PSI - Issue 32

Mikhail Semin et al. / Procedia Structural Integrity 32 (2021) 180–186 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

183

4

In this case, the outer FW boundary b can be determined from the integral equation:

4 2          

b  a

b  a

( ) 

( ) 

c



4 2         

2

tan

1

1

2 tan



P b 

d

b

d

(11)







.





4 2         

2

tan



Clearly, when c is a constant, equation (11) turns into the well- known Vyalov’s solution (9). The specific type of the c(r) function of the radial coordinate is determined by two functions — the c(T) function of temperature and the T(r) function. The actual temperature profile T(r) in the FW can vary greatly depending on the stage of artificial freezing and the FW thickness. The typical T(r) relationships in the FW of a potash mine under construction in the Republic of Belarus are shown in Fig. 2. Although the FW (the zone of negative temperatures) exhibits linear T(r) , obvious non-linearity occurs in the vicinity of the freeze pipes ( r=8.5 m ).

Fig. 2. Radial temperature profiles in the FW from 2D numerical simulation data at 110 days, which corresponds to the end of the active freezing stage (ice growing stage).

The function c(T) can also vary depending on the soil type and the studied temperature range. Vyalov (1965) provided data on the maximum steady limiting strength and internal friction angle for the Callovian sandy loam for the temperature range of [ – 20 ° C, – 5 ° C], explaining the nonlinear c(T) relationship. However, Brovka (2020) studied the long-term strength properties of various types of frozen soils sampled from the freezing interval of a potash mine under construction in the Republic of Belarus. The results showed that, in most cases, the c(T) was linear in the temperature range of [ – 25 ° C, – 4 ° C]. Xu et al. 2016 also showed that the cohesion of silty sand linearly increases with the increase of temperature in range of [ – 6 ° C, – 2 ° C]. Generally, the c(r) function takes the minimum value c min at the FW boundaries and the maximum value c max in the central region (on the circular contour of the freeze pipes). Nevertheless, its performance at other FW points is unknown. Therefore, the regularities of the changes in calculated FW thickness are difficult to analyze because of the non- uniformity of the wall’s strength properties. Such an analysis is possible only for fairly simple problem formulations. In this paper, an analysis is carried out for a case when the radial distribution of cohesion is set as a linear approximation using three points corresponding to the external and internal FW boundaries and the freeze pipe contour. The strength properties of the frozen soil at these extreme points are considered known, and the corresponding cohesions are equal to c min or c max . In this case, the integral (11) shall be as follows: