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The NorSand constitutive model in SIGMA/W can be used to simulate loose and dense sand behaviour in both drained and undrained conditions.

This webinar will introduce you to the NorSand material model and walk you through the parameterisation process using practical examples in GeoStudio.

The NorSand constitutive model is a reference for static liquefaction modelling of loose sands. Through its formulation based on critical state theory, it can adequately simulate loose and dense sand behaviour, both in drained and undrained conditions. This webinar introduces the NorSand soil model and it’s implementation in SIGMA/W. A practical example showing the parameterisation of the model using triaxial compression tests is included.



Vincent Castonguay


45 min

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Video Transcript

<v Vincent>Hello, and welcome to this GeoStudio webinar,</v>

part of our SIGMA/W material model series.

Today’s webinar will focus on a new addition to SIGMA/W,

the NorSand soil model.

I’m Vincent Castonguay,

and I work as a research and development specialist

with the GEOSLOPE engineering team here at Seequent.

Today’s webinar will be approximately 60 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 also be available

so participants can review the demonstration

at a later time.

GeoStudio is a software package

developed for geotechnical engineers

and earth scientists comprised of several products.

The range of products allow users

to solve a wide array of problems

that may be encountered in these fields.

Today’s webinar relates specifically to 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.

There you can find tutorial videos,

examples with detailed explanations,

and engineering books on each product.

The successful use

of numerical tools generally comprises six steps,

which broadly speaking are first conceptualization

of the physical system and expected response.

Second, selection of their relevant physics.

Third, parametrization of appropriate constitutive models.

Fourth, application of boundary conditions.

Fifth, solving the physics subject

to the boundary conditions and considerative loss.

And sixth, verification of the results.

Today’s webinar focuses on step three

of the modeling process.

More specifically, we are going to explore

how to parameterize the NorSand soil model in SIGMA/W.

This webinar will be divided into three parts.

We will first review the most important aspects

of the NorSand model.

We will then explore the calibration

of the model’s input parameters.

Finally, NorSand’s implementation

in SIGMA/W will be presented.

Underlying many aspects of this webinar

will be practical examples of triaxial compression tests.

Users are encouraged to visit Geoslope’s website

to download the triaxial test

on NorSand soil example to serve as a reference.

A set of laboratory data published

by Jefferies and collaborators

is used throughout this webinar

to evaluate NorSand’s performances using real life examples.

This dataset includes seven drained

and five undrained triaxial compression tests performed

on the Fraser River sand.

These drained and non-drained triaxial tests

were performed at a variety

of initial densities and stresses

giving us the possibility to use NorSand for dense

as well as for very loose soils.

Let us start this webinar

by reviewing the important basics of NorSand.

NorSand is a constitutive model created

by Mike Jeffries in 1993,

which has been updated and upgraded considerably

through the years.

It is a soil model specifically designed

to correctly simulate sand behavior for a variety of states

from dense and dilatant samples

to loose and very contractive ones.

Such as what we see here where static liquefaction

is simulated by the model.

Although NorSand’s primary application is geared

towards sand behavior modeling,

it is also well-equipped to deal with other materials

such as soils and tearing materials.

NorSand was effectively used

by the Fundão Tailings Dam Review Panel in 2016

to investigate the possible causes of failure

of the Fundão Dam built using a variety

and complex array

of soil materials showcasing NorSand’s flexibility.

NorSand is a soil model based

on critical state soil mechanics.

Very briefly this theory stipulates

that as the soil is sheared,

for example from point A to point B on this figure,

it will move toward its critical state.

A unique point in mean effective stress,

deviatoric stress, void ratio space,

where it will undergo very large shear deformation

for any small shear stress increment.

The state parameter denoted by the Greek letter psi

is a critical component of NorSand.

The parameter is a measure of the distance

in terms of void ratio

to the critical state line at constant mean stress.

It represents a clever way to express both density

and main stress states in one simple parameter.

By definition, if the state parameter is negative,

the soil is dense compared to its critical state,

meaning that it will have to dilate

to reach its critical state.

Conversely, a positive state parameter denotes a loose soil

that will contract when sheared.

The magnitude of the state parameter

also indicates the density in a relative sense.

A large positive psi

for example indicates a very loosely packed soil.

