Mechanical properties of steel 18: strain hardening

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Mechanical properties of steel 18: strain hardening
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The 18th 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.
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so today we are Starting discussion on strain hardening In I went for it piece of steel if you've ever done this in the media the actual tensile machine you you've noticed that once you pass the yield .period the machine has to apply a ever-increasing of force 2 but continued to forming the material and then there are the basic process that's behind that is we have generate dislocations at a very high rate lots of them and have these dislocations prevent each other from moving freely and so on as the dislocation density increases we also get increasingly stronger obstacle effects From this locations on although dislocations end of both the distribution spatial distribution of the dislocations and their density involves during the strain it and so will try to understand us found in the course of these letters and try to see you know can we developed equations that will allow us to basically the compute stress-strain curves in practice so the
facility can always compute stressed drinkers if you have data on the material but so we've already talked about the whole power along signal is constant stood times Epsilon to the power and and this BC which you get from it stress-strain curve and if somebody's donned the task and tells you would end and they are you can basically recalls but the stress-strain curve and in engineering about talk this week this factor and that's what we call the strain hardening yes but and that contains all these physical processes complex physical processes that I've just talked about In engineering of course this factor is very important and while we have standards to determine their status and and it's just you know this is for instance according to this uh and standard how you computer that lets you just don't do your own thing you have to do it and there are different standards anyway but whenever its computer and you get it from your machine has it's it's computed over a specific length of the stress-strain curve has and I went for a number of discrete stress-strain values and and and this is 1 of these examples are not going to go into debt but that that's that's how you do it you knew you measure discrete points on this curve and the prisoners 5 points here between 10 and 20 per cent and that's how you compute the new engineering value of stress string of the strain hardening yes right and have another thing this that I want to I repeat I think of said it earlier is that this and value certainly in the introductory materials mechanics lecturers is often related to plastic instability because you just don't use combined the cozy dare criteria for plastic instability with the Holleman Paula along yes and if you do that you considered criterion says that the you get to at the end of uniform elongation the start of necking when the derivatives to of stress-strain curves that is equal to the stress and on that's this is reviewed United Engineers transparent curved that state as you know the maximum in the stress-strain curve that that's where you have the uniform loaded so if you apply this speak applied this equation to the whole amount of power law you find that end is equal to the uniform elongated right that's a handy when both trying to understand or see that there is a relation between strain hardening and uniform along gations yes and that's why it's interesting the thing to remember but using it in practice you may not know is definitely not something you you should do it right it's not the way you determine the strain hardening of a material number 1 we abide by by determining the maximum of this stressed engineering stress-strain curve and then recalculating that strain in terms of true strain that if you do that you will you find that as if you could uniforms strain here and a strain Hartmarx exponent there you you you clearly see that if you do this in practice anemic measurements that the uniform irrigation is not equal to the strain hardening yes so I don't use it for To determine strain hardening coefficient having said this it's something important to remember because it basically tells you that having a high strain hardening yes gives you large uniform elongated yes and of course if you want to have a very plastic material but that means you'll have to work on the strain hardening alternatively if you have a certain material that doesn't have much elongate should you know that the problem is has something to do With the waiter materials strain hearts yes OK so but that's
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the exhibition but when we have the wrapped up last week I said to her who agreed to talk about strain hardening and and I also known as safety factor the strain hardening is important and you know we were young we were starting out as a strengthening mechanism that but it's not very often used as a strengthening mechanism in practice and in the rest of it and the reason very simple if you you strain hardening to get stronger material which you can do know you of course have reduced the amount of plastic deformation potentials left and so they're not many products stupid products directly produced which are which use strain hardening as way to strengthen material but at very young you know well known of product corrugated galvanized roofing sheet is this an exception to this rule the stiffness and the hardness of this material is is obtained by AFP basically plastic deformation but once you have this product you officer you don't deform it anymore so you don't