Mechanical properties of steel 19: strain hardening

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Mechanical properties of steel 19: strain hardening
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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.
Keywords The Graduate Institute of Ferrous Technology (GIFT)
Steel Gemstone Slip (ceramics) Train Game Outsourcing Sizing Volumetric flow rate Work hardening Gentleman Gebr. Frank KG Rutschung Plane (tool) Book design Single-cylinder engine Kümpeln Firearm Model building Material Wegfahrsperre
Stagecoach Steel Cut (gems) Engine-generator Disc brake Separation process Brickyard Gentleman Rutschung Slip (ceramics) Automatic Block Signaling Spare part Negation Kümpeln Firearm Material Engine displacement Lumber Tin can Ford Orion Ford Focus Mechanic Steel Stagecoach Slip (ceramics) Ford Transit Multiple birth Outsourcing Zipper Rolling (metalworking) Cartridge (firearms) Work hardening Ford Explorer Leitwerk <Flugzeug> Food storage Remotely operated underwater vehicle Single-cylinder engine Ship of the line Model building Flatcar Wegfahrsperre Firearm
Key (engineering) Mechanismus <Maschinendynamik> Schubvektorsteuerung Engine-generator Fahrgeschwindigkeit Roots-type supercharger Arbeitszylinder Food storage Rutschung Plane (tool) Kümpeln Firearm Material International Space Station Rohrpost Schubvektorsteuerung Key (engineering) Kopfstütze Volumetric flow rate Wood Fahrgeschwindigkeit Work hardening Food storage Plane (tool) Gas turbine Single-cylinder engine Ship of the line Model building Kümpeln Stock (firearms)
Stagecoach Nut (hardware) Key (engineering) Mechanismus <Maschinendynamik> Fahrgeschwindigkeit Saw Roots-type supercharger Replenishment oiler Food storage Ship class Plane (tool) Single-cylinder engine Negation Kümpeln Firearm Ford Focus Steel Schubvektorsteuerung Clothing sizes Sizing Zipper Cartridge (firearms) Fuel economy in automobiles Food storage Single-cylinder engine Stock (firearms) Model building
Friction Steel Kickstand Bomb Lapping Casting defect Roots-type supercharger Turning Rake (tool) Food storage Ship class Single-cylinder engine Kümpeln Watercraft rowing Firearm International Space Station Schubvektorsteuerung Clothing sizes Finger protocol Turning Knitting Sizing Zipper Cartridge (firearms) Work hardening Food storage Remotely operated underwater vehicle Single-cylinder engine Jig (tool)
Friction Tin can Capital ship Fiat 500 (2007) Schubvektorsteuerung Gaussian blur Clothing sizes Turning Sizing Work hardening Cartridge (firearms) Gentleman Remotely operated underwater vehicle Single-cylinder engine Rail profile Kümpeln Jig (tool) Firearm
Fiat 500 (2007) Volumetric flow rate Mechanic Ship class Work hardening Gentleman Single-cylinder engine Tanning Firearm Material Cylinder block
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thank you there was it's the graphite showed on Monday and then after after the lecture I thought on group and when it projected attending remember where you want really wanted to show it to you this afterwards I thought maybe I'd come back to this life to the justified in any 1 of you was wondering if if there wasn't anything wrong with the slide but so when such we we go there to models 2 basic models of strain hardening 1 model but which shot which was the book and the strain warning has to do with it the tendency to dislocations will have to form a cell structures very quickly in the difficult process of defamation so you will find the form the of non-uniform distribution of dislocation and you have found cell walls with lots of dislocations and Ansell interiors which little this few dislocation but that's where the dislocations or of are moving in and that's where you find a mobile dislocations to up now that's an interesting model in the suggestions that that model is his size in or maybe valid under their theories and these series show for instance in the actually based on the fact that there is a direct relation flow stress and the size of the the inverse size of the dislocation self size dynamic so that the smaller the size of of the larger the flow stress and that's that's basically what happens when you do that you to foreign material a steel that you see the sell side because smaller and smaller and the flow stress increase an adult despite facts interesting model nowadays we tend to prefer models which which only use 1 parameter which is a trend which ignores the cell formation basically and the fact is that and parameters just dislocation density having said this year that particular model of I have adapted a model can be made more complex and you can actually introduce something that is similar 2 I dislocation cell formation in in that model so uh so don't get me wrong here is that the model of dislocation density you can do a lot with it actually right and when this so that model says OK so at the beginning You don't have many disillusioned very good annealed material you start generating dislocations from the answers let's just look at the grain here so you can follow what I'm saying let's say it's hiring years and you know that when we start making dislocations we we need have sources of dislocations around in the C C R and that's very readily available you have a dislocation will happen will have the but will have a job for instance and the jocks cringe can be uh can generate dislocations or you will have a process of double-crossed left double-crossed and that will generate frankly yes Frank reading to the Frank Reed sources that have Munich took 1 little a pair of jobs here due to cross slip which is very simple In The because of the high stacking fault energy and you generate very very many dislocation so at the beginning so let's let's say you this is 1 of these Frank Reed sources and which generate lots of dislocated and maybe 1 here also we generate lots of dislocations will generate a lot of dislocations so at the beginning we