# Sodium Metasilicate Cemented Analogue Material and Its Mechanical Properties

Authors: Songlin Yue; Yanyu Qiu; Pengxian Fan; Pin Zhang; Ning Zhang
Journal: Advances in Materials Science and Engineering
 (2016)

---

## Abstract

Analogue material with appropriate properties is of great importance to the reliability of geomechanical model test, which is one of the mostly used approaches in field of geotechnical research. In this paper, a new type of analogue material is developed, which is composed of coarse aggregate (quartz sand and/or barite sand), fine aggregate (barite powder), and cementitious material (anhydrous sodium silicate). The components of each raw material are the key influencing factors, which significantly affect the physical and mechanical parameters of analogue materials. In order to establish the relationship between parameters and factors, the material properties including density, Young’s modulus, uniaxial compressive strength, and tensile strength were investigated by a series of orthogonal experiments with hundreds of samples. By orthogonal regression analysis, the regression equations of each parameter were obtained based on experimental data, which can predict the properties of the developed analogue materials according to proportions. The experiments and applications indicate that sodium metasilicate cemented analogue material is a type of low-strength and low-modulus material with designable density, which is insensitive to humidity and temperature and satisfies mechanical scaling criteria for weak rock or soft geological materials. Moreover, the developed material can be easily cast into structures with complex geometry shapes and simulate the deformation and failure processes of prototype rocks.


---

## Body

## 1. Introduction

Geomechanical model test is one of the most widely used approaches in field of geotechnical and geology research [1, 2]. The accuracy and reliability of geomechanical model tests depend on the similarity of physical process, which involves geometric, boundary, initial conditions and scale criterion of physical material parameters [3–5]. Use of analogue material is the key of ensuring similarity between model and prototype [6, 7]. Thus, the preparation of analogue materials is a fundamental problem of geomechanical model tests.Analogue materials research started in Europe. In the 1960s, Fumagalli [8] pioneered techniques for geomechanical model tests in the Experimental Institute for Models and Structures (ISMES). They developed analogue materials which used gypsum as a binder and lead oxide powder and bentonite clay as filler materials. Then, Stimpson [9] gave a detailed review of various constituents that have been used in the past and group the materials as cemented and noncemented, with plaster and ordinary Portland cement being the most common cementing agents. In order to simulate coal rock, Burgert and Lippmann [10] developed a new analogue material which was made by adding a kind of hardener into epoxy resin. Indraratna [11] presented a synthetic material to simulate soft sedimentary rocks and it was constituted with gypsum cement, fine sand, and water. Glushinkhin et al. [12] developed a series of analogue materials and applied them in model tests of zonal disintegration in deep rock masses, mining disturbances, and so on. Dykeman and Valsangkar [13] presented results of centrifuge modelling of socketed caissons in a weak model rock made of cement, sand, bentonite, and water. Another kind of analogue material (NIOS) was developed by Li et al. [14], which contained magnetite powder, sand, cement or gypsum, water (as a mixing agent), and an additive. Dunham et al. [15] performed series of centrifuge tests by using a model rock made from a mixture of sand, bentonite, cement, and water. Chen and Bai [16] developed a type of analogue material to simulate a rockburst by using a mixture of quartz sand, gypsum, cement, water, glycerine, gelatin, and so forth. This kind of material has a low density and low Poisson’s ratio. Zhang et al. [17] developed analogue material by using iron ore powder, barite powder, and quartz sand as the aggregates, rosin and alcohol as the cementing agent, and gypsum powder as a conditioning agent. The material has the advantages of a high density, a wide range of parameter variation, and a stable performance: it can be used to simulate most rock masses between soft rock and hard rock. Imre et al. [18] provided a recipe of a synthetic cemented sand, together with a comprehensive characterisation for its mechanical properties. The material was named ETH analogue material for rock (ETHAR).Generally, analogue materials can be divided into two types, organic cementitious materials and inorganic cementitious materials, and can be further subdivided into more than a dozen subclasses, shown in Figure1 [8]. Most of organic cementitious materials are shaped by compressing, which cannot cast into large dimensions or structural mould. Inorganic cementitious materials can be easily shaped without external pressure, but most of them have properties with low density and high strength. Specifically, cement mortar and sodium thiosulfate cementing material are casting materials, but they always have high strength. Gypsum mortar and gypsum and cement mixed mortar are sensitive to humidity, which limit their application.Figure 1
Classification of analogue materials for rock [8].According to Indraratna’s suggestions [11], excellent analogue material should meet the following specifications: (a) satisfy mechanical scaling criteria; (b) be mouldable or can be cast into mould; (c) be insensitive to heat and humidity; (d) be economical and environmental friendly; (e) have a short curing time.Geomechanical model test is built on strict similarity law, which should satisfy simulated condition based on equilibrium equations, geometric equations, physical equations, boundary conditions, and displacement conditions [19, 20]. And the single valued similar conditions should meet the following similarity criteria [21, 22]:(1)
α
σ
=
α
ρ
g
α
L
,
α
δ
=
α
E
α
L
,
α
σ
=
α
E
α
ε
,
α
t
=
α
L
,
α
ε
=
1
,
α
φ
=
1
,
α
μ
=
1
,where the operator α is the notation of the ratio/scale of prototype materials’ parameters to analogue materials’. According to similarity criteria above, in the process of preparing analogue materials, bulk density should keep constant, but Young’s modulus, stress, and strength should be scaled as geometric scale [23]. Therefore, analogue materials always have the characteristics of high density, low strength, and Young’s modulus.Although scholars have tried many material preparation methods, most of them cannot satisfy mechanical scaling criteria for weak rock or soft geological materials, or some incur high cost, are complex, and offer low controllability of mechanical properties. Therefore, a new type of analogue material that meets excellent analogue material’s specifications should be developed.In this paper, we present a type of material with high density, low strength and Young’s modulus, easy mouldability, and good stability, which can simulate weak rock or soft geological materials. In order to investigate the properties of the proposed material, orthogonal experiments are conducted and relationships between the physicomechanical properties of the analogue material and influencing factors are established by regression analysis.


## 2. The Curing Mechanism of Anhydrous Sodium Silicate

Sodium silicate sand is a type of material used to make moulds and is widely used in the casting industry. However, the silicate content of sodium silicate cannot be accurately controlled and lead to a wide variation range of material properties. Therefore, sodium metasilicate solution is used here to allow control of the silicate content.Sodium metasilicate often contains water of crystallisation and thus forms sodium metasilicate pentahydrate, or sodium metasilicate nonahydrate, but all these compounds have a low water solubility. Therefore, anhydrous sodium silicate powder was used as the raw material, because of its high water solubility. Sodium metasilicate can react with CO2 and harden, but the hardening process is slow, and what is worse is that a hardened layer will form on the surface and prevent the full reaction of the inner parts. Therefore, sodium fluorosilicate was used as a curing agent to accelerate the process.The chemical formula for sodium fluorosilicate is Na2SiF6: which has a low water solubility and can be mixed with aggregate. Then it is stirred with the sodium metasilicate solution, after which it starts to harden. Specimens were made by casting the unset mixture into a mould and allowing it to harden.The chemical reaction between sodium metasilicate and sodium fluorosilicate is(2)
2
N
a
2
S
i
O
3
+
N
a
2
S
i
F
6
+
6
H
2
O
⟶
6
N
a
F
+
3Si
OH
4There are two reaction products: NaF and Si(OH)4. NaF will separate out from the solution. As shown in Figure 2, Si(OH)4 will gelate when the pH is less than 7 but will solate when the pH is alkaline.Figure 2
Growth of colloidal particle and gelatinization [22].


## 3. Sample Preparation and Testing Programme

### 3.1. The Proportion of Raw Materials

Analogue materials have different characteristics as a result of the differences in aggregates and cementitious materials used [24]. The proportion of raw materials can determine the parameters of analogue materials.The raw materials were anhydrous sodium silicate, sodium fluorosilicate, quartz sand, barite sand, barite powder, and water. Anhydrous sodium silicate and sodium fluorosilicate are both pure granular materials; the sizes of quartz and barite and sand grains were 0.6 to 1.18 mm; and the sizes of the barite powder grains were 0.06 to 0.1 mm.Therefore, material proportion can be determined by four coefficients:A (the proportion of fine powder to aggregates), B (the proportion of barite sand to coarse aggregate), C (the mass ratio of anhydrous sodium silicate to aggregate), and D (the mass ratio of sodium fluorosilicate to anhydrous sodium silicate). According to the orthogonal experimental method, tests can be distributed so as to clarify the relationship between experimental conditions and experimental results. As shown in Tables 1 and 2, the paper put forward an orthogonal experimental design, in which the level distributions of each factor and the proportions of raw materials of each group are listed.Table 1
Levels of each factor.
Level
Factor
A
B
C
D
1
40%
0
1%
1/4
2
50%
1/3
3%
2/4
3
60%
2/3
5%
3/4
4
70%
1
7%
4/4Table 2
The proportions of raw materials of each group.
Number
Factors
A
B
C
D
1
40%
0
1%
1/4
2
40%
1/3
3%
2/4
3
40%
2/3
5%
3/4
4
40%
1
7%
4/4
5
50%
0
3%
3/4
6
50%
1/3
1%
4/4
7
50%
2/3
7%
1/4
8
50%
1
5%
2/4
9
60%
0
5%
4/4
10
60%
1/3
7%
3/4
11
60%
2/3
1%
2/4
12
60%
1
3%
1/4
13
70%
0
7%
2/4
14
70%
1/3
5%
1/4
15
70%
2/3
3%
4/4
16
70%
1
1%
3/4A four-factor, four-level test scheme was designed according to the orthogonal tableL
16
(
4
5
), as shown in Table 2.According to Tables1 and 2, the proportion of raw materials of each group was determined.


### 3.2. Sample Preparation Process

In order to ensure the repeatability of tests [25], 5 uniaxial compressive specimens and 5 Brazil split specimens were prepared for each group simultaneously, and all the specimens were put into a standard curing room at 20°C and a relative humidity of 90% after stripping. The process of sample preparation was shown in the following specifications:(1)
Weighing (shown in Figure3): Raw materials were weighed according to the mix design proportions, and the anhydrous sodium silicate was added into water to form an aqueous solution.
(2)
Agitation (shown in Figure4): First, cast the aggregate and sodium fluorosilicate into the mixer, and keep it dry while mixing until completely combined (3 min). Then cast the sodium silicate solution slowly into the mixer, and keep stirring for 5 min until mixed evenly.
(3)
Casting (shown in Figure5): After mixing, cast the mixture slowly into the mould and then vibrate it to prevent the generation of a honeycombed surface. The casting process must be completed within 20 minutes.
(4)
Stripping (shown in Figure6): In virtue of specimens that were in natural curing condition before stripping, the best time for stripping should be determined by the indoor environment. Due to the experimental time being longer, the temperature range was large. The stripping time was 5 days at 0 to 20°C, and the stripping time is 3 days when the average temperature exceeds 20°C.
(5)
Curing: The curing serves several purposes: to accelerate the development of material strength and to prevent cracking, shrinkage, and damage, which are caused by drying, temperature changes, and other natural factors. Specimens should be conserved under standard curing for 28 days.Figure 3
Weighing.Figure 4
Agitation.Figure 5
Casting.Figure 6
Stripping.


