Although frozen soil is in nature the discrete material, it is generally treated as the continuum material. The mechanical properties of frozen soil are so complex to describe adequately by conventional continuum mechanics method. In this study, the nonlinear microcontact model incorporating rolling resistance is proposed to investigate the particlescale mechanical properties of frozen soil. The failure mechanism of frozen soil is explicated based on the evolution of contact force chains and propagation of microcracks. In addition, the effects of contact stiffness ratio and friction coefficient on stressstrain curve and energy evolution are evaluated. The results show that the nonlinear microcontact model incorporating rolling resistance can better describe the experimental data. At a higher axial strain, the contact force chains near shear band which can give rise to the soil arch effect rotate away from the shear band inclination but not so much as to become perpendicular to it. The propagation of microcracks can be divided into two phases. The stressstrain curve is strongly influenced by contact stiffness ratio. In addition, friction coefficient does not significantly affect the initial tangential modulus. Compared with frictional coefficient, the effect of contact stiffness ratio on stressstrain curve and energy evolution is greater.
In the past 50 years, a large number of engineering constructions in cold regions and artificial ground freezing projects have been built throughout Europe and East Asia [
Although the continuum mechanics method plays an important role and is widely used, the mechanical properties of frozen soil are so complex that the discrete features of frozen soil should not be neglected. The Discrete Element Method (DEM) is regarded as a powerful tool to investigate the particlescale mechanical properties of frozen soil due to the fact that DEM can control the complex responses of an assembly of discrete materials by very simple contact laws [
It should be pointed out that although few comprehensive studies on DEM investigation of particlescale mechanical properties of frozen soil have been conducted, a lot of attempts on particlescale mechanical properties of unfrozen soil have been done. Zhao et al. [
The main objectives of this paper are to (1) present the nonlinear microcontact model incorporating rolling resistance for frozen soil; (2) explicate the failure mechanism of frozen soil based on the evolution of contact force chains and propagation of microcracks; (3) evaluate the effects of contact stiffness ratio and friction coefficient on stressstrain curve and energy evolution.
The microcontact model in this study is similar to that proposed by Jiang et al. [
Microcontact model for frozen soil.
The basic mechanical elements are spring, bond, slider, divider, dashpot, and roller. The elastic relationship between contact force and relative displacement is reflected by spring. The cementation at contact is reflected by bond. The frictional sliding is reflected by slider. The divider reflects that the tensile force between particles is zero once the bonds are broken. The energy dissipation is reflected by dashpot. The roller reflects the rolling resistance when a particle rotates against another particle through a small angle.
The normal contact force, tangential contact force, and rolling contact moment are calculated as follows.
Figure
Mechanical response of normal contact model.
Before the bond breaks, the normal contact force under tension can be also described by (
The mechanical response of tangential contact model is presented in Figure
Mechanical response of tangential contact model.
Before the bond breaks, the tangential contact force can be expressed by
Once the bond breaks, the two contacting particles may slip relative to one another and the tangential contact force under tension abruptly reduces from tangential bonding strength
The rolling contact moment can be divided into two parts: the rolling contact moment before the bond breaks and that after the bond breaks.
For the conventional microcontact model which is widely used to investigate the particlescale mechanical properties of granular materials, two adjacent particles are in contact at a discrete point. The ESEM micrograph of bonded granular materials shows that two adjacent particles are in contact over a width [
Small relative rotation before the bond breaks.
Since
Contact forces at the lefthand side and righthand side before bond breaks.
Figure
Small relative rotation after bond breaks.
The rolling contact moment after bond breaks can be divided into two parts: the rolling contact moment purely due to bond
Maximum and minimum contact forces.
Since
Let
The total rolling contact moment is a linear summation of the rolling contact moment before bond breaks and that after bond breaks for simplicity. So the total rolling contact moment can be calculated as follows:
Equation (
Mechanical response of rolling contact model.
For the calibration of microcontact model for frozen soil, the DEM sample is generated according to the test conditions of our previous laboratory experiment on the mechanical characteristics of frozen soil in [
The microcontact model for frozen soil is implemented in the twodimensional particle flow code (PFC2D) provided by Itasca Consulting Group. The processes of DEM sample generation are performed as follows. (1) Soil particles are randomly generated inside the rectangular frictionless boundaries which are simulated by four rigid walls. The system then arrives at the equilibrium state by using an equilibrium ratio limit. Note that the grain size distribution curve of soil in laboratory tests is adopted as that in numerical tests and the assembly of soil particles has a width of 39.1 mm and height of 80 mm. (2) The DEM sample is subsequently subjected to an isotropic constant confining pressure. In this stage, a slight change in sample height which can be neglected may occur. (3) A few “floating” soil particles are eliminated to obtain a denser network of contact, as shown in Figure
DEM sample.
It is well known that not only the internal structure of frozen soil changes constantly but also the physical and mechanical properties of ice are not constant during the loading process. The strain is considered as an important index reflecting the dynamic process. It is reasonable to assume that the normal bonding strength
A series of triaxial compression tests on the static mechanical properties of frozen soil were performed to provide the experimental data for model calibration. The laboratory tests are introduced briefly here in order to explain the numerical results further. The soil used in laboratory tests is Fujian standard sand. The grain size distribution curve of Fujian standard sand is presented in Figure
Other physical indices of Fujian standard sand.
Maximum dry density  Minimum dry density  Curvature coefficient  Nonuniformity coefficient 

