Mechanical properties of steel 22: precipitation hardening

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Mechanical properties of steel 22: precipitation hardening
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The 22nd 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 precipitation hardening.
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so do we will learn restoring a new chapter and on precipitation hardening In steals
so for those of you who ever taken the introductory
mechanical mental agility or materials science class on only mechanical properties of materials will know that that is the very common way use to hardened alloys and the and Steelers also is also we also use precipitation hardening and steel and in fact and in all types of Steele's we applied the method 2 1 to increased strength In now steals in general with the young carbon so the most of the famous application high-strength low alloy steels although there we have to say that the the amount of strengthening you get from the precipitation is not that large mainly because we have very low levels the volume fractions of precipitates the young the reason why it a list that speeches are laced steals around strong is a combination of both the grain refinement and precipitation hardened but the other owners is steals that our precipitation hardened are typically special steels stainless steels spent pickle so we will talk a little bit about different classes of steel Safran guys who like Martin said extinguished steals in Boston that extremist different extensions that are precipitation hardening it's a little bit of a departure from the types of skills we've been discussing of 2 new flooring so
in general if you want to picture yourself what is happening during the precipitation strengthening so imagine yeah single crystals for grain inside steel and we applied externally certain for staff and we have here is screw dislocations propagating from right to left 1 2 left and we have by certain by certain trick of like rastructure control we've managed to get particles in embedded in the lattice this and I and dislocation mates these the particles on its way through the lattice and interacts with these particles and In general you can say that there are 2 mechanisms if the particles are very large or if they are very hard and they can in that case the very small but very and we have a process which is called back passes this locations not go through the particle that bypassed the particles and in the process of doing so as they bypassed the particle they leave behind a this location loop around the precipitous In every time a new dislocation passes thrown at least another dislocation loop around the precipitate OK so that's 1 thing that can happen but the thing that can happen is that we have small particles there are a soft war this time screw dislocation passes now and it keeps it cuts through the To the particles and I In order to do that uh there needs to be some kind of coherency between the that's harder call and the at the matrix so that the the slips can be transferred rule they the particles a high level of coherency will be required to manage to cut the particle so and and if we go back now a step 2 this bypassing the mechanism yes for lack of coherency between the particle and the the matrix can also be but a way to forests dislocations to bypass the particles so we we have basically 2 extremes to this picture lot or hard or incoherent particles or particles that are all these things at the same time or we can have a small soft coherent particles that there can be a shared but as dislocation perhaps look at the
end of there noticed is a very complex so the precipitates often the precipitates that will impact viii the strength yes volume cutting or bypassing have necessarily to be in the lattice so if you precipitated something in a grain boundary you know grain boundary is by itself they are not the goal of having the precipitating a grain boundary doesn't happen right it's important that you want to achieve the solid solutions to the precipitation strengthening the particle needs to be in the lattice theirs some examples 1st of all there in the corner of vanadium carbide snake example of a precipitate In the a highest-ranked still used as you can see affair I grind clearly and then the particles the vanadium carbide particles embedded in the In the grave I'm very common I precipitates in in steel is used in is car is too is in use Semin tied and said that particular image is of the Martin site the blast which contains small precipitates of Symantec's and you can see indirectly that the semen tight is found coherent with the far-right lactose because it's got this needle shape and you can see the needles are either at this specific somewhere at it's 70 or 80 degree on angle or horizontal red so that tells you there is a high degree of coherency of there's going to be the lattice correspondence orientation relationship between this human tide and the matrix examples here of the other precipitated a that we see in the high-strength low alloy steels and these are Carbide's and nitrites of niobium and Itanium and another way of 2 the precipitation hardened variety is by Copper edition we'll talk about this a little bit more in-depth today and you can see here this is a micrograph you conceal these ball at the docks here are basically copper precipitates again inside you see the grain here grin boundaries here just because the it's inside the drain that's a necessary condition for the precipitation hard so when you look at the S 2 want what can you see obviously there's going to be very important parameters to precipitation strength in me 5 of opposite the properties of the the particles and whether they're hard or soft for instance a copper is definitely is much softer face and fairer of these nitrites are extremely hard yes in comparison to fried that so very different behavior Our so that's 1 thing that the properties of the precipitous 2nd the Folio fractions lawyer you can see here volume fraction of Mike copper here is much larger than the volume fraction I have there both life and agent Carbide volume fraction and this is even higher density the Carbide in the Martin site the and ham and then obviously the size of these particles the the consumer tired particles are the policy which will never do that thing in this case the actually the these Carbide's here and our the smallest of the of these examples to the size of the particle will have and of course has always been a dislocation strengthening mechanism these as