### 3.3. Mechanical Testing Programme

Mechanical testing programme was carried out on pressing machine with the rate of displacement 0.02 mm/min.In order to investigate the uniaxial compressive strength (UCS) (Figure7) and Young’s modulus of the new type of analogue materials, 16 groups of uniaxial compressive strength tests were carried out. There were 5 specimens for each group in uniaxial compressive strength test, but the effective data for each group may be less.Figure 7
Uniaxial compressive strength test.The stress-strain curves from Group 3 are shown in Figure8 where the maxima are the uniaxial compressive strength (UCS).Figure 8
Uniaxial compressive stress-strain plots: Group 3.Besides, the curves of uniaxial compressive stress and strain from Group 7 are shown in Figure9. On the uniaxial compressive stress-strain curve, the elastic region was well-defined and Young’s modulus could be found from the slope of the plot therein.Figure 9
Uniaxial compressive stress-strain plots: Group 7.Young’s modulus can be calculated as the following:(3)
E
=
f
c
1
-
f
c
2
ε
1
-
ε
2
,where f
c
1, ε
1 are, respectively, the stress and strain on elastic starting point and f
c
2, ε
2 are, respectively, the stress and strain on elastic ending point.In order to investigate the tensile strength, 16 groups of flattened Brazilian disk tests (Figure10) were carried out, and there were also 5 specimens for each group test.Figure 10
Flattened Brazilian disk test or tensile splitting test.According to Wang and Wu’s suggestions [26], as shown in Figure 11, the tensile strength can be determined by the following formulae from a flattened Brazilian disk specimen:(4)
f
t
=
1.92
P
c
π
D
t
,where P
c is the critical load, D is the diameter of the specimen, and t is the thickness.Figure 11
Flattened Brazilian disk specimen subjected to uniform diametral compression [26].The load-time curves for the flattened Brazilian disk specimen used for the determination of rock tensile strength are shown in Figure12 where the maxima denote critical tensile strengths.Figure 12
Load-time curve: Brazilian disk specimen used for the determination of rock tensile strength.


## 3.1. The Proportion of Raw Materials

Analogue materials have different characteristics as a result of the differences in aggregates and cementitious materials used [24]. The proportion of raw materials can determine the parameters of analogue materials.The raw materials were anhydrous sodium silicate, sodium fluorosilicate, quartz sand, barite sand, barite powder, and water. Anhydrous sodium silicate and sodium fluorosilicate are both pure granular materials; the sizes of quartz and barite and sand grains were 0.6 to 1.18 mm; and the sizes of the barite powder grains were 0.06 to 0.1 mm.Therefore, material proportion can be determined by four coefficients:A (the proportion of fine powder to aggregates), B (the proportion of barite sand to coarse aggregate), C (the mass ratio of anhydrous sodium silicate to aggregate), and D (the mass ratio of sodium fluorosilicate to anhydrous sodium silicate). According to the orthogonal experimental method, tests can be distributed so as to clarify the relationship between experimental conditions and experimental results. As shown in Tables 1 and 2, the paper put forward an orthogonal experimental design, in which the level distributions of each factor and the proportions of raw materials of each group are listed.Table 1
Levels of each factor.
Level
Factor
A
B
C
D
1
40%
0
1%
1/4
2
50%
1/3
3%
2/4
3
60%
2/3
5%
3/4
4
70%
1
7%
4/4Table 2
The proportions of raw materials of each group.
Number
Factors
A
B
C
D
1
40%
0
1%
1/4
2
40%
1/3
3%
2/4
3
40%
2/3
5%
3/4
4
40%
1
7%
4/4
5
50%
0
3%
3/4
6
50%
1/3
1%
4/4
7
50%
2/3
7%
1/4
8
50%
1
5%
2/4
9
60%
0
5%
4/4
10
60%
1/3
7%
3/4
11
60%
2/3
1%
2/4
12
60%
1
3%
1/4
13
70%
0
7%
2/4
14
70%
1/3
5%
1/4
15
70%
2/3
3%
4/4
16
70%
1
1%
3/4A four-factor, four-level test scheme was designed according to the orthogonal tableL
16
(
4
5
), as shown in Table 2.According to Tables1 and 2, the proportion of raw materials of each group was determined.


## 3.2. Sample Preparation Process

In order to ensure the repeatability of tests [25], 5 uniaxial compressive specimens and 5 Brazil split specimens were prepared for each group simultaneously, and all the specimens were put into a standard curing room at 20°C and a relative humidity of 90% after stripping. The process of sample preparation was shown in the following specifications:(1)
Weighing (shown in Figure3): Raw materials were weighed according to the mix design proportions, and the anhydrous sodium silicate was added into water to form an aqueous solution.
(2)
Agitation (shown in Figure4): First, cast the aggregate and sodium fluorosilicate into the mixer, and keep it dry while mixing until completely combined (3 min). Then cast the sodium silicate solution slowly into the mixer, and keep stirring for 5 min until mixed evenly.
(3)
Casting (shown in Figure5): After mixing, cast the mixture slowly into the mould and then vibrate it to prevent the generation of a honeycombed surface. The casting process must be completed within 20 minutes.
(4)
Stripping (shown in Figure6): In virtue of specimens that were in natural curing condition before stripping, the best time for stripping should be determined by the indoor environment. Due to the experimental time being longer, the temperature range was large. The stripping time was 5 days at 0 to 20°C, and the stripping time is 3 days when the average temperature exceeds 20°C.
(5)
Curing: The curing serves several purposes: to accelerate the development of material strength and to prevent cracking, shrinkage, and damage, which are caused by drying, temperature changes, and other natural factors. Specimens should be conserved under standard curing for 28 days.Figure 3
Weighing.Figure 4
Agitation.Figure 5
Casting.Figure 6
Stripping.


## 3.3. Mechanical Testing Programme

Mechanical testing programme was carried out on pressing machine with the rate of displacement 0.02 mm/min.In order to investigate the uniaxial compressive strength (UCS) (Figure7) and Young’s modulus of the new type of analogue materials, 16 groups of uniaxial compressive strength tests were carried out. There were 5 specimens for each group in uniaxial compressive strength test, but the effective data for each group may be less.Figure 7
Uniaxial compressive strength test.The stress-strain curves from Group 3 are shown in Figure8 where the maxima are the uniaxial compressive strength (UCS).Figure 8
Uniaxial compressive stress-strain plots: Group 3.Besides, the curves of uniaxial compressive stress and strain from Group 7 are shown in Figure9. On the uniaxial compressive stress-strain curve, the elastic region was well-defined and Young’s modulus could be found from the slope of the plot therein.Figure 9
Uniaxial compressive stress-strain plots: Group 7.Young’s modulus can be calculated as the following:(3)
E
=
f
c
1
-
f
c
2
ε
1
-
ε
2
,where f
c
1, ε
1 are, respectively, the stress and strain on elastic starting point and f
c
2, ε
2 are, respectively, the stress and strain on elastic ending point.In order to investigate the tensile strength, 16 groups of flattened Brazilian disk tests (Figure10) were carried out, and there were also 5 specimens for each group test.Figure 10
Flattened Brazilian disk test or tensile splitting test.According to Wang and Wu’s suggestions [26], as shown in Figure 11, the tensile strength can be determined by the following formulae from a flattened Brazilian disk specimen:(4)
f
t
=
1.92
P
c
π
D
t
,where P
c is the critical load, D is the diameter of the specimen, and t is the thickness.Figure 11
Flattened Brazilian disk specimen subjected to uniform diametral compression [26].The load-time curves for the flattened Brazilian disk specimen used for the determination of rock tensile strength are shown in Figure12 where the maxima denote critical tensile strengths.Figure 12
Load-time curve: Brazilian disk specimen used for the determination of rock tensile strength.


## 4. Results and Analysis

### 4.1. Experimental Results

Experimental results including the data of density, Young’s modulus, uniaxial compression strength, and Brazil splitting strength were listed in the Appendix.


### 4.2. Analysis Method

The total effect function of all factors can be expressed as the sum of each factor effect:(5)
y
=
b
0
+
P
A
+
P
B
+
P
C
+
⋯
,where P
A
,
P
B
,
P
C
,
… denote the effect of A, B, C, and so forth, respectively. The effect function of each factor can be expanded according to the orthogonal polynomial:(6)
P
A
=
b
1
a
Φ
1
A
+
b
2
a
Φ
2
A
+
⋯
+
b
n
-
1
a
Φ
n
-
1
A
=
b
1
a
λ
1
a
Ψ
1
A
+
b
2
a
λ
2
a
Ψ
2
A
+
⋯
+
b
n
-
1
a
λ
n
-
1
a
Ψ
n
-
1
A
,
P
B
=
b
1
b
Φ
1
B
+
b
2
b
Φ
2
B
+
⋯
+
b
n
-
1
b
Φ
n
-
1
B
=
b
1
b
λ
1
b
Ψ
1
B
+
b
2
b
λ
2
b
Ψ
2
B
+
⋯
+
b
n
-
1
b
λ
n
-
1
b
Ψ
n
-
1
B
,
P
C
=
b
1
c
Φ
1
C
+
b
2
c
Φ
2
C
+
⋯
+
b
n
-
1
c
Φ
n
-
1
C
=
b
1
c
λ
1
c
Ψ
1
C
+
b
2
c
λ
2
c
Ψ
2
C
+
⋯
+
b
n
-
1
c
λ
n
-
1
c
Ψ
n
-
1
C
.Regression coefficientb
k and constant term b
0 can be calculated as follows:(7)
b
k
=
∑
t
=
1
n
Φ
i
x
t
y
t
r
∑
t
=
1
n
Φ
i
2
x
t
=
B
i
r
S
i
,
(8)
b
0
=
1
n
r
∑
t
=
1
n
∑
j
=
1
r
y
t
j
=
1
n
r
∑
t
=
1
n
y
t
,where y
t
=
y
t
1
+
y
t
1
+
⋯
+
y
t
1, t
=
1,2
,
…
,
n, and “r” denotes repeat test times for the same factor at one level: the regression equation can then be established.Using the following formula, the level of each factor is changed to a standard isometric point:(9)
A
′
=
A
-
30
%
10
%
;
B
′
=
B
-
-
1
/
3
1
/
3
;
C
′
=
C
-
-
1
%
2
%
;
D
′
=
D
1
/
4
.There were four level tests, andn
=
4; therefore each factor can be expanded to three terms. In this case, “x” denotes the effect function of A
′, B
′, C
′, and D
′, in the orthogonal polynomials, and the regression equation can be expressed as(10)
y
=
b
0
+
b
1
a
λ
1
a
a
+
b
2
a
λ
2
a
a
2
-
1.25
+
b
3
a
λ
3
a
a
2
-
2.05
a
+
b
1
b
λ
1
b
b
+
b
2
b
λ
2
b
b
2
-
1.25
+
b
3
b
λ
3
b
b
2
-
2.05
b
+
b
1
c
λ
1
c
c
+
b
2
c
λ
2
c
c
2
-
1.25
+
b
3
c
λ
3
c
c
2
-
2.05
c
+
b
1
d
λ
1
d
d
+
b
2
d
λ
2
d
d
2
-
1.25
+
b
3
d
λ
3
d
d
2
-
2.05
d
,where(11)
a
=
A
-
30
%
10
%
-
2.5
;
b
=
B
-
-
1
/
3
1
/
3
-
2.5
;
c
=
C
-
-
1
%
2
%
-
2.5
;
d
=
D
1
/
4
-
2.5
.The regression coefficients are calculated by using (7), in which Φ
i
(
x
t
) and ∑
Φ
i
x
t
2 values can be directly sought from the orthogonal polynomials table, and b
0 is calculated by use of (8).To establish the optimal regression equation and the effect of the factors on the significance of the decision and to determine the significance of the regression coefficients, first of all, the sum of the squares variation of regression coefficients was evaluated, followed by anF-test. It is well known that(12)
l
y
y
=
U
+
Q
,where l
y
y denotes the sum of the total squared variations, U denotes the regression sum of the squares, and Q denotes the residual sum of the squares. Also,(13)
l
y
y
=
∑
t
=
1
n
∑
j
=
1
r
y
t
j
2
-
1
n
r
∑
t
=
1
n
∑
j
=
1
r
y
t
j
2
,
U
=
∑
i
=
1
k
b
i
B
i
=
∑
i
=
1
k
b
i
2
r
∑
t
=
1
n
Φ
i
2
x
t
=
∑
i
=
1
k
b
i
2
r
S
i
,
Q
=
l
y
y
-
U
=
∑
i
=
1
n
f
i
S
i
2
.S
i
2 is the variance of repeated measurements under the same conditions; f
i is the number of degrees of freedom of the variance. The sum of the squares of the variation of the regression coefficients can be calculated as follows:(14)
S
b
i
=
b
i
B
i
=
b
i
2
r
S
i
.The numbers of degrees of freedom are(15)
f
r
=
n
r
-
1
;
f
u
=
k
;
f
Q
=
n
r
-
k
-
1
;
f
b
i
=
1
;
i
=
1,2
,
…
,
k
.


