Data processing. The calculation formula of axial strain ε1 is :
$${\varepsilon _1}{\text{=}}\frac{{\Delta h}}{{{h_0}}} \times 100$$
1
Where ∆h is axial deformation, unit is mm ; h0 is the height of the sample, and the unit is mm.
During the test, the sample will produce transverse deformation and the cross-sectional area will change. Therefore, the average method is adopted for the sample area when processing the data, that is, the cross-sectional area of the sample is approximately replaced by the corrected cross-sectional area of the equal volume cylinder. The correction formula of the corrected cross-sectional area Aa of the specimen is as follows :
$${A_{\text{a}}}{\text{=}}\frac{{{A_c}}}{{1 - 0.01{\varepsilon _1}}}=\frac{{{V_0} - \Delta V}}{{\left( {{h_0} - \Delta h} \right)\left( {1 - 0.01{\varepsilon _1}} \right)}}$$
2
Where Ac is the cross-sectional area of the sample before the test, the unit is mm2 ; V0 is the volume of the sample before the test, the unit is mm3 ; ∆V is the volume change in mm3.
The axial deviatoric stress ( σ1-σ3 ) is calculated according to the corrected area, and the calculation formula is :
$${\sigma _1}{\text{-}}{\sigma _3}{\text{=}}\Delta \sigma {\text{=}}\frac{F}{{{A_{\text{a}}}}}$$
3
Where F is the axial pressure, the unit is N.
Duncan-Chang constitutive model is used to fit the stress-strain relationship of triaxial shear test by hyperbola, that is:
$$q={\sigma _1} - {\sigma _3}=\frac{{{\varepsilon _{\text{a}}}}}{{a+b{\varepsilon _{\text{a}}}}}$$
4
Where a and b are test parameters. For the conventional triaxial shear test, ε1 / ( σ1-σ3 ) is approximately linear with ε1, and Eq. ( 5 ) can be expressed as:
$$\frac{{{\varepsilon _1}}}{{{\sigma _1} - {\sigma _3}}}=a+b{\varepsilon _1}$$
5
Because dσ2=dσ3=0 in the conventional triaxial shear test, the tangent modulus Et can be reduced to :
$${E_{\text{t}}}=\frac{{{\text{d}}({\sigma _1} - {\sigma _3})}}{{{\text{d}}{\varepsilon _1}}}=\frac{a}{{{{(a+b{\varepsilon _1})}^2}}}$$
6
When ε1 = 0, the initial tangent modulus Ei is Et, then:
$${E_{\text{i}}}=\frac{1}{a}$$
7
If ε1→∞, then the limit deviatoric stress ( σ1-σ3 )ult is:
$${({\sigma _1} - {\sigma _3})_{{\text{ult}}}}=\frac{1}{b}$$
8
In fact, the axial strain ε1 is not infinite. In this paper, the peak deviatoric stress ( σ1-σ3 )f of the specimen is determined according to ε1 reaching 15%, so there is ( σ1-σ3 )f༜( σ1-σ3 )ult, and the failure ratio Rf can be defined as:
$${R_{\text{f}}}=\frac{{{{({\sigma _1} - {\sigma _3})}_{\text{f}}}}}{{{{({\sigma _1} - {\sigma _3})}_{{\text{ult}}}}}}$$
9
Thus, Et can be expressed as a function of stress.
$${E_{\text{t}}}={E_{\text{i}}}{\left[ {1 - {R_{\text{f}}}\frac{{({\sigma _1} - {\sigma _3})}}{{{{({\sigma _1} - {\sigma _3})}_{\text{f}}}}}} \right]^2}$$
10
Table 3
Triaxial compression test scheme.
Sample number | Freezing and thawing times (time) | Moisture content(%) | Confining pressure(kPa) | Freezing temperature(℃) |
D-1 | 0 | 10 | 100 | -10 |
D-2 | 1 |
D-3 | 3 |
D-4 | 5 |
D-5 | 7 |
H-1 | 1 | 6 | 100 | -10 |
H-2 | 8 |
H-3 | 10 |
H-4 | 12 |
W-1 | 1 | 10 | 50 | -10 |
W-2 | 100 |
W-3 | 150 |
D-1 | 1 | 10 | 100 | -5 |
D-2 | -10 |
D3 | -15 |
Changes in the appearance of the sample after loading.