get lots of mobile dislocations right and end and so in this area in the model I love the dislocation density evolution of strain hardening of gradually these but you get into a situation where the dislocations run into forest dislocations stands and it's so they get and these forces location works that workers obstacles related to business location will encounter this locations where which which cut its slipped plane yes when and where did they come from well from other sources in your brain right and these will act as right yes obstacles here In intent so what would we have is we have a production of mobile dislocations in these mobile dislocations be calm mobilized units the become immobile dislocations I know entered about yeah can in fact you could say that that these are actually the forest dislocations that because they don't move don't participate so and these if if if this is destroying yes and let me let me draw the new remember I said that when you strain material dislocation density but increases from but tend to be 11 2 depending on how much strain you can apply by tend to 15 the dislocation density now if we look into details and into they dislocations this actually you can see to this location population you have your mobile dislocation population and have you immobile dislocation of and at the beginning the the mobile dislocation density increases yes faster then the immobile because you don't have mobile dislocations mobile dislocations yet right and so so at low strains we have more mobile dislocations than immobile dislocations but as we did for more and more yes most of the dislocations become immobile dislocation so it hires trains yes we have more mall but the mobile dislocations or forest dislocations then we have mobile locations but and and this and this is where most of maybe this was limit confusing because I don't think stress little but the fact that you know when you strain material on you you have a lot more immobilize dislocations and mobile workers next stroke but at very low levels of the
stressed .period and defamation which is this case like this is for single crystals are at an end and that about this in the results your stress of 20 make Pascal you you barely starting to the of deformed material then of course you forest dislocation that is low OK that's what I meant here but the beginning of the square shoulders and I didn't quite I remember why I had included that slide in the presentation so just just in case you get it confused you that they were so low the day the force dislocation entity was so in them yeah this and so on the right and so what what's nice about this particular model of this location it is that you know it shows very nicely as we will see that but the strain hardening it's basically if if you make the connection with the strain hardening that you observe with single crystals where you have 3 stages to easy glide then staged 2 with a very high rate of strain hardening and that is so what would you do the former steel you actually looking at streamed through the stage 3 strain hardening where the string heartening as a function of the stress is a linear function decreasing linear function OK so and will show that I have with this model we can get stress-strain curves that nicely mimic the experimentally measured data where you haven't shear stress as a the serious strain or you can change this year stressed strain into tensile stress cell strains and you will have been initially high rate the strain rate which will gradually go to 0 0 at the saturation of soaring at that area at the saturation stressed the separation present this is this is steep equations simple equation that you get between the strain hardening and distant stressed in practice so no In reality known as I said the mobile dislocation density and the immobile dislocation density 2 separated populations yes however this is what will follow yes we that most this is 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 the that this looks like less the mobile dislocation density and the mobility so it's only at the very beginning yes that might mobile dislocation densities higher you so In the following week assume what's written on the 1st line here is that the but the total dislocation density is what we will be computing yes and the total densities than the sum of mobile dislocation plus immobile this will be and because the experiments show as this yes we will just say when we compute the it would were actually computing state a mobile dislocation density and we don't really need to worry too much about the mobile dislocation density because it's so small and it's very small yes intent back into society and it said the idea now is we will but we don't need to 2 seperate mobile dislocation from amount mobile this begin just talk about dislocation density that's point number 1 listen important point here right on the 2nd point is OK so how do we compute this dislocation density was again it's simple we sigh all right these sources here these Frank resources that come from a double-cross slip processes in that generate let this locations yeah my stress many dislocations are generated so I generate this location I have a process of dislocation generation right but then these dislocations this is a bit and run into each other and they will form somehow but would we don't need to go into details are actually I'm therefore Forest dislocation that there just stuck there so there is a transition from immobile for mobile this location to mobile actually it's it's the sources that produce the most immobile dislocations yes we will assume also that of the rate at which we produce yes mobile dislocations is equal to the rate at which we story dislocations in and we will assume that's the move of its location the distaste flat doesn't change In the alright so that it is that means this location it's the increases increase increases so you would we have the a strange