So let's get going. So what you basically have to imagine here, you have a ferretic steel, for instance. In the middle of the grain, we have a piece of screw dislocation that has double cross slept,

and that is generating dislocation loops. One after the other, they expand, and then they reach the grain boundary. So you've got a set of dislocations which repel each other.

And so they exert a back stress on the dislocations that are generated by the Frank Reed source. And these are generated by the externally applied force. So obviously, this back stress, the size of this back stress

will be determined by the number of dislocations in the pileup. So as written here, so the Frank Reed source

will stop producing dislocations when this back stress is large enough to prevent further emissions of dislocations from the source. And the number of dislocations in a pileup is determined by the back stress.

So we will use what we know of dislocation theory to get some quantitative values here. So we assume we have an applied stress, which

we call tau a, acting on the slip plane, where you have the source and where you have the pileup. Now, the pileup works as a magnifier of the stress

that you apply. And I'm not proving this. Just take my word for it. That's one of the things I get from dislocation theory,

is that the stress that is, this is the grain boundary, and this is the stress. I'm wondering, what is the stress on this first dislocation if you have n dislocations,

the same dislocation, piled up again? Well, dislocation theory tells me that the effect of this pileup is to magnify the stress, shear stress,

acting on this dislocation. So normally, it would be applied force times Burgers vector. Now, it's n times the applied force times Burgers vector times the length of the dislocation. And we just forget about the length

and drop the Burgers vector. And so we say the stress on the leading dislocation of this pileup is n times the applied shear stress. So now, the question is, so I can know what this stress, shear stress, is if I know n.

So again here, I use a result from dislocation theory about these pileups.

So n is equal to a number of constants, pi, g, b, and k1 are constants in this thing. You have two parameters. One is the applied shear stress,

and l, the length of my pileup. So in that situation, what we calculate is the moment where you have n dislocations forming

a pileup of length l over the distance l at the time that they will prevent the source from creating more dislocations in that equilibrium state.

Yes? OK. So OK, say I have applied less force than n is smaller. Because if I apply a big force,

I will push a lot of dislocations against the barrier before the back stress balances the applied shear stress. If my applied shear stress is very small,

the number of dislocations here will be small, too, because you won't need to have this much back stress to stop the source from generating dislocations. OK?

So the number of dislocations per unit length of the pileup, n over l, is a function of the applied force, shear force. All right? And if you look at this, it's as

if it's equivalent to stating that if you have a pileup of n dislocations, you can pretty much think of it as a single dislocation

with a very large Burgers vector. So this is equivalent to a big dislocation with not a Burgers vector b, but a big B times n. And it kind of makes sense if the dislocations are

very close to each other. You have all these extra half planes. So it kind of magnifies the Burgers factor. So l is the length of the slip planes with these pileup dislocations. T is the applied stress.

k1 is a constant here in this parameter. It's, again, coming from dislocation theory. k1 is 1 for screw dislocations and is 1 minus the Poisson ratio for edge dislocation. So it's a very simple parameter. So the stress at the grain boundary

on this particular dislocation is now, so this was my equation here, n times the applied stress. So I just take this, this here, and I multiply it with tau a simply.

These two equations together give me, not surprisingly, tau 1 is pi k1 divided by g b l times tau a square. And now, obviously, this solves everything.

And this assumption is made in multiple theories. You make the step and you say, well, the length of the pileup is related to the grain diameter.

If I have small grains, I will necessarily have smaller, smaller grain sizes. If I have small grains, excuse me, I will have smaller pileups.

It's kind of reasonable, yes?