#### 4.2.1. Regression Analysis: Density

The density index results are listed in the Appendix (Tables3 and 4). There are four levels (n
=
4) for each factor and four groups of tests for each level. For each group, there are five effective values of the density index. The regression coefficients have been calculated based on formulas (7) and (8).Table 3
Test scheme and density index results.
Test ID
A
B
C
D
Density (g/cm3)
Fine aggregate : coarse aggregates
Barite sand : quartz sand
Anhydrous sodium silicate : aggregates
Anhydrous sodium silicate : sodium fluorosilicate
1
2
3
4
5
∑
y
t
y
-
t
1
L
16
4
5
2.0281
2.0411
2.0299
2.0288
2.0836
10.2116
2.0423
2
2.2802
2.2863
2.3439
2.2448
2.3126
11.4679
2.2936
3
2.2938
2.4721
2.4567
2.4771
2.4946
12.1943
2.4389
4
2.5172
2.5123
2.5198
2.5068
2.5051
12.5612
2.5122
5
2.1153
2.1581
2.2020
2.1711
2.1674
10.8140
2.1628
6
2.0465
2.0597
2.0800
2.0628
2.0517
10.3007
2.0601
7
2.3080
2.3233
2.3008
2.3092
2.3218
11.5631
2.3126
8
2.4797
2.4663
2.4623
2.4667
2.4930
12.3680
2.4736
9
2.1763
2.1450
2.1829
2.1589
2.2079
10.8709
2.1742
10
2.2912
2.2646
2.2476
2.2847
2.2481
11.3362
2.2672
11
2.3146
2.3397
2.3352
2.3891
2.3533
11.7318
2.3464
12
2.5032
2.4217
2.4908
2.4023
2.5109
12.3289
2.4658
13
2.1848
2.1992
2.1698
2.1704
2.1606
10.8847
2.1769
14
2.2467
2.2568
2.2963
2.2503
2.2876
11.3377
2.2675
15
2.1576
2.1905
2.2248
2.2061
2.1538
10.9328
2.1866
16
2.2303
2.2552
2.2518
2.2404
2.2814
11.2734
2.2547
K
1
46.4385
42.7810
43.5175
45.4525
∑
t
=
1
4
∑
j
=
1
4
×
5
y
t
j
=
182.1772
;
∑
y
-
=
36.4365
K
2
45.0570
44.4420
45.5440
46.4525
K
3
46.2680
46.4340
46.7710
45.6180
K
4
44.4285
48.5315
46.3560
44.6655Table 4
Orthogonal polynomial regression analysis of variance: density index.
The source of variance
Square and variation
Degree of freedom
Variance estimate
F-value
F
0.05
(
1
,
69
)
Level of significance
Remarks
b
1
a
5.8057 × 10−2
1
5.8057 × 10−2
28.70
4
∗
∗
A, 1-order term
b
2
a
2.6220 × 10−3
1
2.6220 × 10−3
1.30
A, 2-order term
b
3
a
7.9609 × 10−2
1
7.9609 × 10−2
39.36
∗
∗
A, 3-order term
b
1
b
9.2578 × 10−1
1
9.2578 × 10−1
457.67
∗
∗
∗
∗
B, 1-order term
b
2
b
2.3817 × 10−3
1
2.3817 × 10−3
1.18
B, 2-order term
b
1
c
2.3729 × 10−1
1
2.3729 × 10−1
117.31
∗
∗
∗
C, 1-order term
b
2
c
7.4512 × 10−2
1
7.4512 × 10−2
36.84
∗
∗
C, 2-order term
b
1
d
2.5528 × 10−2
1
2.5528 × 10−2
12.62
∗
D, 1-order term
b
2
d
4.7653 × 10−2
1
4.7653 × 10−2
23.56
∗
∗
D, 2-order term
b
3
d
7.3659 × 10−3
1
7.3659 × 10−3
3.64
D, 3-order term
Error
1.3957 × 10−2
69
2.0228 × 10−3According to (14), the sum of the squares of the variations of the regression coefficients can be calculated (Table 4). Besides, according to (13), other parameters can be calculated as follows:(16)
l
y
y
=
1.600
;
U
=
1.463
;
Q
=
1.377
×
10
-
1
f
r
=
79
;
f
U
=
12
;
f
Q
=
67
.The variance estimates of the error effect can be written asQ
/
f
Q
=
2.055
×
10
-
3. In Table 4, b
3
b and b
3
c are the variance estimates of the regression coefficients which are both less than the error effects, and they can be merged therewith.As shown in Table4, the order of (decreasing) importance of the factors was B, C, A, and D.Regression analysis gives the optimal regression equation as(17)
y
d
=
2.346
+
0.073
a
+
0.096
b
+
0.049
c
-
0.016
d
.


#### 4.2.2. Regression Analysis: Young’s Modulus

There were four levels (n
=
4) for each factor and four groups of tests for each level. For each group of tests, there were three effectual pieces of data of the elasticity modulus index (see Appendix, Tables 5 and 6).Table 5
Test scheme and Young’s modulus index results.
Test ID
A
B
C
D
Young’s modulus (MPa)
Fine aggregate : coarse aggregates
Barite sand : quartz sand
Anhydrous sodium silicate : aggregates
Anhydrous sodium silicate : sodium fluorosilicate
1
2
3
∑
y
t
y
-
t
1
L
16
4
5
21.8962
27.963
19.646
69.5052
23.1684
2
61.9473
93.1975
77.4981
232.6428
77.5476
3
120.2802
100.0844
103.2362
323.6007
107.8669
4
159.9495
173.4145
174.1913
507.5553
169.1851
5
22.2968
27.2367
21.1451
70.6785
23.5595
6
23.5181
15.346
15.5944
54.4584
18.1528
7
74.0988
65.8336
72.6667
212.5992
70.8664
8
69.1296
56.8568
98.6768
224.6631
74.8877
9
102.5573
111.2596
68.2513
282.0681
94.0227
10
89.7668
100.6273
123.138
313.5321
104.5107
11
16.2539
8.0914
9.5063
33.8514
11.2838
12
11.1586
11.443
11.9502
34.5519
11.5173
13
37.2454
50.4401
38.6258
126.3114
42.1038
14
19.3454
16.8179
12.8439
49.0071
16.3357
15
30.0612
35.9429
34.9628
100.9671
33.6557
16
10.7866
10.395
10.6829
31.8645
10.6215
K
1
1133.304
548.5632
189.6795
365.6634
∑
t
=
1
4
∑
j
=
1
4
×
3
y
t
j
=
2667.857; ∑
y
-
t
=
889.2856
K
2
562.3992
649.6404
438.8403
617.4687
K
3
664.0035
671.0184
879.339
739.6758
K
4
308.1501
798.6348
1159.998
945.0489Table 6
Orthogonal polynomial regression analysis of variance: Young’s modulus index.
The source of variance
Square and variation
Degree of freedom
Variance estimate
F-value
F
0.05
(
1
,
35
)
Level of significance
Remarks
b
1
a
2.3480 × 104
1
2.3480 × 104
138.39
4.13
∗
∗
∗
∗
A, 1-order term
b
2
a
9.6348 × 102
1
9.6348 × 102
5.68
∗
A, 2-order term
b
3
a
5.3201 × 103
1
5.3201 × 103
31.36
∗
∗
∗
A, 3-order term
b
1
b
2.4806 × 103
1
2.4806 × 103
14.62
∗
∗
B, 1-order term
b
2
b
1.4674 × 101
1
1.4674 × 101
0.09
B, 2-order term
b
3
b
1.4405 × 102
1
1.4405 × 102
0.85
B, 3-order term
b
1
c
4.6801 × 104
1
4.6801 × 104
275.85
∗
∗
∗
∗
C, 1-order term
b
2
c
2.0670 × 101
1
2.0670 × 101
0.12
C, 2-order term
b
3
c
5.1386 × 102
1
5.1386 × 102
3.03
C, 3-order term
b
1
d
1.4421 × 104
1
1.4421 × 104
85.00
∗
∗
∗
∗
D, 1-order term
b
2
d
4.4916 × 101
1
4.4916 × 101
0.26
D, 2-order term
b
3
d
1.8862 × 102
1
1.8862 × 102
1.11
D, 3-order term
Error
5.9382 × 103
35
1.6966 × 10−2To establish the optimal regression equation, it was important to judge the significance level of each factor, and to determine the significance of the regression coefficient, firstly, the regression coefficient of the variation of the squared sum was found, and then anF-test was conducted.According to (13) and (14), the variation of the regression coefficients can be obtained (see Table 6):(18)
l
y
y
=
1.003
×
10
5
;
U
=
9.439
×
10
4
;
Q
=
5.938
×
10
3
;
f
r
=
47
;
f
u
=
12
;
f
Q
=
35
.The results are listed in Table6: the order of importance (decreasing) of the factors was C, A, D, and B.Regression analysis gives the optimal regression equation as(19)
y
E
=
49.980
+
12.391
a
+
6.430
b
+
27.928
c
+
15.503
d
.


