is used to prepare the sample. Several steps involved in the sample preparation are briefly described. First, the basis materials are weighed,    Figure 3 shows the typical steps during the sample preparation.

Experimental program
Orthogonal experiment is one of the commonly adopted optimal design schemes. In this study, the orthogonal experiment is used to obtain the optimum mix proportion of the similar material for weathered limestone. Four factors, i.e., the mix proportions (or weight ratios) of quartz sand (Q), cement plus gypsum (G + C), red clay plus diatomite (D + R), and limestone powder (L), are considered. The total mass of the specimen is 100%. The mass ratio of cement to gypsum is the same, the mass ratio of red clay to diatomite is also the same, and the amount of mixing water is 3/14 of the total mass. Referring to the mixed proportions of the similar materials developed by other researchers and taking the mechanical behavior of the prototype limestone material into consideration, an orthogonal table with four factors and two levels is designed as shown in Table 5. In addition, the range analysis method is used to analyze the results of the unconfined compression and direct shear tests.

Unconfined compression tests
According to the Engineering Rock Test Method Standard GBT50266-2013 [30], Φ50 × 100 mm size specimens are used for uniaxial compression test. A series of unconfined compression tests are performed on the similar material  Note: Q = quartz sand; G = gypsum; C = cement; D = diatomite; R = red clay; and L = limestone powder.
samples to obtain their unconfined compression strength, σ c , and elasticity modulus, E. The apparatus adopted is the RMT-150C Rock Mass Testing System, as depicted in Figure 4. This testing system, as a digital controlled electrohydraulic servo testing machine, is developed by the Institute of Rock and Soil Mechanics, Chinese Academy of Sciences. By using this testing system, loads can be automatically applied to the sample, and the deformation of the tested sample can be automatically recorded. During the tests, the loading rate is set to be 0.01 mm/s. A full stressstrain curve of the sample can be obtained by using this testing system.

Direct shear tests
Direct shear tests are also performed on the similar material samples to obtain their cohesion, c, and internal friction angle, φ. According to the Test Methods of Soils for Highway Engineering JT3430-2020 [31], ZJ-2 manual strain shear is used for the straight shear experiment, and the specimens are prepared by the ring knife with the inner diameter size of Φ61.8 × 20 mm. After the specimens are demolded and maintained at room temperature for 14 days, the specimens are then placed in the straight shear box with permeable stones on the top and bottom.  The apparatus adopted is shown in Figure 5. By using this apparatus, the similar material samples are horizontally sheared under four different magnitudes of vertical pressure, i.e., 100, 200, 300, and 400 kPa. The shear stress at failure, τ, will be recorded, which will be used to determine the shear strength parameters (i.e., c and φ) according to Coulomb's law (e.g., Figure 6): where τ is the shear stress at failure, and σ is the vertical pressure.

Parametric analysis
In this section, the results of unconfined compression and direct shear tests are analyzed using the range analysis method. The objective of this section is to capture the sensitivity of the unconfined compression strength, elasticity modulus, cohesion, and internal friction angle to the mix proportion of the similar material. Table 6 summarizes the mechanical parameters of the similar materials with various mix proportions. Figure 7 shows the stress-strain curves of the similar materials with different mix proportions. It can be indicated that the mix proportion has an observable effect on the mechanical behavior of the similar material. As shown in Figure 7(a), when the weight ratio of quartz sand equals 66% and the weight ratio of the cementing agent ranges between 20 and 24%, the stress-strain curve is elastoplastic. This is because the cementing agent has the ability of combining tightly coarse aggregates with fine particles. This ability contributes to a larger peak strength and elasticity modulus. The elasticity modulus ranges from 368.5 to 601.2 MPa, and the unconfined compressive strength ranges from 1.77 to 2.57 MPa. At a mix proportion of Q:G:C:R:D:L = 66:10:10:4:4:6, the unconfined compression behavior of the similar material is comparable to the targeted mechanical behavior of the similar material for moderately weathered limestone in karst area. When the weight ratio of quartz sand equals 80%, and the weight ratio of the cementing agent ranges between 6 and 10%, as depicted in Figure 7(b), the stress-strain curve of the similar material exhibits the feature of "plastic-elastic curve." For this phenomenon, the reason is that a reduction in the amount of the cementing agent weakens the binding between the aggregates and the cementing agent. Because of this, a larger strain and a lower peak strength and elasticity modulus of  Table 5).     the similar material will be induced. In detail, the elasticity modulus ranges between 52.8 and 145.7 MPa, and the unconfined compression strength ranges between 0.68 and 1.14 MPa. At a mix proportion of Q:G:C:R:D:L = 80:3:3:6:6:2, the unconfined compression behavior of the similar material is comparable to the targeted mechanical behavior of the similar material for strongly weathered limestone in karst area. Figures 8 and 9 show the two typical failure modes of the similar materials subject to unconfined compression: one is the splitting damage ( Figure 8); the other is the lateral compression damage ( Figure 9). As the particle size of cement and gypsum is relatively uniform, a more homogeneous sample can be achieved when a larger amount of cement and gypsum has been added. In this case, the failure mode of lateral compression damage occurs, as shown in Figure 9. However, when the amount of quartz sand increases, the failure mode of splitting damage is prone to occur, as shown in Figure 8. This is primarily attributed to the existence of a weak structural plane within the sample induced by the irregular shape of quartz sand. Figure 10 shows the range analysis results for the unconfined compression strength and elasticity modulus, where R-value represents the degree of influence of a factor on the unconfined compression behavior of similar material. It is found that at a certain amount of quartz  sand, the degree of influence of the three typical factors (i.e., (C + G):(D + R), (C + G):L, and L:(D + R)) on the unconfined compression strength and elasticity modulus can be sorted from highest to lowest as follows: (C + G):(D + R), (C + G):L, and L:(D + R). In other words, the sensitivity of the unconfined compression strength on these three factors is consistent with that of the elasticity modulus. The factor, (C + G):(D + R), is the dominant factor affecting the unconfined compression strength and elasticity modulus of the similar material.

