Mechanical properties of steel 19: strain hardening
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Cooman, Bruno C. de
The 19th in a series of lectures given by Professor Bruno de Cooman of the Graduate Institute of Ferrous Technology, POSTECH, South Korea. Deals with the theory and practice of strain hardening.
 With today's... There was a graph I showed on Monday, and after the lecture I thought, when I projected it, I didn't remember why I really wanted to show it to you. Afterwards, I thought maybe I'd come back to this slide to... Just if any one of you was wondering if there wasn't anything wrong with the slide. So, we know there are two models, two basic models of strain hardening. One model, which has... The strain hardening has to do with the tendency that these locations will have to form cell structures very quickly in the process of deformation. So, you'll find... A non-uniform distribution of dislocations, and you'll have cell walls with lots of dislocations, and then cell interiors with very little, this few dislocations, but that's where the dislocations are moving. That's where you find the mobile dislocations. So, now, that's an interesting model, and there are suggestions that model is maybe valid. There are theories, and these theories show, for instance, and are actually based on the fact that there is a direct relation between flow stress and the inverse size of the dislocation cell size, diameter. So, the smaller the size, the larger the flow stress, and that's basically what happens when you deform a material, a steel, you see the cell size becomes smaller and smaller, and the flow stress increases. Now, despite the fact it's an interesting model, nowadays we tend to prefer models which only use one parameter, which ignores the cell formation, basically, and that parameter is just the dislocation density. Having said this, that particular model can be made more complex, and you can actually introduce something that is similar to dislocation cell formation in that model. So, don't get me wrong here, that model of dislocation density, you can do a lot with it, actually. So, that model says, okay, so at the beginning you don't have many dislocations, very good annealed material, you start generating dislocations from, so let's just look at the grain here, right? So, you can follow what I'm saying, and let's say it's iron, yes, and you know that when we start making dislocations, we need to have sources of dislocations. In BCC iron, that's very readily available, you have a dislocation, we'll have a, we'll have a jog, for instance, and jogs can generate dislocations, or you will have a process of double cross-slip, yes, double cross-slip, and that will generate Frank loops, yes, Frank, read, excuse me, Frank, read sources, that, you know, one little pair of jogs here, due to cross-slip, which is very simple in BCC iron, because of the high stacking fault energy, and you generate very, very many dislocations. So, at the beginning, so let's say here this is one of these Frank read sources, and we generate lots of dislocations, there may be one here also, we generate lots of dislocations, we generate a lot of dislocations. So, at the beginning we get lots of mobile dislocations, right? And so in this, in the model of dislocation density evolution of strain hardening, gradually these, you get into a situation where the dislocations run into forest dislocations, yes, and so they get, and these forest dislocations work as obstacles, right, so this dislocation will encounter dislocations where are, which cut its slip plane, yes, and where do they come from, well from other sources in your grain, right, and these will act as, right, as obstacles, yeah, okay. So, what we have is we have production of mobile dislocations, and these mobile dislocations become immobilized, yes, they become immobile dislocations, yeah, and in fact you could say that these are actually the forest dislocations, because they don't move, they don't participate in it like that. So, if this is the strain, yes, and let me draw, so you remember, I said that when you strain a material, dislocation density increases from about 10 to the 11 to, depending on how much strain you can apply, you know, 10 to the 15th dislocation density. Now, if we look into detail into the dislocations, there's actually, you can see two dislocation populations, you have your mobile dislocation population, and you have your immobile dislocation population, and at the beginning, yes, the mobile dislocation density increases, yes, faster than the immobile, because you don't have mobile dislocations, immobile dislocations yet, right, and so, so at low strains, we have more mobile dislocations than immobile dislocations, yes, but as we deform more and more, yes, most of the dislocations become immobile dislocations, yeah, so at higher strains, yes, we have more immobile dislocations or forest dislocations than we have mobile dislocations, okay. And this is why I thought maybe this was a little bit confusing, because I think I stressed a little bit the fact that, you know, when you strain material, you'll have a lot more immobilized dislocations than mobile dislocations, that's true, but at very low levels of stress or deformation, which is this case, right, this is for single crystals, and about resolved shear stress of 20 megapascal, you're barely starting to deform the material, then of course, forest dislocation density is low, okay, that's what I meant here, so at the beginning, that's why I showed this, and I didn't quite remember why I had included that slide in the presentation, so just in case it confused you that there were so low, the forest dislocation density was so low, in, let me see here, if I have a, yeah, and so, right, and so what's nice about this particular model of dislocation density is that it shows very nicely as we will see that the strain hardening, it's basically, if you make the connection with the strain hardening that you observe with single crystals, where you have three stages, the easy glide, then stage two with a very high rate of strain hardening, that is, when you do deform a steel, you're actually looking at stage three strain hardening, where the strain hardening as a function of the stress is a linear function, decreasing linear function, okay, so and we'll show that with this model, we can get stress strain curves that nicely mimic experimentally measured data, where you have a shear stress as a sort of shear strain, or you can change this shear strain into tensile stress, tensile strains, and you will have an initially high strain rate, which will gradually go to zero at a saturation, sorry, at a saturation stress, yes, a saturation stress, so this is the equation, simple equation that you get between the strain hardening and the stress in practice, all right, so and now in reality, as I said, the mobile dislocation density and the immobile dislocation density are two separate populations, yes, however, yes, in what will follow, yes, we assume that most, excuse me, when you strain the material, and by the way, this is a log scale, yeah, so if I put this in a linear scale, if I make a linear scale, yeah, basically looks like this, the mobile dislocation density and the immobile, so it's only at the very beginning, yes, that my mobile dislocation density is higher, yes, so in the following, we assume what's written on the first line here is that the total dislocation density is what we will be computing, yes, and the total density is the sum of mobile dislocation density plus immobile dislocation density, and because the experiments show us this, yes, we will just say when we compute the dislocation density, what we're actually computing is the immobile dislocation density, and we don't really need to worry too much about the immobile dislocation density because it's so small and it stays small, yes, okay, right, and so the idea now is we will, we don't need to separate mobile dislocation from immobile dislocation, we can just talk about dislocation density, that's point number one, that's an important point here, right, the second point is, okay, so how do we compute this dislocation density? Well, again, it's simple, we say these sources here, these Frank Reed sources that come from double cross-slip processes, yes, they generate dislocations, yes, I stress, many dislocations are generated, so I generate dislocation, I have a process of dislocation generation, right, okay, then these dislocations, these dislocations run into each other, yes, and they will form somehow, we don't need to go into details actually, they form forest dislocations, they're just stuck there, yeah, so there is a transition from immobile, from mobile dislocation to immobile dislocation, actually it's the sources that produce the immobile dislocations, yes, we will assume also that the rate at which we produce, yes, mobile dislocations is equal to the rate at which we store dislocations, yes, and we will assume that the mobile dislocation density just stays flat, doesn't change, okay, alright, so then, so that means dislocation density increases, increases, increases, so you would have a strain situation that the strain hardening would increase all the time, but it doesn't, we know that the strain hardening goes like this, yes, it goes like this, goes to zero, right, there is no increase anymore, so that means that something must happen to the dislocations, right, something must be taking them away, yes, and that mechanism, those mechanisms are called annihilation, annihilation, so we also have a competing process of annihilation, and those are dislocation-dislocation interactions, yes, whereby