really care about the fact that there is a reduced plastic deformation range In this material but it's a very cheap way of making note of getting strong Steve OK this week we discussed just now ranked so I do want to say that again it depends very much on the crystallography of your steel what this to hardening will be and there is a very pronounced difference between this strain hardening of Austin that text for Riddick states so for addicts deals this is an example of a very nice example where we compare to stainless steels which of has a frantic Mike restricted the other 1 has also phonetic Mike rastructure and it up into different it's very obvious right you can see that the Austin that it still has a a huge uniform along gations in comparison to the forensics to the uniform engage with fatigue still would be here a little less than 20 per cent and because this engineering stress-strain curve and the but also still has what 3 times that amount of uniform the and the reason for this is basically the difference in strain hardening behavior you can see strain hardening behavior of Riddick steel compared to the strain hardening behavior of the strategic for are you but with a few of the students who were involved in the doomed to research on all Semitic steals up or SEC steals this particular great the 304 it also doesn't there is no markings attic transformation involved is purely strain hardening increase of the dislocation density as you and but if you do the status in universal tensile testing machine you will find that by the end value is indeed much larger Florida and Austin that Dick Steele down for favorite exterior and it's about double the map right so any of this is nothing to do with you know special little things in the microscope tour anything effect at all you know boost with the grade boundaries are putting President this is purely dislocation related so you see if you do if you look at a single crystals of these alloys single crystal of a of a BCC IRA and you compare with single crystals of the SEC also made a you see the same thing Avery pronounced differences In strain hardening with the Austin they take a structure crystal structure giving you all a lot more strain hardening that the frantic a microscopic Woods it's nothing to do with no which with the the the micro structure it's got everything to do with the dislocations the way they that falls endurance training in terms of their distribution and in terms of their density you can have look at this hour of the dissolution of the dislocation density just as you take a pure of heart single crystal and strain Our that slices out of this single crystal and look into Mike rastructure To determine the the dislocation and see for instance a by direct observation using TEM or by using indirect methods such as X-ray diffraction also used to measure dislocation density by looking at the peak broadening and this is a typical values you get there well in annealed crystal grains of far-right or single crystals will give you somewhere between 10 to the 11 to 10 to 12 of dislocations per square meters this density and as you straighten This is a test in compression so the basically you take the single crystal and you just roll at an independent laboratory of Mel yes and you measured the dislocation and and you find that interestingly enough you get there initially very strong increases in the dislocation density and then what happens is the dislocation density reaches a saturation value that there is some kind of the maximum value of dislocations that you get in the structure and but and this is a lot of of defamation right you can see the 60 80 90 per cent of compression presents a serious amount of compression have the other thing that's interesting is that that saturation valued around them 2 to 3 times in this case for this particularly single crystal entered the 15th 4 square meters of dislocation density AS Close to the dislocation density you get in site you can see that's the dislocation and you can take it out you know make it thermal treatment of the steel and again by extra diffraction or by direct observation in the video you can measure dislocation density and you obtain values the very close to the huge dislocation densities you get after extensive plastic differ with 1 of the reasons why Martin site this the hard and strong yes but and this is the 2nd reason why Martin cites 1st markers as a hard and strong is because they have a very high density of dislocation and where do these dislocations come from because you just do it thermal treatment while the result of the transformation itself the transformation requires sharing the latter's yes and that is this is the sharing is accomplished by transformation dislocations transformation and you get a very high density of those during the march inside the 1st Martin site formation there the only explanation for the strain hardening again just like in the case of a solid solution strengthening the different schools of thoughts the meeting different ideas different basic theories all V I formation