situation that this strain hardening would increase all the time but it doesn't wait we know that the strain hardening goes like this and we will a the was like this was 2 0 rights there is no increase anymore so that means that something must happen to the dislocations write something must be taking them away yes and and that that the mechanism was magnetical annihilation annihilation show we also have a competing process of annihilation and and those are this location dislocation interactions yes whereby this location actually uh annihilate each other slight as I showed you if I have a positive edge dislocation that means means and negative edge dislocation from this location has got him or I have to dislocations yes that react together to form a 3rd dislocation again delayed sufficient length is a dislocation densities decrease of that but they they don't have to be this close to each other all this perfect to to give this annihilation certainly not in sporadic steals where the yard and those in the new year we would cross the processes of very young simple look at them so did so so that but here so the total the
evolution of dislocation density is result of dislocation multiplication or storage site find a use these words together makes them so when I say multiplications also means storage of arms and dislocation annihilation of the writers this change of dislocation and stick is that there is a term that increases and is a term that causes decrease from and and of course increase of dislocation is accumulation of immobile dislocations because by the arrest of these dislocation and strong obstacles in which they cannot captain and now with a lot of forest dislocations for instance will do that to oblige the dislocation let's have a look here so right to how do we discuss the dislocation but multiplication rate again well in the streets seem strange but the permitted that will use is called the mean free path mean dislocation mean free path and that basically what the name says it's uh the In this stands at this location will move from the source to the place where it's immobilized the dislocation here for instance is created it moves it most of them here meets very strong obstacles formed by Forest dislocation and it stops so that distance Our from it has moved from here to here freely yes and cycles that distance a call that landed and they say that splits the I mean free path of dislocations and so that approach is eventually is a very elegant but is it allows you to put in a lots of the facts into this model does not only have the effect of this locations of this the strain hardening the effect of dislocation density get so this mean free path nothing else and that as some kind of distance between dislocations source and the obstacle to the motion of the dislocation and so on and now it's important also the the increment of mobile dislocation is proportional to the displacement of the dislocation so I generated a number of dislocations part you need time they cross they get that get immobilized so I increased the amount of immobile this location the rate at which I create my store these dislocations has to do with this distance from the longer I take the longer this distance the more dislocations I can generate before they get there as well so so the sedate the increment this is proportional to this displacement of the further they are part the more dislocations I can generate yes take the dilute makes 10 dislocations the 2nd that it and if it takes 10 seconds to move well on May 10 the mobile dislocations the 2nd if they moved 20 I can get more this locations pumped into the uh obstacles right to the increments it we called in deep wrote it is and there is equal to roh them the mobile dislocation density when the mobile dislocation density has traveled a distance London so I I make a certain density it takes that been the move 1 after the other when when the when they have moved a distance equal to the the uh uh mean free path the amount of mobile this mobile this focus has increased by the number of dislocations that was originally created at the source and we assume yes that's it the density of mobile this look is always stays the same to you always have the same amount of mobile dislocation that is fed into the dislocation obstacles right so originally I have a dislocation density wrote immobilize here the source makes roll uh mobile dislocations they travel for a distance land yes and when they have all traveled a distance land that they all become part of of this the mobile dislocation density yes if they only travel a fraction of this distance yes the increase will be a fraction of the total amount right OK so so I can express this simply mathematically by saying so the increase in dislocation densities the increase mobile dislocation as it is Roanne years no this lucrative times the X the distance they travel divided by lavender the mean free path so they say DX is lined up right then the increase is equal to the mobile dislocation and for so the dislocations dissident way you understand this is the dislocation source creates roh and mobile dislocations and dates become immobile dislocations at the obstacles so and of course depends on where you know how about how they are within this this stance this is the exploration the other but the that was why are we interested in in having this DX this distance the move but of course because that's is the X is related to the story the macroscopic strength OK the up so again so that this increase is equal to roh when the X is lined and of course and again I remembered the rate at which act and generate mobile dislocation is constantly so now let's connect this this production of dislocation to the strain that's because as I made dislocations yes as I make this locations I will generate strength yes case so rectum and now we will move will have a just 2 of this is a formal we've already seen by the way writes so if I have no that and dislocations here a number of this look edge dislocation which moved a distance the ax yes what is the sheer that there they will make here OK well the density of this look of 1st the number of dislocations I have here times their burgers factor times the density attempts at the distance that they have traveled do they all traveled DX I will get this amount of share so Our density change was given by the equation just appear and the sheer newness that the motion of these dislocations these mobile disagreed over a distance of the X give me is the gamma role so severe role mobile this revision of the times the Times the acts all right if I combine these 2 equations I can now have the road the down the change mobile