The less important concept to review

before looking at NorSand’s input parameters is dilation.

Let us consider this volumetric deformation plot,

which I borrowed from one

of the drained triaxial tests performed

on dense sand available in our dataset.

As is typical for dense sands,

we see an initial contractive phase associated

with positive volumetric deformations.

At around 2% of vertical strain,

the maximum positive volumetric strain is reached

and the soil starts to dilate or expand if you will.

At around 3% of vertical strain,

the accumulated volumetric strain becomes negative,

meaning that the samples volume

is now larger than it was at the beginning of the test.

Another way to look at the same data

is to calculate dilation.

Dilation is the change of volumetric strain divided

by the change of deviatoric strain.

This plot shows how dilation evolved during the test.

Notice that dilation is initially positive,

meaning that the change in volumetric strain

is positive compared to the accumulating deviatoric strain.

In other words, the sample is contracting.

At around 2% of vertical strain,

we find point A again

where dilation becomes negative

inverting the volumetric trend observed in the first plot.

An interesting point about dilation

is that it closely relates to the critical state.

If we were able to continue to share the sample

in the laboratory to very large deformations,

we would see that dilation will tend towards zero

and that the slope of dilation will also tend towards zero.

These two conditions are the exact definition

of the critical states.

This is in fact one of the several ways

to express the critical state conditions.

The maximum negative dilation you reach

during a test is called D min

and is of particular importance in Norsand.

As we seen this figure, which includes data for 20 sands,

maximum dilation is closely related to the state parameter.

Denser soils will show larger maximum dilation.

Maximum dilation is in fact directly proportional

to the state parameter for any given sand.

It is possible to reduce the scatter of this data set

by normalizing the state parameter using a constant,

which would be calibrated for each sand.

This constant is called chi,

the state dilating parameter,

one of NorSand’s input parameters.

Throughout NorSand formulation,

maximum dilation is used as a driver to pace deformations.

As this webinar only covered some

of NorSand’s most important aspects,

interested users should refer

to the static stress train modeling we choose

to do book available on Geoslope’s website

and the “Soil Liquefaction: A Critical State Approach” book

for a more thorough description

of the model than what is presented here.

Now that we have reviewed some important foundation blocks

of NorSand, let us review the models

and put parameters and their calibration.

Elasticity in NorSand is defined using three parameters,

the elastic shear modulus at their reference pressure,

the elastic exponent, m, and Poisson’s ratio, mew.

The first two are used together

to define a stress dependent elastic shear modulus, G.

The exponent m can take values from zero to one

in order to establish the type of stress dependency desired.

If m is set to zero,

the shear modulus will be constant irrespective of stresses.

Or the opposite,

if m is set to one,

the shear modulus will be directly proportional

to the mean stress.

Real sand behavior tends to fall midway

between these two extremes and m is often taken as 0.5.

Poisson’s ratio is used to define the elastic bulk modulus

using the value of the elastic shear modulus.

Poisson’s ratio has limited impact

on the outcome of large deformation simulations,

where plasticity will govern the soil’s behavior.

For sands, it is generally safe to assume

that this constant will take values

between 0.15 and 0.25.

Specifically for the Fraser River sand from our data sets,

a Poisson’s ratio of 0.2 and an m failure of 0.47 were used.

The position of the critical state line is defined

in NorSand just as it is

for any other critical state soil model,

generally as a straight line

in the void ratio natural logarithm of mean stress space.

The altitude of the critical state line at a mean stress

of one kPa is defined using the parameter capital gamma

while its slope is defined by Lambda.

Note that there are no strict shapes

a critical state line should take

as we will see further on in this webinar.

The final set of NorSand parameters relates to plasticity,

namely the critical state ratio,

the volumetric coupling co-efficient,

the state dilatancy parameter,

the plastic hardening modulus components,

and then optional additional softening index.

Of these six parameters,

three are directly calibrated

using triaxial compression test results.

This is in addition to the critical state line parameters,

which are also calibrated using triaxial test results.

The calibration of these five parameters can easily be done

using an Excel spreadsheet

and doesn’t involve any numerical simulations

as we will see later on.

As we just discussed a moment ago,

we generally simplify our lives by assuming

that the elastic exponent, m

and Poisson’s ratio are taken as constants

when calibrating the model,

rather than undergoing a strict calibration process.