if you have an obstacle you remember me the condition for breakaway from obstacles this is that the externally applied stress this is proportional to the critical the breakaway on goal here's 1 0 for L & L being the distance between the obstacles so they density the radios size of the particles and the interior of particle spacing are important parameters no 1 of the things you have to realize is that the radius density the Inter particle spacing and even then structure Of these precipitates is not these are not single numbers this very much depends on how you engineered the Microsoft tractors so and the differences are very nice example of precipitation hardening In steel it is when you add a few per cent of copper To fair yes this is the work has recently done at at GFT here and found it shows what happens when you precipitate Copper when you for copper indeed far-right matrix so originally the copper which is in Super saturation so you you basically from a supersaturated solution will wield you want to go to much amid the microfracture when you choose Microsoft lost a few words about this but probably later on today so you start making very tiny particles germ or clusters of copper atoms and these clusters you know copper is FCC Mattel will in this case it's actually a BCC precipitated to basically it's it's it's an alloy it does contain Irish yes but it's definitely BCC yes and obviously that's not the natural state of copper and it will evolve too SEC pure fcc copper particles but it will do this in steps has in steps where you make 8 special structure crystal structure which is below 19 are yes which is support to Rumbek which is still iron copper hello you can see the MicroStrategy appear in high the the lactose high-resolution two-year lattice image up there just a magnification of the small particles as small particles about 5 nanometers at this the status this is grown from a tiny cluster to a larger the particles and that this thing will continues to grow Nancy and there is again at a crystal structure change has to SEC Copper all the while the Byron atoms are expelled from the latter's from that the lattice of this growing particle you get nicely twins you can see the twins in these parts and eventually the particle becomes larger yeah it cost since if you wait long enough the particle will you will have to the ripening process it's the particles recoveries larger very much larger at the expense of smaller particles the and and you end up with SEC copper and it's basically pure copper particles so the pending on How far is carried out the aging at what temperature of I've done the aging I will get different particles different types of particles focus and that will impact the press protection strengthening that's an important you ever involved in God research in this area will never forget that it's it's on because you do 1 heat treatment that you have to get it's single the radius composition and distribution and Inter particle spacing of For Your precipitates so it falls a little bit more the work and what will discuss this
but yet but so on also so again well and let let us now have a look at the precipitates an to connect with the previous slide let's look at it crystallography of this precipitous we have when we tried to engineer precipitation hardening in 8 In the integrating inside the grain yes we always have to make sure that there is some new crystal graphic relations possible between the precipitate and The Matrix and so it just to highlight the similarities between the matrix which can be of BCC 4 FCC gamma Byron I show here on the slide the that the matrix and crystallography and then the crystallography of the Maine but types of precipitates that we see for steel so that for instance a C irony you'll also Arab crystal structures goes also goes by the name 8 2 most well 1 of the precipitates that you can use for precipitation the B-2 structure also cesium chloride cynicism chloride you familiar with that and uh that's the structure of as the hell for nickel aluminum has and you can see it's basically very similar to the BCC Byron except that the central pattern is now replaced bye 18 another factor along this so you can have iron aluminum you can also have a nickel aluminum by adding nickel and aluminum too your IRA in a certain way will talk about the details more and then we have the important deal at 3 structure Port mutely structure and if you look carefully enough trying to visualize this as best as I could in this uh Christopher you can see that it's nothing else than the combination of the 8 to structure and that the 2 structure you can see you see for 8 2 blocks and for the 2 blocks the these decisions chloride unit sales of as the normal BCC unit-cell and that's the big units have to do is swing and a very important representative of 2 dealer 3 of structure is Byron 3 aluminum cans so don't mix tyrant 3 aluminum when Nichols free aluminum they have nothing in common alright 3 aluminum then we you can also go 1 step further me and complicates the structure of the deal 3 1 more step by having the 4 units 4 units cells that interview with really yes which still looked like internal yes we now in the center we put another atom for instance all the dark atoms maintaining the lighter items silicon and I get a structure which is called the L 2 1 structure and in steals the you can have the L 2 1 structure that would precipitates as the Titaniums selected this too was last summer faces which Everybody gets into this