dislocation density what's strange yes please speak the change of the storage of dislocations with defamation I so I just make it the ratio of these 2 parameters and I find 1 over Lambda time speed so it's only dependent on the mean free path the respect of his right but that's that's really interesting because it means that if I can make out of it have a way to determine the mean free path in this model I'm you know I can basically get 2 the change the storage In the dislocation density evolution right so I knew how How will this look like while we we do this is how we determined this land or 104 land if you want so the dislocations it the free this is this is obviously a key parameter and so how are we doing to us and so it's not a constant 1st of all obviously you as you can see it's not going to be a constant because it's will be smaller as the dislocation density increases by more dislocations I have the closer they are I so that's 1 thing so only got but that you mean free path is connected to the dislocation density the other thing is it's connected this location is also connected to the interaction with the obstacles so in general the dislocations will not stop at the 1st obstacle right now it's a dislocation dislocations are not that would be that they stop at the 1st obstacle they can cut through a few obstacles before they get stuff so to take into account the fact that not every but it is not the 1st time in a dislocation meets an obstacle of forces location it will stop we introduced a factor take this the parameter is introduced physical meaning of it is simply to say Let's a number of obstacle at this location will need to encounter before its immobilized and it takes years and years of so-so Lander yeah yes equal to these distance that we have between on dislocations and that's we know that's who a proportional to 1 over the square root of the dislocation density and that's that's actually exact you know if we have a square array of edge dislocations as shown many times but it holds ended in general terms also and then we have this key factor numerical factor which takes into account you know this the fact that this locations don't stop at the 1st and uh Forest dislocation of so we can rewrite that the rate of storage of dislocations is proportional to the current dislocation density divided by 2 numerical parameters K and B there are so that's perfect since now we've got the to take into a consideration these annihilation so what would we say here it is well you know what will think of this locations as not as lines but as like flexible cylinders yes wait a certain this far as certain ranges are immense and then no 1 should flexible tubes like this and they can meet other flexible tubes yes inverse of Rangers are and on if the this locations are within this distance far from each other they can interact they can annihilate or they can give junctions then and so we presenters as follows :colon so the dislocation generations of the term I just showed which I'd have to that many people do this so USA generations storage and accumulation that's kind of always the slaying the reason is because they the dislocation you generate are stored so that students determine location storage this this is the same as the storage but we just make sure there's no confusion will win because I I know this may sometimes certainly if it's the 1st time you hear about this concept this may be so this is a storage term In and out again this this is balanced by spontaneous annihilation of dislocations went to dislocation Prince of opposites signed made under favorable conditions and that this process has a name would call a dynamic recovery that is basically you are and the foreign material yes and of the process of defamation itself yes I results in a reduction of the dislocation and that's because dislocations interact with dynamic reap their recovery the so the number of recovery sites on the slick plane is of course the very simple proportional to dislocation density what else because we're talking about this location is looking interaction planted them at but not all sites result annihilation yes and of course the dislocations can interact if they are would then a certain distance from each other right do we need to have a term the kind of takes these things into account so we look at it in a very general idea you have a dislocation segment of length L it moves which a certain velocity the acts and this can capture annihilate all the dislocations of opposite Burgers vector if these are within a radius r of the dislocation call but not necessarily innocent like when Central and for higher the CCI is actually very realistic because we know that the dislocations even if they're there and there they can cross yes they can cross and react if necessary that it's possible so the moving this location will sweep a certain volume the which is equal to To the R. L times the text in that time interval of dt and so now let's have a look right and then we introduced the the fraction of dislocation which actually be annihilated what worked what we need to introduce a speed well no there the dislocation like this can I