dislocation actually annihilate each other, for instance, as I showed you, if I have a positive edge dislocation that means meets a negative edge dislocation, look, the dislocation is gone, or I have two dislocations, yes, that react together to form a third dislocation, again, the dislocation length is, the dislocation density is decreased, okay, they don't have to be this close to each other or this perfect to give this annihilation, certainly not in ferritic steels where the cross-slip processes are very simple, okay, so let me get the pen here, so the total evolution of dislocation density is a result of dislocation multiplication or storage, I kind of use these words together or mix them, so when I say multiplication it also means storage, and dislocation annihilation, so you write this, the change of dislocation density is there is a term that increases and there is a term that causes decrease, okay, and of course increase of dislocation is accumulation of immobile dislocations, it caused by the arrest of these dislocations at strong obstacles, which they cannot cut, and then, no, a lot of forest dislocations, for instance, will do that to a glide dislocation, okay, so let's have a look here, so how do we describe the dislocation multiplication rate, okay, well it may seem strange, but the parameter that we'll use is called the mean free path, the dislocation mean free path, and basically what the name says is the distance at dislocation will move from the source to the place where it's immobilized, this dislocation here, for instance, is created, it moves, it moves, and here it meets very strong obstacles formed by forest dislocation and it stops, so that distance, it has moved from here to here freely, yes, and so I call that distance, I call that lambda, and I say that's the mean free path of the dislocations, yeah, and so that approach is actually been, is very elegant because it allows you to put in lots of effects into this model, yes, not only the effect of dislocation, just strain hardening the effect of dislocation density, okay, so this mean free path is nothing else than some kind of distance between dislocation source and the obstacle to the motion of the dislocation, and so, and now what's important also is the increment of immobile dislocation is proportional to the displacement of these dislocations, so I generate a number of dislocations per unit time, they cross, they get immobilized, so I increase the amount of immobile dislocations, yes, the rate at which I create, I store these dislocations has to do with this distance, the longer I take, the longer this distance is, the more dislocations I can generate before they get, yes, all right, so the increment, yes, is proportional to this displacement, the further they are apart, the more dislocations I can generate, yes, the loop makes 10 dislocations per second, if it takes 10 seconds to move, I'll make 10 immobile dislocations per second, if they move 20, yes, I can get more dislocations pumped into the obstacle, right, so the increment, we call the d rho, yes, and is equal to rho m, the mobile dislocation density, when the mobile dislocation density has traveled a distance lambda, so I make a certain density, yes, it takes, they move one after the other, yes, when they have moved a distance equal to the mean free path, the amount of immobile dislocation has increased by the number of dislocations that was originally created at the source, and we assume, yes, that the density of mobile dislocation always stays the same, so you always have the same amount of mobile dislocation that is fed into the dislocation obstacle, right, so originally I have a dislocation density immobilized here, yes, the source makes rho m mobile dislocations, they travel for a distance lambda, yes, and when they've all traveled a distance lambda, they all become part of this immobile dislocation density, yes, if they only travel a fraction of this distance, yes, the increase will be a fraction of the total amount, right, okay, so I can express this simply mathematically by saying, so the increase in dislocation density is the increase in immobile dislocation density is rho m, yes, the dislocation times dx, the distance they travel divided by lambda, the mean free path, so say dx is lambda, right, then the increase is equal to the mobile dislocation density, all right, so the dislocations, the way you understand this is the dislocation source creates rho m mobile dislocations, and they become immobile dislocations at the obstacles, so, and of course it depends on where, you know, how they are within this distance, that's this dx, all right, okay, the other direction, all right, okay, now, so why are we interested in having this dx, this distance they move, but of course because that's this dx is related to strain, the macroscopic strain, okay, okay, so again, so this increase is equal to rho m when dx is lambda, and of course, and again remember the rate at which I generate mobile dislocation is constant, so now let's connect this, this production of dislocation to the shear strain, yes, because as I make dislocations, yes, as I make dislocations I will generate strain, yes, okay, so, right, so now we'll have a, just to, this is a formula we've already seen by the way, right, so, if I have dislocations here, number of dislocations, which move a distance dx, yes, what is the shear that they will make here, right, okay, well, the density of dislocation, so first the number of dislocations I have here times their burgers factor times the density, excuse me, the distance that they have traveled, yeah, so if they all travel dx, yes, I will get this amount of shear, so our density change was given by the equation I've just written up here, yes, and the shear, yes, that the motion of these dislocations, these mobile dislocations over a distance dx gives me is d gamma rho, so here, rho, mobile dislocation, now b times dx, alright, so if I combine these two equations, I can now have d rho d gamma, the change of the immobile dislocation density with strain, yes, so basically the change of the storage of dislocations with deformation, so I just make the ratio of these two parameters and I find 1 over lambda times b, so it's only dependent on the mean free path and the burgers factor, obviously, right, okay, so that's really interesting because it means that if I can, if I have a way to determine the mean free path in this model, I can basically get to the change, the storage term in the dislocation density evolution, okay, right, so how will this look like? Well, we do, this is how we determine this lambda or 1 over lambda if you want, so the dislocations mean free distances is obviously a key parameter, right, and so how are we going to determine it? So it's not a constant, first of all, obviously, you can see it's not going to be a constant because it will be smaller as the dislocation density increases, right, the more dislocations I have, the closer they are, yeah, so that's one thing, so omega, sorry, lambda, the mean free path is connected to the dislocation density, the other thing is it's connected to dislocation density, but also connected to the interaction with the obstacles, yes, so in general, the dislocations will not stop at the first obstacle, right, if a dislocation, dislocations are not that wimpy that they stop at the first obstacle, they can cut through a few obstacles before they get stopped, so to take into account the fact that not every, not the first time a dislocation meets an obstacle, a forest dislocation, it will stop, we introduce a factor k, this k parameter is introduced, physical meaning of it is simply to say, well, that's the number of obstacles a dislocation will need to encounter before it's immobilized, and it takes, yeah, so lambda is equal to the distance that we have between dislocations, and that's, we know that's proportional to one over the square root of the dislocation density, and that's actually exact, you know, if we have a square array of edge dislocations, as I've shown many times, but it holds in general terms also, and then we have this k factor, numerical factor, which takes into account, you know, the fact that dislocations don't stop at the first forest dislocation that they meet, so we can rewrite this, the rate of storage of dislocations is proportional to the current dislocation density divided by two numerical parameters, k and b, all right, so that's perfect, so now we've got the, to take into consideration these annihilation terms, yes, so what we say here is, well, you know, we'll think of dislocations as not as lines, but as like flexible cylinders, yes, with a certain radius r, yes, a certain radius r, yes, and then, so they're flexible tubes like this, yes, and they can meet other flexible tubes, yes, like this, right, of radius r, yes, and if the dislocations are within this distance r from each other, they can interact, they can annihilate or they can give junctions, right, and so we present this as follows, so the dislocation generation, so the term I just showed, which I have to, but many people do this, so you say generation, storage, and accumulation, that's kind of always the same, yes, the reason is because the dislocation you generate are stored, yes, so that's the term here, the dislocation storage, this is the same as storage, yeah, term, let me just make sure there's no confusion with, because I know this may sometimes, certainly if it's the first time you hear about these concepts, this may be, so this is the storage term, yes, and, okay, so this is balanced by spontaneous annihilation of dislocations, when two dislocations, for instance, of