this unit-cell here right so you've you've got it here once and here once and here in the back and here in the from the rights of the story R B 2 units and then the other units are 1 2 3 and 4 year part 8 into units that makes the youth wing and now it's have basically 2 different unit cells 1 of them and there too the 2 of us so this is a combination of the 2 last the 2 and this is a combination of 2 types of the two's together right and so 1 atom would be silicone and the other at and would be tightening so you can see that simply by making the lot is a little bit more complex having larger unit sells I can make structures that will pretty much be able to match crystallography the then the matrix without too much what this straight that's BCC 4 SEC so are starting structure is again In the basic yourself for FCC governor and that's a 1 year I can do the same thing as what I did here I can I have the crystal structure princess Titaniums aluminum the been held when I replaced the atoms on the horizontal on the made horizontal plane here by and in the other 10 atoms Our tight you we can I get the I of B L 1 step further so that this would be the equivalent once the Fuhrer's nickel 3 aluminum I get this by having all the items on the sex site claims to be aluminum upstart Nick Anderson have won 3 so you have it you can see the formal you have a 6 statins divided by 2 so you have 3 nickels and you have 8 atoms divided by age you have 1 aluminum at poker and and then we can also have Carbide's yes and Carbide's I have I'm also the the basic structure very similar if you look at this very similar to gather our own knows except that In between the main atoms yes I put in interstitial Lee as it words this not an interstitial but if I put carbon atoms here interstitial I would make the structure of the new Carbide's and all the Carbide's look like this and then and we can form Carbide's novel in In FCC Wilson in fairer In the CCI of but you can see the connection between these common precipitates in the BCC and FCC steals at end the and and that's really the reason why you can precipitate them in the lattice because there's going to be some lattice trying for the last 3 it's not that large so you you can you can precipitated in the lattice rather than in the grain boundaries as when the the mismatch is very high you will get precipitation grain boundaries and that's not that's not what you want it's good
right so let's them that's a start asking ourselves is there is the simple way to read relates volume fraction of precipitates the radius has ended the distance between the particles the mean distance between the particles that that's a very useful saying To heck with this kind of relations and I'm so this but nice derivation and an all also add a correction to the formula of it's very nice original and because of the early when you're involved in experiments as you may have you may know what the volume fraction it will be of precipitate just by calculation and you know and how much difference my Oldham you've added how Harbin you've added so you can calculate what the volume fraction of niobium carbide will be there you can also from Francis TEM analysis get to know the radius of the particles 58 if you know these 2 things then you can calculate the 3rd thing that they be me the Inter particle distance and if you know that into mean India article this as you can you have a way To determine the strengthening effect that you that you are trying to achieve that you have a chance to get so so the relation between the volume fraction of precipitates the mean precipitates size and the average spacing between the precipitate what we do you you consider a thin slab of degrading yes and if that slap has has its thickness Of the mean the rain diameter to our people it's a slab it's as thick as the mean particle diameter precipitated diameter and it's got the surface area of aid which is not that we don't necessarily need to specify so the whole volumes the whole value it is 2 are times the positive and if I multiplied its weight the volume fraction of precipitates it gives me the volume of precipitous in that's in that slack business the the volume of precipitating the slab is too R P 8 times so that once we have this we're in business so this is the volume fraction the of the volume of precipitates so then I can calculate the number of precipitates in this slap the answer is simple I have the and while I'm at it basically the problem with the upside down here I have the the year dimensions all of the the precipitates for which I assume to be circular the forces pie are today R P 2 the 3rd divided by the the volume of precipitates I this should be the 1 over the actually along correct this 1 idea but it on their own each of the classes because of the way this is the the answer and I came determine the aerial density of particles hence so that is the number of yesterday this is correct here this is the number of precipitates has divided by 8 that gives me How many precipitates I have heard unit area 10 now if I assume yes if so severely the aerial density tells me How much basically How much Of the area by assigned to 1 precipitous this 1 precipitated on this for this for this area residents the aerial density of course if I know the aerial density 1 over that's the aerial density and the square root from that this means the inter-party spacing can this is what I do here the average distance between 2 particles is 1 for the square root aerial density and it's OK this is important