And very often in this theory, in the theory, people say, well, it's probably close to 2, half a grain size. So l is half a grain size or thereabouts. You don't have to assume it's half. And then you find tau 1 is equal to,

OK, that's the same equation you saw on the last slide. And now what you say is, if the shear stress that's pushing this dislocation into the boundary here,

so n times tau applied, which is equal to the tau square times l, and all the other parameters, which are constants, if that shear stress is high enough,

yes, one theory says, if that's the case, if you reach, if this reaches a critical value,

then you will have breakthrough. This slip will somehow generate dislocation at the grain boundary, and you'll have breakthrough, burst through the grain boundary. So the pile that dislocations can

burst through the grain boundary when you reach a critical stress tau c. So if tau 1, which is n times tau a, is larger or equal than this critical shear stress, then you will have your propagation of the slip

through your grains. So you basically rearrange this. You first get tau a square, and then you tau a, and you find, not surprisingly, 1 over square root

of the grain size. That's reasonable. The thing that's unreasonable about this model is, of course, that we know dislocations will very rarely

cross boundaries. So what other theories, another theory says, and that's the Cottrell theory you may have heard of, they say, well, this large shear stress you have here

is actually not used to burst through the grain boundary, but to activate a source that's into the adjacent grain.

So alternatively, the piled up dislocations cause a Frank Reed dislocation source to generate dislocation as adjacent grain. And the way the original theory is built up is they look at this pile up as a shear crack.

And they then determine, on the basis of the geometry here, the maximum shear stress at a distance r from this crack. And you can show that this shear stress

is equal to the applied stress times the square root of the grain size divided by 4 times r, r being the distance between the grain boundary and the Frank Reed source in the grain.

And so now, again, if this shear stress is larger than a critical stress to activate a Frank Reed source, then I have propagation of slip.

And so simply by rearranging this equation, you can see that the applied stress is, again, proportional to 1 over the square root of d, in this case. So this is, in general, how people

try to make sense of the 1 over square root d relation that we observe in the Hall-Petch equation for steels. But when you go into some details,

it turns out that these models miss a lot of the features of the process. And we'll talk about two features. The first feature is the fact that grain boundaries have properties, too. Yes?

And in fact, grain boundaries can emit dislocations themselves, can be sources of dislocations. That's one thing. And second, grain boundaries are places where you have what we call strain incompatibilities.

So you have grains deform and two adjacent grains deform, and where they meet, there is what we call strain incompatibility. And that gives rise to, we'll talk about this in a moment, geometrically necessary dislocations.

And that may be, in other theories, the source of strengthening, grain size strengthening. So what's the problem with one of the problems with the dislocation pilot models? Well, it does not explain why, for instance, interstitial carbon and nitrogen

are affecting the hull patch slope differently. So carbon and nitrogen have an effect on the stress needed to unlock dislocations from their atmospheres. But carbon has a very noticeable influence

on the hull patch parameter k for ferrite. So basically, that's what you see here. For instance, you have the hull patch relation for an IF steel.

It doesn't contain any carbon. Yes? You get a very low k value, ky value. You add some carbon in solution, so low amounts of carbon, of course. And you see that the more you add carbon,

the higher k becomes. Now, the way you can understand k, if you want a physical explanation for k, k tells you how difficult is it for deformation to pass through the boundary.

So the higher k value, the more difficult it is to propagate slip across the boundary. So why would carbon have that influence? Yes? There's no mechanism in these pile-up theories

that can account for this. You see nitrogen, very low effects of nitrogen on the k value in the hull patch equation. But in austenitic steels, you see the same also

for nitrogen. You see that as we add nitrogen to our austenitic steels, the k value increases dramatically. So it basically tells you that suddenly your grain size effect is improved just because you have some nitrogen.

So obviously, there's something more than just the size of the grain that plays a role. And that is the properties of the grain boundary are also important in the hull patch relation.

So the problem is the effect of carbon and nitrogen decisions. So we have to think of the fact that the role of grain boundaries is also important. And there are some models which takes that into consideration. And so the grain boundaries are assigned a more active role.

And in fact, that's being done by just saying, well, the grain boundaries emit dislocations. They act as sources of dislocations. So you have to, and there are ways to show that.

If you have ledges on grain boundaries, like steps, it is possible from a purely geometrical point of view to have these ledges produce dislocations, be dislocation sources. And in this case, I'm not going into all the details.

But in this case, you make use of the equation we know, which relates the flow stress or the flow shear stress to the dislocation density, square root

of the dislocation density, this being the dislocation densities. And we make use of this plus an experimental observation that dislocation densities are proportional to 1 over the grain size.

