COMSOL 6.4 - Triaxial and Oedometer Test with Hardening Soil Material Model

Triaxial and Oedometer Test with Hardening Soil Material Model

In this example, triaxial and oedometer tests on a cylindrical soil sample are simulated using a Hardening Soil material model.

The triaxial test consists of two stages: an initial isotropic compression is followed by an axial compression. The oedometer test only have an axial compression stage. The analysis can be simplified by taking the axial symmetry of the specimen into account. The results are compared with those presented in Ref. 1 and Ref. 2.

The expected hyperbolic stress–strain relation is recovered by the model. It is also verified that the asymptotic value of the axial stress is equal to the analytical value of the failure stress. In the oedometer test, loading and unloading cycles are conducted to assess the robustness of the numerical implementation.

Model Definition

In this example, a cylindrical soil specimen of 10 cm in diameter and height is loaded as shown in Figure 1.

For the triaxial test, a confinement pressure is applied to create a state of isotropic compression. Thereafter, the soil sample is compressed axially.

For the oedometer test, the soil sample is compressed axially and constrained on the side.

Figure 1: Dimensions, boundary conditions, and boundary loads for the triaxial test.

Soil Properties

The soil properties for loose Hostun sand are inspired from Ref. 1 and Ref. 2. Along with some properties given in the references, the material parameters are presented in Table 1.

Table 1: Material properties for The soil model.
Property	Variable	Value in Ref. 1	Value in Ref. 2	Used value
Poisson’s ratio	ν	0.2	0.2	0.2
Density	ρ	NA	NA	2400 kg/m3
Reference initial stiffness for primary loading	Eiref	NA	68.913 MPa	68.913 MPa
Reference stiffness for unloading and reloading	Eurref	60 MPa	60 MPa	60 MPa
Stress exponent	m	0.65	0.65	0.65
Cohesion	c	0 kPa	0 kPa	0 kPa
Angle of internal friction	ϕ	34°	34°	34°
Dilatation angle	ψ	0°	1.5°	1.5°
Failure ratio	Rf	0.9	0.95	0.95
Initial void ratio	e0	0.89	NA	0.89
Reference pressure	pref	100 kPa	100 kPa	100 kPa
Hardening modulus	Kc	NA	16.5 MPa	16.5 MPa
Initial ellipse centroid	pcc0	NA	NA	2 kPa
Preconsolidation pressure	pc0	NA	NA	5 kPa

Constraints and Loads

•

For the triaxial test, a confinement pressure of 300 kPa is applied using the option in the node to model the isotropic compression step. For axial compression, the soil sample is compressed by applying a prescribed displacement on the top boundary. Allow the top boundary to expand freely in the radial direction and apply a roller boundary condition on the bottom boundary.

•

For the oedometer test, the soil sample is compressed by applying a prescribed displacement on the top boundary. Apply roller boundary conditions at the vertical and bottom boundary.

Results and Discussion

The analytical solution to the axial failure stress for the triaxial test is given by the Mohr–Coulomb criterion

(1)

The axial failure stress in compression is given by

The values of the axial failure stress are shown in Table 2.

Table 2: Failure stress values.
Failure stress	Axial compression
σ1	-1061.1 kPa

For the sake of consistency with the geomechanical convention, the compressive axial stress and axial strain are plotted along the positive axis in all figures.

The axial stress versus axial strain curves for the triaxial test are shown in Figure 2. The stress–strain curve has an hyperbolic shape, which is a characteristic of the Hardening Soil material model; as the axial displacement increases, the axial stress increases hyperbolically and approaches the failure stress given in Table 2. The results presented for the axial compression case in Figure 2 also closely match the results presented in Ref. 1 (see Figure 6 therein).

The same behavior is observed for the von Mises stress in axial compression; it asymptotically matches the ultimate deviatoric stress based on the Mohr–Coulomb criterion; see Figure 3. The results presented for the axial compression case in Figure 3 also closely match the results presented in Ref. 2 (see Figure 12 therein). The shear failure stress is reached at a compressive axial strain of 0.11, which is similar to the value reported in Ref. 2.

Figure 2: Axial stress versus axial strain for the triaxial test.

Figure 3: von Mises stress versus axial strain for the triaxial test.

Figure 4 shows the variations in volumetric strain versus applied axial strain. The volumetric strain shows parabolic behavior with respect to the axial strain, as shown in Ref. 2 (see Figure 13 therein). The volumetric strain matches well with the result presented in the Ref. 1. The volumetric strain is highly dependent on the mobilized dilatation angle formulation.

The dilatancy characteristics of a soil describes its volumetric response to shearing. For the Hardening Soil model presented in Ref. 1, Rowe’s stress dilatancy theory is used, where the mobilized dilatancy angle, ψm, is related to the critical state friction angle, ϕc, and the mobilized friction angle, ϕm:

(2)

The authors in Ref. 2 use the modified Rowe’s formulation which is written as:

(3)

This example uses Rowe’s formulation (similar to Ref. 1), which gives a linear distribution with respect to the mobilized friction angle. The result is shown in Figure 6.

