Isotropic Compression with Modified Cam-Clay Material Model

Isotropic compression is a common material test for soils. In this example, the Modified Cam-Clay (MCC) soil model is examined; in particular the relation between the void ratio and the logarithm of the hydrostatic pressure or mean stress is studied.

In COMSOL Multiphysics, several soil plasticity material models are implemented. With particular choices of parameters, some of these can be reduced to the MCC model. For example, the Extended Barcelona Basic model (BBMx) with zero suction reduces to the Modified Cam-Clay model. By setting the initial structural strength and the additional void ratio to zero and the plastic potential shape parameter to two, the Modified Structured Cam-Clay (MSCC) model reduces to the MCC model. With these choices of material parameters, we can verify that the BBMx and MSCC models replicate the behavior of the MCC model.

Model Definition

In this example, the test specimen is a cylindrical soil sample of 10 cm in diameter and 10 cm in height, see Figure 1. Due to the symmetry, the model is solved in 2D axisymmetry. A boundary load produces isotropic compression conditions.

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

Modified Cam-Clay Material Properties

Table 1: Material properties for THE Modified CAm-Clay material model.
Property	Variable	Value
Density	ρ	2400 kg/m3
Shear Modulus	G	10 MPa
Angle of internal friction		30°
Swelling index	κ	0.013
Compression index	λ	0.032
Void ratio at reference pressure	eref	0.7
Reference pressure	pref	100 kPa
Initial consolidation pressure	pc0	300 kPa

Extended Barcelona basic Material Properties

Common material properties are the same as for the MCC model. Properties that are specific to the BBMx are listed in Table 2.

Table 2: Material properties for The extended Barcelona basic material model.
Property	Variable	Value
Suction	σ	0
Swelling index for changes in suction	κs	0.0013
Compression index for changes in suction	λs	0.0032
Weight parameter	w	0.75
Soil stiffness parameter	m	10 kPa
Plastic potential smoothing parameter	bs	100
Tension to suction ratio	k	0.6
Initial yield value for suction	sy0	0.3 MPa

Note that material parameters related to the suction are chosen arbitrarily and does not affect the results since the suction is set to zero.

Modified structured Cam-Clay Material Properties

The material parameters in common for the MSCC and MCC models are set identical.The properties that are specific to the MSCC model are listed in Table 3.

Table 3: Material properties for The Modified Structured CAm-Clay material model.
Property	Variable	Value
Initial structure strength	pbi	0
Plastic potential shape parameter	ζ	2
Additional void ratio at initial yielding	Δei	0
Destructuring index for volumetric deformation	dv	1
Destructuring index for shear deformation	ds	1
Critical effective deviatoric plastic strain	εdcp	0.02

Note that material parameters related to the structuring are chosen arbitrarily and does not affect the results since the structure strength will be set to zero in the Elastoplastic Soil feature.

Constraints and Loads

•

The left boundary of the computational domain is the axis of symmetry. A roller condition is applied at the lower boundary, and a boundary load is applied on the right and the top boundaries.

•

The boundary load is applied in three steps: First the pressure increases from 0.5p0 to 3p0. Next, the pressure is reduced to 1.5p0, and finally the pressure increases again up to 4p0.

In order to reproduce the analytical results of Ref. 1, the load is controlled in a parametric analysis.

Results and Discussion

The relation between void ratio and pressure is shown in Figure 2. Note that the pressure is plotted on a logarithmic scale.

Figure 2: Void ratio as a function of the pressure in an isotropic compression test.

For the pressures from 100 kPa to 300 kPa, the curve follows the slope defined by the swelling index κ. Once the consolidation pressure is reached (pc0 = 300 kPa), the soil behaves plastically, and the curve follows the slope defined by the compression index λ. During the unloading and reloading of the soil (between the parameters 0.4 and 0.8), the curve in Figure 2 follows the elastic slope defined by the swelling index κ. Finally, the soil is compressed between the parameters 0.8 and 1, and it undergoes plastic deformation until it reaches its final stage at a void ratio e0 = 0.630 at p = 900 kPa.

Figure 2 reproduces characteristic curves called the Normal Compression Line (NCL) and the Swelling Line (or Unloading/Reloading Line URL). The NCL has a slope defined by the compression index λ, and at p = pref on the NCL the void ratio is e = eref.

Figure 3: Void ratio versus pressure in the isotropic compression test for all three material models.

Figure 3 shows the variation in the void ratio with applied pressure for all three models. The behavior predicted is the same for all soil models, which verifies the correctness of the BBMx model for the case of saturated soils and that of the MSCC model for destructured soils.