Another simplifying approach

we generally take just to avoid using Hy and S

when beginning a calibration process

as these parameters are optional in NorSand.

This then leaves us with two remaining parameters

that need proper calibration.

The elastic shear model is at the reference pressure

and the base value of the plastic hardening modulus.

In an ideal scenario,

we will have laboratory measures

to guide the calibration of the shear modulus.

However this is rarely the case,

and we must then proceed to calibrate this value

in tandem with the hardening modulus

using NorSand simulation results.

But before we jump into NorSand simulations,

you must first calibrate the critical state line parameters.

Here are 12 triaxial compression tests

from our dataset expressed

as void ratio mean effective stress plots.

Notice that some tests are straight lines here.

These are the undrained test

during which devoid racial cannot change

because volumetric deformations are prevented

by the undrained boundary condition.

I have identified the end of each test by a black square.

Although our job here is to fit the critical state lines

through this data,

one must be careful not to confuse the meaning

of the end of a test.

It is not necessarily because a test has reached it’s end

in the laboratory

that it has also reached its critical state.

In fact reaching critical state

in a common laboratory setup is quite rare.

Let us look at a particular test

to illustrate this important point.

Here are the test results

for this drained triaxial compression tests performed

on dense sand.

This test ended at 20% of vertical

or axial strain as is common for triaxial tests.

But we’re still short of reaching its critical state.

There are many ways to notice this.

One of my favorite method is to look at the dilation plot

on the bottom right.

Notice that at the end of the test,

dilation is not equal to zero.

Neither is the slope of the dilation curve.

To reach these two conditions,

which is necessary to each critical state,

the test would have had to continue for quite a while.

This is unfortunately impossible in the laboratory

because as large deformations occur,

strain localization takes place

and the sample is no longer deforming homogenously.

As a general rule of thumb,

it would be safe to assume that at 20% of vertical strain,

critical state is very rarely achieved.

Back to our series of results then.

We should expect that this blue curve would still need

to experience some level of the formation

before it really reaches its critical state.

In this case,

this would mean

that it’s void ratio should continue to increase.

We should do a similar exercise of cautious judgment

for each end of tests before attempting

to place a critical state line.

Assuming that the critical state line would be straight

in this void ratio log of mean effective stress plot

would give the following best fit.

We are able to achieve a very good fit

for the lower right part of our body of data.

However notice that two undrained tests

on the upper left portion of the plot are way out of place.

Their end points are far from the critical state line.

These two tests are loose sands,

which underwent static liquefaction.

If we were to use this critical state line position

for simulations, we would inevitably

end up with bad fitting results for these tests

hence negating NorSand’s potential capability

to simulate static liquefaction.

Fortunately, by using a curved critical state line,

we can achieve a much better fit.

As I stated earlier in this webinar,

there is no particular reason to assume

that the critical state line should be straight

other than because it is convenient and easy to adjust.

In some cases, a curved critical state line is desired

because this is what the data simply requires.

For these special cases, SIGMA/W can accommodate a curve

or a linear critical line.

After the critical state line is defined,

we can then revisit each triaxial test to calculate

how the state parameter evolved during shearing.

For this particular test, starting at point A

and ending at point B, the state parameter will show

that the sample was dense since psi is less than zero.

And after 20% of deformation ended up quite close

to its critical state,

which by definition would happen when psi

and the slope of psi are both zero, similar to dilation.

Now that our critical stateline parameters have been chosen,

we can move on to calibrating three

of the plasticity related parameters.

We have already seen that volumetric deformation plots

can be expressed in terms of dilation.

There is however another way to look at the same data.

Instead of looking at dilation

as a function of vertical strain,

let us look at stress ratio, R as a function of dilation.

As a reminder,

the stress ratio is simply the deviatoric stress divided

by the mean effective stress.

At the start of an anisotropic triaxial test,

the stress ratio is zero.

On this plot however,

the first recorded dilation value is point A

which is already at a stress ratio of 0.4.

This is simply caused by dilation being calculated

as a change in deformation,

hence it cannot be calculated on the first step of a test.

As the test progresses,

the stress ratio rises at the same time

as dilation becomes more negative.

Maximum dilation, D min will eventually be reached,

which would also coincide

with the maximum stress ratio, D max.