annihilate this dislocation if they are with their the distance are from each other right however but if this is this dislocation yes this good not going to be annihilation by some not all dislocation dislocations In the meetings as it were lead to annihilation so that's why I I introduced this factor beta which tells me that only a limited number of interactions will lead to annihilation and and the amount of dislocations with is annihilated per unit length of moving location is given by the 2 this is so In my mind this year and annihilation rate it is therefore given by this students it's basically the same thank you multiplied With the mobile dislocation density because this is for per unit length of moving dislocations in the front have multiplied this is for 1 dislocation and the unit length of it so if a multiplied this constant mobile dislocation density has I get the the annihilation rate but now I have the annihilation rate is proportional to the dislocation density for this and now I do the same I and use the same equation 4 D X relations between the expense the sheer strain remember that the shear strain West equal to dislocation density mobile Thompson the terms of the acts quiet and so this gives me the X is the government divided by role at times be so this this is me yeah formula for the rate of annihilation the rate of annihilation yes 2 b The Times role the dislocation and the terms are defined by the soap it is proportional 2 the dislocation density the so you have storage terror which is proportional to the square root and haven't I annihilation term which is proportional to the dislocation density so let's have a look at what we have found no we to combine 2 equations but in these 2 parameters that we the dislocation we find that a fundamental equation for strain hardening no I went to this really important it too says that dislocation in the woods with sheer strength and consists of 2 terms the storage chairman and animation terror and the storage stock is proportional to the square root of the dislocation density and the annihilation term is proportional to the the density of dislocations and what is interesting here is that is to look at what is the implication of this 4 strain hardening no L and we can look at this point for saying Well if we look at this equation there it seems to be that there is a condition years of saturation yes there is a saturation this look at that and you can see it Net deeds there are no roads the divided by the government so the rate of generations of dislocation can become 0 yes but in and of course can become 0 if these 2 thirds are equal to each other right so we have the saturation dislocation generation which is given by for just square the square spots square His which are in fact these parameters of our model but the rest of the saturation dislocation density Brexit now Our why why is this so important in this equation because now that have this equation it gives me the evolution Of the dislocation of which strain yes as a function of they current value of the dislocation density look at and if I can combine this With this relation which give the relation between the share stressed and this square root of the dislocation density I have basically as dress straining relations yes and because if I numerically integrated for numerically integrated by obtaining dislocation density as a function of shares strength and if I take the square root of this at every value of the straight again shear stress strength because good but less so now I have earned 1st perhaps a look at this trying hard and is and I can do this and I don't need to numerically integrate this because I actually need this formula so if I look at the strain hardening its How Djamel expressed in terms of the shear trusted friends it's that is so I knew you wouldn't derivative of this equation that's very simple Alpha G the the Times one-half and then wonder the role of government To the minus 1 house d gamma the role deep down and he really gamma is here you have it so that you can just input this year but this basically gives me indeed strain hardening yes as a function of these the dislocation density that so we can rewrite this this last equation we cannot rewrite this as so that by bringing out In this factor here this factor here I will call the 0 and of course if I look carefully here have Alford GE b square root of them so so I have to here I have a factor yes which includes the sheer strength shares strategy was to the we write this like best the 2 0 0 1 minors this year stressed divided by the shear stress the saturation Sears stressed and In this very simple equation Peter 0 is the initial strain hardening which is Alford Penske divided by 2 times K and the saturation is the saturation stressed as difficult as such so I can plop this strain hardening as a function of the shear stress and it's it's is basically a light the straight line that goes from teachers 0 2 I'm tell saturation of and that's exactly if I go back a few slides here for the last time and we it's
exactly the relationship that we have here Is that had predicted for Pollack
crystallize still and so
that's just a nice result in Paris so again no we are now ready
