Triaxial and Oedometer Test with Modified Cam-Clay Material Model

The Modified Cam-Clay (MCC) model is a popular constitutive model for soft soils and clays. The MCC model has a nonlinear relation between stress and strain with a smooth yield surface. In this example, simulations of triaxial tests are carried out to examine the constitutive relation of the MCC model, as it is originally developed for triaxial loading conditions. The oedometer test is also an important test in the field of geomechanics, which is used frequently to determine the material parameters of soils. In this example, the drained triaxial compression test and the oedometer test presented in Ref. 1 are simulated.

The Cam-Clay family of models do not have any stiffness at zero stress; hence, it always starts with an initial mean stress. In COMSOL Multiphysics, the initial mean stress of the MCC model is equal to the reference pressure. The MCC model comes in two flavors: it either requires the specification of a constant shear modulus or a constant Poisson’s ratio. In this example, the constant Poisson’s ratio formulation is used in order to match the results in Ref. 1. Although the analysis presented in Ref. 1 is transient, a stationary analysis is sufficient to predict the behavior.

Model Definition

In both the triaxial and oedometer tests, a cylindrical soil specimen of 3.91 cm in diameter and 8 cm in height is used. For the triaxial test (see Figure 1), a confinement pressure is applied to create a state of isotropic compression, and later the soil sample is compressed axially. For the oedometer test, the bottom and side boundaries of the cylindrical specimen are constrained in the normal direction and an axial load is applied on the top boundary.

Soil Properties

The soil properties for the MCC material model as given in Ref. 1 are

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Density ρ = 2400 kg/m3, Poisson’s ratio ν = 0.35, slope of critical state line M = 1.2, swelling index κ = 0.02, compression index λ = 0.1, void ratio eref = 1 at a reference pressure pref = 98 kPa, and initial consolidation pressure pc0 = 100 kPa or 500 kPa.

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

Constraints and Loads

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For the isotropic compression stage in the triaxial test, a mean stress of 100 kPa is maintained throughout the test. As the MCC model has an initial mean stress equal to the reference pressure, an additional pressure of 2 kPa is applied using an node with the in-situ stress option.

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For the axial compression stage in the triaxial test, the soil sample is compressed by applying a prescribed displacement on the top boundary. Allow the right boundary to expand freely in the radial direction, and apply a roller boundary condition at the bottom boundary.

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For the axial compression stage in the oedometer test, the soil sample is compressed using a boundary load on the top boundary. A roller boundary condition is applied on the bottom and right vertical boundaries in order simulate confinement (zero radial strain).

Results and Discussion

Note that for the sake of consistency with geomechanics conventions, the compressive axial stress and strain are plotted along the positive axis, whereas the tensile stress and strain are plotted along the negative axis in all the figures. The response of the MCC model with different overconsolidation ratios (OCRs) are plotted in the same figures for comparison purposes.

The OCR is a ratio between the initial consolidation pressure to the initial mean effective stress. For the oedometer test in Ref. 1, the OCR is the ratio of the initial consolidation pressure to the initial vertical load. A soil with an OCR equal to 1 is referred to as normally consolidated. When the OCR is equal to 5 or 50, the soil is instead referred to as highly overconsolidated.

The variation in von Mises stress versus axial strain with different OCRs is shown in Figure 2. The stress-strain curve is nonlinear, which is a characteristic of the MCC model. As the axial displacement increases, the von Mises stress increases hyperbolically and approaches a critical state asymptotically. When the soil attains the critical state, additional loading does not produce any volume changes or hardening. The response of the soil in Figure 2 matches very closely with the numerical results given in Ref. 1 (see Figure 7 in Ref. 1).

Figure 2: von Mises stress versus axial strain.

The variation in the total volumetric strain versus the axial strain with different OCRs can be seen in Figure 3, which matches very closely with the numerical results given in Ref. 1 (see Figure 8 in Ref. 1). For normally consolidated soils (OCR = 1), the total volumetric strain remains compressive. In contrast, the volumetric response for highly overconsolidated soils turns tensile after an initial compressive phase. This counterintuitive behavior can be further explained by Figure 4 and Figure 5.

Figure 3: Volumetric strain versus axial strain.

The evolution of the consolidation pressure and the volumetric plastic strains is shown in Figure 4 and Figure 5, respectively. For normally consolidated soils, the consolidation pressure increases, indicating that the final yield envelope is expanding. This, in turn, gives compressive volumetric plastic strains, see Figure 5. This behavior is called isotropic hardening. For highly overconsolidated soils, the consolidation pressure is decreasing, indicating the shrinking of the final yield envelope, which in turn develops tensile volumetric plastic strains, see Figure 5. This behavior called isotropic softening

Figure 4: Consolidation pressure versus axial strain.