The shape changes of the samples after loading are similar, and the representative samples are selected, as shown in Fig. 4. By analyzing the appearance of the sample, it can be seen that when the strain is loaded to 15%, the smoothness of the side wall of the sample becomes worse, and the middle of the side wall becomes concave and convex. The middle and lower parts of the sample swell obviously, the transverse deformation of the middle part is the largest, the transverse deformation of the top and bottom is small, and the whole sample is drum-shaped. The shear band on the side wall of the sample is not obvious. Cracks can be seen in the upper and middle parts of the sample ; the cracks are distributed obliquely, the angle is greater than 45 °, and the distribution direction is different. Some are distributed in one direction, and some are distributed in two directions, showing ' V ' type and '∧' type.
The effect of freeze-thaw cycles on mechanical properties.
Figure 5 is the stress-strain curve of the thawed subgrade soil after different freeze-thaw cycles ( 0,1,3,5,7 times ) with a water content of 10%, a confining pressure of 100 kPa, and a freezing temperature of-10°C. It can be seen from the figure that with the increase of freeze-thaw cycles, the stress-strain relationship curves of subgrade soil are basically the same, all of which are hardening curves. The strength of subgrade soil gradually decreases with the increase of freeze-thaw cycles. In the initial stage of loading, when the axial strain is less than 0.3%, the curve shape is approximately straight line, indicating that the axial deviatoric stress of the sample increases sharply and linearly, and the sample is undergoing elastic deformation.
For the strain hardening curve, the axial deviatoric stress failure stress corresponding to the axial strain of 15% is taken. Under the condition of freeze-thaw cycle, the change law of failure stress and elastic modulus of subgrade soil is shown in Fig. 6. From the diagram, it can be concluded that with the increase of the number of freeze-thaw cycles, the failure stress of subgrade soil shows an approximate exponential downward trend. After 7 freeze-thaw cycles, the failure stress of subgrade soil decreased from 321.7 kPa to 289.9 kPa, with a total decrease of 9.9%. Among them, after the first freeze-thaw cycle, the failure stress decreases the most, up to 5.5%, which indicates that the effect of the first freeze-thaw cycle is the most significant. In general, the influence of the first three freeze-thaw cycles on the failure stress is more obvious, and the influence of the third − 7 freeze-thaw cycles on the failure stress is smaller. Especially after the seventh freeze-thaw cycle, the failure stress decreased by only 0.3%, which is basically stable.
According to the characteristics of the stress-strain relationship curve in Fig. 5, the ratio of the axial deviatoric stress increment corresponding to 0.3% axial strain to the axial strain increment is used as the elastic modulus of subgrade soil, that is :
$$E{\text{=}}\frac{{\Delta {\sigma _{0.3\% }}}}{{\Delta {\varepsilon _{0.3\% }}}}$$
11
It can be seen from Fig. 6 that the elastic modulus of subgrade soil decreases exponentially with the increase of freeze-thaw cycles, and the attenuation rate gradually slows down. After 7 freeze-thaw cycles, the elastic modulus of subgrade soil decreased from 0.832 MPa to 0.666 MPa, and the attenuation rate reached 20.0%. Among them, the first freeze-thaw cycle has the greatest disturbance to the sample, and the elastic modulus of the subgrade soil is reduced by up to 8.0%. After the seventh freeze-thaw cycle, the elastic modulus decreased by only 1.9%.
The strength of soil will decrease under freeze-thaw action. The main reason is that the pore volume in soil increases due to the freezing of water in soil. When the pore ice melts, the larger pores cannot be restored to the original state, so the soil becomes loose, the density decreases, and the bonding force between soil particles also decreases. After multiple freeze-thaw cycles, the pore volume increases to the limit, so the soil strength will eventually stabilize.
The data of triaxial shear test are sorted out according to the function form of ε1/(σ1-σ3) and ε1. The results show that there are deviations in the initial stage test points. Therefore, according to the principle of curve fitting method, some test points with large deviations are adjusted, and the relationship curve between ε1/(σ1-σ3) and ε1 under different freeze-thaw cycles is obtained, as shown in Fig. 7.
The values of test parameters a and b in the model can be obtained by linear fitting of each curve in Fig. 7.As shown in Table 4. It can be seen that the coefficient of determination R2 > 0.98, the degree of fitting is high.
Table 4
Parameters under different freeze-thaw cycles.