true to see if we can use this equation because it would be really neat if we could use this equation and that and let's try to use it for 4 steals yes I obviously the way I presented here we can only use it for a single phase steel so I I I could use it for I know for a dick steel which very low carbon but I could principal use it for marked site also because Martin side is basically far-right when the number of Michael structural features 1 of the microspectroscopy just that I haven't talked about here yes and you feel that maybe does it should have an effect and that's the brain size right obviously if I want to use this equation for steel and and hand and get a reasonable the prediction a model predictions I will need to say to take into account grain sizes and the reason is because steals have relatively small grain sizes so that the there is in effect virtually important effect from the grain size How do we take this into account well this is a nice example of where this approach but works very well it's actually very simple to put in the Of the effect of the of the grain size or anything else might restrict troll feature that works as a obstacle To dislocation got so we could take the grain size into countered by saying Well the rigid obstacles now the rigid obstacles that 1 dislocation hats of a grain boundary it's finished just cannot go on any more such as biscuits store and I now it's not like meeting a forest locations where I made the dislocation may be able to cut through quite a few of them before it actually stops so on weekend and In the presence of grain boundaries we can expand the definition of our mean free path taking into account the dislocation grain size so instead of using that is so severe that the storage terror the dislocation density deed rode the gamma plus equal to 1 Overland times B in the case of only dislocation is focused interactions this is what we had dislocation density square divided by case if I have an additional obstacle just added 1 over the plus very simple you can understand why I can just we do not have a direct sea and feel that I can just make the some of these 2 but let's just imagine a 2 it the simple cases yes I say I have a few luge grains and humungous crates mm size Sabine is very very large and I'm performing this and you know I have lots of dislocations here and in glide dislocations that gets tangled In and around here is a dislocation saws and this is my eyes and the mean distance as between the dislocations won all 4 of them so in this case this factor will become a lot 1 over the will be very very small tests and this equation will refer to square would be divided by Ken yes because because these forces Association will be them the main obstacle them but let's listen very different situations with 2 very different we have very tiny grains yet so these very very small then it becomes very difficult to actually generate dislocations and we you you remember they have they had to do when you know when you have Frank Reed the smaller and say we are use this picture all of the do double-cross lived here and there and so this is the distance between depending .period If this distance is very small yes it becomes very hard to produce a dislocation because there isn't 1 over relationship so if I make it crystal very very small necessarily the pending points will be closer to each other yes so right now so in the end yeah the term 1 over the is much much larger than the stock because I have a low dislocation densities that will in this case dislocation densities very very low so this term is low mister is very high it's an obviously but the mean free path will be the size of the brain at best that today so I'm glad that I hope this makes this sort of acceptable so now we can you know did our the equation becomes like this 1 over the square root rolled overcame him we can't numerically integrate this I'll try to regain the I promised you already 2 things that I would put on the class was held put that on the class how you can you know if you don't know how to numerically integrated quits like this all Chileans very simple and uploaded on yeah class and I the Ameritech so that gives you this function and role as a function of gamma then an negative year stress Hughes stress as a ferocious restraint is this that's the strain hardening its actually rather simple focus so let's
see what can we use this this year and actually death with you that bomb and I it we can make up for the phase a situation we can actually I in which analysts say the simplify things by just just by noting that we have in in practice we often have situation like this where did mobile dislocation density stays the same so what does this mean if the the mobile dislocation density doesn't change that I can't send idea actually simplified the 1st heard I the 1st 10 years I can change it into so this is the equation would be arriving this week so basically I'd like I can rewrite it as K 1 is this being constant and then minus K 2 times dislocation density and and so in this case came into areas only 2 2 Peter times rule .period divided by by yes and and the without saying in In addition is that this term here this is a constant so the dislocations storage rate is considered columns can it basically means I'm not changing the the rate at which I produce mobile dislocations that I can do this because my the number of location and he's still low and pretty much flat right now if that's the case but we can actually integrate this equation directly 2 the 1 of the things I do 1st this is I'm going to get rid of shares strain and you know I can do this by using Taylor's factor Taylor factor epsilon is gamma divided by an right and so quite obtained very simply but as something they can in that they can integrate directly the row over them Kimes K 1 minus dislocation density is the Epsilon right now if I integrate as it will give me an equation Of the dislocation density as a function of the tensile strained so in indicating to greatest equation and I get the dislocation density strain dependence this basically but you can look at it after class that it's is missus straightforward integration here and I have written down the steps of so this dislocation as he would strain is equal to there's the 1st Earl K-1 overcame 2 times 1 minus the minus Q 2 and along last rose 0 I'm so he might escape to times at and here K-1 and K-2 on these parameters here and it is a tale of factor rose 0 is the initial dislocation density and then the and slowness is of course the strength of let's now have a look at but how we didn't calculate the strain hardening the strain hardening is but system this descended the strain hardening distress shares specification of