And so when we do this, when we combine these two type of equations, we find that the flow stress is proportional to 1 over the square root of the grain size again. So that's this alternative.

The method's a little bit more worked out here, the text. But that's basically the same. Let's read through the text here, Holpech-Kotrel approach was criticized

on the basis of experimental observations, namely the carbon and the nitrogen effect. And there is another effect that reason why people

have objections against the pilot model is because if you take steels, yes, and you deform steels, and you look into the grain boundary region,

you don't see pilates, yes? You don't see burst trues either, but you certainly don't see pilates, yes? The reason is that in ferrite, once you have dislocation which feels a pretty high force building,

it just cross-slips. It just says, forget it. I'm not standing in this row of dislocations that's compressed. I just cross-slip. So there are no pilates in ferrite, basically.

So you cannot explain the carbon and the nitrogen effect. And so this is the relation between the dislocation

density and the size of the grain here. And this is more or less a very strong experimental observation. And then you plug this into the relation

between the shear stress and the square root of the dislocation density. And that's what we just did here. So the properties of the grain boundary are important. And so the question is now, can we explain perhaps what the effect is of the carbon?

Well, the thing is we know that when we have carbon in ferritic steels, the carbon will, the solute carbon, will very likely be segregated in grain boundaries. And there it will strengthen the grain boundary.

And one of the things it's believed to do is make it harder for dislocations to be generated at the boundary. And that's the reason why you get a steeper slope. It becomes harder to generate dislocations at the boundary.

And the more carbon you have, the steeper the slope becomes. Why don't we see it for nitrogen? Because nitrogen doesn't do this. Nitrogen, in contrast to carbon, doesn't particularly favor grain boundaries for some reason.

It will stay pretty much homogeneously distributed. Yes? So that is probably the reason why there is this compositional effect.

Right. So you can read this when you look at the slides. But there is also this idea that is very important, where people have said, but look, when you're deforming a crystal,

and you're deforming another crystal with another orientation, yes? So I have a crystal here. And I let it deform, yes? And say this crystal doesn't care about the other grains around it, yes?

Then these dislocations will arrive at the end of their slip plane. And they'll move out of the crystal, yes? Making steps. Steps that suddenly the boundary will have steps. And the other grain will do the same thing, yes? So soon enough, if you let all the grains do that,

you'll find that necessarily there will be grain boundaries that are holes, have become holes. And other grain boundaries, where I don't know how they fix that, yes?

But there is material, extra material, yes? So obviously that cannot happen, right? So what happens in order to avoid this happening, yes? The person's name who is associated with it, Ashby,

said, well, in order to avoid this, these volumes that are too much or not enough, missing material or too much material, that's just dislocations. Special dislocations. Geometrically necessary dislocations, yes?

They're being introduced in the model for geometrical reasons. If we don't introduce them, your crystals, your polycrystal, will contain overlapping material or holes, yes? And that cannot happen. So I introduce these geometrically necessary dislocations on top of the normal dislocations, yes?

And so we have two groups of dislocations, the statistically necessary dislocations, statistically stored dislocations, are they called, or these geometrically necessary

dislocations. We'll talk more about them and how you can derive their density later on at another stage of the course, because they pop up anytime you have a problem with strain. So one part of the solid deforms.

The other part deforms also, but it's not compatible at the interface. One of the ways to resolve the issue is by introducing geometrically necessary dislocations. So you get small grains, you'll

need more geometrically necessary dislocations. And that's the relation now between why is it that I need, I have more flow stress, is because if the grain size is smaller, I need more of this geometrically necessary dislocation. And we know the flow stress is proportional to the square root of the dislocation density.

So it's basically a dislocation model. And so with smaller grain sizes, we get a higher rate of dislocation accumulation. Because tau is proportional to the square root

of dislocations. And so this becomes increases as the grain size decreases. That becomes the connection, as it were.

So in this model, there's no need to have pileups at all, no need for pileups. So first, the two reasons are you have the strain incompatibility due to the difference

in crystalline orientation leads to the formation of these extra dislocations, because you need these extra dislocations here. In addition to the statistically accumulated dislocations, this is the one that gives you regular slip. And so the total dislocation density

is increased by this density. And second, why do I have more dislocation accumulation with small grains, is that you get a reduction of the average slip distance of dislocations in smaller grains.