Freezing and thawing times (time) | a | b | Adjusted R square |
0 | 0.96465 | 2.74137 | 0.98142 |
1 | 1.07915 | 2.98762 | 0.99929 |
3 | 1.10352 | 3.09856 | 0.99643 |
5 | 1.11787 | 3.16806 | 0.99731 |
7 | 1.13136 | 3.2354 | 0.99144 |
The reciprocals of parameters a and b are the initial tangent modulus and the ultimate deviator stress, which are related to the number of freeze-thaw cycles, as shown in Fig. 8. It can be seen that after the first freeze-thaw cycle, the parameter 1 / a is reduced by 11.0%, and the parameter 1 / b is reduced by 8.2%. After the third freeze-thaw cycle, the parameter 1 / a decreased by 2.0%, and the parameter 1 / b decreased by 3.3% ; after the fifth freeze-thaw cycle, the parameter 1 / a decreased by 1.2%, and the parameter 1 / b decreased by 1.9%. After the seventh freeze-thaw cycle, the parameter 1 / a decreased by 1.1%, and the parameter 1 / b decreased by 1.8%. The above results show that with the increase of the number of freeze-thaw cycles, the initial tangent modulus of the soil gradually decreases, its growth rate gradually decreases, the ultimate deviation stress of the soil gradually decreases, and its reduction rate gradually slows down.
According to formula ( 9 ), the failure ratio of soil is the product of failure stress and b value. The failure ratio Rf of subgrade soil under different freeze-thaw cycles is calculated by calculating the test data, and the results are shown in Table 5.
Table 5
Damage ratio of soil under different freeze-thaw cycles.
Freezing and thawing times (time) | Rf |
0 | 0.88177 |
1 | 0.90806 |
3 | 0.91116 |
5 | 0.92186 |
7 | 0.93801 |
The analysis and calculation results show that with the increase of freeze-thaw cycles, the failure ratio Rf of soil increases gradually, and the increase speed is relatively slow. After 7 freeze-thaw cycles, the damage of soil increased by 6.4% compared with Rf. According to the formula ( 9 ), the larger the failure ratio Rf means that the strength is closer to the ultimate deviation stress. It can be seen that the more the number of freeze-thaw cycles, the closer the strength of the melted soil is to the potential ultimate strength.
Effect of freezing temperature on mechanical properties.
Figure 9 is the stress-strain curve of thawed subgrade soil under different freezing temperatures ( -5°C, -10°C, -15°C ) after one freeze-thaw cycle, with a water content of 10% and a confining pressure of 100 kPa. It can be seen from the figure that as the freezing temperature decreases, the stress-strain relationship curves of the subgrade soil are basically the same, all of which are hardening curves. As the axial strain increases, the axial deviatoric stress gradually increases, and there is no peak deviatoric stress. And with the decrease of freezing temperature, the strength change is not obvious, it can be seen that the influence of freezing temperature is weak.
Taking the axial deviatoric stress corresponding to 15% axial strain as the failure stress, the influence of freezing temperature on the failure stress and elastic modulus of subgrade soil is shown in Fig. 10. It can be seen from the diagram that with the decrease of freezing temperature, the failure stress of subgrade soil shows a slightly decreasing trend. When the freezing temperature is reduced from-5°C to-15°C, the failure stress of subgrade soil is reduced from 303.9 kPa to 290.1 kPa, which is reduced by 13.8 kPa, and the reduction is about 4.5%. The elastic modulus of subgrade soil decreases approximately linearly with the decrease of freezing temperature. When the freezing temperature decreases from-5°C to-15°C, the elastic modulus of subgrade soil decreases from 0.765 MPa to 0.753 MPa, which decreases by 0.012 MPa, with a decrease of about 1.6%. By analyzing the test results, it can be concluded that the freezing temperature has a weak effect on the failure stress and elastic modulus of subgrade soil.
The relationship between ε1/(σ1-σ3) and ε1 under different freezing temperatures. As shown in Fig. 11.
The values of test parameters a and b in the model can be obtained by linear fitting of each curve in Fig. 11. As shown in Table 6. It can be seen from the table that the determination coefficient R2 > 0.98, and the fitting degree is high.
Table 6
Parameters under different loading rate.
