PSI - Issue 32

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

73

3

(a)

(b)

Fig. 1. Geometry of the computational domain and boundary conditions for numerical simulation of stress-strain states in the shaft lining during thawing of the frozen wall (a); after elimination of the frozen wall (b) (1 is the cast iron tubbing, 2 is the concrete shell, 3 is the frozen wall (a) or cement grouted sand (b), 4 is sand) To describe a change in the stress-strain state of the shaft lining during thawing of the frozen wall a thermo mechanical model is developed. The model includes the heat transfer equation and equilibrium equation which can be written by the following way

S





,

(1)

div grad

C T λ 

T Lρ n 

i

eff

eff

i

t

t





div  σ 0 , (2) where T is the temperature, C eff and λ eff are the effective volumetric heat capacity and thermal conductivity of thawing soil, L is the latent heat, n is the soil porosity, S i is the ice saturation, σ is the stress tensor. The ice saturation S i is assumed to be a power function on temperature T [13]. The stress-strain state in the shaft lining, cement grouted soil layer and frozen wall is determined by the following relation   : in   σ C ε ε , (3) where C is the stiffness tensor for the isotropic material, ε is the total strain tensor and ε in is the inelastic strain tensor. The stress tensor of the unfrozen soil σ is expressed through the effective stress tensor σ and the hydrostatic pressure of water. The effective stress σ of unfrozen soil is defined by analogy with (3). Total strain ε is determined by the geometric relation for small strains:   (4) where u is the displacement vector. Creep strain in the concrete shell is written according to the Voigt-Kelvin model for the three-dimensional stress state as 1 2 T     ε u u ,

1 1 exp     

  

1 G t

  

  

s ,

(5)

ε

cr

G

η



1

where s is the deviatoric part of the tensor σ , G 1 and η are material parameters responsible for creep behavior. At the same time, creep strain in the soil surrounding the concrete shell is neglected.