Isotropic Compression with Modified Cam-Clay Material Model

Isotropic compression is a common exercise in soil testing. 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.

The Extended Barcelona Basic (BBMx) model with zero suction reduces to the Modified Cam-Clay model. While, setting initial structural strength and additional void ratio to zero, and plastic potential shape parameter to two, the Modified Structured Cam-Clay (MSCC) model reduces to the Modified Cam-Clay model. With these choice of material parameters, we verify that the Extended Barcelona Basic and Modified Structured Cam-Clay material models replicate the behavior shown by the Modified Cam-Clay material model.

Model Definition

In this example, a soil sample is placed inside a cylinder 10 cm in diameter and 10 cm in height, see Figure 1. Due to the symmetry, the model is solved in 2D axial symmetry. A boundary load produces isotropic compression conditions.

Figure 1: Dimensions, boundary conditions, and boundary load 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 intex	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 Extended Barcelona Basic model 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	104
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

Common material properties are same as for the MCC model. Properties that are specific to the Modified Structured Cam-Clay 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 soil is not structured.

Constraints and Loads

•

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

•

The boundary load is applied in three steps: First the pressure increases from 0.5p0 to 3p0, then the sample is unloaded until the pressure reaches 1.5p0, and finally the pressure increases again until it reaches 4p0.

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

Results and Discussion

The produced void ratio versus pressure is shown in Figure 2. Note that the log operator is implemented in base “e” and not in base “10”.

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

For the parametric sweep between 0 and 0.2 (the boundary load ranges from 100 kPa to 300 kPa), the curve follows the slope defined by the swelling index κ.

Once the user-defined consolidation pressure is reached (pc0 = 300 kPa), the sample of 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 pressure p = 900 kPa.

Figure 2 reproduces characteristic curves showing 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 normal compression line the void ratio is e = eref.

Figure 3: Void ratio versus pressure in the isotropic compression test.

Figure 3 shows the variation of the void ratio versus pressure for Modified Cam-Clay model, the Extended Barcelona Basic model, and Modified Structured Cam-Clay model. The predicted behavior is the same for all soil models. This verifies the correctness of the BBMx model for saturated soils and MSCC model for destructured soil, where it portraits similar curves in the void ratio versus pressure space as shown by the MCC model.

As expected, once the stress level reaches the modified Cam-Clay ellipse in the stress space, (p = pc0, parameter 0.2), the soil sample starts deforming plastically. Isotropic hardening expands the major semi-axis of the ellipse, with the expansion given 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 material models.

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

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New

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Model Wizard

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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
para	0	0	Parameter
p0	200[kPa]	2E5 Pa	Pressure

Definitions

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
0.2	1*p0
0.4	2*p0
0.6	1*p0
0.8	2*p0
1	4*p0

Locate the section. In the text field, type 1.

In the text field, type Pa.

Click

The interpolation function is used to define the boundary load. The boundary load first compresses the soil sample, then it relaxes, and finally it compresses the sample again.

Geometry 1

Rectangle 1 (r1)

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In the window for , locate the section.

In the text field, type 5[cm].

In the text field, type 10[cm].

Click

Definitions

Integration 1 (intop1)

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and choose .

In the window for , locate the section.

From the list, choose .

Select Point 4 only.

Locate the section. From the list, choose .

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].

Roller 1

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and choose .

Select Boundary 2 only.

Boundary Load 1

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and choose .

Select Boundaries 3 and 4 only.

In the window for , locate the section.

From the list, choose .

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

The load function goes from 0 to 4*p0 with an unloading/reloading loop.

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
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

Mesh 1

Mapped 1

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Size

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Study: MCC

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Step 1: Stationary

Set up an auxiliary continuation sweep for the para parameter.

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In the table, enter the following settings:


Parameter name	Parameter value list	Parameter unit
para (Parameter)	range(0,0.01,1)

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Results

Void Ratio (MCC)

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and choose .

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

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

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

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

Select the check box.

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

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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.

In order to plot the e vs. ln (p) characteristic plot, use a log scale for the x-axis.

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Point Graph 1

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Annotation 1

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In the text field, type p=eval(intop1(solid.pm), kPa) kPa.

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Locate the section. In the text field, type intop1(solid.pm)/1000.

In the text field, type intop1(solid.epsm1.evoid).

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

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Annotation 2

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Annotation 3

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Annotation 4

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Annotation 5

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Point Graph 2

In the window, under right-click and choose .

In the window for , locate the section.

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

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

Void Ratio (MCC)

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Table Annotation 1

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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

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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).

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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 .

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

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In the window for , locate the section.

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

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


Legends
Analytical

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Add an Extended Barcelona Basic Material Model with zero suction in order to replicate the behavior shown by the Modified Cam-Clay Model.

Solid Mechanics (solid)

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. From the list, choose .

In the s text field, type 0.

The common material parameters should be kept the same. The additional material parameters in the BBMx model related to the suction are chosen randomly since the suction will be kept zero throughout the analysis.

Materials

Soil Material (mat1)

In the window, under click .

In the window for , 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

Study: MCC

Step 1: Stationary

Disable the BBMx model in the first study.

In the window, under click .

In the window for , locate the section.

Select the check box.

In the tree, select .

Click

Add a second study for the BBMx model and disable the MCC model in this study.

Add Study

In the toolbar, click

to open the window.

Go to the window.

Find the subsection. In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Study: BBMx

In the window, click .

In the window for , type Study: BBMx in the text field.

Locate the section. Clear the check box.

Step 1: Stationary

In the window, click .

In the window for , locate the section.

Select the check box.

In the tree, select .

Click

Locate the section. Select the check box.

Click

In the table, enter the following settings:


Parameter name	Parameter value list	Parameter unit
para (Parameter)	range(0,0.01,1)

In the toolbar, click

In order to replicate the results of the Modified Cam-Clay model with the Modified Structured Cam-Clay Material Model, set the initial structural strength and the additional void ratio to zero. Also set the plastic potential shape parameter to 2.

Solid Mechanics (solid)

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. 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 .

The common material parameters should be kept the same as the MCC model. The additional material parameters in the MSCC model are chosen randomly as they will not affect the analysis.

Materials

Soil Material (mat1)

In the window, under click .

In the window for , 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