#### 4.2.3. Regression Analysis: UCS

The UCS results are listed in the Appendix (Tables7 and 8).Table 7
Test scheme and compressive strength index results.
Test ID
A
B
C
D
Compressive strength (MPa)
Fine aggregate : coarse aggregates
Barite sand : quartz sand
Anhydrous sodium silicate : aggregates
Anhydrous sodium silicate : sodium fluorosilicate
1
2
3
4
5
∑
y
t
y
-
t
1
L
16
4
5
0.2949
0.2984
0.2188
0.1995
0.2912
1.3028
0.2606
2
0.5785
0.5832
0.7068
0.5723
0.6511
3.0918
0.6184
3
2.5366
2.7429
2.4855
2.6060
2.7399
13.1110
2.6222
4
4.1403
3.6729
4.7884
4.0876
4.5456
21.2348
4.2470
5
0.5408
0.4813
0.6027
0.4228
0.5529
2.6005
0.5201
6
0.1228
0.1549
0.1706
0.2080
0.1592
0.8155
0.1631
7
1.7678
1.5183
1.5102
1.4143
1.9126
8.1232
1.6246
8
0.9976
1.0144
1.0713
0.8484
0.9594
4.8912
0.9782
9
1.3242
1.1996
1.9182
1.9821
2.0006
8.4247
1.6849
10
2.3129
1.6285
1.8888
2.4849
1.3878
9.7029
1.9406
11
0.2654
0.1876
0.2182
0.2508
0.2150
1.1370
0.2274
12
0.2049
0.1392
0.2293
0.1864
0.2427
1.0026
0.2005
13
0.8433
0.8210
0.7965
0.8615
0.8251
4.1475
0.8295
14
0.3936
0.3880
0.3782
0.3416
0.3835
1.8849
0.3770
15
0.3292
0.4720
0.5058
0.3780
0.4433
2.1284
0.4257
16
0.2533
0.1871
0.2265
0.2256
0.2369
1.1294
0.2259
K
1
38.7404
16.4755
4.3847
12.3135
∑
t
=
1
4
∑
j
=
1
4
×
5
y
t
j
=
84.7282; ∑
y
-
=
16.9457
K
2
16.4304
15.4951
8.8233
13.2675
K
3
20.2672
24.4996
28.3118
26.5438
K
4
9.2902
28.258
43.2084
32.6034Table 8
Orthogonal polynomial regression analysis of variance: compressive strength index.
The source of variance
Square and variation
Degree of freedom
Variance estimate
F-value
F
0.05
1,69
Level of significance
Remarks
b
1
a
1.7856 × 101
1
1.7856 × 101
245.48
4
∗
∗
∗
∗
A, 1-order term
b
2
a
1.6055 × 100
1
1.6055 × 100
22.07
∗
∗
A, 2-order term
b
3
a
4.1944 × 100
1
4.1944 × 100
57.66
∗
∗
A, 3-order term
b
1
b
4.9177 × 100
1
4.9177 × 100
67.61
∗
∗
∗
B, 1-order term
b
2
b
2.8070 × 10−1
1
2.8070 × 10−1
3.86
B, 2-order term
b
3
b
5.7996 × 10−1
1
5.7996 × 10−1
7.97
∗
B, 3-order term
b
1
c
4.6213 × 101
1
4.6213 × 101
635.31
∗
∗
∗
∗
C, 1-order term
b
2
c
1.3671 × 10
1
1.3671 × 10
18.79
∗
∗
C, 2-order term
b
3
c
9.6450 × 10−1
1
9.6450 × 10−1
13.26
∗
∗
C, 3-order term
b
1
d
1.3744 × 101
1
1.3744 × 101
188.95
∗
∗
∗
∗
D, 1-order term
b
2
d
3.2584 × 10−1
1
3.2584 × 10−1
4.48
∗
D, 2-order term
b
3
d
9.5443 × 10−1
1
9.5443 × 10−1
13.12
∗
D, 3-order term
Error
4.8736 × 100
67
7.2740 × 10−2To establish the optimal regression equation, it was important to judge the significance level of each factor, and to determine the significance of the regression coefficient, firstly, the regression coefficient of the variation of the squared sum was evaluated, and then anF-test was carried out.According to (13) and (14), the variation of the regression coefficients can be obtained, as shown in Table 8:(20)
l
y
y
=
97.877
;
U
=
9.300
×
10
1
;
Q
=
4.874
;
f
r
=
79
;
f
u
=
12
;
f
Q
=
67
.The results are listed in Table8: for the compressive strength, the order of importance (decreasing) of the factors was C, A, D, and B.Thus the optimal regression equation was(21)
y
f
c
=
0.639
+
0.277
a
+
0.482
b
+
1.015
c
+
0.519
d
.


#### 4.2.4. Regression Analysis: Tensile Strength

Tensile strengths and the regression coefficients are listed in the Appendix, Table9.Table 9
Test scheme and tensile strength index results.
Test ID
A
B
C
D
Tensile strength (MPa)
Fine aggregate : coarse aggregates
Barite sand : quartz sand
Anhydrous sodium silicate : aggregates
Anhydrous sodium silicate : sodium fluorosilicate
1
2
3
4
∑
y
t
y
-
t
1
L
16
4
5
0.0973
0.0976
0.0895
0.0949
0.3793
0.0948
2
0.1949
0.1903
0.1917
0.2078
0.7847
0.1962
3
0.7546
0.6716
0.7655
0.7924
2.9841
0.7460
4
0.9333
0.8623
0.8947
0.8466
3.5370
0.8842
5
0.1382
0.1207
0.1675
0.1755
0.6019
0.1505
6
0.0459
0.0407
0.0492
0.0406
0.1763
0.0441
7
0.6694
0.5585
0.4850
0.5193
2.2322
0.5581
8
0.2910
0.2140
0.2459
0.2622
1.0132
0.2533
9
0.4919
0.5025
0.5248
0.4561
1.9752
0.4938
10
0.6960
0.5942
0.5703
0.5589
2.4193
0.6048
11
0.0945
0.0891
0.0888
0.0932
0.3656
0.0914
12
0.0698
0.0647
0.0597
0.0624
0.2565
0.0641
13
0.2543
0.2770
0.2880
0.2876
1.1069
0.2767
14
0.1137
0.1052
0.1032
0.1177
0.4397
0.1099
15
0.1404
0.1426
0.1388
0.1406
0.5625
0.1406
16
0.0603
0.0680
0.0558
0.0651
0.2492
0.0623
K
1
7.6851
4.0633
1.1704
3.3077
K
2
4.0236
3.82
2.2056
3.2704
K
3
5.0166
6.1444
6.4122
6.2545
K
4
2.3583
5.0559
9.2954
6.251To establish the optimal regression equation and the effect of the factors on the significance of the decision and to determine the significance of the regression coefficient, firstly, the regression coefficient of the variation of the squared sum was evaluated, and then anF-test was carried out. According to (13) and (14), the variation of the regression coefficients can be obtained, as shown in the Appendix, Table 10:(22)
l
y
y
=
4.444
;
U
=
4.362
;
Q
=
8.163
×
10
-
2
f
r
=
63
;
f
U
=
12
;
f
Q
=
51
.Table 10
Orthogonal polynomial regression analysis of variance: tensile strength index.
The source of variance
Square and variation
Degree of freedom
Variance estimate
F-value
F
0.05
1,51
Level of significance
Remarks
b
1
a
7.0194 × 10−1
1
7.0194 × 10−1
438.56
4
∗
∗
∗
∗
A, 1-order term
b
2
a
1.5725 × 10−2
1
1.5725 × 10−2
9.82
∗
A, 2-order term
b
3
a
2.1558 × 10−1
1
2.1558 × 10−1
134.69
∗
∗
∗
A, 3-order term
b
1
b
8.7854 × 10−2
1
8.7854 × 10−2
54.89
∗
∗
B, 1-order term
b
2
b
1.1162 × 10−2
1
1.1162 × 10−2
6.97
∗
B, 2-order term
b
3
b
1.1177 × 10−1
1
1.1177 × 10−1
69.83
∗
∗
B, 3-order term
b
1
c
2.5528 × 100
1
2.5528 × 100
1594.97
∗
∗
∗
∗
∗
C, 1-order term
b
2
c
5.3361 × 10−2
1
5.3361 × 10−2
33.34
∗
∗
C, 2-order term
b
3
c
6.3135 × 10−2
1
6.3135 × 10−2
39.45
∗
∗
C, 3-order term
b
1
d
4.3616 × 10−1
1
4.3616 × 10−1
272.50
∗
∗
∗
∗
D, 1-order term
b
2
d
1.7851 × 10−5
1
1.7851 × 10−5
0.01
D, 2-order term
b
3
d
1.1284 × 10−1
1
1.1284 × 10−1
70.50
∗
∗
D, 3-order term
Error
8.1629 × 10−2
51
1.6006 × 10−3For the tensile strength, the order of importance (decreasing) wasC, A, D, and B.Thus the optimal regression equation was(23)
y
f
t
=
0.259
+
0.084
a
+
0.161
b
+
0.275
c
+
0.202
d
.


## 4.1. Experimental Results

Experimental results including the data of density, Young’s modulus, uniaxial compression strength, and Brazil splitting strength were listed in the Appendix.


## 4.2. Analysis Method

The total effect function of all factors can be expressed as the sum of each factor effect:(5)
y
=
b
0
+
P
A
+
P
B
+
P
C
+
⋯
,where P
A
,
P
B
,
P
C
,
… denote the effect of A, B, C, and so forth, respectively. The effect function of each factor can be expanded according to the orthogonal polynomial:(6)
P
A
=
b
1
a
Φ
1
A
+
b
2
a
Φ
2
A
+
⋯
+
b
n
-
1
a
Φ
n
-
1
A
=
b
1
a
λ
1
a
Ψ
1
A
+
b
2
a
λ
2
a
Ψ
2
A
+
⋯
+
b
n
-
1
a
λ
n
-
1
a
Ψ
n
-
1
A
,
P
B
=
b
1
b
Φ
1
B
+
b
2
b
Φ
2
B
+
⋯
+
b
n
-
1
b
Φ
n
-
1
B
=
b
1
b
λ
1
b
Ψ
1
B
+
b
2
b
λ
2
b
Ψ
2
B
+
⋯
+
b
n
-
1
b
λ
n
-
1
b
Ψ
n
-
1
B
,
P
C
=
b
1
c
Φ
1
C
+
b
2
c
Φ
2
C
+
⋯
+
b
n
-
1
c
Φ
n
-
1
C
=
b
1
c
λ
1
c
Ψ
1
C
+
b
2
c
λ
2
c
Ψ
2
C
+
⋯
+
b
n
-
1
c
λ
n
-
1
c
Ψ
n
-
1
C
.Regression coefficientb
k and constant term b
0 can be calculated as follows:(7)
b
k
=
∑
t
=
1
n
Φ
i
x
t
y
t
r
∑
t
=
1
n
Φ
i
2
x
t
=
B
i
r
S
i
,
(8)
b
0
=
1
n
r
∑
t
=
1
n
∑
j
=
1
r
y
t
j
=
1
n
r
∑
t
=
1
n
y
t
,where y
t
=
y
t
1
+
y
t
1
+
⋯
+
y
t
1, t
=
1,2
,
…
,
n, and “r” denotes repeat test times for the same factor at one level: the regression equation can then be established.Using the following formula, the level of each factor is changed to a standard isometric point:(9)
A
′
=
A
-
30
%
10
%
;
B
′
=
B
-
-
1
/
3
1
/
3
;
C
′
=
C
-
-
1
%
2
%
;
D
′
=
D
1
/
4
.There were four level tests, andn
=
4; therefore each factor can be expanded to three terms. In this case, “x” denotes the effect function of A
′, B
′, C
′, and D
′, in the orthogonal polynomials, and the regression equation can be expressed as(10)
y
=
b
0
+
b
1
a
λ
1
a
a
+
b
2
a
λ
2
a
a
2
-
1.25
+
b
3
a
λ
3
a
a
2
-
2.05
a
+
b
1
b
λ
1
b
b
+
b
2
b
λ
2
b
b
2
-
1.25
+
b
3
b
λ
3
b
b
2
-
2.05
b
+
b
1
c
λ
1
c
c
+
b
2
c
λ
2
c
c
2
-
1.25
+
b
3
c
λ
3
c
c
2
-
2.05
c
+
b
1
d
λ
1
d
d
+
b
2
d
λ
2
d
d
2
-
1.25
+
b
3
d
λ
3
d
d
2
-
2.05
d
,where(11)
a
=
A
-
30
%
10
%
-
2.5
;
b
=
B
-
-
1
/
3
1
/
3
-
2.5
;
c
=
C
-
-
1
%
2
%
-
2.5
;
d
=
D
1
/
4
-
2.5
.The regression coefficients are calculated by using (7), in which Φ
i
(
x
t
) and ∑
Φ
i
x
t
2 values can be directly sought from the orthogonal polynomials table, and b
0 is calculated by use of (8).To establish the optimal regression equation and the effect of the factors on the significance of the decision and to determine the significance of the regression coefficients, first of all, the sum of the squares variation of regression coefficients was evaluated, followed by anF-test. It is well known that(12)
l
y
y
=
U
+
Q
,where l
y
y denotes the sum of the total squared variations, U denotes the regression sum of the squares, and Q denotes the residual sum of the squares. Also,(13)
l
y
y
=
∑
t
=
1
n
∑
j
=
1
r
y
t
j
2
-
1
n
r
∑
t
=
1
n
∑
j
=
1
r
y
t
j
2
,
U
=
∑
i
=
1
k
b
i
B
i
=
∑
i
=
1
k
b
i
2
r
∑
t
=
1
n
Φ
i
2
x
t
=
∑
i
=
1
k
b
i
2
r
S
i
,
Q
=
l
y
y
-
U
=
∑
i
=
1
n
f
i
S
i
2
.S
i
2 is the variance of repeated measurements under the same conditions; f
i is the number of degrees of freedom of the variance. The sum of the squares of the variation of the regression coefficients can be calculated as follows:(14)
S
b
i
=
b
i
B
i
=
b
i
2
r
S
i
.The numbers of degrees of freedom are(15)
f
r
=
n
r
-
1
;
f
u
=
k
;
f
Q
=
n
r
-
k
-
1
;
f
b
i
=
1
;
i
=
1,2
,
…
,
k
.


### 4.2.1. Regression Analysis: Density

The density index results are listed in the Appendix (Tables3 and 4). There are four levels (n
=
4) for each factor and four groups of tests for each level. For each group, there are five effective values of the density index. The regression coefficients have been calculated based on formulas (7) and (8).Table 3
Test scheme and density index results.
Test ID
A
B
C
D
Density (g/cm3)
Fine aggregate : coarse aggregates
Barite sand : quartz sand
Anhydrous sodium silicate : aggregates
Anhydrous sodium silicate : sodium fluorosilicate
1
2
3
4
5
∑
y
t
y
-
t
1
L
16
4
5
2.0281
2.0411
2.0299
2.0288
2.0836
10.2116
2.0423
2
2.2802
2.2863
2.3439
2.2448
2.3126
11.4679
2.2936
3
2.2938
2.4721
2.4567
2.4771
2.4946
12.1943
2.4389
4
2.5172
2.5123
2.5198
2.5068
2.5051
12.5612
2.5122
5
2.1153
2.1581
2.2020
2.1711
2.1674
10.8140
2.1628
6
2.0465
2.0597
2.0800
2.0628
2.0517
10.3007
2.0601
7
2.3080
2.3233
2.3008
2.3092
2.3218
11.5631
2.3126
8
2.4797
2.4663
2.4623
2.4667
2.4930
12.3680
2.4736
9
2.1763
2.1450
2.1829
2.1589
2.2079
10.8709
2.1742
10
2.2912
2.2646
2.2476
2.2847
2.2481
11.3362
2.2672
11
2.3146
2.3397
2.3352
2.3891
2.3533
11.7318
2.3464
12
2.5032
2.4217
2.4908
2.4023
2.5109
12.3289
2.4658
13
2.1848
2.1992
2.1698
2.1704
2.1606
10.8847
2.1769
14
2.2467
2.2568
2.2963
2.2503
2.2876
11.3377
2.2675
15
2.1576
2.1905
2.2248
2.2061
2.1538
10.9328
2.1866
16
2.2303
2.2552
2.2518
2.2404
2.2814
11.2734
2.2547
K
1
46.4385
42.7810
43.5175
45.4525
∑
t
=
1
4
∑
j
=
1
4
×
5
y
t
j
=
182.1772
;
∑
y
-
=
36.4365
K
2
45.0570
44.4420
45.5440
46.4525
K
3
46.2680
46.4340
46.7710
45.6180
K
4
44.4285
48.5315
46.3560
44.6655Table 4
Orthogonal polynomial regression analysis of variance: density index.
The source of variance
Square and variation
Degree of freedom
Variance estimate
F-value
F
0.05
(
1
,
69
)
Level of significance
Remarks
b
1
a
5.8057 × 10−2
1
5.8057 × 10−2
28.70
4
∗
∗
A, 1-order term
b
2
a
2.6220 × 10−3
1
2.6220 × 10−3
1.30
A, 2-order term
b
3
a
7.9609 × 10−2
1
7.9609 × 10−2
39.36
∗
∗
A, 3-order term
b
1
b
9.2578 × 10−1
1
9.2578 × 10−1
457.67
∗
∗
∗
∗
B, 1-order term
b
2
b
2.3817 × 10−3
1
2.3817 × 10−3
1.18
B, 2-order term
b
1
c
2.3729 × 10−1
1
2.3729 × 10−1
117.31
∗
∗
∗
C, 1-order term
b
2
c
7.4512 × 10−2
1
7.4512 × 10−2
36.84
∗
∗
C, 2-order term
b
1
d
2.5528 × 10−2
1
2.5528 × 10−2
12.62
∗
D, 1-order term
b
2
d
4.7653 × 10−2
1
4.7653 × 10−2
23.56
∗
∗
D, 2-order term
b
3
d
7.3659 × 10−3
1
7.3659 × 10−3
3.64
D, 3-order term
Error
1.3957 × 10−2
69
2.0228 × 10−3According to (14), the sum of the squares of the variations of the regression coefficients can be calculated (Table 4). Besides, according to (13), other parameters can be calculated as follows:(16)
l
y
y
=
1.600
;
U
=
1.463
;
Q
=
1.377
×
10
-
1
f
r
=
79
;
f
U
=
12
;
f
Q
=
67
.The variance estimates of the error effect can be written asQ
/
f
Q
=
2.055
×
10
-
3. In Table 4, b
3
b and b
3
c are the variance estimates of the regression coefficients which are both less than the error effects, and they can be merged therewith.As shown in Table4, the order of (decreasing) importance of the factors was B, C, A, and D.Regression analysis gives the optimal regression equation as(17)
y
d
=
2.346
+
0.073
a
+
0.096
b
+
0.049
c
-
0.016
d
.


### 4.2.2. Regression Analysis: Young’s Modulus

There were four levels (n
=
4) for each factor and four groups of tests for each level. For each group of tests, there were three effectual pieces of data of the elasticity modulus index (see Appendix, Tables 5 and 6).Table 5
Test scheme and Young’s modulus index results.
Test ID
A
B
C
D
Young’s modulus (MPa)
Fine aggregate : coarse aggregates
Barite sand : quartz sand
Anhydrous sodium silicate : aggregates
Anhydrous sodium silicate : sodium fluorosilicate
1
2
3
∑
y
t
y
-
t
1
L
16
4
5
21.8962
27.963
19.646
69.5052
23.1684
2
61.9473
93.1975
77.4981
232.6428
77.5476
3
120.2802
100.0844
103.2362
323.6007
107.8669
4
159.9495
173.4145
174.1913
507.5553
169.1851
5
22.2968
27.2367
21.1451
70.6785
23.5595
6
23.5181
15.346
15.5944
54.4584
18.1528
7
74.0988
65.8336
72.6667
212.5992
70.8664
8
69.1296
56.8568
98.6768
224.6631
74.8877
9
102.5573
111.2596
68.2513
282.0681
94.0227
10
89.7668
100.6273
123.138
313.5321
104.5107
11
16.2539
8.0914
9.5063
33.8514
11.2838
12
11.1586
11.443
11.9502
34.5519
11.5173
13
37.2454
50.4401
38.6258
126.3114
42.1038
14
19.3454
16.8179
12.8439
49.0071
16.3357
15
30.0612
35.9429
34.9628
100.9671
33.6557
16
10.7866
10.395
10.6829
31.8645
10.6215
K
1
1133.304
548.5632
189.6795
365.6634
∑
t
=
1
4
∑
j
=
1
4
×
3
y
t
j
=
2667.857; ∑
y
-
t
=
889.2856
K
2
562.3992
649.6404
438.8403
617.4687
K
3
664.0035
671.0184
879.339
739.6758
K
4
308.1501
798.6348
1159.998
945.0489Table 6
Orthogonal polynomial regression analysis of variance: Young’s modulus index.
The source of variance
Square and variation
Degree of freedom
Variance estimate
F-value
F
0.05
(
1
,
35
)
Level of significance
Remarks
b
1
a
2.3480 × 104
1
2.3480 × 104
138.39
4.13
∗
∗
∗
∗
A, 1-order term
b
2
a
9.6348 × 102
1
9.6348 × 102
5.68
∗
A, 2-order term
b
3
a
5.3201 × 103
1
5.3201 × 103
31.36
∗
∗
∗
A, 3-order term
b
1
b
2.4806 × 103
1
2.4806 × 103
14.62
∗
∗
B, 1-order term
b
2
b
1.4674 × 101
1
1.4674 × 101
0.09
B, 2-order term
b
3
b
1.4405 × 102
1
1.4405 × 102
0.85
B, 3-order term
b
1
c
4.6801 × 104
1
4.6801 × 104
275.85
∗
∗
∗
∗
C, 1-order term
b
2
c
2.0670 × 101
1
2.0670 × 101
0.12
C, 2-order term
b
3
c
5.1386 × 102
1
5.1386 × 102
3.03
C, 3-order term
b
1
d
1.4421 × 104
1
1.4421 × 104
85.00
∗
∗
∗
∗
D, 1-order term
b
2
d
4.4916 × 101
1
4.4916 × 101
0.26
D, 2-order term
b
3
d
1.8862 × 102
1
1.8862 × 102
1.11
D, 3-order term
Error
5.9382 × 103
35
1.6966 × 10−2To establish the optimal regression equation, it was important to judge the significance level of each factor, and to determine the significance of the regression coefficient, firstly, the regression coefficient of the variation of the squared sum was found, and then anF-test was conducted.According to (13) and (14), the variation of the regression coefficients can be obtained (see Table 6):(18)
l
y
y
=
1.003
×
10
5
;
U
=
9.439
×
10
4
;
Q
=
5.938
×
10
3
;
f
r
=
47
;
f
u
=
12
;
f
Q
=
35
.The results are listed in Table6: the order of importance (decreasing) of the factors was C, A, D, and B.Regression analysis gives the optimal regression equation as(19)
y
E
=
49.980
+
12.391
a
+
6.430
b
+
27.928
c
+
15.503
d
.


