COMSOL 6.4 - Triaxial Test with Hardening Soil Small Strain Material Model

Triaxial Test with Hardening Soil Small Strain Material Model

In this example, a triaxial test on a cylindrical soil sample is simulated using a Hardening Soil Small Strain material model and the results are compared with those presented in Ref. 1. This example consists of a monotonic triaxial compression test and a cyclic triaxial test. In the monotonic triaxial compression test, the hyperbolic stress–strain relation is expected to be recovered by the model. In the cyclic triaxial test, the small-strain stiffness and the hysteresis effects are recovered.

Model Definition

In both triaxial tests, 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 stress is applied to create a state of isotropic compression. Thereafter, the soil sample is compressed axially in the monotonic test, while axially compressed and extended repeatedly in the cyclic test.

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

Soil Properties

For the monotonic triaxial compression test, three different soils are used in Ref. 1: Hostun dense soil, Hostun loose soil, and Kaoln clay. The soil properties are presented in Table 1.

Table 1: Material properties for The soil models.
Property	Variable	Hostun dense	Hostun loose	Kaolin clay
Poisson’s ratio	ν	0.25	0.2	0.2
Density	ρ	2400 kg/m3	2000 kg/m3	1700 kg/m3
Reference failure stiffness	E50ref	30MPa	15MPa	3 MPa
Reference stiffness for unloading and reloading	Eurref	90 MPa	60 MPa	11.5 MPa
Initial Young’s modulus at reference pressure	E0ref	270 MPa	168 MPa	80 MPa
Stress exponent	m	0.55	0.75	0.8
Cohesion	c	1 kPa	1 kPa	1 kPa
Angle of internal friction		42°	34°	20°
Dilatation angle	tψ	16°	0°	0°
Failure ratio	Rf	0.9	0.9	0.9
Initial void ratio	e0	0.65	0.85	0.9
Reference pressure	pref	100 kPa	100 kPa	100 kPa

The reference failure stiffness and cohesion values are slightly altered compared to the Ref. 1 for better numerical convergence. The density is not given in the Ref. 1 so we make some assumption.

For the cyclic triaxial test, soil properties are taken from the Hostun dense soil except the initial shear modulus and reference pressure. The initial shear modulus is assumed to be constant (G0 = 190 MPa) and not derived from the initial shear modulus at reference pressure. This change is consistent with the cyclic test of the Small-Strain Overlay model presented in Ref. 1, where the initial shear modulus is considered as constant. The reference pressure is considered as 300 kPa same as the in situ stresses.

Constraints and Loads

•

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

•

For axial compression in the monotonic triaxial compression test, the soil sample is compressed by applying a prescribed displacement on 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.

•

For axial compression and extension in the cyclic triaxial compression test, the soil sample is compressed or extended by applying a prescribed displacement on 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

In Ref. 1, the monotonic triaxial tests are carried out with three different confinement pressures: 100 kPa, 300 kPa, and 600 kPa. In the current example, the monotonic and cyclic triaxial tests are carried out with a confinement pressure of 300 kPa.

Figure 2 shows the axial stress versus axial strain in the monotonic triaxial test for three different soils. The stress-strain curve is hyperbolic, which is a characteristic of the Hardening Soil Small Strain material; as the axial displacement increases, the axial stress increases hyperbolically and approaches the failure stress. The results of the Hardening Soil Small Strain model with Matsuoka–Nakai octahedral shape matches well with the results presented in Ref. 1.

Figure 3 shows the variations in volumetric strain with applied axial strain for the monotonic triaxial test. The volumetric strain shows nonlinear behavior with respect to the axial strain. The volumetric strain of Hardening Soil model with Matsuoka–Nakai octahedral shape matches well with results presented in the Ref. 1 for Hostun dense and loose sand, but results differ for Kaolin clay. The reason is that a different mobilized dilatation angle formulation is used compared to Ref. 1, which affects the plastic potential and the computed volumetric plastic strains.

Figure 4 shows the axial stress versus axial strain in the cyclic triaxial test for the dense Hostun soil. The hysteresis and stiffness degradation effect is visible in the cyclic loading. The results are consistent with the results presented in Ref. 1. Note that in this example, a different initial shear moduli as well the reference pressure is used than Ref. 1. The stress–strain path in cyclic loading with the Small-Strain Overlay model remains closed, but with the Hardening Soil Small strain model it does not close due to the occurrence of plastic strain, a behavior that is consistent with that shown in Ref. 1.

