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Coupling consolidation problem in a nonlinear formulation. theory and method of solution

Artamonova N.B., Sheshenin S.V.

#### Abstract:

The consolidation problems are related to the study of soil deformation under load in the presence of fluid outflow. In the process of joint deformation of the porous skeleton and the fluid contained in the pores, the solid and liquid phases of the soil interact. The filtration processes in the soil mass are described by a coupling system of differential equations with rapidly oscillating coefficients. To solve such equations, averaging over the representative volume element (RVE) is used. In the paper, the equations of the nonlinear consolidation model are written from the general laws of conservation of continuum mechanics (the equilibrium equation, the law of mass conservation of solid and liquid phases of the soil, and Darcy’s filtration law) using spatial averaging over the representative volume element. The following assumptions were made: the fluid fills the pores entirely, the fluid is Newtonian and homogeneous, the deformation of the fluid with a change in pore pressure obeys the law of barotropy, and the soil skeleton material is incompressible. To determine effective properties, an approach based on solving local problems in a representative volume element is possible. As a result, a coupled physically and geometrically nonlinear formulation of the boundary value problem was obtained using the Lagrange approach with adaptation for the solid phase and the ALE (Arbitrary Lagrangian-Eulerian) approach for the fluid under the assumption of quasistatic deformation of the rock skeleton. In the method of solving the coupled problem, linearization of variational equations is carried out in combination with internal iterations according to the Uzawa method for connecting at each time step. For spatial discretization, the finite element method is used: trilinear type elements for approximating the filtration equation and quadratic elements for approximating the equilibrium equation. An implicit time scheme can be used to take into account the inertia forces.

#### Keywords:

Biot’s model,

Biot’s tensor parameter,

coupling consolidation problem,

cвязанная задача консолидации,

Darcy's law,

iterative solver,

NONLINEAR DEFORMATION,

pore pressure transfer tensor,

saddle system,

variational formulation,

вариационная формулировка,

закон Дарси,

итерационный решатель,

модель Био,

нелинейное деформирование,

седловая система,

тензор передачи порового давления,

тензорный параметр Био
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