So I will need an increase in the dislocation density to achieve a specific strain. Because remember, the strain is a product of dislocation density times the distance that the dislocation move. So if I make the distance the dislocation can move smaller,

but I still have to get the deformation, the only thing I can do is increase the dislocation density. And that's what happens. All right. So the picture that emerges is a very different picture

from the original pile-up model, or dislocation pile-up breakthrough model, is you have grain interiors where you have single slip, certainly at the beginning of the deformation, and an accumulation of geometrically necessary

dislocations at the boundary. So in the notes, you will see some equations,

theories that are based on pile-up theories, or grain boundary dislocations, sources, or geometrically necessary dislocation theories, some equations that are derived from these theories. You can have a look at them.

So how do we fit all this into what we had derived previously?

So now if you say, I have a steel, I have a grain size, and I want to know how do I put in my grain

size in my equations to calculate the strength of the material? So what you do, the strength of the material as a function of strain will be, so you should first write the equations in terms

of shear stress, shear strain. So first you do the lattice friction. That's the critical result, shear stress. So if you do a calculation at room temperature, that is 19 megapascal or 20 megapascal. Then you add your solid solution strengthening effects.

Yes? And then you add the effect of the grain size on the yield strength.

So this ky divided by square root of t. Yes? You have to add this there. And then you have this, so at this point, the fourth term. Yes?

Where you compute, as I showed you earlier this week, where you compute the evolution of the dislocation density with strain. And you remember there, we had a term, yes, for the mean free path of the dislocations.

Yes? And that's where you put the term in. So you have one, so here if you remember, so you numerically integrate this equation, yes?

So here there is a constant. I'm trying to make sure I don't get this wrong. Like this, OK?

This is the grain size. Yes? So that's how you take, so you do the numerical integration of this equation here. You have the grain size as a strong boundary. Yes?

And that's in there, OK? This is how you can take into account the grain size in the theory of strain hardening. But you should not forget to add the strengthening effect

for the yield strength, OK? Because this term here, it just takes into account the effect the grain size will have

on dislocation accumulation, OK? And if you do that, you can actually check if the whole patch equation should hold. Yes?

OK, there's another point that's interesting to look at. Because up to now, what we've discussed are we've talked about constructional steels, about ferrite basically. And in relation to the whole patch equation.

But we have, there are many situations where we don't have, we're not even talking about whether the grains are circular or not. Look at this, martensite. What's the grain size? You know, I mean, is there an equation

for grain size for martensite or for bainite? Bainite just looks very much like it, you know? Can we talk about microstructure refinement? And it's important because martensite

is a very hard material, yes? And it's very widely used in engineering applications. So originally, to make this martensite, you can work with this ferrite and perlite

constructional steel and do thermal treatment. And you get this very homogeneous microstructure, yes? And the question is now, can we look at microstructure refinement in structures like martensite?

And are there equations that resemble whole patch relations for this type of microstructure? And the answer is yes, actually. You can refine martensite and change the properties.

So the microstructure, first of all, we'll need to say something about martensite, in particular, lath martensite. It consists of packets, yes? OK, so you have packets which are formed within the original austenite grain. So you know martensite is, you

get that when you quench austenite. So you have the original PAGB, so prior austenite grain boundaries, yes? And inside, you have, they consist of packets. And in these packets, you have parallel blocks.

And these packets are typically 100 to 150 micron in size. So they can be big. For a prior austenite grain size of, and this is, again, thanks to Microsoft, this is micron, please.

And these packets are subdivided into many parallel blocks. You see these parallel blocks here, which contain groups of narrow laths, yes? And when we talk about this microstructure here being lath martensite, that's the laths we talk about, about these narrow laths.

So the blocks are 1 to 15 micron thick, yes? And the laths are typically less than half a micron. Yes, again? I'll make sure that's corrected when I post it on eClass tonight.