To calibrate some of NorSand’s input parameters,

we must record this pair values

as well as the value of the state parameter

when this maximum dilation is reached.

Another interesting piece of data available

on this plot relates to the critical state ratio, Mtc.

After maximum dilation has been reached,

the soil will fall toward its critical state,

which would be the y-intercept on this plot

at very large deformation.

This is one of the several methods

for determining the critical state ratio.

For each drained triaxial compression test performed

on dense sand in the dataset,

we would want to identify these three important points,

the maximum dilation,

the stress ratio at maximum dilation,

and the state parameter at maximum dilation.

It is important that only dense samples are used

for this step

since there wouldn’t be any maximum dilation

to identify for a loose simple by definition.

For our dataset,

this process would yield the following values.

By plotting these together on two separate plots,

we can see trends emerging,

which will allow us to determine three

of NorSand’s input parameters.

On plot A,

which is the maximum stress ratio

as a function of maximum dilation,

the y-intercept corresponds to the critical state ratio Mtc.

The slope of this best fit line would then be related

to the value metric coupling coefficient, N.

On plot B,

which is the maximum dilation as a function

of the state parameter at maximum dilation,

the slope of the best fit would correspond

to the state dilatancy parameter chi tc.

Notice that both chi tc

and a critical state ratio Mtc have the tc subscript.

This indicates that these parameters are specific

to triaxial compression.

The corresponding internal model parameters vary

when loading conditions depart from triaxial compression.

Also note that the best fit line on plot B

should pass through the origin as by definition,

dilation should be zero

when the state parameter is zero.

Now that most of NorSand’s input parameters were defined,

we are only left with parameterizing

the elastic shear modulus at the reference pressure

and the base value of the plastic hardening modulus.

While the role of the elastic shear modulus is obvious,

it is less so for the hardening modulus.

The plastic hardening modulus, H controls the rate

of hardening of the yield surface.

H is built using two of NorSand’s input parameters,

a base value H zero

and a dependent value Hy which is optional

and used to impose a dependence on the state parameter.

We will see later on how this can be used

to generalize the model.

Notice that when Hy is equal to zero,

which is a default value, H is simply equal to H zero.

The hardening of the yield surface refers to a change

in the yield surface size in position.

Here is NorSand’s yield surface.

It erupts a similar teardrop shape

as the original Cam-Clay model.

The yield surface that emits the elastic

from the elasto-plastic region.

A stress point that is located inside the yield surface

will only produce elastic deformations upon loading.

A stress point located exactly

on the yield surface will produce both elastic

and plastic deformations.

And finally a stress point located

outside the yield surface is inadmissible.

To reach a stress point at sea,

the yield surface would have to expand,

a mechanism called hardening.

It is precisely this aspect of the model

that the plastic hardening modulus, H controls.

How much hardening occurs

for a given plastic shear strain increment

is determined by H’s value.

This parameter is calibrated

using a trial and error procedure

using NorSand’s simulation results.

Now that we have reviewed the input parameters required

to use NorSand,

we can move on to using the model and SIGMA/W

to complete the calibration procedure.

A good practice before jumping into SIGMA/W

to calibrate a model is to properly organize our test data.

This can easily be done using Microsoft Excel.

Here is how I like to set this up.

This workbook contains all the laboratory results

that I have available to adjust NorSand’s parameters

for the Fraser River sand.

I created a summary sheet where each test is identified.

I also identified the initial conditions related

to each test as I will need these values

to set up the SIGMA/W simulations.

Finally, the known input parameters

for NorSand previously determined also indicated.

This will be common to all the simulations.

The parameters identified in red are those that need

to be adjusted using the simulation results from SIGMA/W.

In this other sheet are the various plots

that helped me fit the NorSand parameters

that were already discussed earlier.

You will recognize some of these plots.

Finally, all the other sheets

contain the laboratory test data.

Each sheet corresponds to one single test

where a few plots help me visualize each test.

For a drained test,

I like to view the stress point plot, this trespass plot,

the volumetric deformations,

and the state parameter evolution.

For a non-drained test,

instead of the volumetric deformations,

I plot the pore water pressure.

Let us use the test identified as CID_D_113

to start the parameter adjustment procedure in SIGMA/W.