1.74 g/cm^{3}  1.43 g/cm^{3}  1.014  1.543 
Grain size distribution curve of Fujian standard sand.
The test device is an electrichydraulic servocontrolled triaxial material testing machine which is equipped with an automatic numerical control system and a data collection system, as shown in Figure
The triaxial compression test conditions of specimen DTJwd2
Specimen  Temperature  Loading rate  Confining pressure 

DTJwd2 

1.072 mm/min  0.2 MPa 
The static triaxial material testing machine.
Model calibration.
Due to the restrictions of experiment equipment, the microparameters such as bonding strength cannot be obtained easily even by the advanced technologies such as Xray, the stereophotogrammetric technique and particle image velocimetry. The “trial and error” is considered as the best way to determine the microparameters at present. The microparameters such as normal bonding strength in this paper are determined by “trial and error.” In this study, the normal bonding strength in linear model is 10 kN and the relationship between
Other microparameters of nonlinear and linear models are listed in Table
Other microparameters of nonlinear and linear models.
Normal contact stiffness  Tangential contact stiffness  Tangential bonding strength  Friction coefficient 

620 MN/m  620 MN/m  10 kN  0.3 
Before the deviatoric stress reaches its peak value, for the nonlinear and linear models, the specimens exhibit strainsoftening behavior which can be found from the laboratory test results and the stress values are close to the experimental data for a certain strain. Hence, the nonlinear and linear models can better describe the laboratory data before the deviatoric stress reaches its peak value. However, it is obvious that the stress value of laboratory test for a certain strain is greater than that of linear microcontact model and is close to that of nonlinear microcontact model after the deviatoric stress reaches its peak value. The conclusion that the simulated results of nonlinear microcontact model show a good agreement with the test data is achieved. The reason is that the normal bonding strength does not vary greatly before the deviatoric stress reaches its peak value, as shown in Figure
Relationship between normal bonding strength and strain.
The evolution of contact force chains and propagation of microcracks can well account for the failure mechanism of frozen soil since the evolution of contact force chains can illustrate how the applied loads are transmitted within frozen soil and the formation of shear bands in frozen soil is associated with the propagation of microcracks.
The contact force chains can be divided into two parts: the strong and weak contact force chains. Almost all the applied loads transmit along the strong contact force chains. Figures
Evolution of contact force chains.
It is common knowledge that there are two kinds of microcracks in frozen soil: original cracks and microcracks caused by the breakage of bond (ice). In this paper, the original cracks are not taken into account for simplicity. Figures
Propagation of microcracks.
At the initial state
Figure
Number of microcracks at various axial strains.
The contact stiffness ratio
Relationship between stressstrain curve and contact stiffness ratio.
Three numerical tests with various friction coefficients
Relationship between stressstrain curve and friction coefficient.
The energies considered in this study are boundary, bonding, frictional, rolling, strain, and kinetic energies. The boundary energy is the total accumulated work done by all boundaries on the assembly and can be defined by
The strain energy is the total energy stored at all contacts and can be defined by
The frictional energy is the total energy dissipated by sliding and can be defined by
The rolling energy is the total energy dissipated by rolling and can be defined by
The kinetic energy is total kinetic energy of all particles, accounting for both translational and rotational motion and can be defined by
The bonding energy
The impact of contact stiffness ratio on energy evolution is studied by three numerical tests (1, 1.3, and 1.5), as shown in Figure
Energy evolution under different contact stiffness ratios: (a) boundary energy; (b) bonding energy; (c) strain energy; (d) frictional energy; (e) rolling energy; and (f) kinetic energy.
The influence of friction coefficient on energy evolution is studied by three numerical tests (0.1, 0.3, and 0.5), as shown in Figure
Energy evolutions and microcracks propagations under different friction coefficients: (a) boundary energy; (b) bonding energy; (c) strain energy; (d) frictional energy; (e) rolling energy; and (f) kinetic energy.
The nonlinear microcontact model incorporating rolling resistance for frozen soil is proposed. Then, the failure mechanism of frozen soil is explicated based on the evolution of contact force chains and propagation of microcracks. Finally, the effects of contact stiffness ratio and friction coefficient on stressstrain curve and energy evolution are evaluated. The salient findings are summarized as follows:
Although the nonlinear and linear models can better describe the laboratory data before the deviatoric stress reaches its peak value, only the simulated results of nonlinear microcontact model show a good agreement with the test data after the deviatoric stress reaches its peak value.
The evolution of contact force chains and propagation of microcracks can well account for the failure mechanism of frozen soil since the evolution of contact force chains can illustrate how the applied loads are transmitted within frozen soil and the formation of shear bands in frozen soil is associated with the propagation of microcracks.
The stiffness of sample increases with the increase of contact stiffness ratio. The friction coefficient does not significantly affect the initial tangential modulus. In addition, almost all energies increase with the increasing contact stiffness ratio and friction coefficient. Compared with frictional coefficient, the effect of contact stiffness ratio on stressstrain curve and energy evolution is greater.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This research is supported by the Scientific Research Fund of Institute of Engineering Mechanics, China Earthquake Administration (Grant no. 2017B06), Open Research Fund Program of State Key Laboratory of Frozen Soil Engineering of China (Grant no. SKLFSE201609), the National Natural Science Foundation of China (nos. 51408163 and 51578200), the Natural Science Foundation of Heilongjiang Province (no. ZD201218), China Postdoctoral Science Foundation Funded Project (no. 2012M520751), and the Fundamental Research Funds for the Central University (no. HIT. NSRIF. 2014078).