Direct shear behavior
The internal friction angle and cohesion of the similar materials obtained from the direct shear tests are been summarized in Table 6. It can be indicated that the influence of cementing agent amount is trivial on the internal friction angle, while is obvious on the cohesion of the similar material. Moreover, the range analysis method is also adopted to analyze the sensitivity of the direct shear behavior to the three typical factors (i.e., (C + G):(D + R), (C + G):L, and L:(D + R)). Figure 11 shows the degree of influence of these three typical factors on the cohesion and internal friction angle of similar material. It can be indicated from Figure 11(a) that the sensitivity of cohesion to these three typical factors can be ranked from highest to lowest as: (C + G):(D + R), L:(D + R), and (C + G):L. This indicates that the factor, (C + G):(D + R), is dominant among others in controlling the cohesion of the similar material. However, the change in the amount of (C + G), (D + R), and L has little effect on the internal friction angle of the similar material, as depicted in Figure 11(b).

Conclusions
Based on the theory of similarity and an actual project in Chizhou, China, this article deduces the ratios of similitude for various parameters that are needed for performing scaled model tests on the bearing behavior of pile penetrating weathered limestone in karst area. Taking the influence of mineral composition into account, a new similar material for weathered limestone in karst area is developed by virtue of mixing quartz sand, gypsum, red clay, cement, diatomite, limestone powder, and water. Laboratory investigation including unconfined compression and direct shear testing is then performed on the similar materials with various mix proportions to capture their mechanical behavior and sensitivity to typical factors. The conclusions drawn from this study can be summarized as follows: • The mechanical behavior of the new similar material developed in this study matches well with that of the in-situ limestone in karst area, which ensures the accuracy of the scaled model testing results on the bearing behavior of piles in karst area. • When the weight ratio of quartz sand equals 66% and the weight ratio of the cementing agent ranges between 20 and 24%, the stress-strain behavior of the similar material generally exhibits an "elastic-plastic curve" feature. However, when the weight ratio of quartz sand is increased to 80% and the weight ratio of the cementing agent is reduced to 6-10%, the stress-strain behavior changes to be "plastic-elastic curve." The optimum mix proportions of similar materials for moderately weathered and strongly weathered limestones are, respectively, Q: In contrast, at a larger amount of quartz sand, the samples tend to exhibit the failure mode of splitting damage, which is primarily attributed to the weak structural planes within the sample induced by the irregular shape of quartz sand. • The sensitivity of the unconfined compression strength of the similar material to the mix proportion is comparable to that of the elasticity modulus. At a certain weight ratio of quartz sand, the change in the ratio of cement plus gypsum (i.e., C + G) to diatomite plus red clay (i.e., D + R) is the dominant factor affecting the unconfined compression strength, elasticity modulus, and cohesion of the similar material. However, the internal friction angle of the similar material is not affected significantly by the mix proportion.