This test is a drained test on dense sand.

Since there are already many tutorials available

on Geoslope’s website

on how to set up a triaxial in SIGMA/W,

I will go through these steps swiftly.

Feel free to download one of such examples for our website

if you are unfamiliar with some of the steps featured

in this part of the webinar.

First, I set up the product geometry for a triaxial test.

I used a 2D axisymmetric geometry to take advantage

of the triaxial simple cylindrical shape.

This allows me to only model a quarter of the sample.

I then create the SIGMA/W low deformation analysis

to apply the stresses that exist

at the end of the consolidation phase.

I create the simple isotropic elastic material model here,

as the deformations are irrelevant for this step.

Our goal is simply to establish the proper stresses

that exist at the start of the triaxial test.

I apply boundary conditions

to represent the confining stresses

to which the real sample was subjected.

A single higher order element,

which reduce integration will surface given

that there are no special variability of stress and strain.

I can start the analysis and verify

that the stresses are correctly applied to the sample.

The next step is to create

a new SIGMA/W low deformation analysis

to perform the shearing phase of the triaxial test.

Tolerable errors and number of steps are adjusted

to produce better results when using NorSand.

A NorSand material model is created

and the parameters that are demanded

at the previous part of this webinar are entered.

Then a first try

for defining the four missing parameters is done.

We haven’t discussed the over consolidation ratio yet.

The overall consolidation ratio, OCRp is the ratio

of the current

over the historical maximum mean effective stresses.

The W input displays the subscript P to remind us

that the ratio is defined in terms of mean effective stress.

For sense, the OCR is normally never very high,

but it is generally correct

to assume that a little over consideration exist

in a triaxial sample.

A value of 1.1 is used here.

For sands, values between one and 1.3 are normally expected

and generally will not have an overly important impact

on the simulation results,

especially for drained simulations.

Next is the shear modulus at the reference pressure.

There are many ways to generate the first guess

of this value.

Use whichever method you prefer.

From my geo-technical engineering courses

back in university,

I have always remembered that Young’s modulus

for sense can be approximated using the relationship N,

the SPT low count times 2000.

Young’s modulus divided by two

times one plus Poisson’s ratio

will then yield the elastic shear modulus.

For a Poisson’s ratio roughly estimated to 0.25

to keep digits easy to work with,

this would mean that I could estimate the shear modulus, G

as being N, the SPT blow count times 800.

Let us estimate

that our somewhat dense could have an SPT blow count of 25,

which would yield an estimated shear modulus of 20,000 kPa.

While an N value of 25

would technically not be considered dense in the field,

it would probably correspond to a dense sample built

in the laboratory for a triaxial test.

This is of course just a roughest estimate to get us going.

I like this method because I can genuinely get this estimate

in my head just by remembering this simple relationship.

Next is the plastic hardening modulus, H.

As I said earlier,

it is good practice to first neglect the effect

of the state parameter psi on the hardening modulus

by setting Hy to zero.

We will come back to this value later,

but for now we assume that the hardening modulus

is simply equal to H zero.

The plastic hardening modulus value generally sits

between 20 to 400.

And a good first estimate for it is H is equal

to four divided by Lambda,

the slope of the critical state line.

In our case

since we use a curve critical state line,

Lambda is not defined.

So let us just use a value of 60 as a starting point.

Finally, let us consider S, the additional softening index.

This is a feature

which helps NorSand correctly simulate the response

of loose soils being loaded under undrained condition.

S can take values from zero to one,

but is generally used with either one or zero,

like an on-off switch if you will.

For any drained test, S should always be kept to zero.

For a non-drained test, if the sample is dense,

for example if the state parameter is smaller than zero,

then S should also be kept to zero.

However, if the sample is loose,

for example if the state parameter is larger than zero,

then S could be non-zero.

I will show the effect of S equal one

for some undrained examples later on.

For now, since the test we are calibrating is strain,

S will be set to zero.

Back to SIGMA/W, we can now input the missing parameters

we just discussed to complete our definition

of the NorSand material.

The last step remaining before solving this analysis

is to create the boundary conditions needed

to perform a triaxial compression test

meaning that the top part

of the sample be made to move downward.

