PSI - Issue 17

M. Zhelnin et al. / Procedia Structural Integrity 17 (2019) 316–323 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

319

4

F  

= & & ε

,

(9)

p

σ



2 1 F J aI b = + − ,

(10)

where 0   – the plastic multiplier, 2 J , 1 I – the second and the first invariants of the stress tensor σ , a , b – material parameters. The material parameters are expressed as

2sin 3(3 sin )   −

a

=

,

(11)

2 3 cos 3 sin C −



b

=

,

(12)



where C – the cohesion coefficient,  – the friction angle. Studies of the artificial freezing process of a rock mass are carried out for two cases. In the first case a domain bounded by an excavation and including pair of freezing wells is considered. A mechanical response of the rock mass on frost heave during the freezing process and an unloading caused by the shaft sinking is determined. In the case only elastic deformation is taken into account. In the second case plastic strain of soil in a neighborhood of the freezing well is estimated. In all cases from the rock mass the three soil stratums with the most different material characteristics are chosen. The system of equation (1)-(12) is solved numerically with a use of the finite element method. The computational domains are divided on mapped elements of the first order for variables T , f p and the second order for variable u . In the first case the problem is solved in three-dimensional configuration, in the second one the problem is solved in axisymmetric configuration. The axis of symmetry coincides with the axis of the well. One of the main situation leading to the stress concentration and, as a consequence, to the emergency situation with wells and mine shaft is freezing of the border between two soil stratum with different physical-mechanical properties. To investigate this situation we carried out a simulation of freezing process for three soil stratums: clay, sand and sandstone. The clay stratum is placed at the bottom; the sandstone stratum is placed at the top. Depth of each stratum is 3 m. Hydrodynamic and thermophysical properties of the soils are listed in Table 1, mechanical properties are listed in Table 2. Temperature of a liquid refrigerant is 253 K. The data is given according to design documentation for a vertical mine shafts construction in the Petrikov Site of Starobinsky potash deposit. In Fig 1 distributions of the mechanical pressure p and Mises stress 2 3 J in the first computational domain are 3. Results of the numerical simulations

Table 1. Hydrodynamic properties and thermophysical properties of frozen and unfrozen soils at temperatures T= 263.15 K (index 1) and T =283.15 K (index 2). Soil n k, m 2 ,1 p c , J/(kg·K) ,2 p c , J/(kg·K) 1  , kg/m 3 2  , kg/m 3 1  , W/(m∙K) 2  , W/(m∙K) Sandstone 0.34 4.51·10 -12 1170 1910 2029 2058 6.15 5.30 Sand 0.35 5.07·10 -12 1180 1930 2037 2066 3.85 2.37 Clay 0.40 5.72·10 -12 1280 2140 1920 1954 1.5 1.29