### 4.2.3. Regression Analysis: UCS

The UCS results are listed in the Appendix (Tables7 and 8).Table 7
Test scheme and compressive strength index results.
Test ID
A
B
C
D
Compressive strength (MPa)
Fine aggregate : coarse aggregates
Barite sand : quartz sand
Anhydrous sodium silicate : aggregates
Anhydrous sodium silicate : sodium fluorosilicate
1
2
3
4
5
∑
y
t
y
-
t
1
L
16
4
5
0.2949
0.2984
0.2188
0.1995
0.2912
1.3028
0.2606
2
0.5785
0.5832
0.7068
0.5723
0.6511
3.0918
0.6184
3
2.5366
2.7429
2.4855
2.6060
2.7399
13.1110
2.6222
4
4.1403
3.6729
4.7884
4.0876
4.5456
21.2348
4.2470
5
0.5408
0.4813
0.6027
0.4228
0.5529
2.6005
0.5201
6
0.1228
0.1549
0.1706
0.2080
0.1592
0.8155
0.1631
7
1.7678
1.5183
1.5102
1.4143
1.9126
8.1232
1.6246
8
0.9976
1.0144
1.0713
0.8484
0.9594
4.8912
0.9782
9
1.3242
1.1996
1.9182
1.9821
2.0006
8.4247
1.6849
10
2.3129
1.6285
1.8888
2.4849
1.3878
9.7029
1.9406
11
0.2654
0.1876
0.2182
0.2508
0.2150
1.1370
0.2274
12
0.2049
0.1392
0.2293
0.1864
0.2427
1.0026
0.2005
13
0.8433
0.8210
0.7965
0.8615
0.8251
4.1475
0.8295
14
0.3936
0.3880
0.3782
0.3416
0.3835
1.8849
0.3770
15
0.3292
0.4720
0.5058
0.3780
0.4433
2.1284
0.4257
16
0.2533
0.1871
0.2265
0.2256
0.2369
1.1294
0.2259
K
1
38.7404
16.4755
4.3847
12.3135
∑
t
=
1
4
∑
j
=
1
4
×
5
y
t
j
=
84.7282; ∑
y
-
=
16.9457
K
2
16.4304
15.4951
8.8233
13.2675
K
3
20.2672
24.4996
28.3118
26.5438
K
4
9.2902
28.258
43.2084
32.6034Table 8
Orthogonal polynomial regression analysis of variance: compressive strength index.
The source of variance
Square and variation
Degree of freedom
Variance estimate
F-value
F
0.05
1,69
Level of significance
Remarks
b
1
a
1.7856 × 101
1
1.7856 × 101
245.48
4
∗
∗
∗
∗
A, 1-order term
b
2
a
1.6055 × 100
1
1.6055 × 100
22.07
∗
∗
A, 2-order term
b
3
a
4.1944 × 100
1
4.1944 × 100
57.66
∗
∗
A, 3-order term
b
1
b
4.9177 × 100
1
4.9177 × 100
67.61
∗
∗
∗
B, 1-order term
b
2
b
2.8070 × 10−1
1
2.8070 × 10−1
3.86
B, 2-order term
b
3
b
5.7996 × 10−1
1
5.7996 × 10−1
7.97
∗
B, 3-order term
b
1
c
4.6213 × 101
1
4.6213 × 101
635.31
∗
∗
∗
∗
C, 1-order term
b
2
c
1.3671 × 10
1
1.3671 × 10
18.79
∗
∗
C, 2-order term
b
3
c
9.6450 × 10−1
1
9.6450 × 10−1
13.26
∗
∗
C, 3-order term
b
1
d
1.3744 × 101
1
1.3744 × 101
188.95
∗
∗
∗
∗
D, 1-order term
b
2
d
3.2584 × 10−1
1
3.2584 × 10−1
4.48
∗
D, 2-order term
b
3
d
9.5443 × 10−1
1
9.5443 × 10−1
13.12
∗
D, 3-order term
Error
4.8736 × 100
67
7.2740 × 10−2To establish the optimal regression equation, it was important to judge the significance level of each factor, and to determine the significance of the regression coefficient, firstly, the regression coefficient of the variation of the squared sum was evaluated, and then anF-test was carried out.According to (13) and (14), the variation of the regression coefficients can be obtained, as shown in Table 8:(20)
l
y
y
=
97.877
;
U
=
9.300
×
10
1
;
Q
=
4.874
;
f
r
=
79
;
f
u
=
12
;
f
Q
=
67
.The results are listed in Table8: for the compressive strength, the order of importance (decreasing) of the factors was C, A, D, and B.Thus the optimal regression equation was(21)
y
f
c
=
0.639
+
0.277
a
+
0.482
b
+
1.015
c
+
0.519
d
.


### 4.2.4. Regression Analysis: Tensile Strength

Tensile strengths and the regression coefficients are listed in the Appendix, Table9.Table 9
Test scheme and tensile strength index results.
Test ID
A
B
C
D
Tensile strength (MPa)
Fine aggregate : coarse aggregates
Barite sand : quartz sand
Anhydrous sodium silicate : aggregates
Anhydrous sodium silicate : sodium fluorosilicate
1
2
3
4
∑
y
t
y
-
t
1
L
16
4
5
0.0973
0.0976
0.0895
0.0949
0.3793
0.0948
2
0.1949
0.1903
0.1917
0.2078
0.7847
0.1962
3
0.7546
0.6716
0.7655
0.7924
2.9841
0.7460
4
0.9333
0.8623
0.8947
0.8466
3.5370
0.8842
5
0.1382
0.1207
0.1675
0.1755
0.6019
0.1505
6
0.0459
0.0407
0.0492
0.0406
0.1763
0.0441
7
0.6694
0.5585
0.4850
0.5193
2.2322
0.5581
8
0.2910
0.2140
0.2459
0.2622
1.0132
0.2533
9
0.4919
0.5025
0.5248
0.4561
1.9752
0.4938
10
0.6960
0.5942
0.5703
0.5589
2.4193
0.6048
11
0.0945
0.0891
0.0888
0.0932
0.3656
0.0914
12
0.0698
0.0647
0.0597
0.0624
0.2565
0.0641
13
0.2543
0.2770
0.2880
0.2876
1.1069
0.2767
14
0.1137
0.1052
0.1032
0.1177
0.4397
0.1099
15
0.1404
0.1426
0.1388
0.1406
0.5625
0.1406
16
0.0603
0.0680
0.0558
0.0651
0.2492
0.0623
K
1
7.6851
4.0633
1.1704
3.3077
K
2
4.0236
3.82
2.2056
3.2704
K
3
5.0166
6.1444
6.4122
6.2545
K
4
2.3583
5.0559
9.2954
6.251To establish the optimal regression equation and the effect of the factors on the significance of the decision and to determine the significance of the regression coefficient, firstly, the regression coefficient of the variation of the squared sum was evaluated, and then anF-test was carried out. According to (13) and (14), the variation of the regression coefficients can be obtained, as shown in the Appendix, Table 10:(22)
l
y
y
=
4.444
;
U
=
4.362
;
Q
=
8.163
×
10
-
2
f
r
=
63
;
f
U
=
12
;
f
Q
=
51
.Table 10
Orthogonal polynomial regression analysis of variance: tensile strength index.
The source of variance
Square and variation
Degree of freedom
Variance estimate
F-value
F
0.05
1,51
Level of significance
Remarks
b
1
a
7.0194 × 10−1
1
7.0194 × 10−1
438.56
4
∗
∗
∗
∗
A, 1-order term
b
2
a
1.5725 × 10−2
1
1.5725 × 10−2
9.82
∗
A, 2-order term
b
3
a
2.1558 × 10−1
1
2.1558 × 10−1
134.69
∗
∗
∗
A, 3-order term
b
1
b
8.7854 × 10−2
1
8.7854 × 10−2
54.89
∗
∗
B, 1-order term
b
2
b
1.1162 × 10−2
1
1.1162 × 10−2
6.97
∗
B, 2-order term
b
3
b
1.1177 × 10−1
1
1.1177 × 10−1
69.83
∗
∗
B, 3-order term
b
1
c
2.5528 × 100
1
2.5528 × 100
1594.97
∗
∗
∗
∗
∗
C, 1-order term
b
2
c
5.3361 × 10−2
1
5.3361 × 10−2
33.34
∗
∗
C, 2-order term
b
3
c
6.3135 × 10−2
1
6.3135 × 10−2
39.45
∗
∗
C, 3-order term
b
1
d
4.3616 × 10−1
1
4.3616 × 10−1
272.50
∗
∗
∗
∗
D, 1-order term
b
2
d
1.7851 × 10−5
1
1.7851 × 10−5
0.01
D, 2-order term
b
3
d
1.1284 × 10−1
1
1.1284 × 10−1
70.50
∗
∗
D, 3-order term
Error
8.1629 × 10−2
51
1.6006 × 10−3For the tensile strength, the order of importance (decreasing) wasC, A, D, and B.Thus the optimal regression equation was(23)
y
f
t
=
0.259
+
0.084
a
+
0.161
b
+
0.275
c
+
0.202
d
.