Freezing temperature(℃) | a | b | Adjusted R square |
-5 | 1.07915 | 2.98762 | 0.99929 |
-10 | 1.17915 | 3.04751 | 0.99863 |
-15 | 1.21915 | 3.13235 | 0.98912 |
The relationship between initial tangent modulus, ultimate deviation stress and freezing temperature is obtained by taking the reciprocal of parameters a and b, as shown in Fig. 12. From the diagram, it can be seen that with the decrease of freezing temperature, the initial tangent modulus of soil gradually decreases, and the reduction rate increases. The freezing temperature is reduced from-5°C to-15°C, the parameter 1/a is reduced by 11.5%, and the limit deviation stress 1/b of the soil is approximately linearly reduced, and the value is reduced by about 4.6%.
Further calculation, the failure ratio Rf of soil at different freezing temperatures is shown in Table 7. It can be seen that with the increase of freezing temperature, the failure ratio Rf of soil fluctuates between 0.90 and 0.91, and the change trend is not obvious, which indicates that the freezing temperature has little effect on the strength utilization rate of thawed soil.
Table 7
Damage ratio of soil under different loading rate.
Freezing temperature(℃) | Rf |
-5 | 0.90803 |
-10 | 0.89539 |
-15 | 0.90869 |
The influence of water content on mechanical properties.
Figure 13 is the stress-strain curve of the thawed subgrade soil under the conditions of confining pressure of 100 kPa, freezing temperature of-10°C and different water content ( 6%, 8%, 10%, 12% ) after one freeze-thaw cycle. It can be seen from the figure that with the increase of water content, the softening curve gradually changes into the hardening curve, and the strength of subgrade soil decreases obviously with the increase of water content. When the water content is 6%, the stress-strain curve of subgrade soil shows the characteristics of softening curve, which is a brittle failure form. When the water content reaches 8% and above, the stress − strain curve shows the characteristics of hardening curve.
The influence of water content on the failure stress of subgrade soil is shown in Fig. 14. It can be seen from the diagram that the failure stress of subgrade soil decreases with the increase of water content, from 405.43 kPa to 288.4 kPa, which is reduced by 28.9%. When the water content increases from 6–8%, the failure stress decreases the most, with a decrease of 19.1%. When the water content increases from 8–12%, the failure stress decreases approximately linearly. For every 1% increase in water content, the failure stress decreases by 2.4% on average. The elastic modulus of subgrade soil decreases approximately linearly with the increase of water content. When the water content increases from 6–12%, the attenuation of elastic modulus reaches 50.4%. For every 1% increase in average water content, the elastic modulus decreases by 8.4%. From the above analysis of the test results, it can be concluded that the water content has a strong influence on the failure stress and elastic modulus of the subgrade soil.
The relationship between ε1/(σ1-σ3) and ε1 under different water content. As shown in Fig. 15.
The values of test parameters a and b in the model can be obtained by linear fitting of each curve in Fig. 15. As shown in Table 8. It can be seen from the table that the determination coefficient R2 > 0.98, and the fitting degree is high.
Table 8
Parameters under different water content.
Moisture content(%) | a | b | Adjusted R square |
6 | 0.88516 | 2.04746 | 0.98135 |
8 | 1.00462 | 2.72571 | 0.99972 |
10 | 1.07915 | 2.98762 | 0.99929 |
12 | 1.20416 | 3.14859 | 0.99961 |
The relationship between initial tangent modulus, ultimate deviation stress and water content is obtained by calculating 1 / a and 1 / b, as shown in Fig. 16. It can be seen from the figure that the water content increases from 6–8%, the parameter 1 / a is reduced by 11.9%, and the parameter 1 / b is reduced by 24.9%. The water content increased from 8–10%, the parameter 1 / a decreased by 6.1%, and the parameter 1 / b decreased by 5.4%. The water content increased from 10–12%, the parameter 1 / a decreased by 8.5%, and the parameter 1 / b decreased by 4.7%. The above results show that with the increase of water content, the initial tangent modulus of soil decreases gradually, the decreasing rate decreases gradually, the ultimate deviation stress of soil decreases gradually, and the decreasing rate slows down gradually.
The failure ratio Rf of soil under different water content is calculated as shown in Table 9. It can be seen from the table that with the increase of water content, the damage ratio Rf of soil gradually increases, and the growth rate is relatively slow. When the water content increases from 6–12%, the damage of soil increases by 9.3% compared with Rf. This shows that the increase of water content makes the strength of thawed soil gradually close to its strength limit value, and the utilization rate of strength increases.
