Modeling of a nonlinear coupled consolidation problem | Mekhanika | kompozitsionnykh | materialov i konstruktsii
> Volume 26 > №3 / 2021 / Pages: 341-361

Modeling of a nonlinear coupled consolidation problem



Previously by the authors, 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. The differential formulation in “velocities” includes the equilibrium equation, the filtration equation and the porosity change equation derived from the laws of conservation of continuum mechanics using spatial averaging over the representative volume element. In this paper, we propose a method for solving this problem and present the results of numerical simulation. The system of equilibrium and filtration equations is solved under the assumption of constant porosity, which is recalculated at each time step. To solve the system, a generalization of the implicit scheme with internal iterations at each time step is used according to the Uzawa method. The paper analyzes the stability of a linear problem when approximated by elements Q1-Q1 and Q2-Q1. Numerical examples are given of calculating a nonlinear coupled consolidation problem for a hyperelastic material when approximated by the Mooney, Mooney-Rivlin, Treloar, and Saint-Venant-Kirchhoff potentials, the influence of taking into account geometric nonlinearity is investigated, and the problem with varying porosity and filtration coefficient is solved. For modeling the constitutive relations for elastic-viscoplastic soil deformation under short-term loads, the Grigoryan-Rykov model generalized to large deformations is selected. In this theory, the associated flow law is considered in the five-dimensional Ilyushin space, and the relationship between the first invariants of the stress and strain tensors is determined according to the theory of viscoplasticity. Comparison of the results of calculations of effective elastic moduli by the averaging method based on three-dimensional and two-dimensional models of the real structure of pure limestones and experimental values is presented.