## 4.2.1. Regression Analysis: Density

The density index results are listed in the Appendix (Tables3 and 4). There are four levels (n
=
4) for each factor and four groups of tests for each level. For each group, there are five effective values of the density index. The regression coefficients have been calculated based on formulas (7) and (8).Table 3
Test scheme and density index results.
Test ID
A
B
C
D
Density (g/cm3)
Fine aggregate : coarse aggregates
Barite sand : quartz sand
Anhydrous sodium silicate : aggregates
Anhydrous sodium silicate : sodium fluorosilicate
1
2
3
4
5
∑
y
t
y
-
t
1
L
16
4
5
2.0281
2.0411
2.0299
2.0288
2.0836
10.2116
2.0423
2
2.2802
2.2863
2.3439
2.2448
2.3126
11.4679
2.2936
3
2.2938
2.4721
2.4567
2.4771
2.4946
12.1943
2.4389
4
2.5172
2.5123
2.5198
2.5068
2.5051
12.5612
2.5122
5
2.1153
2.1581
2.2020
2.1711
2.1674
10.8140
2.1628
6
2.0465
2.0597
2.0800
2.0628
2.0517
10.3007
2.0601
7
2.3080
2.3233
2.3008
2.3092
2.3218
11.5631
2.3126
8
2.4797
2.4663
2.4623
2.4667
2.4930
12.3680
2.4736
9
2.1763
2.1450
2.1829
2.1589
2.2079
10.8709
2.1742
10
2.2912
2.2646
2.2476
2.2847
2.2481
11.3362
2.2672
11
2.3146
2.3397
2.3352
2.3891
2.3533
11.7318
2.3464
12
2.5032
2.4217
2.4908
2.4023
2.5109
12.3289
2.4658
13
2.1848
2.1992
2.1698
2.1704
2.1606
10.8847
2.1769
14
2.2467
2.2568
2.2963
2.2503
2.2876
11.3377
2.2675
15
2.1576
2.1905
2.2248
2.2061
2.1538
10.9328
2.1866
16
2.2303
2.2552
2.2518
2.2404
2.2814
11.2734
2.2547
K
1
46.4385
42.7810
43.5175
45.4525
∑
t
=
1
4
∑
j
=
1
4
×
5
y
t
j
=
182.1772
;
∑
y
-
=
36.4365
K
2
45.0570
44.4420
45.5440
46.4525
K
3
46.2680
46.4340
46.7710
45.6180
K
4
44.4285
48.5315
46.3560
44.6655Table 4
Orthogonal polynomial regression analysis of variance: density index.
The source of variance
Square and variation
Degree of freedom
Variance estimate
F-value
F
0.05
(
1
,
69
)
Level of significance
Remarks
b
1
a
5.8057 × 10−2
1
5.8057 × 10−2
28.70
4
∗
∗
A, 1-order term
b
2
a
2.6220 × 10−3
1
2.6220 × 10−3
1.30
A, 2-order term
b
3
a
7.9609 × 10−2
1
7.9609 × 10−2
39.36
∗
∗
A, 3-order term
b
1
b
9.2578 × 10−1
1
9.2578 × 10−1
457.67
∗
∗
∗
∗
B, 1-order term
b
2
b
2.3817 × 10−3
1
2.3817 × 10−3
1.18
B, 2-order term
b
1
c
2.3729 × 10−1
1
2.3729 × 10−1
117.31
∗
∗
∗
C, 1-order term
b
2
c
7.4512 × 10−2
1
7.4512 × 10−2
36.84
∗
∗
C, 2-order term
b
1
d
2.5528 × 10−2
1
2.5528 × 10−2
12.62
∗
D, 1-order term
b
2
d
4.7653 × 10−2
1
4.7653 × 10−2
23.56
∗
∗
D, 2-order term
b
3
d
7.3659 × 10−3
1
7.3659 × 10−3
3.64
D, 3-order term
Error
1.3957 × 10−2
69
2.0228 × 10−3According to (14), the sum of the squares of the variations of the regression coefficients can be calculated (Table 4). Besides, according to (13), other parameters can be calculated as follows:(16)
l
y
y
=
1.600
;
U
=
1.463
;
Q
=
1.377
×
10
-
1
f
r
=
79
;
f
U
=
12
;
f
Q
=
67
.The variance estimates of the error effect can be written asQ
/
f
Q
=
2.055
×
10
-
3. In Table 4, b
3
b and b
3
c are the variance estimates of the regression coefficients which are both less than the error effects, and they can be merged therewith.As shown in Table4, the order of (decreasing) importance of the factors was B, C, A, and D.Regression analysis gives the optimal regression equation as(17)
y
d
=
2.346
+
0.073
a
+
0.096
b
+
0.049
c
-
0.016
d
.


## 4.2.2. Regression Analysis: Young’s Modulus

There were four levels (n
=
4) for each factor and four groups of tests for each level. For each group of tests, there were three effectual pieces of data of the elasticity modulus index (see Appendix, Tables 5 and 6).Table 5
Test scheme and Young’s modulus index results.
Test ID
A
B
C
D
Young’s modulus (MPa)
Fine aggregate : coarse aggregates
Barite sand : quartz sand
Anhydrous sodium silicate : aggregates
Anhydrous sodium silicate : sodium fluorosilicate
1
2
3
∑
y
t
y
-
t
1
L
16
4
5
21.8962
27.963
19.646
69.5052
23.1684
2
61.9473
93.1975
77.4981
232.6428
77.5476
3
120.2802
100.0844
103.2362
323.6007
107.8669
4
159.9495
173.4145
174.1913
507.5553
169.1851
5
22.2968
27.2367
21.1451
70.6785
23.5595
6
23.5181
15.346
15.5944
54.4584
18.1528
7
74.0988
65.8336
72.6667
212.5992
70.8664
8
69.1296
56.8568
98.6768
224.6631
74.8877
9
102.5573
111.2596
68.2513
282.0681
94.0227
10
89.7668
100.6273
123.138
313.5321
104.5107
11
16.2539
8.0914
9.5063
33.8514
11.2838
12
11.1586
11.443
11.9502
34.5519
11.5173
13
37.2454
50.4401
38.6258
126.3114
42.1038
14
19.3454
16.8179
12.8439
49.0071
16.3357
15
30.0612
35.9429
34.9628
100.9671
33.6557
16
10.7866
10.395
10.6829
31.8645
10.6215
K
1
1133.304
548.5632
189.6795
365.6634
∑
t
=
1
4
∑
j
=
1
4
×
3
y
t
j
=
2667.857; ∑
y
-
t
=
889.2856
K
2
562.3992
649.6404
438.8403
617.4687
K
3
664.0035
671.0184
879.339
739.6758
K
4
308.1501
798.6348
1159.998
945.0489Table 6
Orthogonal polynomial regression analysis of variance: Young’s modulus index.
The source of variance
Square and variation
Degree of freedom
Variance estimate
F-value
F
0.05
(
1
,
35
)
Level of significance
Remarks
b
1
a
2.3480 × 104
1
2.3480 × 104
138.39
4.13
∗
∗
∗
∗
A, 1-order term
b
2
a
9.6348 × 102
1
9.6348 × 102
5.68
∗
A, 2-order term
b
3
a
5.3201 × 103
1
5.3201 × 103
31.36
∗
∗
∗
A, 3-order term
b
1
b
2.4806 × 103
1
2.4806 × 103
14.62
∗
∗
B, 1-order term
b
2
b
1.4674 × 101
1
1.4674 × 101
0.09
B, 2-order term
b
3
b
1.4405 × 102
1
1.4405 × 102
0.85
B, 3-order term
b
1
c
4.6801 × 104
1
4.6801 × 104
275.85
∗
∗
∗
∗
C, 1-order term
b
2
c
2.0670 × 101
1
2.0670 × 101
0.12
C, 2-order term
b
3
c
5.1386 × 102
1
5.1386 × 102
3.03
C, 3-order term
b
1
d
1.4421 × 104
1
1.4421 × 104
85.00
∗
∗
∗
∗
D, 1-order term
b
2
d
4.4916 × 101
1
4.4916 × 101
0.26
D, 2-order term
b
3
d
1.8862 × 102
1
1.8862 × 102
1.11
D, 3-order term
Error
5.9382 × 103
35
1.6966 × 10−2To establish the optimal regression equation, it was important to judge the significance level of each factor, and to determine the significance of the regression coefficient, firstly, the regression coefficient of the variation of the squared sum was found, and then anF-test was conducted.According to (13) and (14), the variation of the regression coefficients can be obtained (see Table 6):(18)
l
y
y
=
1.003
×
10
5
;
U
=
9.439
×
10
4
;
Q
=
5.938
×
10
3
;
f
r
=
47
;
f
u
=
12
;
f
Q
=
35
.The results are listed in Table6: the order of importance (decreasing) of the factors was C, A, D, and B.Regression analysis gives the optimal regression equation as(19)
y
E
=
49.980
+
12.391
a
+
6.430
b
+
27.928
c
+
15.503
d
.


## 4.2.3. Regression Analysis: UCS

The UCS results are listed in the Appendix (Tables7 and 8).Table 7
Test scheme and compressive strength index results.
Test ID
A
B
C
D
Compressive strength (MPa)
Fine aggregate : coarse aggregates
Barite sand : quartz sand
Anhydrous sodium silicate : aggregates
Anhydrous sodium silicate : sodium fluorosilicate
1
2
3
4
5
∑
y
t
y
-
t
1
L
16
4
5
0.2949
0.2984
0.2188
0.1995
0.2912
1.3028
0.2606
2
0.5785
0.5832
0.7068
0.5723
0.6511
3.0918
0.6184
3
2.5366
2.7429
2.4855
2.6060
2.7399
13.1110
2.6222
4
4.1403
3.6729
4.7884
4.0876
4.5456
21.2348
4.2470
5
0.5408
0.4813
0.6027
0.4228
0.5529
2.6005
0.5201
6
0.1228
0.1549
0.1706
0.2080
0.1592
0.8155
0.1631
7
1.7678
1.5183
1.5102
1.4143
1.9126
8.1232
1.6246
8
0.9976
1.0144
1.0713
0.8484
0.9594
4.8912
0.9782
9
1.3242
1.1996
1.9182
1.9821
2.0006
8.4247
1.6849
10
2.3129
1.6285
1.8888
2.4849
1.3878
9.7029
1.9406
11
0.2654
0.1876
0.2182
0.2508
0.2150
1.1370
0.2274
12
0.2049
0.1392
0.2293
0.1864
0.2427
1.0026
0.2005
13
0.8433
0.8210
0.7965
0.8615
0.8251
4.1475
0.8295
14
0.3936
0.3880
0.3782
0.3416
0.3835
1.8849
0.3770
15
0.3292
0.4720
0.5058
0.3780
0.4433
2.1284
0.4257
16
0.2533
0.1871
0.2265
0.2256
0.2369
1.1294
0.2259
K
1
38.7404
16.4755
4.3847
12.3135
∑
t
=
1
4
∑
j
=
1
4
×
5
y
t
j
=
84.7282; ∑
y
-
=
16.9457
K
2
16.4304
15.4951
8.8233
13.2675
K
3
20.2672
24.4996
28.3118
26.5438
K
4
9.2902
28.258
43.2084
32.6034Table 8
Orthogonal polynomial regression analysis of variance: compressive strength index.
The source of variance
Square and variation
Degree of freedom
Variance estimate
F-value
F
0.05
1,69
Level of significance
Remarks
b
1
a
1.7856 × 101
1
1.7856 × 101
245.48
4
∗
∗
∗
∗
A, 1-order term
b
2
a
1.6055 × 100
1
1.6055 × 100
22.07
∗
∗
A, 2-order term
b
3
a
4.1944 × 100
1
4.1944 × 100
57.66
∗
∗
A, 3-order term
b
1
b
4.9177 × 100
1
4.9177 × 100
67.61
∗
∗
∗
B, 1-order term
b
2
b
2.8070 × 10−1
1
2.8070 × 10−1
3.86
B, 2-order term
b
3
b
5.7996 × 10−1
1
5.7996 × 10−1
7.97
∗
B, 3-order term
b
1
c
4.6213 × 101
1
4.6213 × 101
635.31
∗
∗
∗
∗
C, 1-order term
b
2
c
1.3671 × 10
1
1.3671 × 10
18.79
∗
∗
C, 2-order term
b
3
c
9.6450 × 10−1
1
9.6450 × 10−1
13.26
∗
∗
C, 3-order term
b
1
d
1.3744 × 101
1
1.3744 × 101
188.95
∗
∗
∗
∗
D, 1-order term
b
2
d
3.2584 × 10−1
1
3.2584 × 10−1
4.48
∗
D, 2-order term
b
3
d
9.5443 × 10−1
1
9.5443 × 10−1
13.12
∗
D, 3-order term
Error
4.8736 × 100
67
7.2740 × 10−2To establish the optimal regression equation, it was important to judge the significance level of each factor, and to determine the significance of the regression coefficient, firstly, the regression coefficient of the variation of the squared sum was evaluated, and then anF-test was carried out.According to (13) and (14), the variation of the regression coefficients can be obtained, as shown in Table 8:(20)
l
y
y
=
97.877
;
U
=
9.300
×
10
1
;
Q
=
4.874
;
f
r
=
79
;
f
u
=
12
;
f
Q
=
67
.The results are listed in Table8: for the compressive strength, the order of importance (decreasing) of the factors was C, A, D, and B.Thus the optimal regression equation was(21)
y
f
c
=
0.639
+
0.277
a
+
0.482
b
+
1.015
c
+
0.519
d
.


