Triaxial Test with Hardening Soil Material Model

In this example a triaxial soil test is simulated using a Hardening Soil material model. The test consists of two main stages: an initial isotropic compression followed by an axial compression or extension. The analysis can be simplified by considering the axial symmetry of the specimen.

The expected hyperbolic stress-strain relation is recovered by the simulation. 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. Initially, a confinement pressure is applied to create a state of isotropic compression, and later the soil sample is compressed and extended axially to simulate either axial compression or extension.

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

Soil Properties

The soil properties for the Hardening Soil material model are

•

Density ρ = 2400 kg/m3, reference stiffness for primary loading, E50ref = 25 MPa, reference stiffness for unloading and reloading Eurref = 75 MPa, bulk modulus in compression Kc = 20 MPa, stress exponent m = 0.5, and Poisson’s ratio ν = 0.2.

•

Cohesion c = 1 kPa, angle of internal friction

, and dilatation angle

•

Void ratio at reference pressure eref = 0.8, reference pressure pref = 100 kPa, and initial consolidation pressure pc0 = 1 MPa.

Constraints and Loads

•

It is sufficient to only model the right half of the domain due to the axial symmetry.

•

The stresses from the isotropic compression stage are considered as in-situ stresses, hence there is no need to model this stage explicitly. Instead a confinement pressure of 100 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 1.

Table 1: Failure stress values.
Failure stress	Axial compression	Axial extension
σ1	-303.46 kPa	-32.18 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 variation of axial stress versus axial strain in compression and extension can be seen 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 then asymptotically matches the corresponding failure stress given in the Table 1. 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.

Figure 2: Axial stress versus axial strain.

Figure 3: von Mises stress versus axial strain.

The variations of equivalent plastic strain and volumetric plastic strain are shown in Figure 4 and Figure 5, respectively. The equivalent plastic strain increases monotonically with the axial strain in the case of axial compression and extension, while the volumetric plastic strain first gradually decreases and then shows an increasing trend in both cases.

Figure 4: Equivalent plastic strain versus axial strain.

Figure 5: Volumetric plastic strain versus axial strain.

The dilatancy characteristics of soil describes its volumetric response to shearing. In the Hardening Soil model, Rowe’s stress dilatancy theory is used, where the mobilized dilatancy angle

and the mobilized friction angle

(2)

Many implementations of the Hardening Soil model use Equation 2 with a lower cutoff, but such an approach gives a bilinear function, see Ref. 1. In COMSOL Multiphysics, a scaled approach is used that provides a single nonlinear function instead:

(3)

When the mobilized dilatancy angle is plotted against the mobilized friction angle it gives a parabolic distribution in both cases; this result, shown in Figure 6, matches results shown in Ref. 1.

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 radians. 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 radians 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 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, with combination of and nodes. The second method is to use the in-situ stress option in the node with single study, which gives initial stresses in the soil sample without any strain. In this example, the second method is used to model the initial/in-situ stresses in the soil sample.

A node is used to apply axial displacement on the top boundary in order to compress or extend the soil specimen. The axial compression and extension analysis does not converge for axial displacement values greater than 0.0425 m and 0.0235 m, respectively. These are the values at which the simulation fails as the axial stress approaches the failure stress.

Reference

1. T. Bower, Constitutive modelling of soils and fibre-reinforced soils, PhD Thesis, Cardiff University, 2017.

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 K0nc list, choose .

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

Apply a confinement pressure of 100 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	Basic
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
Bulk modulus in compression	Kc	kc	N/m²	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