Less than half a micron thick. The lath size is independent of the prior austenite grain. And what's important here is that these laths, these very elongated laths, have the same habit plane. So although they look like very small units,

they're actually highly oriented, yes? And they don't really work as grain boundaries, even though from a first view of them, it would look like them.

In fact, if you put this in a TM and you do diffraction patterns from these laths, groups of laths, you usually see a single crystal diffraction pattern, yes? Despite the fact that you have all these dislocations and interfaces, lath interfaces, and that's

because the misorientation is very small. So the misorientation between laths is very, very small. And then also, within these blocks here, we have the close-packed crystallographic planes of the laths within a packet are also nearly parallel.

So the crystallographic differences within blocks are small, yes? So I also want to stress the dislocation density. The martensite is very high. But because the crystallographic differences here

are very small, we never have very high angle boundaries there. And slip can easily propagate between the laths. So the packets are meaningful structural units

that you can work on to get a refinement effect for martensite. So the packet size rather than lath width or block size are the main grain size strengthening contribution

in lath martensite. And you can see that if you plot, and how can I control the packet size? Well, very simply, by having smaller prior austenite grains. And you can see here, if I have a very small

prior austenite grain size, I will also have a smaller packet size. So it is meaningful to refine the martensite just as much as you can refine ferrite, ferritic steels

or constructional steels to get a whole patch effect. The structural unit that's of importance is the packet size. So the whole patch actually applies to low carbon martensite. But the D here is the packet size.

And this D is controlled by the original austenite grain size. And the reason is because you have high misorientation angles between packets. And it's the packet boundaries that act as very efficient barriers to dislocation motions.

And it works very well because this is a fresh martensite that point to carbon. You can see very nice whole patch relation. And you have a high value of ky of the whole patch parameter.

So transfer of slip across packet boundaries is difficult. And again here, just as in the case of regular steels,

this very high resistance is believed to be related to, of course, the misorientation, but also the presence of carbon atoms at this boundary in the martensite. And the reason why we know that that's the case is because if you temper the martensite,

this is before tempering, this is after tempering. What happened to the slope of the whole patch equation?

Instead of having a nice high k value, we have almost flat. So suddenly, the microstructure hasn't changed, by the way. It's still that martensite. So suddenly something has happened to make the packet boundary much less of an obstacle

to propagation of deformation. And the reason is, well, the carbon. The carbon that used to be in the packet boundaries is now not in the packet boundaries anymore. It's formed carbides. Yes?

It's formed carbides. So what we get is the tempering of the martensite has an effect on the packet size strengthening. It reduces the packet size dependence of the yield strength, so the slope here. And it reduces the work hardening rate

of the martensite as a consequence. So both effects are due to carbide formation. So this pretty much ends the part I wanted to discuss about the grain boundary strengthening.

And just to close this afternoon's session, so what's important for you to realize is that you may have thought that Hall-Petch equations, if everything was settled, it's actually not. There are still things we don't understand and know.

In particular, everything that's related to the grain boundary properties. And that our 1 over square root d relation may not actually hold, in principle, if we ever find the right theory.

Having said this, working on reducing the grain size is very efficient from an engineering point of view. So in the industry, any metals industry, steel industry, aluminum industry, working on refining the grain is always very high priority, because you get

very high strength effect. Having said this, I also illustrate the fact that pushing this to the limit by having very tiny grains doesn't really make sense, because you kill the plasticity. You can make it totally irrelevant.

And finally, with the martensite, very important also, you can also refine. It shows that you can also refine microstructures that look very tiny already, like I said in particular for steels, martensite and bainite. If you are able to refine the starting austenite

microstructure, you will also refine the structural units in martensite and bainite and get an equivalent effect. But don't forget, again, as I said,

the theoretical basis for the strengthening is a little bit weak. So at this stage, best thing to do is use your whole patch equation, yes? But be very careful what ky value you use,

because as I've shown, it's very, very sensitive to the deformation, prior deformation. And it's very sensitive to composition, carbon, no carbon, tempered, not tempered. So be very careful when you select a ky value

for your calculations. OK, thank you very much, and have a very pleasant weekend.