all the on the yet on on the formation of strain hardening of you know how does it if all of this and that the 2 school of stocks 1 school of thought says Wang when you did for steals the you look at the mike rastructure very quickly you see the patterning a patterning of the dislocations and that patterning is described as cell structure formation into 1 group of theory says and in the basic theory is called "quotation mark Mundwiller soft model they say that's the key to the strain hardening and what does this series there of be serious says that the dislocations have a natural yes tendency to form these patents because they carry so much strain energy around so there's lots of lattice straining around this week so they'll form low energy associations Nancy which are called these cell boundaries and all of them organize themselves in cell structures rather than remain uniformly distributed so we have we already described as you know we have to parallel edge dislocations they'd like to 4 energy still some boundaries at 45 degrees if they have of a certain type or Arnold till boundaries if tariff and other types of that there is this tendency for this location to do this note this cell structure formation happens can you can already observe it if you if you're looking at the at that steel so there's not very much not a very strongly deformed like 4 per cent 5 per cent of strain of this small amount of you can already see the emergence of a cell structure With this locations are mostly localized in cell walls and there's so you have concentrated strain this locations in the still more and the dislocation density in the cell was can be 3 to 5 times higher than the average dislocation density and the volume fraction of the cell walls is we intend to 30 per cent of the volume carries a lot of the dislocation the result is that the size of the cell wall as as you increased the dislocation density the size of the cell walls decreases yes and that limits the distance over which dislocations can go I and that's the key to the strain hardening yes because but they didn't do it dimensions over which the smokers freely delight decreases steadily it takes a lot more forests externally applied forced to have been moved over these attitude to generate term and have moved across the South the volume that and there is a very clear of the relation between the flow stress and and the size of the cell in fact that is actually that is considered as the reason why this theory should be right basically there's this occasion cells on the main agent of strain hardening in this area code theory because there is a strong the relation between flow stress and size of the cell and in fact slow the flow stressed issue has been should be proportional to the inverse of the cell diameter and so In the flow stressed that is proportional 2 this equation here and you can write it in a more universal way by writing the flow stress divided by GE to shear stress yes b be divided by D & D is the most important relations 1 over relations and you see here that uh 4 the and Byron alloys here I carbon alloy our entire Tanium alloys we get the hardness increasing when we go from 1 might prompt cell size yes to a 50 nanometers Axelsson a very nice linear relation but 1st of all just for the people who were familiar to hold patch because has nothing this is not related to halt back relations writers as the of grain size it's related to this cell formation so this is what you see inside the grain yes inside the brain you see that if you make small amounts of deformations as you can see here this isn't horrific stillness for Boston that extreme you can see that the regions off and there is 1 big 1 there the region is pretty much empty of dislocations and then regions where very high dislocation tangles we have very high dislocation that this is a cell structure this is an early cell structures somewhat later cell structure and all Semitic steel you can see here is a magnification of this image that very high dislocation density in the cell walls and inside the cell very low dislocation density as when you deformed material you basically have location crossing this this volume yes and so be 1 over the fence in there and on the previous slide is his related to the dimensions of this step not a grain size this is 1 idea is that this location cells and in particular their dimension is the main agent of heart and how does it work well as you add more dislocations in the cell size decreases that's and that gives you the hardening it is another school of thought says well the key player 2 strain hardening is the devolution of the dislocation density period you don't have to worry about the details Of the distribution the spatial distribution of the dislocations and this theory just says that the strain hardening is caused by forest dislocations which intersect this that the slip plane of your primary dislocation and so this is a start playing I have his edge dislocations they move on the slip play and there are some forest dislocations which crossed cut this glide plane and they will be the obstacles so I have a short-range interaction here and this interaction has an effect on you the forests I need to move to this location and the rate of dislocation storage on the glide by I will come back to this this concept here of storage of dislocation and so the year the original form of this small excludes cell formation is no talk about formation in this model the only thing that's of importance is the density the devolution of the dislocation density and intent this this is a picture here last schematic and you can see and you know if if if you're lucky you can see in TM foreign students in dislocations interacting with Forest dislocations in the Microsoft rather common occurrence and so this is the
