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Finite element implementation of a geometrically and physically nonlinear consolidation model

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Abstract

The paper presents a rather general formulation of a porous medium deformation coupled with a fluid flowing through the pores within the framework of physical and geometric nonlinearity. The boundary-value problem is formulated in terms of the solid phase displacement increment, fluid pressure and porosity increments in the form of differential and variational equations. The equations were derived from the general conservation laws of Continuum Mechanics using spatial averaging over a representative volume element (RVE). The model takes into account the porosity and permeability evolutions during deformation process. The equations of filtration and porosity evolution are formulated in the material coordinate system related to the solid phase, according to the idea of Arbitrary Lagrangian–Eulerian (ALE) approach. The linearization of variational equations was done using Gateaux differentiation technique. The proper finite elements were used for the spatial discretization of the saddle system of equations to satisfy well-known Ladyzhenskaya–Babuška–Brezzi (LBB) correctness condition. A generalization of the implicit time integration scheme with internal iterations at each time step according to the Uzawa method is employed. The convergence of the iterative process is partly theoretically studied. The formulation is numerically implemented in the form of a self-made computer code. Examples of calculations are given.

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Acknowledgements

This work was supported by the Russian Foundation for Basic Research (project No. 20-01-00431) and by the Ministry of Education and Science of the Russian Federation as part of the program of the Moscow Center for Fundamental and Applied Mathematics under the agreement No. 075-15-2022-284.

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Authors and Affiliations

  1. Faculty of Geology, Lomonosov Moscow State University, GSP-1, Leninskie Gory 1, Moscow, Russian Federation, 119991

    Nina B. Artamonova

  2. Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, GSP-1, Leninskie Gory 1, Moscow, Russian Federation, 119991

    Sergey V. Sheshenin

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  1. Nina B. Artamonova
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Correspondence to Nina B. Artamonova.

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Communicated by Andreas Öchsner.

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Artamonova, N.B., Sheshenin, S.V. Finite element implementation of a geometrically and physically nonlinear consolidation model. Continuum Mech. Thermodyn. 35, 1291–1308 (2023). https://doi.org/10.1007/s00161-022-01124-5

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