Sears strain equation is known shown here Alfred B. the Times Peter's square root of dislocation and I can get they share the from the shear stress the tensile stress by multiplying this With stand with the tale of In the end I can get the change of the dislocation density with the the strain From here and so this is the rule as a function of tensile Strange so I get tensile stress as a function of of straight and that's stress-strain yes then Of course ever want to have a real stressed finger of the to add other strengthening parameters such as the Pirates stressed the lattice friction and solid solution effect and all the other effects we still have to discuss but let's just say we have these 2 terms so let's have an example of how this works well 1st of all we'll need to Burgos factor in this equation that's that's here 2 4 8 times the nanometers orientation factor the Of the year-round Taylor factor about the shear modulus uh 80 something 84 bigger Pascal here this parameter alpha here it is 135 OK and then we will use this equation to do 2 things will make aid as the stress-strain curve for these I prefer right single face Farai and inform Arkansas and so the big difference here when we do this is that for the Farai brain size yes we basically have In equivalent this does this mean free path for dislocations which is equal to the grain size and say we take here grain size of 35 microbes kids were pretty large grain size location but for the but not unreasonable very well in yields I you for instance Martin side however the we have to there's no grain size and might reside with lifting the lab size into account and that's much smaller it's much smaller yes it is of the order of 100 to 200 nanometers so so for the example of cultural engendered you can recalculate if you want is 128 nanometers in this case because or good so L so what we do here remember what was the 1 the the 1 is this this term here and this and this comes from the start In this story we go back a little bit here here in this that's 1 over the last square root rolled over so that's the equation I'm using here
so k is equal to 1 over times and land can be 1 over this look square root of dislocation density or the the grains such green on Lap size yes depending on the situation in this case yes we know we can use grain size it's because we have a very low the increase in the rates in production increase in the amount of mobile dislocations and there but this year is no increase which stressed Simpson so determined that's important here will be 1 over indeed yes 1 over the grain size so this indicated 1 in in this particular example is this is equal to 2 to 1 0 4 landed deep it's 1 over the time speaking to delight but the ticket this is the general form dense and so so this is the mean free path equation yes and so in this case the land is d to dismiss will be K 1 In this particular case the
goal back here lattice friction 55 pound write to for this particular example 55 so that is yes then I annihilation parameters for fair I and animation parameter for Martin side I give you values something you can calculator but based on what we know for more a capital K and the 2 units are these values are determined on the basis of the experiments experiments with good values yes I know that many people have found so Will you initial dislocation densities in the right is very low initial dislocation densities in markets at a very high I remember but when we are of the form well annealed variety the initially we have very few dislocations in March side you have plenty of transformation as well and then solid solution strengthening we also need to take into account and there I know I can use series or I can use an empirical approach this is the 1 we've chosen and these are the solid solution strengthening effects I've taken into account and this and this and this is what you get this is what you get as stress-strain curve 4 variety stress-strain for Martin side as I you can see that the young and the stress-strain personal very realistic yes and in any case sickly and very well of mobile single phase the steals in in this simple using a simple analytical form of the equation right so the nicest thing you cannot see and even in this relatively simple case that although you have many
parameters I agree that most of these parameters you cannot freely choose the factor is what it is the Berbers factor you cannot touch obviously it's it's it's not really a fitting parameter like so many of these numbers are not setting parameters in your because there were there what our physically deal but you can with the Europe parameters you can play well you know you can change the grain size that you could know you can change a composition that who you could 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 of what you do
on stress thinker yes was very handy theory too but I predict not only do this the strength of the material but also the flow strength at different strengths of that and there are ways to expand this Syrian it's been our note multi-phase steals or to fatal the steals with more complex strain hardening or additional strain hardening mechanisms frightened and little bit over time so sorry about that I thank you for your attention I hope everybody got the e-mail that we have to make up lecture time on Friday from 4 to 6 of its 2 2 lectures are being made up for next week because I would be absent on Tuesday and I will not be able to get back here in time to teach on Thursday so will do to classes and then will have denied this and then will have to be Aquarius on the week after that and so on Monday won a regular quizzes Thursday a it will have an that will cover the whole block of lectures there will be more questions on the slogan more stressful and on the
themselves for a ride as of tomorrow afternoon we will be I'm talking
about grain size strength since the


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