In COMSOL Multiphysics, the reference pressure acts as an initial stress and needs to be nonzero. For the oedometer test in Ref. 1, there seems to be no initial stress. Hence, for the corresponding test in COMSOL Multiphysics, set the reference pressure to 1 kPa. The void ratio at the reference pressure is calculated based on its value at 98 kPa as

The variation in the void ratio versus the logarithm of the vertical load for the highly overconsolidated soil in the oedometer test is plotted in Figure 6, which matches qualitatively with numerical results given in Ref. 1 (see Figure 6 in Ref. 1). The slight difference in the results can be due to the different initial conditions.

The variation in the total volumetric strain versus the axial load for the highly overconsolidated soil in the oedometer test can be seen in Figure 7. When the yield limit is reached, the response is nonsmooth.

Figure 5: Volumetric plastic strain versus axial strain.

Figure 6: Void ratio versus the axial load (log-scale).

Figure 7: Volumetric strain versus axial load.

Notes About the COMSOL Implementation

In COMSOL Multiphysics, the MCC model comes in two flavors: it either requires specification of constant shear modulus or a constant Poisson’s ratio. For the constant shear modulus option, the Poisson’s ratio is computed based on the bulk modulus and the given shear modulus. This Poisson’s ratio depends on deformation, but does not enter into the constitutive relation and remains only as a postprocessing variable. For the constant Poisson’s ratio option, the shear modulus is calculated from the bulk modulus and the Poisson’s ratio. This variable shear modulus does enter into the constitutive relation.

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, with a combination of and nodes. The second method is to use the in-situ stress option in an 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 triaxial test.

Reference

1. G. Ye and B. Ye, “Investigation of the Overconsolidation and Structural Behavior of Shanghai Clays by Element Testing and Constitutive Modeling,” Underground Space, vol. 1, pp. 62-77, 2016.

Geomechanics_Module/Verification_Examples/triaxial_and_oedometer_test_mcc

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.

In the table, enter the following settings:


Name	Expression	Value	Description
disp	0[cm]	0 m	Axial displacement
p0	2[kPa]	2000 Pa	Pressure
OCR	5	5	Overconsolidation ratio
F	10[kPa]	10000 Pa	Axial load
isOedometerTest	0	0	Boolean for oedometer test

Add variables for the reference void ratio and the reference pressure, which are different for the two tests.

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
e_ref	1(1-isOedometerTest)+1.4584isOedometerTest		Void ratio at reference pressure
p_ref	98[kPa](1-isOedometerTest)+1[kPa]isOedometerTest	Pa	Reference pressure

Geometry 1

Rectangle 1 (r1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 1.955[cm].

In the text field, type 8[cm].

Click

Solid Mechanics (solid)

Modified Cam-Clay Material Model

In the window, under right-click and choose .

In the window for , type Modified Cam-Clay Material Model in the text field.

Select Domain 1 only.

Locate the section. From the eref list, choose . In the associated text field, type e_ref.

In the pref text field, type p_ref.

In the pc0 text field, type 100[kPa]*OCR.

The triaxial test is carried out in two steps. The first step is needed to get the initial stress state of the sample, and the second step is an axial compressive loading. The initial stress state can be modeled using the option of the node.

External Stress [Triaxial Test]

In the toolbar, click

and choose .

In the window for , type External Stress [Triaxial Test] in the text field.

Locate the section. From the list, choose .

In the σins text field, type -p0.

Roller [Triaxial Test]

In the toolbar, click

and choose .

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

Select Boundary 2 only.

Prescribed Displacement [Triaxial Test]

In the toolbar, click

and choose .

In the window for , type Prescribed Displacement [Triaxial Test] in the text field.

Select Boundary 3 only.

Locate the section. Select the check box.

In the u 0 z text field, type disp.

Roller [Oedometer Test]

In the toolbar, click

and choose .

Select Boundaries 2 and 4 only.

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

Boundary Load [Oedometer Test]

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 F.

In the text field, type Boundary Load [Oedometer Test].

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	0.35	1	Young’s modulus and Poisson’s ratio
Swelling index	kappaSwelling	0.02	1	Cam-Clay
Compression index	lambdaComp	0.1	1	Cam-Clay
Slope of critical state line	M	1.2	1	Cam-Clay
Density	rho	2400[kg/m^3]	kg/m³	Basic