By creating a dense displacement function,

I can ensure that the top nodes will move down

by 0.01 meter,

which corresponds to 20% of vertical deformation

for the size of our sample.

I can now go ahead and remove the stress boundary conditions

that were created for the stress initialization step

as we don’t need these anymore.

The parent analysis,

which this current analysis depends on,

will transfer it’s stress conditions to its child.

I now apply the newly created displacement boundary

condition to the top of the sample.

As a side note,

the function should be defined over an elapsed time

that is commensurate with the duration of the analysis.

For example a one day.

That way, the number of steps used

to simulate the total vertical straining can be varied

without having to change the boundary condition.

We are now ready to solve the analysis.

Once this is done,

I can create plots to view the data

and to export it to Excel

to compare the NorSand results with the laboratory data.

To do so, I can select all my plots,

which are conveniently expressed in terms of vertical strain

and right click on a plot to select copy graph data.

Once plotted alongside the laboratory results,

I can visually inspect the simulation results

and evaluate the model’s performance.

In this case,

it looks like the model stress strain curve on plot A,

the orange curve is too soft compared

to the laboratory results, the green curve.

It is however worth pointing out that this first estimate

already provides a very acceptable fit compared

to the laboratory data.

This showcases the benefits of being able to adjust most

of NorSand’s parameters straight from laboratory data.

By returning to Joules to do

and adjusting the indicated parameters,

I can obtain what we will call the best fit.

Judging that simulation results

represent the best fit possible

is of course quite subjective

and there might be various combinations of parameters

that will lead to similar fits.

Care must be taken to avoid using unrealistic values.

For example, one can hardly expect OCR

is larger than 1.5 for sands

or shear moduli larger than 200,000 kPa.

When I’m satisfied with the fit I obtained,

I want to leave a trace of the final parameters I use

for this specific test.

To do so, I will return to my Excel summary worksheet

and input the chosen parameters

under the test I was working on.

I will then proceed similarly

with all the tests I have in hand,

using the same steps I just described.

I will use the same geometry,

but we’ll add new stress initialization

and triaxial compression phases for each test.

The stress boundary conditions are specific to each test.

In addition, the NorSand material models

are defined uniquely for each test

representing the appropriate initial conditions indicated

in my Excel summary worksheet.

I will follow the same procedure where I make a first guess

of the missing material parameters

and proceed to adjust them until I reach a satisfying fit.

We will work on normalizing these first guess parameters

at a later stage of the procedure.

I will finally record the best fit parameters

for all the tests in my Excel summary workbook.

You will notice that for three of these tests,

the additional softening index was turned down.

This can be done since these tests respect the requirements

that they must have been performed

in non-drained conditions for loose soils.

Here’s an example of the effector

in the additional softening index can have on such tests.

These other simulation results we obtain

when we use the default S value,

which is zero

for a test where static liquefaction is observed.

We can see that the responses much differ

for NorSand than for the laboratory results.

In particular, notice on plot D how NorSand

is still very far from critical state

whereas that parameter plot will be zero

while the laboratory results

on the contrary fairly quickly reached critical state.

By simply turning the additional softening index on,

we can get these results instead.

NorSand is now able to capture every aspect

of the static liquefaction behavior very nicely

producing large pore water pressure

that drives the main stress to very low values,

generating large deformations for minimal stress increments.

Although the additional softening index should only be used

after thoughtful consideration,

it can greatly help

to properly simulate some more complex aspects

of loose soil behavior.

Back to our Excel summary worksheet,

now that we have defined the NorSand input parameters

for every laboratory test we had available,

we can explore the possibility of generalizing some

of the input parameters to obtain a set of parameters

that would be as consistent as possible

across all the tests studied.

This should be done

in order to render our NorSand material more flexible.

In an ideal situation,

all the input parameters should remain the same,

no matter what the test conditions we encounter are.

Let us focus on the elastic shear

and plastic hardening moduli,

and how these two input parameters seem

to depend on the initial state parameter.

This would help us generalize our inputs

to make NorSand easier to use in real life simulations.

I have plotted the plastic hardening modulus on the left

and the elastic shear modulus at the reference pressure

on the right,

both as a function of the initial state parameter.

These plotted values are the best fit parameters

I have adjusted using SIGMA/W.

As it can be expected,

both moduli show a dependency

on the initial state parameter.

