The well-known Modified Cam Clay material model is used worldwide to model clays and other soils that exhibit yielding and non-linear elastic-plastic deformaiton patterns.
This webinar will review this material model in SIGMA/W.
The Modified Cam Clay material model in SIGMA/W can be used to simulate clays and other soils that exhibit yielding and non-linear elastic-plastic deformation patterns. This webinar will introduce you to MCC model theory and formulation and step you through the analysis and calibration of a triaxial test in SIGMA/W. A comparison of laboratory data to the SIGMA/W results will also be presented.
<v Kathryn>Hello, and welcome to this GeoStudio</v>
Material Model webinars series on Modified Cam Clay.
My name is Kathryn Dompierre,
and I’m a Research and Development Engineer
with the GEOSLOPE engineering team here at Seequent.
Today’s webinar will be approximately 30 minutes long
attendees can ask questions using the chat feature.
I will respond to these questions via email
as quickly as possible.
A recording of the webinar will be available
so participants can review the demonstration
at a later time.
GeoStudio is a software package developed for geotechnical
engineers and earth scientists comprise of several products.
The range of products allows users to solve
a wide array of problems
that may be encountered in these fields.
Today’s webinar will focus on SIGMA/W.
Those looking to learn more about the products,
including background theory,
available features and typical modeling scenarios
can find an extensive library of resources
on the GEOSLOPE website.
Here, you can find tutorial videos,
examples with detailed explanations
and engineering books on each product.
To begin this webinar,
let us consider a soil element
using both the experimental approach
and a numerical approach.
And in experimental approach there’s a real soil sample
that is used for example, to conduct attracts yield test.
The responses of this soil sample are measured
in the laboratory and the soil behavior
can be studied accordingly.
And the numerical approach on the other hand,
it is necessary to perform a calibration process
for the constituent of model
as the first step of numerical modeling,
the measured results from the soil sample
are required at this step.
This calibration procedure provides model constants
for the soil.
These constants are then used as input parameters
for a numerical model
for example in SIGMA/W
and this simulated results will be obtained
by solving the numerical model.
The simulated results can then be compared
to the corresponding laboratory results
to verify the accuracy of the numerical modeling.
In today’s webinar the calibration procedure
for the modified Cam Clay material model will be provided.
This process will be applied to a specific type of Clay
as an example and the numerical values
of the model constants will be calculated.
The main steps of modeling attracts yield test
in SIGMA/W will be reviewed,
and the simulation results will be compared
to the experimental and analytical results
for validation purposes.
This webinar will begin by reviewing the theory
of the modified Cam Clay constituent of model.
Then step-by-step instructions will be provided
for calibrating this material model
using attracts field data series.
A demonstration of the steps of modeling attracts
yield test and SIGMA/W will be provided.
And the simulated results will become paired and verified
using the analytical and laboratory results.
The modified Cam Clay material model is a critical state
based non-linear constituent model
developed by Roscoe and Berlin in 1968.
In this model the author is modified
their previous constituent of model
by changing the geometry of the yield surface
from a bullet shape service to an ellipse.
For an isotropic loading condition
where the stress path is entirely located
on the horizontal axis.
The plastic strain increment vector is also horizontal
note that the plastic strain increment vector
is perpendicular to the yield surface.
The soil state is over consolidated
before reaching the yield surface.
After that the elliptical surface will be scaled up
as loading progresses.
The soil undergoes normal consolidation in this phase.
The soil response is assumed to be linear
in V versus lawn P prime space.
The slope of the normal consolidation line
is called the Virgin compression index or lambda
while the over consolidation and unloading, reloading lines
are assumed to be parallel.
Their slope is called the swelling index or Kappa.
In a deviatoric loading condition
where the vertical principles stress
is different than the horizontal one.
The stress path reaches the elliptical yield surface
at a point away from the horizontal axis.
This means that the plastic strain increment vector
contains deviatoric components as well.
In V versus lawn P prime space
the over consolidation and unloading, reloading phases
are still aligned with a slope of kappa.
However, the normal consolidation phase
is not necessarily a straight line
for a general and isotropic loading condition.
Before moving on to the demonstration,
we need to review some fundamental formulations.
As mentioned before the modified Cam Clay material model
is a critical state based constituent of model.
So different stress paths
eventually tend to the critical state line.
In this model the slope of the critical state line
or the critical state stress ratio
can be expressed in terms of the effective friction angle
in the more coolum failure criteria.
The yield surface is an ellipse
that passes through the origin
and intersects the critical state line
at its peak point.
The over consolidation ratio
is the ratio between the maximum isotropic stress
experienced by the soil
and the current isotropic stress of the soil.