opposite sign meet under favorable conditions, yeah, and this process has a name, we call it dynamic recovery, yes, it's basically you are deforming material, yes, and the process of deformation itself, yes, results in a reduction of the dislocations, and that's because dislocations interact, dynamic recovery. So the number of recovery sites on a slip plane is of course very simple, proportional to dislocation density, what else, because we're talking about dislocation-dislocation interactions, right, so it's, yeah, but not all sites result in annihilation, yes, and of course the dislocations can only interact if they are within a certain distance from each other, right, so we need to have a term that kind of takes these things into account, so we look at the very general idea, you have a dislocation segment of length L, it moves with a certain velocity, the x dt, and this can capture or annihilate other dislocations of opposite burgers vector, if these are within a radius R of the dislocation core, right, so not necessarily on the same glide plane, etc., and for iron, VCC iron is actually very realistic because we know the dislocations, even if they are there and there, they can cross-slip, yes, they can cross-slip and react if it's possible, so the moving dislocation will sweep a certain volume V, which is equal to 2drl times dx in a time interval of dt, so now let's have a look, right, and then we introduce beta, the fraction of dislocation which actually be annihilated, yeah, why do we need to introduce this beta, well, this, the dislocation like this can annihilate this dislocation if they are within a distance R from each other, right, however, if this is this dislocation, yes, there's not going to be annihilation, right, so not all dislocation dislocation meetings as it were lead to annihilation, so that's why I introduce this factor beta, which tells me that only a limited number of interactions will lead to annihilations, and then the amount of dislocations which is annihilated per unit length of moving dislocation is given by beta, so, by this, yeah, and the annihilation rate is therefore given by this here, it's basically the same thing, multiplied with the mobile dislocation density, because this is for per unit length of moving dislocation, right, so if I have to multiply, this is for one dislocation and a unit length of it, so if I multiply this times the mobile dislocation density, yes, I get the annihilation rate, okay, so now I have the annihilation rate is proportional to the dislocation density, okay, and now I do the same, I use the same equation for dx relation between dx and the shear strain, you remember that, the shear strain was equal to dislocation density mobile times v times dx, yes, right, and so this gives me dx is d gamma divided by rho m times v, okay, so this gives me a formula for the rate of annihilation, the rate of annihilation, yes, to v times rho, the dislocation density times r divided by v, so it is proportional to the dislocation density, so I have a storage term which is proportional to the square root, and I have an annihilation term which is proportional to the dislocation density, so let's have a look at what we have found now, so if we combine these two equations, these two parameters that we have now, the dislocation we find a fundamental equation for strain hardening, yeah, right, so this is really important, it says that the dislocation density with shear strain consists of two terms, a storage term, an annihilation term, and the storage term is proportional to the square root of the dislocation density, and the annihilation term is proportional to the density of dislocations, and what is interesting here is that is to look at what is the implication of this for strain hardening, yeah, and we can look at this by first saying well if we look at this equation, it seems to be that there is a condition, yes, of saturation, yes, there is a saturation dislocation density, you can see that the road D divided by D gamma, so the rate of generation of dislocation can become zero, yes, and of course can become zero if these two terms are equal to each other, right, so we have a saturation dislocation generation which is given by this, for beta square times k square times r square, yes, which are in fact these parameters of our model, yeah, but there is a saturation dislocation density, okay, right, so now why is this so important, this equation, because now that I have this equation it gives me the evolution of the dislocation density with strain, yes, as a function of the current value of the dislocation density, okay, and if I can combine this with this relation which gives the relation between the shear stress and the square root of the dislocation density, I have basically a stress strain relation, yes, because if I numerically integrate this, yeah, I obtain the dislocation density as a function of shear strain and if I take the square root of this at every value of the strain I get my shear stress shear strain equation, good, but let's now have first perhaps a look at the strain hardening, okay, and I can do this, I don't need to numerically integrate this because I actually need this formula, so if I look at the strain hardening, that's d tau d gamma expressed in terms of shear stress shear strain, so that is, so you do the derivative of this equation, that's very simple, alpha gb times one half and then one rho gamma to the minus one half d gamma, d rho d gamma, and d rho d gamma is here, you have it, so you can just input this here, okay, so this basically gives me the strain hardening, yes, as a function of this dislocation density, so we can rewrite this, this last equation, yeah, we can now rewrite this as, so I bring out this factor here, this factor here I will call theta zero, and of course if I look carefully here I have alpha gb square root of gamma, so I have, here I have a factor, yes, which includes the shear strain, shear stress, excuse me, so I can rewrite this like this, theta zero one minus the shear stress divided by the shear stress, the saturation shear stress, and in this very simple equation theta zero is the initial strain hardening, which is alpha times g divided by two times k, and t saturation is the saturation stress, yes, I call this saturation, so I can plot this strain hardening as a function of the shear stress, and it's basically a line, a straight line that goes from theta zero to tau saturation, yes, and that's exactly, if I go back a few slides here, from last time we, yes, it's exactly the relationship that we have here, yes, that I had predicted for polycrystalline, polycrystalline steel, all right, so that's a nice result to have, okay, all right, okay, so again, we are now ready to see if we can use this equation, because it would be really neat if we could use this equation, and let's try to use it for steel, yes, obviously the way I presented it here, we can only use it for a single-phased steel, so I could use it for a ferritic steel with very low-carbon ferritic steel, but I could, principle, use it for a martensite also, because martensite is basically ferrite with a number of microstructural features. One of the microstructural features that I haven't talked about here, yes, and you feel that maybe it does, it should have an effect, and that's the grain size, right, obviously if I want to use this equation for steel and get a reasonable prediction, a model prediction, I will need to say to take into account grain sizes, and the reason is because steels have relatively small grain sizes, so there is an effect, actually an important effect from the grain size, so how do we take this into account, well this is a nice example of where this approach works very well, it's actually very simple to put in the effect of a grain size, or anything else, microstructural feature that works as a obstacle to this location glide. So we can take the grain size into account by saying, well, they're rigid obstacles, yeah, they're rigid obstacles, when a dislocation hits a grain boundary, it's finished, it just cannot go on anymore, so it just gets stored, and yeah, so it's not like meeting a forest dislocation where the dislocation may be able to cut through quite a few of them before it actually stops, so we can, in the presence of grain boundaries, we can expand the definition of our mean free path, yes, taking into account the dislocation grain size, so instead of using, so the storage term for the dislocation density, d rho d gamma plus equal to 1 over lambda times b, in the case of only dislocation-dislocation interactions, this is what we had, dislocation density square divided by k, if I have an additional obstacle, I just add it, 1 over d, plus, yes, very simple, you can understand why I can just, maybe not directly feel that I can just make the sum of these two, but let's just imagine two simple cases, yes, say I have huge grains, I mean humongous grains, millimeter size, yes, right, so d is very, very large, yeah, and I'm deforming this, and you know I have lots of dislocations here, and then glide dislocations that get tangled, and here is a dislocation source, and this is my, the mean distance, yes, between the dislocations, 1 over, yeah, so in this case, this factor will become what, yes, 1 over d will be very, very small, yes, and this equation will refer to square root d divided by k, yes, because these four dislocations will be the main obstacle, yeah, right, now let's do a very different situation, let's do a very different, we have very tiny grains, yeah, so d is very, very small, then it becomes very difficult to actually generate dislocations, yeah, and you remember, they