Once the stress level reaches the MCC ellipse in stress space, (p = pc0, parameter para = 0.2), the soil starts deforming plastically. Isotropic hardening expands the major semi-axis of the ellipse, with the expansion determined by the increase in consolidation pressure, see Figure 4. During the unloading–reloading steps (between parameter values 0.4 and 0.8), the consolidation pressure is kept constant.

The changes in consolidation pressure with respect to the boundary load is identical for all three material models, as expected.

Figure 4: Increase in consolidation pressure due to isotropic hardening.

Reference

1. W.F. Chen and E. Mizuno, Nonlinear Analysis in Soil Mechanics, Elsevier, 1990.

Geomechanics_Module/Verification_Examples/isotropic_compression

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
para	0	0	Parameter
p0	200[kPa]	2E5 Pa	Pressure

Boundary Load

In the toolbar, click

and choose .

In the window for , locate the section.

In the text field, type Pressure.

In the text field, type Boundary Load.

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


t	f(t)
0	0
0.4	2*p0
0.6	1*p0
0.8	2*p0
1	4*p0

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


Argument	Unit
t	1

In the table, enter the following settings:


Function	Unit
Pressure	Pa

Click

An interpolation function is used to define the boundary load. First, a pressure of 2*p0 is applied. Next, the load is reduced to p0 followed by a reloading to 2*p0 and finally an increase up to 4*p0.

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].

Array 1 (arr1)

In the toolbar, click

and choose .

Select the object only.

In the window for , locate the section.

In the text field, type 3.

Locate the section. In the text field, type 15[cm].

Click

Click the

button in the toolbar.

Add the three different material models. Start with the MCC model.

Solid Mechanics (solid)

Modified Cam-Clay Model (MCC)

In the window, under right-click and choose .

In the window for , type Modified Cam-Clay Model (MCC) in the text field.

Select Domain 1 only.

Locate the section. From the list, choose .

From the M list, choose .

In the pc0 text field, type 300[kPa].

Next, add an BBMx model with zero suction in order to replicate the MCC behavior.

Extended Barcelona Basic Model (BBMx)

In the window, right-click and choose .

In the window for , type Extended Barcelona Basic Model (BBMx) in the text field.

Locate the section. Click

Select Domain 2 only.

Locate the section. From the list, choose .

In the s text field, type 0.

Last, add a MSCC model. Set the initial structural strength and the additional void ratio to zero. Also, set the plastic potential shape parameter to 2.

Modified Structured Cam-Clay Model (MSCC)

In the window, right-click and choose .

In the window for , type Modified Structured Cam-Clay Model (MSCC) in the text field.

Locate the section. Click

Select Domain 3 only.

Locate the section. From the list, choose .

From the M list, choose .

From the pbi list, choose . From the ζ list, choose . In the associated text field, type 2.

From the Δei list, choose .

Add one feature for each of the three models. Note that the material properties in common should be kept the same between all models. The parameters unique to the BBMx and MSCC models are set arbitrarily, as they have no influence on the solution because of the settings chosen in the corresponding nodes under .

Materials

Modified Cam-Clay Material

In the window, under right-click and choose .

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

Locate the section. Click

Select Domain 1 only.

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


Property	Variable	Value	Unit	Property group
Shear modulus	G	10[MPa]	N/m²	Bulk modulus and shear modulus
Swelling index	kappaSwelling	0.013	1	Cam-Clay
Compression index	lambdaComp	0.032	1	Cam-Clay
Void ratio at reference pressure	evoidref	0.7	1	Cam-Clay
Angle of internal friction	internalphi	30[deg]	rad	Mohr-Coulomb
Density	rho	2400[kg/m^3]	kg/m³	Basic

Extended Barcelona Basic Material

In the window, under right-click and choose .

In the window for , type Extended Barcelona Basic Material in the text field.

Locate the section. Click

Select Domain 2 only.

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


Property	Variable	Value	Unit	Property group
Swelling index	kappaSwelling	0.013	1	Barcelona Basic
Swelling index for changes in suction	kappaSwellings	0.0013	1	Barcelona Basic
Compression index at saturation	lambdaComp0	0.032	1	Barcelona Basic
Compression index for changes in suction	lambdaCompss	0.0032	1	Barcelona Basic
Weight parameter	wB	0.75	1	Barcelona Basic
Soil stiffness parameter	mB	1e4	Pa	Barcelona Basic
Plastic potential smoothing parameter	bB	100	1	Barcelona Basic
Tension to suction ratio	kB	0.6	1	Barcelona Basic
Void ratio at reference pressure and saturation	evoidref0	0.7	1	Barcelona Basic
Initial yield value for suction	sy0	0.3[MPa]	Pa	Barcelona Basic

Modified Structured Cam-Clay Material

In the window, under right-click and choose .