## 4.2.4. Regression Analysis: Tensile Strength

Tensile strengths and the regression coefficients are listed in the Appendix, Table9.Table 9
Test scheme and tensile strength index results.
Test ID
A
B
C
D
Tensile strength (MPa)
Fine aggregate : coarse aggregates
Barite sand : quartz sand
Anhydrous sodium silicate : aggregates
Anhydrous sodium silicate : sodium fluorosilicate
1
2
3
4
∑
y
t
y
-
t
1
L
16
4
5
0.0973
0.0976
0.0895
0.0949
0.3793
0.0948
2
0.1949
0.1903
0.1917
0.2078
0.7847
0.1962
3
0.7546
0.6716
0.7655
0.7924
2.9841
0.7460
4
0.9333
0.8623
0.8947
0.8466
3.5370
0.8842
5
0.1382
0.1207
0.1675
0.1755
0.6019
0.1505
6
0.0459
0.0407
0.0492
0.0406
0.1763
0.0441
7
0.6694
0.5585
0.4850
0.5193
2.2322
0.5581
8
0.2910
0.2140
0.2459
0.2622
1.0132
0.2533
9
0.4919
0.5025
0.5248
0.4561
1.9752
0.4938
10
0.6960
0.5942
0.5703
0.5589
2.4193
0.6048
11
0.0945
0.0891
0.0888
0.0932
0.3656
0.0914
12
0.0698
0.0647
0.0597
0.0624
0.2565
0.0641
13
0.2543
0.2770
0.2880
0.2876
1.1069
0.2767
14
0.1137
0.1052
0.1032
0.1177
0.4397
0.1099
15
0.1404
0.1426
0.1388
0.1406
0.5625
0.1406
16
0.0603
0.0680
0.0558
0.0651
0.2492
0.0623
K
1
7.6851
4.0633
1.1704
3.3077
K
2
4.0236
3.82
2.2056
3.2704
K
3
5.0166
6.1444
6.4122
6.2545
K
4
2.3583
5.0559
9.2954
6.251To establish the optimal regression equation and the effect of the factors on the significance of the decision and to determine the significance of the regression coefficient, firstly, the regression coefficient of the variation of the squared sum was evaluated, and then anF-test was carried out. According to (13) and (14), the variation of the regression coefficients can be obtained, as shown in the Appendix, Table 10:(22)
l
y
y
=
4.444
;
U
=
4.362
;
Q
=
8.163
×
10
-
2
f
r
=
63
;
f
U
=
12
;
f
Q
=
51
.Table 10
Orthogonal polynomial regression analysis of variance: tensile strength index.
The source of variance
Square and variation
Degree of freedom
Variance estimate
F-value
F
0.05
1,51
Level of significance
Remarks
b
1
a
7.0194 × 10−1
1
7.0194 × 10−1
438.56
4
∗
∗
∗
∗
A, 1-order term
b
2
a
1.5725 × 10−2
1
1.5725 × 10−2
9.82
∗
A, 2-order term
b
3
a
2.1558 × 10−1
1
2.1558 × 10−1
134.69
∗
∗
∗
A, 3-order term
b
1
b
8.7854 × 10−2
1
8.7854 × 10−2
54.89
∗
∗
B, 1-order term
b
2
b
1.1162 × 10−2
1
1.1162 × 10−2
6.97
∗
B, 2-order term
b
3
b
1.1177 × 10−1
1
1.1177 × 10−1
69.83
∗
∗
B, 3-order term
b
1
c
2.5528 × 100
1
2.5528 × 100
1594.97
∗
∗
∗
∗
∗
C, 1-order term
b
2
c
5.3361 × 10−2
1
5.3361 × 10−2
33.34
∗
∗
C, 2-order term
b
3
c
6.3135 × 10−2
1
6.3135 × 10−2
39.45
∗
∗
C, 3-order term
b
1
d
4.3616 × 10−1
1
4.3616 × 10−1
272.50
∗
∗
∗
∗
D, 1-order term
b
2
d
1.7851 × 10−5
1
1.7851 × 10−5
0.01
D, 2-order term
b
3
d
1.1284 × 10−1
1
1.1284 × 10−1
70.50
∗
∗
D, 3-order term
Error
8.1629 × 10−2
51
1.6006 × 10−3For the tensile strength, the order of importance (decreasing) wasC, A, D, and B.Thus the optimal regression equation was(23)
y
f
t
=
0.259
+
0.084
a
+
0.161
b
+
0.275
c
+
0.202
d
.


## 5. Application and Discussion

This analogue material was applied in a model test that investigated the reinforcement effect on the upper part of the tunnel. In the model, the parameters of analogue material were determined by surrounding rock that was weak or soft rock in the prototype.According to the method of analogue material preparation and the similarity laws for geomechanical models, analogue materials may be prepared by controlling the mechanical parameters of the materials.Based on the scaling criterion, the scale factors for stress, length, and bulk density of a material have a determinate relationship [27]:(24)
α
σ
α
L
α
γ
=
1
.Given the scale factor for bulk densities is unity, the relationship can be transformed to [1](25)
α
σ
=
α
L
.Therefore, the scales of mechanical parameters match the geometrical scale.The geometric similarity ratio can be set based on the relevant scale of the model and the prototype: a value of 100 was chosen here. Therefore, the corresponding mechanical parameters of the model or analogue materials can be calculated according to the similarity law and the parameters of the prototype material.Then the range of each mechanical parameter of the objective material was calculated (Table11). According to the regression equations, the raw material configuration to meet the mechanical index requirements was calculated.Table 11
The range of mechanical parameters of objective and prototype materials.
Density (g/cm3)
Young’s modulus (GPa)
Compressive strength (MPa)
Tensile strength (MPa)
1# prototype material
2.800~2.900
20~35
500~1000
140~150
1# objective material
2.800~2.900
0.20~0.35
5~10
1.4~1.5
1# prototype material
2.100~2.200
3~4
20~40
10~20
2# objective material
2.100~2.200
0.03~0.04
0.2~0.4
0.1~0.2According to the raw material configuration (Table12), two kinds of materials were made within the property ranges listed in Table 11. Therefore, both materials were able to demonstrate the accuracy of the proposed method and its results.Table 12
Raw material configurations and the measured mechanical indices of analogue materials.
Material ID
Raw material configuration
⁢Measured mechanical index
A (the proportion of fine powder to aggregates)
B (the proportion of barite sand to coarse aggregate)
C (the mass ratio of anhydrous sodium silicate to aggregate)
Density(g/cm3)
Young’s modulus(GPa)
Compressive strength (MPa)
Tensile strength(MPa)
1# analogue material
30%
50%
3%
2.855
0.286
6.058
1.469
2# analogue material
65%
00%
3%
2.181
0.035
0.315
0.180
Note: the mass ratio of sodium fluorosilicate to anhydrous sodium silicate was 3 : 4 (this was the optimal value).Subsequently, number 2 analogue material was applied in a geomechanical model test which investigated the antistrike property of reinforcement layer on the top of tunnel. Some photographs of the specimens made of number 2 analogue material have been shown in Figure13; specifically, there are scaling tunnel models with or without reinforcement layers. Both of the two kinds of scaling tunnel models have been subjected to the same impact loads provided by a drop hammer test machine. It is obvious that reinforcement layers can improve the medium resistance on the top of tunnels.Figure 13
Some specimens made of number 2 analogue material.
(a)
Scaling tunnel model without reinforcement layer
(b)
Scaling tunnel model within reinforcement layersSince it is an example of our material used in application, more details and data of the scaling tunnel model tests are not convenient to disclose.Generally, the properties of the developed analogue materials can be predicted according to the proportions. The experiments and applications indicate that it is a type of excellent analogue material which satisfies mechanical scaling criteria for weak rock or soft geological materials, and it will have broad application prospects.


## 6. Conclusions

(1)
A new type of analogue material is developed, which is composed of coarse aggregate (quartz sand and/or barite sand), fine aggregate (barite powder), and cementitious material (anhydrous sodium silicate). It is a type of low-strength and low-modulus material with designable density, which is insensitive to humidity and temperature and satisfies mechanical scaling criteria for weak rock or soft geological materials.
(2)
In order to establish the relationship between parameters and factors, the material properties including density, Young’s modulus, uniaxial compressive strength, and tensile strength were investigated by a series of orthogonal experiments with hundreds of samples. According to the orthogonal experimental method, a four-factor, four-level test scheme is designed for the new material according to the orthogonal tableL
16
(
4
5
).
(3)
The relationship between parameters and factors was obtained. For the density index, the most important factor isB (the proportion of barite sand to coarse aggregate), followed by C (the mass ratio of anhydrous sodium silicate to aggregate), and A (the proportion of fine powder to aggregates), and the effects of D (the mass ratio of sodium fluorosilicate to anhydrous sodium silicate) could be negligible. For the indices of elastic modulus, compressive strength, and tensile strength, the shared characteristic, where the biggest effect is C (the mass ratio of anhydrous sodium silicate to aggregate), followed by A (the proportion of fine powder to aggregates), D (the mass ratio of sodium fluorosilicate to anhydrous sodium silicate), and B (the proportion of barite sand to coarse aggregate), is seen.
(4)
Regression equations of the parameters including density, Young’s modulus, compressive strength, and tensile strength were obtained by using orthogonal polynomial regression analysis. The experiments and applications indicated that the properties of analogue materials were stable and predictable. It was easy to obtain objective material from the regression equations and trial test.



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*Source: 3716145-2016-08-15.xml*