had to do with, you know, when you have a frank read source, the smaller, and say we, you use this picture of the double cross-lip here, and so this is the distance between the pinning points, if this distance is very small, yes, it becomes very hard to produce a dislocation, yes, because there is a 1 over r relationship, yeah, so if I make a crystal very, very small, necessarily the pinning points will be closer to each other, yes, so, right, so in the end, yes, the term 1 over d is much, much larger than this term, because I have a low dislocation density, yeah, low dislocation, so in this case, this location density is very, very low, so this term is low, this term is very high, yes, and obviously the mean free path will be the size of the grain at best, yeah, okay, so I hope this makes this acceptable, so now we can, you know, get our equation becomes like this, 1 over d plus square root rho over k, yeah, we can numerically integrate this, I'll try to, again, I promised you already two things that I would put on E class, I'll put it on E class how you can, you know, if you don't know how to numerically integrate equation like this, I'll show you, it's very simple, I'll put it on E class, so, so that gives you this function rho as a function of gamma, yes, and that gives you the shear stress, shear stress as a function of shear strain is this, that's the strain hardening, it's very, actually rather simple, okay, so let's see, can we use this, yes, and actually yes, we can use this, and we can make for a single phase situation, we can actually, what shall I say, simplify things, and by just, just by noting that we have, in practice, we often have situation like this, where the mobile dislocation density stays the same, right, so what does this mean? If the mobile dislocation density doesn't change, yes, or, you know, I can actually simplify the first term, I can, the first term here, I can change it into, so this is the equation we derived, yes, so basically, I can rewrite it as k1, this being a constant, and then minus k2 times dislocation density, and so in this case, k2 is, you know, 2 beta times r divided by b, yes, and what I'm saying in addition is that this term here, yes, is a constant, so the dislocation storage rate is considered constant, and it basically means I'm not changing the rate at which I produce mobile dislocations, yes, and I can do this because my mobile dislocation density is low, and pretty much flat, yes, right, now if that's the case, we can actually integrate this equation directly, so one of the things I do first, yes, is I'm going to get rid of the shear strain, yes, and you know I can do this by using Taylor's factor, Taylor factor, epsilon is gamma divided by m, all right, and so I obtain very simply something that I can integrate directly, and basically, d rho over m times k1 minus k2 times dislocation density is d epsilon, right, so now if I integrate this, it will give me an equation of the dislocation density as a function of the tensile strain, so I integrate this equation, and I get the dislocation density strain dependence, it's basically, you can look at it after class, but it's basically a straightforward integration here, and I've written down the steps, so this dislocation density with strain is equal to this, so first term k1 over k2 times 1 minus e minus k2 m epsilon plus rho zero times e minus k2 m times epsilon, and here k1 and k2 are these parameters here, m is the Taylor factor, rho zero is the initial dislocation density, and epsilon is of course the strain, okay, so let's now have a look at how we then calculate the strain hardening, the strain hardening is, oops, yes, okay, yes, so the strain hardening, the stress, shear stress as a function of shear strain equation is shown here, alpha bg times b times square root of dislocation density, I can get the shear, from the shear stress to the tensile stress by multiplying this with m, with the Taylor factor, okay, and I can get the change of the dislocation density with the strain from here, so this is rho as a function of tensile strain, so I get tensile stress as a function of tensile strain, and that's basically a stress strain curve, yes, of course if I want to have a real stress strain curve I need to add other strengthening parameters such as the Pyrall stress, the lattice friction, and the solid solution effect, and all the other effects we still have to discuss, but let's just say we have these two terms, yeah, so let's have an example of how this works, well first of all we'll need the Burgers factor in this equation, that's here, 0.248 times nanometers, the orientation factor, the Taylor factor is about 3, the shear modulus 80 something 84 gigapascal here, this parameter alpha here is 0.35, okay, and then we will use this equation to do two things, we'll make a stress strain curve for the ferrite, single phase ferrite, and for martensite, and so the big difference here when we do this is that for the ferrite grain size, yes, we basically have an equivalent distance, mean free path for dislocations, which is equal to the grain size, and say we take here a grain size of 35 microns, okay, it's pretty large grain size, okay, but for a, but not an unreasonable for a very well annealed IF steel for instance, martensite, however, the, we have to, there is no grain size in martensite, we have to take the last size into account, and that's much smaller, that's much smaller, yes, that is of the order of 100 to 200 nanometers, so for the example I'll show, and you can recalculate if you want, is 120 nanometers in this case, okay, all right, good, so, so what we do here, remember what was K1, K1 is this term here, and this comes from this term, okay, and this term we go back a little bit here, here it is, that's the term, one over D plus square root rho over K, okay, so that's the equation I'm using here, so K is equal to one over B times lambda, and lambda can be one over the square root of dislocation density, or the grain's sub-grain or lat size, yes, depending on the situation, in this case, yes, we know that we can use grain size, yes, because we have a very low increase in the rate, production increase in the amount of mobile dislocations, basically there's no increase, yes, with strength, so the term that's important here will be one over D, yes, one over the grain size, so this K1 in this particular example is equal to one over lambda D, sorry, one over D times P, so this is the general form, yes, and so this is the mean free path equation, yes, and so in this case, lambda is D, so this will be K1, alright, in this particular case, so let me go back here, lattice friction 55, right, so for this particular example I chose 55, so that is, yes, then annihilation parameters for ferrite, an annihilation parameter for martensite, I give you values, okay, you can calculate them based on what we know for capital K and beta, yes, these values are determined on the basis of experiments, experiments, they're good values, yes, that many people have found, so we'll use those, initial dislocation densities in ferrite is very low, initial dislocation densities in martensite are very high, right, remember, when we deform well annealed ferrite, initially we have very few dislocations, in martensite you have plenty of transformation dislocations, and then the solution strengthening, we also need to take into account, and there, you know, I can use theories or I can use an empirical approach, this is the one we've chosen, and these are the solution strengthening effects I've taken into account, and this is what you get, so a stress strain curve for ferrite, stress strain curve for martensite, yes, and you can see that the stress strain curves look very realistic, yes, and basically very well model single phase steels, in this simple, using this simple analytical form of the equation, yeah, alright, so the nice thing you can see, even in this relatively simple case, that although you have many parameters, I agree, most of these parameters you cannot freely choose, yeah, the Taylor factor is what it is, the Bergus factor you cannot touch, obviously it's not really a fitting parameter, so many of these numbers are not fitting parameters, because they are physically for steel, but there are parameters you can play with, you know you can change the grain size, you know you can change the composition, yes, you know you can change the initial dislocation density, that's fine, you can do this, and then of course you can see directly what the impact will be of what you do on the stress strain curve, yes, so it's very handy theory to predict not only the strength of the material, but also the flow strength at different strains, yes, okay, and there are ways to expand this theory and it's been done, you know, to multi-phase steels or to phase steels with more complex strain hardening or additional strain hardening mechanisms, okay, alright, I'm a little bit over time, so sorry about that, thank you for your attention, I hope everybody got the email that we have make-up lecture time on Friday, okay, from 4 to 6, so these two lectures are being made up for next week, because I'm going to be absent on Tuesday, and I will not be able to get back here in time to teach on Thursday, so we'll do two classes, alright, and then we'll have a quiz, the week after that, yes, on a regular quiz Thursday, okay, yes, we'll have, and that will cover the whole block of lectures, okay, so there will be more questions, okay, just a little bit more stressful, and on, so on Friday, so tomorrow afternoon, we will be talking about grain size strengthening in steels. Thank you. Thank you.