For a stress state inside the yield surface
the soil response is elastic.
The compliance matrix is diagonal
and the bulk and shear modular are both proportional
to the mean effect of stress, P prime
and the specific volume V.
After reaching the yield surface and as it expands,
the soil response becomes elasto plastic
and non diagonal coupled components
appear in the compliance matrix.
Incremental strains in this condition
are the summation of elastic and plastic strains.
And so the current stress ratio plays a key role
in the soil response.
According to the framework proposed in the modified
Cam Clay model, the model constants include
the slope of the normal isotropic consolidation line
represented by lambda,
the slope of the over consolidation line or kappa,
the effective poisson’s ratio, new prime
and the effect of critical state friction angle, five-prime.
These four constants determined the soil type
based on the modified Cam Clay model.
Some other parameters will also be needed
to describe the initial conditions
of a specific soil sample.
These include the initial void ratio, the unit weight
and the over consolidation ratio.
In order to find the model constant
corresponding to a specific soil,
a step-by-step calibration procedure is discussed here
based on drain tracks, yield test results.
As the first step, the critical state stress ratio
M sub C or its equivalent,
the effect of critical state friction angle
can be estimated from drain tracks, yield test results.
According to critical state soil mechanics,
stress paths of samples
with different initial confining stresses
eventually tend to the critical state line
at their failure point.
Therefore, the critical state line
is a straight line that passes through the failure points.
The critical state stress ratio MC
is the slope of this line.
This constant can be calculated
using the method of least squares.
Since the failure criteria in the modified Cam Clay model
is the same as the more cooler model,
the effect of critical state friction angle five-prime
is not independent of the critical state stress ratio, MC
and can be expressed directly in terms of it.
To demonstrate the calibration procedure
let us consider three drained tracks yield compression tests
on Bothkennar clay that was conducted by McGinty in 2006.
These three tests have been performed
at different confining pressures.
The deviator compression stress
and these tests continued until failure was achieved.
The critical state line
that passes through the failure points
in P prime versus Q space has a slope of 1.36.
Thus, the critical state ratio MC is 1.36
and consequently, the effective friction angle
is calculated to be 33.7 degrees.
The second step of the calibration procedure
is to estimate the slope
of the over consolidation line or kappa.
For an over consolidated soil
kappa is the slope of the over consolidation line
and represents the elastic response of the sample.
The trends of the measured values of P prime and V
in this branch of the graph
can be projected to a straight line.
The slope of the Y intercept of this line
can be calculated directly
using the method of least squares.
An alternative method for estimating kappa
is to use the corresponding parameters
measured during an unloading, reloading process.
For a normally consolidated soil
the unloading reloading process
is required for estimating kappa.
The method of least squares can again be used
to calculate the slope and the Y intercept
of this straight line.
Let us apply the second calibration step
to the example, tracks yield test results
on Bothkennar clay mentioned earlier.
The over consolidation branch in each of these three curves
is projected to a straight line.
The slope of these three lines is almost the same
So their average 0.084 is used as the constant kappa
for this type of clay.
In the third step of the calibration procedure,
the slope of the normal isotropic consolidation line
or lambda will be estimated.
It should be noted that lambda’s the slope
of the normal consolidation line
only in an isotropic loading condition.
For other loading conditions however,
the normal consolidation branch
in V versus lawn P prime space is not necessarily linear.
And so the slope of its curve is obviously
not equal to lambda.
Based on the modified Cam Clay framework
it can be proved that there is an alternative space
in which the sample response is linear,
even under normal consolidation.
The X and Y axes in this space
are functions of the measured parameters
and constants estimated in the previous steps.
The over consolidation and unloading, reloading branches
are both horizontal in this new space.
And the normal consolidation branch
is a straight to sending line.
The slope of this line represents the difference
between the isotropic normal consolidation slope, lambda,
and the over consolidation slope, kappa.
Given that we already estimated kappa in the previous step,
the constant lambda can be found by applying
the method of least squares on the proposed space.
As shown in this figure,
the data points corresponding
to three example tracks yield test
are plotted in this alternate space.
The average value of the slopes of these three curves
in the normal consolidation branch
indicates the difference between lambda and kappa.
Since the value of kappa was previously obtained as 0.084
the constant lambda is estimated here to be 0.332.
The fourth step of the calibration procedure
is to calculate the over consolidation ratio
for each soil sample.
To do this, it is required to recognize
the mean effective pressure
in which the response of the sample changes
from the over consolidation condition
to the normal consolidation condition.
Let us refer to this pressure as the yield pressure
or P prime Y.
