Triaxial Test with Hardening Soil Material Model

In this example, a triaxial test on a cylindrical soil sample is simulated using a Hardening Soil material model and the results are compared with those presented in Ref. 1 and Ref. 2. The test consists of two stages: an initial isotropic compression is followed by an axial compression or extension. The analysis can be simplified by taking the axial symmetry of the specimen into account.

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, both in extension and in compression.

Model Definition

In this triaxial test, a cylindrical soil specimen of 10 cm in diameter and height is loaded as shown in Figure 1. First, the confinement pressure in terms of in situ stresses is applied to create a state of isotropic compression. Thereafter, the soil sample is either compressed or extended axially.

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

Soil Properties

Soil properties for loose Hostun sand are taken from Ref. 1 and Ref. 2. Note that some properties are different in Ref. 1 and Ref. 2. Along with properties given in the references, the properties used in the COMSOL Multiphysics model are presented in Table 1.

Table 1: Material properties for The soil model.
Property	Variable	Value in Ref. 1	Value in Ref. 2	Value used in COMSOL
Poisson’s ratio	ν	0.2	0.2	0.2
Density	ρ	NA	NA	2400 kg/m3
Reference stiffness for primary loading	E50ref	20 MPa	23.89 MPa	20 MPa
Reference stiffness for unloading and reloading	Eurref	60 MPa	60 MPa	60 MPa
Stress exponent	m	0.65	0.65	0.65
Swelling to compression ratio	Ks/Kc	NA	1.75	1.75
Cohesion	c	0 kPa	0 kPa	1 kPa
Angle of internal friction		34°	34°	34°
Dilatation angle		0°	1.5°	1.5°
Ellipse aspect ratio	Rc	NA	1.0428	1.0428
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
Initial consolidation pressure	pc0	NA	NA	1000 kPa

Constraints and Loads

•

The stresses resulting from isotropic compression are considered as in-situ stresses; therefore, there is no need to model this stage explicitly. Instead, a confinement pressure of 300 kPa is applied using the in-situ stress option in the node. Note that no boundary load is applied in this example.

•

For axial compression or extension, the soil sample is compressed by applying a prescribed displacement to the top boundary. Allow the top-right corner to expand freely in the radial direction and apply a roller boundary condition at the bottom boundary.

Results and Discussion

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

(1)

The axial failure stress in compression is given by

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

Table 2: Failure stress values.
Failure stress	Axial compression	Axial extension
σ1	-1064.9 kPa	-83.7 kPa

Note that for the sake of consistency with geomechanical convention, the compressive axial stress and strain are plotted along the positive axis, while the tensile stress and strain are plotted along the negative axis in all the figures.

The axial stress versus axial strain curves in compression and extension are shown in Figure 2. The stress–strain curve is hyperbolic, 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).

The same behavior is observed for the von Mises stress in axial compression and extension; it asymptotically matches the ultimate deviatoric stress computed internally 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).

Figure 2: Axial stress versus axial strain.

Figure 3: von Mises stress versus axial strain.

Figure 4 shows the variations in volumetric strain with applied axial strain. The volumetric strain shows parabolic behavior with respect to the axial strain in the case of axial compression, as shown in Ref. 2 (see Figure 13). The volumetric strain matches well with one of the experimental curves presented in the reference, but shows larger values than the theoretical results. The reason could be a different dilation angle (compared to Ref. 1) and a different mobilized dilatation angle formulation.

Figure 5 shows the variations in volumetric plastic strain with applied axial strain. The behavior is the same as for the volumetric strain presented in Figure 4.

Figure 4: Volumetric strain versus axial strain.

Figure 5: Volumetric plastic strain versus axial strain.

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

is related to the critical state friction angle

and the mobilized friction angle

(2)

This formulation comes with a lower cutoff of zero, which gives a bilinear function; see Ref. 1. This COMSOL Multiphysics model uses a scaled approach (Soreide), which provides a single nonlinear function:

(3)

When the mobilized dilatancy angle is plotted against the mobilized friction angle it gives a parabolic distribution in both cases; the result is shown in Figure 6.

Figure 6: Mobilized dilatancy angle versus mobilized friction angle.

Figure 7 shows the variation of the Lode angle versus the axial strain. For axial compression, the Lode angle is π/3, while for axial extension it is zero. This result verifies the established convention in COMSOL Multiphysics stating that the Lode angle is π/3 when the stress state is on the compressive meridian and zero when the stress state is on the tensile meridian.

Figure 7: Lode angle versus axial strain.

Notes About the COMSOL Implementation

The in-situ stresses are the stresses in the soil sample 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 in-situ stress option in the node with a single study, which gives initial stresses in the soil sample without any strain. In this example, the second method is used to model the in-situ stresses in the soil sample.

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.

Geomechanics_Module/Verification_Examples/triaxial_test_hardening_soil

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_test_hardening_soil_parameters.txt.

Define variables for the failure stress in axial compression and extension 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
sigmafe	-p0(1-sin(Phi))/(1+sin(Phi))+2c*cos(Phi)/(1+sin(Phi))	Pa	Failure stress in extension

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

Solid Mechanics (solid)

Elastoplastic Soil Material 1

In the window, under right-click and choose .

Select Domain 1 only.

In the window for , locate the section.

From the list, choose .

From the Kc list, choose .

In the Rf text field, type 0.95.

In the pc0 text field, type 1000[kPa].

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.

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.

Select the check box.

In the u 0 z text field, type disp.

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	Young’s modulus and Poisson’s ratio
Reference stiffness for primary loading	E50Ref	E50ref	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	Mohr-Coulomb
Dilatation angle	psid	Psi	rad	Mohr-Coulomb
Swelling to compression ratio	rsc	rc	1	Hardening Soil
Ellipse aspect ratio	Rcap	Rc	1	Hardening Soil
Initial void ratio	evoid0	e0	1	Hardening Soil
Angle of internal friction	internalphi	Phi	rad	Mohr-Coulomb
Density	rho	Rho	kg/m³	Basic