 With today's...  There was a graph I showed on Monday, and after the lecture I thought,  when I projected it, I didn't remember why I really wanted to show it to you.  Afterwards, I thought maybe I'd come back to this slide to...  Just if any one of you was wondering if there wasn't anything wrong with the slide.  So, we know there are two models, two basic models of strain hardening.  One model, which has...  The strain hardening has to do with the tendency that these locations will have to form cell structures  very quickly in the process of deformation.  So, you'll find...  A non-uniform distribution of dislocations, and you'll have cell walls with lots of dislocations,  and then cell interiors with very little, this few dislocations, but that's where the dislocations are moving.  That's where you find the mobile dislocations.  So, now, that's an interesting model, and there are suggestions that model is maybe valid.  There are theories, and these theories show, for instance, and are actually based on the fact that  there is a direct relation between flow stress and the inverse size of the dislocation cell size, diameter.  So, the smaller the size, the larger the flow stress, and that's basically what happens when you deform a material,  a steel, you see the cell size becomes smaller and smaller, and the flow stress increases.  Now, despite the fact it's an interesting model, nowadays we tend to prefer models which only use one parameter,  which ignores the cell formation, basically, and that parameter is just the dislocation density.  Having said this, that particular model can be made more complex, and you can actually introduce something that is similar to dislocation cell formation in that model.  So, don't get me wrong here, that model of dislocation density, you can do a lot with it, actually.  So, that model says, okay, so at the beginning you don't have many dislocations, very good annealed material,  you start generating dislocations from, so let's just look at the grain here, right?  So, you can follow what I'm saying, and let's say it's iron, yes, and you know that when we start making dislocations,  we need to have sources of dislocations.  In BCC iron, that's very readily available, you have a dislocation, we'll have a, we'll have a jog, for instance,  and jogs can generate dislocations, or you will have a process of double cross-slip, yes, double cross-slip,  and that will generate Frank loops, yes, Frank, read, excuse me, Frank, read sources, that, you know, one little pair of jogs here,  due to cross-slip, which is very simple in BCC iron, because of the high stacking fault energy, and you generate very, very many dislocations.  So, at the beginning, so let's say here this is one of these Frank read sources, and we generate lots of dislocations,  there may be one here also, we generate lots of dislocations, we generate a lot of dislocations.  So, at the beginning we get lots of mobile dislocations, right?  And so in this, in the model of dislocation density evolution of strain hardening,  gradually these, you get into a situation where the dislocations run into forest dislocations, yes,  and so they get, and these forest dislocations work as obstacles, right, so this dislocation will encounter dislocations where are, which cut its slip plane, yes,  and where do they come from, well from other sources in your grain, right, and these will act as, right, as obstacles, yeah, okay.  So, what we have is we have production of mobile dislocations, and these mobile dislocations become immobilized, yes, they become immobile dislocations, yeah,  and in fact you could say that these are actually the forest dislocations, because they don't move, they don't participate in it like that.  So, if this is the strain, yes, and let me draw, so you remember, I said that when you strain a material, dislocation density increases from about 10 to the 11 to,  depending on how much strain you can apply, you know, 10 to the 15th dislocation density.  Now, if we look into detail into the dislocations, there's actually, you can see two dislocation populations, you have your mobile dislocation population,  and you have your immobile dislocation population, and at the beginning, yes, the mobile dislocation density increases, yes, faster than the immobile,  because you don't have mobile dislocations, immobile dislocations yet, right, and so, so at low strains, we have more mobile dislocations than immobile dislocations, yes,  but as we deform more and more, yes, most of the dislocations become immobile dislocations, yeah, so at higher strains, yes, we have more immobile dislocations or forest dislocations than we have mobile dislocations, okay.  And this is why I thought maybe this was a little bit confusing, because I think I stressed a little bit the fact that, you know, when you strain material,  you'll have a lot more immobilized dislocations than mobile dislocations, that's true, but at very low levels of stress or deformation, which is this case, right,  this is for single crystals, and about resolved shear stress of 20 megapascal, you're barely starting to deform the material,  then of course, forest dislocation density is low, okay, that's what I meant here, so at the beginning, that's why I showed this, and I didn't quite remember why I had included that slide in the presentation,  so just in case it confused you that there were so low, the forest dislocation density was so low, in, let me see here, if I have a, yeah, and so, right,  and so what's nice about this particular model of dislocation density is that it shows very nicely as we will see that the strain hardening,  it's basically, if you make the connection with the strain hardening that you observe with single crystals, where you have three stages, the easy glide,  then stage two with a very high rate of strain hardening, that is, when you do deform a steel, you're actually looking at stage three strain hardening,  where the strain hardening as a function of the stress is a linear function, decreasing linear function, okay,  so and we'll show that with this model, we can get stress strain curves that nicely mimic experimentally measured data,  where you have a shear stress as a sort of shear strain, or you can change this shear strain into tensile stress, tensile strains,  and you will have an initially high strain rate, which will gradually go to zero at a saturation, sorry, at a saturation stress,  yes, a saturation stress, so this is the equation, simple equation that you get between the strain hardening and the stress in practice, all right,  so and now in reality, as I said, the mobile dislocation density and the immobile dislocation density are two separate populations, yes,  however, yes, in what will follow, yes, we assume that most, excuse me, when you strain the material, and by the way, this is a log scale, yeah,  so if I put this in a linear scale, if I make a linear scale, yeah, basically looks like this, the mobile dislocation density and the immobile,  so it's only at the very beginning, yes, that my mobile dislocation density is higher, yes,  so in the following, we assume what's written on the first line here is that the total dislocation density is what we will be computing, yes,  and the total density is the sum of mobile dislocation density plus immobile dislocation density,  and because the experiments show us this, yes, we will just say when we compute the dislocation density, what we're actually computing is the immobile dislocation density,  and we don't really need to worry too much about the immobile dislocation density because it's so small and it stays small, yes,  okay, right, and so the idea now is we will, we don't need to separate mobile dislocation from immobile dislocation,  we can just talk about dislocation density, that's point number one, that's an important point here, right,  the second point is, okay, so how do we compute this dislocation density?  Well, again, it's simple, we say these sources here, these Frank Reed sources that come from double cross-slip processes, yes,  they generate dislocations, yes, I stress, many dislocations are generated,  so I generate dislocation, I have a process of dislocation generation, right,  okay, then these dislocations, these dislocations run into each other, yes,  and they will form somehow, we don't need to go into details actually, they form forest dislocations, they're just stuck there, yeah,  so there is a transition from immobile, from mobile dislocation to immobile dislocation,  actually it's the sources that produce the immobile dislocations, yes,  we will assume also that the rate at which we produce, yes, mobile dislocations is equal to the rate at which we store dislocations,  yes, and we will assume that the mobile dislocation density just stays flat, doesn't change, okay,  alright, so then, so that means dislocation density increases, increases, increases,  so you would have a strain situation that the strain hardening would increase all the time,  but it doesn't, we know that the strain hardening goes like this, yes,  it goes like this, goes to zero, right, there is no increase anymore, so that means that  something must happen to the dislocations, right, something must be taking them away, yes,  and that mechanism, those mechanisms are called annihilation, annihilation,  so we also have a competing process of annihilation, and those are dislocation-dislocation interactions,  yes, whereby dislocation actually annihilate each other, for instance, as I showed you,  if I have a positive edge dislocation that means meets a negative edge dislocation, look, the dislocation is gone,  or I have two dislocations, yes, that react together to form