The over consolidation ratio however,
is the ratio between the isotropic
pre consolidation pressure P prime C,
and the isotropic initial pressure P prime zero.
As stated earlier the response of the soil changes
from an elastic or over consolidated state
to a elasto plastic or normally consolidated state
only if its stress path touches the yield surface.
As a result for a stress path that is not
necessarily isotropic the yield pressure P prime Y
is less than the isotropic
pre consolidation pressure P prime C
these two pressures however, are not independent.
For example, in a drain tracks yield stress pass
with a slope of three to one.
The deviatoric stress at the yield surface is three times
the difference between the yield and initial pressures.
So the isotropic pre-consultation pressure P prime C
can be expressed in terms of yield stresses
using the yield function of the elliptical surface.
Finally, the over consolidation ratio
is estimated as a ratio between
the isotropic pre consolidation pressure, P prime C,
and the isotropic initial pressure B prime zero.
This approach is applied on the Bothkennar clay
triaxial test data to estimate the over consolidation ratio
for each of the three samples.
First of all,
the yield points of the samples
in V versus lawn P prime space are detected.
The values of P prime Y for tests A1, A2 and A3
are about 83, 107 and 179 KPA respectively.
According to these yield pressures,
three yield surfaces are drawn
and the isotropic pre-consultation pressures
for three samples have been estimated as 115,
120 and 201 KPA respectively.
With this the over consolidation ratios for the samples
are found to be 2.069, 1.224 and 1.339.
Therefore the first sample is more over consolidated
than the other two samples.
This can also be deduced from the response of the sample
in V versus lawn P prime space,
where it has a longer elastic branch than the other samples.
The last step of the calibration procedure
for the modified Cam Clay constituent of model
is to determine the effective Poisson’s ratio, new prime.
Poissons ratio is defined as the ratio
of the change in the elements radio strain
to the change in its axial strain in a drained test.
In the modified Cam Clay model,
the Poisson’s ratio is considered only for the elastic
or over consolidated condition
and its value is assumed to remain constant during loading.
Consequently in a drain tracks yield test,
the method of least squares can be applied
on the purely elastic or over consolidated part of the curve
in radial strain versus axial strain space
to estimate the value of the effective Poisson’s ratio.
Lets us supply this final step to the Bothkennar clay
tracks yield test data.
The yield points of the samples have already been detected
in the previous steps and are marked with an X
in these figures.
As can be seen from the figure on the right
the sample response in this space is almost linear
before reaching the yield surface
while nonlinear behavior begins after yielding.
If the linear response of all three samples
was approximated by only one straight line
passing through the origin,
the slope of that line would be approximately 0.353.
This value is an estimation of the effective Poisson’s ratio
of the Bothkennar clay.
The results of the calibration procedure
for the Bothkennar clay can be summarized as follows
four model constants have been estimated,
including the slope of the normal
isotropic consolidation line,
the slope of the over consolidation line,
the effective Poisson’s ratio
and the effective critical state friction angle.
In addition the sample specific parameters
for these three tests include initial void ratios,
which were measured directly in the lab
and over consolidation ratios,
which were estimated in the calibration procedure.
These constants and parameters will be inputted in SIGMA/W
using the modified Cam Clay material model.
Now that we’ve reviewed the step-by-step
calibration procedure for the modified
Cam Clay material model,
I will describe how two model attract seal test in SIGMA/W
and compare the numerical results
with the previously mentioned laboratory data.
The numerical model configuration is based on the geometry
and boundary conditions of the tracks yield test
conducted in the laboratory.
A typical track yield test sample has a cylindrical shape
with a diameter of five centimeters
and a height of 10 centimeters.
Due to the axisymmetry shape of this geometry
with respect to the central vertical axis,
we can use the 2D axisymmetry geometry type in SIGMA/W.
The geometry is also symmetric with respect
to the horizontal axis.
So I will only model the top half
of the tracks yield sample.
An eight node Single element model
is considered for the resulting rectangle.
The left and bottom sides of this model
are fixed in the X and Y directions respectively,
while loads are applied to the other two sides.
The next step for setting up our SIGMA/W analysis
is the material definition.
This step is performed in the defined materials dialogue.
The material model is set to modified Cam Clay.
The sample parameters including the initial void ratio
and over consolidation ratio
are entered as discussed earlier.
It should be noted that the effect of the soil self weight
can be removed by setting the unit weight to zero.
While the key note effect can be deactivated
by specifying Key note and C as one.
Lastly, the constitutive model constants are entered
based on the results from the calibration procedure
for both Cam Clay.
The simulated tracks yield models are then loaded
under first, the isotropic confining pressure
and secondly the deviatoric pressure,
thus there are two SIGMA/W analysis required
for each of the triaxial tests.
