PSI - Issue 32

M. Zhelnin et al. / Procedia Structural Integrity 32 (2021) 71–78 M. Zhelnin/ Structural Integrity Procedia 00 (2021) 000–000

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Plastic strain in the concrete shell and the cement grouted soil is determined according to the associated flow rule. For the concrete shell the Bresler-Pister yield function FBP is adopted:

2

2 3 BP F J k I k I k     , 1 1 2 1

(6)

where J 2 is the second invariant of the deviator stress tensor s , I 1 is the first invariant of the stress tensor σ , k i ( i = 1,…,3) are material parameters depending on uniaxial tensile σ t , uniaxial compressive σ c , and biaxial compressive σ bc strength. To describe plastic flow in the cement grouted soil the Drucker-Prager yield criterion F DP is used:

2 1 DP F J αI k    ,

(7)

where material parameters α and k can be expressed using cohesion c and friction angle φ . The governing equations of thermo-mechanical model together with the mechanical constitutive relations were implemented in Comsol Multiphysics® software. Numerical solution of the equations is conducted relative to the temperature T and displacement vector u using finite element method. To study creep process in a concreted shell during thawing of the frozen wall a coupled system of heat transfer and equilibrium equations are solved. To determine elasto-plastic behavior in the shaft lining and cement grouted soil after thawing of the frozen wall, only equilibrium equation is solved. 3. Results of numerical simulation Mechanical response in a shaft lining during thawing of the frozen wall is considered in a sand stratum laying at the Petrikov potash deposit at the depth of 190 m. The radius of the mineshaft is 5.25 m. The thicknesses of the cast iron tubbing and concrete shell are equal to 0.06 m and 0.45 m, respectively. At the beginning of the thawing the thickness of the frozen wall is equal to 6.75 m. Applied boundary conditions are presented in Fig. 1(a). At the outer boundaries of the domain the lateral pressure P = 2.13 MPa and natural temperature T 0 = 10 C are given. At the inner side of the shaft lining the convective heat flux with the external temperature of 20 C and the heat transfer coefficient of 4 W/(m 2 K) is set. The boundary is free from any loadings. Symmetry condition is applied to other boundaries. Fig. 2 shows distribution of the equivalent strain after 0, 125, and 275 days of the frozen wall thawing. At the beginning of the thawing, the equivalent strain takes minimum value due to the high stiffness of the frozen soil. As the frozen soil turns into the unfrozen state, the equivalent strain near the concrete shell rises. It should be noted that after 125 days of the thawing the equivalent strain in the thawed soil adjacent to the concrete shell is higher than after 275 days. The thawed soil exhibits smaller stiffness in comparison with the frozen soil and the concrete shell. So, the thawed soil is deformed more uniformly under the loading which reflects in the redistribution of the stress throughout the thickness of the layer.

(a)

(b)

(c)

Fig. 2. Distribution of the equivalent strain after 0, 125, 275 days of the thawing (the white line corresponds to the position of the phase change interface).