1 this month called in the cock lacking extra mobile and if we go into detail yes what does the that models say In In terms of dislocation density evolution right that's the that's the big thing what the dislocation density devolution and determines the strain hardening and the 2 things that occurred 1st of all there is this location creation and storage units the an increase in the dislocation density when you deforming grain but these dislocations Wendy the move aims of the crystals their energy can be reduced by reacting true 4 last energetic dislocations but reaction or by this location the dislocation configuration you remember we talked about the the 80 0 1 1 0 junctions in years and that's an example where you have to dislocations that come together and form a 3rd dislocation with the law be square I parameter prejudice and rejection of energy but the and and so and in this way you get new form obstacles the dislocations interact with each other you can form obstacles in and all pending points if you want and in order to release the dislocation you will need to increase this distress yes To have a forest that's larger than the depending for working dislocation the distressed must be increased to remove the junction and led to the dislocation crossed each other as acting Teresa dislocation density the number of dislocation dislocation interactions will increase and that's the process of strain hardening the number interaction .period increases as the dislocation density increased and this leads to a model that very nicely DEC he warned that a square root independence Of that this look of the this flow stress on the dislocation and isn't flow stress is proportional to the dislocation density the square root of death but 2nd can we make sense of this square room dependence 1st of all let's have a look at the square root dependents In practice so if you look at there a lot of measurements that have been done all Irish yes you find indeed that for instance if you measure critical resolved shear stress Seymour Paula crystals on single crystals that consumers can look at which the flow stressed In materials that have different amounts of strength and so different amounts of dislocation density and you plot stress at which you material starts to deformed "quotation mark Pollack crystal and you blocked this as a function of the square root of the dislocation until you find a pretty good linear relation for Pollack crystals and a very nice clear the linear relation for a single crystal data and the equation this summer the show here the share In stress is proportional to a basic share stresses and that's related friction solid solution hardening etc the times a pleasant wasn't a factor which is strain hardening what will go into the theory in a moment but this is the way very often the you also analyzed the experimental data as Alford Darnstaedt and speak square root of the dislocation density and a dislocation density is a functioning all of the shares striding rights of the more you knew the larger your sutures trainers the larger your or dislocation density but I don't and so in fact you know that I can relate single crystal data to Pollack Crestline data simply by multiplied the shear stress shares strain the relation with at the tail of factor I multiplied this equation with and this gives me distress as a function of straying and this is really interesting if you think about it because it this is the stress-strain curve stress as a function of stress and you see that the only parameter that strain the pennant here it's a dislocation density so if you have a way to get to the to describe the dislocation density evolution with strange if you have a way to 2 if you have a way to determine the evolution of the dislocation density with strange as you can basically generate stress-strain curve on basis of this it what very simple more OK so the 1st all the without going into too much detail at this point let's try to make sense Of this square root dependents can we make sense of the fact that the flow stress would be dependent on proportional to the square root of the matter and dislocation densities of the 1st so let's look at the D. Cox smacking Mobile which says this locations are basically bothered by more experienced forest dislocations as obstacles so let's look at this Reds dislocation here it slip plane is the plane of my projection here and then it encounters Forest dislocations which cutting its slip away yes and inside for To make things easy to make sense of this square root depends I will just say that this is the square lattice of atoms Forest dislocation what if I have a square lattice of locations and the dislocation density is world indeed since then there will be yes 1 dislocation Her 1 forest dislocations purpose bells where where L is 1 of the square root of the dislocation density of the rightists in the previous lecture burial site wall isn't it a little bit simplified here because you you only consider 1
this location here and all the dislocations all the other dislocations of Forrester's locations as you put every well actually that's not a bad picture there are a lot more Forest dislocations then there are glide dislocations yes refused to strengthen material and I don't think I have a slightly but later on and in but his trainer material you find out that there are 2 types of dislocation and you can see it's already from simplified model here I have dislocations that move yes and this occasion that this location that act as obstacles yes and dislocations that glide into the deformation so I have this locations which are mobile and I have dislocation at mobile and in a mobile dislocations of forest which it turns out that we need a foreign materials and very quickly the Forest dislocations growth of forests dislocation density outpaces the growth of the the light dislocation or mobile dislocations but many orders of magnitude I think so so we know that's the distance yeah the beach we are the obstacles Forest dislocations is 1 over the square root of the dislocation density so if we go back now to fundamental equations of for strengthening like strengthening we have basically seeing that so we have part Forest dislocation points here intersecting the glide this is this location is trying to pass these obstacles you know that at breakaway yes these so did that in the forests Head of the dislocations on the obstacles the is to attain "quotation mark critical 1 goal of the breakaway I'm when this reaches maximum value which is the forests of the maximum