0:05

[MUSIC] Hello everyone,

Â welcome to the Materials Data Science and Informatics class.

Â I'm Surya Kalidindi and I am the instructor for this class.

Â Today's lesson is titled Structure-Property Linkages using

Â a Data Science Approach Application Part 1.

Â In this lesson,

Â the learning outcomes are to understand how to apply the data science

Â approach we have been learning in this class, particularly through a case study.

Â This particular case study involves an inclusions/steel composite system.

Â 0:37

Okay, let's start with a basic introduction to the problem we're going to

Â deal with.

Â This particular problem we are going to look at non-metallic inclusions in steel.

Â So this will be considered as a composite system where the matrix

Â is essentially the steel and the inclusions are the second phase.

Â These inclusions are typically impurities in steel, so they are usually carbides,

Â nitrites, etc.

Â 1:15

The main issue with having inclusions is that they have a significant

Â effect on the macro-scale properties of the finished product.

Â In particular, notice that when you have some hard inclusions,

Â you may have these pores or defects.

Â And also, when these hard inclusions actually become soft at a high

Â temperature, the inclusion shape can be quite flat or pancaked,

Â and the sharp edges here in these corners can cause stress concentrations.

Â 2:02

So from a Data Science point of view,

Â what we are asking in this particular case study is can we understand and

Â establish quantify the linkages between the material structure?

Â And in this case, the structure is essentially inclusions and steel.

Â 2:21

The shape, size, and spatial distribution of inclusions in the steel.

Â And can we connect this to properties of interest,

Â effective properties of interest?

Â And these properties could be effective in points,

Â hardening rates, ductility, so on so forth.

Â 2:41

Now, let's remind ourselves the main

Â steps involved in the data science approach for this homogenization problem.

Â We went through these steps in the previous lessons, so

Â here I'm only trying to summarize them and remind you what the steps are.

Â Step 1, it involves generation of the datasets.

Â In this particular case, we will have both sub-tasks in step 1,

Â and the first sub-task will be to generate synthetic microstructures.

Â Of course if you actually have experimental microstructures,

Â you can use them, but in this particular case we are going to make microstructures.

Â In other words, we're going to make them up on the computer.

Â 3:26

Step 1B would be once we generate the microstructures, for

Â each microstructure, we're going to evaluate a mechanical property of interest

Â using a numerical model.

Â In this particular case, we are going to use an abacus model.

Â 3:44

After we generate the data sets, Step 2 is reduced

Â order quantification of the microstructures.

Â Again in previous lessons,

Â we have gone into substantial detail of what would this would contain or involve.

Â And broadly speaking, this has two steps, sub-steps, the first step is to

Â compute n-point correlations or in our case, it'll be two point correlations.

Â And then doing a Principle Component Analysis, so

Â that we end up with the reduced order quantification of microstructure.

Â 4:21

And then the last step would be to use the properties

Â that we generate in the dataset, and

Â representation of the microstructure using the principle scores.

Â And connect up these two using regression models.

Â 4:42

Once we generate these models, we want to evaluate or validate the models, and

Â we will use Leave-One-Out cross validation.

Â Again, all of these individual steps have been discussed in previous lessons, and

Â in this particular lesson,

Â we're going to focus on the application of these concepts to a practical problem.

Â 5:03

So let's start.

Â So we're going to start with generational microstructures.

Â In this case, we're generating a bunch of microstructures where we have different

Â types and shapes and distributions of inclusions in a steel matrix.

Â So we start with a library of possible inclusion shapes of interest and

Â here is a small library of five inclusion shapes.

Â You could add more if you want.

Â And then we also have decided that the volume fractions of

Â interest are in between 0 and 20%.

Â It might sound like 20% is a little high but

Â when you actually look at inclusions in steel,

Â sometimes the local volume fractions where the inclusions cluster can be quite high.

Â So this thing has been selected to cover the entire range of potential interest.

Â 5:55

So once we select the inclusion shapes and

Â we decide on volume fractions, we generate an ensemble of microstructures.

Â In this particular case,

Â we decided to generate an ensemble of 900 microstructures.

Â These 900 microstructures are broadly distributed in these four classes.

