3.1. Database of Uniaxial and True Triaxial Compressive Tests
Referring to
Table 4, it is shown the elastic modulus of plastic concrete under uniaxial compressive stress with a curing age of 90 days and a size of 150 × 150 × 300 mm
3, recording this elastic modulus as
E0. The compressive strength of plastic concrete under uniaxial compressive stress, the compressive strength of plastic concrete under true triaxial compressive stress (
σ3) and the percentage increase of true triaxial compressive strength relative to uniaxial compressive strength with a size of 150 × 150 × 150 mm
3 and a curing age of 540 days are also presented in
Table 4. The confining pressure of true triaxial tests is noted as (
σ1,
σ2). The confining pressure of true triaxial test is noted as (0.2 MPa, 0.4 MPa), (0.4 MPa, 0.6 MPa), (0.4 MPa, 0.8 MPa).
It is presented in
Table 4 and
Figure 2 that the strength of plastic concrete under true triaxial compressive tests is greater than uniaxial compressive tests. The strength of plastic concrete under true triaxial compressive tests increases with confining pressure. In true triaxial compressive tests, lateral strain under uniaxial compressive stress is hindered by the application of confining compressive stress, thereby enhancing the compressive load-bearing capacity of plastic concrete specimens.
The strength of plastic concrete decreases with the increase of water-to-binder ratio, and under different confining pressures and water-to-binder ratios, the triaxial compressive strength of plastic concrete increases by more than 114% compared with uniaxial compressive strength.
The effect of sand-to-total mass ratio on the strength of plastic concrete is not obvious. When the sand-to-total mass ratio is constant, the triaxial compressive strength of plastic concrete basically increases with the increase of confining pressure. When the sand-to-total mass ratio is 0.6, the triaxial compressive strength of plastic concrete increases by 181.85% compared with uniaxial compressive strength.
The true triaxial compressive strength of plastic concrete increases with the increase of cement mass. Under different confining pressures and cement content, the triaxial compressive strength of plastic concrete increases by more than 70% compared with uniaxial compressive strength.
The strength of plastic concrete decreases with the increase of clay mass. Under different confining pressures and clay content, the triaxial compressive strength of plastic concrete increases by more than 95% compared with uniaxial compressive strength.
The true triaxial compressive strength of plastic concrete decreases with the increase of bentonite mass. When bentonite mass is constant, the triaxial compressive strength of plastic concrete basically increases with the increase of confining pressure. Under different confining pressures and bentonite mass, the triaxial compressive strength of plastic concrete increases by more than 95% compared with uniaxial compressive strength.
3.2. Mathematical Model of Constitutive Relationship of Plastic Concrete under Triaxial Compressive Stress
At present, the mathematical model of the constitutive relationship of plastic concrete has mathematical models of linear-elastic, non-linear-elastic, and non-elastic constitutive relations. Plastic concrete is a typical nonlinear material. The mathematical model of linear-elastic constitutive relation cannot describe the constitutive relationship of plastic concrete accurately. The mathematical model of non-elastic constitutive relation is not intuitive in mathematical form, complicated in the derivation process, and not convenient for practical engineering. Therefore, this paper put forward a mathematical model of quartic polynomial conformed to the behavior of plastic concrete under true triaxial compressive stress.
3.2.1. Calculation Formula of Peak Secant Modulus
Researches have been carried out that the elastic modulus of plastic concrete gradually increased with the increase of strength, but had great discreteness. The results of triaxial compressive tests carried out the relationship between secant elastic modulus and peak stress. Their relationship could be expressed by Equation (1). The value of
a1 and
a2 can be calculated by Equations (2) and (3) respectively.
where
Ef is the peak secant elastic modulus of plastic concrete under triaxial pressure (MPa).
is the compressive strength of plastic concrete under true triaxial tests (MPa).
is the confining compressive stress of true triaxial tests (MPa).
According to the empirical formula of elastic modulus under uniaxial compressive stress suggesting by ACI318-08 and GB50010-2002 [
26,
27]. Fitting the experimental data of plastic concrete in this experiment, the empirical formula of peak secant elastic modulus of plastic concrete under true triaxial compressive stress was obtained as Equations (4) and (7):
Parameters
b1 and
b2 can be calculated according to Equations (5) and (6).
Parameters
c1 and
c2 can be calculated according to Equations (8) and (9).
The average values of the ratio of calculated values to test values of Equations (1), (4) and (7) are shown in
Table 5.
From the results shown in
Table 4, the average values of the ratio of calculated value to test the value of Equations (1), (4) and (7) are 1.0604, 1.0624 and 0.9992 respectively, so the fitting effect of Equation (7) is better than Equations (1) and (4). The peak secant modulus calculation formula adopted Equation (7).