Table 9
Damage ratio of soil under different water content.
Moisture content(%) | Rf |
6 | 0.83039 |
8 | 0.89376 |
10 | 0.90803 |
12 | 0.90796 |
Effect of confining pressure on mechanical properties.
Figure 17 is the stress-strain curve of the thawed subgrade soil under different confining pressures ( 50 kPa, 100 kPa, 150 kPa ) after experiencing a freeze-thaw cycle, with a water content of 10%, a freezing temperature of-10°C. It can be seen from the figure that with the increase of confining pressure, the stress-strain relationship curves of subgrade soil are basically the same, all of which are hardening curves. With the increase of axial strain, the axial deviatoric stress also increases gradually, and there is no peak deviatoric stress. With the increase of confining pressure, the strength of subgrade soil increases greatly, and the influence of confining pressure is obvious.
The influence of confining pressure on the failure stress and elastic modulus of subgrade soil is shown in Fig. 18.It can be seen from the figure that with the increase of confining pressure, the failure stress of subgrade soil shows an approximately linear growth trend. When the confining pressure increases from 50 kPa to 150 kPa, the failure stress of subgrade soil increases from 194.7 kPa to 367.7 kPa, an increase of 173 kPa, an increase of about 88.8%. The elastic modulus of subgrade soil increases with the increase of confining pressure. When the confining pressure increases from 50 kPa to 150 kPa, the elastic modulus of subgrade soil increases from 0.366 MPa to 0.931 MPa, an increase of 0.564 MPa, an increase of about 154.1%.
From the analysis of the above test results, it can be concluded that confining pressure has a great influence on the failure stress and elastic modulus of subgrade soil. This is because the freeze-thaw action increases the pore volume inside the soil and the soil becomes loose. Therefore, with the increase of confining pressure, the subgrade soil is compacted and the pores are reduced, so the soil strength increases. At the same time, the increase of confining pressure makes the consolidation degree of subgrade soil increase. Therefore, the ability of subgrade soil to resist deformation is enhanced.
Figure 19 is the relationship curve between ε1/(σ1-σ3) and ε1 under different water content conditions.The values of test parameters a and b in the model can be obtained by linear fitting of each curve in Fig. 19. As shown in Table 10. It can be seen from the table that the determination coefficient R2 > 0.99, and the fitting degree is high.
Table 10
Parameters under different confining pressure.
Confining pressure(kPa) | a | b | Adjusted R square |
50 | 1.68317 | 4.48731 | 0.99814 |
100 | 1.07915 | 2.98762 | 0.99929 |
150 | 0.8669 | 2.54143 | 0.99934 |
By calculating 1 / a and 1 / b, the relationship between initial tangent modulus, limit deviation stress and confining pressure is obtained, as shown in Fig. 20. It can be seen from the figure that the confining pressure increases from 50 kPa to 100 kPa, the parameter 1 / a increases by 56.0%, and the parameter 1 / b increases by 50.2%. When the confining pressure increases from 100 kPa to 150 kPa, the parameter 1 / a increases by 38.2%, and the parameter 1 / b increases by 26.4%. The above results show that with the increase of confining pressure, the initial tangent modulus of soil increases approximately linearly, the ultimate deviatoric stress of soil increases gradually, and the growth rate gradually slows down. The failure ratio Rf of soil under different confining pressures is calculated, as shown in Table 11.
Table 11
Damage ratio of soil under different confining pressure.
Confining pressure(kPa) | Rf |
50 | 0.87404 |
100 | 0.90803 |
150 | 0.93441 |
The analysis results show that with the increase of confining pressure, the failure ratio Rf of soil increases gradually, and the growth rate is relatively slow. When the confining pressure increases from 50 kPa to 150 kPa, the failure ratio of soil increases by 6.9% compared with Rf. This shows that the increase of confining pressure makes the strength of thawed soil gradually close to its strength limit value, and the utilization rate of strength is improved.
The analysis results show that with the increase of confining pressure, the failure ratio Rf of soil increases gradually, and the growth rate is relatively slow. When the confining pressure increases from 50 kPa to 150 kPa, the failure ratio of soil increases by 6.9% compared with Rf. This shows that the increase of confining pressure makes the strength of thawed soil gradually close to its strength limit value, and the utilization rate of strength is improved.