a third dislocation,  again, the dislocation length is, the dislocation density is decreased, okay,  they don't have to be this close to each other or this perfect to give this annihilation,  certainly not in ferritic steels where the cross-slip processes are very simple, okay,  so let me get the pen here, so the total evolution of dislocation density is a result of dislocation  multiplication or storage, I kind of use these words together or mix them,  so when I say multiplication it also means storage, and dislocation annihilation,  so you write this, the change of dislocation density is there is a term that increases  and there is a term that causes decrease, okay, and of course increase of dislocation is accumulation  of immobile dislocations, it caused by the arrest of these dislocations at strong obstacles,  which they cannot cut, and then, no, a lot of forest dislocations, for instance,  will do that to a glide dislocation, okay, so let's have a look here,  so how do we describe the dislocation multiplication rate, okay, well it may seem strange,  but the parameter that we'll use is called the mean free path, the dislocation mean free path,  and basically what the name says is the distance at dislocation will move from the source to the place  where it's immobilized, this dislocation here, for instance, is created, it moves, it moves,  and here it meets very strong obstacles formed by forest dislocation and it stops,  so that distance, it has moved from here to here freely, yes, and so I call that distance,  I call that lambda, and I say that's the mean free path of the dislocations, yeah,  and so that approach is actually been, is very elegant because it allows you to put in lots of effects into this model,  yes, not only the effect of dislocation, just strain hardening the effect of dislocation density, okay,  so this mean free path is nothing else than some kind of distance between dislocation source  and the obstacle to the motion of the dislocation, and so, and now what's important also is the increment  of immobile dislocation is proportional to the displacement of these dislocations,  so I generate a number of dislocations per unit time, they cross, they get immobilized,  so I increase the amount of immobile dislocations, yes, the rate at which I create,  I store these dislocations has to do with this distance, the longer I take, the longer this distance is,  the more dislocations I can generate before they get, yes, all right, so the increment, yes,  is proportional to this displacement, the further they are apart, the more dislocations I can generate, yes,  the loop makes 10 dislocations per second, if it takes 10 seconds to move, I'll make 10 immobile dislocations per second,  if they move 20, yes, I can get more dislocations pumped into the obstacle, right, so the increment,  we call the d rho, yes, and is equal to rho m, the mobile dislocation density,  when the mobile dislocation density has traveled a distance lambda, so I make a certain density, yes,  it takes, they move one after the other, yes, when they have moved a distance equal to the mean free path,  the amount of immobile dislocation has increased by the number of dislocations that was originally created at the source,  and we assume, yes, that the density of mobile dislocation always stays the same,  so you always have the same amount of mobile dislocation that is fed into the dislocation obstacle, right,  so originally I have a dislocation density immobilized here, yes, the source makes rho m mobile dislocations,  they travel for a distance lambda, yes, and when they've all traveled a distance lambda,  they all become part of this immobile dislocation density, yes, if they only travel a fraction of this distance,  yes, the increase will be a fraction of the total amount, right, okay, so I can express this simply mathematically by saying,  so the increase in dislocation density is the increase in immobile dislocation density is rho m, yes,  the dislocation times dx, the distance they travel divided by lambda, the mean free path,  so say dx is lambda, right, then the increase is equal to the mobile dislocation density, all right,  so the dislocations, the way you understand this is the dislocation source creates rho m mobile dislocations,  and they become immobile dislocations at the obstacles, so, and of course it depends on where, you know,  how they are within this distance, that's this dx, all right, okay, the other direction, all right, okay,  now, so why are we interested in having this dx, this distance they move,  but of course because that's this dx is related to strain, the macroscopic strain, okay,  okay, so again, so this increase is equal to rho m when dx is lambda,  and of course, and again remember the rate at which I generate mobile dislocation is constant,  so now let's connect this, this production of dislocation to the shear strain, yes,  because as I make dislocations, yes, as I make dislocations I will generate strain, yes, okay,  so, right, so now we'll have a, just to, this is a formula we've already seen by the way, right,  so, if I have dislocations here, number of dislocations, which move a distance dx, yes,  what is the shear that they will make here, right, okay, well, the density of dislocation,  so first the number of dislocations I have here times their burgers factor times the density,  excuse me, the distance that they have traveled, yeah, so if they all travel dx, yes,  I will get this amount of shear, so our density change was given by the equation I've just written up here, yes,  and the shear, yes, that the motion of these dislocations, these mobile dislocations over a distance dx  gives me is d gamma rho, so here, rho, mobile dislocation, now b times dx, alright,  so if I combine these two equations, I can now have d rho d gamma,  the change of the immobile dislocation density with strain, yes,  so basically the change of the storage of dislocations with deformation,  so I just make the ratio of these two parameters and I find 1 over lambda times b,  so it's only dependent on the mean free path and the burgers factor, obviously, right,  okay, so that's really interesting because it means that if I can, if I have a way to determine the mean free path in this model,  I can basically get to the change, the storage term in the dislocation density evolution, okay,  right, so how will this look like?  Well, we do, this is how we determine this lambda or 1 over lambda if you want,  so the dislocations mean free distances is obviously a key parameter, right,  and so how are we going to determine it?  So it's not a constant, first of all, obviously, you can see it's not going to be a constant  because it will be smaller as the dislocation density increases, right,  the more dislocations I have, the closer they are, yeah,  so that's one thing, so omega, sorry, lambda, the mean free path is connected to the dislocation density,  the other thing is it's connected to dislocation density,  but also connected to the interaction with the obstacles, yes,  so in general, the dislocations will not stop at the first obstacle, right,  if a dislocation, dislocations are not that wimpy that they stop at the first obstacle,  they can cut through a few obstacles before they get stopped,  so to take into account the fact that not every, not the first time a dislocation meets an obstacle,  a forest dislocation, it will stop, we introduce a factor k,  this k parameter is introduced, physical meaning of it is simply to say,  well, that's the number of obstacles a dislocation will need to encounter before it's immobilized,  and it takes, yeah, so lambda is equal to the distance that we have between dislocations,  and that's, we know that's proportional to one over the square root of the dislocation density,  and that's actually exact, you know, if we have a square array of edge dislocations, as I've shown many times,  but it holds in general terms also, and then we have this k factor, numerical factor,  which takes into account, you know, the fact that dislocations don't stop at the first forest dislocation that they meet,  so we can rewrite this, the rate of storage of dislocations is proportional to the current dislocation density  divided by two numerical parameters, k and b, all right, so that's perfect,  so now we've got the, to take into consideration these annihilation terms, yes,  so what we say here is, well, you know, we'll think of dislocations as not as lines,  but as like flexible cylinders, yes, with a certain radius r, yes, a certain radius r, yes,  and then, so they're flexible tubes like this, yes, and they can meet other flexible tubes, yes, like this, right, of radius r, yes,  and if the dislocations are within this distance r from each other, they can interact,  they can annihilate or they can give junctions, right,  and so we present this as follows, so the dislocation generation, so the term I just showed, which I have to,  but many people do this, so you say generation, storage, and accumulation, that's kind of always the same, yes,  the reason is because the dislocation you generate are stored, yes, so that's the term here, the dislocation storage,  this is the same as storage, yeah, term, let me just make sure there's no confusion with, because I know this may sometimes,  certainly if it's the first time you hear about these concepts, this may be, so this is the storage term, yes,  and, okay, so this is balanced by spontaneous annihilation of dislocations, when two dislocations, for instance,  of opposite sign meet under favorable conditions, yeah, and this process has a name, we call it dynamic recovery, yes,  it's basically you are deforming material, yes, and the process of deformation itself, yes, results in a reduction of the dislocations,  and that's because dislocations interact, dynamic recovery.  