formula that but as I said connects the average distance between particles which is an essential feature here with the radius of the particles spent the volume fractions of the particles OK so now we can go along 1st of all by making a correction to this form of the If you ever used as a formal and practice for each a steel to school had it's fine formula because we have very little precipitate fractions but if you're using if you're studying precipitation hardens the Steve's which are not of the beaches the ladies type yes the volume fractions are much larger this case much larger than in the case of its sizzling then it then you better use it enough that a better approximation for the into particles no and that is because this the formerly appeared all estimates the the particle spacing so I'm not going to prove this this is the equation it's actually you can see that the origin of equation is in their best and and the correction is very simple this just numerical values which it's you can as well use so so what with two-way trade at Brown so the way it looks like dead it I think it should be volume fraction of so if the volume fraction is is larger than 0 . 1 then of the top Formula overestimates the distance so you need to have To used to correct form of them for 11 steals there's or volume fractions are of the order of 10 to the minus 3 right so it well within the area where this the the formula on top of this the applies so so you can use that formula In other cases stickers precipitation hardened stainless steels etc. you should you should perhaps news this form of and and here I give an example when Evans large that's the case for instance for precipitation hardened Austin antics steals this is so very common grade 8 286 this is a better approximation look at itself there is let's have a look at this formula 1st just just simply look at the formula yes uh and not that that's not think too many years approximate considered many practical considerations so let's slits what's L as a function of the radius of the precipitates yes but at constant volume fraction right so you want when you change when you have
a constant volume fraction you change the radius in the the particles become larger but you also have fewer particles right you cannot have the particles all becoming larger audiences received discussing coarsening so you have small particles to start with you and these particles become larger at constant volume fraction so I get less particles or so so what happens to the distance between these particles it gets it gets larger at now pursuant to look at it let's look at what lets top 1 is for certain volume fraction so here we have a L values Our large and as I increased the rages particles become larger and a less particles obviously and you can see that the distance In particle that becomes larger and if we increase the volume fractions of have a larger volume fraction obviously have twice as many particles automatically the distance becomes smaller yes and again if I costs the particles that same volume fraction at this higher volume fraction again an increase of the Inter particle space the problem is that when you are I would to problem is but when you do actually do precipitation hardening treatment has both these parameters change at the beginning You have no precipitates yes and then you know precipitated yes you precipitate particles and at the beginning you haven't nuclear nation phenomena and after that you have to growth phenomena usually diffusion controlled growth and then eventually you get coarsening phenomenon so it's a nuclear nation of little particles yes tiny particles far apart piers and the particles in role yes notes eventually of course you do your precipitation treatment at the single temperature there's so that eventually you reach these equilibrium volume fraction yes the volume fraction particles increased but as far as you can your Beijing time runs you volume fraction will saturate particles will keep on coarsening yes the radius continues to increase obviously the distance between these particles yes well let's let's see what happens Is it a simple function of time or not will obviously because of the way FJ injures and are changes in so for instance if but let's avoid 0 because 0 is not not such a nice idea but here at the beginning I have tiny particles the API is very very small ends f is increasing right as is increasing good small particles f is increasing and l is ah small provided by something that's increasing so L is decreasing yet but so that's 1 thing let's look at a lot at long times no when the particles are coarsening In this case as hasn't reached its saturation yes if doesn't change any more so at this constant the only thing that happens to L. it is it increases because the Rangers increases so here L L goes down with time and here L goes up for so what it's clear that there's going to be a soft .period yes where hell is minimal yes where we reach a dead end then we the coarsening causes l to increase again and if so which why is that so important for the mechanics move because L is in the denominator so when l is small yes this will be reach its highest value right so this spot here where L is minimal is very important to us and it will correspond to the situation in which we call heat Beijing Peter that's where you get the maximum strength and again it's a function of time and when wiser to function because the volume fraction API are a function of time but so there let's have a look at I am Simpson real situation so precipitation strengthening models it's kind of interesting when you look at so again remind you you have to situations you have the precipitate cutting can be cut what the precipitate can be but bypassed when you cut a precipitate there's many things can happen yeah depending on the dislocations the Matrix dislocations that