For both, denser soils will yield larger moduli.

By passing a linear regression line through both series

of data, I can define simple relationships

which can help automatically choose moduli values

that will yield good simulation results

no matter the initial conditions.

Specifically for the plastic hardening modulus H,

the two parameters of the linear regression curve

can be used directly in SIGMA/W as input values

that will directly take

into account the initial state of the soil.

The next version of SIGMA/W

to be released during the spring of 2021

will also feature similar state dependent capabilities

for the elastic shear modulus.

Let us take a final look at the Excel summary worksheet.

Now that I have defined state dependent relationships

for both moduli,

I can begin the chosen H zero and Hy values.

I can then use the elastic shear modulus relationship

to calculate the generalized shear modulus

at reference pressure for each test.

This allows me to finally define a complete NorSand model

that is almost entirely dependent

on the initial state for its initial parameters.

A case could very well be made to also adopt a single value

of OCR for all the tests, say 1.15 for example.

The only parameter that would then be needed to be adjusted

by the user would be the additional softening index S

that still needs to be turned on manually

for undrained loading of loose soils.

Other than these caveats,

the NorSand model we established

provides all the necessary parameters

to properly simulate the behavior

of the Fraser River sand for triaxial loadings

that include dense as well as loose initial states

and undrained as well as drained conditions.

To showcase the power of the procedure we just completed,

let us compare simulation results

for the best fit possible against the final fit,

using the generalized expressions for the elastic shear

and plastic hardening moduli for a few tests.

These are the simulation results

for a drained triaxial test performed on dense sand.

The green curves are the lab results

and the red curves are the NorSand simulations

for SIGMA/W with parameters chosen

to produce the best fit possible.

Finally, the dashed blue curves are the SIGMA/W results

using the input relationships developed

in the preceding steps that use the initial state parameter

to initiate the shear and hardening moduli.

The two NorSand simulation sets

are very close to each other.

And overall, very close to the laboratory results.

Even when using the input relationships,

NorSand clearly captures the soil behavior.

As expected, the dense soil shows some initial contraction

followed by important dilation

that will bring the soil closer

to its critical state at large deformation.

Here’s an undrained test performed on loose sand

where as another test showed earlier,

static liquefaction is involved.

Again, the NorSand simulation

using the state dependent input relationship are very close

to the best fit possible.

Very contractive behavior is observed

and the soil is quickly reaching its critical state.

Both simulations closely follow

the laboratory result behavior

and correctly stimulate static liquefaction.

This is a really great example of the strength of NorSand.

With a constant set of input parameters

coupled with state dependent relationships,

one can simulate dilatant

as well as very contracted behaviors.

As a last example,

here are the simulation results

for a non-drained loading on dense sand.

Again, the SIGMA/W simulation results

using the state dependent relationships are very close

to the best fit possible.

Both simulations correctly follow the laboratory results

where an initial contraction phase accompanied

by a positive pore water pressure generation is followed

by a dilatant phase

during which negative pore water pressure are generated.

That concludes this webinar part

of the SIGMA/W material model series.

Three main topics were covered during this presentation.

At first, an introduction to NorSand was presented

where the basics of the soil model were discussed.

The various input parameters needed

to use NorSand in SIGMA/W were examined

with a special focus on how most

of them can be adjusted

using triaxial yield compression results directly

while some others need simulation results

from SIGMA/W to be properly defined.

Finally, we considered how to use NorSand and SIGMA/W.

Toward the end of the presentation,

a special focus was put

on how to generalize NorSand’s input parameters

to create a complete and flexible state dependent model,

which represents one of NorSand’s strength.

Our team of development engineers is already hard at work

to provide new improvements to NorSand in the next version

of SIGMA/W that should be released this spring.

Users will notably be able to define the shear modulus

at reference pressure as a function of void ratio.

It will also become possible to use the state parameter

as an input instead of the void ratio.

If you’d like some new features and capabilities added

to NorSand and SIGMA/W,

don’t hesitate to make suggestions

by submitting support requests.

We have now reached the end of this webinar.

A recording of the presentation will be available

to view online.

Please take the time to complete the short survey

that appears on your screen so that we know what types

of webinars you’re interested in attending in the future.

Thank you very much for joining us

and have a great rest of your day.