Figure 4: Volumetric strain versus axial strain in the triaxial test.

Figure 5: Mobilized dilatancy angle versus mobilized friction angle in the triaxial test.

The axial stress versus axial strain curves in the cyclic oedometer test are shown in Figure 6. The stress–strain curves for four different loading-unloading cycles presented in Figure 6 also match the results presented in Ref. 2 (see Figure 7 therein). Figure 7 shows the linear variation in volumetric strain with applied axial strain.

Figure 6: Axial stress versus axial strain in the oedometer test.

Figure 7: Volumetric strain versus axial strain in the oedometer test.

Notes About the COMSOL Implementation

The in situ stress is the stress in the soil sample represented in the strain-free configuration. There are two methods to account for in situ stresses in COMSOL Multiphysics. One method is to create two stationary study steps or studies, using a combination of and nodes. The second method is to use the option in the node, which gives initial stresses in the soil sample without any strain. In this example, the second method is used to model the in situ stress in the soil samples.

References

1. T. Schanz, P.A. Vermeer, and P.G. Bonnier, “The Hardening Soil Model: Formulation and Verification,” Beyond 2000 in Computational Geotechnics, Rotterdam, 1999.

2. T.A. Bower, P.J. Cleall, and A.D. Jefferson, “A Reformulated Hardening Soil Model,” Proceedings of the Institution of Civil Engineers — Engineering and Computational Mechanics, vol. 173, no. 1, pp. 11–29, 2020.

3. T. Forrister, “Analyzing Triaxial Testing Methods for Geomechanics,” COMSOL Blog, 5 Mar. 2018, www.comsol.com/blogs/analyzing-triaxial-testing-methods-for-geomechanics/.

Geomechanics_Module/Verification_Examples/triaxial_and_oedometer_test_hs

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select > .

Click .

Click

In the tree, select > .

Click

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file triaxial_and_oedometer_test_hs_parameters.txt.

Pressure

In the toolbar, click

and choose > .

In the window for , type Pressure in the text field.

Locate the section. In the text field, type P.

In the table, enter the following settings:


t	f(t)
0	0
1	50
2	10
3	60
4	15
5	110
6	30
7	210
8	50
9	230

Locate the section. In the table, enter the following settings:


Argument	Unit
t	1

In the table, enter the following settings:


Function	Unit
P	kPa

Define variables for the failure stress in axial compression based on the Mohr–Coulomb criterion.

Definitions

Variables 1

In the window, under right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
sigmafc	-p0(1+sin(Phi))/(1-sin(Phi))-2c*cos(Phi)/(1-sin(Phi))	Pa	Failure stress in compression

Geometry 1

Rectangle 1 (r1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 5[cm].

In the text field, type 10[cm].

Click

Click the

button in the toolbar.

Solid Mechanics (solid)

Hardening Soil

In the toolbar, click

and choose .

In the window for , type Hardening Soil in the text field.

Select Domain 1 only.

Locate the section. From the list, choose .

From the ψm list, choose .

Find the subsection. From the Eiref list, choose .

In the Rf text field, type 0.95.

Apply a confinement pressure of 300 kPa using an node.

External Stress 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

In the σins text field, type -p0.

Add a node.

Hardening Soil

In the window, click .

Cap and Cutoff 1

In the toolbar, click

and choose .

Roller 1

In the toolbar, click

and choose .

Select Boundary 2 only.

Prescribed Displacement 1

In the toolbar, click

and choose .

Select Boundary 3 only.

In the window for , locate the section.

From the list, choose .

In the u 0 z text field, type disp.

Roller 2

In the toolbar, click

and choose .

Select Boundaries 2 and 4 only.

Boundary Load 1

In the toolbar, click

and choose .

Select Boundary 3 only.

In the window for , locate the section.

From the list, choose .

In the p text field, type P(para).

Prescribed Displacement 1, Roller 1

In the window, under > , Ctrl-click to select and .

Right-click and choose .

Triaxial Test

In the window for , type Triaxial Test in the text field.

Boundary Load 1, Roller 2

In the window, under > , Ctrl-click to select and .

Right-click and choose .

Oedometer Test

In the window for , type Oedometer Test in the text field.

Materials

Soil Material

In the window, under right-click and choose .

In the window for , type Soil Material in the text field.

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Poisson’s ratio	nu	Nu	1	Critical state model
Friction angle	internalphi	Phi	rad	Critical state model
Dilatation angle	psid	Psi	rad	Critical state model
Density	rho	Rho	kg/m³	Basic
Reference initial stiffness for primary loading	EiRef	Eiref	Pa	Hardening soil
Reference stiffness for unloading and reloading	EurRef	Eurref	Pa	Hardening soil
Stress exponent	mh	m	1	Hardening soil
Cohesion	cohesion	c	Pa	Hardening soil
Initial void ratio	evoid0	e0	1	Critical state model
Preconsolidation pressure	pcp0	Pcp0	Pa	Cap and cutoff
Initial ellipse centroid	pcc0	Pcc0	Pa	Cap and cutoff
Hardening modulus	Kc	Kcc	Pa	Cap and cutoff