dislocations in the precipitate the crystallography of the precipitate you name it complex Of 1 of the things is really important is that the strengthening when you cut particles is proportional to the square root of volume fraction times diameter of part this and there are a number of strengthening mechanisms will discuss some of them some of them will calculate examples yes you can have strengthening due to coherency you can have a frightening due to what what's called chemical harder or the Hardin stacking fault hardening and modulus hardening yes these are different ways in which precipitates clause heartening In the case of a particle the properties of the particle themselves are unimportant the dislocations just don't go through the so it's very simple actually there's only 1 mechanism so In this case which we have a hard large particles we cannot be sheared so that the the the properties of dislocations in or faults In on the precipitous as there are irrelevant and in this case the strengthening is proportional to the square root of F and are defined by our people so so we have a sort of cutting mechanism and the other 1 is to bypassing mechanism and here we have a square root that times are key and here we have square root best defined by our people so you can you can see that the fundamentally different mechanisms in this case it's good to have larger particles yes In this case it's bad to have large particles yes In both cases it's good to have a high volume fraction of particles yes so I'm still here this this is always good to have a higher volume fraction of precipitate particles and so
let's now discussed an example of of precipitation Cutting now and will talk about modulus hardening and will derive the formula which describes the but the hardening while the word already says said why would you What is this strengthening due to instituted a difference in elastic modulus and shear modulus next modulus etc. between the matrix and the precipitous and before will arise tells gives us Delta Tau that's the strengthening the fact From they are precipitate which has a different models from the matrix is 0 . 8 3 times be divided by Al and in the square root of 1 minus the ratio of precipitate Young's modulus defined by the matrix Young's modulus square if you look at this formula you can already see something very interesting 1st of all if he the precipitate modulus is equal to the and there's no hard OK that's it's obvious that I the other thing is if EP is 0 0 yes which means avoid there's nothing there yes I the hardening will have a maximum value yes so it it gives you some kind of gives you something the hardening its candlelit counterintuitive it basically means that when a particle is soft I get hardening when it's softer From an elastic point of view I will get harder so when those who would act basically have no way you have to think about it is this location passes through the precipitate yes I have different it I'm sure modulus In both phases and so dislocation properties will be different in the particles that's is basically what modulus the hardening is all about so 1st of all who would work with trying to do is His would you always try to do when you are considering the strengthening mechanisms is you know you basically looked at this grass so this is a review something we discussed earlier you look ahead to dislocation which is held up that these obstacles and in this case these obstacles are not solutes atoms but they are precipitates knows and again the precipitate exert aid they would hold the this locations for moving yes more than this is balanced In this force is balanced by indeed line tension Of the dislocation and we know but if this although this on goal here is the fight over to you this on August 5 years we would basically have F is 2 times the 2 times line attention turned co-signed fight over to this that's a fundamental equation as this can be rewritten In terms of the geometry in if you think you are you can use overdue or over to hear that's the same when you get down to team scientists teetered over to but if so now with the other thing we know instead that well and this this situation is caused by an externally applied stress as otherwise the smokers should just be standing straight minding their own business and not running into these obstacles and so the the force on the dislocation which causes the situation to happen it is the Times the touted as being times the length of 2 dislocation 2nd so it's touted being an if the dissipation and has the Rangers are yes handle and the distance between the particles ' L and then we know we can write tell the time sells a B as told the Times 2 are scientists teachers over to so now if if I combine this and this yes I get this so tell tales B is left over Al N however is this year the force of the particle S & L is the distance between these particles right so I
will there indeed no and apply this to copper In Copper and iron the uh we know from our analysis of the the strengthening mechanism that this homogenous hardening mechanism that so there we looked at there cutting off precipitate copper precipitates which is softer then the matrix softer in terms of modulus ends and we look at the the precipitate the strengthening of the results of basic whenever the same as what we just looked at on the slide the line this location at the counter randomly dispersed proper particles and they have a mean distance felt that we apply externally an external stress in which gives us the shares stress on the glide plane this In the dislocations about between the neighboring precipitates in and have a curved shape with a certain radius and then what we get is we get a balance whereby the precipitate exerts a driving force on the dislocation and it's balanced by underlying tension and so we can write would you just saw that is to teach and signed 5 right they should be should be fight over no it is a little bit of confusion here and there despite over to his this financial make sure