So the number of recovery sites on a slip plane is of course very simple, proportional to dislocation density, what else,  because we're talking about dislocation-dislocation interactions, right, so it's, yeah,  but not all sites result in annihilation, yes, and of course the dislocations can only interact if they are within a certain distance from each other, right,  so we need to have a term that kind of takes these things into account, so we look at the very general idea,  you have a dislocation segment of length L, it moves with a certain velocity, the x dt,  and this can capture or annihilate other dislocations of opposite burgers vector, if these are within a radius R of the dislocation core, right,  so not necessarily on the same glide plane, etc., and for iron, VCC iron is actually very realistic because we know the dislocations,  even if they are there and there, they can cross-slip, yes, they can cross-slip and react if it's possible,  so the moving dislocation will sweep a certain volume V, which is equal to 2drl times dx in a time interval of dt,  so now let's have a look, right, and then we introduce beta, the fraction of dislocation which actually be annihilated, yeah,  why do we need to introduce this beta, well, this, the dislocation like this can annihilate this dislocation if they are within a distance R from each other, right,  however, if this is this dislocation, yes, there's not going to be annihilation, right, so not all dislocation dislocation meetings as it were lead to annihilation,  so that's why I introduce this factor beta, which tells me that only a limited number of interactions will lead to annihilations,  and then the amount of dislocations which is annihilated per unit length of moving dislocation is given by beta,  so, by this, yeah, and the annihilation rate is therefore given by this here, it's basically the same thing,  multiplied with the mobile dislocation density, because this is for per unit length of moving dislocation, right, so if I have to multiply, this is for one dislocation and a unit length of it,  so if I multiply this times the mobile dislocation density, yes, I get the annihilation rate, okay, so now I have the annihilation rate is proportional to the dislocation density,  okay, and now I do the same, I use the same equation for dx relation between dx and the shear strain, you remember that,  the shear strain was equal to dislocation density mobile times v times dx, yes, right, and so this gives me dx is d gamma divided by rho m times v, okay,  so this gives me a formula for the rate of annihilation, the rate of annihilation, yes, to v times rho, the dislocation density times r divided by v,  so it is proportional to the dislocation density, so I have a storage term which is proportional to the square root, and I have an annihilation term which is proportional to the dislocation density,  so let's have a look at what we have found now, so if we combine these two equations, these two parameters that we have now, the dislocation we find a fundamental equation for strain hardening,  yeah, right, so this is really important, it says that the dislocation density with shear strain consists of two terms, a storage term, an annihilation term,  and the storage term is proportional to the square root of the dislocation density, and the annihilation term is proportional to the density of dislocations,  and what is interesting here is that is to look at what is the implication of this for strain hardening, yeah,  and we can look at this by first saying well if we look at this equation, it seems to be that there is a condition, yes, of saturation, yes,  there is a saturation dislocation density, you can see that the road D divided by D gamma, so the rate of generation of dislocation can become zero,  yes, and of course can become zero if these two terms are equal to each other, right, so we have a saturation dislocation generation which is given by this,  for beta square times k square times r square, yes, which are in fact these parameters of our model, yeah, but there is a saturation dislocation density, okay,  right, so now why is this so important, this equation, because now that I have this equation it gives me the evolution of the dislocation density with strain,  yes, as a function of the current value of the dislocation density, okay, and if I can combine this with this relation which gives the relation between the shear stress  and the square root of the dislocation density, I have basically a stress strain relation, yes, because if I numerically integrate this, yeah,  I obtain the dislocation density as a function of shear strain and if I take the square root of this at every value of the strain I get my shear stress shear strain equation,  good, but let's now have first perhaps a look at the strain hardening, okay, and I can do this, I don't need to numerically integrate this because I actually need this formula,  so if I look at the strain hardening, that's d tau d gamma expressed in terms of shear stress shear strain, so that is, so you do the derivative of this equation,  that's very simple, alpha gb times one half and then one rho gamma to the minus one half d gamma, d rho d gamma, and d rho d gamma is here, you have it,  so you can just input this here, okay, so this basically gives me the strain hardening, yes, as a function of this dislocation density,  so we can rewrite this, this last equation, yeah, we can now rewrite this as, so I bring out this factor here, this factor here I will call theta zero,  and of course if I look carefully here I have alpha gb square root of gamma, so I have, here I have a factor, yes, which includes the shear strain,  shear stress, excuse me, so I can rewrite this like this, theta zero one minus the shear stress divided by the shear stress, the saturation shear stress,  and in this very simple equation theta zero is the initial strain hardening, which is alpha times g divided by two times k, and t saturation is the saturation stress,  yes, I call this saturation, so I can plot this strain hardening as a function of the shear stress, and it's basically a line, a straight line that goes from theta zero to tau saturation,  yes, and that's exactly, if I go back a few slides here, from last time we, yes, it's exactly the relationship that we have here, yes, that I had predicted for polycrystalline,  polycrystalline steel, all right, so that's a nice result to have, okay, all right, okay, so again, we are now ready to see if we can use this equation,  because it would be really neat if we could use this equation, and let's try to use it for steel, yes, obviously the way I presented it here, we can only use it for a single-phased steel,  so I could use it for a ferritic steel with very low-carbon ferritic steel, but I could, principle, use it for a martensite also, because martensite is basically ferrite with a number of microstructural features.  One of the microstructural features that I haven't talked about here, yes, and you feel that maybe it does, it should have an effect, and that's the grain size, right, obviously if I want to use this equation for steel and get a reasonable prediction,  a model prediction, I will need to say to take into account grain sizes, and the reason is because steels have relatively small grain sizes, so there is an effect, actually an important effect from the grain size,  so how do we take this into account, well this is a nice example of where this approach works very well, it's actually very simple to put in the effect of a grain size, or anything else, microstructural feature that works as a obstacle to this location glide.  So we can take the grain size into account by saying, well, they're rigid obstacles, yeah, they're rigid obstacles, when a dislocation hits a grain boundary, it's finished, it just cannot go on anymore, so it just gets stored, and yeah,  so it's not like meeting a forest dislocation where the dislocation may be able to cut through quite a few of them before it actually stops, so we can, in the presence of grain boundaries, we can expand the definition of our mean free path, yes,  taking into account the dislocation grain size, so instead of using, so the storage term for the dislocation density, d rho d gamma plus equal to 1 over lambda times b,  in the case of only dislocation-dislocation interactions, this is what we had, dislocation density square divided by k, if I have an additional obstacle, I just add it, 1 over d, plus, yes,  very simple, you can understand why I can just, maybe not directly feel that I can just make the sum of these two, but let's just imagine two simple cases, yes,  say I have huge grains, I mean humongous grains, millimeter size, yes, right, so d is very, very large, yeah,  and I'm deforming this, and you know I have lots of dislocations here, and then glide dislocations that get tangled, and here is a dislocation source, and this is my, the mean distance, yes, between the dislocations, 1 over, yeah,  so in this case, this factor will become what, yes, 1 over d will be very, very small, yes, and this equation will refer to square root d divided by k, yes,  because these four dislocations will be the main obstacle, yeah, right, now let's do a very different situation, let's do a very different, we have very tiny grains, yeah, so d is very, very small,  then it becomes very difficult to actually generate dislocations, yeah, and you remember, they had to do with, you know, when you have a frank read source,  the smaller, and say we, you