Figure 5 shows the variation of Young’s modulus versus axial strain. The stiffness during primary loading and reloading are different and degrade with an increase in axial strain. This behavior is consistent with the behavior presented in Ref. 1.

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

Figure 3: Volumetric strain versus axial strain in the monotonic triaxial test.

Figure 4: Axial stress versus axial strain in the cyclic triaxial test.

Figure 5: Young’s modulus versus axial strain in the cyclic triaxial test.

Notes About the COMSOL Implementation

The in situ stress is the stress 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 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.

The Hardening Soil model comes with different flavors or versions; the model presented in Ref. 2 is known as the Hardening Soil-Original model, whereas the improved version presented in Ref. 1 is known as the Hardening Soil-Smooth model, and the model presented in Ref. 3 is known as the Hardening Soil-Lusas–Cardiff. The Hardening Soil model in COMSOL Multiphysics is based on Ref. 3.

The Hardening Soil Small Strain model is a combination of the Hardening Soil model and Small-Strain Overlay model, first proposed in Ref. 1. The Hardening Soil and Hardening Soil Small Strain models in the COMSOL Multiphysics software are differentiated based on the octahedral shape used. Three different octahedral shapes that can be used are Circular or Drucker–Prager, Mohr–Coulomb, and Matsuoka–Nakai.

References

1. T. Benz, Small-Strain Stiffness of Soils and its Numerical Consequences, PhD Dissertation, Stuttgart University, 2006.

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

3. 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.

4. 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_test_hardening_soil_small_strain

Modeling Instructions

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New

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Model Wizard

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Click .

Click

In the tree, select > .

Click

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

Click

In the table, enter the following settings:


Name	Expression	Value	Description
disp	0[cm]	0 m	Displacement parameter
para	0	0	Parameter
G_0	190[MPa]	1.9E8 Pa	Initial shear modulus for Hostun dense soil
a	0.385	0.385	Material parameter

Soil Properties

In the toolbar, click

and choose > .

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

Locate the section. Click

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

In the toolbar, click

In the window for , type Hostun Dense Soil Properties in the text field.

In the toolbar, click

In the window for , type Hostun Loose Sand Properties in the text field.

Locate the section. Click

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

In the toolbar, click

In the window for , type Kaolin Clay Properties in the text field.

Locate the section. Click

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

Definitions

Interpolation 1 (int1)

In the toolbar, click

In the window for , locate the section.

In the text field, type appliedDisp.

In the table, enter the following settings:


t	f(t)
0	0
1	5e-3
2	-5e-3
3	5e-3

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


Function	Unit
appliedDisp	cm

In the table, enter the following settings:


Argument	Unit
t	1

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

Set up the first physics interface for the triaxial compression test and the second interface for the cyclic triaxial test.

Solid Mechanics [Monotonic]

In the window, under click .

In the window for , type Solid Mechanics [Monotonic] in the text field.

Click to expand the section. From the list, choose .

Hardening Soil Small Strain

In the toolbar, click

and choose .

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

Select Domain 1 only.

Locate the section. From the list, choose .

From the Γ(θ) list, choose .

From the ψm list, choose .

For the monotonic triaxial test, set the to to simplify the numerical computations and reduce computation time.

Locate the section. From the list, choose .

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.

From the list, choose .

In the u 0 z text field, type -disp.

Solid Mechanics [Cyclic]

In the window, under click .

In the window for , type Solid Mechanics [Cyclic] in the text field.

Locate the section. From the list, choose .

Hardening Soil Small Strain

In the toolbar, click

and choose .

For the cyclic test, set the shear modulus at small strains to be constant by applying the initial shear modulus of the domain material, G0.

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

Select Domain 1 only.

Locate the section. From the list, choose .

From the Γ(θ) list, choose .

From the ψm list, choose .

Find the subsection. From the G0 list, choose .

In the pref text field, type p0.

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.

From the list, choose .

In the u 0 z text field, type -appliedDisp(para).

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 failure stiffness	E50Ref	E50ref	Pa	Hardening soil
Reference stiffness for unloading and reloading	EurRef	Eurref	Pa	Hardening soil
Reference initial shear modulus	G0Ref	G0ref	N/m²	Hardening soil
Reference shear strain	gammaRef	gammaR	1	Hardening soil
Stress exponent	mh	m	1	Hardening soil
Cohesion	cohesion	c	Pa	Hardening soil
Initial void ratio	evoid0	e0	1	Critical state model
Initial shear modulus	G0	G_0	N/m²	Hardening soil