this so In in terms of the which you're going to see In the next slide this angle here is not called file over to this could this article 5 the local
characters but so to the signed 5 L where T is a but the line tension we know we can righted by GB square over to them what we also know that this this is not perfectly correct because it too will depend on whether we are looking at the screw dislocation of edge dislocation but will will just use this for the time being look at me and when I have breakaway yes the 3rd that's when this on the lower reaches a critical value equal to the at the scene the fight scenes to T 5 and a 5 substitute the formula for tea in here I get this equation so f max is GB Square "quotation mark fight scene so what I want to say this standard so I had the situation that goes from this To a situation there goes on this yeah so as I increased the power the town is increasing go from here to here that the the TV vectors always stayed the same right remember the only thing that changes is there's some becomes larger as the binding a between the all the advance of the dislocation between the 2 printing .period increases so In other words this on 0 here the at the scene sought and increases that as we consider it's this valued at every time ask is equal to the to the "quotation mark 5 years this but you can see that as the Arnold decreases yes "quotation mark goes up 10 and I read the study higher when I reach f max yes which defines they obstacle strength yes I also reach the critical of Bill and the dislocation passes the obstacle presidents well known from earlier did so so of course this is this force below is you can also related to this the situation also related to the externally applied force which so this f max is then tomorrow the strengthening times B Michelle is equal to this f max the 2nd I can determine that strengthening is equal to Over tentacles of the critical of so how do how do I look at this In terms of the the the critical angle is basically would be fines the obstacles strength right because t is is always the same for all my dislocations so what defines the obstacle strengthens this critical no I I'm going to look at precipitates the shear modulus of the precipitated this she wanted but she peaches humorless and matrixes G M and so we have different line tensions in the matrix the M end In the precipitate so that it should be please and so these are very simply put into the line tension in the precipitous Alpha G P times in the square and in the matrix it's Alpha G Times Square where alpha is 1 happy years now we you can express the force equilibrium at 2 matrix precipitate interface annulled all show you how this is done that basically this this force equilibrium states that TP times sinus Peter P. is equal to 10 time scientists teachers will see this in a moment what it means yes this is how it works the the ones like Florida so the
dislocation it is 1st outside the particle as moves into it as it moves into it the deed this location assumes the shape outside the particle this and inside the particles it makes a straight line yes and then the same situation on the outside so and if you look inside and outside the from outside and inside the particle I have different yes different why tensions you can see here and I also have different little the different articles here this penalty P here is the on gold at this location makes with the house the normal To the interface here has anteater and is the uncle of the that the external blind tension makes with the normal to do particle here don't know what's important here Is that as lying this dislocation lines moves through this the particle eventually it's a very close to break away yes this under here yes you see when this point most of this 2 days here Close to break away from this angle here goes to 90 degrees yes this and so as I approach breakaway breakaway being the dislocation comes out Of the particle this on gold TDP Prime it is it is close to 90 degrees N the dead end year yes again yes as this this line basically moves up to here the 10 AM and Peter moved towards each other because when it was because this the normal 2 the the surface Will move moves upward here right when they meet in take this and that the 2 s become close move closer to each other at breakaway In at breakaway that's when the dislocation this dislocation goes out of the the ceded it when he goes out as small dislocation line segment so and breakaway it comes out and so that's where this this this whole look at
that another thing that's important here the force equilibrium and the matrix interface yields that is expressing the fact that the and I did this the components of the the line tensions In this in this direction here should be equal so the 2 M times scientists of the T amp amp signers of 52 and should be equal to the TPP times scientists TTP itself so if the uncle is large the people should be small and if the bungled is small the M so should be large let's see there is a difference in TM NTP this and that causes this difference in the the damage the and these 2 components of the that the small arrows here should be equal and opposite that is expressing that because I need to express force equilibrium at the interfaces but it's so notify makes use this expressed equilibrium that is this equilibrium and I express this equilibrium at breakaway conditions so that's where the the it's now goes to 90 degrees yes N T 2 N reaches the critical angle 4 breakaways I get this it's a piece this equal to the 2 times sinus critical founder this is the same 3 steps again now it's it's very simple I use the Croatian news affected signed square was "quotation mark Square is equal to 1 and the fact that there is a simple relations between the the shear modulus and the Young's modulus and shear modulus really have the shear modulus is the youngest 1 is survived by 2 minus 1 2 times 1 minus the points or a show combined these 2 things with With the equation pledges and