3.2.2. Mathematical Model of Constitutive Relation of Plastic Concrete under True Triaxial Compressive stress
According to data obtained from the true triaxial compressive stress tests of plastic concrete, a mathematical model of the quartic polynomial was established, which agreed well with the measured values.
The meanings of
x and
y in Equations (10) and (11) are showing in Equations (12) and (13).
where
is the peak compressive strain.
Both sides of Equation (10) differentiated
simultaneously, getting Equation (15).
The finishing equation is available shown as Equation (16).
Substituting
(
) into Equation (16) can get Equation (17).
where
E0′ is the initial tangent elastic modulus of plastic concrete under true triaxial compressive stress. The expression of
E0′ is shown in Equation (18).
It can be seen from Equation (17) that the coefficient D of the model is related to E0′/Ef. By analogy, the coefficients A, B, C, D are related to E0′/Ef. Because the initial tangent elastic modulus E0′ under triaxial compressive stress is difficult to measure in statistical analysis. Therefore, E0′ is replaced by elastic modulus (E0) measured under uniaxial compressive stress with a curing age of 90d.
Peak secant modulus
Ef calculated by Equation (7). The equation of each plastic concrete specimen is obtained by fitting the measured stress-strain curve, thus 33
A,
B,
C,
D values are obtained. The expression of
A,
B,
C,
D can be obtained. The equations are shown in Equations (19)–(22).
where
E0 is the uniaxial compressive elastic modulus of plastic concrete with a curing age of 90 d.
3.2.3. Comparison of Curves Obtained from Theoretical Model and Experiment Measured
The peak compressive strength obtained from true triaxial compressive tests under three confining stress is compared with the peak compressive strength obtained from Equations (10) and (11) to verify the reliability of the mathematical model. The results of peak compressive stress and the comparisons of calculated results and test results are shown in
Table 6 and
Figure 3,
Figure 4 and
Figure 5.
It can be seen from
Table 6,
Figure 3,
Figure 4 and
Figure 5, the peak compressive strength obtained from the theoretical model is in good agreement with the compressive stress obtained from experimental tests. The ensemble average value of the ratio of
σ3/
σ3f is 1.0037, the standard deviation is 0.0031 and the coefficient of variation is 0.0030, which fully verifies the reliability of the established theoretical model of the quartic polynomial.
3.3. Failure Criteria of Plastic Concrete under True Triaxial Compressive Test
The plastic concrete failure criteria are processing a large number of triaxial test data of plastic concrete, drawing the failure envelope surface in principal stress space, and finding the appropriate mathematical expression according to the geometric characteristics of the envelope surface.
The strength in the triaxial space of each plastic concrete specimens (f1, f2, f3) obtained from the experiment is calibrated to the principal stress coordinate space (σ1, σ2, σ3), then connect the adjacent points with smooth curved surface to obtain failure envelop of plastic concrete.
There is a hydrostatic pressure axis in the coordinate space of principal stress, and the stress of each point on the axis is
σ1=
σ2=
σ3. The distance between the point of the hydrostatic pressure axis and the origin of coordinates is hydrostatic pressure, with a value of
. The angle between each principal stress axis and hydrostatic pressure axis is
. The deviatoric plane is perpendicular to the hydrostatic axis. The first principal stress invariant (
I1) is a constant which is the sum of the three principal stresses at each point on the same deflection plane [
28].
Based on 33 test points of plastic concrete measured by true triaxial compressive stress tests and the general equation of failure criteria of octahedral space for ordinary concrete proposed by reference [
29,
30], this paper proposes a failure criterion in the form of the quadratic polynomial with the dimensionless expression of stress in octahedral space for plastic concrete. Its general expression is shown as Equation (24).
The parameters in Equation (24) calculate according to the following equations.
where
is the normal stress of octahedral plastic concrete.
is the uniaxial compressive strength of plastic concrete at an age of 540 days.
θ is the included angle of the offset plane.
is octahedral plastic concrete shear stress.
The values of parameters a, b(bt, bc), c can be determined by experiment.
Define
as the relative normal stress of the octahedral plastic concrete,
as the relative shear stress of the octahedral plastic concrete, the expressions of
γ and
χ are in Equations (29) and (30).
Equation (31) is the general expression of Equation (24).
In the experiment of plastic concrete under true triaxial pressure, octahedral normal stress, shear stress and Lode’s angle in the stress space are shown in
Table 7,
Table 8 and
Table 9 when three confining pressures are (0.2q, 0.4q), (0.4q, 0.6q), (0.4q, 0.8q) respectively.
Based on the data of true triaxial compressive tests, the general equation of failure criteria has been studied as the expression of Equation (32).
A comparison between the measured values and the calculated values of Equation (32) is shown in
Figure 6. The average value of the ratio of the calculated value to the experimental value is 1.002. The coefficient of variation is 0.0444. The standard deviation is 0.0445. It can be seen that the suggested failure criteria equation is in good agreement with the experimental data.