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

Locate the section. Click

Select Domain 3 only.

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


Property	Variable	Value	Unit	Property group
Swelling index for structured clay	kappaSwellingS	0.013	1	Structured Cam-Clay
Compression index for destructured clay	lambdaCompS	0.032	1	Structured Cam-Clay
Void ratio at reference pressure for destructured clay	evoidrefS	0.7	1	Structured Cam-Clay
Destructuring index for volumetric deformation	dvS	1	1	Structured Cam-Clay
Destructuring index for shear deformation	dsS	1	1	Structured Cam-Clay
Critical equivalent deviatoric plastic strain	epdevc	0.02	1	Structured Cam-Clay

Use the feature to carry out isotropic compression tests on the three different material models.

Solid Mechanics (solid)

Test Material [Modified Cam-Clay Model]

In the toolbar, click

and choose .

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

Select Domain 1 only.

Locate the section. In the Np text field, type 100.

From the list, choose .

Find the subsection. Clear the check box.

Select the check box.

In the p text field, type -Pressure(para).

Click in the upper-right corner of the section. From the menu, choose .

Test Material [Extended Barcelona Basic Model]

In the window, right-click and choose .

In the window for , type Test Material [Extended Barcelona Basic Model] in the text field.

Locate the section. Click

Select Domain 2 only.

Click in the upper-right corner of the section. From the menu, choose .

Test Material [Modified Structured Cam-Clay Model]

In the window, right-click and choose .

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

Locate the section. Click

Select Domain 3 only.

Click in the upper-right corner of the section. From the menu, choose .

Results

In the window, expand the node.

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

Right-click and choose .

Void Ratio (MCC)

In the toolbar, click

and choose .

In the window for , type Void Ratio (MCC) in the text field.

Locate the section. From the list, choose .

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

In the text area, type Void Ratio vs. Pressure during the Isotropic Compression Test.

Locate the section.

Select the check box. In the associated text field, type Pressure (kPa).

Select the check box. In the associated text field, type Void ratio (1).

Locate the section. Select the check box.

In the text field, type 95.

In the text field, type 1300.

In the text field, type 0.62.

In the text field, type 0.706.

Set the x-axis in the e vs. p plot to logarithmic.

Select the check box.

Point Graph 1

Right-click and choose .

Select Point 1 only.

In the window for , click in the upper-right corner of the section. From the menu, choose .

Click in the upper-right corner of the section. From the menu, choose .

Locate the section. From the list, choose .

In the toolbar, click

Annotation 1

In the window, right-click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. In the text field, type p=eval(at3(0,0,0,solid1.pm), kPa) kPa.

From the list, choose .

Locate the section. In the text field, type at3(0,0,0,solid1.pm)/1000.

In the text field, type at3(0,0,0,solid1.epsm1.evoid).

Click to expand the section. In the text field, type 3.

Locate the section. From the list, choose .

Annotation 2

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Annotation 3

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Annotation 4

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Annotation 5

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Point Graph 2

In the window, under right-click and choose .

In the window for , locate the section.

In the text field, type solid1.epsm1.evoidref-solid1.epsm1.lambdaComp*log(solid1.epsm1.p/solid1.epsm1.pref).

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

Void Ratio (MCC)

In the window, click .

Table Annotation 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

In the table, enter the following settings:


x-coordinate	y-coordinate	Annotation
160	0.689	Normal compression line
145	0.67	Swelling line
255	0.653	Swelling line

Locate the section. Clear the check box.

Select the check box.

In the toolbar, click

Void Ratio (MCC), Numerical Vs. Analytical

In the toolbar, click

and choose .

In the window for , type Void Ratio (MCC), Numerical Vs. Analytical in the text field.

Locate the section. From the list, choose .

In the text area, type Void Ratio vs. Pressure during the Isotropic Compression Test.

Locate the section.

Select the check box. In the associated text field, type Pressure (kPa).

Select the check box. In the associated text field, type Void ratio (1).

Locate the section. Select the check box.

Point Graph 1

Right-click and choose .

Select Point 1 only.

In the window for , click in the upper-right corner of the section. From the menu, choose .

Click in the upper-right corner of the section. From the menu, choose .

Locate the section. From the list, choose .

From the list, choose .

Click to expand the section. Select the check box.

From the list, choose .

In the table, enter the following settings:


Legends
Numerical

Point Graph 2

Right-click and choose .

In the window for , locate the section.

In the text field, type solid1.epsm1.evoidref-(solid1.epsm1.lambdaComp-solid1.epsm1.kappaSwelling)*log(solid1.epsm1.pc/solid1.epsm1.pref)-solid1.epsm1.kappaSwelling*log(solid1.epsm1.p/solid1.epsm1.pref).