The confining pressure analysis acts as the parent
and its results are used as the initial data
for the displacement controlled deviatoric analysis.
These conditions are established
using the boundary condition
specified in the defined boundary conditions dialogue.
For example, in test A1 the constant normal stress
of 58 KPA is applied on both the right and top sides
of the model using a normal tan stress boundary type
for the confining pressure analysis.
In the deviatoric analysis,
a displacement type boundary condition
is applied to the top of the model.
A displacement of two centimeters that is equivalent
to 40% strain is applied linearly over time
using a splined data point function.
All the analysis are then solved in SIGMA/W
and the stress, strain, displacement
and other responses of the model
can be extracted and interpreted.
As shown here a 2D axisymmetry geometry
is used to set up the model domain.
In the 2D view only one quarter of the cross-section
running through the center of the tracks yield sample
is modeled given the symmetry of the domain.
In the analysis tree, there are three sets of analysis
each representing attracts yield test
conducted on the Bothkennar clay samples.
In each test the confining phase is the parent analysis
and the deviatoric phase
that use the results of the previous phase
is the child analysis.
The material properties for three soil samples
were obtained using the outlined calibration procedure.
The materials are specified
in the defined materials dialogue.
The same type of clay
was subjected to the tracks yield tests.
Therefore, the Constance of the modified Cam Clay model
are similar for each sample.
However, at the initial void ratio
and over consolidation ratio are not the same
for each sample
and so three different materials were defined.
And isotropic elastic material was used
during the first phase of the tracks yield simulations,
because the nonlinear stress strain response
is inconsequential to establishing the initial stresses.
The accumulated displacements and strains are reset
at the start of the loading phase of the simulation.
The three isotropic elastic materials defined here
are applied to the corresponding, confining analysis.
The defined modified Cam Clay materials
are applied during the deviatoric portion of the tracks
yield test simulations.
The boundary conditions representing the confining pressures
and deviatoric strain are specified
in defined boundary conditions.
Three constant confining pressures were created
for the three tracks yield tests
using the normal 10 stress boundary type.
These boundary conditions
are applied to the corresponding confining analysis.
The deviatoric strain boundary for the displacement
control loading phase of the tracks yield test
uses the force displacement boundary type
with a displacement function defined
such that displacement increases
over the deviatoric strain analysis
from zero to two centimeters.
Before solving an analysis
the final step is to review the finite element mesh.
In the draw mesh properties dialog
we see that the global element mesh size is 0.05 meters
thus, the model domain represents one element.
This analysis has already been solved
so I will move to the results view.
I will go to the first deviatoric analysis
to investigate the results.
In the draw graph dialog, I have created graphs
to illustrate the stress path, stress strain curve,
and void ratio versus mean effect of stress.
I can click through the different analysis
to view the results from each.
Let us now compare these results
with the analytical solution
and also with the corresponding laboratory data.
The different simulation results of test A1
are compared here with the corresponding laboratory results.
Discrete scatter points represent the laboratory results
and the continuous black lines
are the SIGMA/W simulation results.
Analytical results for the modified Cam Clay model
are also shown in these diagrams as orange dashed lines.
The full compatibility of the analytical
and numerical curves shows the reliability
of the implementation process
of this material model in SIGMA/W.
When comparing the laboratory results
with the constituent of model simulations,
some of the plots show a better fit than others.
For instance, a very good match
can be seen in the first curve
where the data in V versus lawn P prime space
is simulated by the modified Cam Clay model.
However, the results from test A1
in some of the other spaces
do not match as nicely as the first curve
because of the very nature of calibration.
The calibration procedure highlighted in this webinar
required multiple samples,
which each have varying degrees of disturbance.
For example, during the calibration process,
the best fit line that was used to determine MC
was below the A1 and A3 data points,
but near the A2 data point,
thus, the simulated deviatoric failure stress
was underestimated for the A1 sample.
Meanwhile, the simulated deviatoric failure stress
better correlates to the laboratory results from test A2
and underestimates the deviatoric failure stress
for test A3.
This demonstrates the nature of the calibration process.
However, as mentioned for the first test,
the analytical and numerical results show a perfect match
verifying that SIGMA/W accurately represents
the modified Cam Clay material model.
In this webinar,
the calibration procedure of the modified Cam Clay
material model was provided in five straightforward steps.
The results of the drain tracks yield test,
where the only laboratory data set required for calibration.
The identified material constants
were used in a SIGMA/W numerical model,
and the simulation results were found to compare favorably
with both the analytical solution
and corresponding laboratory results.
Here are other references discussed in this webinar.
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