ИСТИНА

A physically and geometrically nonlinear formulation of the consolidation problem is developed. Consolidation is the process of compaction of water-saturated soil under the action of an external force with the possibility of fluid outflow. This formulation is convenient for discretization with the help of the Finite Element Method (FEM). The Lagrange approach with updating (UL) for the solid phase and the Euler approach for the liquid are used under the assumption of quasi-static deformation of the skeleton. The ALE (Arbitrary Lagrangian-Eulerian) method is used to combine these approaches. Typically, ALE is used to correct highly distorted Lagrangian mesh. In our case, this method is used to link the two approaches and is not associated with re-meshing. At each time step, a saddle system of variational equations that corresponds to the differential formulation is derived. Further discretization of the variational equations by the FEM leads to a nonlinear system of algebraic equations. A number of iterative algorithms based on the Uzawa method are proposed for solving the latter system. Finally, the solution method is implemented as self-made computer code. Test calculations have been carried out and some model consolidation problems have been solved. The analysis of the stability of the finite element solution of the linear consolidation problem when approximated by the elements Q1-Q1 and Q2-Q1 is carried out. The results of calculating the model problem of the action of the static plate load showed that the approximation of the saddle problem by quadratic finite elements (FE) Q2 for soil skeleton displacements and trilinear FE Q1 for water pressure ensures the stability of the solution, and the use of elements Q1-Q1 leads to a conditionally stable system of equations. Calculations of test problems have shown that taking into account changes in porosity and permeability in the process of deformation of the material leads to a significant increase in the values of pore pressure at the initial stage of the consolidation process. The developed nonlinear consolidation model, which takes into account the change in porosity and permeability, is applied to simulate the hyperelastic deformation of a biological material saturated with blood or plasma. The model problem of elastoplastic deformation of porous water-saturated sandy soil is solved. The deformation theory of plasticity was used to model the constitutive relations. This research is supported by RFBR (project 20-01-00431_a).