force that obstacle ii day the forces of fish can exert on this moving dislocations when I reached its conditions I will have a breakaway now of course you what what provides that the forests on the dislocation is which you externally applied right and that this stout tends to be times the length of the dislocation 2nd OK so the that is critical the source touted as as Beaton's L. and is equal to half Max and breakaway f max is too intense teachers "quotation mark of the critical on OK and so you see here is that L the spacing between the uh to the forest this location appears in the denominator so it if I put it in I did she tells the times the square root of the dislocation density time to numerical factor which is the critical angle so I find something that's very similar to this equation good so that 1 of the earth the question questions the that is interesting to ask at this point and also partly because it was a historic of historically that's how people started to think about strain hardening we assume in the the Cox making model that it is these forests dislocations that interact with moving dislocations you could also argue and say Well what about parallel dislocations on no on on online planes parallel disappeared into this officials seemed likely company they work as an obstacle to each other because after all we know that you know that exerts forces on each other the for that leads to the formation of a low angled boundaries OK so if you analyze this yes this the interaction of parallel dislocations and you know for instance considered of this type of I configuration here dislocations edge dislocations when they're sensitive locations like and you you want is this August to move past each other he but since you know that when you have 2 parallel dislocations and there will be a force need working on so you will have to exert a force to make them past over each other to get the information so and that can be it's as being solved by simply looking at what is the forest that's 1 of these dislocations exerts on the other 1 them and that is no we know what this forces it's in the lecture notes is 1 of the derivations we made and this is equation and you can also calculate what you remember we had a curse that looked like this only this month as you can calculate what is the maximum forests that 1 dislocation will exert on the other 1 as as you try to pass 1 over the other past the other so you can determine what is the maximum force 1st where it which the position of the maximum by making the derivative of this force that's where the maximum occurs and the disagrees can pass each other as if the new applied force is larger than this this maximum here so you just put this value in this equation yes and for certain value of what a certain distance can and and this would you find GB uh a pie one-liners New Times won over the and and here again you used a fact that you have a perfect square at this location configurations so the perfect square dislocation configuration has so that means that you have 1 dislocation if this permit the square surface area here where did is 1 over the square root of the dislocation density so if you put this in this this previous equation here for Tao and you have to multiply with
mn To get these distressed and the 1 over the years change too square root of the dislocation density I didn't see you back here you get again an equation that looks very much like the 1 we just obtained for interaction of gliders sufficient forest dislocations in the new if you use this equation now and you put in the parameters that we have yes In for are the you was already the I'm value you find I would Alpha is when you find what distressed is as a function of the dislocation density yes OK and this is what you find you find this year you find this equation here and if you no experimentally determined distress as a function of the square root of the dislocation and this is what you fight so yes you if you have a parallel dislocations directing the flow stress will depend on the square root of the dislocation density but the effect will be very small yes so there's tells us there the strain the hardening is not due to dislocations moving parallel to each other it's due at 10 Forest dislocations yes it's it's important because the need if it were only parallel dislocations interacting strongly with each other you would not get obstacles in a very sharp obstacles and all the dislocations would remain right and because that's not the case the meeting we in terms of the hardening effect we can basically ignore that kind of interaction with we have to look at Forest dislocations so the yeah it yes Sir William that so it has as we increase distrust we get a higher dislocation densities of that's that's basically with the reverse of what I showdown in you can look at this for some different crystals and so on this this linear relation is the very strongly for now it's OK so we now have no aid relations between stressed and dislocation densities and this location and the other thing we know is a dislocation density increases would strain outlets look a little bit that's what we're doing here because we're looking at this location I am at in a in a crystal up to now we haven't talked about Paula crystal ball across languages but when we are interested in steals will have to say I will have to make that jump from single crystals to so let's have a look at what happens in single crystal materials if you take a single crystal and oriented very nicely for friends Al-Faran was single slipped as you find that as you as you strain material in it you can calculate the sheer slip him out of sheer on the light plane ride and this is the critical resolved shear stress on the flow stress of you Crystal was basically stress strain curves share stressed