Undrained soil column 2D test

Author: Ignasi de Pouplana

Kratos version: 9.3

Source files: Undrained soil column 2D

How to run: from terminal using python or from the GUI using GiD.

Case Specification

This example consists on a 1 x 30 m column of saturated soil subjected to a surface loading that lies on a rigid rock bed. The objective of the test is to capture the pressure distribution along the soil column which, near the undrained-incompressible limit, should equal the applied load. The main challenge is that, under such conditions, the direct formulation for the displacement-pore pressure element exhibits locking of the pressure field due to the violation of Babuska-Brezzi conditions. The reference solution is taken from [1].

The problem geometry and boundary conditions are shown below.

The material properties of the soil column are the following:

• Young’s modulus (E): 2.5E+7 N/m²
• Poisson’s ratio (ν): 0.2
• Solid density (ρ_s): 2000 Kg/m³
• Fluid density (ρ_f): 1000 Kg/m³
• Porosity (φ): 0.3
• Solid bulk modulus (K_s): 1.5E+17 N/m²
• Fluid bulk modulus (K_f): 3.0E+14 N/m²
• Intrinsic permeability (k): 1.0E-14 m²
• Dynamic viscosity (μ): 0.001 s·N/m²

This problem is solved in a 2D configuration under plane strain conditions. The geometry is discretized with a structured mesh of 20 quadrilateral elements. Three different types of elements have been used here: 4-noded quadrilateral elements with bilinear shape functions for both the pressure and the displacements (Q4P4-Direct), 9-noded quadrilateral elements with biquadratic shape functions for the displacements and bilinear shape functions for the pressure (Q9P4), and also FIC-stabilized 4-noded quadrilateral elements of equal order interpolation for the displacements and pore pressure (Q4P4-FIC). The formulation of the latter element is thoroughly described in [2].

The considered load is represented in the next figure:

The time step is 0.02 seconds.

Results

The next graph shows the normalized pore pressure along the normalized height of the soil column for each type of element at a time t = 2 s.

References

[1] O.C. Zienkiewicz, C.T. Chang and P. Bettess. Drained, undrained, consolidating dynamic behaviour assumptions in soils. Géotechnique, vol. 30, pp. 385-395, 1980. DOI: 10.1680/geot.1980.30.4.385. https://doi.org/10.1680/geot.1980.30.4.385

[2] I. de Pouplana and E. Oñate. A FIC-based stabilized mixed finite element method with equal order interpolation for solid-pore fluid interaction problems. International Journal for Numerical and Analytical Methods in Geomechanics, vol. 41, pp. 110-134, 2016. DOI: 10.1002/nag.2550. http://onlinelibrary.wiley.com/doi/10.1002/nag.2550/full