There are many flaws, such as fissures, cavities, and inclusions, in geomaterials, which make their mechanical properties with great randomness and uncertainty. Upon loading, the soil structure gradually losses the bearing capacity due to the transformation from microdefects to macroscopic breakage bands. Based upon the experimental data of frozen sandy soils, a new nonlinear strength equation between the first and third principal stresses was proposed, and then the nonlinear strength properties for frozen sandy soils in
Frozen soils are geotechnical materials, which are mainly formed in cryogenic environment. Frozen soils are defined as those containing some ice and having a temperature at or lower than 0°C [
Currently, a great number of research achievements have been obtained on this field for unfrozen soils and frozen soils. For instance, the mechanical properties of unfrozen soils are obtained with great progress. Matsuoka and Nakai [
At the same time, many research results of the stressstrain curves were obtained, especially in rock damage mechanism, mechanical properties, and strength and deformation characteristics [
Due to the peculiar and complicated characteristics of frozen soils, some strength criteria, such as Mohr–Coulomb, Drucker–Prager, and Hoek–Brown strength criterion, are not suitable for frozen soils. So, the classical strength criterion should be modified or a new strength criterion may be established to investigate the nonlinear strength properties and breakage mechanism for frozen soils. Therefore, in this paper, a relationship between the first and third principal stresses is proposed, and then the nonlinear strength properties of frozen sandy soils are investigated. Furthermore, a new statistical damage constitutive model is also proposed, and the model parameters are determined by cryogenic triaxial compression tests data. Finally, the applicability of the new constitutive model is validated by comparisons between predicted and experimental data.
As is illustrated in Figure
The curves of deviatoric stressaxial strain (a) and volumetric strain (b) of frozen sandy soils (Lai et al. [
The strength envelope in the meridian plane.
In geomaterial engineering, the Mohr–Coulomb criterion is widely used to predict the strength and deformation properties; hence, one of the classical strength criteria can be described as follows:
In the early years, Hoek [
Based upon the experimental data from Figure
The relationship between the first principal stress and the third principal stress of frozen sandy soils.
For convenience, in the
According to experimental results, compared with the classical strength criteria, such as Mohr–Coulomb strength criterion and Hoek–Brown nonlinear criterion, the proposed nonlinear strength results are more close to the experimental data, as is depicted in Figure
Comparisons between predicted results and experimental data.
Figure
Schematic description of Mohr’s circle.
By differentiating (
Due to the independence of
Based on the framework of the envelope theorem by Lai et al. [
Differentiating (
Substituting (
Substituting (
Furthermore, we can easily get the expression of internal frictional angle based upon the triangle relationship shown in Figure
Substituting (
Differentiating (
Finally, substituting (
Based upon experimental data, the nonlinear strength envelope can be obtained from (
Nonlinear strength envelope for frozen sandy soils under various confining pressures. Comparisons between (a) Mohr–Coulomb strength criterion and experimental results, (b) Hoek–Brown strength criterion and experimental results, and (c) the proposed model and experimental results.
The influence of confining pressure on the internal frictional angle is investigated as well, as illustrated in
The evolution law of internal frictional angle.
Similar to unfrozen geomaterials, such as rock, concrete, rockfill material, sand, and clay, the damage process can be mainly accounted for by the same token. That is, the degradation of the material makes the effective areas decrease and the effective stress increase. Accordingly, the intact specimens are easy to damage due to the appearance of fissures, cracks, defects, and shear bands.
Based on the strain equivalent principle [
Based on phenomenological method, basic assumptions by Lai et al. [
Compared with the other probability distribution function [
The yield criterion for frozen sandy soils can be written as
By integrating the microstrength function of (
Substituting (
From (
Based upon the framework of critical state soil mechanics [
Hence, in this paper, based on the research results of Nguyen et al. [
Relationships of stressstrain curves for frozen soils in the meridian plane.
From Figure
Substituting (
As is depicted in Figure
The mean stress
Based upon the assumptions, it is easily accepted that the microstrength of frozen soils satisfies the modified failure state line; so, the strength criterion expression of the effective stress can be obtained as follows:
According to (
Substituting (
The elastic constants
Based upon the generalized Hooke’s law, the expression of axial strain
Hence, the effective mean stress and deviatoric stress can be obtained as follows:
Substituting (
In order to obtain the material parameters, (
Combining (
Based upon triaxial compression test data from Figure
Based upon test data from Figure
Basic physical parameters determined.
Confining pressures, 
Bulk modulus, 
Shear modulus, 

1.0  505.1  440.2 
2.0  1133.1  514.9 
4.0  2104.7  642.2 
5.0  2531.1  694.7 
6.0  3244.2  739.7 
8.0  4116.2  807.6 
10.0  4898.1  845.9 
14.0  5329.2  833.4 
16.0  5455.5  782.7 
In order to describe the changing evolution of the bulk modulus
where
The predicted results of the bulk modulus
According to the cryogenic triaxial compression test data as is illustrated in Figure
Parameter determination of the strength criterion.
Parameter values of frozen sandy soil 
 






1.503  1.494  16.68  0.9479  0.9971 
Comparisons between experimental data and predicted results under different confining pressures.
The values of fitting parameters of (
Parameter determination of the damage constitutive model of (
Confining pressures, 
Parameter determination  


 
1.0  1.1948 
2.4424 
2.0  7.3400 
2.4753 
4.0  4.8047 
5.6572 
5.0  3.5515 
4.6744 
6.0  2.7221 
1.6468 
8.0  2.5790 
1.3385 
10.0  2.0809 
4.0019 
14.0  1.7408 
2.1848 
16.0  1.6381 
4.4111 
The relationship between the parameters
As presented in Figure
The relationship between the parameter
After determining the model parameters, the stress under different confining pressures can be calculated by (
Comparisons between test data and predicted results of deviatoric stressaxial strain under different confining pressures: (a) 1.0–4.0 MPa, (b) 5.0–8.0 MPa, and (c) 10.0–16.0 MPa.
In order to simulate the volumetric strain of frozen sandy soils, we give a damage relationship from the mesolevel, and the expression is shown as follows:
From (
Based on the theory framework of continuum damage mechanics (CDM), the damage variable can also be redefined as
So, the strain tensor of (
Under the triaxial symmetric compression condition, the volumetric strain of (
We make an assumption of
From Tables
Parameter determination of (
Confining pressures, 
Parameter determination  


 
1.0  −105.1  0.4253 
2.0  0.0822  0.8487 
4.0  0.5258  0.5862 
5.0  0.4875  0.7570 
6.0  0.5064  0.7930 
8.0  0.5189  0.8119 
10.0  0.5487  0.8896 
14.0  0.5201  0.9199 
16.0  0.4083  1.059 
Comparisons between test data and predicted results of volumetric strainaxial strain under different confining pressures: (a) 1.0–4.0 MPa, (b) 5.0–8.0 MPa, (c) 10.0 and 16.0 MPa, and (d) 14.0 MPa.
In this paper, a new nonlinear strength criterion is proposed, and a damage statistical constitutive model is also established. The following conclusions can be reached:
The relationship between
The microstrength of frozen sandy soils obeys the Weibull distribution function. The strength criterion, which contains the damage properties of microstrength, is chosen as an independent variable in damage variable
The authors declare that there are no conflicts of interest.
The authors appreciate the funding provided by the CAS Pioneer Hundred Talents Program (Dr. Liu Enlong) and National Science Foundation of China (41771066).