here but so that I need to news sinus we need to have scientists critical on goal so this is the critical that's square root one-liners signed a square of critical Arnold and it ends up being simple equation see the divided the times the square root of one-liners the square of the ratio of the particle modulus over D Matrix models Of those this is the basic theory there is a more refined theory assuming random a right of precipitates sir and uh and it basically gives me slightly lower value 4 Delta Data and I can plotters yes a few blocks DEC In dealt a teacher the only important factor is this year's review plopped a square root parameter as a function of the ratio of EPO free and going from 0 to 1 you find that because from 1 for avoid 2 0 0 1 matrix and precipitate parameters on the site and I can determine what the modulus ratio is 4 Copper and Byron there's it's about 0 . 6 so that means that the this parameter will be the closer 1 about 0 . 8 closed very high value and I can also using the form Swedish discussed determines what the breakaway this breakaway on goal looks like so they say if we have be a particle that has the where it and the modulus that's very similar to that of Byron then the uncle here the breakaway before the dislocation breaks away will be this article here will be very close to 180 degrees yes that's what you get if you P E is equal to the breakaway always close to has on the contrary if this is this is avoided yes so basically EP is 0 then the UN before you have breakaway the columns this angle here becomes very small it was close to 0 and that's what you see here and so what's the breakaway uncle that you'll have 1 you should see for daughter send this on here you should see fit for the ratio the particle model is over matrix modulus that's 0 . 6 2 that's the Eagleton open 6 so you can see it should be between 60 and 80 degrees so little bit less than a degree without 70 degrees that should be the breakaway of let's see what we get on the last few minutes in fractures 4 Baron copper so make measurements I don't then think you are you extract From your measurement the strengthening effect due to precipitation the 1st thing you have to do removed the effect of Copper solid solution because it so happens that when you add copper for precipitation hardening you also Have copper in solution you have copper precipitates and some of the copper remains in solution the coverage remains in solution has a solid solution hardening effect 2 for instance if you add 2 per cent Mass Percent of copper you can see you have an increase in about 75 make a basket case so always a very careful when you add alloy heating elements yes and you're trying to evaluate how much strengthening you get "quotation mark precipitation strangling make sure that you also take into account the fact that that element to make in solution also calls solid solution hardened Bledsoe did so in this case you must remove the the solid solution hardening due to copper so let's have a look that the the data so irreversible how how does it how'd you do this in practice simple this is a but the principle is simple this is the irony Copper phase diagram it's the IRA and rich part of it and so do you have a few per cent goes to a 3 and a half mast Percent of copper so let's say we have 3 per cent of copper we've fully so the 1st step is we solution treat the alloy and in this case means it's going to gamma face region so solubility there at a solution temperature high because so all the coppers some of the of the thing you do it is you .period chips you quench it to room temperature and so the solubility at T at at this temperature is very long I'm so so when we age the material we relocated to 500 degrees you can see the solubility very low yes Copper will precipitous 1st nuclear data in a form clusters and the classes will roll and you'll get this the structural change also so you can look at the
precipitation sequence as a function of time yes so you can look at the precipitate radius as a function of Beijing time you can see it here particles the costs and this you can also see the precipitate spacing of the precipitates facing changes with time and so here had particles ago from 5 nanometers 59 0 meters you can see how much time it takes To get 5 about 28 minutes with is 3 thousand seconds the precipitation spacing increases goes from about 50 nanometers to 250 as the Beijing proceeds Nos of the and it depends on the composition also you could see here that 1 mass sent at stake at around 300 nanometers and if I look at the hardness I see that 4 2 for example 1 the mass sent the Peking Beijing is relatively weak yes relatively weak and occurs at around 10 thousand 2nd and the peak aging for 2 master is occurs earlier at around I would say here that's 200 between 2 and 300 seconds yes so a few minutes that's where have the peak age so why don't I see this the precipitate spacing changing while that's because where these measurements starts yes I'm already passed the peak aging yes I'm already possibly I'm already Of these and this is data to last that I'm already passed the peak aging and we're looking at what's called overreaching in the particles of basically costs and them like I can do something similar here but instead of having In the x axis these precipitate that the aging time I can have the precipitate ridges I'm seeing them over time here all of you will just continue on on Thursday because it would
take me too long and I thank you for your attention and we'll see children on Thursday to continue


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