Â So one class is Randomly Scattered, the other class is Vertical Bands and

Â yet another class is Horizontal Bands and the fourth one is Clustered.

Â Each one has a certain number of microstructures, for

Â example the Randomly Scattered one has 300 microstructures, and

Â what you are seeing here is one of them.

Â This is one of the 300 microstructures.

Â So the way each of the microstructures are generated is you randomly select

Â the inclusions from this library, and you select a volume fraction.

Â And you keep adding inclusions until you get to the volume fraction that you want.

Â 6:53

The strategies we're using in these different classes is different.

Â So depending on the strategy we use, we get a randomly selected one or

Â we get what we call as vertical bands.

Â In this case, the inclusions are still selected randomly, but

Â they are placed in vertical bands.

Â Or we might place them in horizontal bands.

Â Or we might cluster them into clusters.

Â And again, in each cluster, we only place the inclusions in a random way.

Â So this is the way we decided to generate the microstructures.

Â There are a total of 900 microstructures and that's Step 1a.

Â After we generate the 900 microstructures, for

Â each microstructure we do a finite element analysis.

Â In this case again,

Â we decided to do finite element analysis because of the expediency.

Â If you actually have experimental datasets,

Â you could easily replace the digital datasets we generated for

Â this case study with real experimental data sets.

Â So, when we do a finite element simulation,

Â we discretize the mesh, discretize the volume into a grid.

Â And then we are applying periodic bond reconditions.

Â In this case,

Â we're applying plane strain compression type periodic bond reconditions.

Â For these bond reconditions, we're evaluating the stress and strain fields.

Â Again, we are doing this for every microstructure.

Â And then once we do the simulation,

Â from the simulation we extract certain properties of interest.

Â The properties of interest for this study have been selected as follows.

Â The first property of interest is the Effective Yield Strength.

Â This is the overall yield strength of the entire composite.

Â 8:48

And lastly, the third property of interest was selective Localization Propensity.

Â And what this really says is what is the likelihood of forming some sheer bands or

Â localized bands in this composite?

Â Again, notice that because the reinforcement phase is

Â 9:08

different from the matrix, you would have some defects.

Â So in the top picture here, the precipitates or

Â the inclusions are hard and therefore you have some deep bonding in the simulation.

Â And in the bottom picture here, the inclusions are soft and so

Â they change shape and they flow with the material but

Â because they're soft, they also cause localization.

Â And localization propensity is simply defined as the area fraction of the matrix

Â elements experiencing an equivalent strain greater than a prescribed cut-off.

Â So this is a user selected prescribed cut-off.

Â So these are the properties that were selected

Â to be of interest in this case study.

Â So that completes Step 1.

Â Now we are ready for Step 2.

Â For each microstructure, again, this is just an example.

Â For each microstructure, we generate the 2-pt statistics.

Â Again, the mathematical theories of generating the 2-pt statistics were done

Â before.

Â Here is a reminder as to what equations we used to compute this.

Â And in fact we used the DFT methods we discussed in previous lessons.

Â So, the 2-pt statistics for each microstructure looks like this,

Â we are doing this for a total of 900 microstructures in this case study.

Â So, we have all 900 sets of 2-pt statistics.

Â That's 2a.

Â Once we have this ensemble of 900 2-pt statistics,

Â we are performing a principal component analysis.

Â And the first three principal scores are shown here, and

Â they're color coded so that you can see the different classes.

Â And you can easily see that they are automatically segregated into

Â different classes.

Â Again, this is unsupervised classification.

Â So PCA itself doesn't understand that there are four classes.

Â But the result of the PCA shows that you can see the four classes separately.

Â What you're seeing here are the average values of the principle components for

Â each class.

Â 11:21

So now we have a low dimensional representation

Â of the microstructure in terms of these three principal scores.

Â So the clustering can also be seen by comparing

Â the points in the principal component space to the actual microstructures.

Â 11:42

So here is an example of a microstructure from the random placement.

Â Here is an example of the microstructure from vertical bands.

Â Then here is an example from clustered microstructures.

Â And one can see that the PCA automatically classifies these

Â microstructures into their separate classes.

Â [MUSIC]

Â