shares strain curves of a single crystal so we where does it start to yield at the critical result shear stress right around 20 megabytes right to that and then we see that the we have the 1st region of the shear stress serious strain that goes like this and 2 of the small increase in and we have a strong increase Of the slope of stress-strain curve and then the decrease of the slope discovery so except when we usually talk but all about the use of this this single crystal behavior in terms of stage 1 stage 2 and stage 3 you don't always see the stages yes this is a case where the McChrystal was oriented in such a way that when you started the differences only 1 slip system active so this 1st step behavior here is what we call easy glide but when about these 2 delights as hardening behavior is and what is the relation went a ferret Dick Steele OK so let's now schematically plot the the relatives to discover the strain hardening the relative to discover which we call the Dutch capital data that but as a function of this to rest right when we talk about strain hardening we will very often like to plot not the strain hardening is a function of strain but also as a function of stress and we'll see later why that's interesting this is because we can get nice Universal blocks when we do so strain on so here at the beginning we have a low strain hardening then we go into a very high strain hardening and then we go into yeah steadily with whom you can't adherence to a steadily reducing strain hardening value so for a single crystal behavior you get there s no streamlining very high strain hardening and then it curves down into a steady lead lower strain hardening against the materials through still strain hardens rights still takes more force to make the defamation but the amount you need is gets lower and if we plot on top of this strain hardening behavior for a polycrystalline steel for steel we get there's something that looks like this red line so if I would try to compare my single crystal data strain hardening data with the same material but present as a polycrystalline material by wooden seats stage won by Woods the stage to I'd only see staged readings yes so single crystal hardening behavior that is actually the relevant In technical steals even staged to you barely see stage to use mostly stage for right so we need to understand what it is that causes this the
let's let's let's see describes stage 1 stage 2 and states the ring them and then not forget but the reason why we're doing this is because we're interested in stage 3 because that's the stage actually relevant to students and prompt so lost strain hardening stage is once again the single crystal material and here we have it's only observed for a crystals of oriented in single slept of no practical importance which you basically have is the dislocations are generated and move freely on the light planes there's only 1 such system they don't encounter Forest dislocations there not just to just not there yes to very little or no strain hardening stage 2 what happens if you have a single crystal it's oriented in singles deformed the material and as you did for the material so single slipped situation is like that the Senate and use compressed air as we stand in stage 1 if we have more it's more deformation in stage 1 more selective in the so so you can see here more and more strain in stage 1 however note that in this isn't machine yes the access of the compression axis of the machine stays the same right but you can see that this single slipped situation is only possible if if the lower part of the crystal move sideways but obviously the Crystal doesn't do this it's grip there and because it's a graft yeah because it's incorrect this has to be to remain in 1 line yes it means that the crystal starts to rotate yes there will be a out of differently but in this case because of compression here in this part of the crystal starts to rotate so said this sport comes back to where it's supposed to be in the machine right the get crystal rotation as you get crystal rotation the conditions for a single slept are not met anymore and you get multiple slept systems are activated so stage students which you get multiple stations and again you there is a single crystal when the initial orientation has changed you to Chris and Stage 2 is characterized by very high strain hardening and we see this in polycrystalline material like conceals but only in the initial stage of the defamation and only if are initial dislocation density in the technical material and that still is low enough and what happens here this is a strange With this strain hardening case that's a stage in which the creation yes all of this locations in this generation generation of dislocation is much larger than the rate at which we annihilate this and in stage 3 recalls softening stage it's not the way the toward softening has to be here very carefully it doesn't mean that the strain hardening is negative it's still positive but it increases at a lower rate this book it's not negative this it don't understand softening as a decrease In it's not hardening as much right it's not the decrees would be negative right it's not negative visits recall is off to its softening in the sense that the strain hardening is lower than in stage 2 the strain hardening has war less parabolic uh dependence on the stress at this stage and what happens at the Microsoft structural level is important and in particular for steals yes you get this this lower strain hardening because of the Crosslin events the dislocations started run into obstacles on the dislocation and they start to cross they can do this because we have higher stresses which made this possible and there's another thing that is important is the fact that we have a larger dislocation density and this leads to this location annihilation so the disappearance of this look as you create modest locally as you can you have a higher dislocation density you will also get this location actions that lead to removal of dislocations for instance if I have this dislocation His encountering this dislocation you know when