use this picture of the double cross-lip here, and so this is the distance between the pinning points, if this distance is very small, yes,  it becomes very hard to produce a dislocation, yes, because there is a 1 over r relationship, yeah, so if I make a crystal very, very small, necessarily the pinning points will be closer to each other, yes,  so, right, so in the end, yes, the term 1 over d is much, much larger than this term, because I have a low dislocation density, yeah, low dislocation,  so in this case, this location density is very, very low, so this term is low, this term is very high, yes, and obviously the mean free path will be the size of the grain at best, yeah, okay,  so I hope this makes this acceptable, so now we can, you know, get our equation becomes like this, 1 over d plus square root rho over k, yeah,  we can numerically integrate this, I'll try to, again, I promised you already two things that I would put on E class, I'll put it on E class how you can, you know, if you don't know how to numerically integrate equation like this, I'll show you, it's very simple,  I'll put it on E class, so, so that gives you this function rho as a function of gamma, yes, and that gives you the shear stress, shear stress as a function of shear strain is this, that's the strain hardening, it's very, actually rather simple, okay,  so let's see, can we use this, yes, and actually yes, we can use this, and we can make for a single phase situation, we can actually, what shall I say, simplify things,  and by just, just by noting that we have, in practice, we often have situation like this, where the mobile dislocation density stays the same, right, so what does this mean?  If the mobile dislocation density doesn't change, yes, or, you know, I can actually simplify the first term, I can, the first term here, I can change it into, so this is the equation we derived, yes,  so basically, I can rewrite it as k1, this being a constant, and then minus k2 times dislocation density, and so in this case, k2 is, you know, 2 beta times r divided by b, yes,  and what I'm saying in addition is that this term here, yes, is a constant, so the dislocation storage rate is considered constant,  and it basically means I'm not changing the rate at which I produce mobile dislocations, yes, and I can do this because my mobile dislocation density is low,  and pretty much flat, yes, right, now if that's the case, we can actually integrate this equation directly, so one of the things I do first, yes, is I'm going to get rid of the shear strain, yes,  and you know I can do this by using Taylor's factor, Taylor factor, epsilon is gamma divided by m, all right, and so I obtain very simply something that I can integrate directly,  and basically, d rho over m times k1 minus k2 times dislocation density is d epsilon, right, so now if I integrate this, it will give me an equation of the dislocation density as a function of the tensile strain,  so I integrate this equation, and I get the dislocation density strain dependence, it's basically, you can look at it after class, but it's basically a straightforward integration here,  and I've written down the steps, so this dislocation density with strain is equal to this, so first term k1 over k2 times 1 minus e minus k2 m epsilon plus rho zero times e minus k2 m times epsilon,  and here k1 and k2 are these parameters here, m is the Taylor factor, rho zero is the initial dislocation density, and epsilon is of course the strain, okay,  so let's now have a look at how we then calculate the strain hardening, the strain hardening is, oops, yes, okay, yes, so the strain hardening, the stress, shear stress as a function of shear strain equation is shown here,  alpha bg times b times square root of dislocation density, I can get the shear, from the shear stress to the tensile stress by multiplying this with m, with the Taylor factor, okay,  and I can get the change of the dislocation density with the strain from here, so this is rho as a function of tensile strain,  so I get tensile stress as a function of tensile strain, and that's basically a stress strain curve, yes,  of course if I want to have a real stress strain curve I need to add other strengthening parameters such as the Pyrall stress, the lattice friction, and the solid solution effect,  and all the other effects we still have to discuss, but let's just say we have these two terms, yeah, so let's have an example of how this works, well first of all we'll need the Burgers factor in this equation, that's here,  0.248 times nanometers, the orientation factor, the Taylor factor is about 3, the shear modulus 80 something 84 gigapascal here, this parameter alpha here is 0.35, okay,  and then we will use this equation to do two things, we'll make a stress strain curve for the ferrite, single phase ferrite, and for martensite,  and so the big difference here when we do this is that for the ferrite grain size, yes, we basically have an equivalent distance, mean free path for dislocations,  which is equal to the grain size, and say we take here a grain size of 35 microns, okay, it's pretty large grain size, okay, but for a, but not an unreasonable for a very well annealed IF steel for instance,  martensite, however, the, we have to, there is no grain size in martensite, we have to take the last size into account, and that's much smaller, that's much smaller, yes, that is of the order of 100 to 200 nanometers,  so for the example I'll show, and you can recalculate if you want, is 120 nanometers in this case, okay, all right, good, so,  so what we do here, remember what was K1, K1 is this term here, and this comes from this term, okay, and this term we go back a little bit here, here it is, that's the term,  one over D plus square root rho over K, okay, so that's the equation I'm using here, so K is equal to one over B times lambda, and lambda can be one over the square root of dislocation density,  or the grain's sub-grain or lat size, yes, depending on the situation, in this case, yes, we know that we can use grain size, yes, because we have a very low increase in the rate,  production increase in the amount of mobile dislocations, basically there's no increase, yes, with strength, so the term that's important here will be one over D, yes,  one over the grain size, so this K1 in this particular example is equal to one over lambda D, sorry, one over D times P,  so this is the general form, yes, and so this is the mean free path equation, yes, and so in this case, lambda is D, so this will be K1, alright, in this particular case,  so let me go back here, lattice friction 55, right, so for this particular example I chose 55, so that is, yes, then annihilation parameters for ferrite,  an annihilation parameter for martensite, I give you values, okay, you can calculate them based on what we know for capital K and beta, yes,  these values are determined on the basis of experiments, experiments, they're good values, yes, that many people have found, so we'll use those,  initial dislocation densities in ferrite is very low, initial dislocation densities in martensite are very high, right, remember,  when we deform well annealed ferrite, initially we have very few dislocations, in martensite you have plenty of transformation dislocations,  and then the solution strengthening, we also need to take into account, and there, you know, I can use theories or I can use an empirical approach,  this is the one we've chosen, and these are the solution strengthening effects I've taken into account, and this is what you get,  so a stress strain curve for ferrite, stress strain curve for martensite, yes, and you can see that the stress strain curves look very realistic,  yes, and basically very well model single phase steels, in this simple, using this simple analytical form of the equation, yeah,  alright, so the nice thing you can see, even in this relatively simple case, that although you have many parameters, I agree,  most of these parameters you cannot freely choose, yeah, the Taylor factor is what it is, the Bergus factor you cannot touch,  obviously it's not really a fitting parameter, so many of these numbers are not fitting parameters, because they are physically for steel,  but there are parameters you can play with, you know you can change the grain size, you know you can change the composition, yes,  you know you can change the initial dislocation density, that's fine, you can do this, and then of course you can see directly what the impact will be  of what you do on the stress strain curve, yes, so it's very handy theory to predict not only the strength of the material,  but also the flow strength at different strains, yes, okay, and there are ways to expand this theory and it's been done,  you know, to multi-phase steels or to phase steels with more complex strain hardening or additional strain hardening mechanisms, okay,  alright, I'm a little bit over time, so sorry about that, thank you for your attention, I hope everybody got the email that we have make-up lecture time on Friday,  okay, from 4 to 6, so these two lectures are being made up for next week, because I'm going to be absent on Tuesday, and I will not be able to get back here in time to teach on Thursday,  so we'll do two classes, alright, and then we'll have a quiz, the week after that, yes, on a regular quiz Thursday,  okay, yes, we'll have, and that will cover the whole block of lectures, okay, so there will be more questions, okay, just a little bit more stressful,  and on, so on Friday, so tomorrow afternoon, we will be talking about grain size strengthening in steels.  Thank you.  Thank you.
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