PSI - Issue 37

A. Kostina et al. / Procedia Structural Integrity 37 (2022) 431–438 A.Kostina/ Structural Integrity Procedia 00 (2019) 000 – 000

434 4

Hook’s law for the total stress tensor can be written in the following form using effective stress concept proposed by Biot: ( ) ( ) 0 : pl T B por α T T α p = − − − − σ C ε E ε E , (9) where C is the stiffness tensor which ha s two components in case of an isotropic elasticity (Young’s modulus and Poisson’s ratio), B α is the Biot coefficient. To describe shear dilation induced by propagation of the phase change front we have applied associated flow rule together with Drucker-Prager yield criterion:

F =  

,

(10)

ε

pl

σ



2 1 F J AI B = + − ,

(11)

2sin



,

(12)

A =

3(3 sin ) + 

2 3 cos c



 ,

(13)

B =

(3 sin ) +

where  is the plastic multiplier, 2 J is the second invariant of deviatoric stress tensor, 1 I is the first invariant of stress tensor, c is the cohesion,  is the internal friction. 2.4. Coupling equations To ensure coupling between mass transfer, heat transfer and mechanical processes we have utilized a porosity equation which relates porosity with total volumetric strains (Rahmati et al. (2017)): ( ) 0 1 vol vol n n + = +   , (14)

where 0 n is the initial porosity, vol  is the volumetric strains. Absolute permeability and porosity were related by the following equation (Hu et al. (2013)):

3

n

,

(15)

K d =

(

)

2

1

n

−

where 0 1 d K n n = − ensures that initial value of absolute permeability is equal to 0 K . 3. Numerical simulation ) 2 3 0 0 (

Numerical simulation is carried out in the finite-element software Comsol Multiphysics®. The following algorithm was proposed. Water saturation, steam saturation, pore pressure and displacement vector were chosen as primary field variables. Governing equations for three-phase flow (1)-(4) were solved using pressure-saturation formulation with total velocity. Artificial diffusion was added to smooth the oscillations caused by the convective terms. The resulting equations were implemented to Comsol Multiphysics® by Weak Form PDE interface. Energy balance equation (5) was insorporated in Heat transfer module using Heat Source node to take into account steam condensation. Momentum balance equation (9), geometric relation (8) and constitutive equations (9)-(13) were solved in Solid mechanics module