they encounter each other or have this dislocation encounters this dislocation if they meet no modification that is a typical this location annihilation process or I have won this location and another dislocation and they come together and a 3rd dislocation yes what this location I had used to have to dislocation lines now it's so it the density of the Soviet has half why did so and of course these processes are called processes of annihilation and they become more likely as you have a higher dislocation density and this is the stage that's relevant to us and state and stage we concede a little bit of at the low strains and stage 1 is pretty much knew not relevant for 4 technical steals yes the let's let's make the connection now with what do you do this strain hardening behavior of our single crystal With the strain hardening behavior of our public line material investigators the very familiar the shape of the stress-strain curve to you basically you .period and then you deformed material this mistrust that such severe it's presented as shares stressed as a function of sheer strength since this is a curve has and you can see that if you lot these the derivative to discover here it has a high value yes and here it has it while Euratom principles 0 right it's flat in this game so I have an initial strain hardening In the final very low strata and in between I have a continuous variation Of the strain hardening so if I want this the relative to this as a function of distress I find a lot a straight line that starts at the initial value for the strain hardening and that coincides with the yield .period Avery close the deal .period answer and on a value of saturation In strain hardening which is close to 0 in this case at the at the saturation strain you want at not for you .period by disarray on the world ultimate tensile stress test it's a very simple and the relation break with uh this ball across line hardening and the single crystal hardening is that is initially you'll see you may see a little bit of stage to dance but most of what you'll see is what you're seeing is staged brief behavior in terms of Of all the crystallize behavior the so what so let's just do a justify minutes of good I can introduce the at the math the math behind this approach of string harder we we basically when you do for you the steel you have in their brains you have to groups of dislocations you have mobile dislocations and you have the mobile dislocation you can think of it as you have mobile dislocations that glide and the immobile disagrees and basically of forest and as you strain as the density of these dislocations move out in all that will change what obviously increase so mobility but there is a bit a little bit of a different already mentioned this because it as you strained the evolution but the density of mobile immobile as was different than that immobile location when you before you have a steady increase of of their amount so what would we say if we story dislocation the storage location however the the density of the mobile dislocation doesn't change very much and in fact considering the big difference we can assume we can assume that the immobile this look at it pretty much constant mobile dislocations is pretty much calls so this is pretty much flat compared to this you never know this is actually like that if you do the measurements it's actually like that In practice so the evolution of the change the change in the this location so the change
in the dislocation density is as it would strain is mobile dislocation density plus In more well dislocation density because this is small right it's not 0 you basically looking at the immobile on forests location accumulation 10 right but if you look at the just thinking about dislocations there are always 2 competing processes and indeed change of the dislocation density 1 is creation of your wits trying to create this locations and would strain you also remove dislocation so you have to have a rate equation here dislocation density changes would strain yes and the change is the result of the creation of dislocations and annihilation of dislocation multiplication of generation and annihilation and how does that happen well you have this location sources this look as if they generate well dislocations obviously they don't generate immobile degenerate mobile dislocations more well dislocations move for a while and then they get stuck all obstacles they may pass these obstacles yes but eventually they will become immobile dislocation that's important is that that I think the mobile dislocation used to be mobile dislocation and you basically increased the density also forces which just by adding mobile dislocation that run into Forest dislocations and against stock cannot move any more they were they were mobile this again and how are Forest dislocation but that's how it works so the increase of the dislocation density is due to the accumulation of immobile dislocations that these were will Bell this review and nothing else than arrested mobile dislocations that strong obstacles yes it's basically that will see how on on on Thursday House we work out theory and how we can apply the series the basically obtained stress-strain curve in In steals by just finding ways to compute or describe the the change in what devolution in this location
all right so thank you very much for
your attention sorry about hiccup at the start the had the long fight


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